Diffusion in Minerals and Melts 9780939950867

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REVIEWS in MINERALOGY and Geochemistry Volume 72

2010

Diffusion in Minerals and Melts EDITORS Youxue Zhang Daniele J. Cherniak

University of Michigan Ann Arbor, Michigan, U.S.A. Rensselaer Polytechnic Institute Troy, New York, U.S.A.

On the Cover: Top Left: A BSE image showing zonation of zircon (Zhang 2008, Geochemical Kinetics). Lower Right: Ar diffusivity in air, water, melts and hornblende, and heat diffusivity as a function of temperature (data are from various sources).

Series Editor: Jodi J. Rosso MINERALOGICAL SOCIETY of AMERICA GEOCHEMICAL SOCIETY

Reviews in Mineralogy and Geochemistry, Volume 72 Diffusion in Minerals and Melts ISSN 1529-6466 ISBN 978-0-939950-86-7

Copyright 2010

The MINERALOGICAL SOCIETY of AMERICA 3635 Concorde Parkway, Suite 500 Chantilly, Virginia, 20151-1125, U.S.A. www.minsocam.org The appearance of the code at the bottom of the first page of each chapter in this volume indicates the copyright owner’s consent that copies of the article can be made for personal use or internal use or for the personal use or internal use of specific clients, provided the original publication is cited. The consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc. for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other types of copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. For permission to reprint entire articles in these cases and the like, consult the Administrator of the Mineralogical Society of America as to the royalty due to the Society.

TABLE of CONTENTS

1

Diffusion in Minerals and Melts: Introduction Y. Zhang, D.J. Cherniak

INTRODUCTION: RATIONALE FOR THIS VOLUME.........................................................1 SCOPE AND CONTENT OF THIS VOLUME........................................................................2 REFERENCES..........................................................................................................................3

2

Diffusion in Minerals and Melts: Theoretical Background Y. Zhang

INTRODUCTION.....................................................................................................................5 FUNDAMENTALS OF DIFFUSION.......................................................................................6 Basic concepts................................................................................................................6 Microscopic view of diffusion........................................................................................9 Various kinds of diffusion............................................................................................10 General mass conservation and various forms of the diffusion equation.....................14 Diffusion in three dimensions (isotropic media)..........................................................17 SOLUTIONS TO BINARY AND ISOTROPIC DIFFUSION PROBLEMS...........................18 Thin-source diffusion...................................................................................................18 Comments about fitting data.........................................................................................19 Sorption or desorption..................................................................................................20 Diffusion couple or triple.............................................................................................22 Diffusive crystal dissolution.........................................................................................23 Variable diffusivity along a profile...............................................................................25 Homogenization of a crystal with oscillatory zoning...................................................26 One dimensional diffusional exchange between two phases at constant temperature..............................................................................................27 Spinodal decomposition...............................................................................................28 Diffusive loss of radiogenic nuclides and closure temperature....................................29 Diffusion in anisotropic media..............................................................................32 MULTICOMPONENT DIFFUSION......................................................................................35 Effective binary approach, FEBD and SEBD...............................................................36 Modified effective binary approach (activity-based effective binary approach)..........39 Diffusivity matrix approach..........................................................................................40 Activity-based diffusivity matrix approach..................................................................42 Origin of the cross-diffusivity terms............................................................................42 DIFFUSION COEFFICIENTS................................................................................................43 Temperature dependence of diffusivities; Arrhenius relation.......................................43 Pressure dependence of diffusivities............................................................................43 Diffusion in crystalline phases and defects..................................................................45 Diffusivities and oxygen fugacity.................................................................................47 Compositional dependence of diffusivities..................................................................47 iv

Diffusion in Minerals and Melts ‒ Table of Contents Relation between diffusivity, particle size, particle charge, and viscosity...................48 Diffusivity and ionic porosity.......................................................................................50 Compensation “law”.....................................................................................................50 Interdiffusivity and self diffusivity...............................................................................50 CONCLUSIONS......................................................................................................................53 Acknowledgments........................................................................................................53 REFERENCES........................................................................................................................53 Appendix 1. Expression of diffusion tensor in crystals with different symmetry..................................................................................58

3



Non-traditional and Emerging Methods for Characterizing Diffusion in Minerals and Mineral Aggregates E.B. Watson, R. Dohmen

Introduction...................................................................................................................61 The thin-film method and pulsed laser deposition (PLD): Principles and recent developments.........................................................63 Definition of a thin film................................................................................................63 Why use thin films?......................................................................................................64 Fitting of diffusion profiles from thin-film diffusion couples......................................65 Analytical solutions – examples...................................................................................65 Fitting uncertainties......................................................................................................67 Pulsed laser ablation: a versatile method for thin film deposition................................68 Application of PLD to diffusion studies – examples....................................................70 Single layer configurations...........................................................................................71 Double layer configurations.........................................................................................74 The powder-source technique...............................................................................78 Overview and history....................................................................................................78 Rationale and details....................................................................................................79 Analytical considerations, advantages and drawbacks.................................................80 Ion implantation and diffusion experiments................................................82 Introduction..................................................................................................................82 Interactions between energetic ions and solids............................................................83 Ion implantation...........................................................................................................84 Mathematical aspects of implantation and diffusion....................................................85 Complications and examples........................................................................................87 The detector-particle method for studies of grain-boundary diffusion......................................................................................................................90 Context and history.......................................................................................................90 The detector-particle approach: general considerations and examples........................91 Numerical simulation: constant-surface model............................................................94 A simple analysis of the detector-particle method.......................................................99 Concluding remarks on detector particles..................................................................100 Acknowledgments......................................................................................................101 References......................................................................................................................101 v

Diffusion in Minerals and Melts ‒ Table of Contents

4

Analytical Methods in Diffusion Studies D.J. Cherniak, R. Hervig, J. Koepke, Y. Zhang, D. Zhao

Introduction.................................................................................................................107 “Classical” Methods for Measuring Diffusion Profiles Using Radioactive Tracers.............................................................................109 Serial sectioning ........................................................................................................109 Autoradiography.........................................................................................................110 Electron Microprobe Analysis............................................................................111 Principles of EMPA....................................................................................................111 Instrumentation for EMPA.........................................................................................113 Applications and limitations of EMPA.......................................................................120 Summary....................................................................................................................123 SECONDARY ION MASS SPECTROMETRY (SIMS)......................................................123 Basic principles of SIMS............................................................................................123 Using SIMS to measure diffusion profiles.................................................................125 Depth profile analyses................................................................................................129 Ion implantation and SIMS........................................................................................134 Summary comments...................................................................................................134 Laser ablation ICP-MS (LA ICP-MS)........................................................................134 Rutherford Backscattering Spectrometry (RBS).....................................137 Basic principles of RBS.............................................................................................137 Depth and mass resolution.........................................................................................140 Example applications of RBS in diffusion studies.....................................................141 Nuclear Reaction Analysis (NRA).......................................................................143 Elastic Recoil Detection (ERD).............................................................................147 Fourier Transform Infrared Spectroscopy.................................................148 Vibrational modes and infrared absorption................................................................148 Instrumentation for Infrared Spectroscopy.................................................................152 Different types of IR spectra......................................................................................152 Calibration..................................................................................................................153 Applications to geology..............................................................................................155 Synchrotron X-ray fluorescence microanalysis (m‑SRXRF)................156 Instrumental setup, spectra acquisition and data processing......................................156 Sample preparation.....................................................................................................158 Applications of m-SRXRF for measuring trace element diffusivities in silicate melts....................................................................................................158 Acknowledgments......................................................................................................160 References......................................................................................................................160

5

Diffusion of H, C, and O Components in Silicate Melts Y. Zhang, H. Ni

INTRODUCTION.................................................................................................................171 vi

Diffusion in Minerals and Melts ‒ Table of Contents diffusion of the H2O component..........................................................................172 H2O speciation: equilibrium and kinetics ..................................................................172 H2O diffusion literature .............................................................................................178 H2O diffusion, theory and data summary ..................................................................180 Molecular H2 diffusion............................................................................................191 diffusion of the CO2 component..........................................................................197 Oxygen diffusion.........................................................................................................199 Self-diffusion of oxygen in silicate melts under dry conditions.................................200 Chemical diffusion of oxygen under dry conditions..................................................207 “Self” diffusion of oxygen in the presence of H2O....................................................209 “Self” diffusion of oxygen in natural silicate melts in natural environments.............211 Contribution of CO2 diffusion to 18O transport in CO2-bearing melts........................213 Oxygen diffusion and viscosity: applicability of the Eyring equation.......................216 O2 diffusion in pure silica MELT............................................................................217 SUMMARY AND CONCLUSIONS.....................................................................................219 Acknowledgments......................................................................................................219 REFERENCES......................................................................................................................219

6

Noble Gas Diffusion in Silicate Glasses and Melts H. Behrens

INTRODUCTION.................................................................................................................227 EXPERIMENTAL AND ANALYTICAL METHODS.........................................................228 Studies at atmospheric and sub-atmospheric pressure...............................................228 Studies at high-pressure..............................................................................................230 DIFFUSION SYSTEMATICS...............................................................................................232 Temperature dependence of diffusivity......................................................................232 Pressure dependence of diffusivity.............................................................................233 Comparison of different noble gases in the same matrix glass..................................236 COMPOSITIONAL EFFECTS ON NOBLE GAS DIFFUSION.........................................238 He diffusion................................................................................................................238 Ne diffusion................................................................................................................240 Ar diffusion................................................................................................................241 Kr, Xe and Rn diffusion ............................................................................................248 COMPARISON OF NOBLE GASES AND MOLECULAR SPECIES................................249 H2 diffusion.................................................................................................................249 H2O diffusion..............................................................................................................250 O2 diffusion.................................................................................................................250 N2 diffusion.................................................................................................................251 CO2 diffusion..............................................................................................................252 ACKNOWLEDGMENTS......................................................................................................252 RERERENCES......................................................................................................................253 Appendix ...........................................................................................................................257

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Self-diffusion in Silicate Melts: Theory, Observations and Applications to Magmatic Systems C.E. Lesher

Introduction.................................................................................................................269 Additional Terminology.........................................................................................270 Theoretical Considerations.................................................................................271 Self and tracer diffusion.............................................................................................271 Intradiffusion..............................................................................................................276 Polyanionic diffusion..................................................................................................280 Experimental Methods and Data........................................................................283 Thin source method....................................................................................................283 Diffusion couple method............................................................................................284 Capillary-reservoir method.........................................................................................284 Gas exchange method.................................................................................................285 Discussion........................................................................................................................285 Background................................................................................................................285 Ionic charge and size..................................................................................................286 Temperature................................................................................................................288 Viscosity and the Eyring diffusivity...........................................................................291 Pressure......................................................................................................................296 Concluding Remarks.................................................................................................303 Acknowledgments......................................................................................................305 References......................................................................................................................305

8

Diffusion Data in Silicate Melts Y. Zhang, H. Ni, Y. Chen

Introduction.................................................................................................................311 Terminology...............................................................................................................312 General comments about experimental methods to extract diffusivities....................313 Grouping of the elements...........................................................................................315 Data compilation........................................................................................................315 Quantification of D as a function of T, H2O, P, fO2 and melt composition.................317 Diffusion of individual elements......................................................................317 Diffusion of major elements versus minor and trace elements..................................317 H diffusion..................................................................................................................320 The alkalis (Li, Na, K, Rb, Cs, Fr).............................................................................320 The alkali earths (Be, Mg, Ca, Sr, Ba, Ra).................................................................330 B, Al, Ga, In, and Tl...................................................................................................340 C, Si, Ge, Sn and Pb...................................................................................................345 N, P, As, Sb, Bi...........................................................................................................352 O, S, Se, Te, Po...........................................................................................................354 F, Cl, Br, I, At.............................................................................................................356 He, Ne, Ar, Kr, Xe, Rn...............................................................................................360 viii

Diffusion in Minerals and Melts ‒ Table of Contents Sc, Y, REE..................................................................................................................360 Ti, Zr, Hf.....................................................................................................................375 V, Nb, Ta.....................................................................................................................380 Cr, Mo, W...................................................................................................................383 Mn, Fe, Co, Ni, Cu, Zn...............................................................................................383 Tc, Ru, Rh, Pd, Ag, Cd...............................................................................................389 Re, Os, Ir, Pt, Au, Hg..................................................................................................389 Ac, Th, Pa, U..............................................................................................................391 DiSCussion........................................................................................................................393 The empirical model by Mungall (2002)...................................................................393 Effect of ionic size on diffusivities of isovalent ions..................................................395 Dependence of diffusivities on melt composition......................................................397 Diffusivity sequence in various melts.........................................................................398 Concluding Remarks.................................................................................................402 Acknowledgments......................................................................................................404 REFERENCES......................................................................................................................404

9



Multicomponent Diffusion in Molten Silicates: Theory, Experiments, and Geological Applications Y. Liang

Introduction.................................................................................................................409 Irreversible Thermodynamics and Multicomponent Diffusion.......411 The rate of entropy production...................................................................................411 Diffusing species and choice of endmember component...........................................412 General Features of Multicomponent Diffusion......................................414 Solutions to multicomponent diffusion equations......................................................414 Essential features of multicomponent diffusion ........................................................415 Experimental studies of multicomponent diffusion..............................423 Experimental design and strategy...............................................................................423 Inversion methods.......................................................................................................425 Experimental results...................................................................................................428 Empirical Models for Multicomponent Diffusion....................................434 Empirical models .......................................................................................................434 Experimental tests of the empirical models ..............................................................436 Geological Applications.........................................................................................437 Modeling isotopic ratios during chemical diffusion in multicomponent melts..........437 Convective crystal dissolution in a multicomponent melt .........................................438 Crystal growth and dissolution in a multicomponent melt ........................................441 Future Directions.......................................................................................................442 acknowledgments......................................................................................................443 REFERENCES......................................................................................................................443

ix

Diffusion in Minerals and Melts ‒ Table of Contents

10

Oxygen and Hydrogen Diffusion in Minerals J.R. Farver

INTRODUCTION.................................................................................................................447 EXPERIMENTAL METHODS.............................................................................................447 Bulk exchange experiments........................................................................................447 Single crystal experiments..........................................................................................448 ANALYTICAL METHODS..................................................................................................449 Mass Spectrometry.....................................................................................................449 Nuclear Reaction Analysis ........................................................................................450 Fourier Transform Infrared Spectroscopy..................................................................450 Other methods............................................................................................................450 RESULTS...............................................................................................................................451 Quartz.........................................................................................................................451 Feldspars.....................................................................................................................455 Olivine........................................................................................................................461 Pyroxene.....................................................................................................................465 Amphiboles................................................................................................................470 Sheet silicates.............................................................................................................471 Garnet.........................................................................................................................472 Zircons........................................................................................................................474 Titanite........................................................................................................................474 Melilite.......................................................................................................................475 Tourmaline and beryl .................................................................................................476 Oxides ........................................................................................................................477 Carbonates..................................................................................................................480 Phosphates..................................................................................................................482 DISCUSSION........................................................................................................................483 Effect of temperature..................................................................................................483 Effect of mineral structure..........................................................................................485 Empirical methods......................................................................................................486 Anisotropy..................................................................................................................486 Pressure dependence...................................................................................................488 Effect of water............................................................................................................488 Hydrogen chemical diffusion and the role of defects.................................................489 ACKNOWLEGMENTS........................................................................................................490 REFERENCES......................................................................................................................490

11

Diffusion of Noble Gases in Minerals E.F. Baxter

INTRODUCTION.................................................................................................................509 The interpretive challenge of bulk-degassing experiments........................................510 HELIUM................................................................................................................................513 He in apatite................................................................................................................514 x

Diffusion in Minerals and Melts ‒ Table of Contents He in titanite...............................................................................................................520 He in zircon and zircon-structure rare earth element orthophosphates......................520 He in monazite and monazite-structure rare earth element orthophosphates.............523 He diffusion in other minerals....................................................................................523 ARGON.................................................................................................................................527 Ar in micas.................................................................................................................528 Ar in amphibole..........................................................................................................529 Ar in feldspar..............................................................................................................529 Ar diffusion in other minerals....................................................................................530 THE OTHER NOBLE GASES: NEON, KRYPTON, XENON, RADON............................532 THEMES IN NOBLE GAS DIFFUSION IN MINERALS..................................................534 Effect of radiation damage.........................................................................................534 Effect of deformation.................................................................................................535 Multi-domain diffusion..............................................................................................536 Multi-path diffusion....................................................................................................537 Synthesis: relative diffusivities of the noble gases in minerals..................................539 CHOOSING THE “RIGHT” DIFFUSION DATA................................................................542 Role of noble gas diffusion data in Ar/Ar and (U-Th)/He thermochronology...........545 SUGGESTIONS FOR FUTURE STUDY.............................................................................548 Diffusion at high pressures and temperatures ...........................................................548 Diffusion of Ar and He in common mantle minerals ................................................548 In situ depth profile analysis ......................................................................................551 Quantification of noble gas diffusion within “fast paths” .........................................551 Integrated studies with multiple noble gases .............................................................551 Quantification of effects of radiation damage, defects, and deformation ..................551 ACKNOWLEDGMENTS......................................................................................................552 References......................................................................................................................552

12

Cation Diffusion Kinetics in Aluminosilicate Garnets and Geological Applications J. Ganguly

INTRODUCTION.................................................................................................................559 NOMENCLATURE OF DIFFUSION COEFFICIENTS......................................................560 EXPERIMENTAL DETERMINATION OF DIFFUSION COEFFICIENTS.......................561 Experimental methods................................................................................................561 Modeling of experimental data...................................................................................564 EXPERIMENTAL DATA AND DISCUSSION....................................................................566 Self/tracer diffusion coefficients of Mn, Fe2+ and Mg................................................566 Diffusion properties of Ca..........................................................................................573 Tracer diffusion coefficients of trivalent rare earth ions.............................................575 D-MATRIX, UPHILL DIFFUSION AND CHEMICAL WAVES........................................578 COMMENTS ON EXTRAPOLATION AND GEOLOGICAL APPLICATION OF EXPERIMENTAL DIFFUSION DATA.........................................................................580 Change of diffusion mechanism and extrapolation of diffusion data.........................580 Modeling prograde vs. retrograde profiles.................................................................580 xi

Diffusion in Minerals and Melts ‒ Table of Contents Treatment of diffusion data........................................................................................580 A SEMI-EMPIRICAL MODEL OF DIVALENT CATION DIFFUSION............................581 Carlson model.............................................................................................................581 Discussion..................................................................................................................582 GEOLOGICAL APPLICATIONS.........................................................................................585 Modeling multicomponent diffusion profiles using effective binary diffusion formulation..........................................................................................................586 Cooling rates of metamorphic rocks: diffusion modeling of garnet vs. geochronological constraints...............................................................................587 Subduction and exhumation rates...............................................................................587 Modeling partially modified growth zoning of garnets in metamorphic rocks..........589 Interpretation of REE patterns of basaltic magma.....................................................592 Sm-Nd and Lu-Hf geochronology of garnets in metamorphic rocks.........................594 CONCLUDING REMARKS.................................................................................................596 ACKNOWLEDGMENTS......................................................................................................598 REFERENCES......................................................................................................................598 APPENDIX: Combined analytical and numerical method for modeling multicomponent diffusion profiles....................................600

13

Diffusion Coefficients in Olivine, Wadsleyite and Ringwoodite S. Chakraborty INTRODUCTION.................................................................................................................603 olivine...............................................................................................................................603 Structure of olivine and types of diffusion coefficients..............................................603 Diffusion mechanisms in olivine................................................................................605 Diffusion of divalent cations......................................................................................608 Diffusion of Si and oxygen.........................................................................................620 Diffusion of ions that enter olivine via heterovalent substitutions.............................623 Information from olivines other than Fe-Mg binary solid solutions..................................................................................................................627 Spectroscopic measurements..............................................................................628 Computer calculations...........................................................................................628 Wadsleyite and Ringwoodite...............................................................................629 Diffusion of divalent cations......................................................................................630 Diffusion of silicon and oxygen.................................................................................631 Diffusion of ions that are incorporated by heterovalent substitutions........................633 A SUMMARY, AND Applications of diffusion data in olivine, wadsleyite and ringwoodite........................................................................633 Acknowledgments......................................................................................................635 References......................................................................................................................635

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14

Diffusion in Pyroxene, Mica and Amphibole D.J. Cherniak, A. Dimanov

INTRODUCTION.................................................................................................................641 CATION DIFFUSION IN PYROXENES.............................................................................641 Pioneering approaches................................................................................................643 More recent investigations of major element diffusion..............................................644 Diffusion of major element cations in clinopyroxenes...............................................645 Diffusion in synthetic versus natural crystals.............................................................656 Major element cation diffusion in orthopyroxenes.....................................................656 Pyroxene point defect chemistry................................................................................658 Diffusion of minor and trace elements in pyroxene...................................................661 Comparison of diffusion of cations in pyroxene........................................................672 Diffusion in amphiboles and micas....................................................................676 F-OH interdiffusion in tremolite................................................................................677 Sr diffusion in tremolite and hornblende....................................................................678 Sr diffusion in fluorphlogopite...................................................................................679 K and Rb diffusion in biotite......................................................................................679 Acknowledgments......................................................................................................680 References......................................................................................................................680 APPENDIX............................................................................................................................685

15

Cation Diffusion in Feldspars D.J. Cherniak

Introduction.................................................................................................................691 Diffusion of Major Constituents.......................................................................692 Sodium .......................................................................................................................692 Potassium....................................................................................................................695 K-Na interdiffusion ...................................................................................................696 Calcium......................................................................................................................698 Barium........................................................................................................................699 CaAl-NaSi interdiffusion............................................................................................700 Silicon ........................................................................................................................703 Diffusion of Minor and Trace Elements.........................................................705 Lithium.......................................................................................................................705 Rubidium....................................................................................................................705 Magnesium.................................................................................................................707 Iron.............................................................................................................................708 Strontium....................................................................................................................708 Lead............................................................................................................................717 Radium.......................................................................................................................721 Rare Earth Elements...................................................................................................721 Comparison of relative diffusivities of cations in various feldspar compositions......................................................................................723 xiii

Diffusion in Minerals and Melts ‒ Table of Contents Albite..........................................................................................................................723 K-feldspar...................................................................................................................723 Intermediate alkali feldspars......................................................................................725 Anorthite.....................................................................................................................725 Labradorite.................................................................................................................726 Oligoclase...................................................................................................................728 Acknowledgments......................................................................................................728 References......................................................................................................................728

16

Diffusion in Quartz, Melilite, Silicate Perovskite, and Mullite D.J. Cherniak

Introduction.................................................................................................................735 Diffusion in Quartz.....................................................................................................735 Silicon.........................................................................................................................736 Aluminum and gallium...............................................................................................738 Alkali elements – Li, Na, K........................................................................................739 Calcium......................................................................................................................741 Titanium.....................................................................................................................741 Diffusion in quartz – a summary................................................................................742 Diffusion in Melilite..................................................................................................743 Al+Al ↔ Mg + Si interdiffusion................................................................................743 Mg..............................................................................................................................743 Mn, Fe, Co, and Ni.....................................................................................................746 Ca, Sr, and Ba.............................................................................................................748 Potassium ...................................................................................................................749 Diffusion in melilite – a summary .............................................................................750 Diffusion in Silicate perovskite..........................................................................751 Silicon.........................................................................................................................751 Fe-Mg interdiffusion..................................................................................................753 diffusion in Mullite...................................................................................................753 Acknowledgments......................................................................................................754 References......................................................................................................................754

17

Diffusion in Oxides J.A. Van Orman, K.L. Crispin

INTRODUCTION.................................................................................................................757 PERICLASE..........................................................................................................................758 General considerations...............................................................................................758 Oxygen.......................................................................................................................759 Magnesium.................................................................................................................763 xiv

Diffusion in Minerals and Melts ‒ Table of Contents Other group IIA divalent cations ...............................................................................766 Group IIIA and IIIB trivalent cations.........................................................................769 Tetravalent cations......................................................................................................771 Transition metals........................................................................................................771 Hydrogen....................................................................................................................783 SPINEL..................................................................................................................................783 Oxygen.......................................................................................................................784 Magnesium.................................................................................................................785 Fe-Mg interdiffusion..................................................................................................786 Mg-Al interdiffusion..................................................................................................787 Cr-Al interdiffusion....................................................................................................787 Hydrogen....................................................................................................................788 MAGNETITE........................................................................................................................788 Oxygen.......................................................................................................................789 Iron.............................................................................................................................791 Other cations...............................................................................................................794 RUTILE.................................................................................................................................796 Oxygen.......................................................................................................................797 Tetravalent and pentavalent cations............................................................................799 Divalent and trivalent cations.....................................................................................801 Monovalent cations.....................................................................................................803 ACKNOWLEDGMENTS......................................................................................................804 REFERENCES......................................................................................................................804 APPENDIX............................................................................................................................810

18



Diffusion in Accessory Minerals: Zircon, Titanite, Apatite, Monazite and Xenotime D.J. Cherniak

Introduction.................................................................................................................827 Diffusion in Zircon......................................................................................................827 Lead............................................................................................................................828 Rare Earth Elements (REE)........................................................................................832 Tetravalent cations......................................................................................................835 Cation diffusion in zircon - a summary .....................................................................838 Diffusion in Titanite...................................................................................................841 Strontium and Lead....................................................................................................841 Neodynium.................................................................................................................843 Zirconium...................................................................................................................843 Summary of diffusion data for titanite.......................................................................844 Diffusion in Monazite................................................................................................844 Calcium and Lead.......................................................................................................845 Thorium......................................................................................................................847 Diffusion in Xenotime................................................................................................848 Diffusion in Apatite.....................................................................................................850 Lead and Calcium.......................................................................................................850 xv

Diffusion in Minerals and Melts ‒ Table of Contents Strontium....................................................................................................................852 Manganese..................................................................................................................853 Rare Earth Elements (REE)........................................................................................854 Phosphorus ................................................................................................................858 Uranium and Thorium ...............................................................................................858 F-OH-Cl......................................................................................................................859 Comparison of diffusivities of cations and anions in apatite......................................860 Comparison of diffusivities among accessory minerals......................861 Lead............................................................................................................................861 Rare Earth Elements (REE)........................................................................................862 Thorium and Uranium................................................................................................863 Acknowledgments......................................................................................................864 References......................................................................................................................864

19

Diffusion in Carbonates, Fluorite, Sulfide Minerals, and Diamond D.J. Cherniak

Introduction.................................................................................................................871 CaRBONATES.....................................................................................................................871 Carbon........................................................................................................................872 Calcium......................................................................................................................875 Magnesium.................................................................................................................876 Strontium and Lead ...................................................................................................877 Rare Earth Elements ..................................................................................................878 Diffusion in calcite – an overview..............................................................................879 Fluorite............................................................................................................................880 Fluorine......................................................................................................................881 Calcium......................................................................................................................883 Strontium, Yttrium and Rare Earth Elements.............................................................883 Diamond............................................................................................................................885 Sulfide Minerals..........................................................................................................885 Pyrite..........................................................................................................................886 Pyrrhotite....................................................................................................................888 Sphalerite....................................................................................................................889 Chalcopyrite...............................................................................................................891 Galena.........................................................................................................................892 Summary of diffusion findings for the sulfides..........................................................892 Acknowledgments......................................................................................................893 References......................................................................................................................894

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Diffusion in Minerals: An Overview of Published Experimental Diffusion Data J.B. Brady, D.J. Cherniak

INTRODUCTION.................................................................................................................899 ARRHENIUS RELATIONS..................................................................................................900 DIFFUSION COMPENSATION DIAGRAMS.....................................................................904 IONIC POROSITY................................................................................................................911 DIFFUSION ANISOTROPY.................................................................................................913 CONCLUDING REMARKS.................................................................................................917 ACKNOWLEDGMENTS......................................................................................................917 REFERENCES......................................................................................................................917

21

Diffusion in Polycrystalline Materials: Grain Boundaries, Mathematical Models, and Experimental Data R. Dohmen, R. Milke

INTRODUCTION.................................................................................................................921 Geological relevance of grain boundary diffusion.....................................................921 Physical nature of a grain/interphase boundary.........................................................922 Thermodynamic model for interfaces........................................................................925 The isolated grain boundary...............................................................................927 Basic mathematical description..................................................................................927 Kinetic regimes and diffusion penetration distances..................................................928 The MONoPhase Polycrystalline Aggregate................................................932 Models and kinetic regimes........................................................................................932 Bulk diffusion coefficients..........................................................................................934 A geological example.................................................................................................936 Profile analysis – the Le Claire approach...................................................................937 Complexities of real and polyphase systems..............................................................940 Asymmetric grain boundaries/interphase boundaries................................................941 The migrating isolated grain boundary......................................................................942 Presence of dislocations/sub-grain boundaries...........................................................944 Element/isotope exchange mediated by grain boundary diffusion.............................946 EXPERIMENTAL METHODS.............................................................................................947 Setup with bi-crystals.................................................................................................948 Setup with a polycrystalline aggregate.......................................................................949 Source–sink studies....................................................................................................950 EXPERIMENTAL DATA......................................................................................................950 Parameters affecting grain boundary diffusion coefficients.......................................950 Direct measurement of tracer diffusion in polycrystals of geological relevance.......955 Concluding remarks...................................................................................................964 Acknowledgments......................................................................................................966 References......................................................................................................................966 xvii

Diffusion in Minerals and Melts ‒ Table of Contents

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Theoretical Computation of Diffusion in Minerals and Melts N. de Koker, L. Stixrude

INTRODUCTION ................................................................................................................971 THEORETICAL FOUNDATIONS ......................................................................................972 Thermodynamic description.......................................................................................972 Statistical mechanical description..............................................................................974 COMPUTATIONAL APPROACHES ...................................................................................976 Characterization of bonding.......................................................................................977 Adding temperature....................................................................................................978 Computation of diffusion...........................................................................................980 SELECTED APPLICATIONS...............................................................................................981 Liquids and melts.......................................................................................................981 Solids..........................................................................................................................988 A VIEW TO THE FUTURE .................................................................................................991 ACKNOWLEDGMENTS .....................................................................................................991 References......................................................................................................................991

23

Applications of Diffusion Data to High-Temperature Earth Systems T. Mueller, E.B. Watson, T.M. Harrison

Introduction.................................................................................................................997 Deciphering kinetically controlled processes using diffusion....999 Mass transport in geological systems.........................................................................999 Diffusion in minerals................................................................................................1002 Control of solid-state reaction rates and compositions of reaction products by diffusion.........................................................................................1005 Metamorphic example of diffusion-limited uptake: REE behavior during garnet growth.....................................................................................................1011 Chemical diffusive fractionation..............................................................................1014 Diffusive fractionation in a thermal gradient............................................................1017 Thermochronology.................................................................................................1018 Background..............................................................................................................1018 Bulk closure..............................................................................................................1019 Continuous histories.................................................................................................1021 Dating metamorphic events......................................................................................1024 Geospeedometry........................................................................................................1025 The concept of geospeedometry...............................................................................1025 Deciphering timescales from kinetic modeling........................................................1026 Diffusion in two or three dimentions and the effect of geometry............................1027 Example: Deciphering short-term metamorphic events and timescales...................1029 Acknowledgments....................................................................................................1032 References....................................................................................................................1032 xviii

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Reviews in Mineralogy and Geochemistry

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FROM THE SERIES EDITOR The chapters in this volume represent an extensive compilation of the material presented by the invited speakers at a short course on Diffusion in Minerals and Melts held prior (December 11-12, 2010) to the Annual fall meeting of the American Geophysical Union in San Francisco, California. The short course was held at the Napa Valley Marriott Hotel and Spa in Napa, California and was sponsored by the Mineralogical Society of America and the Geochemical Society. At the MSA website, www.minsocam.org/MSA/RIM, the supplemental material associated with this volume can be found and the reader is encouraged to have a look at it. Any errata will also be posted there. The reader will also be able to find links to the electronic copies of this and other RiMG volumes. Jodi J. Rosso, Series Editor West Richland, Washington October 2010

PREFACE Geologists often need to apply diffusion theory and data to understand the degree of mass transfer, infer temperature-time histories, and address a wide range of geological problems. The aim of this volume is to provide practitioners the necessary background and data for such applications. We have made efforts to present a comprehensive overview, with discussion and assessment of diffusion data in a broad range of rock-forming minerals and all geologically relevant melts. Extensive data tables are provided as online supplements (as well as at websites maintained by individual authors), both for general usage by readers, and for experimentalists and theoreticians in the field to develop greater understanding of diffusion and plan future research directions. We would like to take this opportunity to thank the authors of individual chapters, and those who reviewed the chapters. The reviewers are: Don Baker, Harald Behrens, Bill Carlson, Michael Carroll, Fidel Costa, John Farver, John Ferry, Jiba Ganguly, Matt Heizler, Jannick Ingrin, Motoo Ito, David Kohlstedt, Ted Labotka, Chip Lesher, Yan Liang, Thomas Mueller, Jim Mungall, Martin Reich, Rick Ryerson, Jim Shelby, Frank Spera, Jim Van Orman, Yong-Fei Zheng, and anonymous reviewers. This volume and the accompanying short course in Napa Valley were made possible by generous support for student participants from the US National Science Foundation. The preparation of this volume and the short course benefited tremendously from the efforts of Jodi Rosso and Alex Speer. Youxue Zhang Ann Arbor, Michigan 1529-6466/10/0072-0000$05.00

Daniele Cherniak Troy, New York DOI: 10.2138/rmg.2010.72.0

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Reviews in Mineralogy & Geochemistry Vol. 72 pp. 1-4, 2010 Copyright © Mineralogical Society of America

Diffusion in Minerals and Melts: Introduction Youxue Zhang Department of Geological Sciences The University of Michigan Ann Arbor, Michigan, 48109-1005, U.S.A. [email protected]

Daniele J. Cherniak Department of Earth & Environmental Sciences Rensselaer Polytechnic Institute Troy, New York, 12180, U.S.A. [email protected]

INTRODUCTION: RATIONALE FOR THIS VOLUME Because diffusion plays a critical role in numerous geological processes, petrologists and geochemists (as well as other geologists and geophysicists) often apply diffusion data and models in a range of problems, including interpretation of the age of rocks and thermal histories, conditions for formation and retention of chemical compositional and isotopic zoning in minerals, controls on bubble sizes in volcanic rocks, and processes influencing volcanic eruptions. A major challenge in the many applications of diffusion data is for researchers to find relevant and reliable data. For example, diffusivities determined in different labs may differ by orders of magnitude. Sometimes the differences are a result of limitations not recognized in certain diffusion studies due to the materials or methodologies used. For example, diffusivities determined through bulk analyses are often orders of magnitude greater than those obtained from directly measured diffusion profiles; the former are often affected by cracks, extended defects and/or other additional diffusion paths whose influence may not be recognized without direct profiling. Differences in depth resolution of analytical techniques may also contribute to discrepancies among measured diffusivities, as can the occurrence of non-diffusional processes (e.g., convection, crystal dissolution or surface reaction) that may compromise or complicate diffusion experiments and interpretations of results. Sometimes the discrepancies among datasets may be due to subtle variations in experimental conditions (such as differing oxygen fugacities, pressures, or variations in H2O content of minerals and melts used in respective experimental studies). Experts in the field may be able to understand and evaluate these differences, but those unfamiliar with the field, and even some experimental practitioners and experienced users of diffusion data, may have difficulty discerning and interpreting dissagreements among diffusion findings. For those who want to investigate diffusion through experiments, it is critical to understand the advantages and limitations of various experimental approaches and analytical methods in order to optimize future studies, and to obtain a clear sense of the “state of the art” to put their own findings in perspective with earlier work. Two early books were important landmarks in diffusion studies in geology. One was a special publication by Carnegie Institution of Washington edited by Hofmann et al. (1974) titled “Geochemical Transport and Kinetics.” The other was a Reviews of Mineralogy volume edited by Lasaga and Kirkpatrick (1981) titled “Kinetics of Geochemical Processes.” Various recent tomes are available on diffusion theory in metallurgy, chemical engineering, materials 1529-6466/10/0072-0001$05.00

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science, and geology (e.g., Kirkaldy and Young 1987; Shewmon 1989; Cussler 1997; Lasaga 1998; Glicksman 2000; Balluffi et al. 2005; Mehrer 2007; Zhang 2008) and the mathematics of solving diffusion problems (e.g., Carslaw and Jaeger 1959; Crank 1975). There have also been summaries of geologically relevant diffusion data (e.g., Freer 1981; Brady 1995), review articles and book chapters presenting diffusion data for specific mineral phases (e.g., Yund 1983; Giletti 1994; Cherniak and Watson 2003) and for specific species in minerals and melts (e.g., Chakraborty 1995; Cole and Chakraborty 2001; Watson 1994) and applications of diffusion in geology (e.g., Ganguly 1991; Watson and Baxter 2007; Chakraborty 2008). However, there is no single resource that reviews and evaluates a comprehensive collection of diffusion data for minerals and melts, and previously published summaries of geologically-relevant diffusion data predate the period in which a large proportion of the existing reliable diffusion data have been generated. This volume of Reviews in Mineralogy and Geochemistry attempts to fill this void. The goal is to compile, compare, evaluate and assess diffusion data from the literature for all elements in minerals and natural melts (including glasses). Summaries of these diffusion data, as well as equations to calculate diffusivities, are provided in the chapters themselves and/or in online supplements. Suggested or assessed equations to evaluate diffusivities under a range of conditions can be found in the individual chapters. The aim of this volume is to help students and practitioners to understand the basics of diffusion and applications to geological problems, and to provide a reference for and guide to available experimental diffusion data in minerals and natural melts. It is hoped that with this volume students and practitioners will engage in the study of diffusion and the application of diffusion findings to geological processes with greater interest, comprehension, insight, and appreciation.

SCOPE AND CONTENT OF THIS VOLUME This volume begins with three general chapters. One chapter presents the basic theoretical background of diffusion (Zhang 2010), including definitions and concepts encountered in later chapters. This chapter is not meant to be comprehensive, as detailed, book-length treatments of diffusion theory can be found in other sources. Some discussion of advanced topics of diffusion theory and mechanisms can be found in individual chapters throughout the volume, including models for diffusion in melts (Lesher 2010), multi-species diffusion (Zhang and Ni 2010), multicomponent diffusion (Liang 2010; Ganguly 2010), and defect chemistry (Chakraborty 2010; Cherniak and Dimanov 2010; Van Orman and Crispin 2010). Diffusion data for minerals and melts are most commonly obtained through experimental studies which require analyses of the experimental products; these considerations are reflected in the topics of the next two chapters. For readers who are interested in carrying out experimental research or understanding experimental results and diffusion data, the second general chapter (Watson and Dohmen 2010) covers experimental methods in diffusion studies, with focus on nontraditional and emerging methods. Additional discussion of experimental methods in diffusion studies is provided in Ganguly (2010) and Farver (2010). The third general chapter reviews a range of analytical techniques applied in analyses of diffusion experiments (Cherniak et al. 2010). Experimental methods and analytical techniques are also described in other chapters in the context of discussion of specific diffusion studies. The next five chapters are on diffusion in melts (including glasses), focusing on natural melts relevant in geological systems. Zhang and Ni (2010) discuss the diffusion of H, C and O in silicate melts, which involves multi-species diffusion, where one species (such as molecular H2O) may contribute to the diffusion of two elements (such as H and O in this case). They also assess the relative importance of various diffusing species, and extract oxygen diffusion data in hydrous silicate melts from diffusion data for water. Behrens (2010) offers a thorough review and evaluation of noble gas diffusion data for natural silicate melts and industrial glasses. Lesher (2010) elaborates on the various diffusion models for self diffusion, tracer diffusion, isotopic

Introduction to Diffusion in Minerals and Melts

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diffusion and trace element diffusion. Zhang et al. (2010) summarize available diffusion data (focusing on effective binary diffusivities) of all elements in silicate melts. Liang (2010) presents a systematic assessment of multicomponent diffusion studies for silicate melts. The next eleven chapters review and evaluate diffusion data for minerals. Farver (2010) reviews H and O diffusion data for a range of mineral phases and examines the effect of oxygen, hydrogen and water fugacities on diffusion. Noble gas diffusion in minerals, notably diffusion of the important radiogenic nuclides 40Ar and 4He for application in closure temperature determinations and thermochronometry, is reviewed by Baxter (2010). Ganguly (2010) assesses cation diffusion data in garnet, with discussion of multicomponent diffusion in garnet and its geological applications. Chakraborty (2010) focuses on diffusion in (Fe,Mg)2SiO4 polymorphs (olivine, wadsleyite and ringwoodite) with a discussion of the role of defects in diffusion and the effects of pressure on diffusion in these phases. Diffusion of major and trace elements in pyroxenes, amphibole, and mica is discussed by Cherniak and Dimanov (2010). Cherniak (2010a) reviews diffusion data for feldspars, examining the effects of feldspar composition on diffusion in this common crustal mineral. Cherniak (2010d) summarizes diffusion data for the silicate phases quartz, melilite, silicate perovskite, and mullite. Van Orman and Crispin (2010) discuss diffusion in oxide minerals including periclase, magnesium aluminate spinel, magnetite, and rutile, and explore the intricacies of defect chemistry and its effects on diffusion in these deceptively simple compounds. Cherniak (2010b) reviews diffusion in the accessory minerals zircon, monazite, apatite, and xenotime, phases important in geochronologic studies. Diffusion in other minerals, including carbonates, sulfide minerals, fluorite and diamond, is reviewed by Cherniak (2010c). Brady and Cherniak (2010) take a broad overview of extant diffusion data for minerals, examining possible relations among diffusivities for various mineral phases and diffusants to assess trends and correlations that may be of value in developing or refining predictive models and empirical relations. The next two chapters discuss the specialized topics of grain-boundary diffusion and computational methods for determining diffusion coefficients. Dohmen and Milke (2010) present existing data for grain boundary diffusion in polycrystalline materials, discuss theoretical underpinnings and the different types of grain-boundary diffusion regimes, and outline mathematical treatments and experimental approaches for quantifying grain-boundary diffusion. Computation of diffusion coefficients using ab initio methods and molecular dynamics simulations are reviewed by De Koker and Stixrude (2010) with focus on recent progress and what the future may bring for these rapidly-developing techniques. The final chapter is devoted to geological applications of diffusion data (Mueller et al. 2010). The applications outlined include not only forward problems of applying diffusion theory and data to infer rates and extents of diffusion-related processes, but also inverse problems of thermochronology and geospeedometry.

REFERENCES Balluffi RW, Allen SM, Carter WC, Kemper RA (2005) Kinetics of Materials. Wiley-Interscience, Hoboken, N.J Baxter EF (2010) Diffusion of noble gases in minerals. Rev Mineral Geochem 72:509-557 Behrens H (2010) Noble gas diffusion in silicate glasses and melts. Rev Mineral Geochem 72:227-267 Brady JB (1995) Diffusion data for silicate minerals, glasses, and liquids. In: Mineral Physics and Crystallography, A Handbook of Physical Constants, Reference Shelf 2. Ahrens TJ (ed) AGU, Washington, D.C., p 269-290 Brady JB, Cherniak DJ (2010) Diffusion in minerals: an overview of published experimental diffusion data. Rev Mineral Geochem 72:899-920 Carslaw HS, Jaeger JC (1959) Conduction of Heat in Solids. Clarendon Press, Oxford Chakraborty S (1995) Diffusion in silicate melts. Rev Mineral Geochem 32:411-504 Chakraborty S (2008) Diffusion in solid silicates; a tool to track timescales of processes comes of age. Ann Rev Earth Planet Sci 36: 153-190

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Chakraborty S (2010) Diffusion coefficients in olivine, wadsleyite and ringwoodite. Rev Mineral Geochem 72:603-639 Cherniak DJ (2010a) Cation diffusion in feldspars. Rev Mineral Geochem 72:691-734 Cherniak DJ (2010b) Diffusion in accessory minerals: zircon, titanite, apatite, monazite and xenotime. Rev Mineral Geochem 72:827-870 Cherniak DJ (2010c) Diffusion in carbonates, fluorite, sulfide minerals, and diamond. Rev Mineral Geochem 72:871-897 Cherniak DJ (2010d) Diffusion in quartz, melilite, silicate perovskite, and mullite. Rev Mineral Geochem 72:735-756 Cherniak DJ, Dimanov A (2010) Diffusion in pyroxene, mica and amphibole. Rev Mineral Geochem 72:641690 Cherniak DJ, Hervig R, Koepke J, Zhang Y, Zhao D (2010) Analytical methods in diffusion studies. Rev Mineral Geochem 72:107-169 Cherniak DJ, Watson EB (2003) Diffusion in zircon. Rev Mineral Geochem 53:113-143 Cole DR, Chakraborty S (2001) Rates and mechanisms of isotopic exchange. Rev Mineral Geochem 43:83–223 Crank J (1975) The Mathematics of Diffusion. Clarendon Press, Oxford Cussler EL (1997) Diffusion: Mass Transfer in Fluid Systems. Cambridge Univ. Press, Cambridge, England de Koker N, Stixrude L (2010) Theoretical computation of diffusion in minerals and melts. Rev Mineral Geochem 72:971-996 Dohmen R, Milke R (2010) Diffusion in polycrystalline materials: grain boundaries, mathematical models, and experimental data. Rev Mineral Geochem 72:921-970 Farver JR (2010) Oxygen and hydrogen diffusion in minerals. Rev Mineral Geochem 72:447-507 Freer R (1981) Diffusion in silicate minerals and glasses: a data digest and guide to the literature. Contrib Mineral Petrol 76:440-454 Ganguly J (2010) Cation diffusion kinetics in aluminosilicate garnets and geological applications. Rev Mineral Geochem 72:559-601 Ganguly J (ed) (1991) Diffusion, Atomic Ordering, and Mass Transport: Selected Topics in Geochemistry. Advances in Physical Geochemistry, Vol. 8. Springer Giletti BJ (1994) Isotopic equilibrium/disequilibrium and diffusion kinetics in feldspars. In: Feldspars and their reactions. NATO Advanced Study Institutes Series. Series C: Mathematical and Physical Sciences. Parsons I (ed) D. Reidel Publishing, Dordrecht-Boston, 421:351-382 Glicksman ME (2000) Diffusion in Solids: Field Theory, Solid-State Principles, and Applications. Wiley, New York Hofmann AW, Giletti BJ, Yoder HS, Yund RA (1974) Geochemical Transport and Kinetics. Carnegie Institution of Washington Publ., Vol 634. Washington, DC Kirkaldy JS, Young DJ (1987) Diffusion in the Condensed State. The Institute of Metals, London Lasaga AC (1998) Kinetic Theory in the Earth Sciences. Princeton University Press, Princeton, NJ Lasaga AC, Kirkpatrick RJ (eds) (1981) Kinetics of Geochemical Processes. Reviews in Mineralogy, Vol 8. Mineralogical Society of America, Washington DC Lesher CE (2010) Self-diffusion in silicate melts: theory, observations and applications to magmatic systems. Rev Mineral Geochem 72:269-309 Liang Y (2010) Multicomponent diffusion in molten silicates: theory, experiments, and geological applications. Rev Mineral Geochem 72:409-446 Mehrer H (2007) Diffusion in Solids: Fundamentals, Methods, Materials, Diffusion-Controlled Processes. Springer, Berlin Mueller T, Watson EB, Harrison TM (2010) Applications of diffusion data to high-temperature earth systems. Rev Mineral Geochem 72:997-1038 Shewmon PG (1989) Diffusion in Solids. Minerals. Metals & Materials Society, Warrendale, PA Van Orman JA, Crispin KL (2010) Diffusion in oxides. Rev Mineral Geochem 72:757-825 Watson EB (1994) Diffusion in volatile-bearing magmas. Rev Mineral 30:371-411 Watson EB, Baxter EF (2007) Diffusion in solid-Earth systems. Earth Planet Sci Lett 253:307-327 Watson EB, Dohmen R (2010) Non-traditional and emerging methods for characterizing diffusion in minerals and mineral aggregates. Rev Mineral Geochem 72:61-105 Yund RA (1983) Diffusion in feldspars. In: Feldspar Mineralogy, Short Course Notes 2. Ribbe P (ed) Mineralogical Society of America, p 203-222 Zhang Y (2008) Geochemical Kinetics. Princeton University Press, Princeton, NJ Zhang Y (2010) Diffusion in minerals and melts: theoretical background. Rev Mineral Geochem 72:5-59 Zhang Y, Ni H (2010) Diffusion of H, C, and O components in silicate melts. Rev Mineral Geochem 72:171225 Zhang Y, Ni H, Chen Y (2010) Diffusion data in silicate melts. Rev Mineral Geochem 72:311-408

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Reviews in Mineralogy & Geochemistry Vol. 72 pp. 5-59, 2010 Copyright © Mineralogical Society of America

Diffusion in Minerals and Melts: Theoretical Background Youxue Zhang Department of Geological Sciences The University of Michigan Ann Arbor, Michigan, 48109-1005, U.S.A [email protected]

INTRODUCTION Diffusion is due to thermally activated atomic-scale random motion of particles (atoms, ions and molecules) in minerals, glasses, melts, fluids, and gases (Fig. 1). The random motion leads to a net flux when the concentration (more strictly speaking, the chemical potential) of a component is not uniform. Even though diffusion is a microscopic process, it can lead to macroscopic effects. For example, the initial phase of explosive volcanic eruptions (or more commonly encountered champagne eruptions) is powered by bubble growth, which in turn is controlled by diffusion that brings gas molecules into bubbles. This chapter provides a brief review of the theory of diffusion in minerals and melts (including glasses). More complete coverage of diffusion theory can be found in Crank (1975), Kirkaldy and Young (1987), Shewmon (1989), Cussler (1997), Lasaga (1998), Glicksman (2000), Balluffi et al. (2005), Mehrer (2007), and Zhang (2008). Youxue Zhang (Ch 2) Page mechanism 1 In minerals, diffusive transport is the only for particles to move from one location to another. For example, homogenization of a zoned crystal and loss of radiogenic













Figure 1. An example of random motion of particles. Initially (the left panel), all A particles (such as Fe2+ ions in garnet) represented by filled circles are in the lower side, and all B particles (such as Mg2+ ions in 2+ ions garnet) Fig. represented by open circles are in the upperInitially side. (the Dueleft to panel), random motion, fluxinof A 1. An example of random motion of particles. all A particlesthere (suchis as aFenet from the lower side to the upper side, and a net flux of B from the upper side to the lower side (the middle garnet) represented filled circles lower side, and all B particles (such asand Mg2+ ions in garnet) and right panels). As timebyincreases, A are andinBthewill eventually become randomly uniformly distributed in the whole system. This situation for diffusion is often encountered in diffusion experiments and is referred represented by open circles are in the upper side. Due to random motion, there is a net flux of A from the lower side to as a diffusion couple. to the upper side, and a net flux of B from the upper side to the lower side (the middle and right panels). As time

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increases, A and B will eventually become randomly and uniformly distributed in the whole system. This situation for diffusion is often encountered in diffusion experiments and is referred to as a diffusion couple.


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nuclides (such as 40Ar from the decay of 40K) from a mineral are through diffusion. In silicate melts, mass transport can be through either diffusion or flow (or convection). Only diffusion is covered in this chapter. Even when convection is present, it is still necessary to understand diffusion because in the boundary layer mass transport is through diffusion. Diffusion also plays a role during crystal growth and dissolution in a melt, key processes in magma solidification and evolution. One of the most important geological applications of diffusion is the inverse problem, to infer the details of thermal histories and factors such as closure temperature, apparent equilibrium temperature, and cooling rates from diffusion properties (Zhang 2008). Thermochronology and its application to the understanding of tectonic uplift and erosion rates, require a thorough understanding of diffusion in minerals. The mathematics of diffusion is complicated. An excellent reference book is by Crank (1975), which provides analytical solutions to many diffusion problems. The mathematical description of diffusion is similar to that of heat conduction. Hence, analytical solutions to heat conduction problems (e.g., Carslaw and Jaeger, 1959) can also be applied to diffusion. Because the mathematical treatment is in itself specialized and can be found in the aforementioned treatises, in this chapter, I focus on concepts of diffusion relevant to geological and experimental diffusion studies, rather than the mathematical solutions. Solutions for specific diffusion problems will be given without derivations.

FUNDAMENTALS OF DIFFUSION Basic concepts The German physiologist Adolf Fick (1829-1901) investigated diffusive mass transport and proposed the following phenomenological law that describes diffusion by analogy to Fourier’s law of heat conduction J = −D

∂C ∂x

(1)

where J is the diffusive mass flux (a vector), D is the diffusion coefficient (also referred to as the diffusivity), C is the concentration of the component under consideration (in mass per unit volume, such as kg/m3, or number of atoms per m3, or mol/m3), x is distance, ∂C/∂x is the concentration gradient (a vector), and the negative sign means that the direction of the diffusive flux is opposite to the direction of the concentration gradient (i.e., diffusive flux goes from high to low concentration, but the gradient is from low to high concentration). Hence, when the concentration gradient is large (i.e., the concentration profile is steep), the diffusive flux is also large. The unit of D is length2/time, such as m2/s, mm2/s, and µm2/s (1 m2/s = 106 mm2/s = 1012 µm2/s). The value of the diffusivity is an indication of the “rate” of diffusion and, hence, is essential in quantifying diffusion. Diffusivities depend on several factors, including temperature, pressure, composition, and physical state and structure of the phase, and sometimes oxygen fugacity. Some general relations between diffusivities and other parameters will be presented later in this chapter. Diffusivity values in various systems are the main focus of this volume, of which this chapter is a part. When diffusion is mentioned without special qualification, it refers to volume diffusion occurring inside a phase due to thermally activated random motion (in contrast to grainboundary diffusion or eddy diffusion in natural waters). Typical values of diffusion coefficients are (see Fig. 2 for diffusivity of a neutral gas species as a function of temperature; see also Watson and Baxter 2007 for generalized diffusion behaviors in geological materials): •

In gas, D is large, about 10−5 m2/s in air at 300 K;

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Figure 2. Ar diffusion data in air (gas) (calculated using relations in Cussler 1997), water (liquid) (Wise and Houghton 1966), basalt melt (Nowak et al. 2004), rhyolite melt (Behrens and Zhang 2001) and the 2. Ar diffusion data in1981). air (gas) (calculated using relations in Cussler 1997), water (liquid) (Wise and Houghton mineral Fig. hornblende (Harrison 1966), basalt melt (Nowak et al. 2004), rhyolite melt (Behrens and Zhang 2001) and the mineral hornblende

•

In aqueous (Harrison 1981).

solution, D is intermediate, about 10−9 m2/s in water at 300 K;

•

In silicate melts, D is small, about 10−11 m2/s at 1600 K for divalent cations;

•

In minerals, D is extremely small, about 10−17 m2/s at 1600 K for divalent cations.

Grain-boundary diffusion is diffusion along interphase interfaces, including mineralfluid interfaces (or surfaces) or mineral-mineral interfaces. Eddy (or turbulent) diffusion in fluid phases is due to non-thermal random disturbances such as waves, fish swimming, boats cruising, etc. Hence, eddy diffusion is fundamentally different from thermally activated volume diffusion. Both grain-boundary diffusivities and eddy diffusivities are often several orders of magnitude higher than the respective volume diffusivities listed above. In Fick’s first law, the diffusive flux is related to the concentration gradient. In diffusion studies, often we need to determine how a concentration profile would evolve with time given the initial concentration distribution. For this purpose, we need an equation (referred to as the diffusion equation) to describe how the concentration is related to space and time, such as C(x,t) for the one-dimensional case. The one-dimensional diffusion equation often takes the following form ∂C ∂ 2C =D 2 ∂t ∂x

(2)

where D is independent of C and x. Equation (2) is also referred to as Fick’s second law. Below is a derivation of Equation (2) from Equation (1) and the mass balance condition. Consider diffusion across a thin sheet with the left side at x and the right side at x+dx (thickness of dx). Assume that the flux is one-dimensional along the x direction (Fig. 3). Then the total mass variation in the volume defined by thickness dx and an arbitrary area S and equals the flux into the sheet from the left side (x), JxS, minus the flux out of the sheet from the right side (x+dx), Jx+dxS:

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Youxue Zhang (Ch 2)

Page 3

Jx

Zhang

Jx+dx C

x

Figure 3. Sketch of fluxes into and out of an element volume. The flux along the x-axis points to the right (the x-axis also points to the right). The flux at x is Jx, and that at x+dx is Jx+dx. The net flux into the small volume is (Jx − Jx+dx), which causes the mass and density in the volume to vary.

x+dx

Fig. 3. Sketch of fluxes into and out of an element volume. The flux along the x-axis points to the right (the x-axis also points to the right). The flux at x is Jx, and that at x+dx is Jx+dx. The net flux into the small volume is (Jx -

∂J ( x ) ∂C Sdx = J x S − J x + dx S = − x Sdx ∂t ∂x

Jx+dx), which causes the mass and density in the volume to vary.

(3)

where Jx (a scalar) is the flux along increasing x direction (the vector flux J = Jxi where i is the unit vector along the x axis). Hence, ∂J ( x ) ∂C =− x ∂t ∂x

( 4)

Combining the above with Fick’s first law (Eqn. 1) leads to: ∂C ∂ ∂C = (D ) ∂t ∂x ∂x

(5)

If D is independent of C and x, the above is simplified to Equation (2). In three dimensions, the diffusive flux for a component (Eqn. 1) takes the following form: J = – D∇C

(6)

the mass balance equation (Eqn. 4) becomes: ∂C = −∇ ⋅ J ∂t

( 7)

and the diffusion equation (Eqn. 5) becomes: ∂C = ∇ ⋅ ( D∇C ) ∂t

(8)

where ∇ is the gradient operator when it is applied to a scalar C, and the divergent operator when it is applied to a vector ∇C (i.e., ∇ turns a scalar to a vector and a vector to a scalar). From Equation (2), it can be seen that if ∂2C/∂x2 = 0 at a position (e.g., point 1 in Fig. 4), i.e., if C is locally a linear function of x (including the case of constant concentration), then ∂C/∂t = 0, meaning that the concentration at the position would not vary with time. If ∂2C/∂x2 > 0 at the position (point 2 in Fig. 4; concave up), then ∂C/∂t > 0, meaning that the concentration at the position would increase with time. If ∂2C/∂x2 < 0 at the position (point 3 in Fig. 4; concave down), then the concentration at the position would decrease with time. Although we often talk about diffusion “rate”, and the rate is related to the diffusion coefficient, diffusion is a peculiar process in which there is no single diffusion “rate”. From solutions

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2 and the corresponding ∂2C/∂x2 versus x (arbitrary units) to illu Fig.corresponding 4. Concentration profile∂C2C/∂x versus x, Figure 4. Concentration profile C versus x, and the versus x (arbitrary units) to 2 illustrate whether C increases, decreases or stays the same with decreases time. At point 1, with ∂2C/∂x 0 1,and hence 2 = 0 and hence ∂C/∂t = whether C increases, or stays the same time. At = point ∂2C/∂x ∂C/∂t = 0. At point 2 (concave up), ∂2C/∂x2 > 0 and hence ∂C/∂t > 0. At point 3 (concave down), ∂2C/∂x2 (concave up), ∂2C/∂x2 > 0 and hence ∂C/∂t > 0. At point 3 (concave down), ∂2C/∂x2 < 0 and hence ∂C/ < 0 and hence ∂C/∂t < 0.

of the diffusion equation, the diffusion distance is proportional not to duration, but to the square root of duration; the relation is often written as x ≈ Dt

( 9)

This distance may also be referred to as the mid-concentration distance, or half distance of diffusion (Zhang 2008, p. 201-204), which will become clearer later. Defining the diffusion “rate” as how rapidly the diffusion distance advances with time (dx/dt), then the “rate” equals 0.5(D/t)1/2, and is infinity at t = 0 and then decreases gradually with time. Fig. 4. Concentration profile C versus x, and the corresponding ∂2C/∂x2 versus x (arbitrary units) to illustrate

The diffusivity increases rapidly with temperature, following the Arrhenius relation (Fig. 2),

whether C increases, decreases or stays the same with time. At point 1, ∂2C/∂x2 = 0 and hence ∂C/∂t = 0. At point 2

D = D0 e − E / RT

(concave up), ∂2C/∂x2 > 0 and hence ∂C/∂t > 0. At point 3 (concave down), ∂2C/∂x2 < 0 and hence ∂C/∂t < 0.

(10)

where T is the absolute temperature in K, D0 is the pre-exponential factor and equals the value of D at T = ∞, E is the activation energy and is a positive number, and R is the universal gas constant. The pressure dependence of diffusivities can be either positive or negative. The following equation is often used to describe both the temperature and pressure dependence of diffusivity D = D0 e

− ( E + P ∆V ) / RT

(11)

where ∆V is referred to as the activation volume, which can be either positive (leading to a decrease of D with increasing P) or negative (leading to an increase of D with increasing P). Negative ∆V is not rare. From Equation (11), the activation energy depends on pressure (when ∆V ≠ 0). Similarly, the activation volume ∆V may also depend on temperature, which would change the form of the above equation (see later discussion).

Microscopic view of diffusion Microscopically and statistically, diffusion can be quantified using random walk of particles (atoms, ions, or molecules). Consider, for example, diffusion of Mg2+ (counterbalanced by Fe2+ in the opposite direction) in garnet along any direction, labeled as the x direction. (A cubic crystal is used here so that the effect of diffusional anisotropy does not have to be considered.) Consider two adjacent lattice planes (left and right) at distance l apart. If the jumping distance of Mg2+ is l and the frequency of Mg2+ ions jumping away from the original position is f, then the number of Mg2+ ions jumping from left to right is ½nLfdt and that from right to left is ½nRfdt, where nL and nR are the number of Mg2+ ions per unit area on the left and

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right planes. The factor ½ in the expressions is due to the fact that the ions in each plane can jump to both sides, but we consider only one direction. The jumping frequency f is assumed to be the same from left to right or from right to left, i.e., random walk is assumed. Therefore, the net flux from the left plane to the right plane is J=

1 ( nL − n R ) f 2

(12)

Since nL = lCL and nR = lCR where CL and CR are the concentrations of Mg2+ on the left and right planes, then 1 J = l (CL − CR ) f 2

(13)

1 ∂C J = − l2 f 2 ∂x

(14)

Because CL – CR = –l∂C/∂x, we have

Comparing this with Fick’s law (Eqn. 1), we have 1 D = l2 f 2

(15)

Thus, microscopically, in one-dimensional diffusion, the diffusion coefficient may be interpreted as one-half of the jumping distance squared times the overall jumping frequency. Since l is of the order 3×10−10 m (the interatomic distance in a lattice), the jumping frequency can be roughly estimated from D. For D ≈ 10−17 m2/s, as in a typical mineral at high temperature, the jumping frequency is 2D/l2 ~ 220 per second. Because ion jumping requires a site to accept the ion, the jumping frequency in minerals depends on the concentration of vacancies and other defects. Hence, high defect concentrations lead to high diffusivities. In melts, the jumping frequency is much higher (about 108 per second), depending on the flexibility of the liquid structure, and may also be related to viscosity. The above analysis can be carried forward to the full statistical treatment of random walk using either theoretical analysis (Gamow 1961) or computer simulations (Kleinhans and Friedrich 2007). If initially a large number (trillions) of particles were at a single position (defined as x = 0), after more than 100 jumping steps, the distribution can be approximated well by a continuous function. For one-dimensional diffusion, the concentration of particles at x (or the probability of finding a particle at x) follows the Gaussian distribution: C ( x, t ) =

2 M e − x /( 4 Dt ) 1/ 2 ( 4 πDt )

(16)

where M is the total number of particles, all of which were initially at x = 0. This diffusion problem is known as diffusion from an instantaneous plane source.

Various kinds of diffusion There are many kinds of diffusion encountered in nature and experimental studies. The definitions may differ somewhat in the literature, making it less straightforward to deal with the terms. Below is a summary of the various kinds of diffusion described in most of the geological literature. Because diffusion involves a diffusing species in a diffusion medium, it can be classified based on either the diffusion medium or the diffusing species. When considering the diffusion medium, thermally activated diffusion may be classified as volume diffusion and grain-boundary diffusion. Volume diffusion is diffusion in the interior of a phase; an example is the diffusion of Mg and Fe2+ in a garnet crystal, leading to the homogenization of

Theoretical Background of Diffusion in Minerals and Melts

11

a garnet crystal initially zoned in Fe2+ and Mg (Ganguly 2010, this volume). Volume diffusion is what is typically referred to when we simply say “diffusion” without further qualifiers. In volume diffusion, the diffusion medium can be either isotropic or anisotropic. In an isotropic diffusion medium, diffusion properties do not depend on direction. Both melts (and glasses) and isometric minerals are isotropic diffusion media, but non-isometric minerals are in general anisotropic diffusion media (although in some cases, the dependence of diffusivities on the direction is weak). Anisotropic diffusion will be treated later in this chapter. Grain-boundary diffusion is diffusion along interphase interfaces, including mineral-fluid interfaces (or surfaces), interfaces between the same minerals, and those between different minerals. Because many bonds are not satisfied for atoms on the interface, there are generally very high concentrations of defects, leading to very high grain-boundary diffusivities compared with volume diffusivities. For example, at 1473 K, the grain-boundary diffusivity of Si at forsterite-forsterite boundaries is about 9 orders of magnitude greater than the volume diffusivity of Si in forsterite (Farver and Yund 2000). Grain-boundary diffusion will be the subject of a chapter in this volume (Dohmen and Milke 2010, this volume). Considering differences in the diffusing species, diffusion can be classified as self diffusion, tracer diffusion, or chemical diffusion that can be further distinguished as trace element diffusion, binary diffusion, multispecies diffusion, multicomponent diffusion, and effective binary diffusion. Below is a discussion of these terms; first the definition used in this work is shown, then alternative definitions are also mentioned. Self diffusion. There is no chemical potential gradient in the system in terms of elemental composition but there is difference in the isotopic ratios (or chemical potential gradients are present only in isotopes) (Lasaga 1998; Zhang 2008). The diffusion is monitored through difference in isotopic fractions. For example, in a diffusion couple made of basalt melt, one side may have a high 44Ca/SiCa ratio and the other side has normal Ca isotope ratios, but the elemental composition of the melts in both sides of the couple is uniform (e.g., in the experiments of LaTourrette et al. 1996; LaTourrette and Wasserburg 1997). In Figure 1, one may view the solid circles as 44Ca-enriched Ca, and the open circles as normal Ca, and the matrix is the haplobasalt melt. Because there are no chemical (or elemental) gradients, the diffusivity, which often depends on chemical composition of the system, is assumed to be constant. This works well for self diffusion without exceptions. Small differences, ≤1% relative, in the diffusivities of different isotopes have been measured by, e.g., Richter et al. 2008. Such small differences are important in understanding isotopic fractionation but negligible in quantifying the self diffusion coefficient of an element. Other definitions of self diffusion. Some authors consider self diffusion to be the diffusion of the exact same species (not even with isotopic differences) (e.g., Lesher 2010, this volume; Mungall, personal communication). Such self-diffusivities cannot be measured, however. Others may use self diffusivity to mean the binary diffusivity in the hypothetical ideal mixing case (e.g., Lesher 1994; Ganguly 2002, p 275), which was referred to as intrinsic diffusivity by Zhang (1993). Still others may use self diffusivity to mean the diagonal diffusivity in a multicomponent diffusion matrix (e.g., De Koker and Stixrude 2010, this volume); multicomponent diffusion will be explained later. The diffusion described as self diffusion in the preceding paragraph is sometimes referred to as isotopic exchange (e.g., Lesher, personal communication), or tracer diffusion (e.g., Ganguly 2002, p 275; Mungall, personal communication). Tracer diffusion. A tracer is introduced into the system with undefined low concentrations (e.g., Fig. 5). The tracer is often a radioactive isotope, such as 134Cs used to study Cs diffusion in albite melt (Jambon and Carron 1976), but can also be an otherwise detectable trace element as long as there are no major concentration gradients. Some authors distinguish tracer diffusion using a radioactive isotope versus trace element diffusion (Baker 1989). Zhang et al. (2007)

Zhang

12 (a)

(b)

Figure 5. Thin-source diffusion. (a) Set up of thin-source diffusion. Initially there is no or low diffusant in the inside of the cylinder (or disk), but one surface (in the drawing it is the top surface) has a thin layer of the diffusant. “Thin” means much thinner than the diffusion distance. (b) The resulting diffusion profile (C versus x where x is distance from the upper surface vertically downward) after the experiment for two different times. Both the surface concentration and the length of the diffusion profile depend on time.

discussed the difference between 14C tracer diffusivities and CO2 trace element diffusivities but the difference is likely due to limited spatial resolution in b-particle mapping (International Commission on Radiation Units and Measurements 1984; Mungall 2002) when 14C tracer diffusivities were determined. In tracer diffusion, the bulk composition of the system is roughly uniform, with the only variation being the concentration of the tracer, and the dilution effect by the tracer on other elemental concentrations. Hence, the diffusivity is assumed to be constant, meaning that the diffusivity does not depend on the concentration of the component itself at low concentrations. This is true for many components, but at least for H2O diffusion (using a 3 H tracer), it has been found that the H2O diffusivity depends on its own concentration even at low concentration levels of hundreds or even tens of ppm (Drury and Roberts 1963). Hence, there may be exceptions to the assumption that a tracer diffusion profile can be well described by a constant diffusivity. Another complexity is that the deposited component containing the diffusant (e.g., 134Cs is deposited as CsCl in the study of Jambon and Carron 1976) may have very high solubility in the material, which would result in compositional variation, for example in the case of Jambon and Carron (1976), from almost pure CsCl to the albite melt in a short distance, meaning that there could be significant chemical potential gradients. Other definitions of tracer diffusion. In the definition of some authors, tracer diffusion would include self diffusion (or isotopic exchange) defined above (e.g., Ganguly 2010, this volume). Other authors would include the condition that the trace element behaves as Henrian (meaning constant activity coefficient) (e.g., Lesher 2010, this volume), which would mean that diffusion of 3H2O would not be tracer diffusion because its chemical activity is not proportional to its concentration (Drury and Roberts 1963). Either self diffusion or tracer diffusion. When a radioactive tracer is introduced into a system that contains stable isotopes of the tracer, the diffusion may be referred to as either self diffusion or tracer diffusion. For example, 24Na tracer diffusion into an albite melt (Jambon and Carron 1976) can also be said to be self diffusion. Chemical diffusion. There is chemical potential gradient in major and minor components. Among chemical diffusion, trace element diffusion, binary diffusion (also referred to as

Theoretical Background of Diffusion in Minerals and Melts

13

interdiffusion or mutual diffusion), multispecies diffusion, multicomponent diffusion, and effective binary diffusion may be distinguished. Trace element diffusion. Some authors separate trace element diffusion from tracer diffusion (e.g., Baker 1989), where a trace element means that the concentration is no more than thousands of ppm). Baker (1989) discussed tracer versus trace element diffusion in which trace elements diffusion occurs in the presence of concentration gradients of major oxides such as SiO2 and MgO. If the concentration gradient is only in the trace element and the concentration variation of other components is due to the dilution effect of the trace element, the trace element diffusion is similar to tracer diffusion. If there are concentration gradients in major components, trace element diffusion often show uphill diffusion and must be treated in the framework of multicomponent diffusion. In this work, trace element diffusion is arbitrarily limited to the diffusion of an element with concentrations of up to1 wt% (that is, it includes minor elements) when the other concentration gradients are due to dilution by the diffusion component, so that it is similar to tracer diffusion. Binary diffusion (also referred to as interdiffusion, or mutual diffusion) refers to diffusion in a binary system (such as MgO-SiO2 diffusion in MgO-SiO2 binary melts, or FeMg diffusive exchange in olivine). Self diffusion may be viewed as a special type of binary diffusion (no chemical composition gradient), such as 18O-16O exchange in dry quartz. Tracer diffusion may also be viewed as another special type of binary diffusion (the tracer exchanges with the rest of the system). Furthermore, tracer diffusion is the limiting case of binary diffusion. Consider Fe-Mg interdiffusion in olivine. As the system composition approaches pure forsterite, Fe concentration becomes low, and hence the case approaches Fe tracer diffusion in forsterite. As the system approaches pure fayalite, Mg concentration is low, and the case approaches Mg tracer diffusion in fayalite. Multispecies diffusion. When the diffusing component can be present in two or more species, such as diffusion of H2O that is present as H2O molecules and hydroxyl groups (Doremus, 1969, 1995; Zhang et al. 1991a,b), or diffusion of CO2 that may be in the form of carbonate ions and CO2 molecules (Nowak et al. 2004), the diffusion is referred to as multispecies diffusion. The understanding of multispecies diffusion is one of the contributions made by geologists to the theory of diffusion (Zhang et al. 1991a,b; Zhang and Behrens 2000; Behrens et al. 2007). Multicomponent diffusion. If diffusive transport involves three or more components in the system, the diffusion is referred to as multicomponent diffusion (e.g., Cussler 1976; Lasaga 1979; Ghiorso 1987; Trial and Spera 1994; Kress and Ghiorso 1993, 1995; Liang et al. 1997; Mungall et al. 1998). Natural melts and many minerals are multicomponent systems. Because of the complexity in treating multicomponent diffusion, simple treatments, which work well in some cases, but not in others, have often been applied. The most often applied simple treatment is effective binary diffusion, discussed below. Effective binary diffusion. When diffusion of a component in a multicomponent system is treated as simply due to its own concentration gradient (equivalent to either (i) treating all other components as one combined component, or (ii) ignoring the cross diffusivities in the multicomponent diffusion matrix), the diffusion of the component is referred to as effective binary diffusion (Cooper 1968). In the case of tracer or trace element diffusion without major elemental concentration gradients, strictly speaking, if there is a chemical potential gradient in the tracer component, there will also be chemical potential gradients in other components, however small they may be. Therefore, tracer diffusion may be viewed as a special type of effective binary diffusion. Later in this chapter, I will classify effective binary diffusion further into the first kind of effective binary diffusion (FEBD) and second kind of effective binary diffusion (SEBD).

14

Zhang

Equation (2) describes binary (as well as self and tracer) diffusion with a constant diffusivity and along one direction (often in an isotropic system, but can also be an anisotropic system along a principal axis of diffusion, as explained later). Measurable diffusion effects require at least a binary system, because diffusion in a one-component system, even though theoretically conceivable, does not lead to measurable or macroscopic consequences: the system is always uniform in composition (100% of the component). Two measurably different components (e.g., 18 O-enriched versus 16O-enriched materials, or Fe2+ and Mg2+ exchange) are a minimum for diffusion to lead to detectable elemental or isotopic concentration profiles. Binary diffusion is the simplest diffusion when the mathematics of diffusion is discussed (i.e., when diffusion problems are solved). Many other types of diffusion problems, if mathematically more complicated, are often transformed into a binary diffusion equation. The distinction between isotropic versus anisotropic (different diffusivities along different directions on an interphase interface) diffusion and among self diffusion, tracer diffusion, and chemical diffusion can also be made for grain-boundary diffusion (Dohmen and Milke 2010, this volume).

General mass conservation and various forms of the diffusion equation Even though some simple forms of the diffusion equation have been presented earlier (Eqns. 2, 5 and 8), to understand the general diffusion problem, it is necessary to examine general mass conservation and diffusion. Mass conservation means that mass is conserved except during nuclear reactions. Even during nuclear decay, mass loss from a radioactive nuclide is still only < 0.03%. Hence, total mass is conserved for the whole system. On the other hand, mass conservation for an element or nuclide must include sinks due to radioactive decay and sources due to radiogenic growth. For each chemical species, reactions producing or consuming it must be included in the mass balance equations. For a closed system, mass conservation means that total mass in the system is constant. For an open system, mass conservation means that the mass increase or decrease is quantitatively due to mass flux into or out of the system. The differential form for total mass conservation in a representative element volume is as follows: ∂ρ = −∇ ⋅ J ∂t

(17)

where r is density, J is total mass flux (whereas J in Eqn. 7 is the flux of a component), and ∇·J is the divergence of J. The total mass flux J is related to the bulk flow and can be written as ru, where u is the flow velocity of the bulk material. Hence, the above equation can also be written in the following form: ∂ρ = −∇ ⋅ (ρu ) ∂t

(18)

which is the mass conservation equation, also known as the continuity equation in fluid mechanics. For a given species, the flux can be divided into convective (or advective) flux (i.e., flow) and diffusive flux. As clarified by Richter et al. (1998), the distinction between diffusive flux and convective flux is a matter of reference frame. In a barycentric (or mass-fixed) reference frame, the convective velocity u is defined as: N

u = ∑ wi u i

(19)

i =1

where N is the number of components, wi is the mass fraction of component i, ui is the flow velocity of i. (Similarly, volume fixed and molar fixed reference frames can be defined by

Theoretical Background of Diffusion in Minerals and Melts

15

interpreting wi in Eqn. 19 to be volume fraction of i or mole fraction of i.) The diffusive flux of any component i, or more generally, any species k, relative to the motion of the local center of mass, can be written as follows: J k = ρwk ( u k − u ) = Ck ( u k − u )

(20)

The conservation equation for a species depends on whether other species can react to form or consume the species under consideration. Without reaction terms, the mass balance equation is Equation (7). In the presence of reaction terms, the conservation for species k can be written as: m dξ ∂Ck = −∇ ⋅ J k + ∑ ν kj j ∂t dt j =1

(21)

where Ck is the mole concentration of k (such as mol/m3) because reaction terms are expressed in mole concentrations, dxj/dt is the net chemical reaction rate (i.e., rate of forward reaction minus rate of backward reaction) of reaction j, nkj is the stoichiometric coefficient of species k in reaction j, and m is the total number of reactions, including not only the independent but also the dependent reactions. The value of nkj is positive when component k is a product and negative when component k is a reactant. Positive reaction terms are also called sources, and negative reaction terms are also called sinks. Specific examples of the reaction terms can be found below. For a binary system without reaction terms, combining the above equation with Fick’s law in three dimensions, Jk = −D∇Ck, leads to Equation (8). One famous example with a reaction term is mass conservation of radiogenic 40Ar in a mineral (such as hornblende). Because 40K decays to 40Ar at a rate of le40K = le40K0e−lt, where 40 K0 is the initial concentration of 40K, l is the overall decay constant of 40K, and le is the branch decay constant of 40K to 40Ar, the concentration of 40Ar can be expressed as ∂ 40 Ar ∂ 2 ( 40 Ar) =D + λ e 40 K ∂t ∂x 2

(22)

where 40Ar and 40K are atomic concentrations (such as mol/m3). Another example is OH groups in a silicate melt. OH and molecular H2O (H2Om) can convert to each other through the following reaction (Stolper 1982a,b): H 2Om (melt ) + O(melt )  2OH(melt )

(23)

where “melt” indicates the melt phase. Assuming the above reaction is elementary, meaning it is accomplished by a single step (which may not be correct), with forward and backward reaction rate coefficients of kf and kb, respectively, the concentration variation of OH with time may be expressed as ∂COH ∂ 2COH 2 =D + 2 kf CH2 Om CO − 2 kbCOH ∂t ∂x 2

(24)

where CH2Om, COH, and CO are mole concentrations (mol/m3) of H2Om, OH, and anhydrous oxygen, and the factor 2 is the stoichiometric coefficient in Reaction 23. Experimental studies show that the diffusive term above is negligible, and the OH concentration change is almost entirely due to Reaction (23) (Zhang et al. 1991a). One way to avoid dealing with chemical reactions in the diffusion equation is to use components whose concentrations are independent of chemical reactions. One choice is to use elemental concentrations (such as H concentration). For H2O diffusion, the convention is to use total H2O concentration (H2Ot where CH2Ot = COH/2 + CH2Om). Then the diffusion equation would not include reaction terms, but will include diffusive fluxes of different species, as follows:

Zhang

16 ∂CH2 Ot ∂t

∂CH2 Om 1 ∂  = −∇ ⋅ J H2 Om − ∇ ⋅ J OH =  DH2 Om 2 ∂x  ∂x

 1 ∂  ∂COH  +  DOH ∂x  2 ∂ x   

(25)

Hence, in treating the diffusion of a multispecies component, two approaches may be used. In the first approach, the diffusion equation of a non-conservative species is used, with only one diffusion term but with extra reaction terms. In the second approach, the diffusion for the total component is considered, which contains diffusive contributions from different species but does not contain the reaction terms. In the literature, the second approach is often used (e.g., Zhang et al. 1991a,b; Zhang and Behrens 2000; Behrens et al. 2007). The most oftenencountered multispecies diffusion problems in geology are H2O, CO2, and oxygen diffusion in silicate melts, which will be reviewed by Zhang and Ni (2010, this volume). Similar diffusion problems that have been examined to some degree include diffusion of multivalent elements, such as Fe (Fe2+ and Fe3+), S (S2−, S4+ in SO2, and S6+ in SO42−), Eu (Eu2+ and Eu3+), Sn (Sn2+ and Sn4+), etc. (e.g., Behrens et al. 1990; Behrens and Hahn 2009). There are other variants of the diffusion equation. The concentration in the fundamental diffusion equation is mole per unit volume (mol/m3), especially when there are reactions because the stoichiometric coefficients are in terms of moles. If there are no reaction terms, the concentration unit can also be mass per unit volume (such as kg/m3). Often concentrations are measured as mass fractions (or wt%), or mole fractions. If the mass density of the diffusion medium of a binary system is roughly constant, i.e., r = C1+C2 in kg/m3, is constant, then ∂w = ∇ ⋅ ( D∇w ) ∂t

(26)

where w is mass fraction C/r (or wt%) of either component. One rough example is diffusion in silicate melts. If the bulk molar density of a binary system, i.e., r = C1+C2 in mol/m3, is constant, then ∂X = ∇ ⋅ ( D∇X ) ∂t

(27)

where X is mole fraction of either component. One rough example is Fe2+-Mg2+ exchange in olivine. In reality, neither molar density nor mass density is perfectly constant in a system, but if the variation is small (e.g., < 10%), the approximations are often made in literature for simplicity and the errors from such approximations are small. The choice of the equations is based on convenience instead of rigorousness. In minerals, mole fractions are usually used. In melts, mass fractions are often used. When there is large variation in density (e.g., > 10%, such as from a mineral to a melt), concentrations in mol/m3 or kg/m3 should be used. Equation (8) is the general equation for diffusion in a binary system without reaction terms or multiple species. If D is constant, Equation (8) becomes ∂C = D∇ 2C ∂t

(28)

If diffusion is one-dimensional, Equation (8) becomes Equation (5). If diffusion is onedimensional and D is independent of C and x, then Equation (8) becomes Equation (2). All the above equations are for binary systems and isotropic diffusion media. Diffusion in multicomponent systems or anisotropic diffusion media is more complex and will be discussed in later sections (and chapters). Diffusion equations in three dimensions in isotropic media are discussed below.

Theoretical Background of Diffusion in Minerals and Melts

17

Diffusion in three dimensions (isotropic media) In general, three-dimensional diffusion is much more complicated unless the boundary shape is simple (such as spherical surfaces) and there is high symmetry (such as spherical symmetry). The forms of the diffusion equations are summarized below (for details, see Crank 1975; Carslaw and Jaeger 1959). The three-dimensional diffusion equation takes the following form in Cartesian coordinates: ∂C ∂  ∂C  ∂  ∂C  ∂  ∂C  = D + D D + ∂t ∂x  ∂x  ∂y  ∂y  ∂z  ∂z 

(29)

If D is constant, then the above becomes  ∂2 C ∂2 C ∂2 C  ∂C = D 2 + 2 + 2  ∂t ∂y ∂z   ∂x

(30)

In cylindrical coordinates, defining x = r cosq, and y = r sinq, where r is the planar radial coordinate, then the diffusion equation becomes: ∂C 1 ∂  ∂C  1 ∂  ∂C  ∂  ∂C  D = Dr + D + ∂t r ∂r  ∂r  r 2 ∂θ  ∂θ  ∂z  ∂z 

(31)

If (i) concentration is uniform along z (i.e., only two dimensional radial diffusion is considered), (ii) there is rotational symmetry (i.e., C is independent of q), and (iii) D is constant, then the above equation becomes:  ∂ 2C 1 ∂C  ∂C 1 ∂  ∂C  =D r = D  2 +  ∂t r ∂r  ∂r  r ∂r   ∂r

(32)

In spherical coordinates, defining x = r sinq cosf, y = r sinq sinf, and z = r cosq, where r is the three-dimensional radial coordinate, then the diffusion equation becomes: ∂C 1  ∂  2 ∂C  1 ∂  ∂C  1 ∂  ∂C   = 2   Dr + D sin θ + 2 D     ∂t r  ∂r  ∂r  sin θ ∂θ  ∂θ  sin θ ∂φ  ∂φ  

(33)

If there is spherical symmetry (meaning C is independent of q and f) and D is constant, the above equation is simplified to:  ∂ 2C 2 ∂C  ∂C 1 ∂  ∂C  = D 2  r2 = D 2 +   ∂t r ∂r  ∂r  r ∂r   ∂r

(34)

Comparing the last terms in Equations (32) and (34), the difference between cylindrical and spherical diffusion is only the factor of 1 or 2 in front of (1/r)∂C/(∂r), but this seemingly trivial difference leads to completely different solutions. Equation (34) can also be written as (for r > 0): ∂(rC ) ∂ 2 (rC ) =D ∂t ∂r 2

(35)

Defining u = rC, then the above equation becomes: ∂u ∂ 2u =D 2 ∂t ∂r

(36)

Zhang

18

which has the same form of the basic equation for one-dimensional diffusion (Eqn. 2). That is, three-dimensional diffusion in the case of spherical symmetry can be simplified to one-dimensional diffusion. However, two-dimensional diffusion cannot be simplified in a similar way.

SOLUTIONS TO BINARY AND ISOTROPIC DIFFUSION PROBLEMS This section presents solutions to binary diffusion problems in isotropic media, which are often encountered in experimental studies and in natural systems. Solving a diffusion problem requires knowledge of initial and boundary conditions. Experimental studies are often designed so that the analytical solutions to extract diffusivities are simple. Geological diffusion problems in nature are often much more complicated. The solutions below are for relatively simple diffusion problems, and are presented without derivation, but the experimental or geological aspects to which the solution can be applied will be explained. For constant D, two methods are commonly applied to solve diffusion problems. One is the Boltzmann transformation method, which is widely applied to diffusion in infinite or semiinfinite media. The second method is separation of variables, which is applied to boundary value problems (finite diffusion media). In addition, Laplace and other integral transforms, Green’s function methods, and numerical methods can all be employed to solve diffusion equations. These and other methods are covered in Carslaw and Jaeger (1959), Crank (1975), Lasaga (1998), Glicksman (2000), and Zhang (2008). When D is not constant, analytical solutions are often not available, and numerical solution is necessary.

Thin-source diffusion This diffusion problem belongs to the class of problems referred to as “instantaneous source” diffusion in infinite or semi-infinite space in which the source can be a plane (one dimensional diffusion), a line (two dimensional), or a point (three dimensional). Mathematically, this class of solutions is also useful in deriving other solutions. Experimentally, this is the basis of the thin-source diffusion method, also called thin-film method, which was widely used in the past to determine diffusivities (e.g., Jambon and Carron 1976; Hofmann and Magaritz 1977; Behrens 1992). Thin-source diffusion means diffusion proceeds from the surface of a material (a plane source) into the interior of a material, but with the diffusion distance much smaller than the extent of the material (Fig. 5) so that the medium can be treated as semi-infinite. In the jargon of diffusion mathematics, the thin-source diffusion problem is diffusion in a semi-infinite space with the initial condition that all of the diffusant is at a single location of x = 0; and C = 0 at x > 0. There is no additional flux from either side of the sample, which means that ∂C/∂x = 0 at both ends (the end with the thin film is x = 0, and the other end is x = ∞ if diffusion has not reached this end). This mathematical problem is similar to that of random walk in one dimension (Eqn. 16), except that in the thin source problem, diffusion goes in only one direction instead of both directions. Hence, the resulting concentration profile (i.e., the solution to this diffusion problem) is two times that in Equation (16): C ( x, t ) =

2 2 M e − x /( 4 Dt ) = C0e − x /( 4 Dt ) ( πDt )1/ 2

(37)

where x is distance measured from the surface on which the tracer was applied, C is the concentration of the diffusant (e.g., measured by counting the number of decays in the case of a radiotracer), M is the initial mass of the diffusant in the thin film per applied area, C0 is the concentration of the diffusant at the surface (x = 0), which decreases by half as time is quadrupled. Defining the mid-concentration distance (x1/2) as the distance at which C = C0/2, then

x1/2 = 1.6651(Dt)1/2 (38)

Theoretical Background of Diffusion in Minerals and Melts

19

The above is similar to the general form of Equation (9). If the thin film thickness is < 0.1x1/2, then the solution (Eqn. 37) applies well. Otherwise, the solution may not be accurate. If the “thin” film thickness is > 0.2x1/2, the source is not thin any more, and the problem should be treated as extended source diffusion or finite-medium diffusion (e.g., Zhang 2008). If the “thin” film thickness is > 2x1/2, then the tracer diffusion is almost equivalent to a diffusion couple (discussed in a later section), with one half being the “thin” film, and the other half the diffusion medium of interest. In this case, the tracer diffusion becomes chemical diffusion across two very different compositions (effective binary diffusion in a diffusion couple). When a radiotracer is used as the diffusing species, the integrated concentrations are often measured using the residual activity method (e.g., Jambon and Carron 1976; Behrens 1992). After the experiment, the radioactive nuclide on the surface is washed away, and the radioactivity in the whole sample is measured. Then a thin layer of the sample (e.g., 0.005 mm) is polished off, and the total residual radioactivity of the remaining sample is measured. And another layer is polished off, and the residual activity measured, and so on. Hence, every measurement is total radioactivity from x to ∞ where x starts at zero (the first measurement) and gradually increases. Hence, the solution is the integration of Equation (37): ∞

∞

A( x, t ) = ∫ C ( x, t )dx = C0 ∫ e − x x

2

/( 4 Dt )

dx = A0 erfc

x

x 4 Dt

(39)

where A is defined as the measured residual radioactivity, and erfc is the complementary error function. To the uninitiated, the shapes of the two profiles (Eqns. 37 and 39) may appear similar, but there are important differences between the two profiles. For example, the slope is zero at x = 0 for Equation (37), but the slope is the steepest at x = 0 for Equation (39).

Comments about fitting data When analytical data are fit by Equation (37) or (39), one may choose to carry out nonlinear fit using the equations directly. In the past, this was difficult because one would have to write a software program to do so (e.g., Press et al. 1992). More recently, nonlinear fitting has become easier because many commercially available programs can carry out such fitting. Another approach is to linearize the relations and do a linear fitting; an advantage is that it is simple and visually easy to verify such relations. Hence, many authors have used linearized fitting. Equation (37) is linearized as follows: ln C = ln C0 −

x2 4 Dt

( 40)

A plot of lnC versus x2 would be a straight line and D can be found from the slope. Equation (39) is linearized as follows:  A = erfc −1   4 Dt  A0  x

(41)

where erfc−1 means the inverse of the complementary error function. A plot of erfc−1(A/A0) versus x would be a straight line and D can also be found from the slope. If analytical errors are much smaller than every measured concentration (e.g., 1% relative precision for all measured concentrations), linearized fitting will work well. However, the relative uncertainty of measurements at low concentrations is often large. Therefore, the error in lnC (which is the relative error for C) increases as x increases in Equation (40), and error in erfc−1(A/A0) also increases as x increases. One must be careful either to do an errorweighted fitting, or only use data with high relative precision (e.g., Fig. 3-29b in Zhang 2008). Otherwise, the fit might be dominated by data with large errors and D from the fitting would

Zhang

20

not be accurate. Hence, nonlinear fitting has the advantage of handling errors much better (the data with small concentrations and consequently large errors are not emphasized in nonlinear fitting) and is the preferred method, especially since nonlinear fitting programs are now more readily available. The above comments about fitting data also apply to fitting other kinds of diffusion profiles discussed below.

Sorption or desorption Sorption or desorption of gases into or from a mineral occurs often in nature. For example, loss of radiogenic Ar and He (important for thermochronology) as well as other volatiles from minerals can be considered desorption. Sorption of water into minerals and glasses occurs in nature and can change the properties of the mineral and glasses. In diffusion studies, sorption and desorption experiments are often undertaken to obtain effective binary diffusivities of volatile components in melts and minerals (e.g., Dingwell and Scarfe 1985). The method has also been applied to determine 18O diffusivities in melts and minerals under hydrous conditions (e.g., Giletti et al. 1978). In desorption experiments, a mineral or glass initially containing volatiles is heated in a gas medium that is devoid of the volatile component of interest. The surface condition is hence a zero concentration (or some low equilibrium concentration). In sorption experiments, a mineral or glass initially free (or almost free) of the volatile component of interest is heated in a gas or fluid containing the component of interest in the diffusion study. The surface boundary condition is a fixed concentration of the volatile component. Mathematically, the two problems (sorption and desorption) are similar, with the only difference being the initial and surface concentrations. This diffusion problem is known as the half-space diffusion problem with constant initial and surface concentrations. If the diffusivity D is constant, and diffusion from one surface has not reached the center of the sample (hence a semi-infinite medium), the resulting diffusion profile is as follows: C = Cs + (Ci − Cs ) erf

x 4 Dt

(42)

where erf is the error function, Ci is the initial concentration of the volatile component in the sample, and Cs is the surface concentration. Figure 6 shows a diffusion profile during sorption of Ar into a rhyolite melt. For desorption experiments, if the surface concentration is zero, the solution becomes: C = Ci erf

x 4 Dt

( 43)

For sorption experiments, if the initial concentration is zero, the solution becomes: C = Cs erfc

x 4 Dt

( 44)

If concentration profiles can be measured, the above equations can be used to fit data and D can be obtained. The mid-concentration distance for sorption and desorption is:

x1/2 = 0.9539(Dt)1/2 (45)

In sorption or desorption experiments, the concentration of the volatile component often only changes by hundreds or thousands of ppm, meaning the concentration gradients of major components are small. Hence, the diffusivity is often constant across the profile and the above solutions can be applied. For some diffusant such as H2O in glass or minerals, even when the concentration is low (thousands of ppm, even down to tens of ppm), the concentration profiles

Youxue Zhang (Ch 2)

Page 6

Theoretical Background of Diffusion in Minerals and Melts

21

Figure 6. Diffusion profile for Ar sorption into a rhyolite melt (experiment RhyAr4-0 at 1375 K and 0.5 GPa of Ar pressure; Behrens and Zhang (2001).

Fig. 6. Diffusion profile for Ar sorption into a rhyolite melt (experiment RhyAr4-0 at 1375 K and 0.5 GPa of Ar

pressure; Behrens and Zhang 2001).

cannot be fit by the above equations (e.g., Drury and Roberts 1963; Delaney and Karsten 1981; Zhang et al. 1991a; Wang et al. 1996), signifying that D must depend on the concentration itself. After sorption or desorption experiments, sometimes the concentration profiles cannot be measured, but only the total mass gain or loss as a function of time is measured. If D is constant, the total mass gain or loss from both surfaces of a parallel plate (if loss from other surfaces is negligible) can be described by the following equation (Crank 1975): ∞ M t 4 Dt  nL  = 1 + 2 π ∑ ( −1)n ierfc   M∞ πL  2 Dt  n =1

( 46)

where Mt and M∞ are the amount of the volatile component of interest entering (or exiting) the plate of thickness L at time t and time ∞, and ierfc is the integrated complementary error function. For small times (more specifically, when Mt/M∞ ≤ 0.6), diffusion has not reached the center yet and the above equation can be simplified as (Crank 1975): Mt 4 D t ≈ M∞ πL

( 47)

That is, a plot of Mt versus t1/2 is a straight line. If D depends on concentration, the linearity between Mt and t1/2 still holds, but the diffusivity derived from such data is an average diffusivity, and depends on whether sorption data are averaged (from which one obtains the diffusion-in diffusivity, Din), or desorption data are averaged (from which one obtains the diffusion-out diffusivity, Dout). Din and Dout can be different, depending on how D depends on concentration. In some experiments, one single sphere, or more often, many spheres of roughly equal radius a, are investigated for mass gain or loss to obtain diffusivities. The equation to describe such results is (Crank 1975): ∞ Mt Dt  na  Dt =6 1 + 2 π ∑ ierfc −3 2 M∞ a Dt  πa  n =1

( 48)

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22

where Mt and M∞ are the amount of diffusant (for example, 18O in the case of oxygen diffusion studied using an 18O tracer) entering (or exiting) the sphere of radius a at time t and time ∞. The above equation converges rapidly for small times. Furthermore, if Mt/M∞ ≤ 0.9, the above equation can be simplified as (Zhang 2008, p 291): Mt Dt 6 Dt ≈ −3 2 M∞ a π a

( 49)

In the literature (e.g., Muehlenbachs and Kushiro 1974), the following equation is also used to fit experimental data for spheres, which converges rapidly at large diffusion times (Crank 1975):  n2 π2 Dt  Mt 6 ∞ 1 = 1 − 2 ∑ 2 exp  −  M∞ a2  π n =1 n 

(50)

The sorption problem and the thin-source diffusion problem are similar in that in both cases diffusion starts from a surface, but they are different in that in the sorption problem, the surface concentration is constant, whereas in the thin-source diffusion problem, a fixed amount of diffusant is applied on a surface so that the surface concentration decreases with time. The solutions to the two problems are different. In fact, the solution to the sorption problem (Eqn. 44) is similar to the integration (Eqn. 39) of the thin-source problem.

Diffusion couple or triple The diffusion couple problem is also often encountered in nature and in experimental studies conducted to obtain self diffusivities (e.g., LaTourrette et al. 1996), interdiffusivities (e.g., Freda and Baker 1998), effective binary diffusivities (e.g., Koyaguchi 1989; Zhang and Behrens 2000), and multicomponent diffusion matrix (Kress and Ghiorso 1995; Liang et al. 1996). In this method, two cylinders (each is called a half) of the same phase but different composition (for self diffusion studies the difference is only in the isotopes of the element(s) of interest) are joined together at flat and sometimes polished and annealed surfaces (Fig. 7). For studying diffusion in melts, the two halves are oriented vertically so that the interface is horizontal to minimize convection. Assuming diffusion has not reached either end yet, the diffusion problem is one-dimensional diffusion in infinite space. If the diffusivity is constant, the concentration at the interface is the simple (not weighted) average concentration of the two halves. One may view each half as behaving as a sorption or desorption problem with constant concentration at the interface, so the solution would be an error function. Define the vertical direction as z, z = 0 at the interface, and z > 0 in the upper half. The combined solution of both halves is: C=

CU + CL CU − CL z + erf 2 2 4 Dt

(51)

where the CU and CL are the initial concentrations in the upper and lower halves. Measured concentration profiles can be fit to the above equation to obtain D. In such fitting, CU and CL can often be obtained from measured concentrations at the two ends (each can be obtained by averaging many points) and can be fixed in the fitting. Hence, there is essentially only one unknown parameter, D, to be obtained from the fitting. However, often the interface position is not known accurately, although it may be roughly estimated. Hence, the fitting often takes the following form: C=

CU + CL CU − CL z − z0 + erf 2 2 4 Dt

(52)

where z0 (the position of the Matano interface, defined by mass balance so that the diffusant loss from one side is equal to the diffusant gain on the other side; see Eqn. 54 below) is also

Theoretical Background of Diffusion in Minerals and Melts

23

Figure 7. The diffusion couple setup and the resulting concentration profiles. On the left is a drawing of the diffusion couple configuration with high concentration of the component of interest in the upper half, and low concentration in the lower half (see also Fig. 1). The evolution of the concentration with time is shown on the right for three different times (arbitrary unit).

a fitting parameter and allowed to vary to optimize the fitting. The value of z0 does not have much meaning; it only indicates how well one estimated the interface position before the fitting. The definition of the mid-concentration distance takes some thought for a diffusion couple. If it were defined as the mid-concentration between the two halves, then it would not move at all, inconsistent with diffusive flux into a medium. The adopted definition is to consider diffusion in each half as having a constant surface concentration. Then the mid-concentration distance is the same as in sorption or desorption experiments, with x1/2 = 0.9539(Dt)1/2. Some authors use diffusion triples (e.g., Behrens and Hahn 2009), which are essentially two diffusion couples sharing one common half, in one experiment. In a diffusion triple, three glass or mineral cylinders are stacked together as upper, middle and lower thirds, making two diffusion couples. In nature, diffusion between two layers of a crystal differing in elemental or isotopic compositions may be viewed as a diffusion couple, as can diffusion between two layers of melts (though it is difficult to avoid convection in natural systems). For the complete homogenization of a diffusion couple, the initial concentration evolution is similar to Equation (51), but the concentration evolution after the diffusant has reached at least one end of the material depends on the boundary conditions at the ends (e.g., whether the ends are kept at constant concentration or there is no flux from the outside) as well as the dimensions of the initial two layers.

Diffusive crystal dissolution Crystal dissolution and growth are common in magma chambers. Diffusive crystal dissolution has been applied to obtain chemical diffusivities and to treat multicomponent diffusion (Harrison and Watson 1983; Zhang et al. 1989; Liang 1999). Crystal dissolution rather than crystal growth is adopted in diffusion studies because crystal dissolution can be controlled well; for crystal growth experiments, where new crystals form cannot be wellcontrolled. The modifier “diffusive” is also important: it means that convection needs to be avoided to study diffusion.

Zhang

24

In the design of diffusive crystal dissolution, a gem-quality crystal disk and a glass cylinder are joined vertically with a horizontal interface to minimize convection (Fig. 8). If the melt due to the dissolution of the crystal has a higher density than the ambient (or initial) melt, the crystal is placed at the bottom; otherwise the crystal is placed on the top to minimize convection. Thus mass transport is entirely controlled by diffusion. At a fixed high temperature, the dissolution of the crystal often rapidly establishes a constant melt composition at the interface (Zhang et al. 1989; Chen and Zhang 2008, 2009), and diffusion carries the flux into the melt interior. The diffusion is often complicated due to (i) multicomponent effects and (ii) major compositional variation in the melt.

Figure 8. Setup of an olivine dissolution experiment. Because the dissolution of olivine produces a melt (interface melt) with greater density than the initial melt, olivine is placed at the bottom of a melt to minimize convection in the melt. In this case, the olivine crystal is larger in diameter than the melt so that the edge of olivine is preserved for the accurate determination of the olivine dissolution distance (Chen and Zhang 2008).

For the dissolution of low-solubility minerals such as zircon, the concentration gradients in major oxides are often negligible, and the diffusivity of the main mineral component is roughly constant along a profile. The solution to this diffusion problem is (Zhang et al. 1989): ( x − L) 4 Dt C = Ci + ( C s − Ci ) (− L ) erfc 4 Dt erfc

(53)

where Ci is the initial concentration of the main mineral component (such as ZrO2) in the melt, Cs is the concentration of the component in the interface melt, which is a fitting parameter, and L is the melt growth distance, which is often negligible for dissolution of low-solubility minerals (which are also slowly dissolving minerals) such as zircon. For the dissolution of high-solubility minerals such as pyroxenes and olivine, the concentration gradients in major oxides are significant and the above equation does not work well for most components because of the multicomponent effects. However, for the major mineral component (the component whose concentration in the mineral is much higher than that in the melt, such as MgO during olivine dissolution), it is often possible to treat its diffusion as effective binary diffusion. In such cases, Equation (53) may be applied to fit the data to estimate the effective binary diffusivity. Furthermore, for high-solubility minerals (which are also rapidly dissolving minerals), the melt growth distance L must be determined independently (often from the mineral dissolution distance multiplied by the ratio of the mineral density over the melt density) to apply Equation (53) to fit data. In earlier experimental studies of crystal dissolution, convection was often present (e.g., Brearley and Scarfe 1986), but was either not considered or incorrectly treated (see Zhang et

Theoretical Background of Diffusion in Minerals and Melts

25

al. 1989 for more discussion). Hence, the extracted diffusivities based on crystal dissolution experiments in these studies were often incorrect. Theoretically, there is also a short diffusion profile in the crystal, which is too short to be measured. Furthermore, the dissolution of the crystal shortens the diffusion profile in the crystal (Zhang 2008, p 378-389).

Variable diffusivity along a profile In some diffusion experiments, the diffusivity may vary along a concentration profile. This can happen in at least two scenarios. One is when the major element composition changes significantly along a diffusion profile, such as in the case of Fe-Mg interdiffusion in olivine (Chakraborty 2010, this volume), in which diffusion has a strong compositional dependence. The other is in the case of components such as H2O, where the diffusivity varies with its own concentration due to the effects of speciation even when the compositional variation of major components is negligible. To solve the diffusion equation with concentration-dependent diffusivity, numerical methods are necessary (e.g., Crank 1975; Press et al. 1994), which often is only slightly more difficult than working on complicated analytical solutions to a diffusion problem. In experimental studies, however, the interest is in obtaining the diffusivities from the measured concentration profiles, which is an inverse problem. There are two methods to extract diffusion coefficients if the diffusivity varies along a concentration profile. In one method, the functional form of the variation of the diffusivity with concentration is known, even though some parameters in the function are not known. For example, the diffusivity might be proportional to the concentration: D = aC, where a is the value of D when C = 1. Or the diffusivity may be linear in C: D = aC+b. Or the diffusivity might be an exponential function of concentration: D = b exp(aC) (i.e. lnD is linear in C), where b is the value of D when C = 0. If the functional form is known but not the parameters a and b, the diffusion equation can be solved for given values of a and b, and the solution is compared with the experimental profile. By adjusting a and b to fit the concentration profile, the parameters can be found, so that the way in which D varies with C can be determined. The fitting can be complicated but specific programs have been written to accomplish this task (e.g., Zhang et al. 1991a; Zhang and Behrens 2000; Ni and Zhang 2008). If the functional form of the dependence of D on C is not known and cannot be guessed, then Boltzmann-Matano method, based on an application of the Boltzmann analysis by Matano (1933), can be applied to obtain diffusivities at every point along a profile. This method is most often applied to diffusion couples. In the original method, it is necessary to first find the Matano interface between the two halves of the diffusion couple (which may or may not be the physically marked initial interface between the two halves), so that x defined relative to the Matano interface (i.e., x = 0 at the Matano interface) satisfies:

∫

CU

CL

xdC = 0

(54)

where CL and CU are the concentrations at the two ends, x < 0 in the lower half of the couple, and x > 0 in the upper half of the couple. After obtaining the Matano interface, then the diffusivity at any x = x0 (which also means at a C corresponding to x0) can be found (Crank 1975): Dat C ( x0 ) =

∫

CU

C ( x0 )

xdC

2t ( dC / dx ) x = x0

(55)

where t is the experimental duration. The key in minimizing the errors in extracting D using the above expression is to obtain accurate integrals and slopes, which requires smooth concentration profiles. Often the experimental data are smoothed objectively, either manually

Zhang

26

or by some kind of piecewise fitting (because it is not known what function can fit the whole profile). Furthermore, D values obtained using the above method near the two ends often have large errors. If the method is applied carefully, the general trend of D versus C is often acceptable, but small undulations may be artifacts of inaccurate slopes and integrations. A trivial variation of Equation (55) is Dat C ( x0 ) =

−∫

C ( x0 )

CL

xdC

2t (dC / dx ) x = x0

(56)

A modified approach based on the Boltzmann analysis is provided by Sauer and Freise (1962). The advantage of this method is that there is no need to find the Matano interface. Define y=

(C − CL ) (CU − CL )

(57)

which may be referred to as the normalized concentration. D can be found as follows: Dat C ( x0 ) =

1  y| ∞ (1 − y)dx + (1 − y | ) x0 ydx  x x0 ∫−∞   2t (dy / dx ) x = x0  0 ∫x0

(58)

Again, the key in obtaining reliable D is to obtain accurate integrals and slopes, which requires smooth concentration profiles. The Boltzmann analysis can also be adapted for use in studies of diffusive crystal dissolution in order to extract diffusivities. The equation is (Zhang 2008): Dat C ( x0 ) =

1 2t (dC / dx ) x = x0

Ci

∫

( x − L )dC

(59)

C ( x0 )

where the upper limit of the integration Ci is the initial concentration in the melt, x is the distance from the crystal-melt interface, x0 is the position at which the diffusivity is obtained, and L is the melt growth distance.

Homogenization of a crystal with oscillatory zoning In nature, a crystal (such as plagioclase) may be oscillatorily zoned. Idealize the initial oscillatory zones as follows: the concentration in the zones can be described by a sine or cosine function (which also implies constant width of every zone): CAn

t =0

 2πx  = a + bsin    p 

(60)

where a is the average An content in a plagioclase crystal, b is the peak amplitude (or half of peak-to-peak amplitude), and p is the width of each zone (or period of the oscillation), e.g., from one maximum to the neighboring maximum in Figure 9. As diffusion proceeds in a closed system (nothing entering or leaving the system), the concentration profile would evolve as· 2

CAn = a + be −4π Dt

L2

 2πx  sin    p 

(61)

where D is the diffusivity of the coupled cation exchange Ca+Al ↔ Na+Si, which changes the concentration of the An component in plagioclase. That is, both the average An content and the period of the zoning stay the same, but the compositional amplitude of the zoning decreases

Youxue Zhang (Ch 2)

Theoretical Background of Diffusion in Minerals and Melts

27

Page 9

Fig. 9. Homogenization of an oscillatorily zoned crystal with time. Figure 9. Homogenization of an oscillatorily zoned crystal with time.

exponentially with time (Fig. 9). If the initial oscillatory zoning is periodic but has sharp boundaries, the solution would be an infinite series of sine or cosine functions.

One dimensional diffusional exchange between two phases at constant temperature Often, two minerals in contact have common components that may be exchanged. For example, garnet and olivine both contain Fe2+ and Mg2+, and the two cations can exchange through diffusion (Fig. 10). Garnet and spinel may exchange Mn-Fe2+-Mg in divalent sites, and Al-Cr-Fe3+ in the trivalent sites. The following results are from Zhang (2008, p 426-430). Assume (i) each phase is initially uniform in composition, (ii) the exchange is between only two components (binary diffusion), (iii) the contact interface between the two minerals is flat (planar), (iv) either the mineral is isotropic or diffusion in an anisotropic mineral is along a Fig. principal 9. Homogenization of an of oscillatorily zoned crystal withbelow time. axis diffusion (see on diffusion in anisotropic medium), (v) diffusion has not proceeded to the center of either mineral yet, and (vi) D in each mineral phase is constant. Then, the problem is one dimensional and has analytical solutions. Furthermore, assume that there is instantaneous equilibrium at the contact between the surfaces of two minerals and that the equilibrium condition is described by a constant exchange coefficient KD (which depends on temperature): KD =

(X (X

B 2

X1B

A 2

X

) )

x =+0 A 1 x =−0

(62)

where X means mole fractions, superscripts A and B are the two mineral phases (e.g., A = olivine and B = garnet), subscripts 1 and 2 are the two components (e.g., 1 = Mg and 2 = Fe), X1A is the mole fraction of component 1 in mineral A, the interface is at x = 0, mineral B is on the side of x > 0, mineral A is on the side of x < 0, “x = +0” means the surface of mineral B (x approaches zero (interface) from x > 0 side), and “x = –0” means the surface of mineral A (x approaches zero (interface) from x < 0 side). The solution for the concentration evolution as a function of time is:

(

)

(

)

X1A = X1Ai + X1A, −0 − X1Ai erfc B − X1Bi erfc X1B = X1Bi + X1,+0

x 4D A t x 4DB t

(63a ) (63b)

where X1Ai and X1Bi are the initial mole fractions of component 1 in minerals A and B, X1,A− 0 and

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28

Olivine

Garnet

Figure 10. Fe-Mg exchange between olivine and garnet. The figure on the left shows the geometry of the diffusional exchange, with the horizontal direction being along c-axis of olivine. The figure on the right shows the calculated diffusion profile in the two phases using Equations (63a) and (63b). KD = (Fe/Mg)Gt/ (Fe/Mg)Ol = 1.7.

B X1,+0 are the mole fractions of component 1 at the interfaces of minerals A and B, DA and DB are interdiffusivities between components 1 and 2 in minerals A and B. The mole faction of component 2 in each mineral can be found by stoichiometry (e.g., the sum of mole fractions of 1 and 2 in every mineral is 1). Given initial conditions X1Ai and X1Bi, and diffusivities, there are B ) in the above two equations, which can be solved from two still two unknowns ( X1,A− 0 and X1,+0 equations: Equation (62) (surface equilibrium) and the following (mass balance):

(X

A 1, − 0

)

(

)

B − X1Ai ρA D A = X1,+0 − X1Bi ρB D B

(64)

where rA and rB are the molar densities of components 1 and 2 in minerals A and B. For garnet example, ρolivine Fe+Mg = 43.48 mol/L and ρ Fe+Mg = 25.64 mol/L if there are no other divalent cations. Calculated profiles are shown in Figure 10b.

Spinodal decomposition Spinodal decomposition is the spontaneous decomposition of a single phase to two phases. For example, alkali feldspar at high temperature can be a single phase. As the temperature becomes lower, it may spontaneously separate into two phases, albite and orthoclase. The intergrowth of the two phases is called perthite. The separation of a single uniform phase into two phases of similar structure is called spinodal decomposition. It is accomplished by diffusion and thermal fluctuation. In the process, diffusion may transport elements from low concentration to high concentration (referred to as uphill diffusion), opposite to the transport direction during normal diffusion. Spinodal decomposition in a binary system illustrates that diffusion is not simply responding to concentration differences to homogenize the system, but is a response to the chemical potential (or chemical activity) difference. Diffusion reduces the Gibbs free energy of the system. In ideal or close to ideal binary mixtures, the entropy portion of the Gibbs free energy dominates the total Gibbs free energy of mixing. The chemical potential of a component increases as the concentration of the component increases. Hence, diffusion homogenizes the system, which minimizes the Gibbs free energy of the system. In highly non-ideal binary mixtures, the chemical potential (or activity) may decrease as concentration increases when the enthalpy part of the Gibbs free energy dominates the total mixing energy.

Theoretical Background of Diffusion in Minerals and Melts

29

Then, diffusion would still be downhill in terms of the chemical potential (or activity) gradient, but can be uphill in terms of concentration gradient. Hence, Fick’s law is an approximation of the following more accurate diffusion law in a binary system (Zhang 1993): J=−

D ∇a γ

(65)

where a and g are the chemical activity and activity coefficient of either of the two components with a = gC, and D is the “intrinsic” binary diffusivity, different from the normal binary diffusivity D. When g is constant, the above equation reduces to Equation (6). Comparing Equations (65) and (6), the “intrinsic” effective binary diffusivity and the normal effective binary diffusivy are related as: d ln γ   D = D 1 +   d ln C 

(66)

The “intrinsic” binary diffusivity D is always positive, but D calculated from the above equation is negative in the compositional range when the phase is unstable, leading to uphill diffusion. In binary systems, uphill diffusion occurs rarely, only in cases when the phase is unstable. On the other hand, in multicomponent systems, uphill diffusion often occurs even when the phase is stable. Furthermore, in binary systems, diffusion is always downhill of the chemical potential gradient of any given component, even in uphill diffusion. However, in multicomponent systems, the diffusive flux may also be uphill against the chemical potential gradient.

Diffusive loss of radiogenic nuclides and closure temperature Diffusive loss of radiogenic nuclides is probably the most common application of diffusion in geology, with critical implications for geochronology. One example is the loss of 40 Ar (produced by the decay of 40K) from minerals. Another is the loss of 4He (produced by the decay of 238U, 235U and 232Th, and other nuclides) from minerals. 40Ar is used in the following examples for clarity. The decay of 40K is a branched decay, with 10.48% to 40Ar and the rest to 40Ca. Because 40Ar as a noble gas is not readily incorporated in any mineral structure, it is especially prone to diffusive loss from a mineral, and the process is similar to desorption. The diffusive loss occurs at high temperature and during cooling, in which the diffusivity depends on time since the temperature varies with time. At high temperature, the diffusivity is high, leading to extensive 40Ar loss and little 40Ar accumulation in the mineral. As the temperature decreases, the diffusivity becomes lower, until it becomes sufficiently low that there is little 40Ar loss, and essentially all radiogenic 40Ar accumulates. A numerical example is as follows. For hornblende, Ar diffusivity depends on temperature as D = 2.4×10−6 exp[−268000/(RT)] = exp(−12.94 − 32257/T) where R is 8.3144 J·mol−1·K−1, T is in K, D is in m2/s (Harrison 1981). Consider the average hornblende crystal radius to be 0.5 mm. A plutonic rock typically cools at 10 K/Myr (e.g., Harrison and McDougall 1980), or 1 K per 100,000 years. First, consider diffusion at a typical magmatic temperature of 1200 K, at which the Ar diffusivity is 5.1×10−18 m2/s. In 100,000 years, the diffusion distance estimated from x1/2 = (Dt)1/2 would be 4 mm, much larger than the radius of hornblende, meaning that essentially all radiogenic 40Ar would be lost from the hornblende. Next, consider diffusion at 300 K. The extrapolated Ar diffusivity is 5×10−53 m2/s (which may not be accurate because of the large down-temperature extrapolation from the experimental data range of 773-1173 K, but can nonetheless used for an order of magnitude estimate). In 4.5×109 years (the age of the Earth; Patterson 1956), the diffusion distance estimated from x1/2 = (Dt)1/2 is 3×10−18 m, much less than the atomic distance of 10−10 m. That is, Ar atoms would not have moved even one atomic layer by staying at 300 K for the whole history of the Earth, meaning all radiogenic Ar would

30

Zhang

have been kept inside the hornblende. As the rock cools down from 1200 K to 300 K, the hornblende would become closed to Ar loss at some intermediate temperature, which is called the closure temperature (Fig. 11). A rigorous definition of the closure temperature is as follows. For a given cooling history (Fig 11a), the accumulation of 40Ar in a given mineral is a continuous function. Initially, there is little accumulation because of fast diffusion of Ar at high temperatures. Gradually, as the temperature becomes lower, there is some accumulation, but not complete retention of all radiogenic 40Ar. As the rock cools down further, new radiogenic 40Ar is completely retained in the mineral. The accumulation of 40Ar in the mineral as a function of time is shown as the thin solid curve in Figure 11b. After a geochemist collects the rock, the thermal history is not known and the age of the rock is calculated based on the present-day 40Ar and 40K ratio assuming the rock was a closed system from the beginning (the thick dashed line in Fig. 11b). The calculated age is called the closure age. All geochronology methods give the closure age. As can be seen from the difference of the thin solid curve and the thick dashed line in Figure 11b, the difference between the real age and the closure age in this case is about 5 Myr. The temperature corresponding to 2)the closure age is defined Youxue Zhang (Ch Pageto 11be the closure temperature, which is about 850 K in this case. In this case, the thermal history is assumed and forward calculation is

Figure 11. Closure temperature (Tc) and(Tclosure time (closure age, t ). The solid curve in (a) shows the Fig. 11. Closure temperature c) and closure time (closure age,ctc). The solid curve in (a) shows the temperature temperature history of the mineral. The solid curve in (b) corresponds to the accumulation of 40Ar in the mineral with time. Theofdashed curve insolid (b) curve showsin the assumed 40toArtheaccumulation Ar inage the calculation, mineral with time. The history the mineral. The (b) corresponds accumulation ofin40the with the calculated age at tc, which corresponds to the time when the mineral temperature was Tc. dashed curve in (b) shows the assumed 40Ar accumulation in the age calculation, with the calculated age at tc, which corresponds to the time when the mineral temperature was Tc.

Theoretical Background of Diffusion in Minerals and Melts

31

carried out so that the closure temperature can be readily determined. When rocks are studied, the thermal history is not known and is to be determined from geochronology and other studies, which requires the estimation of the closure temperature given the diffusion properties. To treat 40Ar retention, it is necessary to consider diffusive loss during continuous cooling. Solving the diffusion equation in which the diffusivity depends on time is relatively easy in most cases by simply replacing Dt in the solutions by ∫Ddt where the integration is from t = 0 to the time of interest. Hence, the mid-concentration diffusion distance is roughly: x1/ 2 ≈

t

∫ Ddt 0

(67)

Now consider a mineral in a continuously cooled rock. The cooling rate is often high initially at high temperature and then slower at lower temperature. One function to approximate the temperature history is the so-called asymptotic cooling (Ganguly 1982; Zhang 1994): T=

T0 1 + t / τ1

(68)

where T is temperature, T0 is the initial temperature (at t = 0), and t1 is the time for the rock to cool from T0 to T0/2. This function is often used because it leads to simple expressions in treating diffusion during cooling. The initial cooling rate (cooling rate at T0) q is q=

T0 τ1

(69)

The diffusivity as a function of temperature is given by the Arrhenius equation (Eqn. 10), and can be expressed as a function of time: D = D0 e − E /( RT ) = D0 e − E (1+ t / τ1 )/( RT0 ) = DT0 e − t / τ2

(70)

where DT0 = D0e−E/(RT0) is the initial D at T = T0, and t2 = t1(RT0/E) is the characteristic time for D to decrease from the initial value to 1/e of the initial value. Combining Equations (70) and (67), the diffusion distance during cooling can be found as: x1/ 2 =

∫

∞

0

DT0 e − t / τ2 dt = DT0 τ2

(71)

The upper integration limit is set to be infinity because (i) it makes the final result simple, and (ii) the upper limit can be treated as ∞ because diffusion at room temperature and slightly above room temperature is negligible over the whole history of the Earth for Ar and most other species. At the closure temperature, the mineral roughly becomes closed, implying that the mineral size must be significantly larger than the diffusion distance at and below the closure temperature Tc (i.e., treating the initial temperature to be Tc in Eqn. 71) a 2 >> Dc τ2

(72)

where Dc is the diffusivity at Tc, a is the half thickness for a platy mineral, radius for a cylinder and sphere. To quantify, the above condition is written as: a 2 = GDc τ2 =

GD0e − E /( RTc ) τ1RTc GD0e − E /( RTc ) RTc 2 = E qE

(73)

where q is the cooling rate when the temperature of the rock was Tc. G is greater than 1 and the exact value depends on the shape of the mineral and must be determined by solving the diffusion equation. Rearranging the above equation leads to Dodson’s famous equation for closure temperature (Dodson 1973, 1979):

32

Zhang  GD T 2  E = ln  2 0 c  RTc  a qE / R 

(74)

The value of G has been determined by Dodson (1973) to be 55 for spheres (where a is the radius), 27 for infinitely long cylinders (a is the radius) and 8.65 for plates (a is the halfthickness). The closure temperature of a mineral can be estimated using the above equation given the diffusion parameters (D0 and E), the shape and radius of the mineral grains, and the cooling rate. The dependence of the closure temperature on the cooling rate makes it necessary to estimate the cooling rate before inferring the closure temperature, making it more complex to infer thermal histories. For slowly cooled plutonic rocks, the closure temperature and closure age can be obtained using different minerals and different isotopic systems, hence a temperaturetime history (thermal history) may be obtained. For rapidly cooled rocks, the application of the closure temperature concept can confirm that the age obtained from a radiogenic system is the true rock formation age. Below are two examples. Consider a hornblende grain (a = 0.2 mm) in a rapidly cooled volcanic rhyolite lava (Lewis-Kenedi et al. 2005). The estimated cooling rate is 100 K/year. Using D = exp(−12.94 − 32257/T) and Equation (74) with iteration, the closure temperature is 1304 K. That is, if the initial temperature is below 1304 K, there would be essentially no Ar loss. Because a typical eruption temperature for rhyolite lava is about 1200 K or lower, little Ar would be lost from hornblende at this cooling rate and the Ar-Ar age is the eruption age. Such high cooling rates cannot be inferred from geochronology because small age differences cannot be determined from isotopic dating (but cooling rates of rhyolite lavas can be estimated by other methods, e.g., Zhang 1994; Zhang et al. 2000; Xu and Zhang 2002). Now consider a hornblende grain (a = 0.3 mm) in a slowly cooled granitoid (Harrison and McDougall 1980). Start by assuming a cooling rate of 10 K/Myr. Using D = exp(−12.94 − 32257/T) and Equation (74) with iteration, the closure temperature is 825 K. Hence, the Ar-Ar age of hornblende (about 110 Ma) corresponds to a temperature of about 825 K. The zircon age of this rock is about 116 Ma. The Pb closure temperature in 60-µm zircon is about 1200 K (Cherniak and Watson 2000), higher than the placement temperature of about 1050 K (Harrison and McDougall 1980). Hence, the U-Pb age reflects the intrusion age, and the rock cooled from 1050 K to 825 K in about 6 Myr, about a factor of four times the assumed cooling rate of 10 K/Myr. More closure temperature and closure age data can be obtained from other dating systems to further constrain the cooling history and cooling rate. If necessary, better constrained cooling rates can be used to recalculate the closure temperatures to obtain more accurate temperature versus time histories. The above results are for whole mineral grains when the initial temperature is sufficiently high to have full equilibration of the diffusant in the mineral grain. Dodson (1979) has extended the theory to individual points inside a mineral, in anticipation that age profiles may be measured in many different points in a mineral grain. Ganguly and Tirone (1999) extended the theory to cases when the initial temperature is not high enough to achieve full equilibration of the diffusant in the mineral grain at the onset of cooling.

Diffusion in anisotropic media The above solutions are all for isotropic diffusion media, such as melts, glasses and isometric minerals. In such media, the diffusivity is a scalar. Most minerals are not isometric, and hence most diffusion problems geologists encounter are anisotropic diffusion although diffusional anisotropy is not necessarily pronounced in many cases. Such diffusion is

Theoretical Background of Diffusion in Minerals and Melts

33

complicated and almost never treated with full rigor. Rather, simplifications are made so that the problems can be handled with reasonable effort as well as reasonable usefulness. In this section, diffusion in anisotropic media is reviewed. Diffusional anisotropy means that diffusivity depends on the crystallographic directions in a mineral. For example, the diffusivity of 18O in quartz under hydrothermal conditions along the c-axis is about two orders of magnitude greater than that along a direction in the plane perpendicular to the c-axis (Giletti and Yund 1984). In mathematical terms, diffusivity is a tensor, more specifically, a second-rank symmetric tensor representable by a 3×3 matrix. Only when the diffusion medium has high internal structural symmetry, would the matrix be simplified. For melts, glasses and isometric minerals, the diffusivities along all directions are the same, and the tensor can be simplified to a single number, a scalar. Generally, the diffusivity tensor is denoted as D, and its matrix form is:  D11  D =  D21   D31 

D13   D11   D23  =  D12   D33   D13

D12 D22 D32

D12 D22 D23

D13   D23   D33 

(75)

where underlined terms are tensors or elements in a tensor, rather than elements in a “normal” matrix, such as the diffusivity matrix for multicomponent diffusion (to be discussed next). The second equal sign in the above equation utilizes the property that the diffusivity tensor is symmetric. The diffusivity tensor in different crystallographic symmetries and the calculation of diffusivities along a given direction (that is, measured along the direction of the concentration gradient) are summarized in Appendix 1. Fick’s first law of diffusion (Eqn. 1) for an anisotropic medium is written as: J = − D∇C

(76)

where J is the flux vector, and ∇C is the concentration gradient (a vector). In matrix form, the above equation can be written as:  D11  Jx      J y  = −  D12     D13  Jz  

D12 D22 D23

D13   ∂C / ∂x    D23   ∂C / ∂y    D33   ∂C / ∂z 

(77)

Based on mass balance (Eqn. 7), ∂C/∂t can be written as: ∂J ∂J ∂J ∂C = −∇⋅ J = − x − y − z ∂t ∂x ∂y ∂z

(78)

Therefore, the full diffusion equation for a binary system becomes ∂C ∂  ∂C ∂C ∂C  ∂  ∂C ∂C ∂C  =  D11 + D12 + D13 + D22 + D23  +  D12  ∂t ∂x  ∂x ∂y ∂z  ∂y  ∂x ∂y ∂z  ∂ ∂C ∂C ∂C  +  D13 + D23 + D33  ∂z  ∂x ∂y ∂z 

(79)

If all the tensor components are constant, then ∂C ∂ 2C ∂ 2C ∂ 2C ∂ 2C ∂ 2C ∂ 2C = D11 2 + D22 2 + D33 2 + 2 D12 + 2 D13 + 2 D23 ∂t ∂x ∂y ∂z ∂x∂y ∂x∂z ∂y∂z

(80)

34

Zhang The above equation can be simplified using coordinate transformation to become: ∂C ∂ 2C ∂ 2C ∂ 2C = Dα 2 + Dβ 2 + Dγ 2 ∂t ∂α ∂β ∂γ

(81)

where a, b, and g are the principal axes of diffusion, and Da, Db and Dg are the principal diffusivities (diffusivities along the principal axes). For the above equation to be applicable, the boundary conditions must also be transformable. The fluxes along a principal axis of diffusion can be written as: J α = − Dα ∂C / ∂α

(82a )

J β = − Dβ∂C / ∂β

(82 b)

J γ = − Dγ ∂C / ∂γ

(82c)

One-dimensional diffusion equation along each principal axis of diffusion can be written as: ∂C ∂ 2C = Dα 2 ∂t ∂α ∂C ∂ 2C = Dβ 2 ∂t ∂β ∂C ∂ 2C = Dγ 2 ∂t ∂γ

(83a ) (83b) (83c)

Diffusion along each principal axes is hence unaffected by diffusion in other directions. In melts, glasses and isometric minerals, any direction can be taken as the principal diffusion direction. In non-isometric minerals, the crystallographic axes can be taken as the principal diffusion axes if the symmetry is at least orthorhombic. In addition, in hexagonal, tetragonal, and trigonal minerals, diffusion along any direction in the plane perpendicular to the c-axis can be taken as a principal axis and can be treated simply as one-dimensional diffusion. Diffusion along a direction that is not a principal axis cannot be treated as one-dimensional diffusion, because diffusion along other directions would also contribute to the flux along the direction of consideration; i.e., the diffusive flux J is not parallel to −∇C. The three-dimensional diffusion equation (Eqn. 81) can be simplified further with the following axis transformation: α′ = α / Dα

(84a )

β′ = β / Dβ

(84 b)

γ′ = γ / Dγ

(84c)

The transformed diffusion equation becomes ∂C ∂ 2C ∂ 2C ∂ 2C = + + ∂t ∂α′2 ∂β′2 ∂γ′2

(85)

which is identical to the diffusion equation in isotropic media with D = 1. Hence, after convoluted transformations, the complicated diffusion equation in anisotropic media with a constant diffusivity tensor (Eqn. 80) is simplified to the diffusion equation in isotropic media. Hence, in theory, analytical solutions for diffusion in anisotropic media can be obtained using these transformations. However, the three-dimensional initial and boundary conditions must also be transformed, which is not always easy. Furthermore, the transformations may have drastically

Theoretical Background of Diffusion in Minerals and Melts

35

changed the shape of a crystal, from the physical shape in natural coordinates to the effective shape in the transformed coordinates. The effective shape might be highly unintuitive. Figure 12 gives an example. Diffusion in anisotropic media is often treated with major simpliFigure 12. Comparison of the physical shape and effective shape in terms of fications. If total loss or gain of the diffusion. The upper diagram shows the diffusant is of interest, as in the difphysical shape of mica (plane sheet), and fusive loss of radiogenic nuclides in the shape on the right hand side is the understanding closure temperature shape after transformation to lengthen the and closure age, the shape of the vertical axis by a factor of 100 (because the difference in oxygen diffusivities is 4 orders mineral grains (necessary for deterof magnitude). Hence, diffusion along the mining the shape factor) is the efz direction can be ignored, and the mica fective shape. For example, Fortier can be treated as a cylinder in terms of 18O and Giletti (1991) showed that under diffusion. hydrothermal conditions the 18O diffusivity in mica parallel to the c-axis is about 4 orders of magnitude slower than diffusion in the plane perpendicular to the c-axis (even though mica is monoclinic, it is approximately hexagonal in terms of many of its properties). Hence, the physical shape of mica is platy, but the effective shape of mica in terms of 18O diffusion under hydrothermal conditions is an “infinitely” long cylinder, meaning that bulk mass loss or gain is through diffusion in the plane perpendicular to the c-axis (Fig. 12).

MULTICOMPONENT DIFFUSION Diffusion in multicomponent systems (having three or more components) is also complicated. In-depth treatment of multicomponent diffusion will be covered in other chapters (Liang 2010; Ganguly 2010) in this volume. The general aspects and some simple treatments are covered here. Fick’s law (Eqn. 1) is for binary systems only. When three of more components (such as A, B, C and D) are present, experiments show that the diffusion of a given component A depends not only on the concentration gradient of A, but also on the concentration gradients of B and C. Consider a melt with N components. Because the summation of concentrations (such as mole fractions or mass fractions) of all components is 100%, there are only N−1 independent components in an N-component system. (For example, in a binary system, only the concentration of one component is independent.) Let n = N−1. If the N-component system is a stable phase (i.e., no spinodal decomposition), the diffusive flux of components can be written as (De Groot and Mazur 1962): J1 = – D11∇C1 – D12∇C2 ... – D1n∇Cn

(86a )

J 2 = – D21∇C1 – D22∇C2 ... – D2 n∇Cn

(86 b)

 J n = – Dn1∇C1 – Dn 2∇C2 ... – Dnn∇Cn

(86 n )

where Dij’s are diffusion coefficients for component i due to concentration gradients of component j. Dii’s are referred to as the main or on-diagonal diffusivities, and Dij’s when i≠j are referred to as the off-diagonal or cross diffusivities. In matrix notation, the above becomes:

Zhang

36  J1   D11    J  2  = −  D21  ...   ...     Jn   Dn1

D12 D22 ... Dn 2

... D1n   ∇C1    ... D2 n   ∇C2  ... ...   ...    ... Dnn   ∇Cn 

(87)

The diffusivity matrix is not to be confused with the diffusivity tensor; they are different in at least two aspects: (i) the meanings are different, one refers to diffusion in a multicomponent system but isotropic medium, and the other refers to binary diffusion in an anisotropic medium; (ii) the diffusivity tensor in anisotropic system is always represented by a 3×3 symmetric matrix, but the diffusivity matrix (always a square matrix) is n by n where n ≥ 2 and is nonsymmetrical. For diffusion in a multicomponent system in an anisotropic medium, the full rigorous description would require a diffusivity matrix in which every element is a tensor. Such a diffusion problem has not yet been solved. Even if the mathematical complexity can be overcome in the foreseeable future, it would be impossible to obtain all the coefficients in the tensor matrix experimentally. Because multicomponent and anisotropic minerals are common (e.g., mica, hornblende, pyroxenes, etc), diffusion in nature is truly complicated. Hence, simplifications are usually made out of necessity because requiring theoretical rigor would simply mean getting nowhere in understanding geological problems which otherwise can be approximately quantified. Even for diffusionally isotropic natural silicate melts with typically four or more major components and numerous minor and trace components, no reliable diffusivity matrix has been obtained. In the near future, it may be possible to extract diffusivity matrices of the major components. However, unless theories can be developed to calculate the diffusivity matrix for trace elements in natural melts, the diffusivity matrix approach is unlikely workable for trace elements because it is impractical to experimentally obtain diffusivity matrices involving both major and trace elements (about 80 components in total). Hence, in the foreseeable future, the simple effective binary approach, or some modified simple approach, will be necessary. In this section, the various approaches and their advantages and disadvantages are briefly outlined.

Effective binary approach, FEBD and SEBD The effective binary diffusion approach (Cooper 1968) is the most widely used simple treatment of diffusion in a multicomponent system. In this approach, the diffusion of a component in a multicomponent system is treated as diffusion in a binary system, of which one is the component of interest and the other is all the other components combined. That is, the flux of component i in one dimension is simply expressed as (Eqn. 6): Ji = −

Di ∂Ci ∂x

(88)

The diffusion equation is hence Equation (5). Compared to the multicomponent diffusivity matrix approach (one dimensional form of Eqn. 87) in which Ji = −SDij∂Cj/∂x, it can be seen that n

∂C j / ∂x

j =1

∂Ci / ∂x

Di = ∑ Dij

= Dii +

n

∑

j =1, j ≠ i

Dij

∂C j ∂Ci

(89)

Hence, the effective binary diffusivity depends on the on-diagonal and off-diagonal diffusivities involving the component of interest, as well as the concentration gradients of all components. With the effective binary approach, the solutions for binary diffusion are used for diffusion of a component in a multicomponent system and to extract effective binary diffusivities. Almost all chemical diffusion data in minerals and melts are obtained using this approach.

Theoretical Background of Diffusion in Minerals and Melts

37

These chemical diffusivities are called effective binary diffusivities (EBD, which can also Youxue Zhang (Ch 2) Page 13 mean effective binary diffusion). The approach only works when there is no uphill diffusion (e.g., Fig. 13); otherwise, the effective binary diffusivity would change from positive to negative in a single profile.

Figure 13. Uphill diffusion in a diffusion couple experiment (Van Der Laan et al. 1994). Fig. 13. Uphill diffusion in a diffusion couple experiment (Van Der Laan et al. 1994).

Because the EBD approach does not work for uphill diffusion profiles, authors typically ignore uphill diffusion profiles in data treatment, often simply marking them as “uphill”. In binary systems, the presence or absence of uphill diffusion can be predicted from thermodynamics: uphill diffusion would occur when the phase is not stable and undergoes spinodal decomposition. However, in multicomponent systems, whether uphill diffusion would occur for a given element (such as Ca or Ce) in a diffusion couple or during crystal dissolution (Zhang et al. 1989) cannot be predicted yet. Even in the absence of uphill diffusion, the EBD approach is still prone to many limitations. The most serious one is that effective binary diffusivity of an element may depend not only on the major oxide concentrations, but also on their gradients (Liang 2010, this volume). This may sound strange but can be understood in the following example. The SiO2 diffusivity in CaO-Al2O3SiO2 melts along constant CaO concentration (the CaO concentration gradient is nearly zero, and the Al2O3 concentration gradient roughly compensates the SiO2 concentration gradient) is essentially the SiO2-Al2O3 interdiffusivity, but is essentially the SiO2-CaO interdiffusivity along a constant Al2O3 concentration. It is hence understandable that the two are different even at the same bulk composition, as shown by Liang et al. (1996). Along other compositional directions, even though the diffusivity cannot be simply viewed as an interdiffusivity, the effective binary diffusivity is expected to also depend on other compositional gradients. Because compositional gradients can change as diffusion proceeds, especially in a finite system, the EBD can also change as diffusion proceeds (Liang 2010, this volume). Even knowing that it has limitations, EBD approach is still the most widely used approach in treating diffusion in multicomponent systems because other approaches are too complicated. In the foreseeable future, the EBD approach will still likely be the method of choice for trace element diffusion even if the diffusivity matrix for major components could be obtained. Hence, it is necessary to understand under what conditions the EBD approach is more reliable than others. Cooper (1968) summarized that the EBD approach is meaningful when (i) the concentration gradients of all components are in the same direction; and (ii) either a steady state exists or the diffusion media is infinite or semi-infinite. However, there could still be uphill diffusion even when the two conditions are satisfied, and uphill diffusion is difficult to treat using the EBD approach. Below, some specific situations are discussed:

38

Zhang (a) The EBD approach works well if the base composition of a diffusion system is the same, but there is one relatively minor component (or a few minor to trace components) up to several percent that diffuses in or out, such as in cases of sorption or desorption, hydration or dehydration, bubble growth, or a diffusion couple between two rhyolite melts with only small difference in Sr concentration. That is, the compositional difference in the system is due to the dilution effect of the presence (or absence) of this component under consideration. If the component diffuses more rapidly than other components, the effective binary diffusion approach works even better. Diffusivities from such experiments can be applied accurately to similar situations when the base compositions are the same. (b) In a multicomponent system, if the initial compositional difference is only between two components such as Na2O and K2O, the diffusion of each of the two components can be treated as effective binary diffusion. In fact, this case is similar to interdiffusion between two components, and hence the two components should have the same interdiffusivity. Diffusion of other components in the system may not be treated as EBD. (c) Even if there are major concentration gradients in multiple components, as long as they are consistent in the direction and relative magnitude, the EBD approach can be applied to the component with the largest concentration gradient. Examples include dissolution of a specific mineral in a specific melt (such as olivine dissolution in a basalt melt, or quartz dissolution in an andesite melt). In this case, the component with the largest concentration gradient (such as MgO during olivine dissolution in a basalt melt, or SiO2 during quartz dissolution in an andesite melt, referred to as the principal equilibrium determining component by Zhang et al. 1989) can be treated as EBD. Such effective binary diffusivities can be applied to the identical situations in nature under similar boundary conditions such as semi-infinite diffusion media, meaning MgO EBD extracted from olivine dissolution experiments in basalt melt can be applied to predict MgO diffusion in nature during olivine dissolution in a basalt melt, but it may not be applicable to MgO diffusion during clinopyroxene dissolution in a similar melt, or olivine dissolution in a different melt. Diffusion of other components may or may not be treated as EBD. (d) In diffusion couple studies when two different melts are placed together, effective binary diffusivities may be extracted for components showing large concentration differences between the two halves. These EBD values may be applied to diffusion couples with similar compositions in the two halves. However, if the concentration gradient is switched for some major component (e.g., in one diffusion couple, Al2O3 concentration is 13 wt% in the basalt half and 17 wt% in the andesite half, whereas in the other diffusion couple, Al2O3 is 17 wt% in the basalt half and 13 wt% in the andesite half), EBD from one case may not be applied to the other. Liang et al. (1996) (their Fig. 5) showed an example of this.

As can be seen from the above, effective binary diffusivities are a large category and cover many different situations. Some EBD values are more reliable than others. For easy reference, I propose three types of effective binary diffusivities based on their consistency and reliability: (1) Interdiffusivity or interdiffusion (ID). Binary diffusion is interdiffusion. In multicomponent systems, if the concentration gradients are primarily in two components, and concentrations of all other components are roughly uniform (case (b) above), then it can be referred to as multicomponent interdiffusion. In this case, the diffusivities of the two components are roughly the same, and can be treated well and consistently using the effective binary approach. The interdiffusivity would depend on the bulk composition but not on the concentration gradients of other components (these are es-

Theoretical Background of Diffusion in Minerals and Melts

39

sentially zero). Diffusion of other components may not be treated well by the effective binary approach (e.g., often there can be uphill diffusion for other components). (2) FEBD (first type of effective binary diffusion, or first type of effective binary diffusivity). FEBD corresponds to the diffusion situation of case (a) above. Because this situation is often encountered in experiments and nature (especially sorption/ desorption, bubble growth, and explosive volcanic eruptions), and because of the high degree of consistency of this type of effective binary diffusivity, it is considered to be a special type of EBD, called the first type of effective binary diffusion, with the acronym FEBD. In principle, when the concentration of the component of interest becomes low enough, FEBD approaches the tracer diffusivity (or trace element diffusivity in the absence of major concentration gradients). (3) SEBD (second type of effective binary diffusion, or second type of effective binary diffusivity): All other types of effective binary diffusivities are less reliable and are grouped as SEBD, even though some may be more consistent than others. SEBD values can be applied to systems very similar to experimental systems in terms of bulk composition as well as the direction and size of concentration differences. In studies of diffusion in multicomponent systems, often all monotonic concentration profiles (no uphill diffusion) are treated using the effective binary diffusion approach, assuming a constant SEBD. The fits may not be perfect (e.g., the SEBD of MgO during olivine dissolution in an andesite melt seems to increase with increasing SiO2 content; Fig. 14). In such a case, one is tempted to make efforts to determine how the SEBD varies along the profile and associate the variation with SiO2 or other concentration changes. However, this may not be correct, because the SEBD variation might be due Youxue Zhang (Ch 2)to concentration gradient Page 14 variations, rather than the compositional variations. Hence, in treating SEBD profiles, the simplest approach is to fit the profile with a constant SEBD and ignore the small misfits because of the complexity of SEBD.

Figure 14. MgO diffusion profile in the melt during olivine dissolution in andesite melt (Zhang et al. 1989). The fit curve, assuming constant SEBD, does not match the data well, e.g., at the region near x = 1 mm. The slower decrease of the concentration implies higher diffusivity in this region even though the SiO2 concentration here is high (about 57 wt%) compared to SiO2 near x = 0 (about 52 wt%). This misfit is most likely due to the dependence of the SEBD of MgO on concentration gradients rather than on the bulk composition itself.

Modified effective binary approach (activity-based effective binary approach)

Fig. 14. MgO diffusion profile in the melt during olivine dissolution in andesite melt (Zhang et al. 1989). The fit

curve, assuming constant SEBD, does not match thebinary data well, approach e.g., at the region near xcalled = 1 mm. The The modified effective (also theslower activity-based

effective binary approach) was proposed by Zhang (1993) and based on rough chemical activity estimation decrease of the concentration implies higher diffusivity in this region even though the SiO2 concentration here is in silicate melts. It is assumed that the diffusive flux of a component i is proportional to the

high (about 57 wt%) compared to SiO2 near x = 0 (about 52 wt%). This misfit is most likely due to the dependence of the SEBD of MgO on concentration gradients rather than on the bulk composition itself.

Zhang

40

activity gradient of the component alone (Eqn. 65): Ji = −

Di ∇ai γi

(90)

where ai and gi are the chemical activity and activity coefficient of component i with ai = giCi, and Di is the “intrinsic” effective binary diffusivity, different from the normal effective binary diffusivity. The above equation reduces to Equation (88) when gi is constant. The difference between the effective binary approach and the modified effective binary approach is that the concentration gradient is replaced by the activity gradient. From Equation (66) (or comparing Eqns. 88 and 90), the “intrinsic” effective binary diffusivity and the normal effective binary diffusivity are related as:  d ln γ i  Di = Di  1 +  d ln Ci  

(91)

Zhang (1993) showed that the approach could fit and predict uphill diffusion profiles during crystal dissolution experiments (Fig. 15), which cannot be treated by effective binary diffusion. Lesher (1994) developed a similar model to treat uphill diffusion of trace elements in a diffusion couple. Even though the model can handle uphill diffusion profiles and may also fit monotonic diffusion profiles better, the approach has two disadvantages: (i) it is complicated, and (ii) the activity model for silicate melts is uncertain. The applicability (or inapplicability) of the approach needs to be explored further. Youxue Zhang (Ch 2)

Page 15

Figure 15. Fitting an uphill diffusion profile of FeO in an andesite melt during olivine dissolution using the modified effective binary diffusion model (from Zhang 1993).

Diffusivity matrix approach

Fig. 15. Fitting an uphill diffusion profile of FeO in an andesite melt during olivine dissolution using the modified

The diffusivity matrix approach is the classical and rigorous method to describe multicomponent diffusion (Eqns. 86 and 87; see in-depth discussion by Liang 2010, this volume). This approach works well if the N-component mixture is not very non-ideal. The approach fails when the mixture is unstable, leading to spinodal decomposition because some eigenvalues would change from positive to negative, similar to the case of spinodal decomposition in a binary system. When the system is highly non-ideal, the individual diffusivity values would be highly variable with composition even if the mixture is stable. For ideal and nearly ideal

effective binary diffusion model (from Zhang 1993).

Theoretical Background of Diffusion in Minerals and Melts

41

systems, the diffusion equation for an N-component system (n = N−1) is of the following form: n ∂Ci = ∑ ∇ ( Dij ∇C j ) ∂t j =1

(92)

For one-dimensional diffusion and constant Dij values, the diffusion equation becomes: n ∂ 2C j ∂Ci = ∑ Dij ∂t ∂x 2 j =1

(93)

Therefore, the concentration evolution of component i depends on the concentration gradients of other components. Equation (93) contains coupled equations. Hence, it is necessary to solve the concentration evolution of all n components simultaneously in the N-component system. In the matrix form, the concentrations can be solved as follows. The one-dimensional diffusion equation with constant diffusivity matrix can be written as: ∂C ∂ 2C =D 2 (94) ∂t ∂x where C is the concentration vector, or the transpose of (C1, C2, ..., Cn), and D is the diffusivity matrix. If the N-component mixture is stable, it can be shown that all n eigenvalues of the D matrix are real and positive (e.g., De Groot and Mazur 1962), and D can be diagonalized:

D = TlT−1 (95) where l is a diagonal matrix made of the eigenvalues of D, and T is a matrix made of the eigenvectors of D. Replacing Equation (95) into Equation (94) leads to ∂u ∂2u =l 2 ∂t ∂x

(96)

where u = T−1C is the transformed composition vector. Because l is a diagonal matrix, the above equation is equivalent to: ∂ui ∂ 2u = l i 2i ∂t ∂x

(97)

where i can be 1, 2, ..., n. Hence, in the transformed compositional space, the diffusion equation for each ui depends only on its own concentration gradient with a real and positive diffusivity of li. If the initial and boundary conditions can also be transformed, Equation (97) is in the same form as binary diffusion and ui can be solved. After solving for every ui, the final solution is C = Tu (98) When the diffusivity matrix is not constant (e.g., some of the elements in the matrix depend on concentration, which is common), the above analytical solution is not possible, and the multicomponent diffusion equation must be solved numerically (this complexity also exists for the effective binary treatment). In principle, the diffusivity matrix (i.e., diffusion of all elements in a diffusivity matrix) can be obtained from experimental diffusion studies, similar to binary or effective binary diffusivities, by fitting experimental concentration profiles using the diffusivities of elements as fitting parameters. This has been done for some simple melts (e.g., Vignes and Sabotier 1969; Sugawara et al. 1977; Liang et al. 1996; Liang and Davis 2002). However, for natural silicate melts, due to the large number of fitting parameters involved (e.g., the diffusivity matrix of a 10-component system is made of 81 individual values), this is a daunting task. Strategies

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have been proposed (Trial and Spera 1994) and bold attempts have been made (Kress and Ghiorso 1995; Mungall et al. 1998), but the diffusivity matrices have not been verified and more follow-up work is necessary. More on multicomponent diffusion as well as empirical models for multicomponent diffusion matrices can be found in Liang (2010, this volume).

Activity-based diffusivity matrix approach Even the complicated treatment of multicomponent diffusion using diffusivity matrices is not enough to treat phase separation in multicomponent systems. A more fundamental approach is to use the activity-based diffusivity approach as in the binary system (Eqn. 65). The diffusive flux is expressed as (Zhang 2008) n

Dij

j =1

γj

J i = −∑

∇aj

(99)

where Dij are the “intrinsic” diffusivities. The diffusion equation for component i is hence n D  ∂Ci = −∇J i = ∑ ∇  ij ∇a j   γj  ∂t j =1  

(100)

The advantage of this approach is that it can treat highly non-ideal mixtures as well as spinodal decompositions smoothly without invoking negative diffusivities. The disadvantage of this approach is its complexity. Even the concentration-based diffusivity matrix approach is already too complicated to be carried out for melts and many minerals. When accurate activity models become available, it may be possible to contemplate the activity-based diffusivity matrix approach.

Origin of the cross-diffusivity terms In the concentration-based diffusivity matrix, the cross diffusivities are significant, often resulting in uphill diffusion of many components. Uphill diffusion is the most clear demonstration of the cross diffusivities, and the need to include the cross diffusion terms. Furthermore, uphill diffusion is also the best example of the failure of the concentration-based effective binary approach. One fundamental question is what factors contribute to the cross-diffusivity terms of Dij (i≠j). The cross-diffusivities may arise from two sources, one is non-ideality of the multicomponent mixture (thermodynamic effect), and the other is a purely kinetic effect. Some uphill diffusion profiles in a stable phase may be attributed to non-ideal mixing, i.e., the thermodynamic effects, and can be modeled using activity-based diffusivities (e.g., Zhang 1993). It is not clear how much contribution to the cross terms is from kinetic effects, although there has been some discussion (e.g., Liang et al. 1997). One rigorous way to separate the two effects is to consider diffusion in a perfectly ideal mixing system of different isotopes. At least three isotopes (such as 28 Si, 29Si, and 30Si) are needed so as to make a multicomponent system, with each isotope being a component. For example, diffusion in two chemically identical haplorhyolite melt halves, one normal in silicon isotopes, and one enriched in 30Si, can be investigated. The fraction of 29Si out of all Si can be kept constant in both halves. After diffusion experiments, silicon isotope fractions are measured. Because the chemical composition is uniform and mixing between the three silicon isotopes is expected to be perfectly ideal, the cross diffusivities would be zero if cross diffusivities are entirely due to thermodynamic contributions. In such a case, the 29 Si fraction would stay as a constant across the whole profile because there was no initial 29 Si gradient and hence no 29Si flux. However, if there is a kinetic contribution to the cross diffusivities, the cross terms would cause 29Si fluxes, leading to uphill diffusion profiles when the fraction of 29Si is plotted against distance.

Theoretical Background of Diffusion in Minerals and Melts

43

DIFFUSION COEFFICIENTS Over the years, diffusivities in many systems have been determined experimentally. Many authors have examined how the diffusivities vary with experimental conditions, and developed numerous relations. Some of them have strong theoretical basis and are widely applicable, but others are largely empirical and also uncertain. Experimentally determined values of diffusion coefficients in minerals and melts are reviewed and summarized in later chapters of this volume. In this section, various relations relating diffusion coefficients and other parameters are reviewed. Diffusivity values and activation energy depend on the phase in which diffusion occurs (Watson and Baxter 2007). Figure 2 shows Ar diffusivities in gas, aqueous solutions, basalt melt, rhyolite melt, and the mineral amphibole for comparison. The diffusivity decreases and the activation energy increases from air to water to melt to mineral. From one species to another in a liquid or melt, the diffusivity depends on the bond strength as well as the size of the diffusing species. From one liquid to another, the diffusivity depends on both the diffusing component as well as the liquid composition. Similarly, in minerals, from one species to another, diffusion can depend on size and charge of the diffusing species, and the sites on which they diffuse; from one mineral to another, diffusivities of a particular species may depend on properties such as mineral composition, structure and bond strengths.

Temperature dependence of diffusivities; Arrhenius relation The temperature dependence of diffusivities is well known and works well almost without exceptions. This relation is the Arrhenius equation (Eqn. 10) D = D0e−E/(RT), meaning that logD (or lnD) versus 1/T (or 1000/T) is a straight line with a negative slope, as shown in Figure 2 for Ar diffusivities in water, basalt melt, rhyolite melt and the mineral hornblende. (Fig. 17a in a later section also shows some diffusion data in an Arrhenius plot.) The Arrhenius equation can be derived from either the collision theory or the transition state theory (e.g., Lasaga 1998), in which the activation energy is identified to be the necessary enthalpy for forming the activated complex. The preexponential factor D0 is proportional to T1/2 in the collision theory and to T in the transition state theory. There were efforts in the early years to test how the preexponential factor D0 depends on temperature (e.g., Perkins and Begeal 1971; Shelby and Keeton 1974), but it requires high-quality data (e.g., with ≤ 10% relative error in D) in a large temperature range (e.g., 400-1200 K), and the results are inconclusive: high-quality data over a large temperature range may show a small curvature in lnD versus 1/T, indicating that either D0 or E depends on temperature, but the exact relation is not well constrained. On the other hand, for most diffusion data in melts and minerals in the geological literature, the uncertainty in D is often of the order of 30% and the temperature range is not large enough, so that the Arrhenius relation works well within uncertainty. In limited cases when the temperature range is large from glass to melt, the Arrhenius equation fits the data well and it is rarely necessary to invoke dependence of D0 on T, or E on T, or a discontinuity due to glass transition (e.g., Zhang et al. 1991a; Zhang and Behrens 2000; Behrens and Zhang 2009). In minerals, although some authors proposed discontinuities in lnD versus 1/T (e.g., Buening and Buseck 1973 for Fe-Mg interdiffusion in olivine), later studies show no such discontinuity is present (e.g., Chakraborty 2010, this volume). In summary, the temperature dependence of all experimental diffusivity data in the geological literature can be summarized well by the Arrhenius relation in the form of D = D0e−E/(RT) with positive D0 and E.

Pressure dependence of diffusivities The dependence of diffusivities on dry pressure (P) is more complicated than the temperature dependence. (Hydrous pressure, on the other hand, has two effects: one is pressure, and the other is the presence of water, which may accelerate diffusion.) In a small pressure range, lnD is often linear in P. Such a relation is consistent with the transition state theory

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because the enthalpy is linear to pressure in liquid and solid phases, leading to Equation (11), D = D0e−(E+P∆V)/RT, in which E is the activation energy at zero pressure, ∆V is the activation volume (identified as the volume difference between the activated complex and the nonactivated state), and E+P∆V is the activation energy at pressure P. Even though the activation energy for diffusion is always positive, the activation volume can be either positive (meaning D decreases with increasing pressure, such as Ar diffusivity in rhyolite melt — Behrens and Zhang 2001, see Fig. 16a; H2O diffusivity in rhyolite and dacite melts — Zhang and Behrens 2000, Ni and Zhang 2008, Ni et al. 2009; divalent cation diffusivities in garnet and in oxide minerals — Ganguly 2010, this volume, Van Orman and Crispin 2010, this volume) or negative (meaning D increases with increasing pressure, such as oxygen self diffusivity in dacite melt below 4 GPa; Tinker and Lesher 2001, see Figure 16b; Mg self diffusion above 60 GPa, Van Orman and Crispin 2010, this volume). Whereas the activation energy E is a large positive value and hence small changes due to temperature and pressure variations do not affect E noticeably, the activation volume ∆V as the volume difference between the activated complex and the ground state is small and hence can change significantly as pressure or temperature changes. Such changes produce at least two effects (or complexities) when compared with the temperature dependence of D: (1) The dependence of ∆V with temperature means more terms are needed in fitting lnD as a function of T and P. For example, if ∆V is linear to T as ∆V = a + bT, then Equation (11) becomes ln D = a0 −



E + P∆V E + aP + bPT a +bP = a0 − = ( a0 + b0 P ) − 1 1 RT RT T

(101)

where a0 = lnD0, b0 = -b/R, a1 = E/R, and a1 = a/R. That is, the preexponential term would depend on pressure even though this is not in the original equation derived from the transition state theory (Eqn. 11). If ∆V is a more complicated function of T, there may be more terms in fitting lnD versus T and P. When fitting diffusion data as a function of T and P, usually Equation (11) is used. However, if the fitting quality is not good enough, then Equation (101) or even more complicated function may be used.

(2) The activation volume ∆V can also change with pressure, even from positive to negative or vice versa as pressure increases (Fig. 16b). Therefore, at a given temperature, in a large pressure range (several GPa), lnD versus P may not be linear: lnD may first increase with P and then decrease with P, or first decrease with P and then increase with P. Large changes in ∆V, especially from positive to negative or vice versa, may be associated with structural changes in silicate melts. In addition to dependence on total pressure, H2O pressure may have a disproportionally large effect on diffusivity of some components in hydrothermal diffusion experiments. For example, 18O diffusivities in nominally anhydrous felsic minerals are roughly proportional (rather than exponential) to H2O pressure or fugacity (e.g., Yund and Anderson 1978; Farver and Yund 1990), increasing by two orders of magnitude as H2O fugacity increases from 0.001 GPa to 0.2 GPa. This dependence on pressure is different from the dependence on total pressure, and is related to H218O molecules entering a mineral and carrying 18O into the mineral (Zhang et al. 1991b; McCornell 1995). The contribution by H2O diffusion to the diffusion of 18O has been quantified (Zhang et al. 1991b) and is related to the product of H2O concentration and diffusivity in the mineral. For some minerals (e.g., zircon), however, there is little dependence of 18O diffusion on H2O pressure in hydrothermal experiments, although there is a difference between oxygen diffusivities for dry and hydrothermal experiments (e.g., Watson and Cherniak 1997), probably due to the saturation of mobile H2O in the mineral. For cations such as Na and Sr in feldspar, because H2O does not carry them, H2O diffusion would not directly contribute to their diffusion (although H2O in a mineral may weaken the

Theoretical Background of Diffusion in Minerals and Melts

45

 

Figure 16. Different pressure dependences of diffusivities. (a) The diffusivity decreases with pressure (positive activation volume); (b) The diffusivity first increases with pressure and then decreases with pressure.

structure, enhancing diffusion). For example, H2O pressure of 200 MPa does not change the diffusivity of cations in feldspar compared to room pressure dry experiments (Behrens et al. 1990; Cherniak 2010, this volume). In summary, the dependence of D on pressure is more complicated than on temperature, and for many systems such dependence has not been investigated well. Because pressure can vary widely from the surface to the deep Earth, understanding the pressure effect quantitatively so that diffusivities can be predicted remains an important task for geochemists.

Diffusion in crystalline phases and defects Crystals contain various defects where the periodicity of the lattice is locally disturbed. In crystalline phases, diffusion often occurs through defects in the crystalline structure. For example, a cation may jump into a nearby vacancy (one kind of point defect), leaving another vacancy defect in its original position (into which another cation may jump), and then jump into another vacancy as it comes nearby, and so on. Hence, the diffusivity is often proportional to defect concentrations. For a review of defect theory as well as its applications to minerals, readers are referred to Schock (1985) and Lasaga (1998). Defects may be classified as point defects and extended

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defects. Point defects include vacancies (unoccupied sites that are normally occupied), interstitials (atoms occupying normally unoccupied sites), and impurities (atoms occupying sites that are normally occupied by other types of atoms, such as Al occupying an Si site). If a defect is caused by impurities, it is called an extrinsic defect. Otherwise, it is called an intrinsic defect. The number of defects in a material must satisfy the condition of charge neutrality. A nearby pair of vacancy and interstitial is called a Frenkel defect. A stoichiometric proportion of cation and anion vacancies is called a Schottky defect. Extended defects include line defects (such as dislocations), plane defects, domain boundaries, grain boundaries, and bulk defects or impurities (such as a melt inclusion). Point defects play the most important role in volume diffusion, but extended defects may produce fast diffusion paths. In treating defects, the Kröger-Vink notation is conventionally used. The format of the p notation takes the form M b where M is the species, including (1) elements such as Fe, Mg, Si, O, etc, (2) vacancies (either V or v since V is also the element symbol for vanadium), (3) electrons (e), and (4) electron holes (h). The subscript b indicates the lattice site that the species M occupies, such as Na site in NaCl, Fe site in olivine, or an interstitial site (indicated as i). The superscript p indicates the electronic charge relative to the site it occupies (so that if the site is occupied by its normal occupant, the charge is zero, indicated by x), with positive charges indicated by • (the number of • indicates the number of positive charges), and negative charges indicated by ′. At a given temperature, there is an equilibrium concentration of intrinsic defects. For Schottky defects when cation to anion ratio is 1:1, such as MgO, the reaction can be written as (in Kröger-Vink notation): Ø  v′′Mg + v••O

(102)

where Ø means nothing, v′′Mg means a vacant magnesium site with an effective charge (relative to that of the occupied site) of −2, and v••O means a vacant oxygen site with an effective charge of +2. If this kind of defects dominate, the equilibrium constant is K1 = [v′′Mg ][vO•• ] = [v′′Mg ]2 = [v••O ]2

(103)

where brackets mean mole fractions. Hence, the equilibrium concentration of vacancy pairs is: [v′′Mg ] = [v••O ] = K11/ 2 = e − ∆Gf1 /( 2 RT ) = e ∆Sf1 /( 2 R )e − ∆Hf1 /( 2 RT )

(104)

where ∆Gf1, ∆Sf1, and ∆Hf1 are the formation free energy, entropy and enthalpy for the pair of vacancies. For Frenkel defects, the reaction can be written as (using Mg in MgO as an example): Mg xMg  v′′Mg + Mg••i x Mg

(105)

where Mg means a magnesium ion sitting on a magnesium lattice site, and Mg i means an interstitial magnesium ion. If this kind of defects dominate, the equilibrium constant is: ••

K 2 = [v′′Mg ][ Mg ••i ] = [v′′Mg ]2 = [ Mg ••i ]2

(106)

Hence, the expression for the equilibrium concentration of Frenkel defects is similar to the case of Schottky defects: [v′′Mg ] = [ Mg ••i ] = K 21/ 2 = e − ∆Gf2 /( 2 RT ) = e ∆Sf2 /( 2 R )e − ∆Hf2 /( 2 RT )

(107)

where ∆Hf2 is the formation energy for the pair (i.e., moving Mg from a lattice position to an interstitial position, leaving behind a vacancy).

Theoretical Background of Diffusion in Minerals and Melts

47

Because ions with higher valence are more strongly bonded, it is expected that ∆Hf increases with valence. Other factors, such as oxygen fugacity, can also affect defect concentrations (see below).

Diffusivities and oxygen fugacity Variation in oxygen fugacity in a system results in at least two effects, one is to change the oxidation state of multivalent elements, and the second is to change (often induce extra) defect sites in the structure. The two effects are somewhat correlated at the atomic level, but the resulting effects in terms of diffusivity are different. The first effect is important especially in melts and minerals when the element of interest can be present in multiple valences, such as Fe (Fe2+ and Fe3+), S (S2−, S4+ in SO2, and S6+ in SO42−), Eu (Eu2+ and Eu3+), Sn (Sn2+ and Sn4+), etc (e.g., diffusion of Fe in feldspars, Behrens et al. 1990; diffusion of Eu in orthopyroxene, Cherniak and Liang 2007; and diffusion of Eu in melts, Behrens and Hahn 2009). The diffusivity of the lower valent species is often higher than the higher valent species because of weaker bonding between the low valent ion and the rest of the structure. Hence, as oxygen fugacity increases, the diffusivity of the element of interest often decreases. The diffusion of the element can be quantified using multispecies diffusion equations following the approach of Zhang et al. (1991b), e.g., for Fe diffusion: DFe = X Fe2+ DFe2+ + X Fe3 + DFe3 +

(108)

where XFe2+ = Fe2+/(Fe2++Fe3+). The second effect has been observed and examined largely in mafic minerals, and can be understood by examining the defect concentration in a mineral structure. For example, increasing oxygen fugacity would oxidize a small fraction of Fe2+ to Fe3+, producing vacancy defects (because two Fe3+ ions are equivalent to three Fe2+ in terms of charges). In wüstite, the reaction in Kröger-Vink notation can be written as: 6 Fe xFe + O2 (gas)  4 Fe •Fe + 2v’’Fe + 2 FeO

(109)

The equilibrium constant can be written as: K=

[ Fe •Fe ]4 [v′′Fe ]2 [ FeO]2 [ Fe xFe ]6 fO2

(110)

From stoichiometry in Reaction 109, [Fe •Fe] = 2[v′′Fe]. Hence, Equation (110) can be written as: K=

16[v′′Fe ]6 [ FeO]2 [ Fe xFe ]6 fO2

(111)

Because the activities of Fe xFe and FeO in wüstite are roughly constant, it can be seen that the concentration of this type of vacancy is proportional to the 1/6 power of oxygen fugacity. In other ferromagnesian minerals, the fO12/ 6 relation also holds (e.g., Lasaga 1998), which is consistent with the increase of Fe-Mg interdiffusivity in olivine with oxygen fugacity (Dohmen and Chakraborty 2007a,b). If other defects are present, their effect on diffusivity must also be quantified to obtain the full effect of oxygen fugacity.

Compositional dependence of diffusivities Diffusivity of a component depends on the major chemical composition, and occasionally also on the minor element composition. For example, the effect of oxygen fugacity on diffusion may be viewed as a compositional effect. The effects of H2O content (or H2O pressure) on diffusivities in melts and minerals have been investigated: Minor amounts of H2O may affect the diffusivity of some components in minerals significantly as in the case of the dependence of 18O diffusivity in feldspars and quartz on H2O pressure (Farver and Yund 1990, 1991; Zhang

48

Zhang

et al. 1991b), as well as diffusivities in melts, especially silicic melts (Mungall and Dingwell 1997; Behrens et al. 2007). The effects of SiO2 and alkalies on diffusivities in melts have also been investigated to some degree. In general, more effort is needed to understand the compositional effects on diffusivities. Limited data often indicate that D is an exponential function of some concentration (that is, lnD is linear with respect to concentration). For example, Behrens and Zhang (2001) showed that the Ar diffusivity in rhyolite melts increases exponentially with H2O content. Dohmen and Chakraborty (2007a,b) showed that the Fe-Mg interdiffusivity in olivine increases exponentially with the mole fraction of fayalite (lnD is linear to XFa). Other relations have also been found. For example, for 22Na tracer diffusivity in plagioclase melts, lnD is not linear with respect to XAn (Behrens 1992) but requires a second-order term (Zhang et al. 2010, this volume). For Th and U diffusion in a haplorhyolite melt, lnD seems to be linear with respect to the square root of the H2O concentration, rather than the H2O concentration itself (Mungall and Dingwell 1997; Zhang et al. 2010, this volume). Diffusivity of the hydrous component in orthopyroxenes depends strongly on the composition of orthopyroxenes (Farver 2010, this volume). In addition, the relation between diffusivity and concentration may be modified by the role of speciation of the diffusing component. For example, molecular H2O diffusivity in rhyolite and dacite melts conforms well with the exponential relation (Zhang and Behrens 2000; Liu et al., 2004; Ni and Zhang 2008; Ni et al. 2009), but the diffusivity of total H2O (which includes both molecular H2O and hydroxyls) increases first proportionally with total H2O content and then exponentially. Diffusivity of the hydrous component in pyrope shows complicated behavior, due to both the speciation effect and the oxygen fugacity in pyrope affecting the H2O/H2 ratio (Wang et al. 1996). SiO2 is the dominant component in silicate melts, and its effect on diffusivities can be complicated. Diffusivities of most components decrease as SiO2 content increases (and the degree of polymerization increases), and often the relation between lnD and SiO2 concentration is roughly linear (e.g., Lesher and Walker 1986; Behrens et al. 2004; Ni et al. 2009). However, diffusivities of some components (such as He, Li, and Na) increase as SiO2 content increases (Behrens 2010; Zhang et al. 2010). Overall, compositional effects on diffusivities are complicated, especially in silicate melts, but also in complicated minerals such as hornblende and pyroxenes. We have only scratched the surface for such effects. One complexity is due to the numerous components in natural melts and many minerals because we often can only examine the effect of one component at a time. Another complication arises because there are no theoretical relations to guide studies on the compositional effects although empirical models are available (Mungall 2002) in which the diffusivities of non-alkalies are related to viscosity, but those of alkalies are not related to viscosity. It is anticipated that in the future much more effort will be devoted to understanding the compositional effects on diffusion in melts and minerals.

Relation between diffusivity, particle size, particle charge, and viscosity There are famous relations between diffusivity, particle size and viscosity, all of which only have limited applicability to diffusion in silicate melts. One such famous relation is based on Einstein’s analyses of Brownian motion of neutral and spherical particles in a continuous fluid medium (Einstein 1905), resulting in the following relation between diffusivity, viscosity, particle radius and temperature: D=

kB T 6 πaη

(112)

where kB is the Boltzmann constant (1.3807×10−23 J/K), a is the radius of the particle, and h is the viscosity of the fluid. The above equation is often referred to as the Einstein equation, or

Theoretical Background of Diffusion in Minerals and Melts

49

Stokes-Einstein equation. There are other variations of this equation, such as the Sutherland equation (Sutherland 1905) differing from the above equation by a factor of 1.5, and the Glasstone equation (Glasstone et al. 1941), differing from the above equation by a factor of 3π. Another equally famous equation is the Eyring equation (Glasstone et al. 1941), relating self diffusivity, viscosity, diffusive jump distance and temperature: D=

kB T lη

(113)

where l is the diffusive jump distance. The jump distance is often estimated to be interatomic distances or diameter of the diffusing species. If the diameter of the diffusing species is used as the jump distance, then the Eyring equation becomes the same as the Glasstone equation, and the Eyring diffusivity is 3p times the Einstein diffusivity. The assumptions employed in deriving the Einstein equation and the Eyring equation are different: the Einstein equation is derived for the diffusion of large neutral species, whereas the Eyring equation is derived for the case when the diffusion of the species is also responsible for the viscous flow. Both equations claim that D is inversely proportional to viscosity. The Einstein equation seems to work well (within a factor of 2) for diffusion of relatively large neutral molecules (e.g., heavier noble gases) in aqueous solutions, but does not work at all (orders of magnitude difference) for the diffusion of any known neutral species in silicate melts (e.g., Ni and Zhang 2008). The Eyring relation appears to work well for O (and possibly Si) self diffusion in dry silicate melts (but see Lesher et al. 1996; Liang et al. 1996; LaTourrette and Wasserburg 1997), but does not work at all (orders of magnitude difference) for 18O diffusion in hydrous silicate melts (Behrens et al. 2007). Because most natural melts are hydrous, the applicability of the Eyring relation to natural melts is very limited. For other components, especially for rapidly diffusing components, the Eyring relation does not apply. For example, when 3.5 wt% H2O is added to a dry rhyolite at 1000 K, the viscosity decreases by 7 orders of magnitude (Zhang et al. 2003), but 22Na tracer diffusivity only increases by a factor of about 2 (Watson 1981). Not only is the diffusivity of most components in melts not inversely proportional to melt viscosity, but the negative correlation between D and h is also violated sometimes. For example, tracer diffusivities of Li and Na increase as viscosity increases from basalt to dacite to andesite to rhyolite melts (Jambon and Semet 1978; Jambon 1982; Zhang et al. 2010). There is also no universal relation between the size of the diffusing species and the diffusivity, contrary to the Einstein equation. For example, for the alkali elements, the ionic radius increases from Li+ to Na+ to K+ to Rb+ to Cs+, but the tracer diffusivity decreases from Na to Li to K to Rb to Cs by orders of magnitude (e.g., Jambon 1982). On the other hand, for alkaline earth elements, the ionic radius increases from Be2+ to Mg2+ to Ca2+ to Sr2+ to Ba2+, but the tracer diffusivity decreases from Sr and Ba to Ca to Mg to Be (Mungall et al. 1999), with smaller cations having greater diffusivity, completely opposite to the prediction of the Einstein equation. Even though the opposite trend may be rationalized by the diffusing species of a component being different from the cations, it does show that even for isovalent cations, the size effect cannot be predicted a priori. Diffusivities of ionic species in general decrease with increasing positive or negative charge, from 0 valence (neutral atoms or molecules) to ±1, to ±2, etc., although the trend is not perfect either. The relation is covered by Brady and Cherniak (2010, this volume) for minerals and Zhang et al. (2010, this volume) for melts. Mungall (2002) developed empirical relations to predict cation diffusivities in silicate melts as a function of cation radius and compositional parameters for alkali elements, and of melt viscosity, ionic strength, and compositional parameters for other elements. These are

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50

the best predictive models available currently although the uncertainty can still be orders of magnitude (Behrens and Hahn 2009; Zhang et al. 2010, this volume).

Diffusivity and ionic porosity Diffusion is driven by random motion of particles. If there is more “empty” space in a mineral or melt, a given species should diffuse more rapidly, and vice versa. Based on this concept, it has been proposed that diffusivity of a species depends on the ionic porosity (IP), a measure of the “empty” space in a structure. Ionic porosity is defined as the unoccupied volume divided by the total volume: IP = 1 –

Vions Vtotal

(114)

where Vions is volume occupied by the ions. Below is an example to calculate IP of quartz. The molar volume of quartz is 22.69×10−6 m3/mol. In one mole of quartz, there are two moles of O2− and one mole of Si4+. Ionic radius of O2− is taken to be 1.38×10−10 m. Ionic radius of Si4+ in a tetrahedral site is 0.26×10−10 m (Shannon 1976). Hence, volume occupied by the ions is: Vions = (6.02214×1023)(4π/3) × (2×1.383 + 0.263)×10−30 = 13.30×10−6 m3/mol Therefore, IP = 0.414. Ionic porosity of some common minerals can be found in Zhang (2008, p. 310). Fortier and Giletti (1989) applied this approach to relate 18O diffusivity in different minerals under hydrothermal conditions. This issue is discussed further by Brady and Cherniak (2010) in this volume.

Compensation “law” The compensation “law” is an empirical “law” proposed by Winchell (1969). This “law” states that the logarithm of the preexponential factor D0 is linear to the activation energy E for the diffusion of different components in a single phase (Hofmann 1980; Hart 1981), or the same diffusion species in different phases (Bejina and Jaoul, 1997; Zhao and Zheng 2007): lnD0 = a + bE (115) where a and b are two fitting parameters. From the above equation, one may derive the following (Hart 1981; Lasaga 1998)  E  E  1      D = exp  lnD0 – = exp  a + bE – = exp a + b − E   RT  RT  RT      

(116)

Hence, in the context of the compensation “law”, a compensation temperature Tcomp can be defined as Tcomp = 1/(bR). At this temperature, diffusivities of all components would be the same, exp(a). In other words, when plotted in lnD vs. 1/T, all lines would intersect a common point at the compensation temperature. Examination of experimental data shows that the “law” has large uncertainties even for a single group of elements. Figure 17 shows that for tracer diffusion of alkali elements in a rhyolite melt, the compensation “law” does not work well (there is no single compensation temperature). In some other systems, the “law” works better, but the uncertainty in predicting D is still about an order of magnitude.

Interdiffusivity and self diffusivity There are two models relating the interdiffusivity DAB in a binary system between components A and B, and the self diffusivities DA and DB in the same system with the same composition. The models depend on whether diffusion is ionic or neutral atomic. For ionic interdiffusion between isovalent ions (e.g., Na-K interdiffusion between albite and orthoclase melts, or Fe-Mg interdiffusion in olivine), Helfferich and Plesset (1958) and Barrer et al. (1963) derived the following relation:

Youxue Zhang (Ch 2)

Page 17

Youxue Zhang (Ch 2)

Page 17

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Figure 17. Diffusivities ofrhyolite alkalismelts. in rhyolite melts. Fig. 17. Diffusivities of alkalis in Data sources: Li: Data Jambonsources: and Semet (1978) and Jambon (1982). Li: Jambon and Semet (1978) and Jambon (1982).

Fig. 17. Diffusivities of alkalis in rhyolite melts. Data sources: Li: Jambon and Semet (1978) and Jambon (1982).

DAB =

D AD B ( X A + X B )  d ln γ B  1 +  X AD A + X BD B  d ln X B 

(117)

where XA and XB are mole concentrations of A and B, and gB is the chemical activity coefficient of component B. Note that DA and DB are those at the same composition at which DAB is estimated (DA and DB may depend on composition). For ionic diffusion between ions with different valences, there is a relation accounting for the valence difference (e.g., Zhang 2008) but more assumptions on how charges are balanced must be made. Lasaga (1979) and Liang et al. (1997) extended the ionic model to multicomponent ionic diffusion for the calculation of the multicomponent diffusivity matrix. (Note that dlngA/dlnXA = dlngB/dlnXB from thermodynamics, meaning that A and B can be exchanged in Eqn. 117.) For neutral atomic interdiffusion (such as in alloys), the following equation has been formulated (Darken 1948; Shewmon 1989; Kirkaldy and Young 1987) DAB =

X BD A + X AD B  d ln γ B  1 +  X A + X B  d ln X B 

(118)

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Zhang

Equation (118) is often referred to as the Darken-Hartley-Crank equation. Cooper (1965), Richter (1993), and Liang et al. (1997) extended the above model to multicomponent systems for the calculation of the multicomponent diffusivity matrix. By comparing Equation (66) with either Equation (117) or Equation (118), and recognizing that binary diffusivity is the interdiffusivity, it can be seen that the intrinsic diffusivity in Equation (66) is related to the self diffusivities of the two components, and the exact relation depends on whether the diffusion is ionic or neutral atomic. To illustrate the dependence of DAB on composition in ideal and nonideal systems, Figure 18a shows calculated DAB in an ideal solution (equivalent to a regular solution with W = 0 where W is the interaction parameter) and Figure 18b shows that in a regular solution with W/ (RT) = 2.4 using Equations (117) and (118). When W/(RT) > 2 for a regular solution, the binary mixture would decompose into two phases in the compositional region centered at XA = XB = Youxue Zhang Page 18 0.5. It can be seen that (1)(ChD2) AB values from Equations (117) and (118) can be very different Youxue Zhang (Ch 2) 18values depend strongly on how ideal although the two equations may look similar; and (2)Page DAB

Figure 18. Interdiffusivity DAB as a function of composition for (a) ideal binary mixture, and (b) a binary mixture that can be described by a regular solution model with W/(RT) = 2.4. Fig. 18. Interdiffusivity DAB as a function of composition for (a) ideal binary mixture, and (b) a binary mixture that Fig. 18. Interdiffusivity DAB as a function of composition for (a) ideal binary mixture, and (b) a binary mixture that can be described by a regular solution model with W/(RT) = 2.4.

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the mixture is. In nonideal solutions, DAB values are negative in the compositional region in which the mixture would spontaneously decompose into two phases, as expected. Before applying either of the above equations to a mineral or melt, it is necessary to decide whether diffusion is ionic or through neutral species because the two seemingly similar equations produce rather different interdiffusivities. For diffusion in minerals, it is relatively easy to decide whether the diffusion is ionic or neutral. For example, Ca-Mg interdiffusion in garnet is ionic, but Cu-Au interdiffusion in gold is through neutral atoms. For silicate melts, however, it is more difficult to determine which of the two models to apply, or whether any would apply. For example, Kress and Ghiorso (1995) used the diffusivity matrix they obtained for a basalt melt to test the model of Richter (1993) and found that the model does not work.

CONCLUSIONS Diffusion theory is well developed. Nonetheless, diffusion in minerals and melts are complicated and difficult to describe mostly due to (i) multiple components in melts as well as many minerals, (ii) diffusional anisotropy of many minerals, and (iii) highly non-ideal mixing (including the miscibility gap) of some mixtures. For a given geologic problem, the complexities may also be due to complex boundary conditions and unknown or complicated initial conditions. Hence, in treating diffusion in minerals and melts, simplifications are usually made. These simplifications have led to successful understanding of many geological processes, although not complete understanding of the diffusion behavior of the whole system. Some problems (such as crystal growth and dissolution; Liang 1999, 2000) do require more rigorous treatment of diffusion. On the other hand, there is much room to improve theories and models regarding diffusion coefficients, especially how they should depend on the composition and structure of the system. Such work would provide guidelines on future diffusion work. For example, the understanding of the role of speciation in the diffusion of some components has allowed us not only to gain insights into the diffusion mechanism but also to describe the diffusion behavior of these components well so that practical problems such as bubble growth in melts can be addressed (Proussevitch and Sahagian 1998; Liu et al. 2000; Zhang 2009).

Acknowledgments This research is supported by NSF grants EAR-0711050, EAR-0838127, and EAR1019440. I thank Yan Liang and Daniele Cherniak for insightful and constructive comments.

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Zhang Appendix 1. Expression of diffusion tensor in crystals with different symmetry

This appendix gives explicitly the representation of the diffusion tensor in crystals with different symmetry following similar analyses in Nye (1985). Note that the diffusion tensor is a symmetric tensor based on the Onsager Reciprocal Principle (De Groot and Mazor 1962). Hence, a maximum of 6 coefficients are needed to describe the diffusion tensor in crystals with the lowest symmetry. In glasses and crystals of cubic systems, the D tensor in any arbitrary orthogonal axes can be written as a diagonal matrix with equal diagonal elements (or simply written as a scalar D): D 0 0    D= 0 D 0=D  0 0 D  

( A1)

The diffusivity along any direction is D. In hexagonal, tetragonal and trigonal systems, the D tensor can be written as follows (2 independent coefficients) by choosing the z-axis along c direction, and x- and y-axes as any two mutually orthogonal axes in the plane perpendicular to c:  D⊥ c  D= 0  0 

0 D⊥ c 0

0   0  D||c 

( A2)

The diffusivity along the direction of the concentration gradient (diffusion direction) is D(q) = D^csin2q + D||c cos2q, where q is the angle between the diffusion direction and c. In orthorhombic systems, the D tensor can be written as follows (3 independent coefficients) by choosing the x-, y-, and z-axes along the crystallographic a, b, and c directions:  Da  D= 0  0 

0 Db 0

0   0  Dc 

( A3)

The diffusivity along any direction is D = Dacos2qa + Dbcos2qb + Dccos2qc, where qa, qb, and qc are the angles between the diffusion direction and the respective crystallographic axes a, b and c. In monoclinic systems, the D tensor can be written as follows (4 independent coefficients) by choosing the y-axis along b direction, and x- and z-axes to be two mutually perpendicular directions in the a-c crystallographic plane:  D11  D= 0   D13 

0 D22 0

D13   0   D33 

( A4)

Triclinic systems (6 independent coefficients) (x-, y- and z-axes can be any three mutually perpendicular directions):

Theoretical Background of Diffusion in Minerals and Melts  D11  D =  D12   D13 

D12 D22 D23

D13   D23   D33 

59

( A5)

In monoclinic and triclinic systems, diffusion tensors (A4) and (A5) can be transformed to the diagonal form:  D1  D= 0 0 

0 D2 0

0   0  D3 

( A6)

which is similar to Equation (A3) but the axes are the principal diffusion axes 1, 2, 3 for diffusion (not necessarily the crystallographic directions a, b and c), and D1, D2 and D3 are the principal diffusivities along the principal axes. (In the monoclinic system, principal diffusion axis 2 is the same as the crystallographic direction b, and D2 = Db.) The diffusivity along any direction is D = D1cos2q1 + D2cos2q2 + D3cos2q3, where q1, q2, and q3 are the angles between the diffusion direction and the respective principal diffusion axes 1, 2 and 3.

3

Reviews in Mineralogy & Geochemistry Vol. 72 pp. 61-105, 2010 Copyright © Mineralogical Society of America

Non-traditional and Emerging Methods for Characterizing Diffusion in Minerals and Mineral Aggregates E. Bruce Watson Department of Earth and Environmental Sciences Rensselaer Polytechnic Institute Troy, New York, 12180, U.S.A. [email protected]

Ralf Dohmen Department of Earth Sciences, University of Bristol, Wills Memorial Building Queen’s Road, Bristol BS8 1RJ, United Kingdom Present address: Institut für Geologie, Mineralogie und Geophysik, Ruhr-Universität Bochum, 44780 Bochum, Germany [email protected]

Introduction The data summarized in this RiMG volume reflect the growing interest among geoscientists in the characterization of atomic transport in natural materials. Interest in diffusion dates back to the early 20th century (e.g., Bowen 1921), but the need for constraints on mobility of atoms became especially important in the 1950s and 60s with our increasing reliance upon radioactive decay for determination of mineral and rock ages: the question of decay-product retention in relevant phases became vitally important. With a few notable exceptions, however, experimental geoscientists did not take up the cause of diffusion in solids until the 1970s and 80s. By that time, the motivation for acquiring diffusion data had spread to many areas of petrology, mineralogy and geochemistry, and experimental techniques began to blossom. The result is very much in evidence in this volume. All experiments aimed at measuring a diffusion coefficient have two things in common: a technique to introduce the diffusant of interest to the sample, and a method to determine the extent of diffusion. Traditionally, diffusion experiments are designed to take advantage of an existing solution to the non-steady state diffusion equation, and experiments are conducted in such a way as to reproduce as well as possible the boundary conditions specified for the solution. For ease of analysis and interpretation, most experiments are set up to limit diffusion to just one dimension. Most researchers have further settled on one of three general experiment designs: 1) the interdiffusion couple; 2) the “thin-film” geometry; or 3) the “constant surface concentration” design (Fig. 1). Noble-gas geochemists (including those interested in K/Ar, Ar/ Ar and U-Th/He dating methods) have exploited 3-D variants of this last approach by conducting bulk degassing studies of minerals in which radiogenic noble gases have accumulated over time in the natural environment (Baxter 2010, this volume). In this particular case, concentration at the outer boundary of the system is assumed to be zero for all time. The three general approaches listed above have been reviewed elsewhere (e.g., Ryerson 1987), but a brief discussion of diffusion couples and thin-source experiments is warranted here 1529-6466/10/0072-0003$05.00

DOI: 10.2138/rmg.2010.72.3

62

Watson & Dohmen Figure 1. Illustration of three widely implemented, classical approaches to diffusion experiments, showing the initial configuration (left) and the distribution of diffusant after diffusion has progressed for duration t (right). In the diffusioncouple and thin-film cases, the integrated areas under the initial and “final” curves must be the same, reflecting mass conservation. In the constant-surface case, the source reservoir of diffusant (black) is well mixed by either diffusion or advection so the concentration at the interface effectively does not change with time (see arrow in right panel). Note that the thinfilm technique as discussed in the “The Thin-Film Method and Pulsed Laser Deposition” section of this chapter includes situations intermediate between the diffusion couple and the thin-film case shown here. The equations governing the thin-film and constant-surface cases illustrated here are nos. 7 and 11, respectively, in the text.

because these are not treated elsewhere in this volume, and also because one of the techniques we discuss in detail (pulsed laser deposition of thin films) is essentially a hybrid of these two traditional approaches. The constant-source method is discussed in detail by Ganguly (2010, this volume); the mathematics of all three experiment designs are reviewed by Zhang (2010, this volume). The term diffusion couple (top panel of Fig. 1) refers to an experimental setup in which two materials of different composition are placed against one another with the intent of allowing one or more of their components to interdiffuse in the direction perpendicular to the interface joining them. Diffusion is induced by performing a diffusion anneal—that is, an isothermal treatment of predetermined duration conducted at sufficiently high temperature to produce a measurable change. Diffusion progress is evidenced by relaxation of an initial step distribution (Fig. 1), and the extent of relaxation generally provides the basis for calculating the governing diffusion constant or diffusivity (see Zhang 2010, this volume). Broadly speaking, it is implied in the use of the term diffusion couple that that the two halves will not react with one another to form an intervening phase (in this case, the setup would be more appropriately called a “reaction couple”), and that the net flux across the original interface is zero. The interdiffusion process can involve bulk chemical mixing (as in the case of interdiffusing silicate melts), but for the crystalline materials that are the main focus of this chapter, ion exchange is the operative mechanism in most cases. Iron-magnesium interdiffusion in ferromagnesian minerals (e.g., olivine) is a classic example (see the “Powder-Source Technique” section of this chapter). For ease in extracting diffusion information, sample size and experimental conditions are chosen in such a way that both halves of the diffusion couple are effectively infinite in length in the direction of diffusion—that is, the ends of the couple remain unaffected by diffusion. One of the major challenges in performing successful diffusion-couple experiments with minerals lies in the preparation of perfectly flat surfaces to create a mating interface

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with ~100% contact. Indeed, this problem is one of the main incentives for turning to some of the non-traditional techniques described in this chapter. Excellent examples of the use of “traditional” diffusion couples are the studies of cation diffusion in garnets by Ganguly and coworkers (see especially Chakraborty and Ganguly 1992; Ganguly et al. 1998). These authors were successful largely because of their innovative use of the high pressures achievable in a piston-cylinder apparatus to ensure both the stability of the phases of interest and near-perfect contact between the single crystals forming the diffusion couples. The traditional thin-film or thin-source configuration for diffusion experiments is shown in the middle panel of Figure 1. The experimental protocol involves the introduction of a small amount of diffusant on one surface of the diffusion specimen of interest (e.g., the end of cylinder or one side of a plate), followed by a diffusion anneal. In the restrictive usage of the term “thin film,” the words “small amount” have a specific meaning that derives from the initial and boundary conditions assumed for the relevant solution to the non-steady state diffusion equation (see Shewmon 1963, pp. 7-10; Zhang 2010, this volume): the mass of diffusant must be sufficiently small to dissolve instantaneously in the diffusion medium at the start of a diffusion anneal. This technique has been used extensively in experimental geochemistry to characterize diffusion in minerals, melts and glasses. Exhaustive citation is precluded here, so the reader is referred to Brady (1995) for a summary that includes the vast majority of geoscience studies in which the traditional thin-film technique has been used. Examples of influential work specifically on minerals includes studies of Sr and Al diffusion in diopside (Sneeringer et al. 1984; Sautter et al. 1988), Ca, Mn, Fe, Co, Ni, Sr and Ba diffusion in åkermanite (Morioka and Nagasawa 1991), and Si in olivine (Houlier et al. 1990). Thin-film measurements of diffusion in melts and glasses have included studies of Na, K, Ca, Cs and Rb in obsidian (Magaritz and Hofmann 1978; Jambon 1982) and of a wide variety of cations in basaltic melt (Hofmann and Magaritz 1977; Lowry et al. 1982). The most popular method of depositing a thin film (prior to pulsed laser deposition; see the next section) was evaporation of a solution containing the diffusant of interest. High sensitivity analytical techniques are generally required because of the inherently small masses of diffusants involved; the principal techniques have included use of radiotracers (combined with serial sectioning), SIMS, RBS and NRA (see Cherniak et al. 2010, this volume, for a discussion of the latter three analysis techniques). In the following sections we describe four methods for characterizing diffusion in solids that differ in significant ways from traditional methods. Specifically, we discuss: 1) recent advancements in the thin-film method, with emphasis on the new pulsed-laser deposition (PLD) technique; 2) the powder-source approach; 3) ion implantation; and 4) the “particle-detector” method. The first of these has been applied to diffusion in grain boundaries and in single crystals, the second and third to single crystals only, and the last to grain boundary diffusion only (see Dohmen and Milke 2010, this volume). These techniques vary in their maturity, complexity and likely range of application, so we will not attempt uniformity in treatment.

The thin-film method and pulsed laser deposition (PLD): Principles and recent developments Definition of a thin film As noted above, the strategy of a thin-film experiment involves deposition of a limited mass of diffusant on the surface of a sample for the purpose of conducting an “in-diffusion” experiment to characterize the diffusivity. Most textbooks and treatises on diffusion (e.g., Shewmon 1963) consider a thin film to be “infinitely thin”—meaning that it dissolves immediately and completely in the sample at the start of the experiment. In this usage, the term “thin film” has the same meaning as Crank’s (1975) “instantaneous [plane] source.” Here we use a broader definition that includes films that persist as discrete layers on the sample

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surface throughout the entire diffusion anneal. In this more general definition, the experimental setup can appropriately be called a diffusion couple; but unlike the more standard semi- or effectively infinite couple, it is one in which one part of the couple (the thin film) is finite—that is, concentration changes occur across its entire thickness.

Why use thin films? A fundamental goal of any diffusion experiment is to create a geometrically well-defined concentration gradient for the element, isotope, or chemical component of interest. A common method to produce a finite diffusion source is to deposit a thin film (an atomic monolayer or a layer with a thickness up to several 100 nm) on a polished single crystal or polycrystalline aggregate (Fig. 2a). Depending on the objective of the experimental study, the thin film is either enriched in a diffusion tracer (e.g., a stable isotope, a radiogenic isotope, or a trace element) or has a specific composition designed to create a controlled chemical gradient. In all cases, annealing of the sample after film deposition causes the initially sharp compositional step at the film/substrate interface to relax by diffusion (Fig. 2b). Fitting the post-annealing concentration distribution of the element or isotope of interest enables extraction of a diffusion coefficient as described below. Beginning in the 1950s, the thin-film approach was used extensively by both materials scientists and geoscientists to investigate diffusion in melts, metals and polycrystals—that is, materials characterized by sufficiently high diffusivities to enable recovery of diffusion profiles by serial sectioning or autoradiography. More recently, thin-film strategies have seen increasing use in studies of slow diffusion in single crystals (e.g., Sneeringer et al. 1984; Van Orman et al. 2001, 2002) because characterization of short diffusion profiles (tens of nanometers) is now possible using high-resolution depth-profiling methods such as secondary-ion mass spectrometry (SIMS; also referred to as ion microprobe), nuclear reaction analysis (NRA) and Rutherford backscattering spectroscopy (RBS; see Cherniak et al. 2010, this volume).

Figure 2. Schematic illustration of the basic principle of a thin film diffusion experiment and illustration of the convolution effects for the measured profile. (a) Thin film diffusion couple: The polished substrate could be either a single crystal, a bi-crystal with an interface normal to the thin-film surface, or a dense polycrystalline aggregate. The double arrows indicate the exchange between the film and the substrate due to diffusion. (b) Depth profiles before and after diffusion anneal (the product of D and t is indicated at each profile): The dotted lines illustrate a measured depth profile considering the limited depth resolution. The solid profiles were calculated using the analytical solution for the thin film geometry given in Equation (6) and the dotted profiles by convolving these profiles using (8) and (9) with s = 15 nm. Note that the convolution effect is very significant for the initial profile and decreases with increasing product Dt until it is insignificant.

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As described below, high-resolution depth profiling can be particularly advantageous when combined with recent advances in pulsed laser deposition (PLD) to produce the film containing the diffusant of interest. In some cases, a thin film may be preferable to a powder source (see “The Powder-Source Technique” section of this chapter) if difficulties are encountered in identifying a source that does not adhere to the crystal surface or react with it. In many cases the diffusant initially in a thin film dissolves entirely in the crystal under study, leaving no residue or reaction product and creating no topography. In either case, the key is that the surface of the crystal of interest emerges clean and flat from a diffusion experiment so it can be interrogated on a scale of tens of nanometers for diffusion information. The physics of diffusion should not differ on the nanometer scale (relative to, say, millimeters), but the ability to characterize very short profiles means that sluggish diffusion processes—requiring perhaps millions of years to be effective in nature—can be studied in a laboratory time frame. The ability to quantify diffusion profiles only tens of nm in length can circumvent the need to accelerate diffusion by resorting to geologically unrealistic (high) experiment temperatures. Iron-magnesium interdiffusion in olivine, for example, can be explored in the laboratory over the full range of interest in nature (e.g., Dohmen et al. 2007); note, however, that carbon selfdiffusion in diamond (Koga et al. 2005) and diffusion of tetravalent ions in zircon (Cherniak et al. 1997; Cherniak and Watson 2007) are another matter. Large uncertainties arising from extrapolations over several hundred degrees can be avoided in many cases by using the thin film method in combination with RBS, NRA or SIMS. The extraction of diffusion constants from profiles generated in thin-film experiments is by no means standardized at this point, and strategies must be tailored to the boundary conditions imposed by a specific experimental setup. Before discussing thin film deposition and applications, therefore, we will briefly describe the mathematical approach to fitting diffusion profiles from thin-film couples and discuss ways in which the spatial resolution of the analytical method can be explicitly considered.

Fitting of diffusion profiles from thin-film diffusion couples The traditional procedure to obtain a diffusion coefficient is to fit the profiles to a solution to the 1-D, non-steady state diffusion equation (Eqn. 2 in Zhang 2010, this volume) for the given geometry, initial condition, and boundary conditions. A more flexible strategy, however, is to solve the diffusion equation numerically using a finite-difference scheme (Crank 1975, pp. 137-159; Costa et al. 2008). Numerical approaches can accommodate cases in which the diffusion coefficient in the thin film and/or substrate depends on diffusant concentration, or the boundary condition at the surface is a function of time (e.g., due to evaporative loss of the diffusing component). Finite-difference approaches are readily implemented using technical software such as Mathematica, Matlab, Mathcad, Maple, Origin—or even with an Excel spreadsheet (see, for example, the electronic attachment to the review by Costa et al. 2008). The convenience of the more restrictive analytical solutions has led to their widespread use in fitting profiles from thin-film experiments. However, the simplifying assumptions necessary to implement one of these solutions may result in systematic errors in the diffusion coefficients obtained. Below we provide three different analytical solutions for a thin-film diffusion couple that are commonly used to fit profiles from thin film/single crystal diffusion couples. The procedure to model profiles for diffusion couples using bi-crystals or polycrystalline aggregate is described in detail in Dohmen and Milke (2010, this volume).

Analytical solutions – examples The thin-film geometry is such that a 1-D diffusion model is appropriate to describe the resulting diffusion profile. In the following we do not consider multi-component diffusion processes and restrict the discussion to cases where a single diffusion coefficient is sufficient to describe the process (for a discussion of tracer diffusion or binary interdiffusion, see Zhang

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2010, this volume). Usually, the initial conditions include a homogeneous concentration in the crystal of interest (i.e., the substrate) and also in the thin film, denoted in the following as Cs and Cf, respectively, with Cf > Cs (see Fig. 2b). If we treat the film and the substrate as separate diffusion media characterized by the diffusion coefficients, Df and Ds, respectively, four boundary conditions are required as follows: ∂Cf ∂x Df

∂Cf ∂x

=0

(1)

x =− h

= Ds x =−0

∂Cs ∂x

(2) x =+0

C f = K ⋅ Cs Cs ( +∞) = C

(3) 0 s

( 4)

where x is the distance coordinate (x = 0 at the interface between the substrate and the thin film; >0 in the substrate), h is the thickness of the film, and K is the partition coefficient between the substrate and the film. The first boundary condition means there is no flux at the outer surface of the thin film, which is appropriate for a closed-system diffusion couple (this would not hold if evaporation form the surface of the film occurred during an experiment). The second boundary condition is a mass-balance constraint at the interface between the substrate and the thin film. The third is based on the assumption of thermodynamic equilibrium at the thin film/substrate interface where K is the partition coefficient of the element between the film and the substrate. The fourth defines the substrate as in infinite half-space by stipulating that the diffusant concentration at large x is unaffected by diffusion from the thin film. For K = 1 and if Df and Ds are constant an analytical solution is given by Lovering (1936): C f ( x , t ) − Cs

0

C f − Cs 0

0

Cs ( x , t ) − Cs

=−

0

C f − Cs 0

p :=

0

=−

1+ p 2 1− p 2

  d⋅n+ x   x  1+ p ∞  d ⋅ n − x  n −1 + ⋅ ∑ ( − p ) ⋅  erf   − p ⋅ erf    (5a )  2 D t   2 n =1   2 Df t  f    2 Df t  

⋅ erf 

 x  1 − p2 ∞  d ⋅ Ds Df ⋅ n + x  n −1 + ⋅ ∑ ( − p ) ⋅ erf    2 Dt   2 2 Ds t n =1 s    

⋅ erf 

Df ⋅ Ds −

Ds ⋅ Df

Df ⋅ Ds +

Ds ⋅ Df

( 5b )

where d = 2·h and Cf0 and Cs0 represent the initial concentrations at time t = 0 in the thin film and the substrate, respectively. If Df and Ds are equal the solution is greatly simplified (see, e.g., Crank 1975, p. 31) and only one function C(x,t) is required to describe the whole profile where the distance x is now measured from the surface: C ( x, t ) − Cs0 1   h − x = erf  Cf0 − Cs0 2   2 ⋅ Dst 

  h+x  + erf    2⋅ D t s  

   

(6)

If the film thickness is very small compared to the profile length and if diffusion is sufficiently fast within this layer, the solution for an infinitely thin instantaneous source at the surface can be used (see Crank 1975): C ( x, t ) =

 − x2  0 ⋅ exp   + Cs D t 4 2 πDst  s  M

( 7)

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where M is the total amount of diffusant contributed by the source over and above that already present in the layer.

Fitting uncertainties The analytical solutions given in Equations (5)-(7) can be used directly to fit a measured depth profile, either by a simple “visual” fit or by performing a regression minimizing the sum of the squares of the residuals. The error in the diffusion coefficient obtained from fitting a single profile is usually smaller than 0.2 log unit, but this can depend on the statistical error in the individual analyses and the complexities of the specific diffusion couple (Ganguly et al. 2007; Zhang et al. 2010). However, a serious systematic error can be introduced if the effective spatial resolution of the profile analysis is not considered. The effective spatial resolution may be considerably lower than the nominal resolution of the analytical method if the geometry of the film is imperfect owing to, for example, non-uniform deposition, grain growth, reactions, etc. Such analytical artifacts may lead to a smearing of the measured concentration profile. This can be explicitly modeled following the approach of Ganguly et al. (1988), which was originally developed to consider the convolution effect arising from the limited spatial resolution of the electron microprobe. For SIMS depth profiles, Hofmann (1994) presented an analogous approach. The main principle is that the broadening of the measured profile C′(x) can be predicted by convolving the real concentration profile, C(x), with a resolution function g(x – x′): +∞

C ′( x ) = ∫ C ( x′) ⋅ g( x − x′)dx′ −∞

(8)

The resolution function can be often approximated as a Gaussian in which the spatial resolution is characterized by the value of σ, which represents the standard deviation of the Gaussian: g( x − x′) =

 ( x − x′)2  ⋅ exp   2 σ 2π  2σ  1

( 9)

The integration in Equation (8) can be performed numerically to either determine the value for σ on reference samples with a steep concentration gradient (e.g., as illustrated in Fig. 1b) or when this σ is known to simulate the measured profile with a forward modeling. A simple estimate of how much the diffusion coefficient retrieved from directly fitting the profile, denoted as DC, deviates from the true diffusion coefficient, D, was provided by Ganguly et al. (1988): D σ =1−8⋅  DC L

2

(10)

where L is the length of the diffusion profile measured from the interface. This equation is strictly valid for cases in which L = 4(Dt)½, such as a diffusion couple or diffusion within a substrate from a constant course (Fig. 1), but it can be used for approximate calculation of the convolution effect for diffusion from a thin film. It is clear from Equation (10) that for a given value of σ the effect is strongest for very short times and/or very small diffusion coefficients. A simple calculation using Equation (10) demonstrates that to keep the error induced by the spatial averaging below ~10%, the measured profile length L should be at least ten times the value of σ. Otherwise, the convolution should be explicitly considered in the numerical model according to Equations (8) and (9). If the profile length is very close to σ, one should simply run longer experiments. A simplified way to consider convolution is to replace in the analytical solution 2Dt by 2Dt + s2. The procedures described above can be also applied to estimate the errors induced by the roughness of a sample because the morphology of a rough surface can be approximately represented by a Gaussian function in a depth profile analysis. Convolution effects have two general implications for diffusion studies. First, the error from inadequate spatial resolution always leads to an overestimate of the true value of the diffusion coefficient, as one can easily see from Equation (10). Second, the activation energy

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resulting from multiple experiments over a range in temperature tends to be lower than the true one. The latter point is related to the Arrhenian behavior of diffusion. In most diffusion studies the measured profile lengths are smaller at lower temperatures even if the run duration is much longer, which implies that the resulting diffusion coefficients deviate further (in a positive direction) from the true diffusion coefficients.

Pulsed laser ablation: a versatile method for thin film deposition From the above discussion it is clear that a prerequisite for the measurement of small diffusion coefficients by the thin-film method is a film that is geometrically and chemically uniform. The ease of creating such a film depends on the deposition method. Many techniques have been used; these can be broadly subdivided as either chemical or physical in nature. A complete overview is beyond the scope of this chapter; the reader is referred to published monographs for details: e.g., Smith (1995), Soriaga et al. (2002), and Kern and Schuehgraf (2002). For diffusion experiments specifically on minerals, radio frequency sputtering has been used to study diffusion in olivine (Jaoul et al. 1981, 1995), diopside (Dimanov and Jaoul 1998), and monazite (Gardes et al. 2006). Thermal vapor deposition has been used to deposit a variety of diffusants on minerals, including MgO (Schwandt et al 1993; Zhang et al. 2010), KOH (Ito and Ganguly 2004), and Cr2O3 (Ganguly et al. 2007). Evaporation of an aqueous solution containing the tracer has also been successfully employed to deposit layers of simple oxides, phosphates and silicates (e.g., Sneeringer et al. 1984; Cygan and Lasaga 1985; Chakraborty and Ganguly 1992; Van Orman et al. 2001, 2002; Ganguly et al. 1998; Tirone et al. 2005). There are pros and cons to the various deposition methods (for a short summary see Dohmen et al. 2002a), but the common disadvantage is that a new protocol must be developed for a given desired film composition, and complex compositions may be unattainable. These methods do not lend themselves to the study of a diverse and complex set of minerals and diffusants. The difficulty of controlling the composition of a precipitate from aqueous solution means that the resulting film and the substrate may be thermodynamically unstable together. This can lead to the formation of a reaction layer that interferes with the diffusion process of interest. As discussed in the “Fitting Uncertainties” and “Analytical Considerations, Advantages and Drawbacks” sections in this chapter, the presence of a reaction product can degrade the information obtained by depth profiling because mixture and convolution effects must be addressed. Many of the problems of thin-film production can be circumvented by pulsed laser deposition (PLD), which is being used with increasing frequency in materials science to deposit layers of insulators, metals, superconducting materials, semiconductors, polymers and even biological materials (e.g., see review of Norton 2007). The principle of PLD was discovered in the late 1960s (see, for example, the review of Sankur and Hall 1985), but it was not until in the late 1980s that the potential of PLD to produce superconducting thin films was fully recognized (Dijkkamp et al. 1987). The method is very flexible and has the particular advantage that thermal fractionation effects are minimized. This allows production of thin films that contain elements with strongly divergent chemical properties—different volatility, in particular. Sumit Chakraborty was the first geoscientist to recognize the potential applications of PLD to studies in mineral kinetics, and in early 2001 a PLD facility was established at the Institut für Geologie, Mineralogie und Geophysik, Ruhr-Universität Bochum. This setup has been used successfully to deposit thin films of numerous mineralogical compositions, ranging from simple oxides such as TiO2 and ZrO2 (Dohmen 2008; Dohmen et al. 2009) to various complex silicates, including olivine, pyroxenes, and garnet (Dohmen et al. 2002a; Milke et al. 2007) as well as aluminous spinel (Vogt 2008). The principle of PLD is straightforward, but the physical processes involved are complex and not well understood theoretically (see, e.g., Schou 2010): it is, in principle, a physical vapor deposition process that is carried out in a vacuum chamber (Fig. 3a). A short (0.1 – 20 ns) high-energy laser pulse impinges on a target that has the desirable composition for the

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Figure 3. Illustration of a PLD thin film setup: (a) basic principle; (b) photo through the vacuum window of the PLD vacuum chamber in Bochum during a deposition process. Note the strongly directed bright plasma (green in color in visible light), which is formed here from an olivine target.

thin film. Both single crystals and polycrystalline pressed pellets have been used as targets. The ablation process induces a plasma plume of the target composition perpendicular to the target surface. The plume propagates into the vacuum chamber (Fig. 3b) and impinges on the chosen substrate a few centimeters away (in mineralogical applications this is typically a polished and oriented single crystal). A series of laser pulses transfers the desired amount of nominally stoichiometric material from the target to the substrate. For a given material a number of process parameters still must be optimized, but the most critical parameters are the wavelength and the energy flux of the laser pulse into the target, because this determines the amount of energy absorbed per volume of target material. The combination of the very short, high-energy pulses and strongly efficient absorption of this energy by the target leads to a non-equilibrium process that avoids thermal fractionation effects. Typical energy fluences for PLD are 0.1 – 10 J/cm2, which is the optimum range for most materials. A specific threshold fluence exists for each material; above that threshold a value that is too low or too high can lead to non-stoichiometry of the film (e.g., Schou 2010). From experience with laser-ablation ICP-MS, many geoscientists know that wavelengths in the UV range are most appropriate to ablate silicates, carbonates and also most oxides. The setup in Bochum uses an excimer laser (Lambda Physik LPX305i) that can be operated at three different wavelengths (193, 248, and 351 nm) with pulse energies usually of hundreds of mJ. Experience at Bochum over the last decade has demonstrated that for deposition of most silicates a wavelength of 193 nm is optimal. A wavelength of 248 nm often yields thin films characterized by micrometer-sized droplets of quenched molten material from the target (Dohmen et al. 2002a) called splashing (e.g., Chen 1994). Because this problem can now be avoided, PLD provides a way to produce thin films of silicates and oxides that have minimal surface roughness (< 1 nm; Dohmen et al. 2002a). However, the directed plume produces a non-uniform deposition rate with an angular distribution, which can be described by a cosnϑ relationship where the angle ϑ is measured normal to the target surface (e.g., Saenger 1994; Dohmen et al. 2002a). This distribution leads to a central deposition area of at least ~4×6 mm, within which the film thickness varies by no more than a few %. This allows simultaneous deposition on several mm-sized crystals that will have almost the same film thickness; one of these can be kept as a reference sample to measure the initial thickness and composition. The deposition rate is roughly constant, and typical deposition rates are slightly less than 0.1 nm/pulse. For pulse frequencies of 5 – 20 Hz and laser fluencies of a few J/cm2, this means that a 100-nm layer of olivine (for example) is produced within a minute or two. The calibration of the deposition rate is relatively simple. If a substrate is used that has a higher reflectivity than the thin film (nfilm < nsub), then destructive and con-

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structive interference between the light reflected at the substrate film interface and at the thin film surface imparts color to the film. A silicon wafer or even Al foil are suitable substrates for gauging the thickness of a thin silicate or oxide film from its color. The specific color depends on the difference in the optical path length (for a normal incidence of light given by 2nfilmdfilm). On larger substrates interference color fringes appear due to the continuous thickness variation of the thin film (Fig. 4). By comparison with the Michel Levy color chart used for polarization microscopy, the retardation, g, can be inferred from the color. The thickness of the layer is then given simply by the ratio g/2nfilm. In this way a very good estimate of thickness and thickness distribution can be made. This technique is also useful for identifying the optimal location in the vacuum chamber to obtain the highest and most uniform deposition rate.

Figure 4. Si wafer of one inch diameter with interference color fringes (see color image online) from a thin film made from an anorthite pellet. The gradual change in the grey scale indicates the gradual change in the film thickness here from about 150 nm in the central area down to about 50 nm.

For depositions at room temperatures the films are expected to be amorphous, which has been documented, for example, by Dohmen et al. (2002a) for olivine and by Marquardt et al. (2010) for YAG. In most PLD setups the crystal substrates can be heated in situ before, during, and/or after the deposition. An effective way to achieve this heating is by placing the substrates on thin (~1 mm) electrical insulators (e.g., SiO2 glass) positioned above heating elements in the vacuum chamber. Heating during or after deposition may produce crystalline or even epitaxial thin films. Heating before deposition can be used to “prepare” the substrate by evaporating adsorbed components, particularly water. The latter treatment has been shown to be essential for producing geometrically well defined diffusion couples because heating a couple with trapped volatiles leads to degassing and bubble formation within the layer. Additional technical refinements used in materials science are summarized by Chrisey and Hubler (1994) and Norton (2007); these include: (i) background gas (e.g., O2) to form specific molecules or to lower the kinetic energy of the plasma species (this may be required to ensure stoichiometry and epitaxial growth when the film is deposited on a heated substrate); (ii) rotation of the substrate to produce a more homogeneous layer thickness; (iii) real-time characterization of the thin film using reflection high-energy electron diffraction (RHEED) to control the crystallinity and the smoothness of the surface; (iv) real-time monitoring of film thickness using oscillating quartz crystals; and (v) in situ production of multi-layers using an automatic stage to change ablations targets. Only the first of these techniques has been implemented in the setup at Bochum, but the possibilities for future adoption in mineralogical studies are clear.

Application of PLD to diffusion studies – examples Pulsed laser deposition provides a convenient way to produce well-defined contacts between two or more different solids. The possibilities are numerous, and a number of different thin-film geometries have already been produced—including ones with double layers. Below we present a few examples of diffusion studies that make use of various thin film configurations. We focus here on the basic design principles of the experiments as well as the potential pitfalls in the determination of diffusion coefficients. With respect to the latter, it is important to characterize the thin film diffusion couples both before and after the diffusion anneal. Information on the

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surface morphology can be directly obtained by using reflected light microscopy, white-light interference microscopy, scanning electron microscopy (SEM), and, in particular, atomic force microscopy (AFM). In addition, the novel technique focused ion beam thinning (FIB; Wirth 2004) enables preparation of thin slices (~100 nm) of the thin film samples at a desired location. These cross sections can be further investigated with a common SEM or preferably with a TEM that provides information on the geometry, crystallinity, and microstructure of the film. Diffusion profiles could also be analyzed on the cross section using analytical transmission electron microscopy [ATEM] (Meissner et al. 1998; Dohmen et al. 2002a; Marquardt et al. 2010) with a spatial resolution of about 20-50 nm, or the new generation of electron microprobes with a field emission gun (FEG probe). More commonly, however, RBS and SIMS have been used to measure depth profiles of the elements or isotopes of interest.

Single layer configurations To date, PLD thin films have been used mainly to measure crystal lattice and grain boundary diffusion coefficients in the classical configuration (Fig. 2a) where a film is deposited on an oriented single crystal (e.g., Dohmen et al. 2002a,b, 2007; ter Heege et al. 2006; Chakraborty et al. 2008; Dohmen et al. 2009), a bi-crystal (see Fig. 5), or a polycrystalline aggregate (Dobson et al. 2008; Shimojuku et al. 2009). If a polycrystalline aggregate or a bi-crystal is used this enables simultaneous investigation of diffusion in the crystal lattice and in grain boundaries (Dohmen and Milke 2010, this volume). The strategy in this type of experiment is to create a single-phase diffusion couple in which the film serves as a source of components for either chemical diffusion or isotopic exchange. Examples of the former include a fayalite source to measure Fe-Mg interdiffusion in olivine (Dohmen et al. 2007), and TiO2 enriched in Sm2O3, Nb2O5 and Ta2O5 to measure effective tracer diffusion of Nb, Ta, and Sm in rutile (Dohmen et al. 2009). Isotopic exchange examples include use of films containing 29Si and 18O to measure Si and O self diffusion in olivine (Dohmen et al. 2002b), ringwoodite, wadsleyite (Shimojuku et al. 2009), and Mg perovskite (Dobson et al. 2008). A particular strength of PLD is that it readily accommodates simultaneous doping with several isotopes or trace elements, which enables a direct comparison of the diffusivities for various elements and enhances the accuracy of the relative diffusion coefficients. This made it possible, for example, for Dohmen et al. (2009) to reproduce a relative difference of only 50% in the diffusion coefficients of the geochemical

Figure 5. Bright-field TEM images of YAG bi-crystals deposited with (Yb1.17Y1.83)Al5O12 before and after annealing at 1450 °C for 2 hours (a and b, respectively). The film is perfectly epitaxial and the grain boundary continues through the initially amorphous layer. Unpublished images kindly provided by K. Marquardt.

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twins Nb and Ta. This difference is small by some standards—and would be difficult to quantify in independent experiments—but the Dohmen et al. (2009) results demonstrate unequivocally that Nb and Ta can be fractionated significantly by diffusion. In these experiments the thin film was also enriched in Sm, which was shown to be relatively immobile and therefore was used as a tracer to identify analytical artifacts due to convolution effects. Use of a single-phase diffusion couple within its stability field ensures that the film portion of the couple acts purely as a diffusion reservoir. Undesired chemical gradients in tracer diffusion studies are avoided using this approach. A potential downside is that, because the film is initially amorphous, it will tend to re-crystallize during the diffusion anneal. This may lead to problems in the profile measurement and fitting in two ways. First, the chemical potential (mi) of a diffusing component i within an amorphous layer is different from that in a crystalline medium of the same composition. If the crystallization rate is slow compared to the total run duration, the gradient in mi may change or even be inverted during the diffusion anneal, leading to complex evolution of the diffusion profile. In such a case, determination of the diffusion coefficient would not be straightforward (but complications would be recognizable, at least, from a time series). The second complication that may arise due to crystallization of an amorphous film is that different materials recrystallize in different ways (see examples in Fig. 6). This leads to a unique microstructure and geometry of a given film, which, in the case of polycrystalline films, also varies with time during a diffusion anneal due to grain growth. The microstructure of the thin film may complicate depth profiling by RBS or SIMS, because the effective depth resolution depends on the roughness and thin film/substrate interface irregularity. Consequently, the

Figure 6. SEM images of various annealed thin films (top view) on single crystals of clinopyroxene: (a) Sample Cpx1c32, ~100 nm cpx film, 1306 °C, 10 min; (b) Sample Cpx1c56, ~120 nm ol film, 1105 °C, 4 days; (c) Sample Cpx1c35, ~100 nm cpx film, 1200 °C, 18 hours; (d) Cpx1c51, ~200 nm ol film, 1200 °C, 1.5 hours. [Note: The film thicknesses indicated above are those of the initially amorphous layer as measured from the reference sample, which was deposited together with the individual sample.]

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resulting diffusion profile may be strongly affected by convolution effects as illustrated in the “Fitting Uncertainties” section, and this could lead to large uncertainties or even erroneous diffusion data. [Note: The powder-source method combined with profiling by RBS or NRA (see “The Powder-Source Technique” section) is less susceptible to this effect, because, within limits, roughness on the surface of an experimental run product—as opposed roughness at the interface of a diffusion couple—has little effect on the RBS or NRA spectrum]. With respect to the two points discussed above, olivine thin films are almost ideal. They recrystallize to a polycrystalline layer of uniform microstructure and thickness, thereby providing a geometrically well-defined diffusion couple (Dohmen et al. 2002a). The time required for complete recrystallization depends upon temperature, of course, but observations also indicate that the crystallization rate increases with the fayalite content. For example, a Fo80 layer is fully crystalline after only a few minutes at 1000 °C. At lower temperatures, however, the crystallization rate for such an olivine becomes more sluggish relative to the time scale of FeMg diffusion, which has the effect of inverting the chemical potential gradient of the diffusant between the thin film and the crystalline substrate during the initial stage of the experiment [this inversion has been detected by RBS (unpublished observation by RD)]. This problem was solved by Dohmen et al. (2007) for experiments between 700 and 900 °C by making a film with a much higher fayalite content (Fa70) than that of the substrate (Fa10). This has two advantages: (i) recrystallization of the film is accelerated; and (ii) the chemical potential of the fayalite component in the film is always higher than in the substrate, even when the film is amorphous. Sluggish recrystallization appears to be even more problematic for experiments with pyroxene thin films at temperatures below 1000 °C. This may be due to the higher silica content of pyroxenes relative to olivines, which results in a lower diffusion rate within the amorphous phase. For this reason, ter Heege et al. (2006) and Chakraborty et al. (2008) used Fo30 thin films to measure FeMg interdiffusion in orthopyroxene (opx) and clinopyroxene (cpx). Olivine coexists stably with opx or cpx at the experimental conditions, and Fe-Mg partitioning between olivine and either pyroxene is nearly ideal—i.e., KD is fairly close to one (e.g., Perkins and Vielzeuf 1992; Von Seckendorff and O’Neill 1993). The latter point ensures that the chemical potential gradient always has the same sign during the diffusion anneal. The olivine layer is easily recognized by RBS and, as shown by Dohmen et al. (2007) for Fe-Mg diffusion in olivine, an Fe depth profile can be extracted from the RBS spectrum (Fig. 7). Because of these advantages, a consistent data set of data for Fe-Mg diffusion in both opx and cpx were obtained using olivine thin films (ter Heege et al. 2006; Chakraborty et al. 2008). Vogt (2008) demonstrated that thin films having a spinel composition [(Mg,Fe)Al2O4], are even better suited than olivine, which was confirmed by Marquardt et al. (2010) for Yb or Nd-doped YAG. Simple oxides also perform very well as thin films: e.g., TiO2 (Dohmen et al. 2009) and ZrO2 (Dohmen 2008). Epitaxial growth was actually observed in the case of thin films on a YAG bicrystal where the grain boundary in the substrate was continuous through the film at the end of the experiment (Fig. 5). Sluggish recrystallization rates appear to be more problematic for experiments with pyroxene at lower temperatures. An additional problem related to higher chemical potential of the diffusant within the substrate than in the thin film [item (ii) in the preceding paragraph] arises in experiments designed to measure the small diffusivities governing Si in cpx, which requires temperatures approaching the melting point of cpx. The silicate chains in cpx may be the reason for pronounced anisotropic growth leading to isolated micrometer-sized idiomorphic crystals on the surface (Figs. 6a,c). In thinfilm terminology, this type of growth mechanism is called island growth (Venables 2000). Fast surface diffusion may ensure a homogeneous surface concentration on the large single crystal, but the islands (up to a few mm high) produce strong convolution effects in the depth profile. Again, olivine thin films were used as the diffusant source, but even the relatively smooth olivine films show considerable roughness (Figs. 6b,d), which generally increases with increasing temperature due to enhanced grain growth. Quickly annealed samples were

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Figure 7. Fe concentration depth profiles from olivine thin film/orthopyroxene diffusion couples obtained from the RBS spectra using the software RBX (Kotai 1994). Shown are profiles from an un-annealed reference sample and a sample annealed at 900 °C for 73 hours. For opx with 12 mol% Fs a Fe-Mg diffusion coefficient of 5.8×10−21 m2/s is obtained from the best fit.

used to calibrate the effective depth resolution, which was considered in the profile fitting following the procedure illustrated in the “Fitting Uncertainties” section (see Fig. 8). During the short anneals diffusion is insignificant, so any smearing of a sharp concentration step is related to the roughness of the sample and other processes such as atomic mixing effects arising from the ion bombardment during sputtering. Thin films deposited by PLD have also been used successfully in diffusion experiments run in high- and ultra-high pressure experiments such as the piston-cylinder apparatus (Costa and Chakraborty 2008) and a multi anvil setup (Dobson et al. 2008; Shimojuku et al. 2009). An inert material on top of the thin film can ensure the stability of the film source and a low roughness of the surface even at extreme conditions. Metal capsules (Jaoul et al. 1995; Bejina et al. 1997), Au foil (Shimojuku et al. 2009), or an additional thin film of inert material like ZrO2 (Costa and Chakraborty 2008) have been used for this purpose. The effectiveness of an inert layer was demonstrated by Shimojuku et al. (2009), who used this technique to limit the roughness of a wadsleyite sample surface to ~10 nm during an experiment in a multi-anvil device at 16 GPa and 1400 °C for 50 hours.

Double layer configurations For high-pressure experiments on olivine, some workers have deposited an additional layer of ZrO2 on top of the diffusant source to minimize any interaction with the chemical environment and thus limit dissolution/precipitation processes. Double-layer (“sandwich”) setups have also been implemented to study processes other than diffusion in the substrate, including reaction rim growth (Milke et al. 2007) and element exchange mediated through an inert polycrystalline matrix (e.g., Dohmen 2008). In their study of the growth kinetics of enstatite reaction rims, Milke et al. (2007) deposited a 300-500 nm thick olivine layer on top of a 20-100 nm enstatite layer, which itself had been deposited on quartz single crystal (see Fig. 9). The experimental protocol involved two different deposition and annealing stages to prepare uniform polycrystalline thin films of olivine and enstatite as the starting material for the diffusion anneal (Fig. 9a). During diffusion anneals at temperatures between 1000 °C and 1300 °C the enstatite layer grows continuously at the expense of the olivine layer as shown in Figures 9b-d. Because the enstatite layer is present from the beginning, possible nucleation problems with this phase are avoided. Two additional advantages of this thin-film configuration

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Figure 8. Depth profiles of the 29Si/Si fraction measured with SIMS on two clinopyroxene single crystals deposited simultaneously with a 29Si-enriched olivine film: Sample Cpx1c51-5, 200 nm, 1200 °C, 1.5 hours; Cpx1c51-3, 200 nm, 1250 °C, 4 days. The brief anneal of sample Cpx1c51-5 served only to recrystallize the olivine layer. Diffusion of Si was insignificant during this anneal, so the profile was used to calibrate the depth resolution of SIMS, also taking into account the effects of sample roughness and thin-film/substrate interface irregularity. The corresponding fit using Equations (8) and (9) assuming a sharp compositional jump (dotted line) at the interface is indicated as the dashed line. The solid line is the fit to the depth profile of Cpx1c51-3 using the calibrated depth resolution of s = 50 nm and the analytical solution given in Equation (5). The best fit gives a diffusion coefficient for Si tracer diffusion in cpx of 1.7×10−20 m2/s.

Figure 9. Bright-field TEM images of the cross sections of four thin-film samples prepared to investigate the kinetics of enstatite rim formation: a reference sample showing the initial stage before the diffusion anneal and three samples annealed at 1200 °C and an fO2 = 10−5 Pa for different durations. For more details see Milke et al. (2007). [Used with kind permission from Springer-Science+Business Media: Fig. 4 from Milke et al. 2007.]

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over classic rim growth experiments are worth noting (see also Dohmen and Milke 2010, this volume). First, as in the case of studies addressing purely diffusion phenomena, the investigation of the reaction/diffusion processes on the nm scale makes possible the acquisition of data at lower temperatures. In the case involving the growth of enstatite rims (Milke et al. 2007), it was possible to compare rates obtained at a given temperature from experiments performed “wet” at 0.1-0.2 GPa (and yielding micrometer-scale growth) with those obtained under completely “dry” conditions at 0.1 MPa (yielding only nm-scale growth). This comparison documented unequivocally the dramatic effect of even traces of water on the diffusivity of elements at grain boundaries (about four orders of magnitude) and the corresponding acceleration of reaction rates. Note that increasing the pressure often has the opposite effect (compare with Zhang 2010, this volume) and thus cannot account for this observation. The second advantage of the thin-film technique for measuring reaction rates arises from the fact that growth of a reaction rim or corona is a coupled diffusion process controlled by the mobile chemical components of the system (see references in Dohmen and Milke 2010, this volume). In principle, the rate-controlling component(s) may be unique to the specific system under study. By simultaneous doping of the reacting top layer with stable isotopes, the most mobile components can be identified (Abart et al. 2004; Milke et al. 2007). Isotopically enriched materials can be very expensive, so an added practical advantage of combining isotopic doping with the thin-film technique is that thin films use very little material. The double-layer or sandwich configuration can also be used in instances where an intervening polycrystalline layer acts only as a passive medium for the exchange of elements between the top layer and the substrate (Fig. 10a). This setup was used by Dohmen (2008) to study Fe-Mg exchange by grain-boundary diffusion through inert, polycrystalline ZrO2 layers (note that these experiments have conceptually similarities to the detector-particle approach described in “The Detector-Particle Method for Studies of Grain-Boundary Diffusion” section). In this case the top layer was a fayalite-rich olivine (a source for Fe, a sink for Mg), intended to exchange Fe and Mg with a single crystal of San Carlos olivine (a sink for Fe, a source for Mg) mediated through the inert ZrO2 layer. This model system served as a prototype for studying the kinetics of exchange reactions between two minerals sitting in a polycrystalline matrix—a typical situation for many rock types. The depth profiles of the sample were collected using both RBS and SIMS, which gave consistent results. Diffusive transport through the polycrystalline ZrO2 layer could be identified, but grain-boundary transport was not efficient because local equilibrium between the olivine thin film and the surface of the olivine single crystal was achieved during the course of the exchange reaction, thus eliminating the chemical potential gradient across the ZrO2 layer. This kinetic behavior can be classified as rate-controlled by combined grain-boundary and lattice diffusion (Dohmen and Chakraborty 2003). Because of this, Dohmen (2008) was able to obtain the time-integrated diffusive properties of the inert layer from fitting of the concentration depth profiles for Fe and Mg (Fig. 10b). In general, the configuration shown in Figure 10a allows characterization of the diffusion properties of the inert “sandwiched” film as long as the transport through the layer governs the bulk rate of exchange. Such a setup would be particularly useful to measure the diffusivity of incompatible elements that strongly segregate into the grain boundaries (e.g., Watson 2002; Hiraga et al. 2003; Hayden and Watson 2007, 2008). An appropriate source and sink for the element of interest must be chosen as the top layer and substrate; the thickness of the middle layer can be varied to confirm that transport is controlled by diffusion through the layer. Many of the considerations of this double-layer strategy are similar to those pertaining to the detector-particle method (see “The Detector-Particle Method for Studies of Grain-Boundary Diffusion” section). Similarities to the both the single- and double-layer techniques can also be found in the approach used by Hwang et al. (1979) to characterize grain-boundary diffusion of Ag in Au.

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Figure 10. (a) Basic principle of the setup to examine element exchange along the grain boundaries of an inert layer. Depth profiles of the Fe content (b) and the mole fraction of Fe (c) for three olivine-ZrO2-olivine diffusion couples in direct comparison. The solid lines are best fits using a diffusion model as described in Dohmen (2008). The fitting also includes the modeling of the convolution effect as calibrated by the reference sample.

These workers deposited an epitaxial Ag film on a polycrystalline Au layer (substrate) and annealed the setup at temperatures where grain-boundary diffusion was effective but lattice diffusion was not. Silver atoms diffused through the Au layer along grain boundaries and accumulated on the opposite side, where they were detected by Auger spectroscopy. This, then, was a source-sink experiment in which an epitaxial layer served as the source of diffusant (Ag) and the free surface on the opposite side of the diffusion medium of interest (polycrystalline Au) served as the sink. The mathematical framework for interpreting the results of these experiments was provided in a companion paper by Hwang and Balluffi (1979). In summary, the examples of double-layer setups described above effectively illustrate the power of the PLD method. Researchers now have the opportunity to study not only the kinetics but also, in principle, the equilibrium properties of multi-phase systems at a scale that was not possible previously. Clearly, we are still at the beginning of this endeavor and have not yet realized the full potential of this versatile method.

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Watson & Dohmen The powder-source technique

Overview and history The powder-source technique does not rely upon new or sophisticated technologies at the execution stage of an experiment, so its application to the study of diffusion in crystals pre-dates that of the thin-film method implemented using PLD. As its name suggests, the powder-source technique involves the use of finely-ground crystalline material as the source of diffusant in contact with the surface of a large single crystal of interest. The strategy differs from the thinfilm approach described in “The Thin-Film Method and Pulsed Laser Deposition” section in that the source is effectively infinite rather than deliberately finite. Early applications of the powder-source approach in geochemistry involved studies of Fe↔Mg interdiffusion in olivine (Buening and Buseck 1973; Misener 1974), and were conducted by packing millimeter-sized natural olivine crystals in powder sources consisting of either MgO (Misener) or synthetic Fe olivine. The samples were held at high temperatures (up to 1400 °C in the Misener study), which resulted in Fe↔Mg exchange between the olivine crystals and the powder source surrounding them. In this case, because Fe↔Mg exchange is fast at the high temperatures investigated, the samples were analyzable by electron microprobe in traversing mode (the run products were sectioned perpendicular to the surface of the olivine crystal and polished for analysis of both the crystal itself and the powder source bonded to it). Interdiffusion coefficients were calculated from the analytical profiles using solutions to the 1-D diffusion equation for a diffusion couple in which the diffusion coefficient can vary with position and the concentration at the original interface can vary with time (Matano 1933; Wagner 1969). In addition to generating much-needed diffusion data, these pioneering experiments on olivine demonstrated the feasibility of the powder-source method in general. Most importantly, it was clear from these studies that it is not necessary to have a perfectly mated interface between two single crystals for diffusive exchange to occur: the myriad point contacts of fine particles against a crystal surface can also allow effective “diffusive coupling”—that is, migration of ions across an interface. In their study of Fe-Mg interdiffusion in aluminous spinel, Liermann and Ganguly (2002) developed a variant on the approach used by Buening and Buseck (1973) that avoided possible complications from grain-boundary diffusion in the fine-grained powder. These authors prepared diffusion couples by juxtaposing pre-sintered, polycrystalline pellets of Ferich aluminous spinel with natural single crystals of nearly pure MgAl2O4. The pre-sintered pellets were prepared at high P and T in a piston-cylinder apparatus, which produced nearly 100% dense material of relatively large grain size. The coarse grain size of the pellets made it possible to determine interdiffusion profiles across individual grains at the interface with the single crystal in the diffusion couples. The Liermann-Ganguly experiments might stretch the definition of the powder-source method, but their “course-polycrystal” approach introduced added flexibility to diffusion measurements. As implemented over the past two decades by Daniele Cherniak and co-workers, the powder-source method differs from the early approach used by Buening and Buseck (1973) and Misener (1974) in that it incorporates the implicit assumption that the interface concentration of the diffusant does not change with time. In terms of the manner in which diffusion coefficients are extracted from measured concentration profiles, the more recent incarnation of the powder-source method can therefore be viewed as a variant of the “constant-surface” approach reviewed by Ryerson (1987); see also Fig. 1). The constant-surface assumption is necessary in cases where the powder source must be removed after the diffusion anneal to allow analysis of a “clean” mineral surface. An important difference between the pioneering work on olivine and more recent studies relying on the powder-source technique is that most systems of recent interest involve diffusion that is too slow to be resolved by EPMA traverses parallel to the diffusion direction. In general it is necessary to characterize diffusion by depth-

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profiling into the surface of interest using SIMS, RBS, NRA or laser-ablation ICP-MS (see Cherniak et al. 2010, this volume, for discussion of these methods). This requirement places an additional demand on the powder-source technique in such applications: the source must be removable from the surface of interest without introducing damage that would compromise depth profiles possibly as short as a few tens of nanometers. In other words, the powder source cannot bond tenaciously to the crystal of interest during thermal treatment to induce diffusion. Because of this requirement, the powder-source method in its simplest form is effective mainly for measuring diffusivities at 1 bar pressure. Beginning about 20 years ago, Cherniak and co-workers began a program of research on diffusion in minerals that capitalizes on the versatility of the powder-source technique in combination with depth-profiling by RBS, NRA and (to a lesser extent) SIMS.

Rationale and details An ideal powder source must meet the following requirements: 1) an adequate concentration of the diffusant of interest and the ability to accept counter-diffusing ions from the crystal under study; 2) relatively fast diffusion of the ion of interest in the source itself (so it plays no role in regulating diffusive uptake in the crystal); and 3) lack of reaction with the crystal surface that would result in secondary phases, roughening or strong adherence of the source to the surface. The role of the powder source is shown schematically in Figure 11, where it is emphasized that continuous contact between the source particles and the crystal of interest is neither necessary nor expected. Success of the experiment depends upon exchange between the source and the crystal at localized point contacts (transfer may also be mediated by a gas phase in cases where the diffusant has a significant vapor pressure at the conditions of interest). Rapid surface diffusion and surface-parallel diffusion within the crystal lattice tend to disperse the diffusant in the plane of the surface. Even if concentration gradients persist in this plane for the duration of the experiment, post-experiment depth profiling perpendicular to the surface will yield an accurate diffusivity if the resulting concentration profile averages information over an area that is large relative to the individual point contacts with the source

Figure 11. Schematic representation of a powder-source experiment at three scales. Note that continuous contact of source particles with the surface of the sample of interest is not required if surface diffusion is effective (wiggly arrows). See text for discussion.

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grains. Tannhauser (1956) demonstrated that an evenly dispersed diffusant source on a surface is not required for extraction of an accurate diffusivity. Note, also, that crystallographic direction-dependent diffusion in anisotropic minerals can be investigated by choice of the crystal facet or polished section that is exposed to the powder source (Fig. 11). Following a powder-source diffusion experiment at high temperature, the recovered sample is cleaned of adhering material by ultrasonic treatment and/or gentle scraping with a brush or soft implement. At this stage it is ready for depth profiling by SIMS, laser-ablation ICP-MS, RBS or NRA. The latter two analytical techniques are particularly well matched with the powder-source approach because of their inherent averaging of depth-profiling information over an area typically ~1 mm2 or greater. The depth profile is converted to a diffusivity using standard approaches, which generally means invoking the solution to the non-steady state diffusion equation for 1-D diffusion into a semi-infinite medium in which the surface concentration is held constant (Crank 1975, p. 32):  x  Cixtl ( x, t )  = 1 − erf  Csurf  4 Dixtl t 

(11)

where Cixtl ( x, t ) is the concentration of diffusant i in the crystal at depth x and time t—in this case t is the duration of the diffusion anneal—Csurf is the concentration in the crystal at the surface, and Dixtl is the (spatially-invariant) diffusivity of i in the crystal.

Analytical considerations, advantages and drawbacks Depending on the system of interest, the powder-source method can be used to run experiments intended for analysis by a variety of methods, including EPMA, SIMS, RBS, NRA and laser-ablation ICP-MS. Over the past two decades, however, the powder-source method has mainly targeted diffusion measurements on crystals characterized by very low diffusivities, and as a consequence the vast majority of analyses of powder-source experiments have required high-resolution depth-profiling by RBS, NRA, SIMS or some combination of these techniques. A brief discussion of considerations related to the analysis of powder-source experiments by high-resolution techniques is therefore warranted. The focus here will be mainly on RBS and NRA, and will include a brief discussion of differences between samples recovered from thin-film vs. powder-source experiments. As noted above, one of the advantages of RBS and NRA is that these techniques average information over a relatively large area of the sample surface. In some cases, of course, this is also a major drawback, because recovery of large, intact samples from some types of experimental apparatus can be challenging. On the other hand, RBS and NRA can be relatively forgiving when applied to analysis of powder-source diffusion samples in ways that do not apply in case of thin film experiments. This is illustrated in Figure 12, which shows schematic sections through samples produced by thin-film and powder-source experiments. In this figure, roughness has been deliberately introduced to the relevant surfaces and interfaces. Roughness may be present on natural crystal facets, which are sometimes favored in powder-source experiments because of their inherent stability as rational crystal forms. Roughness may also result from difficulties in preparing a perfectly flat surface, or it may develop during a diffusion anneal as a consequence of surface diffusion accompanying changes in curvature of grain-boundaries formed at points of contact with the source particles. In thin-film experiments, unevenness in film thickness may be a result of recrystallization or other factors described in the “Pulsed Laser Ablation: A Versatile Method For Thin Film Deposition” section (see also Fig. 5). In a simplified sense, RBS and NRA obtain information on concentration as a function of depth from energy-loss considerations as incident and backscattered (or reaction-product) particles traverse the lattice of the diffusion medium of interest (see Cherniak et al. 2010, this

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Figure 12. Schematic illustration of the effects of surface and/or interface roughness on depth profiling by RBS or NRA. Panels (a)-(d) are sections through thin-film experiments like those described in “The Thin-Film Method and Pulsed Laser Deposition” section of this chapter. In cases (a)-(c), the incident beam and backscattered or reaction-product ions pass through different film thicknesses depending on position (the paired arrows show thick and thin points in the surface layer). This condition will lead to analytical “spreading” that does not occur in case (d) where, despite thin-film roughness, the thickness traversed by the beam is nearly uniform. Panels (e)-(g) depict samples from powder-source experiments, which are generally less sensitive to analytical spreading (here the dashed lines are concentration contours). In (e) it is clear that the analyzed length of the diffusive-uptake profile depends only on the local deviation in the angle (θ) of the incident beam to the surface, the effect of which is small because it goes as the cosine of θ. Only in case (g), where the concentration contours do not follow the topography of the sample surface will significant analytical spreading occur. The incident beam is coming from the left in all cases. See text for further discussion.

volume). For this reason, the presence of roughness on the surface or interface of a sample being depth-profiled by RBS or NRA has deleterious consequences mainly when the roughness leads to variation in the thickness of a layer having specific scattering and/or energy-loss characteristics. This can be appreciated with reference to Figures 12a-c, where it is clear that depth-profiling through a thin film of variable thickness (due to roughening) must result in apparent “blurring” of even the sharpest compositional boundary between the film and the substrate (see the “Fitting Uncertainties” section; note that this effect will be superimposed on blurring of a compositional boundary due to limited resolution of the analytical method). The varying thickness of the film means that during RBS analysis, for example, incident and backscattered He+ ions pass through different thicknesses of the compositionally-distinct layer (i.e., the film) depending on surface location. If, however, the roughness at the film-substrate interface conforms with surface roughness—i.e., if the film thickness is uniform despite “topographic” variations—then degradation of the measured profile is minimal (Fig. 12d). Depth-profiling of samples having surface roughness but no compositionally-distinct film (as in the case of a powder-source experiment) is subject to similar effects, but to a lesser degree. If the concentration contours in the sample essentially follow the surface roughness (as in Figs. 12e,f), the effect on the RBS or NRA spectrum will be small. The roughness results in local variation in the angle of incidence of the primary beam, but the distance of material traversed by the relevant particles changes only as the cosine of the deviation in the angle of incidence (Fig. 12e). If, on the other hand, the concentration contours do not conform with surface

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roughness (as in Fig. 12g) the spectrum will represent an average depth profile integrated over the entire sample. In the analysis of a powder-source experiment, the importance of surface roughness in degrading depth resolution depends mainly on the wavelength of that roughness relative to the length scale of diffusive penetration into the sample. In principle, the powder-source method can provide information on equilibrium phenomena as well as diffusion. In their study of Ti diffusion in quartz, for example, Cherniak et al. (2007) used fine TiO2 powder as the diffusion source, in which case the TiO2 concentration at the surface of the quartz crystals should represent the solubility of TiO2 in quartz. The values obtained over a temperature range of ~650-1200 °C are in qualitative agreement with solubility determinations made in other types of experiments (e.g., Wark and Watson 2006; Thomas et al. 2010), but there is substantial scatter in the data [see Figure 5 of Cherniak et al. (2007), but note that since the publication of Wark and Watson (2006) a significant pressure effect on TiO2 solubility in quartz has been documented (Thomas et al. 2010)]. This scatter is probably due to incomplete coverage of the quartz surface by TiO2 particles and/or slow surface diffusion of TiO2. One lesson from this comparison is that the assumption of constant surface concentration (Eqn. 11) that is invoked to obtain diffusion coefficients from powdersource experiments is probably not always rigorously correct: imperfect surface coverage may mean that the surface concentration is a weak function of time during a diffusion anneal. This effect may be responsible for some of the inter-experiment variability in experiments done by the powder-source method, which Cherniak and colleagues generally estimate to be ±50%. It is clear, however, that this is not an important source of error. The RPI group always includes a time-series in their studies that ranges over times differing by a factor approaching 10. Systematic variation in the surface concentration over time would lead to systematic changes in the extracted diffusivities, which generally are not observed. Over the years, the consistency of the powder-source results with those from other types of experiments has been demonstrated in many investigations, including studies of Sr diffusion in apatite (Cherniak and Ryerson 1993), Pb diffusion in zircon (Cherniak and Watson 2001), and Ti diffusion in quartz (Cherniak et al. 2007). Several studies from the RPI lab have also shown excellent agreement between (dry) powder-source results and those from experiments at elevated pressures in the presence of aqueous fluids (Cherniak and Watson 2007, 2010).

Ion implantation and diffusion experiments Introduction From the preceding discussion of thin-film and powder-source diffusion methods, it is clear that ion beams play a key role in the measurement of diffusion coefficients. Secondaryion mass spectrometry (SIMS), for example, involves the use of a relatively low-energy beam (usually O− or Cs+) to erode (sputter) material from the sample surface material and acquire information about composition as a function of depth. Higher-energy ion beams are used in RBS (usually ~2- to 4-MeV He+) and NRA to obtain composition vs. depth information based on the energy loss of particles moving through the material of interest (see Cherniak et al. 2010, this volume). Ion beams can play yet another role in diffusion studies: they can be used to introduce diffusant into the crystal of interest at a predetermined depth, providing a possible alternative to thin-film and powder-source methods. This procedure is referred to as ion implantation. [Interestingly, during a typical RBS analysis, incident He+ ions that are not elastically scattered back out through the lattice are effectively trapped in the target material— so RBS itself is a process of He implantation.] Ion implantation has been used since the 1940s to introduce foreign atoms into target materials and by the 1950s and 60s was widely used in the doping of semiconductors, among other applications. The principles are described in a very general way in the following paragraphs.

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Interactions between energetic ions and solids Ions impinging on a solid surface penetrate the solid if they have sufficient energy. The depth of penetration is determined by the energy of the incident beam and the stopping power (or specific energy loss) of the target, defined as -dE/dx, where E is the ion energy and x is the spatial coordinate in the solid. The latter is generally measured in the direction of the incident beam, but in some cases may be taken as the instantaneous direction of travel of the ion. Two main processes contribute to the energy loss of a moving ion: the elastic interactions between the nucleus of the ion and those of the target material (these are Coulombic interactions between electron-screened nuclei); and the inelastic interactions between the ion and the electron orbitals of the target atoms. The total specific energy loss is thus given approximately by: dE  dE   dE  = + dx  dx  n  dx e

(12)

where the subscripts n and e refer to nuclear and electronic effects. Relatively small contributions to dE/dx may also arise from electron-transfer processes occurring between the moving ion and neighboring atoms. The nuclear and electronic contributions are not independent of one another because energy is transferred by both mechanisms during a single close collision. However, the relative magnitude of these two contributions depends on the velocity (energy) of the moving ion, as shown schematically in Figure 13a. At low ion velocities, most of the energy loss occurs through nuclear collisions, which have the effect of scattering the incident ions and dislodging target atoms. At high ion velocities the energy transfer is dominated by inter-electron effects, which do not dislodge target atoms. In general, high-energy ions are decelerated in a solid mainly by electronic interactions with target atoms, and scattering events are infrequent. Quantification of these broad generalizations requires consideration not only of the incident-ion energy but also the masses of both the incident ion and the target atoms. Figure 13b provides a good feeling for the relative importance of electronic and nuclear stopping in zircon—a material familiar to geochronologists. Implantation energies of ~60-500 keV are typical for samples to be analyzed by RBS, NRA or SIMS. It is clear from the figure that 100-keV Kr or Pb ions are stopped mainly by nuclear scattering, whereas He ions are decelerated mainly by electron-electron interactions. This means that heavy 100-keV ions do significant damage to the zircon structure, but 100-keV He ions do not (see section on “Complications and Examples”).

Figure 13. (a) Schematic illustration after Dearnaley et al. (1973; p. 11) of the relative effects of nuclear and electronic stopping on incident ions of varying velocity (energy). The axes are only relative because the scales depend on the identity of the incident ion and the atoms in the target material. In (b) are shown actual contributions of electronic and nuclear stopping for specific ions of varying mass and energy impinging on zircon (ZrSiO4), computed using the Monte Carlo simulation program SRIM (Ziegler and Biersack 2006).

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Electronic stopping and nuclear stopping are well understood phenomena, having been targeted theoretically and empirically for nearly a century [the first calculation of electronic stopping was done by Bohr (1913) for the specific case of ions moving faster than the K-shell electron velocity]. In amorphous materials (dE/dx)n and (dE/dx)n are independent of direction, but it stands to reason that this is not necessarily the case in crystals. Specific directions in a crystal lattice may offer “easier passage” of ions in a process called channeling. Channeling has some interesting applications in geochemistry and mineralogy because it can be used to determine siting of impurity ions, but for ion implantation it is deliberately avoided, typically through use of sample holders and implantation chambers designed to minimize the likelihood of this phenomenon. Further discussion of energy-loss considerations is beyond the scope of this chapter, but the preceding introduction will serve as an adequate basis for introducing the concept of range theory. The reader is referred to Dearnaley et al. (1973), Ryssel and Ruge (1986), Ziegler (1992) and Nastasi et al. (1996) for more information on ion-solid interactions.

Ion implantation A critical consideration in ion implantation is the range of the ion in the target material. This refers to this distance traveled through the target material as it is decelerated (within ~10−14 s) from its initial energy to ~20 eV, where it is effectively stopped by interatomic bonding forces. Due to the statistical aspects of ion energy loss and scattering by nuclear collisions, the individual ions of a mono-energetic beam do not all travel exactly the same distance in the target. An ion beam impinging on a target results in a probability distribution of ion penetration that can be described in terms of a probable range, a median range, a mean range and (sometimes) a maximum range. Decelerating ions follow a random path in the target, so in addition to the total range (Rtot) it is advantageous for implantation purposes to describe a projected range (Rp), which is the range measured in the direction of the incident ion beam. The total range is given by Rtot = ∫

E

0

dE −( dE / dx )

(13)

The stopping power (-dE/dx) is generally a function of E, so calculation of Rtot requires knowledge of this dependence in most instances. For a simplified case of a monatomic target in which electronic stopping is ignored and inverse square law potentials are used to describe elastic scattering, Rtot can be estimated from Rtot =

0.6( Z12 / 3 + Z 22 / 3 )1/ 2 ( M1 + M 2 ) M 2 E Z1Z 2 M1

(14)

where Z1 and Z2 are the atomic numbers of the incident ions and target atoms, respectively, and M1 and M2 are their masses (the units of Rtot are mg of target per cm2). For the simplified case described above, Lindhard and Scharff (1961) showed that Rtot M ≅ 1+ 2 Rp 3 M1

(15)

from which it is clear that if the moving ion and target atoms are of similar masses, Rp does not differ greatly from Rtot. In most implantation applications a defocused ion beam is directed at a large target area, and Rp is the parameter of primary interest. The above discussion should give the reader a feeling for the factors involved in ion implantation, but today most actual calculations of stopping power and range in complex materials are done using open-source software such as SRIM-2006 (Ziegler and Biersack 2006) to perform Monte Carlo simulations.

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Mathematical aspects of implantation and diffusion Figure 14 shows a typical Gaussian profile of implanted ions in the direction parallel to the ion beam. This distribution is described by C ( x ) = Cmax e

−X2 2

(16)

where C(x) is the concentration of implanted ions at distance x from the target surface, Cmax is the concentration at x = Rp, and X = (x-Rp)/ΔRp. The significance of ΔRp is clear from Figure 14; this is referred to as the range straggle. For monatomic targets in which nuclear stopping dominates and M1 > M2, the range straggle is given by the theory of Lindhard et al. (1963) as  2( M1M 2 )1/ 2  ∆R p ≅ 0.44 R p    M1 + M 2 

(17)

(see Nastasi et al. 1996, p. 130). Formulae for estimating range straggle in polyatomic materials can be found in Nastasi et al. (1996; ch. 6). The total number of implanted atoms per unit area of the target can be computed by integration of Equation (16) over the relevant depth interval (see Dearnaley et al. 1973, p. 496), and is given by N S = Cmax ⋅ ∆R p 2 π

(18)

The quantity NS is referred to as the implantation dose; this would pertain to a given

Figure 14. Distribution of implanted ions as a function of distance from the target surface. Graphs (a) and (b), respectively, illustrate the cases in which the mass of the incident ion (M1) is either less than or greater than the mass of the target atoms (M2) (after Nastasi et al. 1996; p. 4). The near-Gaussian profiles shown are expected for an amorphous target material or a crystalline one in which the lattice was oriented so as to preclude channelling. Definitions of Rp and ΔRp are given in the text.

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implantation session utilizing an ion beam of a specific energy and mass [Note: In their study of He diffusion in apatite and zircon, Cherniak et al. (2009) used a 3He ion dose of 5×1015 atoms/cm2]. It is clear from Figure 14 that the projected range, Rp, determines the “centroid depth” of the implanted layer. From our earlier discussion it is also clear that this is a strong function of both ion mass and beam energy. For the purposes of this chapter, ion implantation is a means to set up a diffusion experiment: the implanted layer given by Equation (16) essentially defines the initial conditions of the experiment. In most diffusion studies, the initial distribution would be characterized by SIMS, RBS or NRA. Subsequent heating to a temperature sufficient to mobilize the implanted atoms results in spreading of the initial Gaussian distribution, accompanied by loss from the surface, given sufficient annealing time and volatility of the diffusant. Provided the diffusivity is not a function of concentration or position in the sample, the governing diffusivity can be calculated by characterization of one or more of the following after a diffusion anneal: 1) the relaxed concentration profile, obtained by a depth-profiling technique; 2) the integrated amount of diffusant remaining in the sample; or 3) the total amount of diffusant lost from the sample surface (the fractional loss; nos. 2 and 3 are clearly complementary: for any given diffusion anneal these must sum to NS×A, where A is the implanted area). In principle, the first approach provides the most accurate estimate of the diffusivity through fitting of the measured profile to the appropriate solution to the 1-D, non-steady state diffusion equation. The general equation describing diffusive spreading of a Gaussian distribution in an infinite medium is (Ryssel and Ruge 1986):  ( R − x )2  p  C ( x, t ) = exp  −  2 ∆R p2 + 4 Dt    2 Dt    1 +  ∆ R p2   Cmax

(19)

In many post-implantation diffusion experiments, however, outward-diffusing implanted atoms will reach the surface of the sample at x = 0. In this case, the relaxing profile of a diffusant is “reflected” at x = 0 if the diffusant is refractory (i.e., if it is not volatilized at the surface). In the case of a volatile diffusant such as a noble gas, the surface concentration goes to zero concentration at x = 0 (that is, the gas escapes the sample). Addressing these situations requires more complex solutions to the diffusion equation that apply to a semi-infinite medium in which x = 0 is either a zero-flux boundary (refractory diffusant) or C(x = 0) = 0 for all t (volatile diffusant). The appropriate solution is (Ryssel and Ruge 1986) C C ( x, t ) = max 2

   Rp 4 Dt + x∆Rp 2    ( x − R )2     4 Dt   p  × 1 + erf  ∆Rp 2  exp  − 2 2  2 ∆R p + 4 Dt    2 ∆R p + 4 Dt               Rp 4 Dt − x∆Rp 2     ( x + R )2    4 Dt    p   × 1 + erf  ∆Rp 2 + or − exp  −   2  2 ∆R p2 + 4 Dt    2 ∆R p + 4 Dt             

 2 Dt  ⋅ 1 +   ∆R p2   

−1 / 2

(20)

For the volatile diffusant case, the lower line of Equation (20) is preceded by a “+” sign, and for the refractory diffusant case (zero-flux boundary), it is preceded by a “−” sign (so at t = 0 there is no difference between the two cases). An excellent example of the implementation of Equation (20) is the study by Cherniak and Ryerson (1993) of Sr diffusion in apatite. Figure 15a shows the time evolution of a model system from the initial implanted Sr distribution. Actual profiles measured by RBS following high-temperature diffusion anneals are shown in Figure 15b; fits to Equation (20) (also shown on the figure) were used to calculate diffusion

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Figure 15. (a) Evolution of an initial implantation profile with time (dotted curve) computed using Equation (20) and reasonable estimates of Rp, ΔRp and diffusivity of Sr in apatite (after Cherniak and Ryerson 1993; see that paper for more details). In (b) are shown actual RBS depth profiles obtained by Cherniak and Ryerson after annealing apatite samples implanted with Sr+ at 70 keV. The Arrhenius plot in (c) illustrates the excellent agreement between the data obtained by ion implantation and high-temperature data obtained in an earlier electron-microprobe study by Watson et al. (1985).

coefficients for Sr. This study is a valuable contribution to the geochemical diffusion literature not only because of the strategic use of ion implantation, but also because the authors were able to show consistency of diffusion results obtained by Sr implantation with those obtained from SrO powder-source experiments. Moreover, both the implantation and powder-source results are in very good agreement with an earlier study at higher temperatures in which the diffusion profiles were characterized by electron microprobe (Fig. 15c). Accurate measurement of an implantation profile that has relaxed by diffusion is not always possible due to limitations in the analytical sensitivity or spatial resolution of the profiling method, in which case it is necessary to use approach no. 2 or 3. Fractional-loss measurements are particularly well suited to diffusion studies using implanted noble gases because the released gas can be captured and analyzed by mass spectrometer (e.g., Futagami et al. 1993).

Complications and examples Ion implantation is a straightforward way to establish a compositional gradient in a crystal that will lead to diffusion of the implanted atoms if the crystal is heated. Interestingly, however, ion implantation has not seen extensive use by geoscientists in diffusion measurements. The

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reason may be in part that ion-beam facilities are not widely available, but there are other complications that may be of special concern when working with crystalline ionic solids (minerals): 1) ions implanted in a crystal alter the composition of the target in an additive way (that is, without ion exchange); and 2) they can cause damage to the lattice—which could affect the very parameter (the diffusion coefficient) one seeks to measure. The first concern may be unimportant if the material of interest is a metal or if the implantation dose can be kept low. Unfortunately, however, little is known about the possible effects on diffusion of the non-stoichiometry resulting from the presence of implanted ions. Low-dose implantation of a cation in a silicate would result in a slightly cation-rich (non-stoichiometric) lattice, presumably creating either anion vacancies or cation interstitials. Although vacancy concentration is well known to affect diffusion, the vacancies resulting from cation implantation—being anion vacancies—would not necessarily enhance cation diffusion. We speculate that once the implanted atoms are mobilized (as in a diffusion experiment) they will occupy lattice sites in competition with the major elements already present in the crystal, and the slight nonstoichiometry may not affect the measured diffusivity. This suggestion could be tested by evaluating the conformance with theory of the diffusive evolution of the implanted profile (Eqn. 20), which has been done (with time series experiments) in several of the studies that have used ion implantation in measuring cation diffusion in minerals (see below). Failure of the diffusion behavior to conform with Equation (20) might suggest a concentration-dependent diffusivity attributable to non-stoichiometry. The reliability of ion implantation as a means of introducing diffusant into minerals is confirmed in a limited way by two early geochemical examples: that by Cherniak and Ryerson (1993) discussed previously (see the “Mathematical Aspects of Implantation and Diffusion” section) and the earlier study by Cherniak et al. (1991) of Pb diffusion in zircon and apatite. In the 1991 study it was shown that implanted Pb in apatite diffuses at rates consistent with those obtained from earlier Pb-Ca interdiffusion experiments. Cherniak et al. (1991) were somewhat less successful in their efforts to constrain Pb diffusion in zircon using an implanted source. These authors realized that damage to the zircon lattice during implantation of the heavy Pb atoms had resulted unrealistically high Pb diffusivities in their experiments. Confirmation of the magnitude of implantation-damage effects on Pb diffusion in zircon had to await highquality Pb diffusion measurements made using the powder source (see below and Cherniak and Watson 2001). Damage issues aside, the introduction of “excess” atoms by ion implantation should not be a concern in the case of noble gases, which play no role in charge balance, and—in the case of smaller atoms like He and Ne—may occupy and diffuse through interstitial sites anyway. Cherniak et al. (2009) measured diffusion coefficients for He in zircon and Durango apatite by implanting 3He and subjecting the samples to systematic isothermal anneals to induce diffusive spreading and loss from the surface. Diffusion coefficients were extracted by characterizing 3 He remaining in the sample with NRA (the 3He(d,p)4He reaction) and applying Equation (20). The results are broadly consistent with diffusion measurements based on thermal release of He accumulated in natural crystals by radioactive decay (e.g., Farley 2000; Reiners et al. 2004). Ouchani et al. (1998) and Miro et al. (2006, 2007) used similar techniques to measure He diffusion in other apatites (including Nd-silicate apatite and polycrystalline apatite ceramic), obtaining results consistent with those of Cherniak et al. (2009). As discussed in the “Interactions Between Energetic Ions and Solids” section and portrayed in Figure 13b, lattice damage by 3He implantation at the ~100 keV energies used in the studies cited above is minor and limited mainly to ionization, which is transient even with modest heating. Heavy-ion implantations are another matter: in this case, elastic collisions between energetic heavy ions and the constituent atoms of an implantation target can result in extensive atomic displacements. Generally speaking, if an energy greater than ~20-30 eV (the displacement

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energy, Ed) is transferred to an atom during a collision, that atom may be dislodged from its site and come to rest elsewhere in the lattice, most likely a nearby interstitial location. The site from which the atom is dislodged becomes a vacancy, and the vacancy-interstitial pair is referred to as a Frenkel defect. Such defects are of obvious concern in diffusion studies because vacancies provide a mechanism for diffusion. If implantation-induced vacancies were to persist during a subsequent diffusion experiment, the resulting diffusivity would not reflect the value relevant to the undamaged lattice. The value of Ed depends upon the nature of the target (mass and identity of atoms, bond energy, etc.) and also upon the direction in which an atom is struck, due to lattice anisotropy. Figure 16a is a schematic representation of the sequence of events leading to lattice damage in a crystalline solid. The first event is the elastic collision of the incident particle with a target atom; this is followed by a cascade of subsequent collisions that displace many more lattice atoms than the one initially struck (the latter is called a primary knock-on atom, PKA; see Fig. 16a). The overall extent of damage to a crystal during ion implantation is determined by the energy and mass of the incident ions and the masses of the target atoms. It is also affected by the temperature and nature of the target material (discussed below). The extent of overall damage and its consequence for diffusion are also determined by the implantation dose. Whereas low doses may cause localized damage that creates high-diffusivity pathways, high doses may result in continuous overlap of damaged regions to produce a completely amorphous material characterized by entirely different diffusion properties (see Fig. 16b). Geochronologists are familiar with this type of phenomenon in the case of α-recoil damage in zircon and other minerals containing U and Th. In this case, heavy nuclei are accelerated “internally” by the ejection of an α-particle from the parent nucleus; these accelerations can be sufficient to impart the displacement energy to other lattice atoms. For more information on lattice damage by energetic ions, the reader is referred to Nastasi et al. (1996). For the present (geochemical) purposes, the main concern is whether lattice damage

Figure 16. Schematic illustration of the interactions between ions and target material. The sequence of events following an elastic (nuclear) collision is depicted in (a) at an atomic scale (after Dearnaley et al. 1973). These collisions create Frenkel defects, as discussed in the text. A more general view is shown in (b), where the distinction is made between low doses creating non-interconnected damage regions and high doses leading to overall amorphization of the target (after Nastasi et al. 1996).

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resulting from implantation of the diffusant might affect the measured diffusion coefficient. This depends on the extent and nature of the damage and its thermal stability relative to the effectiveness of the diffusion process of interest. It is clear that introduction of heavy-atom diffusants (e.g., Xe, Pb) by implantation can affect the resulting diffusion coefficient in at least some minerals. In pioneering studies of noble-gas diffusion in minerals, for example, Melcher et al. (1981, 1983) implanted Xe in feldspar and olivine and profiled the annealed samples by RBS. These studies demonstrated that retention of implanted Xe depends on the implantation dose for both minerals. The same is true of Pb in zircon. Cherniak and Watson (2001) used the powder-source technique to revisit the topic of Pb diffusion in zircon that had been explored previously by Cherniak et al. (1991) using ion-implanted Pb. Comparison of the two data sets reveals that the Arrhenius relation from the implantation study lies well above the powder-source line, with the two converging only at T > 1500 °C. In their powder-source study, however, Cherniak and Watson (2001) also reported the results of experiments done by hot implantation of Pb at 800 °C (the 1991 implantation results were obtained from samples implanted at room temperature). Maintaining the sample at elevated temperature during implantation enables constant healing of damage as mobile displaced atoms are able to return to normal lattice sites. For the case of zircon specifically, 800 °C is hot enough for the lattice to resist amorphization even at quite high doses of heavy ions (Wang and Ewing 1992; Weber et al. 1994); other materials will have different critical amorphization temperatures depending on bonding and other crystal characteristics. Cherniak and Watson (2001) were able to show that diffusion results in the 1200-1350 °C range obtained from hot-implanted samples are consistent with those obtained by the powder-source method. In any given diffusion study conducted using an ion-implanted source, care must be taken to ensure that the results reflect diffusion in the material of real interest, which is usually the undamaged crystal lattice. If lattice damage is introduced at the implantation stage, then subsequent diffusion anneals may involve the simultaneous occurrence of two separate kinetic processes: healing of implantation-induced lattice damage and diffusion of the implanted species. In some instances, diffusion in radiation-damaged material may be the process of actual interest. In this context, we note that ion implantation can be used to simulate the damage caused by natural radioactive decay: for example, heavy Xe ions of appropriate energy can cause lattice damage (atom displacements) similar to those resulting from α-recoil in Uand Th-bearing minerals. In principle, ion implantation thus provides the means to introduce a controlled amount of lattice damage—albeit over a relatively localized region within the crystal—and evaluate the consequences for diffusion.

The detector-particle method for studies of grain-boundary diffusion Context and history Compared with the other techniques described in this chapter, the detector-particle method for grain-boundary diffusion is relatively new: in fact, it is still undergoing development to enable extraction of more quantitative information. Some of the ongoing refinements are described here for the first time. Because of its wide-ranging technological and scientific applications, the study of grainboundary diffusion has a long history that has led to both a robust mathematical basis (e.g., Fisher 1951; Whipple 1954; Harrison 1961; Leclaire 1963) and numerous experimental measurements (see Dohmen and Milke 2010, this volume). A conceptually valuable classification of grainboundary diffusion into discrete “types” (kinetic regimes) was provided by Harrison (1961) and augmented by Mishin and Herzig (1995), as discussed by Dohmen and Milke 2010 (this volume; see their Fig. 5). The case of type C grain-boundary diffusion involves essentially

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no diffusion in the lattice (i.e., transport occurs in grain boundaries only)—a situation that could arise either (transiently) because the grain-boundary diffusivity (Dgb) is overwhelmingly larger than the lattice diffusivity (Dlat) or because the diffusant of interest is highly incompatible in the grains (for more quantitative constraints, see Dohmen and Milke 2010, this volume). This situation may be commonplace in geological settings, including instances such as grainboundary diffusion of siderophile or large-ion lithophile elements in the mantle. Diffusion of rare-earth and high field-strength elements along grain boundaries of major crustal phases such as quartz and feldspars is a plausible crustal scenario. Experimental characterization of type C kinetics presents a particularly difficult problem because the existing 2-D solution to the non-steady state diffusion equation developed for grain-boundary diffusion (e.g., that of Leclaire 1963) assumes uptake of diffusant in the crystal lattice. In the case of type C kinetics, in contrast, the diffusant remains exclusively in the grain boundaries essentially by definition. Accordingly, for a grain boundary resembling a thin, infinite slab and having a source of diffusant uniformly distributed along one edge (see Fisher 1951; Leclaire 1963; Dohmen and Milke 2010, this volume), transport along the boundary conforms to 1-D diffusion into a semi-infinite medium. However, the lack of diffusive “leakage” into the contacting grains means that traditional grain-boundary diffusion experiments involving type C kinetics require an analytical technique capable of quantitative determination of diffusant at challenging length scales and/or very low concentrations (see Dohmen and Milke 2010, this volume; Farver and Yund 2000).

The detector-particle approach: general considerations and examples Watson (1986) put forth an alternative approach for characterizing type C grain-boundary diffusion—logically called the “detector-particle” method—which is contrasted with the type B case in Figure 17. The approach was originally conceived for the purpose of quantifying oxygen transport along grain boundaries in fluid-absent rocks. Strictly speaking, this is not the best geochemical example of type C kinetics (because oxygen is compatible in silicate minerals), but grain boundary diffusion of oxygen was expected to be many orders of magnitude faster than lattice diffusion under dry conditions. The detector particles in Watson’s (1986) experiments were CuO or Fe2O3 grains dispersed in synthetic dunite or quartzite. The synthetic rocks were encapsulated in graphite or Fe metal, and grain-boundary diffusion of oxygen was assessed from the width of the zone in contact with the container where the detector particles were reduced to Cu or FeO. Watson (2002) later used the detector method to quantify transport of incompatible siderophile elements (Pt, Pd, Au, W) along grain boundaries of mantle rock analogs (MgO and forsterite). The extreme incompatibility of these elements in oxide minerals precludes the use of conventional grain-boundary diffusion experiments. The proposed alternative approach was to synthesize polycrystalline MgO samples containing dispersed small particles in which the diffusants of interest were compatible, even though they were excluded to the point of being immeasurable in the major-phase (MgO) grains. If elements of interest—say Pt and Pd—are mobile along MgO grain boundaries, then small Pd particles dispersed in polycrystalline MgO should “communicate” (i.e., form alloys) with distant Pt particles by diffusion along MgO grain boundaries, provided diffusion in the particles themselves is sufficiently fast (see the “Numerical Simulation: Constant-Surface Model” section). A sectioned sample and analytical results from the original experiments of Watson (2002) are shown in Figure 18. To date, the main application of the detector-particle method has been to estimate grainboundary diffusivities of 10 siderophile elements in polycrystalline MgO (Hayden and Watson 2007) and also of carbon in MgO and dunite (Hayden and Watson 2008). In these studies, chemical potential gradients were set up in experimental samples by placement of mutually soluble “sources” and “sinks” separated by dense polycrystalline aggregate of the bulk material of interest. In most instances the sources and sinks were small (~10 mm) metal particles, but

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Figure 17. Characterization of grain-boundary diffusion in the “type B” regime (top) relies upon uptake of the diffusant of interest in the crystal lattice (small flux arrows). The detectorparticle strategy (bottom) addresses situations where the diffusant is incompatible in the lattice and therefore immeasurable in the bulk sample. A particle of another phase in which the grainboundary diffusant is compatible is placed on the grain boundary; diffusion in the boundary is estimated from the diffusant uptake in this particle. See text for further discussion.

Figure 18. Backscattered-electron images of polycrystalline MgO from a 5-hour piston-cylinder experiment at 1600 °C and 2.3 GPa. The sample contains two horizons of metal particles, initially pure Pt and Pd, that have alloyed at run conditions by grain-boundary diffusion in the MgO. Images are from Watson (2002); see Hayden and Watson (2007) for details of similar detector-particle experiments.

metal foils and wires were also used as sinks for carbon, and massive graphite was used as a carbon source. Dohmen (2008) designed a conceptually similar approach to examine Fe↔Mg interdiffusion in olivine and through the grain boundaries of (inert) polycrystalline ZrO2 (see Fig. 10). In that case, the PLD technique was to deposit thin layers of ZrO2 and Fo30 olivine on the polished surface of a single crystal of San Carlos olivine (Fo89). Interdiffusion of the two olivine compositions occurred by transport through the ZrO2 grain boundaries, and the grainboundary diffusivity was obtained by mathematical analysis of the Fe depth profile—obtained

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by RBS and SIMS—in the single-crystal olivine. Dohmen was able to extract information not only on diffusion but also on the “storage capacity” of ZrO2 grain boundaries for the impurity diffusants Fe and Mg. In the Hayden and Watson studies, approximate grain-boundary diffusion coefficients were calculated in two ways. The simple approach was to estimate the length scale (x) over which distant particles communicated with one another by diffusion. The grain-boundary diffusivity was then computed as Dgb ≈ x2/t [Note: the tortuosity was not considered in this approximate calculation; in principle, this should result in underestimation of the diffusivity by a factor of ~3 (Watson’s 1991 study suggested a tortuosity of ~1.7)]. Electron microprobe analysis of sink particles (e.g., Pt) at various distances from the source of a diffusant of interest (e.g., Pd metal) provided an estimate of the diffusive length scale and hence an order-of-magnitude diffusivity. The second approach used by Hayden and Watson involved conversion of diffusant concentrations in the detector particles to time-integrated grain-boundary fluxes. This strategy can be appreciated with reference to Figure 17, which depicts a single detector (sink) particle intersecting a grain boundary with a diffusant source at one end. The diffusant atoms enter the grain boundary at the source (left) and diffuse along the boundary until they encounter the sink phase at the right. The time-integrated grain-boundary flux (Jgb) can be calculated by measuring the total number of diffusant atoms (n) accumulated in the sink over a given time; then t

n = A∫ J gb dt

(21)

t =0

where A is the effective cross-sectional area of the grain boundary over which the diffusant is intercepted by the sink phase (for this calculation the grain-boundary width is taken as 1 nm; Hiraga et al. 2002). The flux alone is valuable as a qualitative indicator of the effectiveness of grain-boundary diffusant transport, but additional information or assumptions are needed to calculate an actual diffusivity. Assuming the system quickly approaches a steady-state condition (see the “Numerical Simulation: Constant-Surface Model” section), the diffusivity in the grain boundary is given simply by Dgb = − J gb / dc dx

(22) gb

where (dc / dx ) gb is the concentration gradient of the diffusant in the grain boundary. Complications arise in this simplistic approach because neither the absolute concentration in the grain boundary nor (dc / dx ) gb is likely to be directly measurable or strictly constant. Another approach is to assume that local partitioning equilibrium exists between sink grains and grain boundaries (see, e.g., Hiraga et al. 2004; Hiraga and Kohlstedt 2007), so the diffusant concentration in sinks at various distances from the diffusant source will reflect (i.e., be proportional to) the concentration profile of the diffusant in the grain boundaries. Such a profile can be fit to a simple diffusion model as was done by Hayden and Watson (2008) to obtain an effective bulk diffusivity that has practical value in addressing geochemical problems. The effective length scale of grain-boundary diffusion is well constrained even if the absolute grainboundary diffusivities are not. The detector particle method as described above could be implemented to yield qualitative results for a wide variety of systems, the main requirements being: 1) diffusion in the detector particles themselves must be reasonably fast; 2) the diffusant of interest must partition into them sufficiently to enable measurement; and 3) the detector particles must be in chemical equilibrium with the major phases of the rock of interest. In view of the fact that many problems involving chemical transport in the Earth are completely unconstrained at present—is it plausible, for example, that the mantle and core communicate by diffusion?— even qualitative results have significant value.

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Numerical simulation: constant-surface model Since the initial efforts of Watson (2002) and Hayden and Watson (2007, 2008), additional insight into the behavior of detector particles during grain-boundary diffusion has been obtained at RPI through numerical simulations. The geometry and boundary conditions are set up in a manner much like that used by Fisher (1951) in his pioneering analysis. Figure 19a is a realistic, 2-D representation of a polycrystalline aggregate containing dispersed detectors particles and a source of diffusant at the left (x = 0). The model system (Fig. 19b) contains regularly-spaced detector particles of uniform size, and the grain boundaries are in three orthogonal sets: one parallel to the diffusion direction (solid lines) and another perpendicular to these (dashed lines). The third (implied) set is parallel to the plane of the figure (i.e., in the x-y plane). Given a uniform planar source, diffusion in the x direction advances in an identical manner along parallel grain boundaries, so there are no sustained grain-boundary fluxes in the y or z directions once the y- and z-parallel grain boundaries are locally “filled” with diffusant. For this reason, the behavior of an entire aggregate can be captured with reasonable accuracy by modeling a single, tabular grain boundary as was done in the case of type B kinetics (Fisher 1951; Leclaire 1963). Such a grain boundary, depicted in Figure 20, has a finite width (δ) in the y direction, and detector particles of radius r are spaced along it at interval Dx (the distance between the first particle and the source, Dx1 = Dx − r —i.e., the same as the distance from the surface of one particle to the center of the next). For the numerical simulations, the grain boundary of the model system shown in Figure 20 is divided into discrete volume elements (δx = δz; δy = δ) that are allowed to interact diffusively. Each volume element is treated individually by calculation of fluxes to or from the four neighboring elements with which it shares a side. Relative to the center of the volume element under consideration at any moment, the coordinates of the volume elements with which it interacts are −δx, +δx, −δz, +δz; the node-to-node fluxes in dt are given by the product of the governing diffusivity, Dgb, and the concentration gradients, ΔC/Δx and ΔC/Δz [Note that this approach amounts to an explicit finite difference method in the terminology of Crank (1975)]. The detector particles are imbedded in the grain boundary as shown in the figure, and local equilibrium between the grain boundary and the particle surface is assumed to be governed by a partition coefficient: K dprt / gb =

srf C prt

Cgb

(23)

srf where C prt is the concentration of the diffusant of interest at the surface of the particle and Cgb is the local concentration in the grain boundary near the particle surface. The latter quantity is taken as the average concentration in the grain-boundary volume elements contacting the particle (see Fig. 20). This concentration is assumed to be in equilibrium with the entire (spherical) surface area of the particle, which amounts to assuming infinitely fast diffusion around the particle surface. Diffusion within the particle itself is assumed to be governed by the lattice diffusivity, Dprt, and is treated, independently for each particle, as diffusion in a sphere with a locally time-dependent surface concentration. The reader is referred to Watson et al. (2010) for more detailed discussion of the finite-difference approach using discrete volume elements. The intent here is to illuminate the essential features of type C grain-boundary diffusion as captured by concentration changes in detector particles. In order to assess consistency with actual measurements, simulations were run using durations, particle spacings and particle sizes similar to those used in ongoing laboratory experiments at RPI.

Figure 21a shows the results of a 7-day numerical simulation made using plausible input parameters as indicated on the figure. The bulk concentration of diffusant (Cprt) in the first of 5 particles spaced at ~200-mm intervals along the grain boundary is shown as a function of time. For the input parameters used, Cprt depends almost linearly upon time for all values of Dgb (1×10−10 to 3×10−9 cm2/s). The linearity indicates an essentially steady-state condition, as was

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Figure 19. 2-D illustration of a “real” polycrystalline sample (a) containing dispersed detector particles, compared with a model system (b) having a uniform grid of grain boundaries and regularly-spaced detector particles. Because all grain boundaries oriented parallel to the x direction are the same, the diffusion behavior of the bulk sample is captured by treating just one of them (e.g., the region enclosed in the dotted rectangle). See text for discussion.

Figure 20. A single grain boundary (upper panel) isolated from the model system in Figure 19 and assigned properties and boundary conditions. The concentration in the grain boundary at the source (left) is held constant; the detector particles are assigned a specific radius and spaced at regular intervals along the boundary. The lower panel illustrates the 2-D array of volume elements in the grain boundary and imbedded particles used for numerical computation (see text and Fig. 25).

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Figure 21. (a) Results of numerical simulations showing the time-evolution of the bulk diffusant concentration (Cprt) in the particle nearest the source (no. 1 in graph b) for several values of the grainboundary diffusivity, Dgb. Note that the plots are almost linear, indicating a near steady-state condition was reached early in the simulations. Graph (b) shows the time-evolution of diffusant concentration in five particles as a function of their distance from the source. Three values of the particle/grain-boundary partition coefficient Kdprt/gb are considered, but the influence of this parameter is minor. Note that the main effect of increased time is to raise Cprt in the first and second particles: penetration further along the boundary is limited, as evidenced by the minimal changes in particles 3-5 with increasing time. See text for details and discussion, and compare (b) with actual data shown in Figure 22.

assumed by Hayden and Watson (2007) in interpreting their experimental results for siderophile element diffusion in polycrystalline MgO. Note, however, that all Cprt vs. t lines intersect the time axis at positive values, especially at Dgb < 10−9 cm2/s. The non-zero intercept represents an initial transient during which diffusion in the grain boundary reaches the first particle. Figure 21b shows the time evolution of diffusant concentration in five particles spaced at 200-mm intervals for three values of Kdprt/gb (10, 1, and 0.1). This figure demonstrates that, within the limited range of parameters examined, uptake of diffusant in the particles is insensitive to Kdprt/gb. There is essentially no difference in behavior for values of Kdprt/gb > 1, and even modest incompatibility (Kdprt/gb = 0.1) in the particles relative to the grain boundary affects diffusant uptake in the particles by only ~30%. The reason for this insensitivity to Kdprt/ gb is that the capacity of 12-mm particles to incorporate diffusant overwhelms the capacity of 1-nm grain boundaries to supply that diffusant: the overall process is regulated by the diffusive flux in the grain boundary. Despite a high diffusivity, this flux is small because the cross-sectional area of the grain boundary is minute. Figure 21b also reveals that concentration changes are most apparent in the particles nearest the diffusant source: these intercept most of the diffusant, so particles further from the source show only minor changes in concentration with time. This means that the time evolution of the overall system is characterized not by progressive penetration deeper into the bulk sample

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(as is the case with type B grain-boundary diffusion), but by progressive steepening of particle concentration profiles like those shown in the figure. Use of the bulk-sample penetration depth (i.e., diffusive length scale) to estimate grain boundary diffusivities would yield a deceptively low apparent grain-boundary diffusivity. This principle is illustrated in Figure 22, which shows unpublished data from the RPI lab pertaining to Mg↔Fe interdiffusion along grain boundaries in synthetic quartzite. In this study, the surface source of Mg2+ at one edge of the quartzite was MgF2 and the detector particles dispersed in the quartzite were fayalite (Fe2SiO4). The summary panel of Figure 22 clearly shows only minor increases in Mg penetration depth with increasing time, but the profile of MgO in the particles vs. depth in the quartzite steepens markedly as the experiment duration progresses from 1 to 8 days. In future experiments using the detector particle approach, the optimal strategy might be to choose particles in which the diffusant of interest is only slightly compatible—enough to result in measurable diffusant concentrations in the particles, but not so much that the particles near the source act as “infinite sinks” for the diffusant. This goal might be achieved by using particles having Kdprt/gb 10−17 m2s−1, e.g., Carroll 1991), such as those for diffusion in melts and inter-diffusion of rapidly diffusing elements in minerals, or diffusion at relatively high temperatures, can be measured using EMPA. Smaller diffusivities require other methods, such as Rutherford Backscattering or SIMS in depth-profiling mode. Figure  9a shows an Ar concentration profile measured by EMPA (Behrens and Zhang 2001). The profile is a result of Ar diffusion into a dry Ar-free rhyolite melt at high Ar pressure and high temperature (Behrens and Zhang 2001). The profile is long and well-resolved. The data can be fit well by an error function (solid curve), indicating that the Ar diffusivity is independent of Ar concentration. Figure  9b shows an interdiffusion profile of FeO measured by EMPA between a high-silica rhyolite (77 wt% SiO2) and a lower-silica rhyolite (73 wt% SiO2) melts (Van Der Laan et al. 1994). These data show con-

Figure 9. Two concentration profiles measured by EMPA. (a) Ar

diffusion profileconcentration in dry rhyolitic melt profiles (Behrens and Zhang 2001);by EM Fig. 9. Two measured (b) FeO interdiffusion profile for a rhyolite diffusion couple (Van rhyolitic (Behrens andtemperatures, Zhang 2001); der Laan etmelt al. 1994). Experimental pressures(b) and FeO i durations arecouple shown in (Van the figures. diffusion der Laan et al. 1994). Exper durations are shown in the figures.

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siderably more scatter than the example of Figure 9a. Within uncertainties, the data can also be fit well by an error function (solid curve), allowing the extraction of an effective binary diffusivity. Points analyzed by EMPA can be seen optically after the analyses, and can be used to confirm the spots analyzed (e.g., distance from the surface or an interface) and the distance between points. The application of EMPA in diffusion studies may be affected by factors such as the phase boundary fluorescence effect (Zhao 1998; Reed 2005). As illustrated in Figure 4, the electron beam-sample interaction volume is limited to a few micrometers in depth. However, the fluorescence volume, in which secondary X-ray fluorescence occurs, is much larger than the interaction volume in which the primary X-rays are generated. When the interaction volume or fluorescence volume crosses a phase boundary, fluorescence corrections of the matrix effects for quantitative analysis may not be correct. For example, in a common mafic and ultramafic assemblage with chromite-olivine pairs (olivine is a common inclusion in chromite hosts), analyses of Cr2O3 concentrations in tiny olivine inclusions in the chromite host could be affected by secondary fluorescence (Zhao 1998). When the electron beam bombards the olivine, the primary Fe Ka X-ray generated from the olivine will penetrate into a much larger fluorescence volume and may cross the olivine-chromite grain boundary. Since the chromite host has a high Cr2O3 concentration and the chromium’s critical ionization energy, 5.989 keV, is slightly lower than the energy of Fe Ka X-rays (6.403 keV) the Cr in the chromite will be excited by the primary Fe Ka X-rays, and secondary Cr Ka X-rays will be generated and detected. Therefore, the secondary fluorescence of Cr Ka in chromite by the Fe Ka from olivine will dramatically increase the apparent Cr2O3 concentration of the olivine inclusion, making a correction of Cr2O3 concentration in the olivine necessary to account for this effect. Figure 10 shows that the apparent increase in Cr2O3 concentration from the center of the olivine inclusion to the olivine-chromite boundary, illustrating that the apparent Cr2O3 concentration could be artificially increased even at the center of an olivine inclusion of 50 mm in size. The secondary fluorescence effect was confirmed when higher apparent Cr2O3

Figure  10. Variations of Cr content across an olivine inclusion in a chromite host. The samples were analyzed at two different accelerating voltages — 10 and 30 kV. The apparent increase in Cr concentration toward the chromite-olivine boundary due to the effect secondary fluorescence, where X-rays Fig.10. Variations of Criscontent across an of olivine inclusion in a chromite host. The generated from Fe in the olivine X-raysatintwo the different adjacent accelerating chromite. The effect becomes more with samplesexcite wereCr analyzed voltages - 10 and 30 kV.pronounced The increasing accelerating when nearer thetoward olivine-chromite boundaryboundary (Zhao 1998). apparentvoltage increaseand in Cr concentration the chromite-olivine is due to the effect of secondary fluorescence, where X-rays generated from Fe in the olivine excite Cr X-rays in the adjacent chromite. The effect becomes more pronounced with increasing accelerating voltage and when nearer the olivine-chromite boundary (Zhao,

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concentrations were obtained for same sample analyzed at 30 kV, a higher energy where the effects of secondary fluorescence are more pronounced. The effects of secondary fluorescence became more evident when the olivine was analyzed after removal of the chromite host, which resulted in Cr2O3 concentrations for the olivine inclusion of ~0.10±0.02 wt%. Therefore, the phase boundary fluorescence effect must be considered when small mineral inclusions are analyzed by electron microprobe, as well as when performing analyses near interfaces where there are compositional differences and the potential for secondary excitation.

Summary Electron microprobes can analyze almost all elements (from Be to U) with a spatial resolution of a few micrometers and a typical detection limit of 100 ppm (with detection limits of 10 ppm achievable for some elements under certain circumstances). They are the most widely used instrument for major and minor element analyses in the geological sciences, but are unable to analyze most elements at trace levels. Furthermore, they are unable to detect the lightest elements (H, He and Li). For example, hydrous components in minerals cannot be analyzed using EMPA. There are also overlaps of X-ray lines for some elements in WDS detection, making quantitative analysis difficult due to these interfering signals. In addition, EMPA cannot distinguish between the different valence states of an element, as in the case of Fe3+/Fe2.

SECONDARY ION MASS SPECTROMETRY (SIMS) Secondary ion mass spectrometry (SIMS), also known as the ion microprobe, is a powerful microanalytical tool. It has found considerable application in characterizing the concentration and distribution of dopants in semiconductors, and is a crucial instrument in quality control (and research and product development) in this industry. The capabilities that make it essential in this context (high sensitivity and depth resolution) also make it well-suited to the characterization of diffusion profiles in geological materials.

Basic principles of SIMS In SIMS, atoms from a sample are ejected, or “sputtered” by the impact of energetic (several keV) primary ions on the sample surface. A schematic of this process (based on the Cameca SIMS design; http://www.cameca.fr/) is shown in Figure 11. The analyst has a choice of primary ion species and the polarity of the secondary ion beam; different primary beams enhance the ion signal for different elements. For example, elements with high electron affinities (e.g., F, S, As) show the best sensitivity when negative secondary ions are detected and an electropositive primary beam (such as Cs+) is used, while elements with low ionization potentials (e.g., Li, Be, Y) yield low levels of detection when positive secondary ions are sputtered by an electronegative primary species such as oxygen. The schematic (Fig.  11) shows the effect of changing the primary ions from 16O− (most commonly used in geological studies) to 16O2+ (most commonly used in the semiconductor industry). Most sputtered atoms are ejected as neutral species, some as molecular ions, some secondary electrons are generated, but some atoms in the sample form elemental ions in the sputtering process. Placing a high voltage on the sample accelerates sputtered ions away from the sample, and into a mass spectrometer where the ions are separated by their mass:charge ratio and then counted. Changing the primary beam polarity has several effects on the analysis, but the two most obvious from Figure 11 are the impact angle and the impact energy. Tutorials on SIMS are available on the web, with one from Evans Analytical Group (http://www.eaglabs. com/training/tutorials/sims_theory_tutorial/) being commonly used. Sputtered neutral mass spectrometry (SNMS) is a related technique that ionizes the neutral species after ejection from the sample using electron impacts, ion impacts, or lasers. While not discussed in this chapter, the interested reader is referred to Williams and Streit (1986) and Vad et al. (2009).

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a)

b) Figure 11. Schematic diagram of the sputtering process inprocess SIMS. in (a)SIMS. Sample(a) held at +4500 V at and+4500V sputtered Figure 11. Schematic diagram of the sputtering Sample held and 16 − with asputtered primary beam O ions accelerated fromaccelerated a duoplasmatron at −12.5 kV. held The primary with aofprimary beam of 16O- ions from aheld duoplasmatron at -12.5 ions kV. strike The the sample an angle fromatthe (b) Same as (a) normal. except that primary primaryations strikeof the~25° sample an sample angle ofnormal. ~25 from the sample (b) the Same as (a) beam is 16O2+ with an impact angle ~39° from the sample normal. Sputtered particles are indicated in except that the primary beam is 16O2+ with an impact angle ~39 from the sample normal. light shades. The sample is ~5 mm from the grounded plate in front of it. This creates a strong extraction Sputtered particles areinindicated in light shades. The sample is also ~5 mm fromthe theprimary grounded plate field (~1 V/mm) that results efficient collection of secondary ions but deflects beam as in front ofthis it. This creates a strong extraction field (~1V/µm) that results in steering efficient(not collection it encounters voltage gradient, requiring substantial changes in primary beam shown) of as secondary ionsisbut also deflects the primary beam as it encounters this voltage gradient, the primary polarity changed.

requiring substantial changes in primary beam steering (not shown) as the primary polarity is changed.

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Using SIMS to measure diffusion profiles There are two approaches to using SIMS for diffusion studies: 1) line scans and 2) depth profiling. Line scans (or “step” scans) are used in those cases where the diffusion length is many times larger than the diameter of the primary ion beam. In such cases, chemical, isotopic, and tracer diffusion can be determined. Depth profiles are used most often when the diffusion length is less than a few micrometers. Line scans. One example of a diffusion experiment characterized by a line scan is shown in Figure 12. For this study, a platinum capsule was filled with rhyolite powder and melted at 1 atmosphere to produce a nominally dry cylinder of glass. The cylinder was then sealed in a larger capsule filled with D2O and taken to 750 bars pressure, 850 °C in a cold seal pressure vessel (where the pressurizing medium was also D2O). Deuterium diffused into the molten rhyolite cylinder for 12 hours, after which the experiment was quenched. The quenched glass cylinder was embedded in epoxy, cut longitudinally, and polished (see Fig. 12). Step scans were conducted using SIMS to determine how far the deuterium had diffused into the silicate melt, and the resulting analytical craters are shown on Figure  12 while Figure  13 displays the analytical results). The different spacings between craters in earlier line scans represent attempts to examine the zoning of different elements and/or to test different analytical parameters. For example, while the experiment was designed to measure the diffusion of D2O (and compare to related experiments on H2O diffusion) it was realized in the mid-1990s that noble metal capsules are often contaminated with Li and B, so that these samples might show diffusion profiles for other light elements. Subsequent analyses revealed high boron only at the epoxy/glass interface of this 12-hour duration experiment, but the diffusion of Li is pronounced, and reaches the baseline Li concentration in the starting rhyolite at about the same distance as D2O (Fig. 13; Hervig, unpublished data).

Figure 12. Figure Polished ofsample rhyolite showing several step scans light image). 12.sample Polished ofglass rhyolite glass showing several stepacross scans glass across(reflected glass (reflected light mounting image). Note the epoxy mounting medium the left. The liquid at 750 bars was Note the epoxy medium on the left. The liquidon meniscus at 750 barsmeniscus was convex against the D2O most recent step The scansscan showatup dark craters. The scan thesteps convex against thescans D2O fluid. fluid. The most recent step show The up as dark craters. theasbottom represents 25 at mm represents 25 um steps ~1400 microns (~20 µmscans beam(illustrating diameter). Most of the olderof step ~1400 mm bottom long (~20 mm beam diameter). Most of thelong older step a wide range step scans (illustrating a wide range of step sizes and primary diameters as specified by the sizes and primary beam diameters as specified by the operator) showbeam brighter craters because the sample operator) showprior brighter craters sample re-coated with gold prior to the was re-coated with gold to the mostbecause recent the work scanswas (sample from Stanton 1989). Allmost craters were recent work beam scans (sample made using16 a primary of 16O−.from Stanton, 1989). All craters were made using a primary beam of O.

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Figure 13. Variation in D2O and Li contents as determined by the step scan shown in Figure 2 above. The by the step scan shown in Fig. 2 13.converted Variation D2O and Li contents as determined 2 Figure 2 + H+ signal was to Din 2O (wt%) by comparison with the H signals obtained on bulk-analyzed 2 + 2 + H signal waspresence converted D2atOdifferent (wt.%)pressures by comparison withThethe above. Thesynthesized rhyolitic glasses in the of D2Oto fluid (Stanton 1989). Li H signals ob concentration in the starting glass was ~30 ppm (Hervig, unpublished data). on bulk-analyzed rhyolitic glasses synthesized in the presence of D O fluid at different pre 2

(Stanton, 1989). The Li concentration in the starting glass was ~30 ppm (Hervig, unpublish The results in Figure 13 illustrate the capability of SIMS in that both volatile and trace data).

light elements can be determined with a lateral resolution typically around 5-30 mm. The lateral resolution limitations are dominated by the performance of the primary ion source (how small one can focus the beam) and the amount of sample one needs to consume to obtain statistically meaningful analyses on a reasonable time scale (i.e., the useful ion yield; Hervig et al. 2006). The results are also interesting from a diffusion standpoint in that D2O shows the well known concentration-dependent, concave-down diffusion profile (Behrens et al. 2007; Ni and Zhang 2008) while the lithium profile is not concentration-dependent, and concave up.

SIMS line scans can also be used for isotopic diffusion (Lesher et al. 1996; Richter et al. 2003). An example from Lesher et al. (1996) is displayed in Figure 14. These experiments required the synthesis of basaltic glasses doped with 30Si and 18O and creation of a diffusion couple by placing these nominally anhydrous samples against basaltic glass with normal isotopic abundances. After treating the couple at high pressure and temperature, the experiment was quenched, and the resulting isotopic gradient was characterized using a primary beam of Cs+ and detection of negative secondary ions. As the results show, there are gaps in the line scan. These are caused by the development of cracks in the glass quenched from high pressure, and point out the need for flat surfaces when making SIMS analyses. If the sample surface is variably tilted, measured ion intensities of different secondary species can change significantly (these outliers were removed from Fig. 14). Careful optical study of run products prior to analysis can reveal areas likely to suffer from such problems. In the case of the data in Figure 14, the results were used to document similar diffusion of silicon and oxygen in highpressure silicate melts (Lesher et al. 1996). SIMS line scans of the run products from Richter et al. (2003) were used to document gradients in Li content and the Li isotope ratio in a chemical diffusion experiment.

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Figure  14. Isotopic diffusion of silicon and oxygen in basaltic liquids at 1450 °C, 1 GPa (35 minute Figure 14.scans Isotopic diffusion and basaltic liquids at 1450 � C, 1 GPa duration). The line used 50 mm steps, of butsilicon overlap of theoxygen beam on in numerous cracks that developed in duration). scans usedanalyses. 50 µmThe steps, thepublished beam on the run minute product during quenchThe led toline some unusable figurebut wasoverlap redrawn of from datanumerous (Lesherthat et al.developed 1996). in the run product during quench led to some unusable analyses. The figu

redrawn from published data (Lesher et al., 1996).

Limitations of SIMS line scans. Despite the advantages of using SIMS in line-scan mode to measure diffusion profiles, there are prices to pay: 1) analyses take a long time, 2) calibrations may not be linear, 3) molecular ion interferences may be difficult to resolve and may change along the profile, and 4) the diffusion length scale should be much longer than the lateral resolution as defined by the diameter of the primary ion beam. Analysis duration. The time required to collect a measurement of sufficient precision at each analysis point depends on several parameters. One important value is the sputter yield, S, defined as the atoms ejected from the sample surface/primary ion impact. The sputter yield changes as a function of the target density, primary beam species, and primary beam impact energy. When an analysis is initiated, the addition of the primary beam to the sample changes the chemistry of the target. This leads to significant changes in the observed secondary ion intensities for a few minutes. For each analysis in a line scan, waiting until the ion signals of interest reach steady-state conditions (known as the “pre-sputter” time) can take as few as 1 or as many as 10 minutes (depending on primary species and intensity). For a ~60-step profile such as shown in Figure 13, this would translate into a total analysis time of ~10 hours or more, depending on how long the analyst sets the actual data collection portion within each crater. Linearity of secondary ion signal with concentration. Yields of ions are generally linear (i.e., ion intensity increases with element abundance) in constant matrices (where the species of interest represents a dilute component) making measurement of concentration profiles relatively straightforward in many applications of SIMS in diffusion studies. However, there are examples where this is not true. For example, major elements (Steele et al. 1981), hydrogen in high-H glasses (Hauri et al. 2002; Tenner et al. 2009), and Li isotope ratios in olivine (Bell et al. 2009) can show non-linear effects. These are typically related to matrix effects (i.e., major element variations influencing the ionization probability of other elements, which

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can also be related to changes in the way the primary beam interacts with the sample). This problem can be avoided if experiments are designed to examine diffusion of multiple elements at trace concentrations. Interfering molecular ions. As indicated in Figure  11, the sputtered flux includes molecular as well as elemental ions, and some molecular ions may have the same mass/charge ratio as the diffusing species. For many potential diffusion experiments, this might not pose a problem. For example, if one wanted to measure the isotope diffusion of 40Ca vs. 44Ca across two otherwise identical basaltic glass samples, the most important interferences on the calcium isotopes would be 24Mg16O and 28Si16O, at the same nominal mass/charge ratios, respectively. If the Si and Mg contents were constant across the diffusion couple, these molecular species would add a constant amount to the ion signals for both calcium isotopes throughout the profile, allowing accurate diffusion coefficients to be derived in spite of the background signal. In contrast, if the Mg or Si concentration were also changing across the sample, it would be critical to eliminate these molecular ions from the mass spectrum before interpreting the SIMS analyses in the context of calcium diffusion. The analyst has two choices for separating molecular ions from the elemental ion signal: 1) high mass resolution or 2) energy filtering. The former solution takes advantage of the fact that there is a difference in mass between an isotope of interest and a molecular ion at the same nominal mass/charge ratio. In the case of 40Ca+ (mass 39.962 u) vs. 24Mg16O+ (mass 39.979 u), the mass difference is 0.017 u (the exact masses of the isotopes, based on defining 12C =12.000, can be found in charts of the nuclides, textbooks, and web sites). Magnetic-sector secondary ion mass spectrometers allow the operator to close the entrance slits to the mass spectrometer to resolve these two ion signals at a nominal mass/charge ratio of 40. The mass resolution required is calculated as M/DM; in the case of the present example, 40/0.017 yields a mass resolving power requirement of ~2300. Operating the SIMS at high mass resolving powers unambiguously eliminates the molecular ion, but operation at these conditions also decreases signal intensity and decreases the width of the peak of interest, placing high demands on the stability of the secondary magnet. Planning any diffusion experiment that will utilize SIMS for characterization requires the experimentalist to carefully consider the potential molecular ion interferences and how to separate these ions from the species of interest. In some cases, energy filtering of the secondary ion beam (Schauer and Williams 1990; Shimizu et al. 1978) will reduce molecular ion intensities to negligible levels while still leaving sufficient signal from the elemental ion to allow high-quality analyses to be obtained. This approach selects only those secondary ions that are sputtered from the sample with energies higher than can be accounted for by the potential placed on the sample. That is, some ions are ejected via the energetic collisions (initiated by primary ion impact) and have energies tens to hundreds of eV higher than the ions that are accelerated only by the potential placed on the sample. An example of an energy spectrum is shown in Figure 15. Energy filtering is generally effective at removing molecular ions composed of 3 or more atoms (e.g., 28Si216O2+ on 88Sr+) but does not eliminate all molecular ions from the mass spectrum. Most molecules composed of two atoms are not removed by energy filtering, such as the 28Si2 molecule in Figure  15 (which would overlap with 56Fe), 24Mg16O and 28Si16O interfering with Ca isotopes, and the light rare earth element oxide ions which interfere with mid- to heavy rare earth elemental ions (Zinner and Crozaz 1986). However, if very high-energy secondary ions are examined (Eiler et al. 1997; Hervig et al. 1989, 1992; Lesher et al. 1996; Schauer and Williams 1990) even these molecular ions can be removed, but at great expense to the signal of the elemental ion. Regardless, care must be taken to ensure that there is an appropriate approach for elimination of these molecular ions when designing diffusion experiments meant for SIMS characterization. The examples shown for characterizing diffusion using SIMS line scans are only for silicate melts. As a general rule, diffusivities of elements in crystalline materials are orders of magnitude

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30 + 15. Secondary energy spectra showing the variation in intensity Figure Figure 15. Secondary ion energy ion spectra showing the variation in intensity of elemental Si ions of elem +a function 28 of + + + 28 28 28 compared to Si and Si ions as their initial kinetic energy. In this case, +10000 V was energy. 2 3 ions compared to Si2 and Si3 ions as a function of their initial kinetic applied to a sample of quartz, and ions with a range of ~40 eV were allowed into the mass spectrometer +10000V applied tobya the sample of The quartz, and ionswas with a range of V~40 (the energy range is awas variable selected operator). sample voltage varied from 1050 (far eV were mass spectrometer (the energy range is initial a variable selected by the operator). Th left sidethe of the figure) to ~9900 V (right side). Ions with positive kinetic energy represent those ions ejectedvoltage with high was relative energies from 1050V energetic (far collisions the near-surface of theto sample. The signal varied from left inside of the figure) ~9900V (right side). for the elemental ion decreases with initial kinetic energy, but remains relatively intense. The complex + positive kinetic energy represent those ions ejected with high relative energ molecular ion, 28Siinitial 3 shows a decrease in intensity by >100× when high-energy ions are selected. In this case, these collisionscollisions also cause the ion to breakof apart. that molecular ions composed energetic in molecular the near-surface theNote sample. The signal for the of elemental only two atoms, such as the dimer 28Si2+, are not efficiently removed by this approach. Energy filtering with initial kinetic energy, but remains relatively intense. The complex molecular cannot completely remove multiply-charged species, but becomes more effective as the number of atoms shows aion decrease >100x when areforselected. in the molecular increases. in Theintensity box on theby figure represents the high-energy range of energiesions selected typical In thi analyses using energyalso filtering. Thethe zeromolecular point on the figure wasbreak definedapart. by the Note center of mass of the 30Si+ ions co collisions cause ion to that molecular energy spectrum. + 28

two atoms, such as the dimer Si2 , are not efficiently removed by this approach. filtering cannot completely remove multiply-charged species, but becomes more e smaller than in melts, and so the scale of diffusion is not likely to be significantly larger than the number of atoms in the molecular ion increases. The box on the figure represents diameter of the primary beam. In this case, it may be necessary to abandon line scans in favor of energies selected forhowever, typical that analyses using energy filtering. The by zero point on the depth profiling analyses. Note, one SIMS (NanoSIMS manufactured Cameca) Si+ this energy spectrum. by the center massdiameters, of the 30and providesdefined sub-micrometer primaryofbeam instrument may be suitable for the characterization of short diffusion profiles in line scan mode.

Depth profile analyses Depth profile analyses take advantage of the fact that SIMS samples atoms derived from the surface of the target. As the analysis progresses, atoms from deeper and deeper in the sample are collected, providing a measure of changing chemistry with depth. The typical analysis involves focusing a primary beam of ions to a point, where those primary beams may be Cs+, O+, O−, O2+, or O2−. Alternative beams are more rarely employed (Hervig et al. 1989, 2004; Hervig and Williams 1986). The primary beam is swept, or rastered, over the sample surface, generally in a square pattern from ~100 to 250 mm on a side, although the primary beam can also be shaped by an aperture to make a nearly cylindrical crater without rastering, as shown in Figure 16 (Clement and Compston 1990; Genareau et al. 2007). The process of

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Figure  16. Sputtered crater in surface of volcanic plagioclase 40 mm deep by 70 mm in diameter obtained on the 3f SIMS in ~6 hours using “aperture illumination,” the O2+ primary beam, and the normal-incidence electron gun for charge neutralization (analysis and image by Dr. Kimberly Genareau, ASU). Note volcanic glass adhering to crystal surface, and the steps in the bottom of the crater. The overlap between these steps and the region of the crater floor allowed into the mass spectrometer will define the depth resolution of this profile.

sputtering sample crater erodesinthe surface a rate depending on40 theμm sputter (defined Figure 16. the Sputtered surface ofatvolcanic plagioclase deepyield by 70S μm in diameter + above) together the area of hours the crater and“aperture the primary current used.the ToOmake certainbeam, that and obtained on the with 3f SIMS in ~6 using illumination”, 2 primary the normal-incidence data from these depth profiles used for the accurate(analysis characterization of diffusion, the electron guncan forbecharge neutralization and image by Dr. Kimberly several requirements must be fulfilled: the floor to of crystal the crater must and be flat, crater depth Genareau, ASU). Note volcanic glass adhering surface, the the steps in the bottom of must be measured to calibrate thethese rate of erosion, anyregion ions originating from theallowed walls of into the the the crater. The overlap between steps and the of the crater floor crater must be eliminated, the properties of the secondary ion beam should be tailored to the mass spectrometer will define the depth resolution of this profile. problem, and the effects of sputtering on the diffusion profile itself must be considered.

Crater depth measurement. After the analysis, the total crater depth can be measured using stylus profilometry or interferometry. An example of stylus profilometry on a sputtered crater produced by a rastered primary ion beam is shown in Figure 17. The profile shows that the crater floor is flat (unlike the SEM image of the crater in Fig. 16) with a depth of 2 mm. It is difficult to

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measure the crater depth during the analysis (although the Cameca Wf SIMS instrument offers an option for in situ crater depth measurements using laser interferometry). The analyst must assume that the erosion rate is constant over the duration of the analysis, which is not a bad assumption when the crater is several hundred nm deep, the sample is uniform in major element chemistry, and the primary beam is stable. In fact, since many geological samples are insulators, it is necessary to coat the sample surface with a thin film of carbon or gold, and the sputtering rate of these films is much different than typical silicates (Au shows a high sputter yield, while carbon has a low sputter yield). Because the Au or C coats are ~20-40 nm thick and the total crater depth is several hundred nm, the error in the sputtering rate (e.g., nm/second) is small (in addition, a gold coat can be easily removed prior to measurement of the crater depth; e.g., Van Orman et al. 2001). However, if the major element chemistry of the sample varies (for example, if diffusion of major elements occurs in the profile), the assumption of linear erosion rates must be tested (at the minimum, by determining the sputter yield on a range of homogeneous targets with appropriately varying major element chemistry).

Contribution from crater walls. During a depth profile, the sides (as well as the bottom) of a crater are sputtered by primary ions and so secondary ions also originate from the walls (which represent all depths in the crater). It is important to allow only those ions derived from the flat-bottomed floor of the crater into the mass spectrometer over the analysis time (= depth). The Cameca design uses stigmatic ion optics to form an image of the sample surface beyond the entrance slits of the mass spectrometer. By placing a variably-sized “field” aperture at this plane, ions from the edges of the crater floor and the crater walls can be eliminated. For those SIMS instruments that are not ion microscopes, electronic gating of the secondary ion detector can be used to eliminate ions sputtered when the primary beam is near the crater walls (note that electronic gating is not appropriate for craters formed by aperture illumination; Fig. 16). A test of whether the selected analysis condition permits crater walls to contribute a signal can be performed by sputtering an ion-implanted sample (Williams 1983). Such samples have approximately Gaussian distributions of the implanted species with depth. 105 The baseline signal should be ap4 proximately a factor of 10 below 11 + the peak intensity (this ratio is B 104 known as the “dynamic range”). Higher background intensity for the Cameca instruments most like103 ly suggests either an instrumental Counts/s “memory” of the isotope of inter102 est, but could also indicate that there is a contribution from the crater walls (see Fig. 18 for an exam101 ple of a good depth profile through an ion-implanted silicon wafer). 100 The above “memory effect” is a 0 1000 2000 3000 4000 5000 6000 consequence of the design of the Depth (Å) high-transmission Cameca SIMS. Figure 18. Depth profile into silicon wafer implanted with 1014 atoms B/cm2 (O2+ pri If a sample rich in a particular elpositive ions profile detected). The silicon background signal at ~6000 Åwith depth correspo Figure secondary 18. Depth into wafer implanted ement is analyzed, some sputtered 2 indicates ppm This lowB/cm signal that the depth profile was conducted in a manner li 1014B.atoms (O2+ primary beam, positive secondary atoms of the element may be deexcluding the contribution of boron from memory or the Å walls of the sputtere ions detected). The background signal effects at ~6000 depth (Hervig, 1996). posited on the electrically groundcorresponds to ~0.1 ppm B. This low signal indicates that the depth profile was conducted in a manner limiting or excluding ed plate in front of the sample (see the contribution of boron from memory effects or the walls of Fig.  11). Later analyses of other the sputtered crater (Hervig 1996). samples exploring the diffusion of

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this same element may result in these deposits being “sputter-cleaned” from the plate and potentially landing on the floor of the crater in the new sample. Subsequent primary impacts may sputter these atoms as ions, showing up as a background signal derived from a previous sample. In addition, background signal may also be encountered if molecular ions at the same mass/ charge are present. The latter topic was covered above under “Interfering molecular ions”. Selection of secondary species. The yields of positive or negative ions during sputtering scales exponentially with the ionization potential or electron affinity of the element (Williams 1983). As such, the sensitivity of SIMS varies over a wide range (Hervig et al. 2006) and the analyst can select the polarity to maximize the signal for the study of trace element diffusion. Yields of secondary ions can be strongly influenced by variations in the bulk composition of the solid; such “matrix effects” were briefly described above. A typical diffusion experiment should not be strongly influenced by matrix effects, as the diffusing species is ideally contained within an unchanging matrix. However, it is common to apply a thin layer very rich in this species to the surface of a phase, and this represents a different composition than the phase of interest. Thus the secondary ion signals may fluctuate dramatically during the beginning of sputtering. These transient variations can make locating the position of the original sample surface in SIMS depth profiles difficult. This is discussed in more detail later in this volume by Ganguly (2010). For some studies, it may not be necessary to maximize the sensitivity for a particular diffusing species if interfering molecular ions are minimal and the gradient is large. In such cases, the analyst may consider conducting the profile using simple, reproducible conditions instead of maximizing sensitivity (for example, operating at the conditions for highest sensitivity may lead to difficulties in controlling sample charging during the depth profile, as discussed below). Effect of sputtering. When considering the effect of the sputtering process on a diffusion profile, there are two important items to consider. One is that the ions sputtered from the sample are dominantly derived from the top two monolayers of the sample surface (Dumke et al. 1983). The initial impression one gets from this observation is that SIMS profiles should have resolution on a similar scale. However, the other item of note is that energetic primary ions will penetrate beneath the surface and thus cause a collision cascade (multiple collisions of the atoms in the sample). The primary beam is added to the initial chemistry of the surface with the result being a more-or-less uniformly mixed chemistry over a depth corresponding to the projected range of the primary ions (Nastasi and Mayer 1994), where the projected range represents the most common depth achieved by primary ions in a sputtered sample (highest concentration of the implanted primary species). The range is a function of the impact energy and mass of the primary beam and the mass of the target. The range shows large variations; consider the two situations shown in Figure 11. In the first case, a primary beam of 16 − O , accelerated to −12.5kV in the duoplasmatron primary ion source, is attracted toward the sample at +4.5 kV and has a total impact energy of 17 keV. In contrast, Figure 11b portrays a beam of O2+, initially accelerated to +12.5 keV in the duoplasmatron but which is decelerated as it approaches the sample, striking with a total energy of 8 keV. In addition, the molecular beam will break into two separate oxygen projectiles upon impact, each having 4 keV energy. The decrease in impact energy translates into a much lower projected range (average depth of penetration) for O2+ compared to O−. Note that the angle of impact is different (Fig. 11), so the normal component of the impact energy is smaller for the positive primary species. The result is two-fold: the range for an O− impact is deeper, so that the sample atoms get mixed to a greater depth than when using O2+, and, because the energy of the molecular O2+ beam is restricted to near-surface interactions (and only atoms from the top two atomic layers are sputtered), more atoms are ejected per primary ion impact (higher sputter yield). The conditions presented in Figure 11b result in a sputter yield for Si metal ~4× greater than in Figure 11a (Sobers et al. 2004). The implications of these changes in the instrument parameters on ion beam mixing and depth resolution are shown in Figure 19. Other modifications to the parameters are possible:

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− iron ions Figure 19. into Depth Sancomparing Carlos olivine comparing intensity using Figure 19. Depth profile Sanprofile Carlos into olivine the intensity of ironthe ions using Oof and O2+ + primary Only the first 200 nm ofinitial the profiles arerepresent shown. sputtering The initial 20-40 n O2the primary beams.and Only first 200beams. nm of the profiles are shown. The 20-40 nm through the overlying gold coat. Thethrough secondary signal for ironcoat. appears in theion profile when represent sputtering theion overlying gold Theearlier secondary signal for iron app O− is used because thisinprimary species penetrates deeper into thethis sample and, through beam mixing, earlier the profile when O- is used because primary speciesion penetrates deeper into th churns iron (and other and, elements in theion olivine) the surface where canother be sputtered and sample through beamtomixing, churns ironthey (and elements indetected. the olivine) to th In comparison, the smaller projected range of O2+ mixes atoms at a shallower level, improving the depth surface where they can be sputtered and detected. In comparison, the smaller projected ran 54 resolution of the profile as revealed by the Fe signal appearing at a greater depth. The spacing between + + a shallower level, improving the depth resolution of the profile as revea mixes atoms at O 2 data points is ~3× greater when O2 is used because the sputter yield of the molecular primary beam is ~3× 54 signal appearing a greater depth. The spacing between data larger than for by O− the on thisFe phase. Note the largerattransient peak for iron between 30-40 nm when O−points is used.is ~3X gre + Varying initialwhen signalsOfor diffusing) elements is aofcommon observation in depth profiles of larger tha used (and because the sputter yield the molecular primary beam is ~3X 2 ismatrix silicates using O an- O species (Ganguly et al.transient 1998; Vanpeak Orman al. 2001; Ito and Ganguly 2006).O- is used on− primary this phase. Note the larger foretiron between 30-40 nm when Using the O2+ Varying beam requires auxiliary gun for charge neutralization al. 2007) in dept initialansignals forelectron matrix (and diffusing) elements is (Genareau a commonetobservation and adds significant complexity to the depth profiling analysis. It also delivers the best depth resolution. profiles of silicates using an O- primary species (Ganguly et al., 1998; Ito and Ganguly, 20 However, most diffusion experiments can be designed so+ that the most important part of the profile is al., 2001). Using the Ostraightforward electron gun for charge 2 beam requires observed belowVan the Orman transientetsputtering region, allowing usean of auxiliary the O− species.

neutralization (Genareau et al., 2007) and adds significant complexity to the depth profiling analysis. It also delivers the best depth resolution. However, most diffusion experiments ca designed so thatcan thebe most important part12.5 of the profile is observed below the transient sputt the primary beam potential decreased from kV, and in some SIMS instruments, region, can allowing of control the O- species. the sample potential also straightforward be decreased touse help the impact energy (and hence the penetration depth of the primary beam) and maximize the depth resolution. Regardless of the variations in instrumental set-up, most diffusion experiments can be engineered to make the diffusion profile long compared to the short-range artifacts of sputtering.

Sample charging. A significant number of geologic phases are bulk insulators, and so addressing the problem of sample charging is very important. If the degree of charging varies throughout the depth profile, the secondary ion signals will vary, and different elements may be affected differently. Charge compensation is most simply addressed when a primary beam of 16O− is used with the detection of positive secondary ions. In this case, accumulation of negative charge in the crater is accommodated because of the abundant secondary electrons produced during sputtering. These electrons can “hop” to the conducting gold or carbon coat surrounding the crater to drain excess negative charge. If it is necessary to use a positive primary beam, the analyst will need to use an auxiliary electron gun to alleviate positive charge build-up in the crater. This increases the complexity of the analysis, but useful depth profiles can be obtained (Genareau et al. 2009).

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Ion implantation and SIMS As discussed in this volume’s chapter on experimental methods by Watson and Dohmen (2010), ion implantation represents one possible experimental approach to introducing diffusants into materials for the determination of diffusion coefficients. Figure  18 shows an example of how SIMS can be used to characterize the distribution of boron in a silicon wafer after implantation. If this sample is heated, the boron will diffuse, and subsequent SIMS depth profiles in aliquots of the implanted wafer treated for different times and temperatures could be used to quantify the diffusion. While ion implantation has been coupled with analyses of profiles by RBS, NRA and ERD for diffusion studies in minerals (e.g., Cherniak et al. 1991, 2009; Ouchani et al. 1998), this type of characterization (i.e., ion implantation with SIMS analyses) has mostly been limited to the semiconductor industry, where it is important to activate the electrical properties of dopants by annealing and to know how these trace elements (e.g., Be, B, and S) have migrated during the annealing step (e.g., Tsai et al. 1979; Oberstar et al. 1982; Wilson 1984). These studies demonstrate that in general, the damage to the crystal lattice from the implantation step results in more rapid diffusion of the implanted species to the surface than into the bulk. In the case of geological samples, if the depth of the implant can be made deep enough to avoid transient effects observed during the first few nm of sputtering, and damage to crystal lattices (in the case of diffusion in minerals) can be annealed faster than the implanted species can diffuse, SIMS could be used to characterize subsequent diffusion.

Summary comments Secondary ion mass spectrometry has been an effective analytical tool for characterizing diffusion profiles. Active dialog between experimenter and SIMS laboratory is essential for designing experiments to make best use of the unique capabilities of this tool.

Laser ablation ICP-MS (LA ICP-MS) Laser ablation ICP-MS (LA ICP-MS) is a microanalytical technique for the determination of trace elements in solid materials. The sample to be analyzed is placed in a sample chamber with a lid transparent to UV light, and a pulsed laser beam is used to ablate a small quantity of sample material. The fine particles produced in the ablation are transported into the Ar plasma of an inductively coupled plasma mass spectrometer (ICP-MS) instrument by a stream of carrier gas (typically Ar and/or He), where they are ionized and then mass-analyzed. In the ablation process, the laser beam leaves behind an ablation crater (typically on the order of a few to tens of mm in diameter) where the analyzed material has been removed. Additional information on ICP-MS instrumentation and application can be found in numerous references (e.g, Montasser 1998; Taylor 2000; Sylvester 2001, 2008; Nelms 2005; Thomas 2008). Lasers were first used with ICP-MS instruments in the 1980s. The initial ablation systems used solid-state ruby lasers (operating at 694 nm, in the infrared region), but these were found to be unsuccessful for applications in trace-element analysis due to poor laser stability, large beam diameters, and low power density, among other factors. Over the next decade, commercial laser ablation systems were developed employing Nd-YAG lasers, which produce IR laser light with a fundamental wavelength of 1064 nm. However, these faced continued limitations because IR laser light does not interact very efficiently with most solids, so further developments in the field explored the use of UV laser light (which couples more efficiently with most materials) for ablation systems given its greater potential for effective use in trace-element analysis. The interaction between UV laser light and most solids tends to involve mostly mechanical breakup of the ablated area, whereas IR laser light may produce a greater degree of sample heating and melting, which can contribute significantly to elemental fractionation and limit capabilities for analysis. More recently, some laboratories have begun to use gas-filled (excimer) lasers operating in the UV region of the electromagnetic spectrum. These lasers, including XeCl (308

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nm), KrF (238 nm), and ArF (193 nm) exhibit better absorption capabilities for UV-transparent materials (e.g, silicates, calcite, fluorite), and possibly less elemental fractionation in ICP-MS than longer wavelength lasers because particles produced in ablation are smaller and easier to volatilize. Since the UV range is the fundamental wavelength for the excimer lasers, there is higher energy transfer than for Nd-YAG lasers, where UV wavelengths are produced by directing laser beams through crystals to quadruple or quintuple their frequency. The less coherent nature of the excimer beam produces better optical homogenization and therefore cleaner, flatter ablation craters, a critical factor in depth profiling. Improvements in laser optics have also provided more controlled, smaller beam spots, important in step scans and depth profiling of materials, the two methods used to obtain diffusion profiles. Laser ablation instruments require an accurate optical system of lenses, prisms and mirrors to direct and focus the laser beam onto the sample. Several parameters are typically adjusted in analyses to optimize for a particular material and the species of interest. These parameters include laser power, laser pulse repetition rate and the number of laser pulses fired in succession. In addition, there is generally a system of apertures of different diameters that can be used to vary the beam diameter and hence the diameter of the ablation crater generated. Although the size of an ablation crater can be made very small, and small size would be advantageous in step scans of diffusion profiles, the usefulness of small volumes of material for quantitative analysis is strongly dependent on the sensitivity of the ICP-MS instrument to the elements of interest, since signal intensities for small amounts of ablated material may be low. In addition, the walls of ablation pits may yield ionic concentrations that differ from the center region because energy interactions are slightly different at the edge of the ablation spot. Further, the physical barrier of the pit edge becomes increasingly significant as ablation proceeds deeper into a sample. Because smaller spot sizes have a larger wall-to-center ratio, a greater fraction of the signal in small spot sizes will be due to contributions from the walls than would be the case for larger spots. In depth profiling of diffusion samples, apertures and rastering may be used to vary the size of the ablated area, with a larger areas removed near the sample surface (typically containing the highest concentration of diffusant) and smaller ablated areas at depth. This helps to avoid contamination at depth in the sample from the ablated material in the upper layers of the sample and from the sides of the ablation crater. Sample chambers for laser ICPMS are typically mounted on a stage that allows the sample to be positioned relative to the laser beam in x-y-z coordinates and move to a particular region of interest or to perform step or area scans. In diffusion measurements, concentration profiles can be measured either through depth profiling or step scans in the direction normal to the interface between the sample and diffusant source. It should be noted that step scans are only practical for relatively fast diffusivities (down to ~1×10−16 m2/sec) given typical sizes of the ablated area, so they are most useful in studies of diffusion in glasses or melts, or for some fast-diffusing species in crystalline materials. In depth profiling, since depth resolutions may range from several tenths to several mm, diffusion profiles will generally need to be on order of several to tens of micrometers in length, thus limiting accessible diffusivities to ~1×10−18 m2/sec. Hence LA-ICPMS measurements using current instrumentation cannot access the very slow diffusivities that may be measurable by depth profiling with RBS or SIMS, and because of larger spot sizes cannot measure step scans on as fine a scale as EMPA or nanoSIMS. In most instances, sample preparation for LA-ICPMS analysis is very simple. Samples used are typically in the form of epoxy mounts or petrographic thin sections similar to those used in electron microprobe analysis. For diffusion studies, samples would generally need to be flat and well-polished to avoid any effects on depth resolution due to surface roughness when depth profiling, and to avoid sampling problems in step scans. While sample ablation cell (the region above the sample where the carrier gas removes the ablated material) design continues to evolve and improve, it is critical that samples be configured in such a manner that they do not create eddies or pockets that could trap or fractionate the analyte, which may

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compromise the ablation cell’s ability to efficiently transport ablated material to the mass analyzer so that an optimal signal can be produced. Standards are required for analytical calibration of LA-ICPMS, with standardization in most cases using various synthetic or natural reference materials of known composition. Materials commonly employed as standards are the set of silicate glasses produced by the US National Institute of Standards and Technology (NIST) containing various trace elements in a range of concentrations, standards from the US Geological Survey, or various other natural or synthetic minerals or glasses of well-determined composition. Appropriate standards will depend on the type of materials and elements to be analyzed, with ideal calibration standards matching the composition of the bulk chemical matrix of the sample (e.g., Koenig 2008). It has been found that for many crystalline silicates and glasses that 193 nm (ArF excimer) laser systems are less matrix-dependent than 213 nm (Nd:YAG) systems, and these are less matrix-dependent than longer-wavelength laser systems. Nonetheless, all LA-ICPMS analyses require calibration, and optimal reference materials that are reasonably close in bulk chemical composition to the sample, well characterized for all species of interest, homogeneous at the required scale of analysis, and available in sufficient quantities to last more than a few analyses. Along with this external standardization, internal standardization (using elements in the sample analyzed) is typically also required in LA-ICPMS to correct raw data for differences in the ablation characteristics among standards and samples and between different elements, as well as to correct for general instrumental drift. Different groups of elements may need to be normalized against different internal standard elements to achieve high quality analyses; this will depend largely on the relative volatility of the elements. It should be noted that the elements used as internal standards must be quantified by an alternate analytical technique (e.g., electron microprobe or SXRF) in both the sample material and the external reference materials. This adds another level of complexity to the process of standardization. Like SIMS and solution ICPMS, LA-ICPMS is subject to various mass spectrometric interferences, including isobaric, molecular and doubly-charged ion interferences. Mass spectrometric interferences are more of a concern for light elements and most of the interferences in ICP-MS systems arise from compounds made from the torch, carrier gas, and gases in the chamber atmosphere. Of secondary concern are interfering compounds generated from the major components of the matrix. Interferences may be minimized or avoided through judicious selection of isotopic species to analyze, and optimization of instrumental operating conditions. It should also be noted that when considering mass interferences in analysis that even state-of-the-art quadrupoles used in ICPMS instruments still cannot match the mass resolution of magnetic sector machines. LA-ICPMS detection limits vary with laser power and the volume of material analyzed. Theoretical detection limits for most elements are typically in the ppb to low ppm range. LA-ICPMS provides better trace element sensitivity than the EPMA or XRF for much of the periodic table, and analyses are generally fast, with detection of the entire periodic table (except for the noble gases, H, N, O and F) at low concentrations possible in less than a minute. Despite the generally excellent trace element sensitivity of LA-ICPMS there are still cases where interferences, poor ionization in the ICP and other problems lead to compromised sensitivity, especially with small ablation spot sizes. While instrumentation has improved significantly and rapidly, questions still remain regarding control of ablation of certain types of materials, mobilization of some species in material beyond the ablated area, and, in the case of depth profiling, the effects on measured concentrations of contamination from pre-ablated material and mixing of components from several depths. Although LA-ICPMS is finding wide application for trace-element analysis in the geosciences (e.g., Sylvester 2001, 2008), it is has not yet been used extensively in diffusion studies of geological materials. Some examples include studies of Ar diffusion in K-feldspar

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(Wartho et al. 1999) using depth profiling for profiles less than 50 mm and step scans for longer profiles, Ar diffusion in quartz (Baxter 2010, this volume) with depth profiling using Nd-YAG (213 nm) and ArF excimer (193 nm) lasers, diffusion of a suite of siderophile elements (Cu, Co, Ni, Ge, Ga, As, Ru, Pd, Pt, Ir, Au) in FeNi metal (Righter et al. 2005) measured with step scans, and diffusion of rare earth elements in olivine and chromite (Spandler et al. 2007) also measured using step-scanning. With advances in instrumentation and refinements of analytical protocols, the application of this technique to investigations of diffusion in geological materials is likely to become increasingly common.

Rutherford Backscattering Spectrometry (RBS) Rutherford Backscattering Spectrometry (RBS) is a method for determining chemical composition and elemental distributions in the outer few micrometers of a material. It is based on elastic collisions from interactions between light energetic ions (typically helium, produced in a small particle accelerator) and nuclei in a sample material. 4He in the energy range of 1 to 4 MeV is the most common beam used, but other ions, including protons, deuterons, 3He, as well as those heavier than He, are also used for backscattering in certain applications. For example, protons can achieve larger depth ranges but at the expense of mass and depth resolution, and heavy ions can achieve high mass resolution, but can probe only shallow depths in a material. General overviews of RBS can be found in Chu et al. (1978) and Leavitt and McIntyre (1995). RBS has been used to investigate diffusion in minerals for over a quarter of a century. The earliest-reported applications include the work by Melcher et al. (1983) in measuring diffusion rates of Xe in forsteritic olivine, and the study of Sneeringer et al. (1984), in which Sr diffusion in diopside was measured and results from RBS, SIMS, and radiotracer methods compared. RBS and Nuclear Reaction Analysis (NRA, to be discussed in the next section), have depth resolutions typically ranging from a few to several tens of nm, thus permitting the measurement of diffusion coefficients down to relatively low temperatures (e.g., < 700 °C) applicable in a wide range of geologic settings, and diffusivities down to ~10−23 m2sec−1, avoiding in many cases the uncertainties of large down-temperature extrapolations. In addition, these methods are essentially “non-destructive” since there is no physical removal of material during the analysis.

Basic principles of RBS In RBS analysis, the energies of ions from scattering events are measured with a detector which is usually positioned at an angle of nearly 180° with respect to the incident beam, so it is referred to as “backscattering.” These collisions can be described by the equations of classical kinematics. The energies of the detected backscattered particles (En) will depend on the masses of the target atoms, related to the incident energy (E0) by KnE0 = En, where Kn is the kinematic factor:

(

)

 M 2 − M 2 sin 2 θ 1/ 2 + M cos θ  n i i  Kn =    Mi + M n  

2

( 4)

Here, q is the laboratory angle through which the ion is scattered with respect to its incident direction, and Mi and Mn are the masses of the incident particle and the target atom from which it is scattered, respectively. It can be clearly seen that backscattered particle energies will be higher for heavier target atoms by simplifying the expression in taking the limiting case of q = 180°:  M − Mi  Kn =  n   M n + Mi 

2

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The probabilities of occurrence of scattering events, or Rutherford scattering cross-sections, can also be determined from first principles. The scattering cross-sections are functions of target and incident particle mass and charge, as well as the scattering angle and the incident particle energy at the time of scattering. The differential scattering cross-section, or crosssection per unit solid angle W, is described by the following:

(

)

1/ 2 2 4  M n2 − Mi2 sin 2 θ + M n cos θ  dσ  Z i Z ne2    =  × 1/ 2 4 2 2 2 dΩ  4 E  M n sin θ M n − Mi sin θ

(

)

2

(6)

Rutherford scattering cross-sections are larger for higher-Z elements (the cross-section varies as Z2), so the technique is more sensitive for high-Z (and larger mass) species. Detection limits can be down to a few tens of (atomic) ppm, but are considerably poorer for light elements. Since scattering cross-sections can be readily quantified and directly relate atomic concentrations to scattered particle yields, it is possible with RBS to perform quantitative elemental analysis without reference to standards. Only a fraction of the ions comprising the ion beam incident on the sample will be backscattered from target atoms, and detected in the solid angle subtended by the detector. The detected spectrum for a particular element will include signal from scattering at the sample surface (which will have energy KnE0, where Kn is the K-factor for the element n, and E0 is the energy of the incident ion [i.e., initial energy of the ion beam]), as well as signal produced by scattering events with this element from greater depth in the material. Those ions not scattered from the surface will lose energy traveling deeper into the sample, primarily through inelastic collisions with the electrons of the target material, at a rate dependent on the material’s density and composition. These lower-energy ions can then scatter from target atoms at depth, losing additional energy as they make their way out of the sample to the detector. If the backscattered particle energies are measured, and the energy loss rates are known or can be determined, information on the depth distribution of this element in the sample can be evaluated as follows:   dE   1   dE   ∆E = [S ] = Kn   +    ∆x   dx in  cos θ   dx out 

( 7)

where (dE/dx)in and (dE/dx)out represent the energy loss per unit distance in the material for the ions going into the sample (before scattering) and out of the sample (after scattering), also referred to as the stopping power, of the material. Stopping powers have been evaluated for ions in all elemental targets based on semi-empirical fitting of experimental data (Ziegler et al. 1985), with stopping powers for compound targets obtained through application of the Bragg Rule, which weights the contribution to the stopping power of each element in the target according to its mole fraction in the material. These values can be calculated for compounds for a range of ions and energies using, for example, the software SRIM-2006 (Ziegler and Biersack 2006; www.srim.org). An example RBS spectrum, from an experiment on Dy diffusion in fluorite, is shown in Figure 20. The RBS spectrum is a superposition of signals from all of the elements present in the outer several micrometers of the sample. Those elements with uniform distribution in the sample will each appear as a “step” (with a parabolic curve upward at lower energies because of the E−2 dependence of scattering cross-section) with their edge (at highest energy) at KnE0. In this spectrum, examples of these are the Ca and F steps for the fluorite matrix. The edge of the step for Ca, the higher mass species, is at higher energy (greater channel number) than the step edge for F. The step height is also smaller for F despite its higher molar abundance than Ca, due to its smaller scattering cross-section. Dy, of considerably greater mass than either of the major matrix elements, is at an even higher energy position, and the signal from this distribution

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Channel Figure 20. An example of an RBS spectrum taken of a sample from a diffusion experiment, in this case measuring Dy diffusion in fluorite. The “steps” toward the lower channel numbers (lower energies) represent contributions from the major elements (Ca and F). The energies of the step rises are proportional to the elemental masses, and step heights depend on Rutherford scattering cross sections, proportional to the square of the atomic number. The contribution from Dy (also shown in the inset figure) indicates a nonuniform distribution of this element, a result of it diffusing a few hundred nm into the fluorite. See text for additional details.

(which only extends a few hundred nm into the material) is well-separated from the RBS spectra constituents due to the major elements. The Dy peak maximum is about one-fifth of the nearsurface height of the Ca step, but its actual concentration is much lower, as the scattering crosssection for Dy is nearly 11 times that of Ca. Elements with non-uniform distribution, like Dy in this case, will have their leading edge at KnE0 provided there is significant concentration of the species near the sample surface, but yields will not be step-shaped. For example, a profile for a species diffusing into a material will appear as a peak with count yield decreasing with depth, extending only to the depth of penetration of the diffusing species. In order to obtain depth distributions and elemental concentrations from RBS spectra, channel number (for the detection of particles recorded in a multichannel analyzer as in this example) is directly related to detected particle energy through calibration with major element edge positions from the sample itself or from standards analyzed in the same session. The energies in turn can be related to depth in the material (with lower detected energies of particles scattering from a given element corresponding to greater depth in the material) by using the energy loss rates in the material, as outlined above (e.g., Eqn. 7). The count yields for the element of interest can be directly related to concentrations through the scattering cross-sections for that element. As this example illustrates, RBS is well-suited for analysis where there is a heavy impurity or species in a relatively light matrix. In contrast to some other depth profiling methods, the entire profile, including all elements present above detection limits, will be obtained in a single

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analysis, so potential interferences must be considered. In diffusion studies, an ideal case would be a material composed of relatively light elements with a heavy diffusant. Diffusion profiles under best circumstances would be sufficiently short such that they would be separated from the contributions to the RBS spectrum from the major elements comprising the sample material, leaving the diffusion profile with little or no background, as is the case in the example shown in Figure 20. Although the simultaneous collection of signals from all elements may be a drawback under some circumstances, it can also be a benefit, as sample stoichiometry can be monitored, as well as changes in concentrations when multiple species are involved in chemical diffusion.

Depth and mass resolution Depth resolution of standard RBS (a few MeV energy incident helium beam, standard solid-state surface barrier detector to detect backscattered ions) in the near-surface region is about 10-20 nm. Solid-state surface barrier detectors are most common for RBS analysis given their low cost and reliability, but other types of detectors may also be used. Time-of-flight detectors, for example, may be used for backscattering when heavy ions are the incident beam, as ions heavier than Li seriously degrade solid-state surface barrier detector performance, and electrostatic analyzers have found application in various areas, including lower-energy ion scattering studies (e.g., Leavitt and McIntyre 1995). With magnetic spectrometers, electrostatic analyzers and other high-resolution detection systems (e.g., Lanford et al. 2000), near-surface resolution can be improved by about an order of magnitude over that provided by typical surface barrier detectors. At greater depth in materials, depth resolution will degrade, primarily due to energy spread of the incident beam as it travels through the sample, referred to as “straggle.” The potential for degradation of depth resolution in this manner also has to be considered in designing experiments to be analyzed by RBS. For example, diffusion couples or “thick film” (greater than several tens of nm) sources are generally precluded because the beam either may not “see through” a thick later to the profile beneath, or depth resolution in measurements may be seriously compromised if the beam must go through a very thick surface layer to reach the profile. In addition, samples should be well-polished (or good natural mineral growth or cleavage faces used) to avoid additional loss of resolution due to surface roughness. However, when films on sample surfaces are thin, these sources can be employed quite successfully in diffusion studies using RBS (and NRA) (e.g., Dimanov et al. 1996; Bejina and Jaoul 1996). Other studies have used powder sources (e.g., Cherniak and Watson 1992, 1994), diffusion couples where materials can be readily separated (e.g., Cherniak et al. 2007), ion implantation of diffusants (e.g., Cherniak et al. 1991; Martin et al. 1999), gas sources (e.g., Watson and Cherniak 2003), and fluid media (provided care is taken to avoid dissolution of the sample and precipitation on sample surfaces; e.g., Watson and Cherniak 1997); in short, a range of source types can be used, provided a clear, smooth sample surface or surface area can be retained for analysis following diffusion experiments. Beam spots for standard RBS are typically in the range of 0.5-2 mm in diameter. Smaller beam spots (down to a few micrometers) can be obtained on specially designed microbeam lines with additional focusing, but usually with poorer detection limits because of much lower beam currents typical of these configurations. Detection limits for heavy elements are good with RBS (typically to a few tens of ppm), but mass resolution becomes poorer in higher mass ranges because of the nature of the mass dependence for the kinematic factor. Therefore, there can be difficulties in separating out signals in RBS spectra from elements whose masses are in close proximity (e.g., the rare earths), and isotopes cannot be readily distinguished under most circumstances; some exceptions are described below. However, high resolution detection systems can improve mass, as well as depth, resolution. As noted above, detection limits for light elements are relatively poor in RBS,

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but in some cases non-Rutherford scattering (to be discussed below) can be used to improve these detection limits, and another ion beam technique, nuclear reaction analysis (NRA), to be discussed in the next section of the chapter, may be used in some circumstances for profiling of low to medium Z atomic number elements. Above certain energies, depending on the projectile, scattering cross-sections will depart from classical Rutherford values as a result of short-range nuclear forces. For a standard backscattering energy for 4He of 2 MeV, cross-sections for elements with Z greater than ~6 will be Rutherford, while for 2 MeV protons this value will be Z ~20; for 4He of 4 MeV energy non-Rutherford behavior will be observed for elements with Z less than ~15. Departure from Rutherford scattering behavior often results in large increases in scattering cross-sections; a good example is the non-Rutherford scattering cross section of 4He from 16O for a 4He energy just above 3 MeV (e.g., Cheng et al. 1993; Leavitt and McIntyre 1991; Demarche and Terwagne 2006) where scattering cross-sections are more than an order of magnitude greater than Rutherford values. While these enhanced cross-sections have not yet found significant direct use in diffusion studies in minerals, they have been exploited to supplement other measurements, as in the study of Cherniak (2000), where non-Rutherford a scattering from 28Si at 6.6 MeV was used to explore substitutional processes involved in REE diffusion in fluorapatite.

Example applications of RBS in diffusion studies Diffusion of elements of geochronologic interest, such as Pb, can present optimal circumstances for RBS analysis, since these elements are generally of high atomic mass. Much of the focus on measuring diffusivities of atomic species of importance in geochronology has been directed toward diffusion in accessory minerals, including zircon, apatite, and monazite. Given the very slow diffusion rates of most atomic species in many accessory phases, RBS, with its superior depth resolution, has found effective application in Pb diffusion measurements (e.g., Cherniak et al. 1991, 2004a; Cherniak and Watson 2001; Gardes et al. 2006, 2007). Since the work of Sneeringer et al. (1984) on diopside, Sr diffusion in other minerals has been investigated with RBS, including feldspars (Cherniak and Watson 1992, 1994; Cherniak 1996), apatite (Cherniak and Ryerson 1993), fluorite (Cherniak et al. 2001), and calcite (Cherniak 1997). In some of these studies, insight into substitutional processes has been obtained by monitoring changes in major constituents in RBS spectra; for example, changes in near-surface Ca in K in calcic plagioclase and K-feldspar, respectively, following Sr diffusion experiments (Cherniak and Watson 1992, 1994) suggest exchange of Sr for these species in feldspars (Fig. 21). Diffusion of a range of other minor and trace elements, including rare-earth elements (e.g., Cherniak et al. 1997a; Martin et al. 1999; Cherniak 2000), high field strength elements (e.g., Cherniak et al. 2007) and actinides (Cherniak et al. 1997b; Cherniak and Pyle 2008) have been characterized with RBS. Measurements of diffusion of noble gases were among the earliest applications of RBS in diffusion studies in minerals, where Melcher et al. (1983) measured Xe diffusion in minerals present in meteorites (olivine, feldspar and ilmenite) in order to better understand the early chronology of the solar system. More recently, the diffusion of Ar has been measured in quartz (Watson and Cherniak 2003) as well as olivine, enstatite and corundum (Thomas et al. 2008; Watson et al. 2007) with RBS, to evaluate the potential of these major rock-forming minerals as reservoirs for noble gases within the earth. RBS measurements of Ar diffusion in quartz have also recently been coupled with measurements of Ar by laser ablation ICPMS (Baxter et al. 2006). Since these analytical methods address different length scales, their application to measurements of diffusion in the same samples can increase understanding of transport processes for noble gases in minerals. Unfortunately, because of the close energetic proximity of signals from Ar to those from K and Ca in RBS spectra, measuring Ar diffusion in minerals of interest in 40Ar/39Ar dating (e.g., K-feldspar) with this method is problematic.

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Figure  21. RBS spectrum of Sr diffusion in anorthite. In (a), the full spectra from a Sr diffusion experiment (grey line) and an untreated specimen of anorthite (black line) are plotted. The figure in (b) illustrates complementary Ca-Sr exchange in the process of Sr chemical diffusion, where the spectrum from the diffusion experiment shows Sr uptake accompanied by a decrease in near-surface Ca.

Counts

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Channel In certain cases, RBS can be used for measurements of major element diffusion and interdiffusion in minerals. For example, isotopic tracers can be used when there is sufficient mass separation between them and the dominant natural isotope of the element of interest (as in the case of 30Si or 44Ca), and interdiffusion may be studied when one of the species significantly differs in mass from the other (and the heavier species is introduced through a thin film or removable source), as in the case of Fe-Mg interdiffusion. Ca self-diffusion has been measured by RBS in natural and synthetic diopside (Dimanov and Ingrin 1995; Dimanov et al. 1996; Dimanov and Jaoul 1998) using a 44Ca tracer deposited in a RF-sputtered isotopically enriched thin film of diopside composition. Also in diopside, RBS has been used to measure (Fe,Mn)-Mg interdiffusion (Dimanov and Sautter 2000), while Fe-Mg interdiffusion has been measured by RBS in both orthopyroxene (ter Heege et al. 2006) and olivine (Bertran-Alvarez et al. 1992; Jaoul et al. 1995a; Dohmen et al. 2007). Silicon diffusion has been measured by RBS, using 30Si tracers, in a range of mineral phases, including olivine (Houlier et al. 1988, 1990), pyroxene (Bejina and Jaoul 1996), zircon (Cherniak 2008), quartz and feldspars (Bejina and Jaoul 1996; Cherniak 2003). Recently, S diffusion in pyrite and sphalerite (Watson et al. 2009) has been measured by RBS using a 34S tracer.

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In addition to single crystal studies, RBS has also been used in investigations of diffusion in polycrystalline materials, for example in characterizing diffusion of heavy elements such as cadmium, the rare earths, and actinides with applications to environmental problems and radioactive waste storage. RBS has been used by Toulhoat et al. (1996) to measure Cd diffusion in hydroxyapatite; diffusion of colloids in granitic rocks has been studied by RBS (Alonso et al. 2007a,b) to better understand transport from waste repositories and migration of other contaminants. Although geologic applications of RBS in diffusion studies have largely focused on diffusion in crystalline materials, RBS has also been used to investigate diffusion controlledprocesses in glasses, including dynamic oxidation in Fe-bearing aluminosilicate glasses and basaltic melts (Cooper et al. 1996; Cook and Cooper 2000).

Nuclear Reaction Analysis (NRA) In certain respects, NRA is similar to RBS in that a beam of ions produced in an accelerator is used to probe the chemical composition of the outer few micrometers of a material. However, for NRA, the ions are sufficiently energetic to overcome the Coulomb barrier and interact with specific target nuclei, resulting in a nuclear reaction. Products of these reactions (which may be gamma rays or light charged particles) are then detected, providing information about the depth distribution and concentrations of specific species in the material. Because this method relies on inducing nuclear reactions, it is isotope-specific. NRA has been applied in studies of diffusion in geologically significant materials over several decades. Among the earliest studies was an investigation of oxygen diffusion in quartz using the nuclear reaction 18O(p,a)15N (Choudhury et al. 1965). By the 1980s, applications of Nuclear Reaction Analysis to diffusion problems became more frequent. Given the beam energies accessible to accelerators commonly used for these techniques (typically up to several MeV), NRA is most often exploited for the detection of light elements. Many useful nuclear reactions, especially those induced by protons, require beam energies of only a few hundred keV, so relatively low-energy accelerators can be used for some types of measurements. Instrumentation required is generally similar to that for RBS, although nuclear reaction analysis methods that rely on the measurement of gamma rays induced in the reaction require scintillation detectors rather than the charged particle detectors used for RBS, and for NRA when charged particles are the product from the nuclear reaction detected. The isotopespecificity of NRA makes it suitable for a range of applications involving a tracer species, including diffusion studies. A good example is the use of an 18O tracer and the nuclear reaction 18 O(p,a)15N to measure oxygen diffusion in minerals. The expression in the previous sentence is the abbreviated way of writing the reaction p + 18O → 15N + a (8) where a proton beam is used to induce a nuclear reaction with an 18O atom. The products of the reaction are a 15N atom and an alpha particle, with the latter species being detected. NRA can be done in either resonant or non-resonant modes, depending on the reaction used and its cross-section. The reaction cross-section describes the probability of occurrence of a specific reaction as a function of incident particle energy and the angle between the incident particle and measured reaction product. As a function of energy, reaction cross-sections can have narrow energy-width regions of large cross-section (“resonances”), with regions of relatively small cross section in energy regions above and below these resonances; for some reactions there may also be broader-width energy regions with enhanced cross-sections (e.g., Fig. 22). For application of NRA in the resonant mode, the ideal reaction would have a large cross-section at the resonance energy Er and negligible or comparatively small cross-section

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O(p,α) N

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165o

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Figure 22. An example of a reaction cross section as a function of incident ion energy, in this case the reaction 18 O(p,a)15N at a laboratory angle (angle between incident beam and detector) of 165°. The reaction cross-section (which describes the probability of the reaction occurring) shows a broad region of enhanced cross-section in the range of 800 keV, and a narrow energy region of enhanced cross-section (a resonance) at ~629 keV. The cross section is plotted as millibarns (where a barn is 10−24 cm2) per unit of solid angle (in steradians). Cross section data are from Amsel and Samuel (1967).

at other nearby energies. Perhaps the best example is the 1H + 15N → 12C + 4He + g resonant reaction (resonance energy 6.385 MeV), which is used to profile 1H (with a 15N beam), one of the few means to obtain quantitative depth profiles of H in materials. This method was used by Laursen and Lanford (1978) in their pioneering study on the hydration of natural obsidian, in which they explored the mechanism of hydration and developed an interdiffusion model. With an incident proton beam, this same reaction can also be employed to profile 15N. The application of resonant profiling is illustrated schematically in Figure 23. When the incident beam is at the resonance energy Er, the concentration of the species of interest at the sample surface is detected. Depth profiling is done by increasing the incident beam energy, thus increasing the depth in the material at which the resonance (and enhanced yield of the products of the nuclear reaction) occurs. Depth scales are determined by the difference between the beam energy and the resonance energy, and the rate of energy loss for the incident ions in the material. Concentrations of the element or isotopic species of interest at a particular depth are determined from the detected yields of the product of the nuclear reaction for the number of particles of the incident beam delivered to the sample, with detected yields of products of the nuclear reaction dependent on the reaction cross-section at the resonance energy, beamdetector geometry, and efficiency of the detector. Resonant profiling is also used with a range of other nuclear reactions that have found application in diffusion studies, such as 27Al(p,g)28Si (using the 992 keV resonance, e.g., Sautter et al. 1988) to measure Al diffusion in diopside, 30Si(p,g)31P (using the 620 keV resonance, e.g., Jaoul et al. 1995b; Béjina and Jaoul 1996; Cherniak 2003, 2008) to measure Si diffusion in quartz, diopside and zircon, and 48Ti(p,g)49V (Cherniak and Watson 2007) to measure Ti diffusion in zircon. When reaction cross-sections are sufficiently large and vary smoothly over an extended energy range, the entire depth profile may be obtained using a single incident beam energy. This is referred to as non-resonant profiling. In non-resonant mode, the depth scales are determined by energy loss rates of incident and outgoing product particles from the nuclear reaction. Non-resonant profiling can be used with many reactions in which particles (e.g., protons, deuterons, 3He, a particles), rather than g rays, are the detected reaction product, since g rays do not experience the energy losses particles do in passing through the sample material and thus cannot provide depth information. The measurement of 18O using the nuclear reaction 18O(p,a)15N is a typical application of non-resonant profiling. 18O profiling is most often done in non-resonant mode at incident

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Figure  23. Schematic drawings illustrating the method of resonant NRA. This example shows the use of the reaction 1H(15N,ag)12C to measure 1H. An incident 15N beam is used, and g rays of characteristic energy produced in the reaction are detected. When the 15N beam is at the resonant energy (Er), hydrogen concentrations at the sample surface are measured. At higher incident 15N energies, H concentrations at depth in the material are sampled, with the depth (x) determined by the difference in energy between the incident beam and the resonant energy, divided by the energy loss rate for 15N in the sample material.

energies of around 800 keV (e.g., Reddy et al. 1980; Ryerson et al. 1989), taking advantage of a region of fairly large and smoothly varying reaction cross-section (Amsel and Samuel 1967) (Fig. 22). A spectrum of a particles over a range of energies is collected, representing contributions from 18O at various depths in the material (Fig. 24). This reaction has been used for oxygen diffusion studies in many mineral phases, including olivine (Reddy et al. 1980; Gérard and Jaoul 1989; Ryerson et al. 1989; Jaoul et al. 1980), zircon (Watson and Cherniak 1997), rutile (Derry et al. 1981; Moore et al. 1998), monazite (Cherniak et al. 2004b) and titanite (Zhang et al. 2006). It should be noted that 18O also can be profiled using the resonant technique, with, for example, the sharp resonance at 629 keV (Fig. 22). In addition to resonant and non-resonant approaches described above, “hybrid” methods are sometimes used in which energies are varied, but the cross-section is sufficiently broad that the signal is comprised of integrated contributions from a significant range of depths in the material at each energy step. This approach is usually employed in cases of faster diffusing species where diffusion distances are comparatively long. Some examples are measurement of He diffusion using the 3He(d,p)4He reaction (e.g., Cherniak et al. 2009, Miro et al. 2006). Interpretation of NRA spectra and conversion to concentration profiles is slightly more complicated than for RBS for a few reasons. In RBS, Rutherford scattering cross-sections can be described analytically; in contrast, cross-sections for nuclear reactions depend on nuclear structure and can vary quite dramatically with incident particle energy and with elemental and isotopic species. Cross-sections must be empirically determined through careful measurement as functions of incident energy and beam-detector angle. Fortunately, cross-sections for many useful reactions have been measured and tabulated for analytical purposes. Tables of reactions

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1500

2000

Channel Figure 24. An example spectrum for non-resonant nuclear reaction analysis, in this case measuring oxygen diffusion in olivine using the 18O(p,a)15N reaction. The count yields in the alpha peak (toward the right of the figure) are proportional to the 18O concentration in the material, with yields at lower channel numbers (and lower energies) corresponding to 18O at greater depths in the material. Also detected in the spectrum are protons backscattered from the constituents of the olivine; these are toward the lower channel number (lower energy) end of the spectrum. Because the energy released in the nuclear reaction is relatively large (3.98 MeV) the alpha particles produced in the reaction will be fairly energetic, with much larger energies than the backscattered protons. However, reaction cross-sections are generally much smaller than Rutherford scattering cross-sections, so the count yields of alpha particles produced in the reaction are considerably less than those of the backscattered protons. Samples from Ryerson et al. (1989).

used in NRA can be found in Cherniak and Lanford (2001) and Tesmer and Nastasi (1995); the online database IBANDL (the Ion Beam Analysis Nuclear Data Library, at http://www-nds. iaea.org/ibandl/) has a frequently updated compilation of reaction cross-sections as a function of energy for a range of nuclei and projectiles. An additional complication in interpreting spectra is that the incident particles and product species from the nuclear reaction will not be the same, as they are in elastic scattering, and thus can have quite different energy loss rates in the sample material. Using the example of the 18O(p,a)15N reaction, the incident protons will lose energy at a much smaller rate while traversing a given distance in the material than the alpha particles that are the product of the reaction. When gamma rays are the species measured, for example when profiling H and Al with the reactions 1H(15N, ag)12C and 27Al(p, g)28Si, respectively, depth information can only be obtained through varying the energy of the incident beam since the gammas will travel through the material without losing energy as will product particles such as protons, alphas, or deuterons. In some cases, a multiple ion beam techniques can be employed in analysis of a single sample or set of samples to provide added information about the diffusional process. For example, RBS measurements of Pb and Sr diffusion in feldspars (Cherniak and Watson 1992, 1994; Cherniak 1995a) were supplemented by NRA measurements of Al and Na in order to provide insight into substitutional mechanisms involved in Pb and Sr exchange in alkali feldspars and sodic plagioclase. Similarly, measurements of phosphorus using the reaction 31P(a,p)34S were made to accompany RBS measurements of REE diffusion profiles in zircon to investigate the role of the substitution REE+3 + P+5 → Zr+4 + Si+4 in rare-earth element diffusion in zircon

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(Cherniak et al. 1997b). In addition, in some studies of Si diffusion, both RBS and NRA (using the 30Si(p,g)31P reaction) have been employed to measure Si diffusion on the same samples, with good agreement among results (e.g., Béjina and Jaoul 1996; Cherniak 2003, 2008).

Elastic Recoil Detection (ERD) ERD (Elastic Recoil Detection) is another ion-beam method for measuring depth profiles of light elements (Z = 1 to ~10). It is like RBS in that it relies upon elastic scattering; however, the incident ions must be heavier than the element to be profiled. Typical incident beams employed (e.g., Petit et al. 1990; Cookson 1991) are 4He at a few MeV (for profiling hydrogen) or heavier ions such as C, Si or Cl at higher energies (≈ 1 MeV/amu). The light target atoms will recoil in a forward direction upon collision with the heavier incident ions; if the sample is tilted these recoiled atoms will escape the sample and can be detected. ERD can also be performed in transmission mode, where the detector is placed behind the sample and recoiled atoms exiting the back of the sample are detected, but this can only be done for thin (typically a few micrometers) self-supporting films (such as polymers), so this approach is of limited use in geological studies. The energy of the detected recoils Er will provide information about the depth distribution of the recoiling species, as described in the expression: Er = E0

4 M1M 2 cos2 φ

( M1 + M2 )

2

= K r E0

( 9)

where Er is the energy of the recoiled atom, E0 is the energy of the incident ion, M1 and M2 are the masses of the incident and recoiled species, respectively, and 180° − f is the angle between the incident beam and the detector used to detect the recoiled atoms. In general, f will be set at angles between 10 and 30°. To prevent scattered incident ions from interfering with the signal from the light recoils, an absorber foil is typically placed in front of the detector to stop these heavier ions. When the absorber foil is used, the detected particle energy is not Er, but a lower value Edet, with Edet = Er − δ f S f

(10)

where df and Sf are, respectively, the thickness of the foil, and “stopping power” or energy loss for the ions in the foil per unit thickness. As with RBS and NRA, depth scales for the detected particles are constructed from information on energy loss in the sample material. If an incident ion penetrates to a specific depth (measured normal to the sample surface) before a recoil event occurs, the incident ion will have lost energy in inelastic collisions while traveling through the material. Similarly, the recoil atom will lose energy traveling out of the sample to its surface. If the energy-averaged stopping powers for the incident and recoil species are Si and Sr, respectively, the initial depth of the recoil species can be determined from the expression KS S  depth = ( Edet + δ f S f − K r E0 )  r i + r  sin α sin β 

−1

(11)

where a and b are the acute angles between the sample surface and the incident beam, and the sample surface and the detector. Depth resolution will depend on the energy loss variations of both the incident and recoiling atoms in the material and the recoiling atoms in the foil, and will typically be in the range of tens of nm. Better resolution (by up to an order of magnitude) can be obtained by eliminating the stopper foil and using other detection systems, such as a magnetic spectrometer, rather than standard surface barrier detectors. Although it holds considerable promise for analysis of light elements, thus far ERD has been used to only a limited extent in diffusion studies of geological materials. An example is

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measurement of He diffusion in fluorapatite by Ouchani et al. (1998). Additional information on the technique (along with schematics of experimental configurations for ERD analysis) can be found in Barbour and Doyle (1995).

Fourier Transform Infrared Spectroscopy Infrared (IR) spectroscopy uses infrared radiation to probe the chemical species and molecular clusters in solids, liquids or gases by exciting the vibrational modes (which absorb in the infrared region) in molecules or clusters. Infrared radiation is electromagnetic radiation with frequencies lower than visible light but higher than radio wave frequencies. The infrared spectrum covers the wavelength range from 0.75 to 100 mm, longer than visible light (400-750 nm) but shorter than the radio wave. The infrared spectrum is often subdivided into the nearinfrared (NIR, wavelength 0.75-3 mm), mid-infrared (MIR, 3-30 mm), and far-infrared (FIR, 30-100 mm) regions. Both the frequency and wavelength are related to the infrared radiation energy. The most common way to express infrared energy in plots of infrared spectra is by wavenumber, which in spectroscopy equals the number of waves per unit length (most often expressed in units of cm−1, but mm−1 will be used here). Infrared spectroscopy is widely applied in identification of specific chemical bonds, species and clusters. Its application in quantitative analyses (the focus of this chapter) is mostly in the measurement of H2O and CO2 concentrations, as well as concentrations of the individual species (molecular H2O and the hydroxyl ion for total H2O, and molecular CO2 and the carbonate ion for total CO2). In the geological literature, these analytical capabilities were pioneered by Stolper (1982a,b), Aines and Rossman (1984), Fine and Stolper (1985, 1986), Newman et al. (1986), Paterson (1986), Rossman (1988), and Bell and Rossman (1992). In addition to the aforementioned species, NH4+ concentrations in mica (Busigny et al. 2003, 2004) have also been measured by IR spectroscopy. Although the number of components that can be analyzed by infrared spectroscopy is limited, H2O and CO2 are major volatile components in the Earth and in melts and minerals, and they cannot be analyzed easily by other methods. Furthermore, infrared spectroscopy is a non-destructive method (but preparation of doubly-polished sections for analysis will cause sample loss) with high sensitivity, high precision, and high spatial resolution. Recent development has resulted in nanoSIMS having even better sensitivity and spatial resolution than infrared spectroscopy (e.g., Saal et al. 2008), but infrared spectroscopy is still the only method available to quantitatively determine the concentrations of species, which may provide critical structural information and further insight into diffusional processes. Below we introduce the basic principles describing the molecular vibrational modes that often cause infrared absorption, the instrumentation used in infrared spectroscopy, and applications to measurement of H2O and CO2 concentrations and their use in diffusion studies.

Vibrational modes and infrared absorption Basic principles of vibrational motion in multi-atom “molecules” (including ionic clusters) have been discussed in numerous texts and papers (e.g., Colthup et al. 1990; Ihinger et al. 1994; Beran and Libowitzky 2004). The vibrational motion absorbs in the infrared region. For an OH group in a structure, there is a stretching mode (Fig. 25), absorbing near 355 mm−1. For an H2O molecule (H2Om), there are 3 vibrational modes (symmetric stretching at 365 mm−1, asymmetric stretching at 375 mm−1, and bending at 163 mm−1; Fig. 26). For a CO2 molecule, because it is a linear molecule, there are 4 vibrational modes (symmetric stretching at 139 mm−1 that is IR inactive, asymmetric stretching at 235 mm−1, and two identical bending modes at 67 mm−1; Fig. 27). For a carbonate ion (CO32−, a triangular ion), there are 6 independent vibrational modes (symmetric stretching at 106 mm−1 that is IR inactive, two asymmetric stretching at about 142 mm−1 that may split into a pair of peaks, two in-plane bending modes at about 68 mm−1, and one out-of-plane bending mode at 88 mm−1).

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Figure 25. modeofofanan cluster, where the Fig. 25.Stretching Stretching mode OHOH cluster, smallwhere solidthe circle and the H, large smallrepresents solid circleH,represents and open circle represents O.open Thecircle diagram only O. shows the large represents The one direction of motion during vibration. Bydirection reversing the direction of all diagram only shows one of motion during By reversing direction arrows, onevibration. gets the other type of the motion in this and other of all below. arrows, one gets the other type of examples motion in this and other examples below.

Symmetric stretch

Asymmetric stretch

Bending

Fig. 26. VibrationalVibrational modes of themodes H2O molecule, where the small solid circles H, andcircles large open Figure 26. of the H where therepresent small solid 2O molecule, circles represent O. represent H, and large open circles represent O.

Symmetric stretch

Asymmetric stretch

Bending

Fig. 27. Vibrational of the CO solid circles C, and large open circles Figure 27.modes Vibrational modes of thewhere CO2small molecule, whererepresent small solid circles 2 molecule, represent C, and large open circles represent O. represent O.

Based on quantum mechanics, the quantized energy level of a harmonic oscillator (an approximation for a vibrational mode) can be expressed as follows (Fig. 28):

E = (n+1/2)hn, where n = 0, 1, 2, ...

(12)

where E is the quantized energy level, n is the quantum number, h is the Planck constant, and n is the characteristic vibrational frequency, which is related to the force constant k as: ν=

k m*

(13)

where m* is the reduced mass, for a diatomic cluster defined as m* =

1 1 1 + m1 m2

(14)

where m1 and m2 are the masses of atom 1 and atom 2 in the diatomic cluster. Based on this definition, the reduced mass m* is smaller than either m1 or m2. For example, for O-H stretching, m1 = 1.0079 u, and m2 = 15.9994 u, leading to a reduced mass m* = 0.9482 u. On the basis of Equation (12), the lowest energy (ground state) of a harmonic oscillator is (1/2)hn when n = 0, which is called the zero-point energy. By absorbing an energy of hn (corresponding to the fundamental vibrational band in an IR spectrum), the harmonic oscillator can be excited to the energy of 1.5hn (Fig. 28). By absorbing an energy of 2hn (corresponding to the first overtone in an IR spectrum), the harmonic oscillator can be excited to the energy of 2.5hn, and so on. If the oscillator is perfectly harmonic, the selection rule states that overtones should not occur. Because the energy versus distance relation for a chemical bond is slightly different from that of a harmonic oscillator (Fig.  29), (1) the overtones do occur with an

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Figure 28. Quantized levelsoscillator. of a harmonic oscillator. Fig. 29. Quantized energy levels ofenergy a harmonic

Figure 29.Fig. Potential energyenergy of a diatomic molecule as aasfunction of of distance atoms. The Thesolid 28. Potential of a diatomic molecule a function distancebetween betweenthe the two two atoms. solid curvecurve is theisactual energy andand the the dashed curve is the harmonic approximation. the actual energy dashed curve is the harmonic approximation.The Ther rthat thatcorresponds corresponds to to the energy minimum is the bond (or equilibrium distance re). re). the energy minimum is the length bond length (or equilibrium distance

intensity about two orders of magnitude lower than the fundamental modes, and (2) the first overtone may occur at a slightly different wavenumber from exactly 2 times the fundamental band. A photon may also simultaneously excite two modes (combination modes). The intensity of the combination modes is between those of the fundamental modes and overtones. Not all vibrational modes can be detected by infrared spectroscopy. If a molecule has a zero dipole moment (or a center of symmetry), and if the vibrational mode is symmetric so that it does not generate a dipole moment, it would be IR-inactive, meaning that in theory it cannot be excited by infrared radiation. In practice, however, IR-inactive modes still absorb in the IR region but the absorption is very weak. For example, H2, N2 and O2 are all IR-inactive. On the other hand, if a molecule does not have a center of symmetry (such as the case of a CO molecule), or if the vibration is asymmetric for a molecule with center of symmetry

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(such as asymmetric stretching of the O=C=O molecule), the vibrational mode is IR-active. When a vibrational mode is IR-inactive, it is Raman-active. When a vibrational mode is IRactive, its absorption in Raman is weak. Hence, IR spectroscopy and Raman spectroscopy are complementary. For example, H2 is IR inactive but can be easily detected by Raman for diffusion studies (e.g., Shang et al. 2009). Furthermore, the spatial resolution of Raman spectroscopy is much better than IR spectroscopy, especially because the Raman beam can be focused to a specific spot (including a specific depth in a sample) with 1-2 mm resolution (e.g., Di Muro et al. 2006). However, the reliability of using Raman spectroscopy for quantitative analyses has been debated due to complications in its calibration when internal calibration has been used (e.g., Thomas 2000; Arredondo and Rossman 2002). Recent attempts using external calibration (or the “comparator technique”) show promise for determining total H2O contents (Di Muro et al. 2006; Thomas et al. 2008). Even though determining total H2O contents may be possible, extracting concentrations of hydrous species (H2Om and OH) does not seem to be possible (Behrens et al. 2006). Due to its high spatial resolution and its ability to be focused at specific depths in a material, future development may potentially make Raman spectroscopy a much more powerful tool in quantitative analyses of volatile species and Fig 30. Two IR spectra, one for low H2O content and one for in diffusion studies.

mm-1 is the asymmetric stretching of molecular CO ; the pea

2 Two infrared spectra are OH (which indicates the sum of H O and OH because H2O m 2 m shown in Figure  30, one for -1 mm is a combination mode indicating OH, the peak at 523 m a low-H2O glass in which the fundamental OH stretching and the peak at 710 mm-1 is the first overtone of OH fundame band (at 355 mm−1) can be seen (2000) and Ni and Zhang (2009a). clearly, and one for a highH2O glass, in which the first overtone (about 710 mm−1) of the fundamental OH stretching band, and the combination bands at 452 mm−1 for X-OH (this Figure  30. Two IR spectra, one for low H2O content and band is characteristic of OH not one for high H2O content. The sharp peak at 235 mm−1 is the associated with H2O molecules) asymmetric stretching of molecular CO2; the peak at 355 mm−1 and at 523 mm−1 for H-O-H the fundamental stretching of OH (which indicates the sum of Two IR spectra, one for low H2O content and oneis for high H O content. The sharp peak at 235 H2Om and OH2because H2Om also contains OH bonds); the peak (this peak is characteristic of −1 -1 is the fundamental at 450 mm is mm a combination mode indicatingstretching OH, the peak s the asymmetric stretching of molecular CO ; the peak at 355 of at molecular H2O) can be observed. 2 523 mm−1 is a combination mode indicating H2Om, and the peak fundamental stretch hich indicates The the sum of H2Om and OHpeak because Hat2O also−1 contains OH bonds); the peak at 450 m mm 710 is the first overtone of OH fundamental stretching. is mode too high (over-scale) to peak be at 523 -1 is Frommm Zhang and Behrens (2000) and Ni et al. (2009a).H2Om, s a combination indicating OH, the a combination mode indicating shown -1 in Figure 30b.

peak at 710 mm is the first overtone of OH fundamental stretching. From Zhang and Behrens and Ni and Zhang (2009a).

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Instrumentation for Infrared Spectroscopy The classical infrared spectrometer uses the wavelength dispersive method to measure infrared spectra. Even though it is no longer used, we introduce the principles of the wavelength dispersive infrared spectrometer because their explanation is straightforward. The dispersive infrared spectrometer consists of four parts: energy source, prism, sample chamber, and detector. An energy source (which depends on whether one is interested in MIR or NIR), emits energy at all wavenumbers in the MIR or NIR region. The energy passes through a prism (or diffraction grating or monochromator) so that only the energy in a narrow wavenumber range is allowed through. The energy then passes through an aperture to limit the beam diameter, and then into the sample chamber with or without a sample. Finally, the remaining intensity is determined by a detector. To measure a full spectrum, the monochromator must be able to adjust so that radiation of different wavelengths can be let through. By adjusting the monochromator, the entire wavenumber range can be scanned. If no sample is present, the intensity is denoted as I0. If a sample is present, the detected intensity is denoted as I. Transmission is defined as I/I0. Absorbance is defined as –log(I/I0) = log(I0/I). Hence, if the sample does not absorb the energy at a given wavenumber, the transmission is 100%, and the absorbance is 0. If the samples absorbs significantly, the transmission becomes much less than 1 and absorbance would have a positive value. When the absorbance is greater than 2 or 3, there is very little energy detectable after passing through the sample; the error then becomes very large and absorbance loses its quantitative meaning. Since the 1990s, Fourier transform infrared spectrometers have become more widely available and classical infrared spectrometers are no longer commonly used. Nowadays, when IR instrumentation is discussed, it is implicitly FTIR, not the classical wavelength-dispersive IR. Classical IR spectrometers have distinct disadvantages compared with FTIR: (i) they only use a small fraction of the total energy from the source at every instant of time, wasting most of the energy and leading to large noise/signal ratios, (ii) one scan takes a long time (several minutes to an hour, depending on the precision needed), (iii) they require intensity calibrations, and (iv) their signals are affected by stray light. Fourier transform infrared (FTIR) technology eliminates all of the above disadvantages. In FTIR, the monochromator is replaced by a unit (Michelson interferometer) that uses a beamsplitter to split the input beam into two equal parts, which are reflected back by mirrors and recombined. The recombined light then goes to the sample and then to the detector. The recombination leads to interference, so that the recombined signal is equivalent to the Fourier transform of the input signal. That is, in the input beam the intensity depends on the wavenumber, which cannot be detected by the detector, whereas in the recombined beam the intensity depends on time, which can be detected by the detector. After the detector records the intensity versus time signal, the digital data are then Fourier transformed back through software into a function of intensity versus wavenumber. Without the sample, the same signal as the input is obtained. With absorption by the sample, the spectrum would display the absorption bands.

Different types of IR spectra IR spectra can be taken either on powders mixed and pressed with KBr into a disc, or on glass or single crystal wafers. Powder spectra are for qualitative identification, but for quantitative analyses of geological materials, spectra of glass plates or single-crystal wafers are most often obtained. If the sample is isotropic (including glass and isometric minerals), then a typical FTIR configuration with a nonpolarized (or partially polarized because the mirrors and beamsplitter can cause polarization) beam works well for quantitative analyses. If the sample is anisotropic (all minerals with symmetry lower than isometric minerals), the absorption band intensity will depend on the orientation of the crystal as well as how the E (electric field) vector is oriented.

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Libowitzky and Rossman (1996) discussed the principles of quantitative IR measurements of anisotropic minerals. In addition to transmission spectra, in which light passes through the sample and goes to the detector, reflectance spectra can also be taken (e.g., Moore et al. 2000), in which light strikes the surface of a sample, is reflected, and then reaches the detector. Peaks in reflectance spectra are usually weak and reflectance spectra are more difficult to quantify. For quantitative analyses, transmission spectra have been the method of choice except for remote sensing. Depending on the infrared wavenumber region one is interested in, it is necessary to decide whether to obtain MIR spectra (typically 40-400 mm−1), using the energy source, beamsplitter and detector for the MIR region, or to obtain NIR spectra (typically 200-900 mm−1). A typical IR instrument is set up for MIR analyses, with a Globar (a silicon carbide rod heated electrically to high temperature so that it glows) energy source, a KBr beamsplitter, and a DTGS (Deuterated Triglycine Sulfate) detector. If one needs to obtain NIR spectra, a special energy source (tungsten light source), beamsplitter, and detector (LN2-cooled InSb) must be added. For CO2 analyses, MIR bands are used. For H2O analyses, NIR bands are often used (except for the bending mode) for analyzing water-rich glasses (e.g., ≥ 0.3 wt% H2Ot), whereas either NIR or MIR can be used for water-poor glasses (e.g., ≤ 0.5 wt% H2Ot) and most minerals. To measure small samples, an FTIR microscope system is necessary, typically equipped with a LN2-cooled MCT (mercury-cadmium-telluride) detector that can be used with either MIR or NIR source and beamsplitter. If large samples are available and there is no need to measure the spatial variation of species concentration, then the main chamber of the spectrometer is often used. An advantage of the main chamber is that the rays in the incidence beam are roughly parallel and the beam is perpendicular to the sample surface. Hence, the light will not bend much and the light path length is similar to the sample thickness so results are more reproducible from one lab to another. On the other hand, if the sample is small (e.g., < 0.1 mm in diameter) or one needs high spatial resolution (e.g., measurement of a diffusion profile), then a FTIR microscope must be used, in which the incidence beam is focused (converged) to a thin beam. However, the rays in the beam are not parallel but are first converging, reaching a minimum beam diameter, and then diverging. Therefore, the beam will be refracted in the sample, and the beam path is longer than the sample thickness, so reproducibility between different laboratories is not as good as in the case when main chambers are used. Furthermore, because light rays in the beam focused by a microscope are not parallel, they diverge in the sample, leading to a spatial resolution inferior to that indicated by the aperture size. For example, Ni and Zhang (2008) showed that with a 20 mm wide aperture and a sample thickness of 200 mm, the actual spatial resolution (FWHM) is about 30 mm.

Calibration In order to convert an IR absorption peak intensity from an infrared spectrum to a concentration, it is necessary to carry out a calibration using samples with known concentrations of the species of interest. The calibration for total H2O content (H2Ot) and species concentrations is used as example here. No method is available to directly determine the species concentrations in Fe-bearing glasses (i.e., natural glasses), which is a difficulty in the calibration. (For Fe-free glasses, NMR is able to determine the concentration ratio of the two species based on peak area ratio without calibration so that once H2Ot content is known, concentrations of both H2Om and OH can be determined, see Schmidt et al. 2001; Yamashita et al. 2008.) Concentrations of H2Ot may be determined by manometry (e.g., Epstein and Taylor 1970; Newman et al. 1986), or by Karl-Fischer titration (e.g., Turek et al. 1976; Westrich 1987; Behrens et al. 1996). According to Beer’s law, for dilute solutions, the concentration of a species i (Ci) and the absorbance at a peak for species i (Ai) are related as follows:

154

Cherniak, Hervig, Koepke, Zhang, Zhao Ci =

18.015 Ai ρdεi

(15)

where r is the density, d is the thickness, and ei is the molar absorptivity at this peak. The thickness of the sample is measured using a micrometer. Glass density is measured as a function of H2Ot: for glasses with the same anhydrous composition but different H2Ot, the density can often be expressed as r = r0(1 − aC) where C is the mass fraction of H2Ot and a is a constant (often about 0.6 , e.g., see the summary in Table 3 of Zhang 1999). A more accurate expression uses the partial molar volume of H2O of 12.0(±0.5)×10−6 m3 (Richet et al. 2000). Either the peak height (linear absorbance) or the peak area (integrated absorbance) may be used for A. In the literature, peak height is more often used because it is simpler and the precision is about the same (at least for H2O species). If C is the H2Ot concentration (mass fraction or weight percent), C1 is the H2Om concentration, and C2 is the concentration of H2O present in the form of OH (i.e., it is the mass of two OH groups minus one oxygen), H2Ot concentration as the sum of H2Om and OH can be expressed as follows, since the H2Om peak occurs at ~523 mm−1, and the OH peak occurs at ~452 mm−1: C = C1 + C2 =

18.015 A523 18.015 A452 + ρdε523 ρdε 452

(16)

where A523 and A452 are the absorbance for the 523 mm−1 and 452 mm−1 peaks, e523 and e452 are the molar absorptivities for the 523 mm−1 and 452 mm−1 peaks. Based on Equation (16), from the measurement of H2Ot concentration by an absolute method, and measurement of peak heights A523 and A452 in an infrared spectrum, one equation relating C, A523 and A452 can be obtained. Because there are two unknowns (e523 and e452) in Equation (16), it is necessary to conduct the analyses over a large range of values of C to yield numerous linearly independent equations (meaning that C1 and C2 concentrations must not be proportional as C increases) so that both e523 and e452 can be determined. A feature of the species equilibrium in the melt (quenched to glass) between H2Om and OH is that the concentrations of the two species are not proportional to each other, allowing the determination of both e523 and e452. One method to obtain e523 and e452 from data is through direct multi-linear regression of C versus A523 and A452 (Newman et al. 1986) by making the intercept be zero. Another method is to rewrite Equation (16) in the following form (Behrens et al. 1996): 18.015 A523 ε 18.015 A452 = ε523 − 523 ρdC ε 452 ρdC

(17)

and plot 18.015A523/(rdC) on the y-axis versus 18.015A452/(rdC) on the x-axis. The y-intercept is e523, the x-intercept is e452, and the slope is −e523/e452. The above treatments assume that e523 and e452 are constant (independent of H2Ot). Because H2Ot in glasses is relatively high (typically a couple of wt%), the assumption may or may not be accurate. For example, Zhang et al. (1997) showed that the e523/e452 ratio in rhyolite glasses derived from a calibration based on different samples and the ratio from heating the same sample to different temperatures are different, indicating either the calibrations are not accurate, or e523 and e452 depend on H2Ot. The uncertainty on e523 and e452 affect the species concentrations more than H2Ot content. By combining manometry data and heating data, Zhang et al. (1997) carried out a calibration by treating e523 and e452 to be a function of A523 and A452. The calibration by Zhang et al. (1997) works well at H2Ot ≤ 3.0 wt%, but the error increases as H2Ot increases above 3.0 wt% (Zhang and Behrens 2000). Schmidt et al. (2001) found that the molar absorptivities in alkali aluminosilicate glasses are independent of H2O concentrations, whereas Yamashita et al. (2008) showed that the molar absorptivities in sodium silicate glasses depend on concentrations. The issue of whether and how molar absorptivities

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in various hydrous glasses vary with H2Ot at greater H2Ot remains to be resolved. At low H2Ot concentrations, the 355 mm−1 peak (fundamental stretching of OH) can be used, which is about a factor of 50 larger than the NIR peaks (452 mm−1 and 523 mm−1). At low H2Ot, OH is the dominant species. Hence, the calibration involves only one molar absorptivity (e.g., Dobson et al. 1989; Dixon et al. 1995). For a sample of about 1 mm thickness, the detection limit of H2Ot using the 355 mm−1 peak is about 1 ppm. Examples for the use of the 355 mm−1 peak include investigation of water diffusion in basalt melt (Zhang and Stolper 1991). For the IR measurement of molecular CO2 and CO32− concentrations, the calibration procedures are similar when only one of the two species is present in the given glass. Sample preparation for infrared analyses is in general straightforward. The sample must be doubly polished and the thickness well-determined. For transmission infrared spectra, the surface of the sample must be cleaned without leaving a residue that may contribute to the species to be measured. The surface along the IR path must not be covered by plastic tape or double-sided tape. Cracks, especially cracks filled by epoxy, must be avoided because epoxy has a large IR signal that would superimpose on the bands of interest.

Applications to geology The most important geological applications of FTIR are to analyze total H2O concentrations, concentrations of H2O species, total CO2 concentrations, and concentrations of CO2 species in natural silicate glasses and minerals. Absorption bands involving H and C are often at much higher wavenumbers and far separated from structural IR bands due to silicate network vibrations and are hence easily quantified. This is because the masses of H and C are small (leading to small values of reduced mass, Eqn. 14), so that at the same bond strength (force constant k), the vibrational frequencies (or wavenumbers) are much higher than those for other species (Eqn. 13). Stolper (1982a,b) pioneered infrared studies of dissolved water in natural silicate melts and glasses, and was first in the earth sciences to discover the presence of both molecular H2O and hydroxyl groups in silicate melts and glasses and to study the equilibrium between H2O molecules and OH groups. At high total H2O contents (such as > 4 wt%), the concentration of H2Om becomes higher than that of OH. With the advancement of IR microbeam techniques, H2O diffusion profiles can be measured, along with species concentrations and total concentration using step scans (Zhang et al. 1991a; Zhang and Stolper 1991) or automatic line scans (e.g., Zhang and Behrens 2000). The analyses of data on species concentration demonstrated that H2Om is the diffusing species and that OH is largely immobile compared to H2Om (Zhang et al. 1991a;, Doremus 1995; Zhang and Behrens 2000; Behrens et al. 2004; Liu et al. 2004b; Ni and Zhang 2008; Ni et al. 2009a,b; Wang et al. 2009), an inference reached earlier by some glass scientists based on the shape of diffusion profiles representing total H2O at low H2Ot contents (Doremus 1969, 1973; Ernsberger 1980; Smets and Lommen 1983; Nogami and Tomozawa 1984). The comparison between H2O diffusion and 18O “self” diffusion also led to the realization that oxygen diffusion in melts and minerals in the presence of H2O often occurs through H2O diffusion (Zhang et al. 1991b; Behrens et al. 2007; see also ab initio calculation results by McConnell 1995). Infrared spectroscopy has also been applied to investigate diffusion of hydrous components in olivine (Mackwell and Kohlstedt 1990), pyroxene (Skogby and Rossman 1989; Ingrin et al. 1995), and garnet (Wang et al. 1996). Other applications include speciation studies (e.g., Ihinger et al. 1999; Liu et al. 2004a; Hui et al. 2009), solubility studies (e.g., Blank et al. 1993; Tamic et al. 2001; Liu et al. 2005), and kinetic and geospeedometry studies (Zhang et al. 1997b; Zhang et al. 2000; Wallace et al. 2003). IR measurements of H2O species concentrations and profiles are much more difficult in glasses with high FeO concentration because (i) the degree of transparency of the glass is

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reduced and (ii) the broad Fe absorption band in the NIR overlaps with the 523 mm−1 peak (e.g., Ohlhorst et al. 2001; Liu et al. 2004a,b; Ni et al. 2009b). One way to get around this difficulty is to use Fe-free melts using equimolar amounts of MgO and CaO to replace FeO (e.g., Ni et al. 2009a). Stolper and coworkers also pioneered the infrared study of CO2 in silicate melts and glasses (Fine and Stolper 1985, 1986). It was found that dissolved CO2 in polymerized (or silicic) melts is present as molecular CO2. As the melt composition changes from rhyolite to andesite to dacite to basalt, increasingly more of the dissolved CO2 is present in the form of carbonate ions (CO32−). Infrared measurements of CO2 and CO32− concentrations have been applied to investigate total CO2 diffusion (Blank 1993; Sierralta et al. 2002; Nowak et al. 2004) and solubility (e.g., Blank et al. 1993; Tamic et al. 2001; Behrens 2010).

Synchrotron X-ray fluorescence microanalysis (m‑SRXRF) The m-SRXRF (“microscopic”-synchrotron radiation X-ray fluorescence) technique is a well-established microanalytical technique for trace element analysis, which in the last decades has been continuously improved in its instrumentation and in the quantification of X-ray spectra (see reviews by Smith and Rivers 1995, Haller and Knöchel 1996, Janssens et al. 2000, Hansteen et al. 2000, Sutton et al. 2002). Due to the outstanding properties of synchrotron radiation, such as high brightness, high degree of polarization and extremely low divergence, m-SRXRF offers a number of interesting advantages in comparison with other trace element analytical tools: (1) non-destructive analysis which enables long-time acquisitions under steady-state conditions, even for fragile biological materials, (2) easy quantification by using a standard-free fundamental-parameter approach, (3) the possibility of calculating matrix effects by using fundamental-parameter approaches, (4) the collection of multi-element spectra with one acquisition; (5) low detection limits (ppm level) at a high spatial resolution, and (6) sensitivity ranging over six orders of magnitude. In the last decades, the m-SRXRF technique has been applied to many disciplines of the earth sciences (see the overviews by Smith and Rivers 1995 and Sutton et al. 2002). Baker and Watson (1988) and Baker (1989, 1990) were the first to apply m-SRXRF to study trace element diffusion in silicate glasses, measuring diffusion profiles of a few selected elements. Koepke and Behrens (2001) improved the technique by using a modern energy dispersive detector to simultaneously analyze 18 trace element diffusivities in a single experiment. With this multielement approach, the relative errors of the diffusion coefficients are minimized, resulting in “internally consistent” data sets (Mungall et al. 1999; Koepke and Behrens 2001).

Instrumental setup, spectra acquisition and data processing Most studies on diffusion in silicate glasses using m-SRXRF as an analytical tool have been performed at the HASYLAB synchrotron source of the DESY in Hamburg, Germany (Koepke and Behrens 2001; Koepke et al. 2003; Hahn et al. 2005; Behrens and Haack 2007; Behrens and Hahn 2009). Therefore, the experimental m-SRXRF setup installed at beamline L of the HASYLAB is described here in detail (Fig. 31). Similar setups can be found in other synchrotron radiation facilities around the world (Brown et al. 2006) For the measurements, the “white” X-ray continuum of a bending magnet is used, which shows maximum brilliance at 16.0 keV (critical energy). During routine analysis, the resulting X-ray spectral distribution enables simultaneous K-shell excitation of elements with atomic numbers from 25 (Mn) to 82 (Pb). The excitation conditions are normally optimized for single elements or element groups by using suitable absorbers. For minimizing the background radiation caused by Rayleigh and Compton scattering, the “horizontal” geometry is used, (for details see Haller and Knöchel 1996; Janssens et al. 2000), for which the radiation is detected

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HPGe detector Microscope + Camera Collimator

Cross slits

Beam monitor 45°

Capillary Synchrotron source

Absorber

Pinhole

Beam monitor

Sample

45°

Beam stop

Figure 31. Schematic drawing of the m-SRXRF analytical setup installed at beamline L of DORIS III synchrotron radiation source at HASYLAB (view from the top).

in the plane of maximum polarization at an angle of 90° (Fig. 31). For this geometry, the angle Fig. 31 of incidence of the synchrotron beam on the sample is 45°. For a sufficiently high spatial resolution, a reduction of the diameter of the incoming synchrotron beam is necessary, which can be accomplished by using different types of glass capillaries as collimators (e.g., Haller et al. 1995; Janssens et al. 2000). For most diffusion studies, the beam size of the incoming synchrotron beam on the sample varies between 2 and 20 mm. Samples analyzed are typically thin sections of diffusion couples, usually oriented so that the incoming synchrotron beam is perpendicular to the direction of diffusion (Fig. 32). Because the high-energy synchrotron beam penetrates through the whole thickness of the

Diffusion Couple

Glass doped with trace elements

50

µm

Excitation volume 45°

r to ec

Trace elementfree glass

to

Ca

.b

nc

Sy

de t

ry

a pill

45°

m ea

Figure 32. Sample geometry for the analysis of trace element diffusion profiles in the m-SRXRF spectrometer. Fig. 32

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sample without significant loss of energy, the variation with depth cannot be resolved. This characteristic is similar to transmission infrared spectroscopy, but differs from EMPA, LA-ICPMS or SIMS, where volumes from near-surface spots are analyzed. Therefore, for diffusion studies using m-SRXRF analyses, the contact plane of the diffusion couple is aligned parallel to the plane of the incoming beam and the direction of detection. Fluorescence-derived X-ray photons, the Ka lines of elements, are detected with an energydispersive high-purity germanium detector. Due to the high brilliance of the synchrotron source, acquisition times are generally low, e.g., 240 seconds in the studies of Koepke and Behrens (2001) and Hahn et al. (2005) resulting in detection limits in the range of 1 to 15 ppm for the measured elements. Peak fitting and determination of the net peak areas are routinely performed using commercial software (e.g., Van Espen et al. 1977), with the option of processing multi-element spectra (Fig. 33a). For example, in the study of Hahn et al. (2005) 24 trace elements were measured simultaneously. Net intensities are routinely normalized to an internal standard (typically Ca or Fe) to correct for variations in synchrotron beam intensity, dead time and thickness of the sample. For most trace element diffusion studies using diffusion couples, the major element matrix is identical throughout the sample. Therefore, the normalized intensities are directly proportional to the concentration of trace elements in the glass, and the determination of absolute concentrations is not necessary. Typical profiles of trace elements with different geochemical behaviors are presented in Figure 33b. Normalized peak areas may be used directly for the calculation of diffusion coefficients.

Sample preparation Samples for typical trace element diffusion studies using m-SRXRF are couples of trace element-doped and undoped silicate melts (for experimental details see Koepke and Behrens 2001; Koepke et al. 2003; Hahn et al. 2005). After experimental runs, capsules are cut perpendicular to the contact plane of the couple for preparation of doubly polished thin sections. Due to the complete penetration of the incoming beam through the sample, the spatial resolution of this method is strongly dependent on the sample thickness. Moreover, since self-absorption is relatively high for the elements used for internal standardization, and practically negligible for elements of high atomic numbers, it must be ensured that the samples are of homogeneous thickness. Therefore, samples for m-SRXRF studies have to be prepared as reasonably thin sections; for example, 50 ± 1 mm thick sections were used in the studies of Koepke and Behrens (2001) and Hahn et al. (2005). It should be noted that a decrease of sample thickness will also reduce the intensity of the fluorescence lines, resulting in lower precision and poorer detection limits of the analyzed elements.

Applications of m-SRXRF for measuring trace element diffusivities in silicate melts Pioneering studies using m-SRXRF to investigate diffusion in silicate glasses were conducted by Baker and Watson (1988), who measured diffusion profiles in complex Cl- and F-bearing silicate melts, Baker (1989) who investigated tracer versus trace element diffusion using Sr isotopes, and Baker (1990) who analyzed Rb, Zr, Sr and Nb concentration profiles generated during chemical interdiffusion between a dacite and a rhyolite melt. Carroll et al. (1993) measured Kr diffusion in different silicic melts including pure SiO2 and rhyolitic compositions. All of these studies considered only a few selected elements, which were analyzed by m-SRXRF. Ten years later, Koepke and Behrens (2001) benefited from advancements in m-SRXRF permitting the collection of multi-element spectra with one acquisition, and started to obtain “internally consistent” data sets for trace element diffusivities in a synthetic haploandesitic melt with and without added water. This approach minimizes the relative errors of the diffusion coefficients, and such datasets provide the opportunity to explore systematic dependencies of diffusivities on ionic charge and radius. Doped elements can be grouped geochemically as LFSE (Rb, Sr, Ba), REE (La, Nd, Sm, Eu, Gd, Er, Yb, Y), HFSE (Ti, Zr, Nb, Hf, Ta), and transition elements

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Zr Nb

10 000

Sr Y Rb

(a)

Ba La

Intensity [counts]

Nd Sm Gd Eu Er

1 000

Yb Hf Ta

Ag

100

12

17

22

27

32

37

42

47

52

57

Energy [keV]

Intensity, normalized

1.0

(b)

0.8 0.6 Zr

0.4

La Zn

0.2

Sr

0.0 0

1000

2000

3000

4000

5000

Distance [µm] Figure  33. (a) m-SRXRF spectrum of a haploandesitic glass doped with various trace elements at the ~ 300 ppm level, which was used as starting material in the trace element diffusion study of Koepke and Behrens (2001). Only the Ka-lines are indicated. Above ~32 keV both Ka1 and Ka2 lines are visible. Peaks not labeled correspond to Kβ-lines. Conditions for the m-SRXRF measurements: 20-mm capillary; acquisition time of 400 seconds (real time); AlFig.33 absorber of 1 mm thickness. The presence of Ag is a due to a contamination resulting from sample polishing. (b) Diffusion profiles for selected trace elements measured with m-SRXRF obtained from a diffusion couple of haploandesitic melt containing 5 wt% H2O. Experimental conditions: Temperature = 1583 K; pressure = 500 MPa; run time = 1 hour. Conditions for the m-SRXRF measurements: Acquisition time = 240 seconds (real time) for each spectrum, resulting in ~ 4 h for the whole profile; absorber = 1 mm thick Al; capillary = diameter of 20 mm, non-focusing. The profiles are normalized to the peak net area of Ca. Shown are examples for trace elements of different chemical behavior with fast (Sr), medium (Zn, La), and slow (Zr) diffusivity. For details see Koepke and Behrens (2001).

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(Sc, Cr, Fe, Ni, Zn). Most of the investigated elements showed a linear relation between log diffusivity and log viscosity, permitting the prediction of diffusivities in hydrous andesite systems at various conditions. This relation could potentially be used to estimate trace element diffusivities for silicate melts with different compositions provided viscosity data are available. Due to the simultaneous measurement of all trace elements, ratios between diffusion coefficients can be obtained with high precision. The precision of the analytical method is determined mainly by the scatter of data points along a profile, which decreases with increasing intensity of the fluorescence signal of the element (e.g., high fluorescence signal for elements like Y and Zr; and low signals for elements like Ta of Hf; see Fig. 33). This results in a slightly different fit parameter when evaluating duplicate analyses of concentration profiles from the same sample. Based on duplicate profiles in different samples, the elements can be divided into three groups: those with high reproducibility of diffusion coefficients (within 3-5 %): Rb, Sr, Y, Zr, Nb; those with intermediate reproducibility (within 10-15 %): Ba, Cr, Fe, Ni, Zn, light REE; and those with low reproducibility (within 30-50 %): Yb, Er, Hf (for details and individual errors of estimated diffusion coefficients see Koepke and Behrens 2001). Difficulties arise in systems with many trace elements in complex matrices due to spectral interferences between elements of interest. Behrens and Hahn (2009) overcame this problem by separating the conflicting trace elements into two sets. Hahn et al. (2005) used the same analytical equipment in measuring 24 trace elements, representing different geochemical groups, in hydrous rhyolitic glasses. These authors applied both m-SRXRF and SIMS in analyzing the concentration profiles, and found that multiple diffusivities derived from both techniques are in very good agreement for most elements. They showed that some trace elements could not be reliably quantified with m-SRXRF. These include Ta and Pb (these elements are used for detector collimator material), Ti, V (low energy of Ka lines), Co (Ka lines for Co have overlaps with Fe Ka lines) and Cr, Ni, Cu, Zn (overlaps with L-lines of REEs). In contrast to m-SRXRF, in SIMS analyses elements are measured sequentially, which is much more time-consuming, so fewer trace elements could be analyzed in a single session. In addition, some elements are difficult to analyze with SIMS due to isobaric interferences (for example, interference of NaSi with V, and interference of CaO with Ni) or very low yields (e.g., Sn). Recently Behrens and Haack (2007) and Behrens and Hahn (2009) obtained “internally consistent” data sets for trace element diffusivities using m-SRXRF in soda-lime-silicate glass melts and in potassium-rich trachytic/phonolitic melts, respectively.

Acknowledgments We thank Harald Behrens for his careful review of the chapter and insightful comments. RLH acknowledges NSF (EAR-0622775) for support of the ASU SIMS facilities and helpful advice from J. Ganguly (University of Arizona), J. Boyce (UCLA) and Y. Guan (Caltech). YZ acknowledges support by NSF (EAR-0838127). DJC thanks Jon Price for helpful advice and comments on various sections of the chapter.

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Reviews in Mineralogy & Geochemistry Vol. 72 pp. 171-225, 2010 Copyright © Mineralogical Society of America

Diffusion of H, C, and O Components in Silicate Melts Youxue Zhang Department of Geological Sciences University of Michigan Ann Arbor, Michigan 48109-1005, U.S.A. [email protected]

Huaiwei Ni Bayerisches Geoinstitut University of Bayreuth D-95440 Bayreuth, Germany INTRODUCTION Silicate melts are complicated in structure and composition. Hence, diffusion in a melt (in this chapter, a melt includes both liquid and glass) can be very complicated due to the presence of many components, each of which can be present in different species, as well as various kinds of diffusion, such as self-diffusion, tracer diffusion, interdiffusion, effective binary diffusion and multi-component diffusion. In this chapter, the diffusion of H, C and O components is reviewed. H may be in the form of H2O molecules (hereafter referred to as H2Om), OH groups (hereafter referred to as OH), H2 molecules (hereafter referred to as H2), and other species. H2Om and OH will be collectively referred to as the H2O component, or the hydrous component. C may be present as CO2 molecules (hereafter referred to as CO2,molec), CO32− groups (hereafter referred to as CO32−), and other species (such as CO and CH4). CO2 molecules and CO32− groups will be collectively referred to as total CO2 (CO2,total) or the CO2 component. O may be present in the form of network oxygen (such as bridging oxygen, non-bridging oxygen, and free O2−), H2Om, OH, CO2,molec, CO32−, O2 molecules (hereafter referred to as O2), etc. H2O and CO2 are the major volatile components in melts and their diffusion plays a critical role in bubble growth (e.g., Proussevitch and Sahagian 1998; Gardner et al. 2000; Liu and Zhang 2000; Wang et al. 2009), magma degassing (e.g., Bottinga and Javoy 1990; Navon and Lyakhovsky 1998; Proussevitch and Sahagian 1998), magma fragmentation (Zhang 1999a), volcanic eruptions (e.g., Proussevitch and Sahagian 1996), and welding of erupted volcanic particles (e.g., Sparks et al. 1999). Hence, understanding H2O and CO2 diffusion is important in terms of volcanic hazard mitigation and global volatile budgets. Diffusion of H2O may also play a role during diffusive and convective growth and dissolution of hydrous minerals (e.g., Chen and Zhang 2008, 2009; Zhang 2008). In glass industry, H2O diffusion is important to glass stability and strength (e.g., Doremus 1973). On the other hand, oxygen is the most abundant element in silicate melts, and its diffusion controls many reactions and transport properties of melts and is related to the melt structure. The diffusion of H2O, CO2 and oxygen is assessed together because of the following: (1) The diffusion of these three components is all affected by the presence of multiple species. Hence, the diffusion belongs to the same category of multi-species diffusion. (2) To understand oxygen diffusion, it is often necessary to understand the diffusion of the individual oxygen species including especially H2O (as well as other oxygen-bearing species such as bridging oxygen, non-bridging oxygen, free O2−, O2 molecules, OH, CO2 and CO32−). (3) As the major volatile components in silicate melts, 1529-6466/10/0072-0005$10.00

DOI: 10.2138/rmg.2010.72.5

Zhang & Ni

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H2O diffusion and CO2 diffusion often must be considered together. For completeness of this review, molecular H2 and O2 diffusion is also included. Below, we first review H2O (including OH) and molecular H2 diffusion, then CO2, and finally oxygen diffusion (including molecular O2 diffusion). Other possible species of H, C, and O include CO and CH4. However, in natural silicate melts, no data are available for the diffusion of these species, and the concentrations of these species are low, meaning that they are unlikely to play a major role in the transport of H, C and O in natural silicate melts.

diffusion of the H2O component A review covering H2O diffusion was published recently (Zhang et al. 2007). Nonetheless, major progress has been made since then. As will be seen below, syntheses of experimental data to formulate more general models in several silicate melts have been carried out since 2007. Because H2O speciation is the key in understanding the complicated behavior of H2O diffusion, we first discuss both the equilibrium and kinetic aspects of H2O speciation to prepare for the discussion of H2O diffusion.

H2O speciation: equilibrium and kinetics In natural silicate melts, H is present in the oxidized form as the hydrous component. Dissolved H2O in silicate melts is present in at least two species: one is H2Om, and the other is OH (Stolper 1982a,b). Total H2O content will be referred as H2Ot. Under highly reducing conditions, there may also be molecular hydrogen (H2) that may transport the hydrogen component. In natural and experimental melts, the presence of H2 has not been detected yet. H2 diffusion will be discussed in a later section. H2Om and OH can interconvert through the following reaction in melts (Stolper 1982a,b):

H2Om(melt) + O(melt)  2OH(melt)

(1)

where O means an anhydrous oxygen ion, ionic charge is ignored, and the phase is indicated inside the parentheses. The two hydrous species, H2Om and OH, can be detected easily by nearinfrared (NIR) spectroscopy (Stolper 1982a; Cherniak et al. 2010, this volume). To convert IR peak heights or areas to H2Om and OH concentrations requires independent calibration to determine molar absorptivities. The equilibrium constant of the above reaction can be written as: K=

[OH]2e [H 2 Om ]e [O]e

(2)

where brackets mean activities approximated by mole fractions, and the subscript “e” means at stable or metastable equilibrium. The mole fractions are calculated as: Cw 18.015 [H 2 O t ] = (100 − Cw ) Cw + 18.015 W C1 [H 2Om ] = [H 2O t ] Cw

(3)

[OH] 2([H 2O t ] − [H 2Om ]) = [O] = 1 − [H 2Om ] − [OH]

where Cw is wt% of H2Ot (at 2 wt%, Cw = 2), C1 is wt% of H2Om, and W is the molar mass of the dry melt on a single oxygen basis (Stolper 1982a; Zhang 1999b; see Table 1). The composition

Trachyte

Basalt 1

Basalt 2

12

13

14

46.12

50.60

59.90

62.20

62.54

57.21

67.54

65.03

76.38

75.45

72.26

77.30

76.14

76.59

SiO2

1.50

1.88

0.39

0.70

0.84

0.77

0.67

0.20

0.19

0.04

0.10

0.08

TiO2

16.11

13.90

18.00

20.10

16.74

17.50

15.74

16.63

10.32

10.05

15.83

13.00

13.53

12.67

Al2O3

10.84

12.5

0.89

0.03

5.55

7.58

4.28

4.16

4.37

4.29

0.61

0.50

1.00

FeO

0.20

0.23

0.12

0.02

0.10

0.11

0.12

0.12

0.06

MnO

7.60

6.56

3.86

2.32

2.97

4.27

1.43

1.96

0

0.02

0.1

0.03

MgO

13.32

11.4

2.92

10.11

6.48

7.59

4.40

5.10

0.17

0.22

0.5

0.52

CaO

3.56

2.64

4.05

4.30

3.20

3.31

3.58

3.95

5.11

5.27

4.14

3.8

4.65

3.98

Na2O

0.76

0.17

8.35

0.98

1.69

1.60

2.15

2.70

4.66

4.56

3.66

4.7

5.68

4.88

K2O

0.21

0.21

0.02

P2O5

37.15

36.59

34.55

33.57

34.13

34.98

33.49

33.84

33.19

33.26

32.37

32.39

32.60

32.55

W

k

n

m

l

k

j

i-k

h-j

g

f

f

e

d

a-c

Ref

References: a. Shaw (1974), Delaney and Karsten (1982), Karsten et al. (1982), and Zhang et al. (1991a); b. Zhang and Behrens (2000); Behrens et al. (2007); c. Ni and Zhang (2008); d. Nowak and Behrens (1997); e. Okumura and Nakashima (2004); f. Behrens and Zhang (2009); g. Wang et al. (2009); h. Liu et al. (2004b); i. Ni et al. (2009b); j. Behrens et al. (2004); k. Okumura and Nakashima (2006); l. Ni et al. (2009a); m. Freda et al. (2003); n. Zhang and Stolper (1991).

The compositions are listed on the anhydrous basis. W is weight of the dry melt per mole of oxygen (g/mol) assuming all Fe is ferrous. (Including ferric Fe would lower W values slightly.) For simplicity in treatment, small difference in W between similar compositions is ignored in literature. For example, for compositions 1-4, W = 32.49; for compositions 5 and 6, W = 33.24 (Wang et al. 2009 used a slightly different value of 33.14 by considering Fe2O3), etc. MAC is a peraluminous rhyolite; NSL and CBS rhyolite are peralkaline rhyolites.

Haploandesite

Dacite 1

7

11

CBS rhyolite

6

Andesite 2

NSL

5

10

MAC

4

Dacite 2

Rhyolite 2

3

Andesite 1

AOQ

2

9

Rhyolite 1

1

8

Composition

ID

Table 1. Chemical composition (wt%) of melts for H2O diffusion studies.

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and W for some melts are listed in Table 1. The temperature dependence of K takes the following form (e.g., Zhang et al. 1997a):

lnK = A + B/T (4)

where A and B are constants related to the standard state entropy and enthalpy of Reaction (1). The equilibrium speciation of rhyolite, dacite and andesite melts has been investigated. Before an understanding of equilibrium speciation, it is necessary to understand (i) the kinetics on how long it would take to reach equilibrium (e.g., Zhang et al. 1991a, 1995, 1997b, 2000; Withers et al. 1999; Liu et al. 2004a; Hui et al. 2008), and (ii) the kinetic effect on the species concentrations during cooling (Dingwell and Webb 1990; Zhang 1994) so as to understand whether species concentrations can be preserved upon quench. These have been reviewed in Zhang (1999b), Zhang et al. (2007) and Hui et al. (2008) and only a brief summary is given below. Speciation data include both quench data and in situ data. Both the quench data and in situ data suffered from problems in the early stages as shown below. The experimental studies of H2O speciation are probably the most debated experimental topics concerning H2O in silicate melts, with two major issues (the quench effect on quench experiments, and the temperature dependence of molar absorptivities on in situ IR analyses) summarized below. Quench effect. The equilibrium speciation data obtained by the quench technique before 1990 (e.g., Silver and Stolper 1989; Stolper 1989; Silver et al. 1990) at temperatures above ~1000 K suffered from the quench effect. The data obtained at significantly lower temperatures in these papers are generally acceptable. The time for Reaction (1) to reach equilibrium decreases as temperature and H2Ot increase. When the experimental temperature is high (typically ~1000 K, but depending on H2Ot and cooling rate), it often takes less than a second (shorter at higher H2Ot) for Reaction (1) to reach equilibrium. That is, there is continuous reaction as the sample cools down. To facilitate discussion, we define quotient Q as: Q=

[OH]2 [H 2Om ][O]

(5)

The expression of Q is the same as that of K except that K is restricted to the case when equilibrium is reached. If Q > K, the reaction goes to the left to form H2Om. Otherwise, it goes to the right to form OH. For a sample cooled to room temperature, the measured species concentration can be used to calculate Q, which is the apparent equilibrium constant Kae of this sample (it is called “apparent equilibrium” since the sample did not exactly reach equilibrium at the corresponding temperature). This apparent equilibrium constant corresponds to an apparent equilibrium temperature (Tae) as follows, lnQ= A +

B Tae

(6)

That is, the apparent equilibrium temperature is the temperature calculated from the quotient Q assuming Q were the same as K even though equilibrium is not reached at any temperature (comparing Eqns. 4 and 6). During cooling at various quench rates, the variation of Q and Tae as a function of temperature is illustrated in Figure 1 (Zhang 1994, 2008; Zhang et al. 1997b, 2000). Starting from a high temperature, if cooling rate is high, the resulting Tae after cooling down is also high, and vice versa. Even at a high cooling rate, species concentrations may change upon cooling. For a rhyolite melt containing 2.5 wt% H2Ot equilibrated at a temperature of 1000 K and cooled down at 70 K/s, the resulting Tae is only about 800 K (Zhang et al. 2000), rather than the experimental equilibrium temperature of 1000 K. Dingwell and Webb (1990) were the first to analyze this problem using the concept of cooling-rate-dependent glass transition temperature, and assumed that the apparent equilibrium temperature for Reaction (1) is similar to the glass transition temperature, which was verified later (e.g., Zhang et al.

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Figure 1. Schematic evolution of Q and Tae for rapid cooling and slow cooling of a hypothetical reaction. Fig 1. Schematic evolution of Q and Tae for rapid cooling and slow cooling of a hypothetical reaction.

1997b, 2003; Behrens and Nowak 2003; Hui et al. 2008). Because the temperature was high for many early experiments using the quench method, equilibrium H2O speciation was not fully quenchable even with the fastest achievable quench rate. That is, the reported data did not reflect true speciation at the experimental temperature; only Tae can be obtained. On the other hand, when the experimental temperature is low enough (usually below 900 K, but depending on H2Ot and quench rate), reaction during quench is negligible and Tae is roughly the same as the experimental temperature. However, the experimental temperature cannot be too low (such as < 600 K) either, otherwise it would require too long a duration to reach equilibrium. Based on the above considerations, the appropriate temperature range for quench method is typically 650-900 K (which are referred to as intermediate temperatures). Dependence of molar absorptivities on temperature. The first in situ IR measurements indicated that the molar absorptivities depend on temperature (e.g., Keppler and Bagdassarov 1993). However, for some years afterwards, temperature effects were not considered in interpreting in situ IR speciation data (Nowak and Behrens 1995; Shen and Keppler 1995; Sowerby and Keppler 1999) and the results were inconsistent with speciation data quenched from intermediate temperatures (e.g., Zhang et al. 1991a, 1995; Ihinger et al. 1999). To resolve the inconsistency, two groups with different views collaborated (Zhang and Behrens 1998; Withers et al. 1999). They demonstrated that (i) molar absorptivities vary with temperature even below the glass transition, which led to an apparent change in species concentrations as well as inaccuracy of K values in Nowak and Behrens (1995), Shen and Keppler (1995), and Sowerby and Keppler (1999); and (ii) after accounting for the temperature effect of the molar absorptivities, the speciation data from quench experiments from intermediate temperatures and those from in situ experiments are consistent, as also demonstrated by other studies (Zhang 1999b; Withers and Behrens 1999; Nowak and Behrens 2001; Behrens and Nowak 2003).

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However, the two data sets cover different temperatures (with only small overlap) and seem to indicate a slightly different standard state enthalpy change for Reaction (1). Summary of speciation data. After two vigorous debates discussed above, one on the quench effect, and the other on the in situ measurements, it is now known that the quench method can be used to obtain reliable speciation data when the experimental temperature is low enough (usually below ~900 K, but the exact temperature depends on H2Ot and quench rate, and must be determined experimentally for a given system), and the in situ method can also be used to obtain reliable speciation data when the temperature dependence of the molar absorptivities is accounted for. In addition, the speciation constant as a function of temperature may be inferred by estimation of the glass transition temperature from viscosity models and assuming the glass transition temperature is the apparent equilibrium temperature (Dingwell and Webb 1990; Zhang et al. 2003); referred to as the fictive temperature method hereafter. Subsequently, speciation in the following melts have been investigated further: albite and some synthetic sodium-calcium aluminosilicate melts (Ohlhorst et al. 2000), AOQ (Qz28Ab38Or34 where Qz means quartz, Ab means albite and Or means orthoclase, and the values indicate wt%) melt (Nowak and Behrens 2001; Behrens and Nowak 2003), dacite melt (Liu et al. 2004a), andesite melt (Botcharnikov et al. 2006), NS4 (Na2O·4SiO2, or Na2Si4O9) and NS6 (Na2O·6SiO2, or Na2Si6O13) melts (Behrens and Yamashita 2008), high-pressure rhyolite melts (Hui et al. 2008), and haploandesite melt (Ni et al. 2009a). The precision of the speciation data depends on the method (the 2σ precision in lnK is often about 0.025 for the quench method, 0.1 for the in situ method, and 0.1 to 0.5 for the fictive temperature method). The accuracy, on the other hand, depends on the accuracy of the infrared calibration for species concentrations, which is not well characterized. For example, even for rhyolite glass, which has been studied most extensively (Newman et al. 1986; Zhang et al. 1997a; Withers and Behrens 1999), the possible variation of molar absorptivities with H2Ot and quench rate is still not accurately known. Studies on hydrous speciation in Fe-rich melts such as dacite melt have another difficulty: the high iron concentration leads to significant and broad absorption bands that overlap with hydrous species bands in the NIR. What makes the problem worse is the variation of the shape and intensity of the iron-related bands before and after heating. Hence, many authors have opted to investigate iron-free analogs of natural silicate melts. With the above preamble and precaution, some speciation data are shown and compared in Figure 2. The pressure effect on speciation is examined by Hui et al. (2008); K may increase or decrease with pressure, but the effect is not very large and not included in Figure 2. From Figure 2, several observations can be made: (i) K in sodium silicates (NS4 and NS6) is much greater than in natural aluminosilicate melts and their analogs (as well as albite melt), supporting the suggestion of formation of NaOH clusters (Kohn et al. 1989), especially in the absence of Al; (ii) K in a natural rhyolite is similar to that in haplorhyolite (AOQ); the similarity also holds for natural andesite and haploandesite; (iii) K increases with temperature (meaning Reaction (1) is an endothermic process); (iv) K increases slightly from rhyolite to dacite to andesite melts, which means that K values do not simply increase with Na2O because the Na2O (as well as the K2O) content decreases from rhyolite to dacite to andesite melts. The speciation models (Zhang et al. 1997a; Liu et al. 2004a; Ni et al. 2009a) that are often used in diffusion studies (Zhang and Behrens 2000; Behrens et al. 2007; Ni and Zhang, 2008; Behrens and Zhang 2009; Wang et al. 2009; Ni et al. 2009a,b) are: 3110   = K rhyolite exp  1.876 − T   2634   = K dacite exp  1.49 − T   2453   = exp  1.55 − K haploandesite T  

(7a) (7b) (7c)

Youxue Zhang and Ni (Ch 5)

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Figure 2. Speciation data (with 2σ errors) and regression lines in different melts at ≤ 0.8 GPa. For rhyolite melts, the data are also limited to H2Ot ≤ 3 wt% because the given calibration (Zhang et al. 1997a) is best 2. Speciation data (with 2σ errors) and regression lines in different melts at ≤ 0.8 GPa. For rhyolite melts, the applied to HFig. 2Ot ≤ 3 wt%. Error bars for lnK in rhyolite and haploandesite melts are smaller than the size of the symbols. Data sources: using the calibration quench method: Zhang al. applied (1997a); et al. data are also limited to H2rhyolite Ot ≤ 3 wt%melt because the given (Zhang et al. 1997a) et is best to H2Ihinger Ot ≤ 3 (1999); Hui et al. (2008); AOQ1: AOQ melt (Table 1) using the in situ method (Nowak and Behrens 2001); wt%. Errorusing bars forthe lnK fictive in rhyolite and haploandesite melts are smaller thanand the size of the symbols. sources: AOQ2: AOQ melt temperature method (Behrens Nowak 2003); Data dacite melt using the quench method: Liu et al. (2004a); haploandesite melt using the quench method: Ni et al. (2009a); rhyolite melt using the quench method: Zhang et al. (1997a); Ihinger et al. (1999); Hui et al. (2008); AOQ1: AOQ andesite melt using the fictive temperature method: Botcharnikov et al. (2006); NS4 (Na2O·4SiO2) and NS6 (Na2O·6SiOmelt fictive temperature method: Behrens and Yamashita (2008). (Table using 1) usingthe the in situ method (Nowak and Behrens 2001); AOQ2: AOQ melt using the fictive temperature 2) melts method (Behrens and Nowak 2003); dacite melt using the quench method: Liu et al. (2004a); haploandesite melt using the quench method: Ni et al. constants (2009a); andesite using the fictive temperature method: Botcharnikov al. Other expressions of equilibrium aremeltavailable (e.g., Ihinger et al. 1999; etNowak and Behrens 2001; Hui 2008), but they have not been much used in diffusion studies either (2006); NS4 et (Na2al. O·4SiO ) and NS6 (Na O·6SiO ) melts using the fictive temperature method: Behrens and 2 2 2 because they are new, or because they are complicated, or because of the need to maintain Yamashita (2008). internal consistency with other studies. Using different K values would impact the retrieval of the diffusivities of individual H2O species but only negligibly the retrieval of DH2Ot because the latter is essentially constrained by the length and shape of an H2Ot concentration profile.

More detailed studies of hydrous species reaction kinetics have been carried out through isothermal experiments or controlled cooling rate experiments (Zhang et al. 1995, 1997b, 2000; Liu et al. 2004a; Hui et al. 2008). However, in treating H2O diffusion, for simplicity, quasi-equilibrium is often assumed for the species reaction, and complicated kinetics are not incorporated. Hence, the kinetics is not discussed in detail here. The variations of H2Om and OH concentrations with H2Ot for a fixed equilibrium constant K are shown in Figure 3. It can be seen that H2Om is not proportional to H2Ot (that is, the Xm/X curve in Fig. 3 is not a horizontal line). Instead, at low H2Ot, H2Om is roughly proportional to the square of H2Ot (in other words, H2Om/H2Ot ratio is proportional to H2Ot), meaning a more rapid increase of H2Om concentration as H2Ot increases. Below about 0.2 wt% H2Ot, almost all H2Ot is OH (which does not mean that OH would play a main role in H2O diffusion, as will be clear later in this chapter). This behavior arises from the stoichiometric coefficient 2 for OH in Reaction (1), which means the square of OH concentration is proportional to H2Om concentration, leading to the complicated relation between H2Om and H2Ot concentrations as well as the complicated behavior of H2Ot diffusion.

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Zhang Page& 3 Ni

Figure 3. Species fractions, dXm/dX, Xm/(dXm/dX) and (Xm/X)/(dXm/dX) versus H2Ot at a fixed K of 0.5 where Xm = [H2O X fractions, = [H2O XmX/X =m[H ]/[H (fraction Hwhere Fig. Species dXt]. /dX)2O andm(X versus H2of Ot atmolecular a fixed K of 0.5 m]3.and 2Ot]m/dX) 2O).XF m/dX, m/(dX m/X)/(dX m =is fraction of OH. DH2Ot and DH2Om are related through dXm/dX; D18O and DH2Ot are related through Xm/(dXm/dX); and 2H[H2Om] and X = [H2Ot]. Xm/X = [H2Om]/[H2Ot] (fraction of molecular H2O). F is fraction of OH. DH2Ot and 1 H self-diffusivity in hydrous melts and DH2Ot (chemical diffusivity) are related through (Xm/X)/(dXm/dX). DH2Om are related through dXm/dX; D18O and DH2Ot are related through Xm/(dXm/dX); and 2H-1H self diffusivity in hydrous melts and DH O are related through (Xm/X)/(dXm/dX). H2O diffusion literature 2 t

Technological improvements and theoretical modeling were the keys in the development of H2O diffusion studies. Most H2O diffusion experiments can be characterized as follows: hydration of an originally homogeneous and almost water-free sample by exposure to water vapor or fluid, dehydration of an initially homogeneous hydrous glass by exposure to dry N2 or Ar, and diffusion couple (two samples with different water contents juxtaposed against each other). In the early years of H2O diffusion studies in silicate melts, diffusion profiles could not be measured, so either the bulk H2O mass loss from dehydration or mass gain from hydration was determined by weight change or mass spectrometry or infrared spectroscopy, from which the diffusivity was inferred (e.g., Shaw 1974). Later, advancements in ion microprobe analyses (Coles and Long 1974; Hofmann 1974) allowed the microanalytical determination of H2O diffusion profiles (Delaney and Karsten 1981), but species concentrations could not be determined. Afterwards, advancement in infrared spectroscopy (Stolper 1982a,b; Acocella et al. 1984) led to the measurement of concentration profiles of both H2Om and OH species (Zhang et al. 1991a). Coupled speciation and diffusion studies (Zhang et al. 1991a) helped establish the basic mechanism of H2O diffusion. Later studies, especially those at high temperatures, used only H2Ot concentration profiles from micro-IR to model H2O diffusion because species concentrations cannot be quenched from high temperatures. In the glass and materials science literature, H2O diffusion studies began in the 1960’s. The studies were limited to low H2O concentrations, with H2Ot typically no more than 0.1 wt% (e.g., Drury and Roberts 1963; Cockram et al. 1969; Burn and Roberts 1970; Lanford et al. 1979; Houser et al. 1980; Tsong et al. 1980; Nogami and Tomozawa 1984a,b; and the recent review by Shelby 2008). The average OH concentration (proportional to H2Ot) in a thin wafer upon heat treatment is monitored by IR and the diffusivity is obtained from the diffusive mass loss equation for a thin wafer (Crank 1975): F=

4 D πL

t

(8)

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where F is the fractional mass loss (H2Ot loss divided by initial H2Ot), L is the thickness of the wafer, t is time, and D is apparent diffusivity (more specifically, D is diffusion-out diffusivity of H2Ot, denoted as Dout, see below). It is often observed that Dout is proportional to H2Ot, from which it was suggested that the diffusing species is H2Om (because the H2Om concentration is proportional to the square of the H2Ot concentration: a rigorous explanation can be found below in Eqn. 11) (Doremus 1969, 1973; Ernsberger 1980; Smets and Lommen 1983; Nogami and Tomozawa 1984a,b), but other explanations included the interdiffusion of hydronium H3O+ and cations (e.g., Cockram et al. 1969; Doremus 1975; Lanford et al. 1979; Houser et al. 1980; Tsong et al. 1980), or the depolymerization of the silicate network (e.g., Haller 1963; Roberts and Roberts 1966; Delaney and Karsten 1981). These other explanations gradually become less cited as later direct measurements of species concentration profiles and modeling showed that assuming H2Om as the diffusing species can model the detailed diffusion profiles almost perfectly. Shaw (1974) was the first to investigate H2O diffusion in a geologically relevant silicate melt. He determined total mass gain due to hydration, and inferred that H2Ot diffusivity depends strongly on H2Ot content. Jambon (1979) carried out dehydration experiments and measured the mass loss from the sample. This study was later corrected by Jambon et al. (1992): because of an error in the estimated initial H2Ot in the sample (0.38 wt% in Jambon 1979 versus the correct 0.114 wt% in Jambon et al. 1992), the diffusion-out H2Ot diffusivity was corrected upward by a factor of 11. Delaney and Karsten (1981) and Karsten et al. (1982) were the first to measure H2O concentration profiles using the ion microprobe after hydration experiments. The profiles were fit by numerical solutions assuming some dependence of DH2Ot on H2Ot. Good fits were obtained for DH2Ot = D0exp(bCw) where Cw is the H2Ot concentration and D0 and b are two fitting parameters. Hence, it was assumed that DH2Ot depends exponentially on H2Ot. However, the expression does not work at low H2Ot (such as below 0.5 wt%) because it implies that DH2Ot approaches a constant as Cw approaches zero, but prior experimental data in the glass science literature showed that DH2Ot is proportional to Cw even down to very low Cw. Lapham et al. (1984) used a similar technique and compared diffusion of 1H2O and 2H2O (or D2O) in rhyolite melts using ion microprobe measurements. Their data apparently showed that the 1H2O diffusivity was 2 times the 2H2O diffusivity. However, a later publication from the same laboratory (Stanton et al. 1985) retracted this claim and found no measurable difference between 2H2O and 1H2O diffusivities. Wasserburg (1988) analyzed the role of H2O speciation in H2O diffusion. Zhang et al. (1991a) were the first to utilize FTIR to measure concentration profiles of both H2Om and OH species in dehydration studies of natural obsidian, at 676-823 K, 0.1 MPa, and ≤ 1.7 wt% H2Ot. By considering the interconversion reaction between H2Om and OH and the diffusion of both H2Om and OH in concert, they concluded that OH diffusivity is negligible compared to H2Om diffusivity (i.e., H2Om is the diffusing species), and H2Om diffusivity is roughly independent of H2Ot in the samples studied. At low H2Ot, a constant H2Om diffusivity leads to proportionality between the H2Ot diffusivity and H2Ot content, consistent with the glass science literature. The much higher diffusivity of H2Om than OH is understandable because H2Om is a neutral molecule hence its diffusion does not require breaking strong bonds, whereas OH is bonded to other cations (Si4+, Al3+, Na+, etc) and its motion requires breaking strong bonds in the silicate structure. In terms of particle size, H2Om is similar or slightly smaller than OH (Shannon 1976; Zhang and Xu 1995). Zhang et al. (1991a) also distinguished H2Ot diffusivities during diffusion-out (dehydration) versus diffusion-in (hydration) experiments (Moulson and Roberts 1961). Because of the dependence of DH2Ot on H2Ot content, the diffusion-in diffusivity is 1.78 times the diffusion-out diffusivity when DH2Ot is proportional to H2Ot under otherwise identical conditions (see also Wang et al. 1996). Zhang and Stolper (1991) investigated H2O diffusion in basalt melt at 1573-1773 K, 1.0 GPa, and ≤0.42 wt% H2Ot, and found that the H2Ot

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diffusivity is roughly proportional to H2Ot, consistent with H2Om being the diffusing species and the H2Om diffusivity being independent of H2Ot. Jambon et al. (1992) re-interpreted their earlier dehydration data and found that (i) the data are consistent with Zhang et al. (1991a) and (ii) disequilibrium in Reaction (1) can significantly affect DH2Ot. Nowak and Behrens (1997) carried out diffusion couple experiments and investigated H2O diffusion in a synthetic rhyolite melt (AOQ) at 1073-1473 K, 0.05-0.5 GPa, and ≤ 9.0 wt% H2Ot. They demonstrated that the H2Ot diffusivity at H2Ot > 3 wt% increases exponentially with H2Ot, implying that the H2Om diffusivity cannot be constant but must increase with H2Ot content at H2Ot > 3 wt%. They fit logDH2Ot as a polynomial function of H2Ot. Zhang and Behrens (2000) studied H2O diffusion in rhyolite melts at 673-1473 K, 0.0001-0.81 GPa, and ≤ 7.7 wt% H2Ot. They confirmed the observations of Nowak and Behrens (1997) and found that the diffusion profiles can be accurately modeled by assuming H2Om is the diffusing species and DH2Om increases exponentially with H2Ot, DH2Om = D0eaX, where X is mole fraction of H2Ot, D0 is the DH2Om value as X approaches zero, and a is a parameter characterizing how rapidly DH2Om increases with H2Ot. The exponential increase means that at low H2Ot, DH2Om increases only slowly with H2Ot and can be treated roughly as a constant (e.g., for a = 25 and from 0 to 1 wt% H2Ot, eaX varies from 1 to 1.56, only slightly outside the experimental uncertainty of diffusivity determinations at high temperatures), consistent with the observations of Zhang et al. (1991a). The exponential increase is also consistent with the diffusivities of neutral molecular species such as Ar (Behrens and Zhang 2001) and CO2 (Watson et al. 1982; Watson 1991). Later experimental studies in general followed the framework established in Zhang et al. (1991a), Zhang and Stolper (1991), and Zhang and Behrens (2000). Freda et al. (2003) investigated H2O diffusion in a trachyte melt at 1334-1601 K, 1 GPa, and ≤ 2.0 wt% H2Ot. Okumura and Nakashima (2004, 2006) developed the in situ FTIR measurements of H2O loss due to dehydration to obtain diffusivities, and reported results on rhyolite melts at ≤ 4.1 wt% H2Ot and on dacite, andesite and basalt melts at ≤ 1.1 wt% H2Ot. Their method works well for rapidly obtaining diffusion data but cannot resolve how DH2Ot depends on H2Ot. Liu et al. (2004b) explored H2O diffusion in dacite melts by dehydration experiments at 824-910 K, ≤ 0.15 GPa, and ≤ 2.5 wt% H2Ot. Behrens et al. (2004) studied H2O diffusion in dacite and andesite melts at 1458-1858 K, 0.5-1.5 GPa, and ≤ 6.3 wt% H2Ot; some of their data are consistent with concentration-independent DH2Ot. Behrens et al. (2007) obtained some H2O diffusion data in rhyolite melts in a study comparing oxygen and H2O diffusion. Ni and Zhang (2008) further resolved the pressure effect on H2O diffusion in rhyolite melts and constructed a general H2O diffusivity model over a large range of T, P and H2Ot. Behrens and Zhang (2009) and Wang et al. (2009) quantified H2O diffusion in peralkaline rhyolite melts. Ni et al. (2009a,b) examined H2O diffusion in dacite and haploandesite melts. All the melt compositions that have been studied for H2O diffusion are listed in Table 1.

H2O diffusion, theory and data summary Strictly speaking, silicate melts are multicomponent systems, and H2O diffusion should fall into the category of multicomponent diffusion. However, the anhydrous melt composition generally does not vary along a diffusion profile in the typical design of the experiments, and addition of H2O mainly causes a dilution effect. Therefore, H2O diffusion in the literature is treated as effective binary diffusion, or the first kind of effective binary diffusion defined by Zhang (2010, this volume) and Zhang et al. (2010, this volume). Because DH2Ot depends on H2Ot concentration, the general equation for one-dimensional diffusion is as follows: ∂X ∂  ∂X  =  DH2 Ot ∂t ∂x  ∂x 

(9)

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where t is time, x is distance, DH2Ot is the diffusivity (effective binary diffusivity) of H2Ot, and X = [H2Ot] (see Eqn. 3a). The above diffusion equation can be solved given how DH2Ot varies with H2Ot, and boundary and initial conditions. For hydration and dehydration experiments, the equilibrium surface concentration and the initial concentration (usually uniform) are the boundary and initial conditions, respectively. For diffusion couple experiments, no boundary condition is needed and the initial concentrations in the two starting halves are the initial condition. If DH2Ot were independent of H2Ot (or X), then the diffusion profile could be fit by an error function solution. However, this is rarely the case: the variation of DH2Ot with H2Ot turns out to be complicated. Experimental results in the glass and geological literature show that when H2Ot is sufficiently low (≤ 2 wt%, but depending on temperature), DH2Ot is proportional to H2Ot (or proportional to X in Eqn. 9). Such an equation can be solved numerically (e.g., Crank 1975) and the solution has been applied to fit diffusion profiles. If a profile can be fit within experimental uncertainty (e.g., Fig. 4), then the proportionality relation is assumed to describe how DH2Ot varies with H2Ot. Youxue Zhang and Ni (Ch 5)

Page 4

Figure 4. Experimental H2Ot diffusion profile (points) in a peraluminous rhyolite melt and two fits. The dashed curve is an error function fit (assuming DH2Ot is independent of H2Ot). The solid curve is a fit assuming DH2Ot is proportional to H2Ot. The solid curve agrees well with experimental data. From Behrens and Zhang (2009).

Fig. 4. Experimental H2Ot diffusion profile (points) in a peraluminous rhyolite melt and two fits. The dashed curve

At high H2OtD(such as >3 wt%, especially at relatively low temperatures such as 800 K), is an error function fit (assuming H O is independent of H2Ot). The solid curve is a fit assuming DH O is the proportionality relation often does not work well (Nowak and Behrens 1997; Zhang and proportional to H2Ot. The solid curve agrees well with experimental data. From Behrens and Zhang (2009). Behrens 2000; Liu et al. 2004b; Ni and Zhang 2008; Ni et al. 2009a,b; Wang et al. 2009). One example is given in Figure 5. Hence, more complicated relations are proposed to describe how DH2Ot varies with H2Ot. Based on our knowledge of H2O speciation, the general and mechanistic approach is to consider the diffusion of both species: 2 t

2 t

∂X  ∂X ∂  ∂X m 1 = DH O + DOH OH  ∂t ∂x  2 m ∂x 2 ∂x 

(10)

where Xm is the mole fraction of H2Om, XOH is the mole fraction of OH, and DH2Om and DOH are the diffusivities of H2Om and OH. The factor 1/2 is due to the fact that one mole of H2Om reacts to form two moles of OH (Reaction 1). Experimental data show DOH 1073 K are scattered and do not follow the trend defined by lower-temperature data. This was explained by a possible reaction of H2 or D2 with silica network at high temperature. They also found that the diffusivity determined at the decay stage of the permeation experiments is more reliable than that determined from the build-up stage, likely because the sample is more fully annealed at the decay stage. Lee (1963) investigated the effect of impurities in the silica glass and obtained new H2 and D2 diffusivity data at 600-1300 K. The effect of impurities including Al2O3 (ranging from 0.1 to 160 ppm) and OH (10 to 3000 ppm) in silica glass turned out to be small as long as care is taken to avoid possible reactions and to use only the steady state and post-steady-state data. The data in Lee et al. (1962) are not discussed further because the work was superseded by Lee (1963). Perkins and Begeal (1971) reported D2 (and noble gas element) diffusivities in silica glass at 298-448 K. Shelby (1977) determined D2 diffusivities in various silica glasses at 450-1000 K, and the differences were found to be no more than 0.17 log units (0.38 ln units) (these are highly reproducible experiments with internal precision better than 10% relative, or 0.04 log units). Shang et al. (2009) developed a new technique and obtained H2 diffusivities at 296-523 K. All of these studies were conducted at low pressures (often less than one atmosphere pressure). Figure 12 compares all the H2 and D2 diffusion data in Lee (1963), Perkins and Begeal (1971), Shelby (1977), and Shang et al. (2009). Data scatter is small; the maximum minus minimum at a given temperature is about 0.4 in lnD. At high temperatures, H2 and D2 diffusion was investigated by the same author (Lee 1963) on the same silica glass, and it was found that H2 diffusivity is greater than D2 diffusivity, as expected, but by only about 20%, not the theoretical 41% difference based on the square root of mass relation. The difference of 20% is

Page 12 Diffusion of H, C, O Components in Silicate Melts

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193

Figure 12. Diffusion data of H2 and D2 from literature (Lee 1963; Perkins and Begeal 1971; Shelby 1977; Shang et al. 2009). Data of Lee (1963) and Shelby (1977) are read from figures in these papers. Nominal 12. data Diffusion data of H2than (in red) D2 (in black) fromNe literature (Leedata 1963;inPerkins Begeal 1971;etShelby errors Fig. on the are smaller theand symbols. He and diffusion silicaand glass (Swets al. 1961; Frank et al. 1961; Perkins and Begeal 1971; Behrens 2010), and molecular H2Om diffusion in nominally 1977; Shang et al. 2009). Data of Lee (1963) and Shelby (1977) are read from figures in these papers. Nominal dry rhyolite melt (Ni and Zhang 2008) are shown for comparison. errors on the data are smaller than the symbols. He and Ne diffusion data in silica glass (Swets et al. 1961; Frank et al. 1961; Perkins and Begeal 1971; Behrens molecular H 2O m diffusion in nominally rhyolite melt (Ni large enough to be distinguished in the2010), sameandglass by the same author whendry self-consistency is high, but roughly within error when results from different silica glasses and different authors and Zhang 2008) are shown for comparison. are compared. At lower temperatures ( 4 GPa. Rubie et al. (1993) and Poe et al. (1997) studied oxygen self-diffusion in dry Na2Si4O9 (or Na2O·4SiO2, NS4) melt. The data (Fig. 19) covering 1893-2800 K and 2.5-15 GPa can be fit by ln D18NaO2 Si4 O9 melt = −17.19 −

12693 − 360 P T

(46)

where D is in m2/s, T is in K. and P is in GPa. The activation energy is surprisingly low, only about 100 kJ/mol (depending on pressure). The activation volume is negative and small, about −3.0×10−6 m3/mol. The maximum error of the above equation in reproducing the experimental data is 0.22 in terms of lnD. Three papers explored oxygen self-diffusion in dry basalt melt (Muehlenbachs and Kushiro 1974; Canil and Muehlenbachs 1990; Lesher et al. 1996). However, the earlier data by Muehlenbachs and Kushiro (1974) and Canil and Muehlenbachs (1990) are scattered (Fig. 16). Lesher et al. (1996) reported three data points at 1 GPa and one datum at 2 GPa. Oxygen self-diffusivities at 1593-1873 K and 1 GPa (solid squares in Fig. 16) can be represented by the following equation (Lesher et al. 1996): basalt melt

ln D18 O

= −12.5 −

20447 T

(47)

where T is in K and D is in m2/s. The activation energy is 170 kJ/mol, smaller than those obtained Youxue Zhang and Ni (Ch 5) Page 19 from other 18O diffusion data in basalt melt at room pressure (377 kJ/mol by Muehlenbachs -20

Oxygen self-diffusion in Na 2Si 4O9 melt

lnD (D in m 2/s)

-20.5 -21 -21.5 -22 -22.5 -23

2.5 GPa 4.0 GPa 6.0 GPa 8.0 GPa 10 GPa 12.5 GPa 15 GPa

-23.5 0.35

0.4

0.45

0.5

0.55

1000/T (T in K) Figure 19. Oxygen self-diffusivity in Na2Si4O9 melt with data from Rubie et al. (1993) (with > in0.45) Poe et al. (1997) (with 1000/T < 0.45). Fig. 19. Oxygen self1000/T diffusivity Na2Siand 4O9 melt with data from Rubie et al. (1993) (with 1000/T > 0.45) and Poe et al. (1997) (with 1000/T < 0.45).

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and Kushiro 1974, and 251 kJ/mol by Canil and Muehlenbachs 1990). The large difference in activation energy is likely due to data uncertainty rather than being real. Because the activation volume is highly variable from dacite to Di58An42 melts (−11×10−6 to −2.0×10−6 m3/mol; see also Tinker et al. 2003), it is not possible to use information from other melts to constrain the pressure effect on oxygen diffusivity in basalt. Three papers investigated oxygen self-diffusion in diopside melt (Dunn 1982; Shimizu and Kushiro 1984; Reid et al. 2001). However, the temperature dependence was not constrained: only at two pressures (0.1 MPa by Dunn 1982 and 3 GPa by Reid et al. 2001) were there diffusion data at different temperatures. The data at 0.1 MPa are scattered and those at 3 GPa are strange in that the diffusivity does not change much from 2073 to 2273 K, with an implied activation energy of only 4 kJ/mol (Fig. 20). An error of a factor of 2.5 on individual D values is needed to allow a more reasonable activation energy of about 250 kJ/mol. Hence, the precision of the oxygen self-diffusion data by Reid et al. (2001) is not high. The reason is not clear. Shimizu and Kushiro (1984) reported oxygen self-diffusion data in jadeite melt at 1673 to 1883 K at 1.5 GPa, and 0.5 to 2.0 GPa at 1673 K (Fig. 21). The diffusivity at 1.5 GPa can be expressed as: melt ln D18jadeite = −10.84 − O

31815 T

(48)

where D is in m2/s, and T is in K. The activation energy is 265 kJ/mol. At 1673 K, the activation volume is −6.4×10−6 m3/mol. Assuming the activation volume is independent of temperature, the P-T dependence of 18O diffusivity in jadeite melt may be written as: melt ln D18jadeite = −10.84 − O

32970 − 770 P T

(49)

where D is in m2/s, T is in K, and P is in GPa. Liang et al. (1996) investigated the compositional effect of 18O self-diffusion in various Youxue (Ch 5) K and 1 GPa. PageThey 20 meltsandatNi 1773 found that 18O self-diffusivity increases CaO-Al2O3-SiO 2 Zhang -20

lnD (D in m2/s)

-21 -22 -23 -24 -25

0.1MPa 1.0GPa 1.7GPa 3GPa 6GPa 7GPa

9GPa 11GPa 12GPa 13GPa 14GPa 15GPa

-26

O self-diffusion in dry diopside melt -27 0.38

0.42

0.46

0.5

0.54

0.58

1000/T (T in K) Figure 20. Oxygen self-diffusivities in dry diopside melt. Data sources are: 0.1 MPa from Dunn (1982); 1.0 GPa and 1.7 GPa from Shimizu and Kushiro (1984); and the rest from Reid et al. (2001). Fig. 20. Oxygen self diffusivities in dry diopside melt. Data sources are: 0.1 MPa from Dunn (1982); 1.0 GPa and 1.7 GPa from Shimizu and Kushiro (1984); and the rest from Reid et al. (2001).

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-27.5

Oxygen self-diffusion in jadeite melt

lnD (D in m2/s)

-28 -28.5 -29 -29.5 -30 -30.5 0.53

0.5 1.0 1.5 2.0

0.54

GPa GPa GPa GPa

0.55

0.56

0.57

0.58

0.59

0.6

1000/T (T in K) Figure 21. Oxygen self-diffusivities in jadeite melt (Shimizu and Kushiro 1984). Fig. 21. Oxygen self diffusivities in jadeite melt (Shimizu and Kushiro 1984).

with decreasing silica and alumina content, as expected. As NBO/T increases from 0.3 to about 0.9 in CaO-Al2O3-SiO2 system, 18O self-diffusivity increases by about an order of magnitude. Figure 22 compares oxygen self-diffusivity in all melts for which the temperature dependence of diffusivity has been determined well at 1 GPa. Oxygen self-diffusivities increase with decreasing SiO2 and Al2O3 contents, or from polymerized to depolymerized melts. From dacite melt (NBO/T ≈ 0.1) to basalt melt (NBO/T ≈ 1), oxygen self-diffusivity increases by two orders of magnitude. It may be inferred that oxygen self-diffusivities depend on oxygen speciation, increasing from bridging oxygen (BO) to non-bridging oxygen (NBO) and then to free oxygen (O2−). The diffusivity of each oxygen species may depend on the overall melt composition. In terms of oxygen self-diffusion, basalt melt (NBO/T ≈ 1) is similar to diopside melt (NBO/T = 2) and Na2Si4O9 melt (NS4 melt, NBO/T = 0.5), whereas dacite melt (NBO/T ≈ 0.1) is similar to jadeite melt (NBO/T = 0) (Fig. 22). But the similarity in each group is not close enough for diffusivities to be merged for a combined fitting. For different melts, there does not seem to be a single compensation temperature where 18O diffusivities in all melts are the same. There are not enough data yet to contemplate a general relation between oxygen selfdiffusivity and natural melt composition under dry conditions. To achieve such a goal, it is necessary to investigate 18O diffusion in dry basalt, dry andesite, and dry rhyolite as a function of temperature and pressure systematically (on par with the investigation of dacite melt), and to examine the compositional dependence. In highly polymerized melts (rhyolite and pure silica), it will be important to make the dry system very dry, e.g., less than 50 ppm H2Ot (depending on temperature), so that the diffusive flux is due to true 18O self-diffusion, not H2O chemical diffusion (see later sections). One may try to use self-diffusivities of other elements such as Si to constrain those of oxygen. However, even though self-diffusivities of oxygen and silicon (both are structural elements) in dry melts are often similar (e.g., Lesher et al. 1996; Poe et al. 1997; Tinker et al. 2003), they may also be significantly different (e.g., Tinker and Lesher 2001; see review by Lesher 2010 and Zhang et al. 2010). It has been shown that oxygen self-diffusivity in dry melts is similar to the Eyring diffusivity defined as D = kT/(λη) where λ is the jump distance and η is the melt viscosity (e.g., Shimizu

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Figure 22. Oxygen self-diffusion data in different melts at 1 GPa. Data sources are: Basalt: Lesher et al. (1996); Diopside and Jadeite: Shimizu and Kushiro (1984); Di58An42: Tinker et al. (2003); and Dacite: Fig. 22. Oxygen self diffusion data in different melts at 1 GPa. Data sources are: Basalt: Lesher et al. (1996); Tinker and Lesher (2001). The long dashed line is for Na2Si4O9 extrapolated using Equation (46). Diopside and Jadeite: Shimizu and Kushiro (1984); Di58An42: Tinker et al. (2003); and Dacite: Tinker and Lesher long dashed line (in green) is for Na2Si4O9 extrapolated using Eq. 46. and Kushiro (2001). 1984;The Tinker et al. 1994; Fig. 27a in this chapter). Therefore, even though 18O self-diffusion in dry basalt, andesite and rhyolite melts has not been investigated extensively, 18 O self-diffusivity in these dry melts may be estimated using the Eyring equation with a jump distance of 2.8×10−10 m as long as the melt viscosity can be estimated. In the last 10 years, viscosity models for specific melts have been advanced to high precision (e.g., Zhang et al. 2003 and Hui et al. 2009 for rhyolite melt; Whittington et al. 2009 for dacite melt; Vetere et al. 2006 for andesite melt) and general viscosity models are also available (Hui and Zhang 2007; Giordano et al. 2008) although with larger uncertainties (see Wang et al. 2009 for a comparison of the models of Hui and Zhang 2007 and Giordano et al. 2008). Hence, 18O self-diffusivities in dry melts can now be estimated with a precision similar to the precision of the viscosity models. For hydrous melts, 18O “self” diffusivity is much greater than the Eyring diffusivity, and must be understood and predicted in the context of H2O diffusion and the role of H2O in oxygen diffusion (see discussion in a later section).

Chemical diffusion of oxygen under dry conditions Chemical diffusion of oxygen under dry conditions is not well understood. Dunn (1983) reported oxygen chemical diffusion data in nephelinite, alkali basalt and tholeiite melts at 1553 to 1723 K and 0.4 to 2 GPa. Oxidized spheres of glass with Fe3+/Fe2+ ratio of 3.7 to 5.8 were packed with graphite powder in a graphite capsule in piston-cylinder experiments. After an experiment, the spheres were retrieved and average FeO concentration in the bulk sample was determined by wet chemical analyses (the Fe2O3 concentration is obtained by difference). Then the diffusivity is estimated using an equation similar to Equation (42). The experimental data of Dunn (1983) on basalt melt are shown in Figure 23. Dunn and Scarfe (1986) used the same approach to obtain diffusivities in an andesite melt at 1623 K and 0.35 to 2 GPa. One possible mechanism for the process in the two reports is as follows: Fe2O3 on the surface of the melt sphere reacts with graphite as:

2Fe2O3(melt) + C(graphite)  4FeO(melt) + CO2(melt) (50)

Reduction of 1.1 wt% Fe2O3 to produce 1 wt% FeO is accompanied by the production of

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0.1 0.4 0.6 0.8 1.2 1.4 1.6 2.0 0.1 1.0

-21

lnD (D in m2/s)

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MPa (W) GPa (D) GPa (D) GPa (D) GPa (D) GPa (D) GPa (D) GPa (D) MPa (C) GPa (L)

-23

-24

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Oxygen chemical & self-diffusion in basalt melt

-26 0.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

1000/T (T in K) Figure 23. Chemical diffusion and oxygen self-diffusion in basalt melt. Oxygen self-diffusion data (open and solid circles) are: 0.1 MPa (C) from Canil and Muehlenbachs (1990); 1.0 GPa (L) from Lesher et al. 23.chemical Chemical diffusion and oxygen self diffusion basalt melt.diffusion Oxygen self data (open and solid (1996).Fig. The diffusivities are related to O2inmolecular ordiffusion FeO effective binary diffusion (see text for discussion): 0.1 MPa (W) from Wendlandt (1991) based on weight gain (filled diamonds; circles in red) are: 0.1 MPa from Caniland anddehydration), Muehlenbachs (1990); (L) a from Lesher al. greater (1996). The likely reflecting molecular O2(C)diffusion which1.0isGPa about factor ofet30 than the self-diffusivities; other data are from Dunn (1983), likely reflecting FeO diffusion. chemical diffusivities are related to O molecular diffusion or FeO effective binary diffusion (see text for 2

discussion): 0.1 MPa (W) from Wendlandt (1991) based on weight gain (filled diamonds; likely reflecting molecular

0.153 wt% CO2. Then CO2 and FeO diffuse into the melt, and Fe2O3 in the interior of the melt and dehydration), is about a factor of 30 greater thanchemical the self diffusivities; other data are from diffusesO2todiffusion the surface of thewhich melt. In this scenario, the diffusivity determined by Dunn (1983) and Dunn and Scarfe (1986) is related to interdiffusion between CO2+FeO and Dunn (1983), likely reflecting FeO diffusion. Fe2O3. Data scatter is considerable in these studies. Wendlandt (1991) reported oxygen chemical diffusion data in basalt and andesite melts at 1433 to 1633 K and 1 atmosphere with controlled oxygen fugacity (Fig. 23). Mass gain or loss of the spherical sample at an imposed oxygen fugacity was monitored by a high precision electrobalance. Diffusivities were obtained by fitting the variation of the total mass with time using Equation (42). The reaction at the surface is likely 2FeO(melt) + (1/2)O2  Fe2O3(melt), then FeO and Fe2O3 would interdiffuse in the melt. The diffusivity is hence oxygen chemical diffusivity related to FeO and Fe2O3. The data precision is high because weight determination is highly reproducible. However, the initial basalt sample contains 1.03 wt% H2Ot (the initial andesite sample contains 0.44 wt% H2O). The presence of H2O in the initial sample may cause two effects: (1) Oxygen diffusivities may be enhanced due to H2O diffusion, which can carry oxygen (see next section). Based on H2Ot diffusivity in basalt (Eqn. 22) and the effect of H2O diffusion on oxygen diffusion, with 1.0 wt% H2Ot, the contribution of H2O diffusion to oxygen self-diffusion would yield a lnDoxygen of −24.9 at 1633 K, and −25.6 at 1533 K, and −26.5 at 1433 K, which is not enough to account for the high oxygen diffusivities by Wendlandt (1991) (Fig. 23). (2) Because of the presence of H2O in the initial sample, there would be simultaneous dehydration as O2 diffuses in or out of the sample. The effect of dehydration on the weight data was not assessed by Wendlandt (1991). Hence, the accuracy of results in this study is uncertain even though the precision is high.

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In summary, the much higher apparent oxygen chemical diffusivity by Wendlandt (1991) might be due to complications that are unaccounted for. Furthermore, even if only the data by Dunn (1983) are considered, the scatter is still considerable, which makes it impossible to infer how chemical oxygen diffusivity depends on pressure, or to infer the activation energy. The large scatter in the data of Dunn (1983), Dunn and Scarfe (1986) and Wendlandt (1991) may be partially related to the use of the bulk mass gain or mass loss method, which is prone to complications. Furthermore, part of the difficulty is likely related to the different meanings of oxygen chemical diffusion (such as molecular O2 diffusion, or FeO and Fe2O3 diffusion, or CO2 or H2O diffusion, etc).

“Self” diffusion of oxygen in the presence of H2O In experimental studies or natural systems, H2O is often present: 18O-enriched H2O may enter the sample of interest from a fluid phase, or H2O may be initially in the sample. Because H2O contains oxygen, the transport of H2O as molecular H2O (or as OH groups) would result in an oxygen flux. Hence, oxygen diffusivity can be affected or controlled by H2O diffusion. The relation between apparent 18O diffusivity in silicate melts and the presence of H2O vapor (as well as H2O in the melt) has been explored. DeBerg and Lauder (1980) conducted 18 O-16O exchange experiments between melt spheres (composition: 62.2 wt% SiO2 and 37.2 wt% K2O prepared under a vacuum of 0.013 Pa) and an 18O-enriched O2 or H2O gas, and then extracted 18O diffusivity. At 1175 K, 18O diffusivity does not depend on pressure of O2 from 7.5 to 56.3 kPa. On the other hand, the presence of 1.3 kPa of H2O vapor (in addition to 7.5 kPa of O2) yielded an 18O diffusivity about 2 times the dry 18O diffusivity at 1093 to 1143 K. The solubility of H2O in this melt is not known. If we assume that the solubility in the potassic silicate melt is roughly the same as that in rhyolite melt (Liu et al. 2005), the solubility at 1.3 kPa of H2O vapor pressure would be about 0.011 wt% (110 ppm). Hence, the observation by DeBerg and Lauder (1980) would imply that when H2Ot concentration is of the order 100 ppm, the contribution to 18O flux by H2O chemical diffusion is about the same as that of dry oxygen diffusion for this potassic silicate melt at about 1100 K. As H2O partial pressure increases, dissolved H2Ot in the melt increases, and 18O diffusivity increases. DeBerg and Lauder (1980) concluded that the dissolution and diffusion of O2 in the melt do not play an important role (due to the small solubility and small diffusivity of O2 molecules in the melt), but the dissolution and diffusion of H2O in the melt play a critical role in 18O diffusion in addition to dry 18O diffusion. Pfeffer and Ohring (1981) investigated 18O diffusion in silica in controlled water steam pressure and reached a similar conclusion. Zhang et al. (1991b) developed the general theory of diffusion of a multi-species component, using the diffusion of H2O and 18O as specific examples. For 18O diffusion in the presence of H2O in the sample (either high or low H2Ot), because both H2Om and OH carry an oxygen atom, H2O diffusion also carries an 18O diffusion flux. Because it has been shown from H2O diffusion studies that in silicate melts OH diffusion is negligible compared to H2Om diffusion under almost all conditions, 18O diffusivity may be viewed as being enhanced by H2Om diffusion alone. In such a case, the general equation for 18O diffusion can be written as (Eqn. 6 in Behrens et al. 2007 plus the anhydrous diffusion term): ∂Ri ∂  ∂Ri  ∂  ∂( Ri ⋅ X m )  ∂  ∂X m  = D 18 + DH O  − Ri ∂x  DH2 Om ∂x  ∂t ∂x  O,anhydrous ∂x  ∂x  2 m ∂x   

(51)

where Ri is the isotopic fraction of 18O, 18O/(16O+17O+18O), Xm is the mole fraction of H2Om on a single oxygen basis, and D18O,anhydrous is 18O diffusivity of oxygen species not associated with H in the presence or absence of H2O. The first term on the right hand side accounts for 18 O diffusive flux due to network oxygen diffusion (not associated with H); the second term accounts for 18O diffusive flux due to H2Om diffusion; and the third term accounts for mass

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balance (so that at constant Ri there is no 18O flux, meaning that the sum of the second and third terms is zero). D18O,anhydrous is expected to depend on H2Ot concentration because addition of H2O would loosen the melt structure, which would likely cause a higher diffusivity of even anhydrous oxygen species, such as those bonded to Si and Mg. Based on the above equation, 18 O diffusivity can often be estimated as (from Eqn. 15 in Zhang et al. 1991b by removing the OH term): D18 O ≈ D18 O,anhydrous + DH2 Om X m

(52)

The DH2OmXm term in the above equation is the apparent 18O “self” diffusivity contributed by H2O diffusion. Using the relation between DH2Om and DH2Ot (Eqn. 13), the above can be written as: D18 O ≈ D18 O,anhydrous + DH2 Ot X m

dX dX m

(53)

where X is the mole fraction of H2Ot on a single oxygen basis. The expression XmdX/dXm is shown in Figure 3 for K = 0.5. At low H2Ot (e.g., when X ≤ 0.01), the above can be simplified as (Eqn. 16 in Zhang et al. 1991b): X DH O 2 2 t Youxue Zhang and Ni (Ch 5)

D18 O ≈ D18 O,anhydrous +

(54) Page 24

Zhang et al. (1991b) summarized literature data to evaluate the role of H2O diffusion in 18O “self” diffusion. Behrens et al. (2007) experimentally investigated H218O sorption into a rhyolite melt, from which both H2Ot and 18O diffusion profiles were measured. Their results show that both H2Ot and 18O profiles indicate the same H2Om diffusivity (Fig. 24) under the assumption that H2Om is the diffusing species and there is chemical and isotopic equilibrium. Thus, their experimental data confirm that H2Om is the diffusing species in both H2Ot diffusion and 18O “self” diffusion, as well as the quantitative theory of Zhang et al. (1991b) presented above. In summary, H2O chemical diffusion (or effective binary diffusion) carries an 18O flux, which contributes to an apparent 18O diffusivity. If the H2O content is uniform in the sample and there is no chemical diffusion of H2O (e.g., diffusion couple with similar starting H2Ot but different 18O/16O in the two halves, or sorption of 18O-enriched H2O into a sample already containing the equilibrium concentration of H2O), 18O-16O exchange can still be due to the Youxue Zhang and Ni (Ch 5) Page 24

Figure 24. Experimental data on H2Ot and Ri = 18O/(16O+17O+18O) profiles (data points) during hydration using 18O-enriched H2O. The solid lines are fit by Equation (11) (for the H2Ot profile) and Equation (51) Fig. 24. Experimental data on H Ot and Ri = 18O/(16O+17O+18O) profiles (data points) during hydration (for the Ri profile) assuming H2Om is the diffusing species and DH2Om = D0eaX2 with the same a and D0 values enriched H2O. The solid lines are fit by Eq. 11 (for the H2Ot profile) and Eq. 51 (for the Ri profile) assum for both profiles. From Behrens et al. (2007).

is the diffusing species and DH2Om = D0eaX with the same a and D0 values for both profiles. From Behren (2007).

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mobility (self-diffusion) of H2O, and the diffusion may be said to be self-diffusion. In both cases, the diffusing species is H2Om. The resulting 18O diffusivities from H2O chemical diffusion and from H2O self-diffusion are not much different. In the next section, we examine the conditions for 18O diffusion in natural silicate melts to be true oxygen “self” diffusion or through H2O diffusion, and quantify apparent 18O diffusivity in hydrous rhyolite and dacite melts.

“Self” diffusion of oxygen in natural silicate melts in natural environments Using the theory presented above, if we know both the true 18O self-diffusivity under dry conditions and the H2O diffusivity, we can compare D18O,anhydrous with DH2OmXm, or D18O,anhydrous with DH2OtXmdX/dXm, to determine whether the true 18O self-diffusion or the apparent 18O flux due to H2O diffusion dominates the 18O flux. Below, we employ data on H2O diffusion and dry 18 O self-diffusion to evaluate quantitatively the conditions when H2O diffusion dominates 18O “self” diffusion. (1) Dacite melt. This melt is considered first because extensive data are available. Figure 25 compares dry 18O self-diffusivity (Tinker and Lesher 2001) and apparent 18O diffusivity contributed by H2O diffusion. First consider a numerical case at 1600 K and 1 GPa (which are close to the covered experimental conditions of both 18O self-diffusion and H2O diffusion). From Equation (44), dry 18O self-diffusivity is 3.5×10−14 m2/s. Based on Equations 18 and 53, about 0.32 wt% H2Ot is required to contribute 3.5×10−14 m2/s to the oxygen diffusivity. Because synthesized experimental dry melts typically contains only ≤ 0.1 wt% H2Ot, the experimental dry 18O self-diffusivities at such high temperatures by Tinker and Lesher (2001) are true D18O,anhydrous. Now consider 1200 K (a more reasonable magmatic temperature for natural dacite) and 1 GPa. Data by Tinker and Lesher (2001) do not cover such a low temperature (metastable melt). Dry 18O self-diffusivity extrapolated using Equation (44) is 2.1×10−17 m2/s. Based on Equation (18), H2Ot is required to contribute 2.1×10−17 Youxue Zhang and Niabout (Ch 5) 0.05 wt% (500 Pageppm) 25

Figure 25. Oxygen and H2Ot diffusion at 1 GPa in dacite melt. The open circles and the solid line are H2Ot diffusivity at 0.95 to 1 GPa from Behrens et al. (2004) and Ni et al. (2009b). The two dashed lines Fig. 25. Oxygen and H2Ot diffusion at 1 GPa in dacite melt. The open circles and the solid black line are H2Ot are calculated oxygen diffusivities due to H2O diffusion based on the H2O diffusivity of Ni et al. (2009b). The filled diamonds (in red in the lineThe(indashed red black in the online version) are diffusivity at 0.95 to 1 GPa fromonline Behrens version) et al. (2004)with and Niaetbest-fit al. (2009b). lines are calculated experimental 18O self-diffusion data at 1 GPa from Tinker and Lesher (2001). The dotted line is Eyring based of on the O diffusivity Ni et al.at (2009b). The filled (in oxygen diffusivities H2O diffusion diffusivity calculated usingdue thetoviscosity model HuiH2and Zhangof(2007) 0.1 MPa to 1 diamonds GPa assuming a jumping distance of 2.8×10−10 m. 18 red) with a best-fit line (in red) are experimental

O self diffusion data at 1 GPa from Tinker and Lesher (2001).

The dotted line (in blue) is Eyring diffusivity calculated using the viscosity model of Hui and Zhang (2007) at 0.1 MPa to 1 GPa assuming a jumping distance of 2.8x10-10 m.

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m2/s to the oxygen diffusivity. The difference at the two temperatures is due to lower activation energy for H2O diffusion and higher activation energy for dry 18O diffusion. As the temperature is lowered further, H2O diffusion would dominate the apparent 18O “self” diffusion (Fig. 25) at even lower H2Ot. Because most natural dacite melts are expected to contain more than 0.2 wt% H2Ot, we conclude that apparent 18O diffusion in natural dacite melts is almost always dominated by H2O diffusion. (2) Basalt melt. Both 18O and H2O diffusion data are limited on basalt melt. Consider 1600 K and 1 GPa. From Equation (47), dry 18O self-diffusivity is about 1.1×10−11 m2/s. Based on Equation (22), about 1 wt% H2Ot is necessary to contribute 1.1×10−11 m2/s to apparent 18 O diffusivity. That is, for basalt melt, due to high dry 18O self-diffusivity and magmatic temperatures, about 1 wt% H2Ot is needed for H2O diffusion to dominate 18O “self” diffusion. Therefore, in almost all mid-ocean ridge basalt melts (typically containing 0.2 to 0.7 wt% H2Ot, but can be up to 1.2 wt%; Dixon et al. 1988; Michael 1988; Workman et al. 2006) and most ocean island basalt melts (often more volatiles than MORB; Dixon et al. 1997; Hauri 2002; Workman et al. 2006), 18O diffusivity is most likely dominated by true 18O self-diffusion. However, in preeruptive island arc basalt (IAB) melts, and even in melt inclusions in mantle megacrysts likely influenced by subduction, H2Ot concentration are 2-6 wt% (Stolper and Newman 1994; Wang et al. 1999; Newman et al. 2000; Gurenko et al. 2005; Wallace 2005), meaning apparent 18O self-diffusion in these melts is dominated by H2O diffusion. (3) Rhyolite melt. There are only limited 18O diffusivity data in dry rhyolite melt at 1553 K and 0.1 MPa (Muehlenbachs and Kushiro 1974) and the quality of the data is not high (see discussion in an earlier section). On the other hand, H2O diffusion in rhyolite melt has been investigated extensively (e.g., comprehensive model by Ni and Zhang 2008). Figure 26 compares the contribution to 18O diffusion by the H2O flux with that by true network 18O self-diffusion. Oxygen self-diffusivities by Muehlenbachs and Kushiro (1974) are higher by a factor of about 24 than the Eyring diffusivity for dry melt calculated from the viscosity model of Zhang et al. (2003). If the data by Muehlenbachs and Kushiro (1974) are accurate and the melt was indeed dry (much less than 0.1 wt% H2Ot), oxygen flux due to chemical diffusion of H2O when H2Ot content is about 0.1 wt% would be roughly the same as the oxygen flux due to true oxygen self-diffusion. On one hand, oxygen self-diffusivities obtained by Muehlenbachs and Kushiro (1974) are close to effective binary diffusivity of P, which seems to imply the Eyring diffusivity limit. On the other hand, typical “dry” rhyolite glasses (either natural or synthesized) contain ≥ 0.1 wt% H2Ot. If the rhyolite used by Muehlenbachs and Kushiro (1974) also contained about 0.1 wt% H2Ot, their measurements would actually mean apparent 18O “self” diffusivity due to H2O diffusion, implying true dry 18O self-diffusivities are yet to be determined. Regardless how this issue is resolved, at typical rhyolite melt temperatures (such as 1200 K), H2O diffusion would dominate oxygen transport at much lower H2Ot, such as 10 to 100 ppm level. That means, in natural rhyolite melt in which H2O content if often 4-6 wt% (e.g, Wallace et al. 2003), diffusive transport of oxygen isotopes is through H2O diffusion. There are no 18O diffusivity data in dry andesite melts, and hence similar quantitative comparisons cannot be carried out. Nonetheless, it is expected that H2O diffusion plays a more important role in transporting 18O flux as temperature is lowered and as the melt becomes more silicic. Hence, the behavior of andesite melt is expected to be between that of basalt melt and dacite melt. When H2Ot is high enough (e.g., ≥ 0.5 wt%), the apparent 18O diffusivity in andesite melt is likely dominated by H2O diffusion. Because pre-eruptive natural andesite melt often contains ≥ 0.5 wt% H2Ot, we expect the 18O flux in natural andesite melt to be often dominated by H2O chemical diffusion. In summary, in nature, true 18O self-diffusion is the diffusion mechanism only for relatively dry basalt melt (such as MORB and OIB). The realization that diffusive transport of 18O in natural rhyolite and dacite melts is almost always due to H2O diffusion means it is possible to predict 18O diffusive transport using Equation

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Figure 26. Oxygen and H2Ot diffusion at 0.1-100 MPa in rhyolite melt. The pressure effect in this small pressure range is smaller than 0.2 lnD units. The solid line is the calculated H2Ot diffusivity. The two 26. Oxygen and H2Otto diffusion at 0.1-100 rhyolite melt. The pressuredue effect small pressure dashed linesFig.(roughly parallel the solid line)MPa areinoxygen diffusivities toinHthis 2O diffusion calculated at 0.1 GPa range based on Ni and Zhang (2008). The Eyring diffusivity at 0-0.1 wt% H O from 2 t is calculated is smaller than 0.2 lnD units. The solid black line is the calculated H2Ot diffusivity. The dashed black lines viscosity data (solid circles; Neuville et al. 1993; Schulze et al. 1996) and the viscosity model (lines; Zhang −10 18 et al. 2003)areassuming a jumping of 2.8×10 two on filled diamonds arewith experimental O calculated atm. 0.1 The GPa based Ni and Zhang (2008) a 2σ oxygen diffusivities due todistance H2O diffusion self-diffusion data at 0.1 MPa from Muehlenbachs and Kushiro (1974). The open squares and short dashed is calculated fromHviscosity data (points; uncertainty of 0.49diffusion in lnD. Thedata Eyring at 0-0.1 wt%GPa H2Otand line are effective binary ofdiffusivity P and fit at 0.8 0.1 wt% 2Ot (Harrison and Watson 1984), which are somewhat higher than the Eyring diffusivity. Neuville et al. 1993; Schulze et al. 1996) and the viscosity model (lines; Zhang et al. 2003) assuming a jumping

distance of 2.8x10-10 m. The two filled diamonds (in red) are experimental 18O self diffusion data at 0.1 MPa from

(52) or (53) because H2O diffusion in these melts has already been investigated well (e.g., Muehlenbachs and Kushiro (1974). The open squares and short dashed line (both in red) are effective binary Behrens et al. 2004; Liu et al. 2004b; Ni and Zhang 2008; Ni et al. 2009b). For the convenience 18 0.1 wt% H2Ot (Harrison and Watson, 1984), which is somewhat higher diffusion data of P and fit at 0.8 GPa and O diffusivities in hydrous rhyolite and dacite melts as of the readers, calculated apparent O are listed in Tables 3 and 4. In the calculation, the diffusivity of a function of T, P and H 2 t than the Eyring diffusivity. oxygen not associated with H is approximated by Eyring diffusivity. The calculated diffusivities have a 2σ uncertainty of about 0.6 in lnD and may be applied in rough estimation of diffusion rates. For calculation of the 18O diffusion profile, it is more accurate to use Equation (51). Prediction of 18O diffusion in basalt and andesite melts require more experimental work in terms of both H2O diffusion and dry 18O diffusion.

Contribution of CO2 diffusion to 18O transport in CO2-bearing melts Because carbon species (CO2 molecule and CO32−) also carry oxygen, oxygen transfer in natural systems may also be realized through the diffusion of these species, especially molecular CO2. One-atmosphere experiments on 18O diffusion often use 18O-enriched CO2 gas as the source for 18O (Muehlenbachs and Kushiro 1974; Canil and Muehlenbachs 1990). Not withstanding the quality of such data, one may wonder whether the extracted diffusivity is true 18O self-diffusivity, or just a reflection of CO2 diffusion carrying 18O into the sample. No simultaneous investigation of 18O and CO2 diffusion has been carried out yet. Although experimental data are lacking, the role of CO2 diffusion in transporting 18O can be treated similarly as the role of H2O diffusion in transporting 18O. However, it is more convenient to consider total CO2 diffusion rather than the diffusion of CO2 molecules because DCO2,total is roughly independent of melt composition. The diffusion equation for the isotopic fraction of 18O can be written as follows (comparing with Eqn. 51):

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Table 3. Calculated lnD18O (D in m2/s) in hydrous rhyolite melt. H2Ot (wt%) 0.1 0.3 0.5 1.0 1.5 2.0 3.0 4.0 5.0 6.0 7.0 8.0

0.1 GPa

0.5 GPa

1 GPa

1000 K 1300 K 1600 K 1000 K 1300 K 1600 K 1000 K 1300 K 1600 K −37.36 −35.11 −34.02 −32.45 −31.43 −30.63 −29.34 −28.26

−35.02 −32.78 −31.73 −30.24 −29.32 −28.63 −27.58 −26.76

−33.43 −31.31 −30.28 −28.86 −28.02 −27.40 −26.51 −25.85

−38.06 −35.80 −34.72 −33.15 −32.13 −31.33 −30.03 −28.95 −27.99 −27.1 −26.26 −25.47

−35.38 −33.14 −32.08 −30.60 −29.68 −28.99 −27.94 −27.12 −26.42 −25.79 −25.22 −24.69

−33.56 −31.46 −30.43 −29.01 −28.16 −27.55 −26.66 −25.99 −25.46 −25.00 −24.59 −24.22

−38.93 −36.67 −35.59 −34.02 −33.00 −32.19 −30.90 −29.82 −28.86 −27.97 −27.13 −26.34

−35.83 −33.59 −32.53 −31.05 −30.13 −29.44 −28.39 −27.56 −26.86 −26.24 −25.67 −25.14

−33.72 −31.64 −30.61 −29.19 −28.34 −27.73 −26.84 −26.17 −25.64 −25.18 −24.77 −24.41

Each cell lists lnD18O values where D18O is calculated total 18O diffusivity in m2/s using Equation (52), with DH2Om from Equation (14), K from Equation (7a), and D18O,anhydrous approximated by DEyring calculated using the viscosity model of Zhang et al. (2003). Values of lnD at other temperatures can be obtained using the Arrhenius relation.

Table 4. Calculated lnD18O (D in m2/s) in hydrous dacite melt. H2Ot (wt%) 0.1 0.3 0.5 1.0 1.5 2.0 3.0 4.0 5.0 6.0 7.0 8.0

0.1 GPa

0.5 GPa

1 GPa

1000 K 1300 K 1600 K 1000 K 1300 K 1600 K 1000 K 1300 K 1600 K −38.62 −36.33 −35.20 −33.54 −32.43 −31.54 −30.08 −28.83

−34.60 −32.47 −31.42 −29.94 −29.02 −28.34 −27.30 −26.48

−29.51 −28.54 −27.91 −26.93 −26.32 −25.87 −25.21 −24.72

−39.21 −36.92 −35.79 −34.13 −33.02 −32.13 −30.67 −29.42 −28.30 −27.25 −26.27 −25.32

−34.95 −32.86 −31.82 −30.34 −29.43 −28.75 −27.72 −26.91 −26.21 −25.58 −25.01 −24.47

−29.53 −28.60 −27.99 −27.06 −26.47 −26.05 −25.42 −24.96 −24.59 −24.27 −23.99 −23.74

−39.95 −37.65 −36.53 −34.87 −33.76 −32.87 −31.40 −30.16 −29.04 −27.99 −27.00 −26.06

−35.33 −33.31 −32.28 −30.80 −29.90 −29.23 −28.22 −27.42 −26.74 −26.13 −25.56 −25.03

−29.54 −28.65 −28.07 −27.18 −26.63 −26.22 −25.64 −25.21 −24.86 −24.57 −24.31 −24.07

Each cell lists lnD18O values where D18O is calculated total 18O diffusivity in m2/s using Equation (52), with DH2Om from Equation (18), K from Equation (7b), and D18O,anhydrous approximated by DEyring calculated using the viscosity model of Whittington et al. (2009).

∂ 2 ( R ⋅ X C0.5 Ototal ) ∂ 2 X C0.5 Ototal ∂R ∂2 R D 18O,noncarbonated 2 + DCO2,total = − DCO2,total R 2 ∂t ∂x ∂x ∂x 2

(55)

where XC0.5Ototal is mole fraction of total C0.5Ototal (each C0.5Ototal contains a single oxygen and two C0.5Ototal makes one CO2,total) on a single oxygen basis, and D18O,anhydrous is diffusivity of 18 O unassociated with carbon. XC0.5Ototal, is calculated as (Ccd/22.005)/[Ccd/22.005+(100Ccd)/W] where Ccd is wt% of CO2,total and W is the molar mass of the dry melt on a single oxygen basis (Table 1). XCO2,total may be used instead of XC0.5Ototal, but a factor of 2 would have to be incorporated because each CO2 molecule carries two oxygen atoms. DCO2,total does not depend on CO2 concentration or anhydrous melt composition and hence can be taken out of

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the differentials. Based on the above equation, the 18O diffusivity can often be estimated as (comparing with Eqn. 52): D18 O ≈ D18 O,noncarbonated + DCO2,total X C0.5 Ototal

(56)

The DCO2,totalXC0.5Ototal term in the above equation is apparent 18O “self” diffusivity contributed by CO2 diffusion. Because DCO2,total can be estimated from Equation (41), we examine quantitatively possible contributions by CO2 to 18O diffusion below. For rhyolite melt at 1553 K and 0.1 MPa (the condition at which 18O diffusivity was determined in a CO2 gas by Muehlenbachs and Kushiro 1974), DCO2,total is 1.2×10−11 m2/s from Equation (41). The measured 18O diffusivity is 4.5×10−15 m2/s. For CO2 diffusion to contribute to such an 18O diffusivity, a CO2 concentration of 260 ppm is needed (equivalent to a CO2 partial pressure of 52 MPa). Typical natural rhyolite does not contain this much CO2. We hence conclude that in this case, 18O diffusivity is not due to chemical CO2 diffusion. This is consistent with the similarity of 18O diffusivities in CO2 gas and in O2 gas. For the convenience of readers, contribution to apparent D18O by CO2 chemical diffusion are calculated and listed in Table 5. Note that this table does not give the apparent D18O values as in Tables 3 and 4 where the dry 18O self-diffusivity has been added or is assessed to be negligible. Table 5 only lists the part of apparent D18O contributed by CO2 chemical diffusion when the CO2 concentration is 100 ppm. The 2σ uncertainty in the calculated values is about 1.1 lnD units. For the full apparent 18O diffusivity, the anhydrous and non-carbonated true 18O selfdiffusivity and that contributed by CO2 diffusion must be added. Because DCO2,total (Equation 41) does not depend on the anhydrous melt composition, Table 5 is roughly applicable to all silicate melts. The contribution to the apparent 18O diffusivity is proportional to CO2,total concentration (Eqn. 56), which can be used to estimate the contribution to D18O by CO2 chemical diffusion at other CO2,total concentrations. Comparing Table 5 with Tables 3 and 4, the contribution by CO2 chemical diffusion to 18O diffusion at the 100 ppm CO2 level is not important to dacite melt or rhyolite, with the possible exception of very dry rhyolite melt (≤ 0.2 wt% H2Ot) at very high temperatures (≥ 1500 K).

Table 5. Calculated lnD18O (D in m2/s) contributed by CO2 diffusion at 100 ppm CO2,total. 0.1 GPa 0.5 GPa 1 GPa H2Ot (wt%) 1000 K 1300 K 1600 K 1000 K 1300 K 1600 K 1000 K 1300 K 1600 K 0 0.1 0.3 0.5 1.0 1.5 2.0 3.0 4.0 5.0

−39.27 −39.19 −39.01 −38.83 −38.39 −37.95 −37.51 −36.63 −35.74

−35.22 −35.15 −35.02 −34.88 −34.54 −34.2 −33.86 −33.18 −32.51

−32.69 −32.63 −32.52 −32.41 −32.14 −31.86 −31.59 −31.03 −30.48

−40.05 −39.95 −39.75 −39.56 −39.06 −38.57 −38.07 −37.08 −36.09 −35.1

−35.82 −35.74 −35.59 −35.44 −35.06 −34.68 −34.3 −33.53 −32.77 −32.01

−33.17 −33.11 −32.99 −32.86 −32.56 −32.25 −31.94 −31.32 −30.7 −30.08

−41.02 −40.91 −40.69 −40.46 −39.9 −39.33 −38.77 −37.64 −36.52 −35.39

−36.57 −36.48 −36.31 −36.13 −35.7 −35.27 −34.83 −33.97 −33.1 −32.24

−33.78 −33.71 −33.57 −33.43 −33.08 −32.73 −32.37 −31.67 −30.97 −30.26

Each cell lists lnD18O values contributed by CO2 chemical diffusion with D calculated as DCO2,totalXC0.5Ototal at 100 ppm CO2,total and DCO2,total from Equation (41). To estimate D18O at other CO2 concentrations, multiply D18O (not lnD18O) by the concentration ratio. For example, for 500 ppm CO2, multiply D18O values in this table by 5. Add D18O here to D18O in Tables 3 or 4 (do not add lnD18O) would give the full D18O in hydrous and carbonated rhyolite and dacite melts.

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Oxygen diffusion and viscosity: applicability of the Eyring equation Oishi et al. (1975) showed that the 18O diffusivity in 4Na2O · 3CaO · 18SiO2, , and Na2O · 4SiO2 melts is inversely proportional to melt viscosity. 2CaO · Al2O3 · 2SiO2  Shimizu and Kushiro (1984) showed that the Eyring equation relates 18O diffusivity well with viscosity for dry diopside and jadeite melts (Fig. 27a). 18O diffusion data in other dry melts (e.g., Liang et al. 1996; Tinker and Lesher 2001; Tinker et al. 2004) also roughly follow the Eyring relation (Fig. 27a) even though in detail there are differences (e.g., Liang et al. 1996; Tinker et al. 2004). Figure 25 shows that the Eyring relation roughly applies to dry dacite melt at 1 GPa (rough agreement between solid diamonds and dotted line). However, Figure 26 indicates either limited experimental 18O diffusion data in dry rhyolite melt do not represent true D18O,anhydrous, or the Eyring relation does not apply to dry rhyolite (comparing solid diamonds and solid circles). Overall, available data largely indicate that the Eyring equation is applicable to 18O diffusion in dry silicate melts. For hydrous silicate melts, the available data indicate that the apparent 18O diffusivity is often orders of magnitude greater than the Eyring diffusivity (or Stokes-Einstein diffusivity or Youxue Zhang and Ni (Ch 5) Page 27 any of the other models of inverse proportionality between diffusivity and viscosity) (Behrens

Figure 27. Comparison of the Eyring diffusivity with 18O diffusivity in (a) dry melts, and (b) hydrous rhyolite melts. Note that the scale is base-10 logarithm, not base-e logarithm. Data sources for (a): Shimuzu84 = Shimizu and Kushiro (1984); Tinker04 = Tinker et al. (2004); Liang96 = Liang et al. (1996). Data sources for (b): Viscosity model used for the calculation of rhyolite melt viscosity is from Zhang et al. (2003). H2Om, H2Ot and 18O diffusivities are from Behrens et al. (2007) and Ni and Zhang (2008).

Fig. 27. Comparison of the Eyring diffusivity and Einstein diffusivity with 18O diffusivity in (a) dry melts, and (b)

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et al. 2007; Fig. 27b). Because natural silicate melts often contain a significant amount of H2O, the results indicate that the application of the Eyring relation in geological melts is very limited. 18 O diffusion in terrestrial silicate melts in natural environments is often dominated by H2O flux and cannot be predicted by the Eyring equation (but can be predicted from H2O diffusivities). The inapplicability of the Eyring equation to 18O diffusion in hydrous silicate melts may be explained as follows. In deriving the Eyring equation, one assumption is that the diffusion and viscous flow are controlled by the same mechanism (motion). For the case of dry silicate melts, viscous flow and diffusion of oxygen (the dominant ion in the melt) are both related to structural mobility, and are hence controlled by the same mechanism. However, for the case of wet silicate melts, viscous flow requires network oxygen mobility, but 18O diffusive flux is carried by the mobility of neutral H2Om molecules, whose motion is not necessarily related to network oxygen relaxation. In the context of this explanation, the Eyring equation cannot be applied as a universal relation; it is necessary to know the diffusion mechanism and diffusing species before assessing whether the equation can be applied. Because most natural melts are hydrous, the applicability of the Eyring relation to natural melts is limited.

O2 diffusion in pure silica MELT Dissolved molecular O2 in silicate melts may also be referred to as physically dissolved oxygen (meaning no reaction between O2 molecule and other components in the melt or glass because essentially all melts are based on oxygen anions). In natural melts, the oxygen fugacity is low. For example, oxygen fugacity at NNO buffer (about the average oxygen fugacity in typical mantle derived silicate melts), fO2 is only about 3×10−7 Pa at 1200 K (calculation based on thermodynamic data of Robie and Hemingway, 1995), lower than that in air by almost 11 orders of magnitude. Therefore, the dissolved O2 concentration in natural silicate melts is expected to be extremely low. However, in industry and in high-temperature experiments carried out in air or in pure oxygen gas, dissolved O2 concentration may be noticeable. Under such conditions, pure molecular O2 diffusion may be studied, and molecular O2 diffusion may contribute significantly to network oxygen diffusion. We have shown that oxygen diffusion data in natural or nearly natural silicate melts are complicated, but it is still possible to make sense out of most data in terms of contributions by various oxygen species. Oxygen diffusion data in the glass and materials science literature are even messier (e.g., Fig. 9 in the review by Lamkin et al. 1992), likely due to the contributions by the various oxygen species discussed earlier (especially the hydrous component), plus the additional complexity due to molecular O2. Because such data are on melts very different from geological silicate melts, they are not the focus in this volume. Hence, we make no attempt to quantitatively explain the complicated behavior of oxygen diffusivity in various silica and silicate melts in the glass and materials science literature. Below, we briefly discuss molecular O2 diffusion in silica melt for two purposes. First, such data allow us to roughly know the molecular O2 diffusivity (for the completeness of this volume). Secondly, we may use the O2 diffusivity to assess the role of molecular O2 diffusion in oxygen diffusion in natural melts (under oxygen fugacity of terrestrial igneous processes), which turns out to be unimportant due to extremely low oxygen fugacity in natural silicate melts (see below). Molecular diffusion of O2 in silica melt is best understood compared to other silicate melts due to the simple composition of silica. Even though there have been numerous studies on oxygen diffusion in silica melt/glass, only a limited number of authors reported genuine molecular O2 diffusivities. Norton (1961) carried out molecular O2 permeation experiments in silica glass and obtained molecular O2 diffusivities at 1223 and 1351 K. Hetherington and Jack (1964) made an order of magnitude estimate of molecular O2 diffusivities in silica glass, and the values are consistent with the

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results of Norton (1961). Susa et al. (1990) reported diffusivities of molecular O2 in variously prepared silica films at 1073-1273 K. Lamkin et al. (1992) reviewed oxygen diffusion data in silica and silicate glasses. Kajihara et al. (2004, 2005) systematically investigated molecular O2 diffusion in silica glass with different OH contents. Kajihara et al. (2008) reviewed diffusion and reaction of interstitial oxygen species in silica glass. Molecular O2 diffusion data in silica glass are shown in Figure 28. Molecular O2 diffusivity in low-OH silica glass (Kajihara et al. 2004) can be expressed as follows:   10727 ± 482   melt D silica =exp  − (19.22 ± 0.39 ) –  O2  T   

(57)

where D is in m2/s, T is in K and errors are given at the 2σ level. The activation energy for molecular O2 diffusion is 89±4 kJ/mol, greater than that for Ne diffusion but slightly smaller than that for H2Om diffusion. Network oxygen diffusivities (such as 18O diffusivities) in silica melts are 5 to 9 orders of magnitude lower than molecular O2 diffusivity at 900-1700 K (Fig. 1A in Lamkin et al. 1992) and the activation energy is much higher. To evaluate the role of molecular O2 in transporting oxygen in natural silicate melts, we compare 2DO2[O2] and DO[O] where O2 means molecular oxygen and O means network oxygen, and the factor 2 is because each molecular O2 carries two oxygen atoms. Because molecular O2 solubility in silica melt is about 10−6 mol/m3 at 1 Pa of O2 pressure (Norton 1961), the solubility at a typical oxygen fugacity of 10−7 Pa in natural silicate melt would be of the order 10−13 mol/m3. The network oxygen concentration in silica melt is about 76560 mol/m3. Hence the concentration ratio of 2[O2]/[O] is about 3x10−18. Because molecular O2 diffusivity is no more than 9 orders of magnitude greater than network oxygen diffusivity, at typical oxygen fugacities of natural silicate melts, the ratio of 2DO2[O2] to DO[O] is smaller than 3×10−9. Hence, the contribution from molecular oxygen diffusion to oxygen transport is negligible in natural silicate melts. For molecular O2 to play a significant role in transporting network oxygen, an oxygen fugacity of about 100 Pa or greater is needed. Youxue Zhang and Ni (Ch 5)

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Figure 28. Molecular O2 diffusivities in silica melts. Susa et al. (1990) and Kajihara et al. (2005) each contain two data sets for slightly different silica melts; these are plotted separately. Fig. 28. Molecular O2 diffusivities in silica melts. Susa et al. (1990) and Kajihara et al. (2005) each contain two data sets for slightly different silica melts; these are plotted separately.

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SUMMARY AND CONCLUSIONS Through decades of research, fairly extensive experimental data have been obtained on H2O, CO2, and oxygen diffusion in silicate melts. The variation of diffusivities with temperature, pressure, and melt composition (including water content) has led us to fundamental understanding of diffusion mechanisms and melt structure. H2O diffusion and CO2 diffusion are both dominated by the diffusion of the respective molecular species. The diffusivity of these two major volatile components increases with temperature and water content, but decreases with pressure. Total CO2 diffusivity increases exponentially with H2Ot, whereas H2Ot diffusivity first increases with H2Ot proportionally, and then exponentially. While CO2 diffusivity does not depend much on melt composition except for the H2Ot concentration, H2O diffusivity increases with decreasing silica content at superliquidus temperatures, but the trend at intermediate temperatures is less well defined. H2O diffusivity also increases slightly with increasing alkalinity. H2O speciation causes proportionality between H2Ot diffusivity and H2Ot concentration at low H2Ot, whereas CO2 speciation leads to independence of total CO2 diffusivity on CO2 concentration, and also causes total CO2 diffusivity to be independent of the anhydrous melt composition. One complexity that needs to be examined in detail in the future is the possible role of molecular H2 diffusion in transporting the hydrous component under reducing conditions. Oxygen chemical diffusivity depends on the specific diffusion mechanisms and more systematic studies are necessary to understand the various chemical oxygen diffusivities. In the presence of H2O, oxygen “self” diffusion is often controlled by H2O diffusion. In typical natural silicate melts, especially rhyolite, dacite and andesite melts, 18O diffusion is often carried by H2O diffusion, and hence can be predicted from experimental H2O diffusivities that have been well characterized in rhyolite and dacite melts. Predicted 18O diffusivities in hydrous rhyolite and dacite melts are listed in tables and can be many orders of magnitude greater than the Eyring diffusivity. Only in fairly dry basalt melts (such as MORB and OIB), would 18O diffusion occur through true oxygen self-diffusion, in which case the 18O self-diffusivity is not much different from the Eyring diffusivity. 18O flux carried by CO2 chemical diffusion is also quantitatively evaluated, and the conclusion is that CO2 diffusion does not significantly enhance 18O diffusion except under very special conditions. The contribution from molecular O2 diffusion is negligible to network oxygen diffusion under oxygen fugacity encountered in natural silicate melts.

Acknowledgments This research is supported by NSF grants EAR-0711050 and EAR-0838127 and NASA grant NNX10AH74G. Huaiwei Ni acknowledges financial support by the visitors program of Bayerisches Geoinstitut. We thank Don Baker and an anonymous reviewer for careful and constructive reviews, and Harald Behrens and Daniele Cherniak for comments.

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Reviews in Mineralogy & Geochemistry Vol. 72 pp. 227-267, 2010 Copyright © Mineralogical Society of America

Noble Gas Diffusion in Silicate Glasses and Melts Harald Behrens Institut für Mineralogie and ZFM Center for Solid State Chemistry and New Materials Leibniz Universität Hannover Callinstr. 3 D-30167 Hannover, Germany [email protected]

INTRODUCTION Studies of noble gas diffusion and solubility provide information on the internal structure of silicate glasses, i.e., the size distribution and interconnection of cavities within the framework (Doremus 1994). Additionally, such diffusion data are of interest for understanding degassing kinetics of glasses and melts. Water and carbon dioxide, the most important volatiles in magmas, can be present in silicate melts both as unreacted (molecular) species and as dissociated species (hydroxyl groups or carbonate groups, respectively) (e.g., Holloway and Blank 1994; Kohn 2000). The contributions of the different species to the overall transport of the volatiles are difficult to separate. In the case of rhyolitic melts it was found that the diffusivity of Ar is similar to that of molecular CO2 (Watson 1994; Behrens and Zhang 2001). Assuming that this similarity is a general property for silicate melts, Nowak et al. (2004) analyzed the diffusion behavior in synthetic rhyolitic to basaltic melts and estimated the relative abundance of carbonate and molecular CO2 in the melt. Knowledge of noble gas diffusivity is also of importance for interpretation of geochemical findings both on short and on large scales. For instance, the isotopic compositions of noble gases have been extensively used to identify long-lived heterogeneities within the Earth’s mantle (Allégre et al. 1986; Graham 2002). The noble gases He, Ne and Ar are highly incompatible elements, which will be preferentially incorporated into a melt during melting of the mantle, so at partial melt fractions relevant to mid-ocean ridges a primary melt will inherit the relative noble gas abundance pattern of the mantle source (Burnard et al. 2004). Helium, on the other hand, is a geochemical tracer for mantle signatures in rocks. Measurements of both helium abundance and the 3He/4He ratios in volcanic rocks, gases and waters (e.g., Allègre et al. 1986; Notsu et al. 2001; Stuart et al. 2003; Starkey et al. 2009) yield estimates of the gas composition of the mantle. Noble gas relative and absolute abundances in oceanic basalts are difficult to interpret due to the considerable fractionation that occurs during magmatic degassing. During degassing of a magma, the He/Ar of the residual volatiles increases because Ar is less soluble than He (Jambon et al. 1986; Lux 1987; Carroll and Stolper 1993; Carroll and Webster 1994), thus the He/Ar ratio can be used as an index of degassing. For the interpretation of such data, knowledge of fluid/melt partitioning of helium and diffusivity of helium in the melt are required (Jambon and Shelby 1980). Burnard (2004) and Matsuda and Marty (1995) pointed out that the large differences in He and Ar diffusivities in silicate minerals could result in fractionation of the He/Ar ratio during melting of the mantle, producing He/Ar ratios in the primary mantle melts that are higher than those of the bulk mantle. A technical aspect of application of diffusivities is in the separation and purification of gases due to the different permeability of gases in glasses. The permeability is proportional to 1529-6466/10/0072-0006$05.00

DOI: 10.2138/rmg.2010.72.6

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the diffusivity and the solubility of the gases in the melts. Solubilities of noble gases are out of the scope of the present review and the reader is referred to reviews in Carroll and Webster (1994), and for simple and technical glasses in Doremus (1994) and Shelby (1996). Although the present paper is focused on studies of geological interests, some relevant information from the glass sciences is included as well, but no complete overview of the numerous studies done in that field can be given here. Furthermore, the review is restricted to SiO2-bearing melts. The majority of experimental studies has been carried out on glasses but may extend in temperatures towards the melt state. In the framework of this review I distinguish only between stable melts (i.e., temperatures above the melting temperature, or in the case of multicomponent systems the liquidus temperature) and supercooled melts and glasses. In this paper experimental and analytical methods used to study noble gas diffusion in glasses and melts are first summarized. Due to the volatile character of noble gases, these techniques differ from those commonly used in diffusion studies (described in chapter 3 – Watson and Dohmen 2010). Next, different noble gases diffusing in the same melt (or glass) are compared, and compositional effects on noble gas diffusion are discussed for the different noble gases. Finally, noble gas diffusion is compared to diffusion of molecular species. Here few data are available and most comes from the glass sciences. However, having some idea of the diffusivities of H2 and O2 is important for redox processes in glasses and melts, and nitrogen diffusivities might be of interest for degassing processes.

EXPERIMENTAL AND ANALYTICAL METHODS Specific techniques have been developed to measure the diffusivity and solubility of gases in glasses and melts at low- and at high-pressures. Studies related to glass sciences usually were performed at near-atmospheric pressures, while in studies of relevance to the geosciences, the knowledge of the effect of pressure is very important because of large increases in solubility with pressure (Carroll and Webster 1994).

Studies at atmospheric and sub-atmospheric pressure The majority of studies at near-ambient pressure focus on glasses or supercooled melts, while studies on high-temperature melts are scarce. In glass sciences, the membrane technique or the powder technique have commonly been used for measurement of diffusivities of noble gases in glasses. Details of these techniques and equations for the evaluation of the data are given in Doremus (1994) and references therein. In the membrane technique, a thin glass membrane (typical thickness of several micrometers) in the form of a tube or a bulb is mounted in a vacuum system and one side of the system is loaded with a certain gas pressure while the second side is evacuated. The transfer of gas through the membrane is measured as a function of time. At short times the diffusivity can be directly calculated from the rate of pressure increase on the initially gas-free side. At long times a stationary flux is reached when pressures on both sides are held approximately constant. In that case the flux is determined by the permeability (product of solubility and diffusivity). Combining short-term and longterm measurements allows determining the solubility, the diffusivity and the permeability of noble gases. Additionally, if the gas on the inlet side is pumped out rapidly after steady state is attained, the diffusion coefficient can be determined from the rate of gas release from the membranes. The membrane technique was found to be very useful, especially for small noble gases (He and Ne) and hydrogen in glasses with relatively open structures such as silica where permeation rates are high (Frank et al. 1961; Swets et al. 1961; Lee et al. 1962; Perkins and Begeal 1971; Shelby 1972a-d, 1973a,b, 1974a,d , 1975, 1977a,b, 1979; Shelby and Eagan 1976; Shelby et al. 1976), but was employed to measure Ar and Kr in silica glass as well (Perkins and Begeal 1971). Application of the technique to geologically-relevant glass compositions is rare. Jambon and Shelby (1980) measured the diffusivity and solubility of helium in obsidians and

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basalt glass at 200-300 °C. Data for helium diffusion in tektite glasses from Reynolds (1960) are questionable because of technical limitations, e.g., the use of Pyrex plumbing and containers in the vacuum system, as outlined in Jambon and Shelby (1980). For slowly diffusing gases the powder method is more suitable. Here the diffusion coefficient is calculated from the rate of adsorption or desorption of gas in a powdered sample, and the solubility is determined from the total amount of gas adsorbed in the glass after heating for a certain time at some temperature and gas pressure (e.g., Woods and Doremus 1971). A general problem in the powder method is the relatively large intrinsic error of such a bulk extraction method due to variation in particle shape and diameter. Other complexities are: (1) The presence of cracks (before glass anneals) would produce apparently high diffusivities. (2) There may also be a surface effect, especially when powder size is small. Wortman and Shackelford (1990) have outlined a glass fiber saturation technique which accounts for the significant range of small fiber diameters in the analysis of diffusivities. This approach was used by Nakayama and Shackelford (1990) to analyze the kinetics of the uptake of argon by silica fibers with diameters around 8 mm at pressures below 1 bar. Diffusion of noble gases in glass melts relevant for the glass science community was investigated by desorption from the melt, monitored by gas chromatography (Mulfinger and Scholze 1962b) or by measuring the shrinkage rate of gas bubbles (Frischat and Oel 1965; 1967). Both techniques are potentially affected by convection in the melt. Microgravity experiments in space shuttles carried out by Frischat and coworkers (Rosenkranz et al. 1985; Jeschke and Frischat 1987; Wendler et al. 1995; Goß et al. 2000) show little deviation from experiments conducted on the Earth’s surface, indicating that for the studied soda-lime silicate melt the bubble shrinkage method is not affected much by convection. Roselieb and coworkers (Roselieb et al. 1992, 1995) used the powder method to study noble gas diffusion and solubility in glasses of the system SiO2-NaAlSi2O6. The gases (He, Ne, Ar, and Kr) were dissolved in the glasses at high-pressure and temperature, while the gas release experiments were performed under vacuum using Knudsen cell mass spectrometry. Diffusion relations were obtained by modeling the gas release curves at a constant heating rate. In the case of albite glasses, the PT conditions for loading with gas and the heating rates were found to have little influence on the Arrhenius parameters for diffusion, and diffusion data for various glass compositions and various noble gases show a good internal consistency. As noted by Roselieb et al. (1992) the activation energy for diffusion has high reproducibility from experiment to experiment, while the pre-exponential factor, which depends strongly on size and shape distribution of the grains, has large uncertainty. Discrepancies with other data imply that the technique of Roselieb and coworkers is not applicable to helium diffusion (see Fig. 4b). Helium release occurs readily at very low-temperatures from He-rich high-pressure glasses (Fig. 5 in Roselieb et al. 1992) and helium loss between the saturation experiment and the analyses may have invalidated the measurements. Gauthier et al. (1999, 2000) measured the diffusivity of radon in a dry natural andesitic melt at atmospheric pressure and at temperatures ranging between 1623 K and 1773 K. 222Rn was produced by radioactive decay of 226Ra present in the natural material. The procedure is based on the determination by gamma-ray spectrometry of radon losses undergone by a diskshaped melted sample during heating. Surprisingly, the derived diffusion coefficients for Ra are larger than for the smaller noble gas Ar, measured by Nowak et al. (2004) using a diffusion couple technique. However, the composition of the andesites used in both studies differed, and Nowak et al. (2004) performed the experiments at 500 MPa pressure so a direct comparison of the data is not possible. Another radiogenic technique was applied by Hazelton et al. (2003) who produced in situ Ar from 39K in potassium-bearing glasses by irradiation with fast neutrons. Ar diffusivity

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was measured by the release of argon during step-heating of glass powder with well-defined grain-size fractions in a furnace at ambient pressure. After purification, the 39Ar was analyzed with a noble gas mass spectrometer. Stepwise heating coupled with mass spectrometry was also applied to study neon release from obsidian powder (Matsuda et al. 1989).

Studies at high-pressure High-pressure studies on noble gas diffusion in silicate glasses or melts are restricted to Ar and Kr, if one ignores that the Xe-bearing samples in the study of Roselieb et al. (1995) were pre-equilibrated at several hundreds of MPa. In most studies the sorption technique was used (Carroll 1990, 1991; Carroll and Stolper 1991; Carroll et al. 1993; Draper and Carroll 1995; Roselieb et al. 1996; Behrens and Zhang 2001; Behrens 2010). This method involves a diffusion annealing of glass plates or spheres in an autoclave with the sample being directly exposed to the pressure medium followed by analyses of quenched samples. In some experiments the glasses were sealed together with the noble gases in noble metal capsules (Roselieb et al. 1992, 1995, 1996). This technique is more demanding but it has the advantage of saving costs for the expensive heavy noble gases and in avoiding contamination present in the pressure medium. The most severe contamination is probably H2O. Although being only a minor component in the pressure medium with concentrations at the per mill level, it might nevertheless affect noticeably experiments with nominally dry glasses due to the high solubility of water in silicate glasses and melts (Behrens and Gaillard 2006). For instance in-diffusion of water during such experiments was recently demonstrated for H2O-poor silica glasses by infrared microspectroscopy (Behrens 2010). Draper and Carroll (1995) tested the effect of the pressure medium on argon diffusion in silicic melts using equimolar Ar/He mixtures as the pressure medium. No significant difference in diffusivities was found compared to experiments with pure argon byFigures: Carroll and Stolper This is surprising because argon profiles fit Noble gas(1991). diffusion innot silicate glasses and the melts well to a model that assumes DAr is independent of argon concentration. After diffusion experiments, concentration-distance profiles of noble gases were usually measured on sample cross sections using an electron microprobe. Examples are shown in Figure 1 a-b. Carroll and Stolper (1991) also used Rutherford Backscattering Spectrometry 0.6

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100

Distance from surface (µm)

Examples of argon diffusion profiles glassesmeasured measured by after sorption Fig. Figure 1. 1. Examples of argon diffusion profiles in inglasses by electron electronmicroprobe microprobe after sorption experiments. a) Peralkaline to peraluminous haplogranite melts at 924 K, 500 MPa. b) Synthetic rhyolite experiments. a) Peralkaline to peraluminous haplogranite melts at 924 K, 500 MPa. b) Synthetic to andesite melts at 855 K, 200 MPa. Lines are fits to the respective data assuming constant diffusion rhyolite to andesite melts at 855 MPa.axis Lines are fits totothe data assuming coefficient. The intercepts withK,the200 vertical correspond therespective solubility of argon in the constant melt in diffusion coefficient. Thefluid. intercepts with the vertical axis correspond to the solubility of argon in the equilibrium with argon melt in equilibrium with argon fluid.

Noble Gas Diffusion in Silicate Glasses and Melts

231

(RBS) to measure the Ar concentration in silica samples and found good agreement of the RBS-derived diffusion data with data from electron microprobe measurements. RBS involves irradiation of the samples with an ion beam and detection of the backscattered ions, the energies of which are related to the mass of the scattering atom and the depth in the sample at which the backscattering event occurred (see chapter 4 – Cherniak et al. 2010, this volume). Synchrotron X-ray fluorescence microanalysis (m-SRXRF, see chapter 4 – Cherniak et al. 2010, this volume) was also applied for measurement of profiles of Kr because of the much greater sensitivity of this method compared with the electron microprobe (Carroll et al. 1993). When using glass sections with planar surfaces in the experiments, the profiles were evaluated assuming a constant surface concentration C0,i (infinite gas reservoir, no influence of surface reactions), a concentration-independent diffusion coefficient Di and a semi-infinite diffusion medium, where i denotes the specific noble gas. If the initial concentration of i in the glass is zero, the solution of Fick’s second law for one-dimensional diffusion is (Crank 1975):

  x = Ci C0,i  1 - erf   4D t  i  

   

(1)

where Ci is the concentration of i at the distance x from the surface and t is the run duration. To fit the profiles, C0,i and Di were adjustable parameters. C0,i can be interpreted as the equilibrium solubility of the noble gas at the experimental conditions (Carroll 1990). Roselieb et al. (1996) measured Ar and Kr diffusion in jadeite melt (NaAlSi2O6) using single spheres of well-defined radius. The spheres were first exposed for a preset time to a gas (Ar or Kr) at known pressure and temperature and then the sorbed gas was extracted and quantitatively determined by gas chromatography (GC). The diffusivity is calculated from the ratio of the sorbed gas in the sphere to the saturation solubility for a given heating time. Good agreement was found with diffusion data based on profiles determined with electron microprobe on cross-sections of spheres. The sorption technique described above is limited to temperatures below the softening point of the glasses, otherwise samples deform during experiments towards a spherical shape. This problem might be avoided when melting glasses in a pan with large diameter/height ratio as done in the past by glass scientists, e.g., to study diffusion of noble gases or water in glass melts (e.g., Scholze and Mulfinger 1959; Mulfinger and Scholze 1962b). However, this technique has not been explored at high-pressure conditions. Studies of noble gas diffusion in melts at high-temperature and pressures are restricted to argon (Behrens and Zhang 2001; Nowak et al. 2004; Spickenbom 2010). Experiments were performed by the diffusion couple technique in which an Ar-bearing and an Ar-free cylinder with same bulk compositions were placed together in a platinum capsule with the polished surfaces in contact. Ar-bearing samples were produced by firstly heating glass powder in an open platinum container at high-PT conditions using argon as pressure medium, then crushing the glasses and sealing the compacted glass powder either dry or together with water (in order to produce hydrous samples) into platinum capsules which were processed again in an internally heated gas pressure vessel (IHPV) to re-melt the charge. The capsule with the diffusion couple was manually squeezed to minimize air and free space in the capsule, welded shut, and compressed in Ar gas at room temperature in a cold seal pressure vessel (CSPV). The compressed capsule was then placed in an IHPV or in a piston cylinder apparatus (PCA), pressurized, and heated to the experimental temperature for a specific duration. After the experiments, argon profiles are measured along the cylindrical axis by electron microprobe. Assuming concentration-independent diffusion coefficients, the solution of Fick’s second law for one-dimensional diffusion between two semi-infinite media for the given boundary conditions is (Crank 1975):

Behrens

232

 x-x - CAr,min  C M  1 - erf  CAr = CAr,min + Ar,max  4D t  2 Ar  

   

(2)

where CAr,min and CAr,max are the initial concentrations in both halves of the diffusion couple and xM corresponds to the Matano interface, defined through a mass balance condition, where the solute loss on one side is equal to the solute gain on the other side. Equation (2) was fit to the measured concentration distance profiles of Ar. Adjustable parameters are CAr,min, CAr,max, DAr and xM.

DIFFUSION SYSTEMATICS In glass sciences much data on noble gas permeation has been determined, i.e., for helium and neon in a variety of glasses, summarized by Doremus (1994) and Shelby (1996). According to these studies the permeation rate is negatively correlated with the concentration of network modifying cations. However, a large part of these variations in permeability are due to the change in solubility. The solubility of noble gases in silicate glasses and melts is strongly correlated with the ionic porosity, which is defined as the difference between the volume of a material’s unit cell and the sum of the volume of anions and cations in that cell divided by the volume of this unit cell (Carroll and Stolper 1993). The diffusivity of atoms or ions in glasses and crystals is also affected by the ionic porosity (Dowty 1980), since the number of possible diffusion paths through the material depends on its free volume. However, recent papers by Tournour and Shelby (2008a,b) on He and Ne solubility in simple silicate melts conclusively demonstrate that the ionic porosity model for solubility does not adequately predict the experimental results. These finding raise questions regarding the reliability of this model in predicting the diffusivity of gases as well. Much more important than the overall ionic porosity is the energy barrier for the transition of the particle from one stable site to another one, which is related to the size of the “doorway” between the two sites and the elastic modulus of the glass (Anderson and Stuart 1954; Carroll 1991; Roselieb et al. 1992, 1995). In high-temperature melts the dynamics of the silicate network play an important role as well, both in assisting hopping processes and in opening and closing paths for noble gas diffusion.

Temperature dependence of diffusivity The temperature dependence of noble gas diffusion in silicate glasses and melts often can be described well by a simple Arrhenius relationship

 -E  D =i Di ,o ⋅ exp  a   RT 

(3)

where Di,o is the pre-exponential factor, Ea is the activation energy for diffusion of the species i, R is the universal gas constant and T is the absolute temperature. A compilation of Arrhenius parameters for noble gas diffusion in silicate glasses and melts can be found in the Appendix Table A. Experimental data for argon and krypton show good linearity in plots of log D vs. reciprocal temperature and, hence, the Arrhenius law is appropriate for these elements. However, a small but significant deviation from linearity was observed for helium and neon diffusion in some glasses when very precise data could be measured over a large temperature range (Fig. 2). To account for the non-linearity, a temperature dependence of the pre-exponential factor was proposed. For helium and neon diffusion in silica glass (Doremus 1994; data from Swets et al. (1961) for He, Frank et al. (1961) and Perkins and Begeal (1971) for (Ne) and helium diffusion in TiO2-SiO2 (Shelby 1973b) and K2O-SiO2 (Shelby 1974a)

Noble Gas Diffusion in Silicate Glasses and Melts

233

-8

SiO2

-9

log D [D in m2/s]

He -10

Figure 2. Arrhenius plots for He and Ne diffusion in SiO2 glass. Data sources: He: Swets et al. (1961); Ne, low-T: Wortmann and Shakelford (1990); Ne, high-T: Frank et al. (1961). Note the small but systematic deviation from linearity which could be resolved by high precision permeation experiments.

-11

Ne

-12

-13

-14 1.0

1.5

2.0

2.5

3.0

3.5

-1

1000/T (K ) Fig. 2. Arrhenius plots for He and Ne diffusion in SiO2 glass. Data sources: He: Swets et al. (1961); Ne, low T: Wortmann and Shakelford (1990); Ne, high T: Frank et al. (1971). Note the small but systematic deviation from linearity which could be resolved by high precision permeation experiments.

glasses a proportionality of the pre-exponential factor to absolute temperature was found to give a good description of the temperature dependence;

 -E  Di = Di ,o ⋅ T ⋅ exp  a   RT 

(4)

Parameters describing noble gas diffusion by Equation (4) are listed in the Appendix Table B. It is noteworthy that activation energies based on Equation (4) are systematically lower than those based on Equation (3), e.g., by 10-14% for He and by 7-8% for Ne and Ar (see data in the Appendix). Activation energies reported in this paper always refer to Equation (3). In terms of diffusion theory the temperature dependence of the pre-exponential factor can be explained by a rise in the attempt frequency for diffusional jumps with temperature. Alternatively, the deviation from Arrhenius behavior can be described assuming a distribution of activation energies (Shelby and Keeton 1974). In amorphous materials a variety of possible diffusion paths exists with different energy barriers for migration of particles, and with increasing temperature, less favored paths contribute increasingly to the overall transport.

Pressure dependence of diffusivity Few data are available to characterize the pressure dependence of diffusivity of noble gases and often the applied pressure range in the experimental studies was too small to resolve any pressure effect. Argon diffusivity in silica glass was found to be pressure independent in the range 48-371 MPa at 973 K and in the range 120-373 MPa at 873 K (Carroll and Stolper 1991). For rhyolite at 1073 K, Carroll (1991) observed a systematic decrease in argon diffusivity from 1.9×10-13 m2/s to 4.2×10-14 m2/s with pressure increase from 136 to 370 MPa. Using a modified Arrhenius equation

 - ( Ea + P ⋅ Va )  D =i Di ,o ⋅ exp   RT  

(5)

he derived an activation volume Va of (36±8)×10-6 m3/mol from the isothermal data set. Fitting the whole data set of Carroll (1991) and Draper and Carroll (1995) for rhyolite to Equation (5) yields a slightly smaller Va of (27.9±6.1)×10-6 m3/mol, see Table  1. Similar activation volumes were derived for peralkaline to peraluminous haplogranites ((25.3-32.5)×10-6 m3/mol,

771-1118 50-5000 28 -4.45±0.52 158.3±9.2 25.3±5.0 0.11

771-1118 50-5000 28 -5.57±0.26 141.0±4.5 32.5±3.5 0.13

772-1101 50-4000 28 -6.62±0.23 133.8±4.0 27.9±3.3 0.15

1.501 -0.071

4.14 4.21 0.022

17.07

74.58

(6)

peralum. HG AOQAE

873-1173 116-373 28 -5.81±0.12 148.5±3.7 27.0±4.6 0.09

1.051 -0.008

77.17 0.06 12.98 0.75 0.03 0.05 0.52 4.12 4.16 0.17

(1), (2)

Obsidian

753-1375 50-801 41 -5.74±0.13 147.3±2.6 18.6±2.3 0.13

(1), (2), (3)

Obsidian

772-1773 200-500 13 -6.06±0.09 143.7±2.0 26 (fixed) 0.11

Notes. All iron is given as FeOtotal. Arrhenius parameters are based on data from the references given on top of the columns. Compositions of albite glasses used in the different studies differ slightly, see appendix. Different obsidians were used in Behrens and Zhang (2001) with slightly different compositions (see appendix) compared to the one used in the studies of Carroll. The alumina saturation index is defined as ASI = Al/(Na + K + 2Mg + 2Ca). NBO/T = (Na + K + 2Mg + 2Ca – Al)/(Al + Si) represents the ratio of non-bridging oxygens to tetrahedral cations. In the calculation of NBO/T FeO and H2O components were not considered. Note that negative values of NBO/T have no physical meaning. In fitting of diffusion data for synthetic rhyolite and rhyodacite the average activation volume based on all haplogranites and obsidians was used as a constraint for fitting. f.s.d. gives the fit standard deviation.

772-1773 200-500 12 -5.58±0.09 158.7±1.8 26 (fixed) 0.09

0.458 0.091

2.57 2.76 4.60 3.06

1.57 1.37 3.82 4.13

0.598 0.039

71.11 0.42 15.51

(5), (6)

Syn. Rhyodacite

74.74 0.23 14.15

(5), (6)

Syn. Rhyolite 

Data Sources: (1) Carroll (1991); (2) Draper and Carroll (1995); (3) Behrens and Zhang (2001); (4) Spickenbom et al. (2010); (5) Nowak et al. (2004); (6) This study.

773-1673 118-500 29 -5.81±0.19 153.0±2.8 9.6±7.1 0.15

0.531 0.087

ASI NBO/T

T range (K) P range (MPa) Experiments log D0 (D0 in m2/s) Ea (kJ/mol) Va (10−6 m3/mol) f.s.d.

4.77 5.69 0.024

4.78 5.87 0.010 0.980 0.004

13.73

7.55

(6) 75.81

(6)

81.79

(1), (2), (3), (4)

metal. HG AOQ

SiO2 TiO2 Al2O3 FeOtotal MnO MgO CaO Na2O K2O H2O, initial

Data sources

peralk. HG AOQPB

Albite NaAlSi3O8

Table 1. Compilation of Arrhenius parameters for several compositions.

234 Behrens

Noble Gas Diffusion in Silicate Glasses and Melts

Ar and Kr diffusion in a jadeite melt was studied over a large pressure range from 20 to 600 MPa at 1073 K by Roselieb et al. (1996) using the sorption technique. Although data show large scatter, a systematic decrease of diffusivity with pressure is evident for both elements (Fig. 3) and activation volumes of (11.4±1.3)×10-6 m3/mol for Ar and (17.3±4.3)×10-6 m3/mol for Kr were computed by the authors.

-13.0

jadeite Ar

-13.5

log D [D in m2/s]

see Table 1) by combined PT fitting, these new data are discussed in detail below. On the other hand no significant value of Va was found for dry albite glasses, neither from the data set of Carroll (1991) nor from combined data sets of Carroll (1991), Draper and Carroll (1995), Behrens and Zhang (2001) and Spickenbom et al. (2010), see Table 1.

235

-14.0

-14.5

-15.0

-15.5

Kr

0

200

400

600

pressure (MPa)

Fig. 3. Pressure effect on diffusion of Ar and Kr in jadeite melts at 1073 K. Solid lines are linear regressions to the respective data. From the slope of the lines activation volumes of 11.4 ± 1.3

Figure 3. ArRoselieb and Kr jadeite cm3/mol for Ar Pressure and 17.3 ± 4.3effect cm3/molon for diffusion Kr are inferredof (after et al.in 1996). melts at 1073 K. Solid lines are linear regressions to the respective data. From the slope of the lines activation volumes of 11.4 ± 1.3 cm3/mol for Ar and 17.3 ± 4.3 cm3/mol for Kr are inferred (after Roselieb et al. 1996).

The activation volume derived from a small pressure range as done in the studies reported above is expected to have large uncertainties. Zhang et al. (2007) combined data from various authors (including Carroll and Stolper 1991; Draper and Carroll 1995; Roselieb et al. 1996; Behrens and Zhang 2001; Nowak et al. 2004) and fit 152 Ar diffusivity data points in rhyolitic, dacitic, albitic, and jadeitic melts covering a wide T-P-H2O space (pressure from 20 MPa to 1500 MPa) and obtained an activation volume of 16×10-6 m3/mol for Ar in dry melts, about half of the values by Carroll (1991) and Draper and Carroll (1995). This value is consistent with the estimates reported above for individual melt compositions, i.e., it averages between small variations of DAr with pressure for albite and large variations for rhyolite. For the interpretation of the activation volumes one needs to consider that the compressibility of the melts/glasses changes strongly at the glass transition temperature Tg. Below Tg the glasses show only an elastic response upon pressurization while above Tg the melts can be densified efficiently by structural relaxation. For instance the molar volume of albite glasses decreases by 1.6% when, at constant cooling from above Tg, pressure increases from 0.1 to 500 MPa (Wondraczek et al. 2009). Hence, when the duration of a diffusion experiment is short compared to the relaxation time scale of the melt (i.e., at low-T), then only a relatively small elastic compression of the glass is induced while at sufficiently high-temperature a more pronounced plastic compression also occurs. That means different pressure dependencies (and thus activation volumes) can be expected at low- and high-temperature. However, available experimental data sets are commonly not suitable to resolve such effects. A significant effect might be expected for larger noble gases, for which the diffusivity becomes too slow to be measured reliably far below the glass transition. A variation of the density of glasses can also be achieved by changing the cooling rate from the melt. As pointed out by Shelby (1996), glasses with steep volume/temperature curves near the glass transition, such as vitreous boric acid and sodium borates, are particularly sensitive to changes in thermal history, and significant variations in He diffusivity and permeability were observed, i.e., a decrease upon relaxation anneals at low-temperature. On the other hand,

236

Behrens

glasses with small changes of volume with cooling rate, such as silica glass, exhibits only small effects of thermal history on noble gas diffusion.

Comparison of different noble gases in the same matrix glass The most extended data set on noble gas diffusion is available for silica glass (Fig. 4a). At 1250 K the diffusivity decreases with increasing size of the atoms from 5.2×10-9 m2/s (He; Swets et al. 1961) to 2.2×10-10 m2/s (Ne; Frank et al. 1961) to 6.8×10-14 m2/s (Ar; data for Infrasil from Behrens (2010)) to 5.2×10-16 m2/s (Kr; Carroll et al. 1993) to 3.6×10-18 m2/s (Xe; Roselieb et al. 1995). Data from other authors are similar (see Appendix). The activation energy increases in the same order (Fig. 5), hence, the difference in diffusivities between the noble gases grows with decreasing temperature. A complete set of diffusion data from He to Xe was also measured for albite composition (Fig. 4b). However, the temperature ranges covered by the experiments is often smaller than for silica glasses, and direct comparison of different noble gases is therefore limited. Diffusion data for helium from Roselieb et al. (1995) based on Knudsen cell mass spectrometry fall off the trend and do not match the measurements of Shelby and Eagon (1976) who used the membrane technique. The activation energy of 67 kJ/mol determined by Roselieb et al. (1976) for helium diffusion in albite glasses is also significantly higher than in most other studies on silicate glasses (23-38 kJ/mole, see Appendix Table A), except for the work of Kurz and Jenkins (1981) who obtained an Ea value of 83 kJ/mol from He release in stepwise heating experiments on sieved basalt glass powder. For rhyolite the trends of the diffusion data for noble gases are in line with those for silica and albite compositions (Fig. 4c). The variation of diffusivity and activation energy for diffusion with the radius of noble gases has been used to discuss the mechanism of noble gas diffusion in silicate glasses (Carroll 1991; Roselieb et al. 1992, 1995; Doremus 1994; Zhang and Xu 1995). Anderson and Stuart (1954) postulated that the activation energy for diffusion is the energy required to deform the silicate network enough to allow the atom to pass from one interstice to another. They assume that the interstices are connected by “doorways” of an average radius R and that the activation energy is the elastic energy which is needed to enlarge the “doorway” the radius r of the diffusing species. This energy can be estimated by E= 8πGR(r - R )2

(6)

where G is the elastic modulus of the glass. According to this relationship a square root dependence of Ea on the particle radius is expected. Indeed, some of the data sets for activation energies of particles with the same charge are consistent with such a relationship, e.g., for alkalis and noble gases in rhyolite and albite glass (Jambon and Carron 1976; Jambon 1982; Carroll 1991) or for molecular species in silica glass (see Doremus 1994). However, the choice of the data set for noble gas radii is crucial for testing the validity of Equation (6). Zhang and Xu (1995) critically reviewed atomic radii of noble gases dissolved in condensed matter and suggested an internally consistent set of radii. When using these radii, the activation energy for the diffusion of He, Ne, molecular H2O, Ar, and Kr in silica, rhyolitic, and albitic glasses might be even better described assuming a linear variation of Ea with atomic radius (Zhang and Xu 1995). In Figure 5 the range of radii is enlarged by adding data for Xe diffusion from Roselieb et al. (1995). While data up to Kr might be described well assuming a linear or a square root dependence of Ea on atomic radius, the data for Xe are far off the predictions of the trends based on data for smaller radii. Thus, although the “doorway” concept has some support from experimental data and may roughly describe the diffusion trends, one should not overemphasize its importance. It contains several simplifications, i.e., that different atoms use the same paths, and that a single value of the elastic modulus represents the forces needed to enlarge an opening for migration.

Noble Gas Diffusion in Silicate Glasses and Melts

-10

a

silica

6

-9

log D [D in m2/s]

-10 -11

O2

-12 -13

4

-14

2 5

3 0.5

3

-13

1.0

Ar

Xe Kr

-15

4

2

-17

1 1.5

Ne

-14

-16

Xe Kr

-17 -18

Ne

He H2

-12

Ar

-15 -16

He

D2

albite

b

-11

log D [D in m2/s]

-8

237

2.0

2.5

3.0

1

-18

3.5

1.0

1.5

1000/T (K-1)

2.0

2.5

1000/T (K-1)

-7 -8

c

rhyolite

H2

-9

log D [D in m2/s]

-10 -11

He

4 3

-12

H2O

-13

-15

Kr 2

5

-16 -17 0.5

Ne

Ar

-14

1 1.0

1.5

2.0

2.5

3.0

-1

1000/T (K )

Comparison representative data data for (solid lines) and molecular diffusion Fig. Figure 4. 4a-c. Comparison ofofrepresentative fornoble noblegas gasdiffusion diffusion (solid lines) and molecular (dashed lines) in glasses meltsand at near-ambient pressure, if not specified. diffusion (dashed lines) in and glasses melts at near-ambient pressure, if not specified. a) Fused DataHe: sources: Swets et al.Ne: (1961); Ne: Wortmann and Shakelford a. Fused silica. Datasilica. sources: SwetsHe: et al. (1961); Wortmann and Shakelford (1990),(1990), including data from Frank et al. (1971); Ar 1: 20 -373 MPa, Carroll and Stolper (1991); Kr 3: 152 -315 including data from Frank et al. (1971); Ar 1: 20 -373 MPa, Carroll and Stolper (1991); Kr 3: 152 MPa, Carroll et al. (1993); Ar 2, Kr 4, Xe: Roselieb et al. (1995); D2 1: Perkins and Begeal (1971); D2 2: 315 MPa, et O al.: (1993); Ar 2, Kr 4, Xe: Roselieb et al. (1995); D2 1: Perkins and Begeal Lee et Carroll al. (1962); 2 Kajihara et al. (2004), Toutnour and Shelby (2005). (1971); D2b)2:Albite Lee et(NaAlSi al. (1963); O2: Kajihara et al. (2004), andNe, Shelby sources: He: Shelby and Toumour Eaton (1976); Ar 1;(2005). Kr 3: Roselieb et al. 3O8). Data b. Albite (NaAlSi Data sources: Shelby (1976); Ne,et Ar 1; Kr 3:Xe: Roselieb (1992); Ar 2: 118 MPa, Carroll He: (1991); Kr 4:and 23 –Eaton 315 MPa, Carroll al. (1993); Roseliebetetal. al. 3O– 8).371 (1995); H2118 : 180–MPa, et al. (1985). (1992); Ar 2: 371 Chekmir MPa, Carroll (1991); Kr 4: 23 – 315 MPa, Carroll et al. (1993); Xe: Roselieb c) Water-poor rhyolite glasses et andal. melts. Data sources: He: Jambon and Shelby (1980); Ne 1: tektite, et al. (1995); H2: 180 MPa, Chekmir (1985). Reynolds (1960); Ne2, Ne3 two obsidians, Matsuda al. (1989); et al. Ne (1991a); Ar 2O: Zhang c. Water-poor rhyolite glasses anddifferent melts. Data sources: He: et Jambon andHShelby (1980); 1: tektite, 4: 116 – 373 MPa, Carroll (1991), Draper and Carroll (1995); Ar 5: 200 MPa, Behrens and Zhang (2001); Reynolds (1954); Ne2, Ne3 two different obsidians, Matsuda et al. (1989); H O: Zhang et al. (1991a); 2 Kr: Matsuda et al. (1989). Ar 4: 116 – 373 MPa, Carroll (1991), Draper and Carroll (1995); Ar 5: 200 MPa, Behrens and Zhang (2001); Kr: Matsuda et al. (1989).

Behrens

238 400

silica albite rhyolite

350

Ea (kJ/mol)

300 250 200 150 100 50 0 1.0

1.1

He

1.2

Ne

1.3

1.4

1.5

1.6

1.7

H 2O Ar atomic radius (Å)

1.8

Kr

1.9

2.0

Xe

Fig. 5. Comparison of activation energies for gasgas diffusion in silicain (SiO (NaAlSi 2), albite 3O8) (NaAlSi O ) Figure 5. Comparison of activation energies fornoble noble diffusion silica (SiO 2), albite 3 8 and rhyolite glasses and melts. H2O assumed to be a spherical molecule is included for comparison. and rhyolite glasses and melts. H O assumed to be a spherical molecule is included for comparison. Data 2 Data sources: see Table A in the appendix. The solid line is a parabolic fit (Eqn. 6) to all data for sources: see Table A inlines the are appendix. The solid line data is aforparabolic fit All (Eqn. data for silica. Other silica. Other linear regressions excluding Xe diffusion. data 6) are to for all nominally anhydrous melts. excluding data for Xe diffusion. All data are for nominally anhydrous melts. lines are linear regressions

COMPOSITIONAL EFFECTS ON NOBLE GAS DIFFUSION He diffusion A compilation of selected He diffusion data in silicate glasses and melts is shown in Figure 6. For simplicity, data are represented by straight lines in this plot, although some of the compositions display non-Arrhenius behavior (see Fig. 2). When adding TiO2 to SiO2 glass the diffusion of He becomes slightly faster (~50% increase when adding 10 mol% TiO2, Shelby 1972d). In contrast, the addition of alkali oxide strongly reduces the mobility of helium, e.g., at 500 K by a factor of 7 when adding 32 mol% K2O (Shelby 1974a) and by a factor or 38 when adding 25 mol% Na2O (Shelby and Eagan 1976). According to the study of Shelby and Eagan (1976), diffusion in the fully polymerized NaAlSi3O8 composition is about one order of magnitude slower than in silica glass, which can be explained by the filling of interstices in the network with sodium. Compared to sodium silicate glasses with similar mol% of Na2O, the diffusivity of He is faster by a factor of two in albite glass. This increase reflects denser packing (lower ionic porosity) in the silicate compared to the aluminosilicate. Data for He diffusion in natural glasses are scarce. The data of Jambon and Shelby (1980) indicate similar He diffusivity in obsidian and in albite glass, but much lower diffusivity in basalt glass, which could not be measured reliably with the membrane technique. Stepwise heating experiments on glass powder and analyses of the released helium confirmed the low He diffusivity in basalt (Kurz and Jenkins 1980). At 500 K, the He diffusivity is about 3 orders of magnitude slower in basalt than in obsidian, and the activation energy for He diffusion is very large for basalt (83±4 kJ/mol, Kurz and Jenkins 1980) compared to other silicate glasses

Noble Gas Diffusion in Silicate Glasses and Melts

239

-7

8

-8

log D [D in m2/s]

-9

He

12 6,7

-10 -11

11

2

10 9 4

5

1 3

-12 -13 -14 -15 -16 0.5

1.0

1.5

2.0

2.5

3.0

3.5

-1

1000/T (K )

Fig. 6. Compilation of selected He diffusion data in silicate glasses (solid lines) and melts (dashed lines). Figure  6. Compilation of selected He diffusion data in silicate glasses (solid lines) and melts (dashed Assignment of curves SiO22 glass, et al. 2: 90 SiO – 10 TiO , Shelby (1972), lines). Assignment of curves andand dot:dot: 1)1:SiO glass,Swets Swets et (1961); al. (1961); 2) 290 SiO 2 2– 10 TiO2, Shelby – 25 and Eagan (1976); 5: 3: Pyrex 7740, borosilicateglass glass (Rogers 75 SiO24) 2O, Shelby (1972d), 3) Pyrex 7740, borosilicate (Rogers1954); et al.4:1954); 75Na SiO 2 – 25 Na2O, Shelby and Eagan 68 SiO2 – 32 K2O, Shelby (1974); 6: Float glass melt, Frischat and Oel (1965); 7: Soda-lime silicate (1976); 5) 68 SiO – 32 K O, Shelby (1974a); 6) Float glass melt, Frischat and Oel (1965); 7) Soda-lime melt2 74.1 SiO22 - 15.8 Na2O - 10.1 CaO in mol%, Wendler et al. (1995), micro gravity experiments; 8: silicate melt 74.1 SiO Na2O melt, - 10.1 CaO inand mol%, Wendler (1995), micro gravity experiments; 2 - 15.8silicate same soda-lime Mulfinger Scholze (1962);et 9:al. NaAlSi 3O8 (albite), Shelby and Eagan 8) same soda-lime silicate melt, Mulfinger Scholze (1962b); 9) NaAlSi O8 (albite), and Eagan (1976); 10: obsidian from Iceland,and Jambon and Shelby (1980); 11: Mid 3Atlantic Ridge Shelby basalt glass (1976); 10) obsidian from Iceland, Jambon and Shelby 11) Mid Ridge glass ALV ALV 519, containing olivine microlites, Kurz and(1980); Jenkins (1980). 12:Atlantic estimate for basaltbasalt melt from (1987). 519, containingLux olivine microlites, Kurz and Jenkins (1981). 12) estimate for basalt melt from Lux (1987).

(range of 23-37 kJ/mol, see Appendix Table A). A preliminary value for He diffusivity in basalt at 1623 K based on a noble gas sorption experiment of Lux (1987) is in good agreement with the extrapolation of the Arrhenius law from Kurz and Jenkins (1980). However, experimental details are not given by Lux (1987) and other experiments on basalt melts are missing to verify the quality of this diffusion coefficient. Lux (1987) also reported preliminary diffusion coefficients for Ne, Ar, Kr, and Xe in basalt at the same temperature. These data are included in subsequent figures, but will not be discussed further because of the experimental uncertainties (i.e., possible influences of convection). Data for He diffusion in tektite from Reynolds (1960) are not shown in Figure  6 because most likely the experiments have systematic errors, see discussions in Jambon and Shelby (1980). Diffusion of He in silicate melts at high-temperature was studied only in a few older papers using the desorption technique (Mulfinger and Scholze 1962a,b) or the bubble shrinkage technique (Frischat and Oel 1965; Jeschke and Frischat 1987; Wendler et al. 1995). Mulfinger and Scholze (1962b) observed in alkali silicate melts and soda lime silicate melts an increase of He diffusivity with increasing concentration of network modifiers and found a positive correlation with the size of the network modifying cations, i.e., DHe increases in the order Li – Na – K. The effect of the concentrations of network modifiers is contrary to what was observed for glasses. They conclude that in addition to the free volume the melt viscosity is an important parameter which controls He diffusion in the melt.

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240

Data from Frischat and Oel (1965) for a commercial float glass composition (see Appendix Table  A) are consistent with the findings of Mulfinger and Scholze (1962b). However, microgravity experiments in a space shuttle (Jeschke and Frischat 1985; Wendler et al. 1995) yield diffusion coefficients for soda lime silicate glass melts which are lower by a factor of two compared to the experiments of Mulfinger and Scholze (1962b) at normal gravity. The difference suggests that convection might have contributed to the transport of helium in the experiments of Mulfinger and Scholze (1962b).

Ne diffusion Few data are available for neon diffusion in silicate glasses and melts. Data from different authors (Frank et al. 1961; Perkins and Begeal 1971; Wortman and Shackelford 1990) for silica glass show very good consistency (Curves 1,2,3 in Fig. 7). Ne diffusion in a float glass melt studied by the bubble shrinkage method (Frischat and Oel 1967) is about the same as in silica glass at ~1250 K, but the activation energy is noticeably higher in the multicomponent glass melt (57 kJ/mol) than in silica glass (40-48 kJ/mol). Neon diffusion in aluminosilicate glasses (albite, obsidian, tektite) covers a narrow range in the Arrhenius plot. The diffusivities are significantly smaller and the activation energies for Ne diffusion are higher in the aluminosilicates (59-95 kJ/mol) than in silica glass. Again, these differences can be rationalized by the filling of holes suitable for migration of neon by akali ions.

-9

9

Ne

4

log D [D in m2/s]

-10 -11 -12

5 6

2 8

-13 -14

3

-15

7

-16 -17 0.5

1.0

1.5

2.0

1

2.5

3.0

3.5

4.0

-1

1000/T (K )

Fig. 7. Compilation of selected Ne diffusion data in silicate glasses (solid lines) and melts (dashed lines). Figure 7. Compilation of selected Ne diffusion data in silicate glasses (solid lines) and melts (dashed lines). Assignment of curves andSiO dot: 1: SiO2 glass, combination of low T data from Wortman and Assignment of curves and dot: 1) 2 glass, combination of low-T data from Wortman and Shackelford Shackelford (1990) with high T data from Frank et al. (1961) 2: SiO2 glass, Frank et al. (1961); SiO2 (1990) with high-T data from Frank et al. (1961) 2) SiO2 glass, Frank et al. (1961); 3) SiO2 glass, Perkins glass, Perkins and Begeal (1971); 4: Float glass melt, Frischat and Oel (1967); 5: NaAlSi3O8 (albite), and Begeal (1971); Float glass melt, Frischat Oel et(1967); 5) 7:NaAlSi et al. 3ONE2, 8 (albite), Roselieb4) et al. (1992); 6: obsidian 7473104,and Matsuda al. (1989); obsidian MatsudaRoselieb et al. (1992); 6) obsidian 7473104, Matsuda(1960); et al.9:(1989); Matsuda (1989); 8: Tektite, Reynolds estimate 7) forobsidian basalt meltNE2, from Lux (1987).et al. (1989); 8) Tektite,

Reynolds (1960); 9) estimate for basalt melt from Lux (1987).

Noble Gas Diffusion in Silicate Glasses and Melts

241

Ar diffusion The largest data set relevant to the geosciences among the noble gases is available for Ar. One reason is the particular interest in the diffusivity of argon for dating of rocks because the radiogenic isotope 40Ar, formed by decay of 40K, may escape from condensed matter via diffusion, which may result in incorrect age determinations. Another reason is that argon diffusion experiments on glasses are relatively easy to do at elevated pressure, and concentration-distance profiles can be measured by electron microprobe. Zhang et al. (2007) reviewed Ar diffusion data in silicate melts. New experimental data were collected over several years in student projects at University of Hannover and are included in this review. The experimental strategy in these works was the same as described in Carroll (1990) and Behrens and Zhang (2001), i.e., polished glass plates were heated for specific times in cold seal pressure vessels using argon as the pressure medium. To resolve compositional effects on argon diffusion, typically three glass chips were placed side by side in open gold containers so that the P-T-t path was the same for all samples. In the following, we will first discuss diffusion data for polymerized glass. Next the effect of the aluminum/alkali ratio will be considered and then the specific role of divalent cations. Finally, studies investigating the effect of dissolved water on argon diffusion in glasses and melts are presented. Argon diffusion in polymerized glasses. A compilation of Ar diffusion data for systems in which the molar concentration of alkali and alkaline earth elements correspond to the amount needed for charge compensation of aluminum is shown in Figure 8. Not all available data are displayed for sake of clarity. Data for silica glass from Perkins and Begeal (1971), Nakayama and Shackelford (1990), Draper and Carroll (1995) and Behrens (2010) cover a narrow range in log D – 1000/T space limited by the curves Qz1 (Roselieb et al. 1995) and Qz2 (Carroll and Stolper 1991). Different methods were used in these studies (permeation through a thin membrane, fiber saturation, sorption in glass plates, Knudsen cell mass spectrometry) as well as different pressures from near-ambient pressure to 370 MPa. Small differences between the reported diffusivities most likely originate from the experimental and analytical techniques employed and from differences in water contents of the glasses (see below) rather than from a pressure effect. Carroll and Stolper (1991) demonstrated negligible influence of pressure on argon diffusivities in silica glass. The activation energy for Ar diffusion in silica glass (96-120 kJ/mol in the studies noted above) is much smaller than in most of the aluminosilicate glasses, which is obvious when comparing the slopes of the lines in Figure 8. Only for orthoclase (KAlSi3O8) glass was a value of Ea comparable to that for silica glass found (121 kJ/mol; Carroll 1991), but diffusion is much faster in orthoclase glass than in fused silica. It appears that the expansion of the network structure by incorporation of the large potassium ion facilitates the diffusion of argon. Compared to orthoclase glass Ar diffusion is much slower in sodium-bearing glasses, e.g., by about one order of magnitude in albite glass (Carroll 1991; Roselieb et al. 1992). Data for a haplogranitic composition (AOQ, Table  1) containing similar amounts of Na and K are intermediate between orthoclase and albite glasses, confirming this trend. The reduction of argon diffusivity in glasses from albite to jadeite (NaAlSi2O6) composition can be explained by greater filling of interstices in the network in the more alkali-rich composition. It is interesting to note that argon diffusion in haplogranitic glasses is systematically faster than in obsidians (Figs. 8 and 9). The haplogranite system (NaAlSi3O8-KAlSi3O8-SiO2) has been often used successfully as a model system for rhyolite and no significant difference was observed, for instance, in water diffusion (Zhang and Behrens 2000) and viscosity (Zhang et al. 2003) between natural rhyolite and haplogranite. Argon diffusivity in synthetic iron-free rhyolite glass (Syn. rhyolite, Table 1) resembles that in obsidians (see Fig. 13). Different water

Behrens

242

-11

Ar polymerized systems

-12

log D [D in m2/s]

Ab4

AOQ5

Qz2

-13

Qz1

-14 Or3 -15

Jd Jd86Qz14 Jd72Qz28 Jd52Qz48

-16

-17

Rh3 Ab3 Jd1

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

-1

1000/T (K ) Fig. 8. Comparison of argon diffusivities various polymerized glasses and melts. selected Figure  8. Comparison of argon diffusivities ininvarious polymerized glasses and Only melts. Onlydata selected are with shownpreference with preference high pressure data. sources: Symbols and solid lines correspond to data are shown for for high-P data. Data Data Symbols lines correspond to high temperature melts of the joinlines Qz –Jdare at Arrhenius 500 high-T melts of thesources: join Qz –Jd atand 500solid MPa studied by Spickenbom (2010). Non-solid MPa studied by Spickenbom (2010). Non-solid lines are Arrhenius relationships for glasses and relationships for glasses and super-cooled melts. Qz1, Jd1: silica and jadeite composition at near-ambient super-cooled melts. Qz1, Jd1: silica and jadeite composition at near-ambient pressure, Roselieb et al. pressure, Roselieb al. (1995); Qz2: silica composition ~200 MPa, Carroll (1991); Ab3, (1995);etQz2: silica composition at ~200 MPa, Carrollatand Stolper (1991); Ab3,and Or3, Stolper Rh3: albite, Or3, Rh3: albite, orthoclase and rhyolite compositions, at ~200 MPa, Carroll orthoclase and rhyolite compositions, respectively, respectively, at ~200 MPa, Carroll (1991); Ab4: albite (1991); Ab4: composition at near-ambient pressure, Roselieb Roselieb etet al. al. (1992); AOQ5: metaluminous haplogranite albite composition at near-ambient pressure, (1992); AOQ5: metaluminous haplogranite AOQ this at 200 MPa, this study. AOQ at 200 MPa, study.

contents of the glasses cannot explain this observation because the obsidians contain more water than air-melted synthetic glasses (Table 1). When comparing these three compositions it appears that the most significant difference is the absence of divalent cations in the haplogranite (Table  1). Since the real composition of rhyolite-like melts affects argon diffusivities we distinguish in the following between haplogranite (free of iron and alkaline earth elements), synthetic rhyolite (free of iron) and obsidian (natural rhyolite). In polymerized melts at high temperature, the effect of alkali content on Ar diffusion is contrary to that observed for glasses. With increasing alkali content Ar diffusion is accelerated along the join SiO2(Qz) – NaAlSi2O6(Jd) (Spickenbom et al. 2010). Extrapolating data for silica glasses towards high temperature implies that the effect of alkali content is particularly strong for silica-rich compositions. This trend is coincident with the variation of viscosity with melt composition (Toplis et al. 1997a,b) implying that the melt dynamics play a predominant role for migration of argon in the melt, while at low-temperature in the glasses the topology of the silicate network structure is the most important parameter. Argon diffusion in glasses and melts with varying alkali/Al ratios. No systematic data is available on the effect on argon diffusion when adding alkali or alkaline earth elements to SiO2 glasses. Some conclusions can be drawn by data for potassium lime silicate glasses (Reynolds 1957) in which the diffusivity of argon is several orders of magnitude faster than in silica glass, e.g., diffusion coefficients are 1.4×10-14 m2/s and 5.8×10-17 m2/s, respectively,

Noble Gas Diffusion in Silicate Glasses and Melts -12

-12

a

b

P (MPa) 50-54 100-105 200 300 380-400 500

P (MPa) 50-54 100-105 200 300 380-400 500

-13

log D [D in m2/s]

-13

log D [D in m2/s]

243

-14

-15

-14

-15

metaluminous haplogranite

peralkaline haplogranite -16 0.8

0.9

1.0

1.1

1.2

1.3

-16 0.8

1.4

0.9

1.0

-1

1.1

1.2

1.3

1.4

-1

1000/T (K )

1000/T (K ) -12

d

P (MPa) 50-54 100-105 200 300 380-400

-14

-15

-16

0.8

1.0

-14

-15

-16

peraluminous haplogranite 0.9

ASI 0.53 0.98 1.50

-13

log D [D in m2/s]

log D [D in m2/s]

-13

c

natural rhyolite

200 MPa 1.1

1.2 -1

1000/T (K )

1.3

1.4

-17 0.8

0.9

1.0

1.1

1.2

1.3

1.4

-1

1000/T (K )

Arrhenius Ar diffusion for haplogranite glasseswith withdifferent different alumina/alkali Fig. 9.Figure 9. Arrhenius plot of plot newofArnew diffusion data data for haplogranite glasses alumina/alkali ratios. a) peralkaline melts (ASI = 0.53; AOQPB) at various pressures; b) metaluminous melts (ASI = 0.98; ratios. AOQ) a) peralkaline melts (ASI = 0.53; AOQPB) at various pressures; b) metaluminous melts (ASI = at various pressure; c) peraluminous melts (ASI = 1.50; AOQPB) at various pressures. Solid lines 0.98; AOQ) at various pressure; c) peraluminous melts (ASI = 1.50; AOQPB) at various pressures. are regressions to data at 380 – 400 MPa, dashed lines are regressions to data at 200 MPa. d) Comparison Solid lines are regressions to data 380 400 line MPa, lines are regressions data at 200 MPa. of diffusivities at 200 MPa. Noteatthat the–solid fordashed dry natural rhyolite after Behrenstoand Zhang (2001) is systematically below the data for MPa. metaluminous haplogranite. d) Comparison of diffusivities at 200 Note that the solid line for dry natural rhyolite after Behrens and Zhang (2001) is systematically below the data for metaluminous haplogranite.

at 743 K (Reynolds 1957; Carroll and Stolper 1991), and even much faster than in orthoclase glass (2.0×10-15 m2/s at the same temperature; Carroll 1991). Due to the high activation energy in potassium lime silicate glass (175 kJ/mol; Reynolds 1957), the difference grows with increasing temperature. The results of Reynolds (1957) imply that depolymerization of the network might have a different effect on argon diffusion than on helium diffusion, i.e., He mobility is reduced while Argon mobility is enhanced when adding alkali ions to polymerized glasses. However, when considering the much lower Ar diffusivities in sodium aluminosilicate glasses than in orthoclase glass (Fig. 8) it is obvious that it is particularly the incorporation of the large potassium ions in the glass which enhances Ar diffusion. An explanation could be that the radius of argon atoms is similar to that of potassium ions, meaning that more interstices are available for migration of argon atoms in K-bearing glasses than in Na-bearing glasses.

Behrens

244

New Ar diffusion experiments were conducted on alkali aluminosilicate glasses with different alumina to alkali ratios in cold seal pressure vessels at temperatures of 771-1118 K and at pressures from 50 to 500 MPa using argon as the pressurizing gas (Fig. 9a-d). Examples of diffusion profiles measured by electron microprobe are shown in Figure 1a. A compilation of the diffusion data for haplogranitic melts is given in the Appendix Table C. The alumina saturation index, defined as ASI = Al/(Na + K+ 2Ca + 2Mg), of the glasses varies between 0.53 and 1.50. The glasses were not specifically designed for the diffusion study, and the compositional variation is rather complex and involves changes in the proportions of all four oxide components (Table 1). One must emphasize that a single parameter such as ASI cannot cover all compositional variations, i.e., the same argon diffusivities would be expected for orthoclase and albite glass, which is obviously not the case (Fig. 8). The Na/K ratio of the three aluminosilicate glasses is similar (1.20-1.50) and resembles that of natural obsidians Table 1. Thus we suppose that the observed trend of argon diffusion with ASI applies to natural rhyolitic melts as well. Typically, the diffusion coefficient of argon increases from the peraluminous glass (ASI = 1.53) to the metaluminous glass (ASI ≈ 1) to the peralkaline glass (ASI = 0.53). The only exception is the experiment at the lowest temperature 772 K, 200 MPa (#AS06) in which argon diffused faster in the metaluminous melt than in the peralkaline melt (Fig. 9d). The entire data set for each composition was fitted to Equation (5) to determine the activation energies and activation volumes for argon diffusion. The activation energy increases with ASI from 133±4 kJ/mol (ASI = 1.53) to 141±5 kJ/mol (ASI ≈ 1) to 158±9 kJ/mol (ASI = 0.53) while the activation volumes show no systematic trend (2.8±0.3 cm3/mol for ASI = 1.53, 3.3±0.4 cm3/ mol for ASI ≈ 1, 2.5±0.5 cm3/mol for ASI = 0.53). The argon diffusivity in the three studied compositions is well fit as a function of temperature (in K), pressure (in MPa) and ASI by a simple Arrhenius-type equation:

log D = -4.84 -

( 7362 + 1.54 P ) - 0.879 ⋅ ASI T

(7)

with a fit standard deviation of 0.15 log units, similar to the experimental error (Fig. 10). The acceleration of Ar mobility by an excess of alkalis is consistent with the results of Reynolds (1957) for potassium-lime silicate and can be explained by increasing dynamics of the network structure, i.e., a decrease in melt viscosity. The effect of excess alumina, however, probably has a different origin. The viscosity of sodium-aluminosilicate melts shows only minor variation with Al2O3 for ASI > 1 (Toplis et al. 1997a; Webb et al. 2004). It was shown by NMR and Raman spectroscopy that aluminum is incorporated in peraluminous glasses either as tetrahedral Al3+ associated with a three-coordinated O, often referred to as a tricluster, or as Al3+ coordinated to more than four O atoms (e.g., Stebbins and Xu 1997; Toplis et al. 1997a; Stebbins et al. 2000; Toplis et al. 2000; Stebbins et al. 2001; Neuville et al. 2004; Mysen and Toplis 2007). The formation of such species may close some of the doorways suitable for the migration of the argon atom. High-temperature melts along the join NaAlSi3O8 – Na2O (Spickenbom et al. 2010; Spickenbom 2010) also exhibit a systematic increase in argon mobility with melt depolymerization (Fig. 11) while the activation energy for argon diffusion (121-145 kJ/mol) does not change noticeably in the studied compositional range. It is common to characterize the degree of melt depolymerization with the parameter NBO/T = (Na + K + 2Mg + 2Ca – Al)/(Al + Si) as defined by Mysen (1988). In the system NaAlSi3O8 – Na2O the change in argon diffusivity with NBO/T is particularly pronounced for near-albite compositions. This observation supports the important role of melt dynamics on argon mobility because the effect of excess alkalis on melt viscosity is also more dramatic at low than at high alkali excess, as shown, for instance, in the haplogranite system (Hess et al. 1995).

Noble Gas Diffusion in Silicate Glasses and Melts

245

-12

log Dmeasured [D in m2/s]

-13

-14

-15

ASI 0.53 0.98 1.50

-16

-17 -17

-16

-15

-14

-13

-12

log Dpredicted [D in m2/s] Fig. 10. Comparison between the measured Ar diffusivity in haplogranite glasses with the calculations Figure 10. Comparison between the measured Ar diffusivity in haplogranite glasses with the calculations using the equation log D = -4.84 - (7362 + 1.54 · P)/T - 0.879 · ASI, with T in K and P in MPa. The using the equation logreproduces D = -4.84 - (7362 data + 1.54P)/T - 0.879 with equation the experimental with a standard deviation×ofASI, 0.15 log units.T in K and P in MPa. The equation reproduces the experimental data with a standard deviation of 0.15 log units.

-9.5

Ab + x Na2O

NBO/T

log D [D in m2/s]

-10.0

-10.5

0.198 0.160 0.127 0.100 0.077 0.048 0.019

-11.0

0.000

-11.5

-12.0 0.55

0.60

0.65

0.70

0.75

-1

1000/T (K )

Fig. 11. Effect of melt depolymerisation on argon diffusion in melts of the system Ab + x Na2O, with

x between and 7 wt% (after Spickenbom 2010). Activation energy of Figure 11. Effect of melt0 depolymerisation on argon diffusion in melts ofAr thediffusion system(proportional Ab + xNato 2O, with x slope(after in the plot) is roughly constant in this system but diffusivity with ratio of(proportional nonbetween 0 and 7thewt% Spickenbom et al. 2010). Activation energyincreases of Ar diffusion to bridging oxygen to tetrahedral cations (NBO/T). the slope in the plot) is roughly constant in this system but diffusivity increases with ratio of non-bridging oxygen to tetrahedral cations (NBO/T).

Behrens

246

Argon diffusion in rhyolitic to basaltic glasses and melts. Nowak et al. (2004) studied argon diffusion in simplified iron-free melts with compositions from rhyolite to basalt at temperatures of 1350-1500 °C and a pressure of 500 MPa. Along this pseudo join the increase in NBO/T is accompanied by an increase of divalent cations and a decrease in silica content. Similar to the system NaAlSi3O8 – Na2O, DAr increases with NBO/T, but the variation is much smaller with 0.4-0.8 log units from NBO/T = 0.04 to NBO/T = 0.59 (Fig. 12). There seems to be an increase in Ea towards the most basic composition from ~180 kJ/mol to ~250 kJ/mol, but the errors in the activation energies are too large due to the narrow T range to prove such a trend. New diffusion experiments were carried at low-temperatures (772-894 K) using the sorption technique in order to better resolve the temperature dependence of the Ar diffusivity. While the rhyolitic and the rhyodacitic composition show Arrhenian behavior in the lowtemperature range, diffusion data for andesite scatter widely (Fig. 13). The scatter is due to very short diffusion profiles and very low argon solubility in the andesitic melts, which makes the precise measurement of the profiles difficult (Fig. 1b). Although the argon diffusion data for the andesite have large uncertainty, all experiments consistently show a decrease in diffusivity from rhyolite to rhyodacite to andesite, opposite to the high temperature trend reported by Nowak et al. (2004). For technical reasons the low-T experiments were performed only at a pressure of 200 MPa. To combine both data sets we have assumed that the activation volume for argon diffusion does not vary with melt composition and can be approximated by the average Va value of 26×10-6 m3/mol derived for haplogranite and natural rhyolite melts using the data in Table 1. The resulting Arrhenius lines for synthetic rhyolite and rhyodacite are displayed for 200 MPa pressure in Figure  13. The combination of both data sets gives evidence that the activation energy for argon diffusion indeed increases with melt depolymerization in this system, similar to that observed for water diffusion in rhyolitic to basaltic melts (Behrens et al. 2004). Effect of dissolved water on argon diffusion. So far the effect of dissolved H2O on argon diffusion in melts was investigated only for rhyolite and albite compositions (Behrens and Zhang 2001). Using Ar-bearing and Ar-free glasses pre-equilibrated with water the authors

-9.5

rhyolite-basalt

log D [D in m2/s]

-10.0

-10.5

-11.0

-11.5

-12.0

-12.5 0.55

Figure  12. Variation of Ar diffusivity along the pseudo join rhyolite – basalt at 500 MPa. Sample names and data are from Nowak et al. (2004). Ar diffusivity in natural dry rhyolite after Behrens and Zhang (2001) is shown for comparison.

nat. rhyolite NBO/T 0.593 0.546 0.439 0.342 0.204 0.089 0.039

Ha Th AnTh An DaAn Da Rhy 0.60

0.65

0.70

0.75

-1

1000/T (K ) Fig. 12. Variation of Ar diffusivity along the pseudo join rhyolite – basalt at 500 MPa. Sample names and data are from Nowak et al. (2004). Ar diffusivity in natural dry rhyolite after Behrens and Zhang (2001) is shown for comparison.

Noble Gas Diffusion in Silicate Glasses and Melts -10

nat. rhyolite -11

log D [D in m2/s]

-12

metalum. haplogranite

-13 -14 -15 -16

500

-17 -18 0.5

0.6

200 MPa syn. rhyolite syn. rhyodacite syn. andesite 0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

247

Figure  13. Arrhenius plot for Ar diffusion in synthetic rhyolitic to andesitic melts. High-T data at 500 MPa are from Nowak et al. (2004), low-T data at 200 MPa are from this study. Solid lines represent trends for syn. rhyolite (top) and syn. rhyodacite (bottom) at 200 MPa. For comparison representations of data for dry natural rhyolite (Behrens and Zhang 2001) and metaluminous haplogranite (this study) at 200 MPa are shown as dashed lines. Low-T data for synthetic andesite are affected by experimental problems and are shown only to document the compositional trend.

-1

1000/T (K ) Fig. 13. Arrhenius plot for Ar diffusion in synthetic rhyolitic to andesitic melts. High temperature data at 500 MPa are from Nowak et al. (2004), low temperature data at 200 MPa are from this study. Solid lines represent trends for syn. rhyolite (top) and syn. rhyodacite (bottom) at 200 MPa. For comparison 2 representations of data for dry natural rhyolite (Behrens and Zhang 2001) and metaluminous haplogranite (this study) at 200 MPa are shown as dashed lines. Low T data for synthetic andesite are affected by experimental problems and are shown only to document the compositional trend.

demonstrated by diffusion couple experiments that H O dramatically enhances the mobility of Ar in the melt. A 4.0 wt% increase in water content increases the Ar diffusivity by approximately one order of magnitude in both rhyolitic and albitic melts at 1000 °C. The major reason for the strong influence of water content on Ar diffusion is the dramatic decrease in melt viscosity upon hydration in particular for polymerized melts, such as rhyolite. In contrast to viscosity and total water diffusion, an exponential dependence of Ar diffusivity on water content was observed over the entire range of water contents. Combining their data with previous data from Carroll (1991), Behrens and Zhang (2001) derived the following equation for Ar diffusion (in 10-12 m2/s) in rhyolitic melts:   17913 P   35936 P = DAr exp  14.627 - 2.569  +  + 27.42  X water  T T  T T  

(8)

where T is in K, P in MPa, and Xwater is the mole fraction of water on a single oxygen basis. This equation implies an activation volume of 21×10-6 m3/mol for anhydrous melts, slightly lower than the values reported in Table 1. Except for two outliers, the error of estimates is ≤ 0.2 in terms of log D for all data, covering a wide range of temperature (480-1200 °C), pressure (0.001-1500 MPa), and water contents (0.1-5 wt%). However, because hydrous melts were only studied at high temperature, Equation (8) is poorly constrained for melts containing >1 wt% water at temperatures below 800 °C. Zhang et al. (2007) combined data from various authors (including Carroll and Stolper 1991; Draper and Carroll 1995; Roselieb et al. 1996; Behrens and Zhang 2001; Nowak et al. 2004) and fit 152 Ar diffusivity data for rhyolitic, dacitic, albitic, and jadeitic melts covering a wide T-P-H2O space and assuming negligible influence of anhydrous melt composition on argon diffusivity in silicic melts:  17367 P   855.2 P  DAr = exp  -13.99 - 1.9448  +  + 0.2712  Cw  T T  T T  

(9)

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where DAr is in m2/s, P is in MPa, T is in K, and Cw is the wt% of H2Ot. The 2s uncertainty of the fit was 0.31 log units. It has to be emphasized that this equation is supported only by few data for hydrous albitic melts and no data for hydrous dacitic and jadeitic melts. Considering the effects of anhydrous composition discussed above, the equation is probably only a first estimate for hydrous melts other than metaluminous rhyolites. In glasses the effect of water on argon mobility might be different than in melts. In a recent study on silica glasses, argon diffusivities were found to be systematically lower by 0.17 log units in water-rich silica glass (~0.05 wt% H2O, Suprasil®) than in water-poor silica glasses (< 0.01 wt% H2O, Infrasil® and Herasil®)) while the activation energy for argon diffusion for both data sets agrees within error. Although the effect is small, it could be unambiguously demonstrated in experiments in which different silica glass were run in the same experiment under identical P-T-t conditions (Behrens 2010). This decrease in argon mobility may be attributed to partial closure of “doorways” for argon in the silicate network by the formation of hydroxyl groups. It is not only that OH groups fill holes in the network, in addition the silicate structure can locally relax when Si-O-Si bonds are broken. A reduction of the mobility of particles in glasses by hydration has been observed also in other studies. The ionic conductivity of sodium silicate glasses (Accocella et al. 1984) and lithium aluminosilicate glasses (Kappes 2002) is lowered upon hydration as well. In both systems the charge transport occurs via diffusion of alkali ions. Concluding remarks on argon diffusivity. Although numerous data are now available for argon diffusion in silicate melts, we are nevertheless far away from being able to construct general equations for the prediction of argon diffusivity as a function of melt composition, temperature and pressure. Some general trends are evident, i.e., the increase in diffusivity in the melts upon depolymerization. Simple parameters such as NBO/T or ASI might be suitable to describe compositional trends for some specific systems, but fail as general parameters for larger compositional variations. In view of application to natural magmas, more diffusion data are needed, particularly for hydrous melts. However, such diffusion experiments are very demanding due to the need to control carefully the water content in the diffusion samples.

Kr, Xe and Rn diffusion The diffusion data for Kr, Xe and Rn are summarized in Figure  14. Both Kr and Xe diffusivities increase with increasing replacement of SiO2 by NaAlO2. This trend is contrary to that for the small noble gases helium and neon. Roselieb et al. (1995) argued that structural relaxation of the network is a dominating factor for the diffusion of heavy noble gases in such melts. This hypothesis is corroborated by melt viscosity, which is much higher for SiO2 than for the aluminosilicate compositions. The high diffusivities of Kr and Xe and in basaltic melts support this idea, although the preliminary data from Lux (1987) has to be considered with caution because they have never been reproduced. The relatively fast diffusion of Rn in andesitic melts measured by Gauthier et al. (1999, erratum in 2000) also suggests that melt relaxation is the controlling parameter. However, for both the basaltic melt studied by Lux (1987) and the andesitic melt studied by Gauthier et al. (1999) diffusion of the heavy noble gases is much faster than the Eyring diffusivity, which reflects the migration of network formers such as Si and Al and, in the case of dry melts, oxygen as well. The Eyring diffusivity was calculated as D = kT/(lh) where k is the Boltzmann constant, l is the jump distance and h is the viscosity. Assuming a value of 3 Å for l and using viscosities of 140 Pa·s for dry andesite (Vetere et al. 2008) and 13 Pa·s for dry basalt (Misiti et al. 2009) at 1623 K, one computes Eyring diffusivities of 5.4×10-13 m2/s and 5.7×10-12 m2/s, respectively, whereas Rn diffusivities of 6.0×10-12 m2/s in andesitic melts (Gauthier et al. 2000) and Xe diffusivity of 3.5×10-10 m2/s m2/s in tholeiitic basalt (Lux 1987) were measured. This calculation suggests that either Xe and Rn diffusivities are still partially decoupled from network relaxation, or that the experiments were affected to certain extent by convection.

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249

-8

Kr Xe

8

-9 -10

7

Rn

log D [D in m2/s]

-11 -12 -13 -14

4

2a 1a

-15

3 5

10 11 1b

-16

2b 6

-17 -18

9

0.6

0.8

1.0

1.2

-1

1000/T (K ) Fig. 14. Compilation of diffusion and Rn silicate glasses and melts. Figure  14. Compilation of diffusion data data for for Kr,Kr,XeXeand Rninin silicate glasses and melts. Assignment Carroll al. (1993); Kr, SiO et al. Assignment of curves: 1a: Kr,etSiO 2 glass, Roselieb of curves: 1a) Kr, SiO2 glass, Carroll al.2 glass, (1993); 1b)et Kr, SiO2 1b: glass, Roselieb et al. (1995); 2a) Kr, (1995); 2a: Kr, NaAlSi3O8 (albite) glass, Carroll et al. (1993); 2a: Kr, NaAlSi3O8 (albite) glass, NaAlSi3O8 (albite) glass, Carroll et al. (1993); 2b) Kr, NaAlSi O (albite) glass, Roselieb et al. (1992); 3 8 Roselieb et al. (1992); 3: Kr, NaAlSi2O6 (jadeite) glass, Roselieb et al. (1995); 4: Xe, SiO2 glass, 3) Kr, NaAlSi2ORoselieb Roselieb et3Oal.8 (albite), (1995); 4) Xe,et SiO Roselieb (1995) 5) Xe, 6 (jadeite) 2 glass, et al. glass, (1995) 5: Xe, NaAlSi Roselieb al. (1992); 6: Xe, NaAlSi2et O6 al. (jadeite) glass,Roselieb Roselieb etetal.al. (1995); 7: Rn, Gauthier al. (1999, 2000) NE2, et 7) Rn, NaAlSi3O8 (albite), (1992); 6)andesite Xe, NaAlSi glass,; obsidian Roselieb etMatsuda al. (1995); 2Oet 6 (jadeite) al. (1989); andesite Gauthier et al. (1999, 2000); obsidian NE2, Matsuda et al. (1989); Black symbols indicate Kr Black symbols indicate Kr and open symbols indicate Xe data: circles, 8: estimate for basalt melt and open symbols indicate Xe data: circles, 8) estimate for basalt melt from Lux (1987); triangles down, from Lux (1987); triangles down, 9: obsidian, Carroll et al. (1993); square, 10: KAlSi3O8 (orthoclase), 9) obsidian, Carroll et etal.al.(1993); square,up,10) Carroll et al. (1993); triangle up, 11) 3O 8 (orthoclase), Carroll (1993); triangle 11: KAlSi obsidian, Matsuda et al. (1989). obsidian, Matsuda et al. (1989).

COMPARISON OF NOBLE GASES AND MOLECULAR SPECIES Hydrogen, oxygen, nitrogen, carbon dioxide and water can be dissolved in silicate glasses both as molecular species and, after reaction with the silicate network, as ionic species or species which are bound to the network. The former case may be considered as physical dissolution and the latter one as chemical dissolution. It is a joint feature of physically dissolved species that van der Waals forces determine the interaction between the species and the silicate network. It can be expected that diffusion mechanisms for molecules such as H2, O2, N2, CO2 and H2O resemble those of noble gases. Comparison of diffusion data of molecular species and noble gases can give insight into how non-spherical shape of particles affects their mobility. A compilation of diffusion equations cited below can be found in the Appendix Table D.

H2 diffusion Diffusion of hydrogen and deuterium in a couple of commercial and simple glasses was studied by glass scientists using the membrane technique (see review by Shelby 1996). Diffusion coefficients for H2 and D2 in glasses are intermediate between those for He and Ne, as shown for silica glass in Figure 4a. For silica glass Lee et al. (1962) found D2 diffusion on average slower by 20% than H2 diffusion in the range of 673-1273 K, which is less than the 41% reduction expected from the square root ratio of masses of the diffusing molecules. However, the discrepancy might originate from the imprecision of the employed experimental

250

Behrens

technique. Recently, using Raman spectroscopy as an analytical tool, Shang et al. (2009) studied the out-diffusion of hydrogen from a silica capsule, which was pre-loaded with hydrogen at elevated pressures. Their data are in very good agreement with the study of Lee et al. (1962). Data for hydrogen diffusion for geo-relevant materials are scare. Chekhmir et al. (1985) measured H2 diffusion in albite glasses near the glass transition using the color change after reduction of CO2 to graphite as an indicator of the hydrogen profiles in the glasses (more details about these experiments are given in Watson (1994)). The obtained diffusion coefficients in dry albite glass are intermediate between those for neon and argon (Fig. 4b). Gaillard et al. (2003a) derived H2 diffusivities by modeling the kinetics of redox exchange of iron with inward-diffusing hydrogen in natural rhyolitic obsidians. In the experiments of Gaillard et al. (2003a) glass cylinders were exposed to reducing mixtures composed of H2-ArCO2-CO in 1-atm furnaces and H2-Ar in a cold seal pressure vessel for durations between 3 hours and 7.5 days at temperatures 300 to 1000 °C. The growth of the reduced layer, which was accompanied by an increase in reaction-derived OH-group concentration, was fitted, considering that the reaction rate is controlled by the migration of a free mobile species (H2) immobilized in the form of OH subsequent to reaction with ferric iron. The derived H2 diffusion coefficients are in the range of those of helium diffusion, if extrapolated to lower temperature. It is noteworthy that the bulk hydrogen diffusivity is much slower than the molecular H2 diffusion in melts with reactive sites (e.g., ferric iron) due to diffusion-reaction coupling. Including results from Gaillard et al. (2003b), the authors infer that the melt/glass structure (degree of polymerization) does not seem to significantly affect the solubility or the diffusivity of H2 and suggests that their model allows the prediction of oxidation–reduction rates in the presence of hydrogen for a wide range of compositions of amorphous glasses and melts. However, this statement is based only on data for a metaluminous and a peralkaline obsidian and should be treated with caution.

H2O diffusion Diffusion of H2O molecules in silicate melts cannot be directly measured because H2O molecules always coexist with OH groups and both hydrous species interconvert rapidly. Thus molecular H2O diffusivity has been derived indirectly by modeling of bulk water diffusion profiles (e.g., Zhang et al. 1991a,b; Zhang and Behrens 2000; Ni and Zhang 2008). The modeling requires the knowledge of the relative abundance of hydrous species in the melt. Such data are rare and, hence, robust data for H2O molecule diffusion exists only for rhyolite (Zhang et al. 1991a; Zhang and Behrens 2000). Attempts were also made to model molecular H2O diffusion in dacite melt (Liu et al. 2004; Ni et al. 2009) and peralkaline rhyolite (Wang et al. 2009), but hydrous species concentrations are less well constrained and, hence, the molecular H2O diffusion data have some uncertainty. The activation energy for diffusion of H2O molecules in rhyolite melts is intermediate to that for the noble gases neon and argon (Fig. 5), as expected when comparing the effective radius of H2O molecules of 1.37 Å with the radii of Ne of 1.21 Å and Ar of 1.64 Å (all radii are for six-fold coordination after Zhang and Xu (1995)). As a rough estimate for molecular H2O diffusion in other melts, the average of neon and argon diffusivities may be used if such data are available (see Fig. 4c).

O2 diffusion Diffusion of oxygen plays an important role in redox processes in silicate melts and in oxygen isotope exchange between melts and coexisting phases. There are only few cases in which diffusion of molecular oxygen could be clearly identified as the rate-controlling mechanism. Oxygen transfer to the melt’s interior can also be achieved also by other mechanisms, e.g., oxidation of basalt glasses was found to be controlled by a counter-flux of electron holes and cations (Cooper et al. 1996). In dry high-temperature melts oxygen isotope exchange is controlled by the melt viscosity, and oxygen diffusion coefficients can be related to

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251

melt viscosity by the Eyring relationship (e.g., Poe et al. 1997; Reid et al. 2001). The dramatic enhancement of 18O diffusion in silicate melts in the presence of water is due to H2O molecules, which act as carriers for the oxygen isotope (Behrens et al. 2007). Studies of molecular oxygen diffusion in glasses are rare and restricted to silica glass. According to Doremus (2002) the effective radius of O2 molecules is about 1.6 Å, similar to the radius of argon atoms. Hence, one might expect similar diffusivities for both species, if oxygen diffuses through silicates without reaction with the network. The first diffusion coefficients of O2 determined by the membrane technique (Norton 1961) are roughly one order of magnitude larger than argon diffusivities but are more than 5 orders of magnitude larger than oxygen diffusion data derived by heterogeneous isotope exchange between silica glass and gas (Williams 1965). In latter studies using different experimental techniques the diffusion coefficients for molecular oxygen in silica glass from Norton (1961) were confirmed and extended to cover a temperature range from 800 to 1200 °C (Hetherington and Jack 1964; Kajihara et al. 2004; Tournour and Shelby 2005).

Molecular oxygen diffusion in melts is even less well studied. Doremus (1960) determined O2 diffusivities in barium aluminum alkali silicate, sodalime silicate and borosilicate glass melts using the bubble shrinkage technique. Oxygen diffusion in soda-lime silicate melts is 1.5 orders of magnitude faster than the Eyring diffusivity but much slower than neon diffusion (Fig. 15).

N2 diffusion The knowledge of diffusion of nitrogen in silicate melts is extremely limited. Frischat et al. (1978) compared diffusion of physically (as N2) and chemically (as nitride) dissolved nitrogen in soda-lime silicate melts

log D [D in m2/s]

Activation energies for O2 diffusion of 76 kJ/mol (Tournour and Shelby 2005) and 79 kJ/ mol (Kajihara et al. 2004) in the most recent studies (Arrhenius representations of their data are shown in Fig. 4c) are relatively low compared to argon diffusion (96-120 kJ/mol, see electronic attachment). The low activation energy and the high diffusivities imply that O2 indeed diffuses through the network as an entity, but the effective radius is probably smaller than the estimate of 1.6 Å by Doremus (2002). From the linear trend shown in Figure 5 the effective radius of O2 molecules is probably closer to 1.45 Å. According to Norton (1961) the solubility of molecular O2 per network oxygen in silica glass is ~ 2×10-6 at 1 bar oxygen pressure and temperatures around 1000 °C. Hence, molecular oxygen diffusion data (Norton 1961; Hetherington and Jack 1964; Kajihara et al 2004; Tournour and Shelby 2005) and diffusion coefficients based on oxygen isotope exchange rates (Williams 1965) are consistent, provided that oxygen molecules rapidly exchange isotopes with the silicate network. At elevated pressures, traces of water will be usually much -9 more efficient in transporting oxygen soda lime silicate isotopes in silicate glasses and melts O2 Ne -10 than oxygen molecules, due to the low concentration of molecular oxygen in these phases. -11

N2 nitrid

-12

-13

-14 0.6

Eyring diffusivity

0.7

0.8

0.9

-1

1000/T (K )

Fig. 15. Comparison of Ne, O2, N2 and nitride diffusion with the Eyring diffusivity in soda lime silicate glass melts. The Eyring diffusivity was calculated as D =kT/λη, where k is the Boltzmann constant, λ is the characteristic jump distance and η is the melt viscosity. In2 the calculation we assume 2 a jump distance of 3 Å and use viscosity data from Bornhöft and Brückner (1999).

Figure  15. Comparison of Ne, O , N and nitride diffusion with the Eyring diffusivity in soda lime silicate glass melts. The Eyring diffusivity was calculated as D = kT/lh, where k is the Boltzmann constant, l is the characteristic jump distance and h is the melt viscosity. In the calculation we assume a jump distance of 3 Å and use viscosity data from Bornhöft and Brückner (1999).

252

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between 1273 and 1673 K. N2 diffusion was studied by degassing from a glass slab, which was pre-saturated in nitrogen atmosphere. The experiments were performed under oxidizing conditions in an oxygen atmosphere, and gas chromatography was used for measuring the rate of nitrogen release. In order to measure the diffusivity of chemically dissolved nitrogen, glass powder was mixed with silicon nitride and melted under reducing conditions. After annealing a glass plate in a Ar-H2 atmosphere the remaining nitrogen was analyzed chemically. N2 diffusion was found to be slightly faster than oxygen diffusion, while nitride diffusion was slightly slower (Fig. 15). However, direct comparison of the data remains uncertain since the composition of the soda-lime silicate glass used by Doremus (1960) is unknown.

CO2 diffusion Diffusion of carbon dioxide in melts of the joins rhyolite – basalt (Nowak et al. 2004) and NaAlSi3O8 – Na2O (Spickenbom et al. 2010) is always controlled by the mobility of CO2 molecules, although the migration of carbonate groups becomes increasingly important with increasing melt depolymerization. To my knowledge rhyolite and silica are the only compositions which dissolve CO2 in glasses only in molecular form, as evidenced by IR spectroscopy (Blank and Brooker 1994; Behrens 2010). In high-temperature melts the equilibrium speciation of carbon dioxide in silicate melts shifts to molecular CO2 (Morizet et al. 2001; Nowak et al. 2003) so it can be expected that carbonate is insignificant in rhyolite melts as well. Comparison of diffusion data from different studies (Watson 1991; Blank 1993; Behrens et al. 2000; Zhang et al. 2007) shows that the diffusivities of CO2 and Ar are almost identical in water-poor rhyolitic melts over a wide range of temperature and pressure and have similar dependence on water content. Recent experiments on silica glasses confirm the similarity of Ar and CO2 diffusion (Behrens 2010). This finding implies that the smaller radius compensates for the greater halflength of the CO2 molecule and that the linear CO2 molecule moves as an oriented molecule to minimize expansion of the “doorways” of the silicate network. CO2 is a long cylindrical molecule with base diameter of 1.4 Å and a half-length of about 2.5 Å, and the Ar atom is spherical with a diameter of 1.64 Å (Zhang and Xu 1995). However, simultaneous experiments with both volatiles in silica glasses reveal that Ar and CO2 diffusivities might not be absolutely the same (Behrens 2010). In these experiments, run in gas pressure vessels at 200 MPa, glass plates in noble metal containers open to the Ar atmosphere were placed side by side to sealed capsules loaded with 2-3 glass plates and silver oxalate, which decomposes to CO2 and silver upon heating. The time-temperature-pressure path was the same for all the samples and, hence, direct comparison of the diffusion data is possible. CO2 and Ar diffusivities agree within experimental error for water-rich silica glass (Suprasil®, 464 ppm dissolved H2O) but in water-poor silica glass (Infrasil®, Na + K), Ca will also serve this role. Al in excess 0.5Ca + Na + K (i.e., peraluminous melt) is treated as a network modifier, as are any alkalis in excess of Al (perakaline melt). Ti+4, Cr+3, Fe+3 and P+5 are typically considered together and under certain conditions may act as network formers. However, it is well established that these elements tend to cluster to form molecular species largely independent of the aluminosilicate network (e.g., Hess 1980). The prevalence of relatively high-temperature liquid immiscibility in Ti- , P-, Fe+3-rich systems is a reflection of this tendency and attests to the large excess entropies of mixing for these systems. For this reason many authors simply ignore these elements when summing up NF. Two compositional indices commonly used as proxies for melt structure/polymerization are %NF [=100×T/(T+M), where T is the total number of network formers and M is the total number of network modifiers] and the M/O ratio, where O is the total number of oxygen atoms. Now consider the role of oxygen in silicate melts. Toop and Samis (1962) proposed that oxygen can be relegated to one of three primary roles: bridging (bonded) to two network former cations (i.e., bridging oxygen, O°), linking a network former to a network modifier cation (non-

Self-diffusion in Silicate Melts

271

bridging oxygen, O-) or linking two cations existing outside the network (free oxygen – O2-) governed by the homogenous equilibrium: O° + O2- = 2O-. The abundance of oxgyen species is given by the equilibrium constant (K), where K = [O-]2/([O°][O2-])and the values in brackets are species activities. Based on theoretical considerations (e.g., Toop and Samis 1962; Masson et al. 1970; Hess 1980) and nuclear magnetic resonance (NMR) studies of analog melts (e.g., Lee et al. 2003) it is shown that 90-95% of the oxygen in rhyolite melt occurs as O°, compared to 6065% in basalt liquid. For melts and compositions between these endmembers the abundance of free oxygen (O2-) is predicted to be ~1%, with the balance existing as O-. This requires values of K for natural silicate melts greater than ~1; thus for most compositions of interest to igneous petrology (barring systems rich in Ti, P, Fe+3, etc.) it is reasonable to assume that oxygen occurs either as bridging or non-bridging species. An estimate of the fraction of non-bridging oxygen (xNBO) at low pressure is given by the ratio NBO/(NBO+BO), where on the basis of 100 oxygens NBO = 200 - 4T and BO = 100 - NBO (assuming negligible amounts of free oxygen) (Mysen et al. 1981; Lee et al. 2004). It is prudent to introduce some additional notations describing how network formers and oxygen associate. Much of this insight comes directly from spectroscopic studies of high temperature and quenched melts, as well as from simulations. Let Qn be any tetrahedrally coordinated species, where n is the number of oxygens shared with neighboring tetrahedra. Q4 represents Si or Al having only Si or Al as neighboring cations, while smaller values of n correspond to Si and Al residing at the edges or the ends of polyanions (Q3, Q2 or Q1) or having only network modifiers as nearest neighbors (Q0). Toop and Samis’ (1962) classic polymerization reaction thus leads to the evolution from Q0 species to Q1 through Q3 to Q4 as the proportions of network formers and BO increase. The relative abundance of these species for any given melt composition will be governed by disproportionation reactions such as 2Q3 ↔ Q2 + Q4 or 2Q2 ↔ Q3 + Q1 (e.g., McMillan and Wolf 1995). Lastly, while Si and Al are predominately in 4-fold coordination by oxygen at low pressure, it is now well established that coordination number increases in response to compression (as well as heating based on simple entropy considerations). Similarly, at low pressure oxygen participating in the network is most commonly linked to just two NF cations, while there is mounting evidence that at high pressure oxygen may bond with three NF cations with further distortion of the network structure. Lee et al. (2008) surmised that formation of oxygen triclusters accompanies the increase in Si coordination at high pressure and that both coordination changes facilitate an increase in the proportion of bridging oxygen, i.e., polymerization in the classical sense.

Theoretical Considerations Self and tracer diffusion To appreciate the factors governing self, tracer and isotope, as well as, chemical diffusion in silicate melts it is useful to first consider the phenomenological basis for most treatments of diffusion. One generally assumes that the velocity of diffusion is linearly proportional to the driving force. For diffusion of chemical constituents driving forces are (negative) gradients in chemical potential. We can define the diffusion velocity for any species i (vi) in the presence of other chemically distinct species as vi =

Mi ∇mi N

(1)

where N is Avogadro’s number and the grad operator refers to chemical potential gradients in any direction that species i diffuses at velocity v (deGroot and Mazur 1984). The constant of proportionality Mi is the velocity of species i per unit force and is referred to as the intrinsic mobility. In most treatments Mi is assumed to be independent of the magnitude of the driving

Lesher

272

force, ∇m. This is a reasonable assumption for systems close to equilibrium, but may not hold for systems far from equilibrium. The flux Ji describes the amount of species i passing through a unit area normal to the diffusion direction per unit time, i.e., J i = ci vi

(2)

where ci is the concentration of the species in grams per unit volume. Combining Equations (1) and (2) gives Ji = -

ci Mi ∇mi N

(3)

Importantly, Equation (3) provides clear and explicit separation of the kinetic and thermodynamic controls on diffusion. More commonly though Equation (3) is written J i = - Li∇mi

(4)

where Li is the Onsager coefficient that depends on both the magnitude of Mi and species concentration (deGroot and Mazur 1984), or Ji = - Dic ∇ci

(5)

st

known as Fick’s 1 law. In the latter case the driving force for diffusion is expressed solely in terms of gradients in concentration so that the diffusion coefficient Dic depends on kinetic and thermodynamic factors. Equation (5) is the classic definition of the chemical diffusion coefficient, which by comparison is related to Li and Mi, as follows dmi ci Mi dmi = Dic L= i dci N dci

(6)

Equation (6) is a good illustration of how difficult it can be to use Fick’s 1st law to account for diffusion phenomena occurring over a large range of concentrations and when complex activity-composition relations prevail. Assuming ci = xiC, where C is the total number of moles in the system, and substituting dmi RTd ln( g i xi ) where gi and xi are the activity for the definition of chemical activity, e.g.,= coefficient and mole fraction of species i, respectively, Equation (6) leads to  d ln g i  = DiDarken kTMi 1 +   d ln xi 

(7)

Equation (7) is credited to L. S. Darken (1948), where DiDarken is the Darken diffusivity for species i, k is the Boltzmann constant, T is temperature, and the remaining variables are as defined above. The term in brackets is often referred to as the thermodynamic factor and takes different forms depending the compositional dependence of the activity coefficient g. For a regular symmetric binary solution with x1 + x2 =1 the term in brackets becomes [1 - (2WG/RT) x1x2], where WG is the Margules parameter. Equation (7) is particularly useful in the present context. For example, if composition is uniform throughout the system the ratio d ln g i / d ln xi will be zero and thus the limiting value of the diffusion coefficient will be kTMi. Thus, in the absence of any chemical flux the diffusivity is simply a measure of intrinsic mobility reflecting random walk of species i. This provides us with an explicit definition of self-diffusion, where kTMi in Equation (7) can be replaced by the self-diffusion coefficient, Di* to give

Self-diffusion in Silicate Melts

273

 d ln g i  = Di* 1 + DiDarken   d ln xi 

(8)

Now consider species i present in dilute concentration and obeying Henry’s law, i.e., gi = constant. In this case, while dlnxi is non-xero, dlngi remains zero and thus DiDarken = Di*. Herein lies the common assumption that the tracer diffusion and self-diffusion coefficients are equivalent measures of intrinsic mobility. The validity of Henry’s law behavior is critical in making this connection and thus caution must be exercised when making laboratory measurements to insure that the tracer is not too concentrated, or that gradients, and thus fluxes, of other components do not lead to non-Henrian behavior of the tracer. In most situations encountered in magmatic systems the ratio d ln g i / d ln xi will have a value other than zero, DiDarken ≠ Di*, and often varies as diffusion occurs. The Darken diffusivity defined by Equation (8) can be either positive or negative depending on whether the activity of species i increases or decreases with concentration, respectively. While the former case is usually encountered and certainly prevails in the absence of gradients in other components, the latter situation arises due to strong non-ideality accompanying gradients in other components, particularly silica content. The occurrence of uphill diffusion in this context can then be appreciated as the consequence of negative Darken diffusivities. The relationships between the Darken diffusivity, self-diffusion coefficient and activitycomposition relations leading to normal (downhill) diffusion and anomalous (uphill) diffusion for the trace element Sr during diffusive exchange between mafic and felsic melts are shown in Figure 1. It is important to note that this model of Sr diffusion between mafic and felsic melts, while rigorously a multicomponent diffusion problem (see Liang 2010, this volume), assumes effective binary diffusion of the major constituents by ignoring cross coefficients and assuming that chemical homogenization is rate-limit by silica diffusion. Lesher (1994) used independent constraints for the variation in Sr self-diffusivity with temperature and melt composition. The compositional dependence of the activity coefficient is derived from twoliquid and thermal diffusion experiments (e.g., Lesher 1986). The numerical simulations solve the 1D diffusion equation for a composite semi-infinite medium, where interdiffusion of SiO2 EBDC ), and the Darken diffusivity is treated as a pseudobinary process (see Fig. 1 caption for DSiO 2 Darken for Sr (DSr ) is recalculated after each time step across the diffusion couple. Richter (1993) took a similar approach using data from Lesher (1990) to constrain both D* and the activitycomposition relations to best fit the experimental isotopic and concentration profiles developed by interdiffusion of basalt and rhyolite. While not an independent verification for Darken’s formulation, the work provides further support for the general approach. Finally, Zhang (1993) modeled uphill diffusion profiles of major and trace elements during crystal dissolution using Equation (8), again ignoring multicomponent effects, but with good success. Figure 1 shows model results for two initial conditions: Case 1 resulting in downhill diffusion (mafic reservoir = 1000 ppm Sr; felsic reservoir = 500 ppm Sr) and Case 2 leading to uphill diffusion (750 ppm Sr in both reservoirs). For the simulation, a diffusion couple halflength of 0.5 cm, temperature of 1300 °C, and duration of 1000 s were assumed. The diffusion gradient for SiO2 (Fig. 1a), variation in the activity coefficient relative to its initial value in the mafic reservoir (Fig. 1b) and variation in DSr* (Fig. 1c) are identical in both cases, but vary for the DSrDarken (Figs. 1d, e) and Sr concentration profiles (Figs. 1f, g). For Case 1, Sr diffuses from high concentration in the mafic reservoir towards lower concentration in the felsic (Fig. 1f). This behavior is consistent with a positive DSrDarkenalong the entire diffusion profile, although this variation in the Darken diffusivity for Sr depends on both DSr* and dlnγi/dlnxi as prescribed by Equation (8) (Fig. 1d). The variation in DSrDarken is more complex for Case 2 where similar initial Sr concentrations in the two reservoirs lead to negative DSrDarken within the interdiffusion zone and eventually to uphill diffusion of Sr into the mafic reservoir (Fig. 1e). To the left of

Figure 1. Model results of normal “downhill” diffusion (Case 1) and anomalous “uphill” diffusion (Case 2) of Sr during interdiffusion of mafic (50 wt% SiO2) with felsic (70 wt% SiO2) melts based on the model of Lesher (1994) for a composite semi-infinite diffusion couple. Initial Sr concentrations: Case 1 - 1000 ppm Sr in the mafic and 500 ppm in the felsic melt; Case 2 -750 ppm in both reservoirs. Initial SiO2 and Sr concentrations are shown as dotted lines in panels a, f and g. Simulation assumes a half-length of 0.5 cm, temperatures of time 1300 º C and duration of 1000 s. SiO2 interdiffusion is simulated assuming an effective binary diffusion coefficient (EBDC) given by lnDSiO2EBDC = -6.70 - (31195/T) + 12.28(0.52 - xSiO2), where T is temperature in kelvin and xSiO2 is the weight fraction of silica. The Darken diffusivity for Sr is computed using Equation 8 where lnDSr* = -16.25 - (16348/T) + 8.02(0.73 - xSiO2) and g/g0 calculated from the parameters given in Table 4a of Lesher (1994). The three panels to the right cover Cases 1 and 2 and are from top down (a) the silica interdiffusion profile, (b) g/g0 variation for Sr across the diffusion couple and (c) the computed DSr* along the composition gradient. Panels d and e below show the variation in the Darken diffusivity across the interdiffusion zone as given by Equation (8) for both cases, while panels f and g are the Sr concentration profiles for Case 1 (downhill diffusion into the felsic reservoir) and Case 2 (uphill diffusion into mafic reservoir), respectively.

274 Lesher

Self-diffusion in Silicate Melts

275

the Sr maximum diffusion is normal (downhill toward the mafic end of the couple). Likewise, diffusion in the felsic reservoir to the right of the minimum is downhill toward the mafic couple. The Sr concentration profiles are distinctly different for Cases 1 and 2 that is a direct result of two factors: 1) the increase in the activity coefficient for Sr with silica content and 2) DSr* EBDC . Figure 2a illustrates this decoupling by the non-linear covariations in SiO2 and Sr

Diffusion melts< 0). Equation (17) fit experimental data to within 0.72 lnD units including the following dry melts: albite and jadeite of Roselieb and Jambon (1997), basalt1 and andesite2 of Lowry et al. (1982), dacite4 and two points of

Figure 7. Cs diffusivities in (a) synthetic silicate melts (a) and (b) natural silicate melts or analogs. The Fig.Equation 7. Cs(17) diffusivities in diffusion (a) synthetic melts (a) and (b)wet natural melts or analo lines are from (fit to all dry data) silicate except for orthoclase and melts. silicate Data sources: albite1 (Jambon and Carron 1976); albite2 and jadeite (Roselieb and Jambon 1997); HR7+Na (Mungall et al. 1999); orthoclase 1976); HR7 (a haplorhyolite melt)and andwet wet melts. HR7 (with 3.6sources: a from Eq. 7b(Jambon (fit to alland dryCarron diffusion data) except for orthoclase Data wt% H2O) (Mungall et al. 1999); rhyolite5 (rhy5) and rhyolite15 (rhy15) (Jambon 1982); dacite4 and pantellerite1 (Henderson al. 1985); andesite2 basalt1 and (Lowry et al. 1982). Carron 1976);et albite2 and jadeite and (Roselieb Jambon 1997); HR7+Na (Mungall et al. 1999)

and Carron 1976); HR7 (a haplorhyolite melt) and wet HR7 (with 3.6 wt% H2O) (Mungall et a

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three on pantellerite1 (Henderson et al. 1985), dry HR7 and HR7 + Na (Mungall et al. 1999), and rhyolite15 at 0.1 GPa (Jambon 1982). One D value in pantellerite1 at 1575 K is higher than that predicted by Equation (17) by 1.0 lnD unit. However, diffusion data for albite by Jambon and Carron (1976) are off by 0.1 to 1.6 lnD units, and those for orthoclase by Jambon and Carron (1976) are off by 4.5 to 7.8 lnD units. Cs diffusivities in rhyolite5 melt at 0.1 GPa (Jambon 1982) are higher than those predicted by Equation (17) by 0.6 to 1.9 lnD units, likely due to higher H2O in the melt. Fr diffusion. No Fr diffusion data in silicate melts are known. Some comments on alkali diffusivities in silicate melts. Diffusivities of alkali elements in rhyolite5 melt at 0.0001-0.1 GPa are summarized in Figure 8. The small pressure difference is not expected to cause noticeable variation in the diffusivities. In the alkali series, smaller cations usually diffuse more rapidly (Li ≈ Na > K > Rb > Cs) in silicate melts. This is similar to noble gas diffusion: the diffusivity of noble gases increases as the size decreases, from Xe to Kr to Ar to Ne to He (e.g., Perkins and Begeal 1971; Roselieb et al. 1995; Behrens 2010, this volume). This trend may sound intuitive, but cannot be explained by the StokesEinstein equation (see Zhang 2010, this volume) because the diffusivity variation is orders of magnitude, much larger than the relative change in the ionic radius. Furthermore, this trend cannot be generalized to other elemental series: e.g., in alkali earth elements, REE, and most other series, the trend can be opposite.

The alkali earths (Be, Mg, Ca, Sr, Ba, Ra) Be diffusion. Only one study (Mungall et al. 1999) reported Be tracer diffusion data on three melt compositions: HR7, wet HR7 with 3.6 wt% H2O, and HR7 + Na (Table 1). The data are shown in Figure 9, together with the line for Cs diffusivities in dry HR7 for comparison. Even though Be is a small cation, Be diffusivities are very low, much lower than the lowest diffusivities of alkali elements (Cs). As will be seen later, Be diffusivities are also the smallest among the alkali earth elements. Be tracer diffusivity in dry HR7 at 1410-1873 K and 0.1 MPa can be expressed as follows: (38690 ± 4236)   HR7 DBe TD = exp ( -8.38 ± 2.62)  T  

(18)

The maximum error of Equation (18) in reproducing the three experimental diffusivities is 0.21 in lnD. The activation energy is high, 322±35 kJ/mol. Mg diffusion. Mg is a major element in most natural silicate melts. Extensive Mg diffusion data in silicate melts are available, with 13 papers and 166 points (of which 15 are for hydrous melts). Zhang et al. (1989) determined Mg SEBD in dry andesite1 melt during crystal dissolution experiments. Kubicki et al. (1990) studied SEBD of Mg in dry diopsideanorthite melts. Sheng et al. (1992) explored Mg self diffusion in dry CMAS melts at 15341826 K and in air. Baker and Bossanyi (1994) examined the effect of H2O and F on Mg SEBD in rhyolite7-dacite1 diffusion couple at 1373-1673 K and 1 GPa. van der Laan et al. (1994) obtained Mg SEBD in dry rhyolite3-rhyolite16 and andesite3-rhyolite16 diffusion couples at 1523 K and 1 GPa. LaTourrette et al. (1996) and LaTourrette and Wasserburg (1997) studied Mg self diffusion in dry HB1 melt at 1623-1773 K and in air. Mungall et al. (1999) examined Mg tracer diffusion in dry HR7 and dry HR7 + Na at 0.1 MPa and wet HR7 (3.6 wt% H2O) melts at 1 GPa. Van Orman and Grove (2000) reported a datum of Mg SEBD in lunar basalt (LB1 in Table 1) during clinopyroxene dissolution. Roselieb and Jambon (2002) investigated Mg tracer diffusion in dry albite and jadeite melts at 1073-1293 K and 0.1 MPa. Lundstrom (2003) acquired SEBD of Mg in dry basalt7-basanite couple at 1723 K and 0.9 GPa. Chen and Zhang (2008, 2009) determined MgO SEBD in basalt11 (MORB) melt during olivine and clinopyroxene dissolution. Mg diffusion data are shown in Figure 10.

Zhang et al. (Ch 8) Diffusion data in silicate melts

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331

Figure 8. Li, Na, K, Rb and Cs tracer diffusivities in rhyolite and albite melts. See Figures 1-7 for data 8. and Li, Cs Na,diffusion K, Rb and tracer diffusivities in rhyolite and albite melts.andSee Figs.1976; 1-7 for data sou sources. In Fig. (b), Rb in Cs albite melts were studied in two papers (Jambon Carron Roselieb and Jambon 1997), with small differences. (b), Rb and Cs diffusion in albite melts were studied in two papers (Jambon and Carron 1976; Roselieb a 1997), with small differences. Examination of the data shows the following. First, the pressure effect at ≤ 2 GPa is negligible (e.g., open circles, triangles, squares and diamonds in Fig. 10a). Second, differences between self-diffusivities and SEBD are less than 1 lnD unit (e.g., self diffusivities in HB1 and SEBD in basalt11 in Fig. 10a) when SiO2 concentration difference across the profile is ≤ 6 wt%, although there are no data at exactly the same bulk composition for a direct comparison. (LB1, even though also a basalt, is compositionally very different from terrestrial basalts.) Thirdly, Figure 10b shows that Mg diffusivities vary significantly with melt composition (and 8 temperature), especially from andesite1 to HR7.

Mg SEBD in dry basalt11 (MORB) melt (containing 0.2 to 0.4 wt% H2O) during olivine

Zhang et al. (Ch 8) Diffusion data in silicate melts

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Figure 9. Be diffusivities HR7at at MPa, wet(containing HR7 (containing 3.6 Fig.tracer 9. Be tracer diffusivitiesinindry dry HR7 0.10.1 MPa, wet HR7 3.6 wt% H2O) at 1wt% GPa, H and dryat 1 GPa, and 2O) dry HR7 + Na at 0.1 MPa. All data are from Mungall et al. (1999). The short dashed (and purple) line with HR7+Na at 0.1 MPa. All data are from Mungall et al. (1999). The short dashed (and purple) line with no no corresponding points are for Cs diffusion in dry HR7, shown for comparison. corresponding points are for Cs diffusion in dry HR7, shown for comparison.

dissolution at 1543-1753 K and 0.5-1.4 GPa can be expressed as (Chen and Zhang 2008): (26222 ± 2470)   basalt11 (oliv diss) DMg = exp ( -7.92 ± 1.50) SEBD  T  

(19)

The maximum error of Equation (19) in reproducing the diffusion data is 0.27 in lnD. Mg SEBD values in the same basalt11 melt during clinopyroxene dissolution at 15441790 K and 0.5-1.9 GPa are lower than that during olivine dissolution by about 0.37 in lnD and can be expressed as follows (Chen and Zhang 2009) (28922 ± 2420)   basalt11 (cpx diss) DMg = exp ( -6.65 ± 1.46) SEBD  T  

(20)

The maximum error of Equation (20) in reproducing the diffusion data is 0.16 in lnD. Although the same initial melt composition (basalt11) is used in the olivine and clinopyroxene 9 dissolution studies by Chen and Zhang (2008, 2009), the interface melt during clinopyroxene dissolution is more SiO2-rich than that during olivine dissolution. Therefore, it is expected that Mg SEBD values during clinopyroxene dissolution are lower than those during olivine dissolution. In andesite1 melt, Mg SEBD during olivine dissolution at 1488-1673 K and 0.5-1.5 GPa can be expressed as: (35049 ± 6283)   andesite1 (oliv diss) DMg = exp ( -3.53 ± 3.96) SEBD  T  

(21)

The maximum error of Equation (21) in reproducing the experimental diffusivities is 0.34 in lnD. Mg SEBD values in the same andesite1 melt during clinopyroxene dissolution are not significantly different from those during olivine dissolution. In highly silicic melt HR7, Mg tracer diffusivities can be expressed as follows:

Zhang et al. (Ch 8) Diffusion data in silicate melts

Diffusion Data in Siliate Melts

333

Figure 10. Mg diffusivities in silicate melts. basalt11(ol) means D in basalt11 melt during olivine 10. Mg diffusivities silicate melts. basalt11(ol) means dissolution; D in basalt11b-b meltmeans duringbasalt7olivine dissolution dissolution;Fig. LB1(cpx) means D ininlunar basalt during clinopyroxene basanite couple. Data sources can be found in the text and Table 1. Pressure is not shown in (b) because (a) LB1(cpx) means D in lunar basalt during clinopyroxene dissolution; b-b means basalt7-basanite couple. D shows that the pressure effect is negligible. sources can be found in the text and Table 1.

Pressure is not shown in (b) because (a) shows that the press

(29341 ± 7457)   HR7 DMg TD = exp ( -11.77 ± 4.51)  T  

effect is negligible.

(22)

The maximum error of Equation (22) in reproducing the experimental diffusivities at 14101873 K and 0.1 MPa is 0.49 in lnD. 10 We made an effort to model the compositional effect on Mg diffusivities (including self and tracer diffusivities, and SEBD) in natural and nearly natural dry basalt to rhyolite melts (rather than melts with mineral compositions), including SEBD in andesite1 melt (Zhang et

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al. 1989), self diffusivities in HB1 melt (LaTourrette et al. 1996; LaTourrette and Wasserburg 1997), tracer diffusivities in HR7 (Mungall et al. 1999), SEBD in lunar basalt (Van Orman and Grove 2000), SEBD in basalt7-basanite melts (Lundstrom 2003), and SEBD in basalt11 (Chen and Zhang 2008, 2009). We also included the SEBD data by Kubicki et al. (1990) on Di-An melts and self diffusion data by Sheng et al. (1992) on “basic” melts because the compositions of these melts are not far from the HB1 melt of LaTourrette et al. (1996) and LaTourrette and Wasserburg (1997). Data by van der Laan et al. (1994) are not included because the H2O content is fairly high (0.3-1.2 wt%). For SEBD values, there is a range of compositions along a profile. For the modeling purpose, the melt composition is taking to be the average of the two diffusion ends if a diffusion couple profile is fit by assuming a constant diffusivity, or the average of the far-field and interface melts if the experiment method is crystal dissolution. The diffusion data can be well fit by the following: dry basalt to rhyolite ln DMg -5.17 - 11.37 XSi - 2.16 X FM SD, TD & SEBD =

10993 + 17839 XSA T

(23)

where X is cation mole fraction, XFM = Fe  +  Mn  +  Mg, and XSA = Si  +  Al. Fits of the data by Equation (23) are shown in Figure 11, with a reproducibility within 0.76 lnD units (or 0.33 logD units) except for the four outlier points (one point on basalt7-basanite couple by Lundstrom 2003 is off by 0.9 lnD units, one point for lunar basalt melt by Van Orman and Grove 2000 is off by 1.9 lnD units, one single point for andesite melt during quartz dissolution by Zhang et al. 1989 is off by 1.2 lnD units, and one point in Kubicki et al. 1990 is off by 1.2 lnD units). The maximum error in using Equation (23) to predict Mg tracer diffusivities is 0.7 lnD units for HR7 + Na melt, 1.1 lnD units for jadeite melt, 1.6 lnD units for albite melt. Mg diffusion data in hydrous melts are limited, with only 15 points. Adding 1 wt% H2O increases the diffusivity by about 2 lnD units (van der Laan et al. 1994); adding 3.6 wt% H2O increases the diffusivity by about 4 lnD units (Mungall et al. 1999); and adding 3.2 Zhang et al. (Ch 8) Diffusion data in silicate melts

Figure 11.Fig. Fitting MgMg diffusion natural and nearly basalt melts to rhyolite melts by Equation (23). 11. Fitting diffusion data data inin natural and nearly natural natural basalt to rhyolite by Eq. 11. basalt11(ol) basalt11(ol) means D in basalt11 melt during olivine dissolution. Data sources: LB1 (LaTourrette et al. means D in basalt11 melt during olivine dissolution. Data sources: LB1 (LaTourrette et al. 1996; LaTourrette and 1996; LaTourrette and Wasserburg 1997); HR7 (Mungall et al. 1999); basalt11(ol) (Chen and Zhang 2008); basalt11(cpx) (Chen1997); and HR7 Zhang 2009); (Zhang et al. 1989); Dibasalt11(cpx) etZhang al. 1990). Wasserburg (Mungall et al.andesite1(ol) 1999); basalt11(ol) (Chen and Zhang 2008); (Chen and 50An50 (Kubicki (2009); andesite1(ol) (Zhang et al. 1989); Di50An50 (Kubicki et al. 1990).

Diffusion Data in Siliate Melts

335

wt% and 6.0 wt% H2O increases the diffusivity by 3.1 and 3.5 lnD units, respectively (Baker and Bossanyi 1994). The limited data indicate that adding H2O increases MgO diffusivity significantly, but the effect cannot be quantified yet. Ca diffusion. Sixteen papers reported Ca diffusion data in silicate melts (129 points, of which 20 are for hydrous silicate melts). Medford (1973) explored Ca SEBD in mugearite1 and mugearite2 melts (Table 1) using the diffusion couple method at 1503-1696 K and 0.1 MPa. Hofmann and Magaritz (1977) studied Ca tracer diffusion in dry basalt3 melt at 15231723 K and 0.1 MPa. Jambon (1982) acquired Ca tracer diffusivities in rhyolite5 melt at 905-1204 K and 0.1 GPa. Harrison and Watson (1984) reported Ca SEBD in rhyolite12 melt during apatite dissolution at 1473-1673 K, 0.8 GPa, and 0.1 and 1.0 wt% H2O. Zhang et al. (1989) extracted Ca SEBD in andesite1 melt during diffusive clinopyroxene dissolution experiments. Kubicki et al. (1990) studied SEBD of Ca in binary systems of dry diopsideanorthite melts at 1633-1963 K and 0.0001 to 2 GPa, with most data at 0.2 GPa. Behrens (1992) reported Ca tracer diffusivities in dry Ab40An60 melt at 993-1123 K and 0.1 MPa. Baker and Bossanyi (1994) examined the effect of H2O and F on Ca SEBD in the same diffusion couple at 1373-1673 K and 1 GPa. van der Laan et al. (1994) obtained Ca isotopic effective binary diffusivities in rhyolite3-rhyolite16 and andesite3-rhyolite16 diffusion couples at 1523 K and 1 GPa. LaTourrette et al. (1996) characterized Ca self diffusion in dry HB1 melt (Table 1) at 1623-1773 K and in air. Liang et al. (1996a) explored Ca self diffusion in various dry CaO-Al2O3-SiO2 melts at 1773 K and 1 GPa. Mungall et al. (1999) examined Ca tracer diffusion in dry HR7 and dry HR7 + Na at 0.1 MPa and wet HR7 (3.6 wt% H2O) melts at 1 GPa. Roselieb and Jambon (2002) studied tracer diffusion of Ca in dry jadeite and albite melts at 923-1293 K and 0.1 MPa. Lundstrom (2003) obtained SEBD of Ca in dry basalt7-basanite couple at 1723 K and 0.9 GPa. Gabitov et al. (2005) examined Ca SEBD in two haplorhyolite (HR6 and HR9 in Table 1) melts during fluorite dissolution at 1173-1273 K, 0.075-0.1 GPa, and 1.2-4.8 wt% H2O. Morgan et al. (2006) measured Ca SEBD in low-Ti and high-Ti lunar picrite melts (labeled LP1 and LP2 in Table 1) during anorthite dissolution at 1673 K and 0.6 GPa. Chen and Zhang (2008, 2009) determined Ca SEBD in basalt11 melts during olivine and clinopyroxene dissolution. During olivine dissolution, the Ca concentration gradient is small, and Ca diffusion is dominated by cross-diffusivities, and sometimes even show uphill diffusion (e.g., Zhang et al. 1989). Hence, Ca effective binary diffusivities extracted by Chen and Zhang (2008) may be useful in some aspects but are not comparable with Ca diffusion controlled by its own concentration gradient. Ca SEBD values during olivine dissolution are hence not modeled quantitatively below; only Ca SEBD values during clinopyroxene dissolution are considered. Melt compositions are listed in Table 1 except for diopside-anorthite and CaOAl2O3-SiO2 melts. Ca diffusion data in dry natural and nearly natural melts are shown in Figure 12. SEBD data of Ca (close to FEBD) in mugearite melt at 1503-1696 K and 0.1 MPa by Medford (1973) are scattered and do not define a good Arrhenius trend (Fig. 12a), reflecting the experimental and analytical difficulties in the early years of diffusion studies. Ca self diffusivities in HB1 melts at 1623-1773 K and 0.1 MPa (LaTourrette et al. 1996), SEBD in lunar picrite melts during anorthite dissolution at 1673 K and 0.6 GPa (Morgan et al. 2006), and SEBD in basalt11 during clinopyroxene dissolution at 1509-1790 K and 0.5-1.9 GPa (Chen and Zhang 2009) are similar and independent of pressure, and can be fit by the following equation: basalt ln DCa -(11.79 ± 1.43) SD & SEBD =

(19138 ± 2349) T

(24)

The maximum error of Equation (24) in reproducing diffusivities in the three experimental studies (LaTourrette et al. 1996; Morgan et al. 2006; Chen and Zhang 2009) is 0.30 in lnD.

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Figure 12. Ca diffusivity in dry (a) mafic melts (SiO2 ≤ 60 wt%) and (b) silicic melts (SiO2 > 60 wt%). 12. Ca diffusivity in dry (a) mafic melts (SiO wt%) and(Mu (b) means silicic mugearite; melts (SiO2 > 60 wt%). T 2 ≤ 60 The legend Fig. contains information about the type of diffusion, the melt composition b-b means basalt7-basanite couple; and1=andesite1; rhy5=rhyolite5; rhy12(ap) means D in rhyolite 12 during apatite dissolution; r-r means rhyolite3-rhyolite16 couple;the a-r melt meanscomposition andesite3-rhyolite16 couple), contains information about the type of diffusion, (Mu means mugearite; b-b mean pressure, and authors (M73: Medford 1973; L96: LaTourrette et al. 1996; L03: Lundstrom 2003; M06: Morgan et al. 2006; C09: Chenand1=andesite1; and Zhang 2009; rhy5=rhyolite5; Z89: Zhang et al. 1989; J82: Jambon H84: Harrison basanite couple; rhy12(ap) means 1982; D in rhyolite 12 during apatite disso and Watson 1984; V94: van der Laan et al. 1994; M99: Mungall et al. 1999).

means rhyolite3-rhyolite16 couple; a-r means andesite3-rhyolite16 couple), pressure, and authors (M73

However, two CaL96: SEBD values in basalt7-basanite diffusion couple 2003) 1973; LaTourrette et al. 1996; L03: Lundstrom 2003; M06:(Lundstrom Morgan et al. 2006;are C09: Chen and Zh greater than those predicted by Equation (24) by 1.5 to 2.0 lnD units (Fig. 12a), and those in mugearite melt 1973) lower by up to 1.2 lnD units. and Watson 1984; V94: Van Der Laan et al. 1 Z89:(Medford Zhang et al. 1989;are J82: Jambon 1982; H84: Harrison Ca diffusivities in rhyolite melts are more variable than in basalt melts. In dry HR7 melt Mungall et al. 1999). (among the most silicic rhyolite), Ca tracer diffusivity at 1410-1873 K and 0.1 MPa (Mungall et al. 1999) can be expressed as:

12

Diffusion Data in Siliate Melts dry HR7 ln DCa -(10.58 ± 2.63) TD =

(29830 ± 4356) T

337 (25)

The maximum error Equation (25) in reproducing the four experimental diffusivities is 0.28 in lnD. Adding H2O enhances Ca diffusivity significantly. Sr diffusion. Ten papers reported Sr diffusion data in silicate melts (110 points, all except for one are tracer or self diffusivities). Hofmann and Magaritz (1977) studied Sr tracer diffusion in dry basalt10 melt at 1532-1719 K and 0.1 MPa. Magaritz and Hofmann (1978a) determined Sr tracer diffusivities in dry rhyolite15 melt at 948-1226 K and 0.1 MPa. Lowry et al. (1982) explored tracer diffusion of 85Sr in dry basalt1 and andesite2 melts at 1566-1676 K and 0.1 MPa. Lesher (1994) investigated Sr self diffusion in dry rhyolite2 melt at 1528-1738 K and 1 GPa, and in dry basalt3 melt at 1528 K and 1 GPa. Perez and Dunn (1996) examined Sr tracer diffusion in rhyolite8 melt (containing < 0.8 wt% H2O, but H2O was not reported for every sample) at 1273-1723 K and 1 GPa. Nakamura and Kushiro (1998) reported tracer diffusivities of Sr in jadeite and diopside melts at a single temperature (1673 K for jadeite and 1863 K for diopside) and some pressures. Mungall et al. (1999) investigated Sr tracer diffusion in dry HR7 at 1410-1873 K and 0.1 MPa, wet HR7 (3.6 wt% H2O) at 1 GPa and two temperatures (1573 and 1873 K), and dry HR7 + Na at 1083-1773 K and 0.1 MPa. Roselieb and Jambon (2002) examined Sr tracer diffusivities in albite and jadeite melts at 918-1293 K and 0.1 MPa. Lundstrom (2003) obtained a single SEBD of Sr between basalt7-basanite melt at 1723 K and 0.9 GPa. Behrens and Hahn (2009) characterized Sr tracer diffusion in dry and wet trachyte and phonolite melts at 1323-1573 K and 0.5 GPa. Sr diffusivity data are shown in Figure 13. Sr diffusivities are the lowest in dry HR7 melt among the melts investigated. Sr self diffusivities in dry rhyolite2 are similar to Sr tracer diffusivities in rhyolite8 containing < 0.8 wt% H2O and in dry rhyolite15. Self and tracer diffusivities of Sr increase from dry HR7, to dry natural metaluminous rhyolite and dry trachyte, then dry andesite, then dry basalt and dry phonolite, then wet trachyte, and then wet phonolite. Some of the equations are given below. Self and tracer diffusivity of Sr in rhyolite2 at 1 GPa (Lesher 1994), rhyolite8 at 1 GPa (Perez and Dunn 1996) and rhyolite15 at 0.1 MPa (Magaritz and Hofmann 1978a) can be fit as follows: (18745 ± 824)   ≤1 GPa DSrdrySDrhyolites, = exp ( -14.80 ± 0.65) & TD  T  

(26)

The maximum error of Equation (26) in reproducing the diffusion data is 0.89 in lnD (0.39 in logD). The activation energy is 156±7 kJ/mol. Sr tracer diffusivities in dry basalt1 and basalt10 melts at 1567-1675 K and 0.1 MPa (Hofmann and Magaritz 1977; Lowry et al. 1982) can be fit as follows: (20361 ± 3881)   DSrdryTDbasalt = exp ( -11.40 ± 2.40)  T  

(27)

The maximum error of Equation (27) in reproducing the diffusion data is 0.30 in lnD. In dry and wet trachyte melts, Sr tracer diffusivities at 1323-1527 K, 0.5 GPa and ≤ 1.7 wt% H2O (Behrens and Hahn 2009) are roughly linear to H2O content, and can be fit as follows: (22435 ± 7571) - (3430 ± 545)w   DSrdryTDtrachyte1 = exp ( -12.71 ± 5.23)  T  

The maximum error of Equation (28) in reproducing the diffusion data is 0.75 in lnD.

(28)

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Zhang et al. (Ch 8) Diffusion data in silicate melts

13. Srin diffusivities melts.isThe default pressure is be 0.1found MPa. Fig. 13.Figure Sr diffusivities silicate melts. in Thesilicate default pressure 0.1 MPa. Data sources can in the text. Data sources can be found in the text.

There are not enough data to resolve how Sr tracer diffusivities in phonolite melts depend on H2O. Assuming the dependence is similar to trachyte melts, then Sr tracer diffusivities in dry and wet phonolite at 1373-1528 K, 0.5 GPa and ≤ 1.9 wt% H2O (Behrens and Hahn 2009) can be fit as follows: (24708 ± 6624) - (1644 ± 299)w   DSrphonolite1 = exp ( -8.98 ± 4.56) TD  T  

(29)

The maximum error of Equation (29) in reproducing the diffusion data is 0.44 in lnD. Ba diffusion. T welve papers reported Ba diffusion data in silicate melts (117 points), most of which are tracer or self diffusivities. Hofmann and Magaritz (1977) explored Ba tracer diffusion in dry basalt10 melt at 1523-1723 K and 0.1 MPa. Magaritz and Hofmann (1978a) determined Ba tracer diffusivities in dry rhyolite15 melt at 973-1226 K and 0.1 MPa. Jambon (1982) reported Ba tracer diffusivities in rhyolite5 melt at 905 and 1083 K and 0.1 GPa. Lowry et al. (1982) characterized Ba tracer diffusion in basalt1 and andesite2 melts at 1572-1673 K 13 and 0.1 MPa. Henderson et al. (1985) studied Ba tracer diffusion in dacite2 (1573-1672 K) and pantellerite1 (1470-1575 K) melts at 0.1 MPa. LaTourrette et al. (1996) examined Ba self diffusion in dry HB1 melt at 1623-1773 K and in air. Nakamura and Kushiro (1998) obtained Ba tracer diffusivities in dry jadeite and diopside melts at 1573-1723 K and 0.75-2.0 GPa. Mungall et al. (1999) characterized Ba tracer diffusion in dry HR7 and dry HR7 + Na at 0.1 MPa and wet HR7 (3.6 wt% H2O) melts at 1 GPa. Koepke and Behrens (2001) measured Ba tracer diffusivities in wet HA1 melt (4.5-5.2 wt% H2O) and one datum for dry HA1. Roselieb and Jambon (2002) investigated Ba tracer diffusion in dry albite and jadeite melts at 10731293 K and 0.1 MPa. Lundstrom (2003) acquired SEBD of Ba in dry basalt7-basanite couple at 1723 K and 0.9 GPa. Behrens and Hahn (2009) examined Ba tracer diffusion in dry and wet trachyte and phonolite melts. Ba diffusion data are shown in Figure 14. Ba diffusivities increase with decreasing viscosity (from polymerized silicic melt to depolymerized melt, and from dry to wet melt). Ba tracer and self diffusivities in dry basalt1 (Lowry et al. 1982), basalt10 (Hofmann and Magaritz 1977) and HB1 (LaTourrette et al. 1996)

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melts are consistent and can be fit by the following equation: (15691 ± 3800)   dry basalt; 0.1 MPa DBa = exp ( -14.68 ± 2.31) SD and TD  T  

(30)

Equation (30) can reproduce the data to within 0.56 lnD units. Experimental data in trachyte and phonolite melts are not enough to resolve how D depends on H2O. We assume that lnD increases linearly with H2O to fit the data. For dry and Zhang et al. (Ch 8) Diffusion data in silicate melts

Figure 14. Ba diffusivities in selected melts. Melt compositions can be found in Table 1. (a) Dry melts. (b) Fig.of14. diffusivities in selected melts. in Melt compositions can bewet found Table 1. Dry melts. Comparison dryBa and wet melts. The H2O contents wet melts are as follows: HA1:in4.5-5.2 wt%;(a)wet tra (trachyte1): 1.1-1.7 wt%; wet pho (phonolite1): 1.6-1.9 wt%; wet HR7: 3.6 wt%. Comparison of dry and wet melts. The H2O contents in wet melts are as follows: wet HA1: 4.5-5.2 wt% (trachyte1): 1.1-1.7 wt%; wet pho (phonolite1): 1.6-1.9 wt%; wet HR7: 3.6 wt%.

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hydrous trachyte1 and phonolite1 melts, Ba tracer diffusivities at 1323-1528 K, 0.5 GPa, 0-1.9 wt% H2O can be fit as: (20885 ± 6797) - (3065 ± 489)w   trachyte1; 0.5GPa DBa = exp ( -14.34 ± 4.67) TD  T  

(31)

(25996 ± 3412) - (1805 ± 190)w   phonolite1; 0.5GPa DBa = exp ( -8.63 ± 3.35) TD  T  

(32)

Equations (31) and (32) can reproduce the data to within 0.60 and 0.23 lnD units, respectively. Ra diffusion. No Ra diffusion data in silicate melts are known. Summary of alkali earth diffusion. For the diffusion of alkali earth elements in dry HR7 + Na melt (Mungall et al. 1999), the diffusivity decreases from Ba to Sr, to Ca, to Mg, and to Be. That is, the diffusivity sequence for the alkali earth elements is: Ba > Sr > Ca > Mg > Be. In dry HR7 melt (Mungall et al. 1999), the diffusivity sequence is: Ba ≈ Sr > Ca > Mg > Be (Fig. 15a). In wet HR7 melt containing 3.6 wt% H2O (Mungall et al. 1999), the diffusivity sequence is: Sr > Ba ≈ Ca > Mg > Be (Fig. 15b). In basalt10 melt (Hofmann and Magaritz 1977), the diffusivity sequence is: Ca ≈ Co > Sr > Ba, opposite to the trend in HR7 + Na. In HB1 melt (LaTourrette and Wasserburg 1996), the diffusivity sequence is: Mg > Ca > Ba, similar to the trend in basalt10. In albite and jadeite melts (Roselieb and Jambon 1995), the diffusivity sequence is: Sr ≈ Ca > Ba > Mg. Hence, for the alkali earth elements, there is no simple relation between diffusivity and cation size: the diffusivity may increase or decrease with increasing size, depending on the melt composition and other factors. That larger cations with the same valence may diffuse more rapidly in some melts may be counterintuitive to some. Nonetheless, this trend will be encountered in diffusion of other isovalent series (REE, B, Al, Ga, Si-Ge, etc.). This and other trends will be discussed in a later summary section.

B, Al, Ga, In, and Tl B diffusion. Three papers reported B diffusion data in silicate melts (70 points). Baker (1992a) investigated B tracer diffusion in dacite1 and rhyolite14 melts at 1573-1873 K and 1 GPa. Chakraborty et al. (1993) studied FEBD of B in diffusion couples with one half made of HR7 and the other half made of HR7 plus 5 wt% or 10 wt% B2O3 at 1473-1873 K and 0.1 MPa. Mungall et al. (1999) examined B tracer diffusion in dry HR7 melt at 0.1 MPa and 1410 and 1673 K, dry HR7 + Na melt at 1083-1473 K and 0.1 MPa, and wet HR7 (3.6 wt% H2O) at 1573-1873 K and 1 GPa. The data are shown in Figure 16. As shown by Chakraborty et al. (1993), FEBD values of B decrease as SiO2 concentration increases along a diffusion couple. When FEBD values of B at the HR7 end with 75-79 wt% SiO2 (Chakraborty et al. 1993) are compared with tracer diffusivities of B in HR7 (Mungall et al. 1999), they are consistent (Fig. 16). B tracer diffusivities in rhyolite14 (76 wt% SiO2) are similar to those in HR7 + B melts containing 70-75 wt% SiO2. B diffusivities (FEBD by Chakraborty et al. 1993 and tracer diffusivities by Mungall et al. 1999) in dry HR7 melt at 1410-1873 K, 0.1 MPa, and B2O3 concentration ≤ 10 wt% can be fit as: (39664 - 1162C )   HR7, 0.1 MPa DBdryTD&FEBD = exp  -8.56 - 0.486C  T  

(33)

where C is wt% of B2O3. All B diffusion data in Chakraborty et al. (1993) and Mungall et al. (1999) can be reproduced by Equation (33) to within 0.62 lnD units. B diffusivity in rhyolite14 melt at 1573-1873 K and 1 GPa (Baker 1992) is about 6 times the diffusivity calculated using Equation (33).

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Figure 15. Comparison of Be, Mg,BaCa, Sr and Baindiffusivities Fig. 15. Comparison of Be, Mg, Ca, Sr and diffusivities dry and wetinHR7 melt (Mungall et al. 1999). dry and wet HR7 melt (Mungall et al. 1999).

When compared with other cations, B diffusivities are similar to diffusivities of trivalent cations Lu and Ga in HR7 melts. Al diffusion. Twelve papers reported Al diffusion data in silicate melts (127 points). Cooper and Kingery (1964) explored Al diffusion (SEBD) in CaO-Al2O3-SiO2 melt during sapphire dissolution at 1618-1823 K and 0.1 MPa. Cooper and coworkers also investigated diffusion in other synthetic systems of interests to ceramic and glass scientists, such as K2O-SrO-SiO2, etc. These works are not covered here because we focus on geologically relevant silicate 15 melts. Baker and Watson (1988) investigated Al diffusion (SEBD) in rhyolite1-rhyolite8 and HD2-rhyolite8 couples (Table 1) at 1171-1273 K and 0.01 GPa and 1373-1473 K and 0.2-1 GPa. Zhang et al. (1989) determined Al SEBD in andesite1 melt during dissolution of olivine, clinopyroxene, spinel and quartz at 1488-1673 K and 0.55-2.15 GPa. Because the interface

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Fig. 16. B diffusivities in silicatemelts. melts. Data Data sources: FEBD at 71-79 wt% SiO MPa et al. Figure 16. B diffusivities in silicate sources: FEBD at 71-79 wt% SiO 0.1 MPa (Chakraborty 2 at 0.1 2 at(Chakraborty et al. 1993); tracer diffusivities in dry HR7 at 0.1 MPa, in HR7 + Na at 0.1 MPa, and in wet HR7 (3.6 wt% 1993); tracer diffusivities in dry HR7 at 0.1 MPa, in HR7+Na at 0.1 MPa, and in wet HR7 (3.6 wt% H2O) at 1 GPa H2O) at 1 GPa (Mungall et al. 1999); and in rhyolite14 and dacite1 at 1 GPa (Baker 1992a). (Mungall et al. 1999); and in rhyolite14 and dacite1 at 1 GPa (Baker 1992).

melt composition changes from basalt (during olivine dissolution) to rhyolite (during quartz dissolution), it is important to include the interface melt composition variation to understand the data. Kubicki et al. (1990) obtained Al SEBD in diopside-anorthite (Di-An) melts at 16331923 K and 0.1-2 GPa. Most of their data are at 0.2 GPa, and compositional change from Di100-Di80An20 couple to Di60An40-Di40An60 couple does not affect Al diffusivities in a major way. Baker and Bossanyi (1994) examined the effect of H2O and F on Al SEBD in rhyolite8dacite1 diffusion couples at 1373-1673 K and 1 GPa. van der Laan et al. (1994) reported Al SEBD in rhyolite3-rhyolite16 and andesite3-rhyolite16 diffusion couples. Liang et al. (1996a) investigated Al self diffusion in CaO-Al2O3-SiO2 systems at 1773 K and 1 GPa. Van Orman and Grove (2000) obtained a single datum for Al SEBD in a lunar basalt melt (LB1 in Table 1) during clinopyroxene dissolution at 1623 K and 1.3 GPa. Lundstrom (2003) obtained two data points for Al SEBD in basalt7-basanite couple. Morgan et al. (2006) determined Al SEBD in LP1 and LP2 melts during anorthite dissolution at 1673 K and 0.6 GPa. Chen and Zhang (2008, 2009) obtained Al SEBD data in basalt11 melt (with 0.3-0.4 wt% H2O) during olivine 16 and diopside dissolution at 1543-1790 K and 0.5-1.9 GPa. Despite the numerous papers on Al diffusivities, most Al diffusivity data were obtained as side-products. Except for Liang et al. (1996a) who investigated Al self diffusion in a ternary system at a single temperature (1773 K), other studies are all on SEBD of Al in various melts. No tracer diffusivities or FEBD on Al are available. Furthermore, Al diffusion data in some papers are scattered. Hence, Al diffusion in natural silicate melts is not very well constrained. Experimental Al SEBD data in dry melts are shown in Figure 17. Al diffusivities decrease from lunar picrites and basalt-basanite couple, to basalt11, and to andesite1. For basalt to andesite melts, the data are less scattered and the trends with SiO2 content is consistent. Al SEBD data for Di-An melts (ranging from Di100-Di80An20 couple to Di60An40-Di40An60 couple) vary more widely, but are roughly the same as those in basalt11 melt. Al diffusivities (SEBD) in basalt11 melt (containing about 0.3-0.4 wt% H2O) during both olivine and clinopyroxene dissolution at 1509-1790 K and 0.5-1.9 GPa (Chen and Zhang 2008, 2009) are similar and roughly independent of pressure, and can be expressed as:

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Figure 17.Fig. SEBD of Al in in dry legends melt composition Table 1 (b-b 17. SEBD of Al drymelts. melts. Indicated Indicated inin thethe legends are theare meltthe composition listed in Tablelisted 1 (b-b in means means basalt7-basanite couple with 46 wt% average SiO2), pressure, and references (C08: Chen and Zhang basalt7-basanite couple with 46 wt% average SiO2), pressure, and references (C08: Chen and Zhang 2008; C09: 2008; C09: Chen and Zhang 2009; L03: Lundstrom 2003; M06: Morgan et al. 2006; V00: Van Orman and Grove 2000; Kubicki al.Lundstrom 1990; Z89: et al.et1989). ChenK90: and Zhang 2009; et L03: 2003;Zhang M06: Morgan al. 2006; V00: Van Orman and Grove 2000; K90: Kubicki et al. 1990; Z89: Zhang et al. 1989).

(31293 ± 2599)   basalt11 DAl SEBD = exp ( -5.75 ± 1.71)  T  

(34)

Equation (34) reproduces the experimental data to within 0.44 lnD units. Furthermore, Equation (34) can also roughly predict Al SEBD in Di-An melts (Fig. 17) with a maximum error of 1.1 lnD units. Al diffusivities (SEBD) in andesite1 melt (containing about 0.04 wt% H2O) during olivine, clinopyroxene and spinel dissolution at 1488-1673 K and 0.5-2.1 GPa (Zhang et al. 1989) are similar and roughly independent of pressure, and can be expressed as: (37649 ± 5848)   andesite1 DAl SEBD = exp ( -2.52 ± 4.95)  T  

(35)

Equation (35) reproduces the experimental data17to within 0.79 lnD units. Al diffusivity during quartz dissolution (Zhang et al. 1989) is lower than that predicted by Equation (35) by 2.05 lnD units, attributed to a large difference in the interface melt composition during quartz dissolution compared to dissolution of the other minerals. Data in Figure 17 show systematic dependence of Al SEBD on SiO2, although the data are scattered. We fit all SEBD data on natural silicate melts as follows: (23111 + 5918 XSi )   basalt to andesite DAl = exp ( -0.88 - 18.02 XSi ) SEBD  T  

(36)

Equation (36) reproduces the experimental data of Zhang et al. (1989), Van Orman and Grove (2000), Lundstrom (2003), Morgan et al. (2006), and Chen and Zhang (2008, 2009) to within 1 lnD unit (or 0.43 logD units). Al is a network-forming element. When compared to the diffusion of the most common major network-forming element Si, Al diffusivity is often slightly higher than Si diffusivity,

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about 1.5 to 2.5 times the Si diffusivity (Zhang et al. 1989; Liang et al. 1996a; Chen and Zhang 2008, 2009). Hence, when Al diffusivity in a melt is not known but Si diffusivity in the melt is known, Al diffusivity can be roughly estimated to be two times the Si diffusivity. Ga diffusion. Two papers reported Ga tracer diffusion data in silicate melts (18 points). Baker (1992a) investigated Ga diffusion in dacite1 and rhyolite14 melts (Table 1) at 1573-1873 K and 1 GPa. Baker (1995) examined Ga diffusion in albite melt at 1427-1775 K and 0.1 MPa. The data are in Figure 18.melts Mungall (2002) dismissed Ga diffusion data by Baker Zhangsummarized et al. (Ch 8) Diffusion data in silicate (1992a).

Figure 18. Ga tracermelts. diffusivities in silicate melts. Data sources: Fig. 18. Ga tracer diffusivities in silicate Data sources: albite (Baker 1995); rhyolite14 and dacite1 (Baker albite (Baker 1995); rhyolite14 and dacite1 (Baker 1992a).

1992).

In diffusion. No In diffusion data in silicate melts are known. Tl diffusion. Only one paper reported Tl diffusion data in silicate melts. MacKenzie and Canil (2008) studied Tl tracer diffusion in dry CMAS1 and NMAS1 (Table 1) melts using devolatization (desorption) experiments at 1473-1623 K and 0.1 MPa. The diffusion data are summarized in Figure 19. There is considerable scatter in the data. Comparison of diffusivities of B, Al, Ga, In, and Tl. There are not enough data to compare diffusivities of B, Al, Ga, In and Tl because (i) there are no data on In diffusion; (ii) the only Tl diffusion data are for CMAS1 and NMAS1 systems, different from melt compositions studied for the other elements; (iii) the only common melts that have been investigated for B and Ga diffusion are dacite1 and rhyolite14 melts; and (iv) Al diffusion data in dacite and rhyolite melts are SEBD data and are highly scattered. Hence, only B and Ga diffusivities are compared in Figure 20. In dacite1 melt, diffusivities of the two elements are similar. However, in rhyolite15 melt, B diffusivity is lower than Ga diffusivity by a factor of 2 to 4. The latter is consistent with 18 smaller cations having smaller diffusivity. the trend shown by the alkali earth elements with Nonetheless, the difference in diffusivity is small. It is likely that Al diffusivities lie between those of B and Ga, meaning that B, Al and Ga all have similar diffusivities. Because Tl is volatile, its diffusivities are expected to be higher than those of B, Al and Ga.

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Figure 19. Tl diffusivities in silicate melts. Data source: MacKenzie and Canil (2008).

Zhang et al. (Ch 8) Diffusion data in silicate melts

Fig. 19. Tl diffusivities in silicate melts.

Data source: MacKenzie and Canil (2008).

Figure 20. Comparison of B and Ga diffusivities in dacite1 and rhyolite14 melts.

Fig. 20.

19

Comparison of B and Ga diffusivities in dacite1 and rhyolite14 melts.

C, Si, Ge, Sn and Pb C diffusion. C diffusion in silicate melts in the form of dissolved molecular CO2 and carbonate ion CO32− is reviewed in another chapter (Zhang and Ni 2010). Data for carbon diffusion in other forms (such as dissolved molecular CO) are not available. Si diffusion. Si is the second most major element in silicate melts (next to oxygen), and it controls the melt structure. Hence, its diffusion has been investigated extensively, with about 23 papers and more than 262 diffusivity values (some of the diffusivities are reported as smooth trends rather than data points). Lesher and Walker (1986, 1991) obtained Si SEBD from Soret diffusion experiments on nephelinite, basalt8, limburgite, basalt11, leuconorite, nordmarkite, trachyte, pantellerite2, FaLcQ (fayalite-leucite-quartz), rhyolite4, and rhyolite10 melts (Table 1). Baker and Watson (1988) investigated Si SEBD in rhyolite1-rhyolite8 and HD2-rhyolite8 couples at 1171-1273 K and 0.01 GPa and 1273-1473 K and 0.2-1 GPa. Koyaguchi (1989) reported Si SEBD in basalt-dacite and basalt-rhyolite couples at 1473-1773 K, 1 GPa, and 0.39-

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13.77 wt% H2O. The compositions used by Koyaguchi (1989) include basalt2, basalt9, dacite3, and rhyolite13. Zhang et al. (1989) obtained Si SEBD in andesite1 melt during dissolution of olivine, spinel, rutile and quartz at 1488-1673 K and 0.55-1.5 GPa. Because the interface melt composition changes from basalt (during olivine dissolution) to rhyolite (during quartz dissolution), it is important to include the interface melt composition variation to understand the data. Baker (1991, 1993) and Baker and Bossanyi (1994) examined the effect of H2O, F and Cl on Si SEBD in rhyolite8-dacite3 diffusion couples at 1373-1673 K and 1 GPa, and found that the effect of F and Cl is negligible (D increases by less than a factor of 2 when F and Cl concentrations are 1 wt% or less), and that of H2O is significant. Kubicki et al. (1990) reported Si SEBD in diopside-anorthite melts at 1873 K and 0.2 GPa. Baker (1992a) studied Si tracer diffusion in rhyolite8 and dacite1 melts at 1573-1773 K and 1 GPa. In these experiments, a layer of sodium silicate glass powder (with 73 mol% SiO2 and 27 mol% Na2O, or close to Na6Si8O19) with thickness < 0.1 mm enriched in 30Si (a stable silicon isotope, meaning that the experiments are more like self diffusion experiments) is loaded on a dacite or rhyolite glass cylinder. The sample is loaded into a piston-cylinder assemblage and heated up. Because the typical Si diffusion length in the experiments of Baker (1992a) is about 0.05 mm (Fig. 1 in Baker 1992a), which might be thinner than the thickness of the loaded film (< 0.1 mm), the experiments may not be true tracer diffusion experiments, but are close to diffusion couple experiments between a synthetic sodium silicate on one half and either dacite or rhyolite on the other half, with an Si isotopic gradient between the two halves. The extracted diffusivities are likely isotopic effective binary diffusivities (IEBD), although such diffusivities based on isotopic fraction profiles are often not too far off self diffusivities (Zhang 1993; Lesher 1994; van der Laan et al. 1994). van der Laan et al. (1994) reported Si SEBD in rhyolite3-rhyolite16 and rhyolite16dacite3 diffusion couples. Baker (1995) investigated Si SD in albite melt at 1438-1831 K and 0.1 MPa. Lesher et al. (1996) investigated Si self diffusion in basalt6 melt (Table 1) at 15931873 K and 1 GPa, and 1673 K and 2 GPa. Liang et al. (1996a) investigated Si self diffusion in the CaO-Al2O3-SiO2 system. Liang et al. (1996b) extracted multicomponent diffusivity matrices in the CaO-Al2O3-SiO2 system (see review in Liang 2010, this volume), and also SEBD of Si, but the SEBD data were only shown in a figure without the melt composition. Poe et al. (1997) measured Si self diffusivity in NS4 (Na2Si4O9) melt at 2100 to 2800 K and 10-15 GPa. Van Orman and Grove (2000) estimated Si SEBD in a lunar basalt melt (LB1 in Table 1) during clinopyroxene dissolution at 1623 K and 1.3 GPa. Reid et al. (2001) studied Si self diffusion in diopside melt at 2073-2573 K and 3-15 GPa. Tinker and Lesher (2001) examined Si self diffusion in dry HD2 melt at 1628-1935 K and 1.0-5.7 GPa. Lundstrom (2003) obtained one datum of Si SEBD in basalt7-basanite couple. Tinker et al. (2003) investigated Si SD in diopside-anorthite (Di52An48) melt at 1783-2037 K and 1-4 GPa. Morgan et al. (2006) determined Si SEBD in two lunar picrite melts (LP1 and LP2 in Table 1) during anorthite dissolution at 1673 K and 0.6 GPa. Chen and Zhang (2008, 2009) obtained Si SEBD in nominally dry basalt11 melt (about 0.3-0.4 wt% H2O) during olivine and clinopyroxene dissolution at 1543-1790 K and 0.5-1.9 GPa. Si self diffusivities. In dry silicate melts, silicon self diffusivity is similar to and often slightly lower than oxygen self diffusivity (Fig. 21). The maximum difference is about 1.5 lnD units and this difference occurs in silicic melts with 69 wt% SiO2. The limited data apparently indicate that when diffusivities are low ( 4 GPa to constrain how Si self diffusivity varies with pressure. Hence, restricting our attention to 1-4 GPa, Si self diffusivity in dry HD2 melt at 1628-1935 K can be fit as follows: (54245 ± 690) - (8523 ± 2984)P   DSidrySDHD2 = exp ( -3.913 ± 1.603)P  T  

(37)

Equation (37) can reproduce the experimental data at 1-4 GPa to within 0.66 lnD units. It can be extrapolated to 0 GPa, but cannot be extrapolated to ≥4 GPa. Effective binary diffusivities of Si. Most Si21diffusivities are SEBD values from diffusion couples made of rhyolite-basalt, rhyolite-dacite, or basalt7-basanite, or mineral dissolution experiments in basalt and andesite melts. In these experiments, as well as in nature, the SiO2 gradient is often the largest gradient, or one of the largest. Hence, even though it is expected that the gradients of other concentration gradients would affect SiO2 diffusion due to cross diffusion effects, such effects are not expected to completely overwhelm SiO2 diffusion. Figure 22a compares different kinds of Si diffusivities and the maximum difference is 1.6 lnD units (0.7 logD units). Therefore, SiO2 SEBD in most geological cases is expected to be roughly the same as the self diffusivity. However, for synthetic melts (such as the CaO-Al2O3SiO2 system, Liang et al. 1996a,b) where SiO2 gradient may be small compared to gradients of other components, the difference between self diffusivities and SEBD can be very large. Experimental SEBD values of Si in diffusion couple or Soret diffusion experiments usually decrease from basalt to rhyolite, and lnD versus SiO2 for typical natural melts is roughly linear (Fig. 22b) (Lesher and Walker 1986, 1991; Koyaguchi 1989). Lesher and Walker (1986, 1991) showed based on Soret diffusion experiments that lnDSi decreases linearly with increasing

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Figure 22. (a) Comparison between SD (self diffusivities) and SEBD of Si. Data sources: SD in basalt6 Fig. 22. (a) Comparison between SD (self diffusivities) and SEBD of Si. Data sources: SD in basal (Lesher et al. 1996); SEBD in basalt12 (Lesher and Walker 1986); SEBD in b-d (basalt2-dacite3; Koyaguchi 1989); SEBD in b-b (basalt7-basanite; Lundstrom 2003); SEBD in basalt11 (Chen and Zhang 2008, 2009). SEBD in2 basalt12 (Lesher and LW86 Walker(Soret 1986); SEBDresults in b-dof(basalt2-dacite3; Koyaguchi 1989); (b) SEBD 1996); of Si versus SiO content. Data sources: diffusion Lesher and Walker 1986); K89 (diffusion couple results of Koyaguchi 1989).

(basalt7-basanite; Lundstrom 2003); SEBD in basalt11 (Chen and Zhang 2008, 2009). (b) SEBD of Si

SiO2: in dry melts atData 1748sources: K, lnDSiLW86 (D in(Soret m2/s) diffusion decreasesresults from of −24.2 in aand basalt with1986); 51 wt% content. Lesher Walker K89 (diffusion cou SiO2 to −26.8 in a rhyolite with 75 wt% SiO2 (a factor of 13, or 1.1 orders of magnitude). 1989). from diffusion couple experiments that lnDSi decreases linearly KoyaguchiKoyaguchi (1989) showed with increasing SiO2: in dry melts at 1773 K, lnDSi (D in m2/s) decreases from −24.2 in a basalt with 50 wt% SiO2 to −27.6 in a rhyolite with 74 wt% SiO2 (a factor of 30, or 1.5 orders of magnitude).

22

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349

Baker (1993) and Baker and Bossanyi (1994) examined the effect of H2O, F and Cl on Si SEBD. Adding 3 wt% H2O increases Si SEBD by a factor of about 10. In dry and wet melts, adding 1 wt% of F or Cl increases Si SEBD by less than a factor of 2, within experimental error of the diffusion data by Baker and coworkers. Even though the data seem to indicate a significant effect on the activation energy of Si SEBD by F and Cl, the activation energy based on a small temperature range and scattered data is not very reliable (e.g., see discussion in Zhang and Ni 2010). Overall, the effect of F and Cl can be ignored unless Si diffusion data accuracy can be significantly improved or F and Cl concentrations are much higher than 1 wt%. Ge diffusion. Only one paper (Mungall et al. 1999) reported Ge tracer diffusion data in silicate melts (9 points), in dry HR7 and HR7 + Na, and wet HR7 (Table 1). The data are summarized in Figure 23. Ge diffusivity in HR7 can be expressed as: (37414 ± 8558)   dry HR7 DGe TD = exp ( -11.27 ± 5.16)  T  

(38)

In dry HR7Zhang melt, diffusivities similar to those of Hf, and they are similar to Eyring et al.Ge (Ch 8) Diffusion data in are silicate melts diffusivities (Fig. 23).

Figure 23. diffusion dataininsilicate silicate Mungall al. (1999). Eyring Fig. Ge 23. Ge diffusion data melts.melts. Data Data source:source: Mungall et al. (1999).et Eyring diffusivity line isdiffusivity line is calculated from the viscosity data of Hess et al. (1995) on dry HR7 melt. calculated from the viscosity data of Hess et al. (1995) on dry HR7 melt.

Sn diffusion. Only one paper reported good-quality Sn tracer diffusion data in silicate melts (Behrens and Hahn 2009). Two other papers reported highly scattered SEBD of Sn (Linnen et al. 1995, 1996) obtained from cassiterite dissolution in haplorhyolite melts, but it was likely that the data were compromised by convection. Sn in silicate melts may be in the form of Sn2 +  or Sn4 + . Hence, Sn diffusivity may depend on oxygen fugacity. Because the data of Linnen et al. (1995, 1996) are the only data examining the effect of fO2, they are shown in Figure 24a. Their data indicate a decrease of Sn diffusivity with increasing fO2, meaning that Sn2 +  diffuses more rapidly than Sn4 + , as expected. Behrens and Hahn (2009) characterized Sn tracer diffusivities in dry and wet trachyte1 and phonolite1 melts (Fig. 24b). The oxygen fugacity in these experiments was near the MnOMn3O4 buffer but somewhat variable. Assuming DSn increases linearly with H2O, Sn tracer

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diffusivities in dry and wet trachyte1 and phonolite 1 melts at 1323-1528 K, 0.5 GPa and ≤ 1.9 wt% H2O (Behrens and Hahn 2009) can be fit as follows: (27290 ± 10275) - (2994 ± 740)w   trachyte1 DSn = exp ( -11.08 ± 7.10) TD  T  

(39)

(30799 ± 8555) - (1492 ± 478)w   phonolite1 DSn = exp ( -6.34 ± 5.98) TD  T  

(40)

Zhangand et al.(40) (Ch 8) Diffusion data in silicate melts Equations (39) reproduce experimental data to within 0.88 and 0.57 lnD units, respectively.

Figure 24. Sn diffusivities in silicate melts. Data in (a) are from Linnen et al. (1995, 1996) and might be incorrect Fig. 24. Sn diffusivities in silicate melts. Data in (a) are from Linnen et al. (1995, 1996) and might be incorrect due to convection in the mineral dissolution experiments. Data in (b) are from Behrens and Hahn (2009). to convection in the mineral dissolution experiments.

Data in (b) are from Behrens and Hahn (2009).

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Pb diffusion. Four papers reported Pb self and tracer diffusion data in silicate melts (40 points). Jambon (1982) obtained one datum of Pb tracer diffusivity in dry rhyolite15 melt (Table 1). LaTourrette et al. (1996) reported one datum of Pb self diffusion in HB1 melt. Perez and Dunn (1996) investigated Pb tracer diffusivities in rhyolite8 melt (containing < 0.8 wt% H2O, but H2O is not always measured) at 1273-1723 K and 1 GPa. MacKenzie and Canil (2008) characterized tracer diffusion of Pb on dry CMAS1 and NMAS1 at 1473-1623 K and 0.1 MPa. The diffusion data are shown in Figure 25. For dry melts, the diffusivity increases from rhyolite to basalt and CMAS1 to NMAS1 melts. Pb tracer diffusivities in dry and wet rhyolite8 melts at 1273-1723 K, 1 GPa, and ≤ 2.6 wt% H2O (Perez and Dunn 1996) can be expressed as: 28512 - 7900 w   rhyolite8, 1 GPa DPb = exp  -9.08 - 4.32 w TD  T  

(41)

The fits by Equation (41) to Pb diffusivities in dry and wet rhyolite are shown in Figure 25. Except for two outliers, Equation (41) reproduces experimental data to within 0.72 lnD units. Pb diffusion data of Perez and Dunn (1996) on rhyolite8 at 1 GPa and those of Jambon (1982) on rhyolite5 at 0.1 MPa lie roughly on the same trend. Comparison of diffusivities of C, Si, Ge, Sn and Pb. Diffusion data of C, Si, Ge, Sn, and Pb are not directly comparable. C in silicate melt is tetravalent and diffuses as a linear molecule (CO2) in most situations even when there is significant CO32− (Nowak et al. 2004; Zhang et al. 2007), whereas Si and Ge are tetravalent and often in tetrahedral sites, but Sn and Pb are often divalent. Hence, one may compare Si and Ge diffusivities as a group, and Sn and Pb diffusivities as another group. For Si and Ge, the only Ge diffusion data are tracer diffusivities in dry HR7, wet HR7, and dry HR7 + Na melts, for which Si diffusion data are not available. Hence, no direct comparison can be made. However, because both Ge and Si diffusivities areetsimilar to the Eyring diffusivity, Si and Ge diffusivities are similar. For Sn and Zhang al. (Ch 8) Diffusion data in silicate melts Pb diffusion, no direct comparison can be made.

Pb diffusion datain in silicate silicate melts. notnot indicated, the meltthe is dry, the is pressure is 0.1pressure MPa and the Figure 25.Fig. Pb25. diffusion data melts.If If indicated, melt dry, the is 0.1 MPa and the diffusivities are tracer diffusivities. Data sources: dry and wet rhyolite8 at 1 GPa (Perez and Dunn 1996); diffusivities are tracer diffusivities. Data sources: dry and wet rhyolite8 at 1 GPa (Perez and Dunn 1996); rhyolite15 (Jambon 1982); self diffusivity in HB1 melt (LaTourrette et al. 1996); CMAS1 and NMAS1 rhyolite15 (Jambon 1982);The self diffusivity HB1 melt (LaTourrette et al. 1996); CMAS1 and and NMAS1 (MacKenzie and Canil 2008). lines areincalculated from Equation (41) for dry wet(MacKenzie rhyolite8. and Canil 2008).

The lines are calculated from Eq. 25 for dry and wet rhyolite8.

Zhang, Ni, Chen

352 N, P, As, Sb, Bi

N diffusion. No N diffusion data in natural silicate melts are known. P diffusion. Four papers reported P diffusion data (SEBD) in silicate melts (35points). Harrison and Watson (1984) investigated SEBD of P in dry and wet (up to 10 wt% H2O) rhyolite12 melt (Table 1) during apatite dissolution at 1373-1773 K and 0.8 GPa. Rapp and Watson (1986) examined SEBD of P in wet rhyolite12 melt (containing 1-6 wt% H2O) during monazite dissolution at 1273-1673 K and 0.8 GPa. Pichavant et al. (1992) obtained two SEBD values of P in wet synthetic haplorhyolite melt during apatite dissolution at 1173 K and 0.2 GPa; the data are likely compromised by convection. Lundstrom (2003) reported two SEBD values in basalt7-basanite diffusion couple at 0.9 GPa. The data are summarized in Figure 26. Interestingly, as noted by Rapp and Watson (1986), P diffusivity during apatite dissolution in rhyolite melt is significantly faster than that during monazite dissolution in the same melt (comparing solid and open symbols in Figure 26; e.g., at 1273 K, 0.8 GPa, and 6 wt% H2O, P diffusivity in rhyolite melt is ~4.3×10−14 m2/s during apatite dissolution, but only 1.5×10−15 m2/s during monazite dissolution, with a difference of a factor of 28), even though one might expect that they would be similar. The data are from the same laboratory, and are hence free of inter-laboratory inconsistencies. One possibility is that the difference in the diffusing species of P during monazite versus apatite dissolution may lead to the difference in the diffusivity. The dependence of P diffusivities on H2O content is shown in Figure 27a. It is not clear whether the relation between lnDP and H2O is linear or curved; some of the scatter is likely due to uncertainty in determining H2O content in the early years. Because P is a network former and diffuses slowly, one might expect that DP is the same as DEyring calculated from viscosity η. Diffusivity of P is indeed among the lowest of all Zhang et al. (Ch 8) Diffusion data in silicate melts

Fig.26. 26. P P diffusivity in silicate melts. melts. The various at 1000/T = 0.978 for hydrous rhyolite melts Figure diffusivity in silicate Thepoints various points at are 1000/T = 0.978 are for(Wolf hydrous rhyolite melts and (Wolf and London 1994). Open symbols represent data during apatite dissolution; solid symbols London 1994). Open symbols represent data during apatite dissolution; solid symbols represent data during represent data during monazite dissolution. Data sources: “rhy12(Ap)”: Apatite dissolution in rhyolite 12 monazite and dissolution. sources:“rhy12(Mon)”: "rhy12(Ap)": Apatite dissolutiondissolution in rhyolite 12 by and Watson by Harrison WatsonData (1984); Monazite inHarrison rhyolite12 by Rapp and Watson (1986); b-b couple: basalt7-basanite couple by Lundstrom (2003); “Haplorhyolite(Ap)”: apatite dissolution (1984); "rhy12(Mon)": Monazite dissolution in rhyolite12 by Rapp and Watson (1986); b-b couple: basalt7-basanite in haplorhyolites by Pichavant et al. (1992). couple by Lundstrom (2003); "Haplorhyolite(Ap)": apatite dissolution in haplorhyolites by Pichavant et al. (1992).

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353

elements, but is different from the Eyring diffusivity. For example, Figure 27b compares lnDP versus lnDEyring, showing that there is no unique correlation between the two: DP is equal to or greater than DEyring during apatite dissolution, but is often smaller than DEyring during monazite dissolution. The difference between DP and DEyring can be almost two orders of magnitude. Hence, it is impossible to use viscosity or Eyring diffusivity to predict P diffusivity. Although DP ≠ DEyring, it is possible to relate P diffusivity to viscosity or Eyring diffusivity under specific example, P diffusivity in wet rhyolite containing 6 wt% Zhangconditions. et al. (Ch 8) For Diffusion data infor silicate melts rhyolite12 H2O during monazite dissolution, ln DPdrySEBD; apatite diss = (20.64±2.26)  +  (1.509±0.064)lnDEyring

Figure 27. P diffusivities in rhyolite12 melts. In the legend of (b), A means apatite and M means monazite; Fig. 27. P diffusivities in rhyolite12 melts. In the legend of (b), A means apatite and M means monazit solid symbols represent data during apatite dissolution; open symbols represent data during monazite dissolution. DEyring is calculated using the viscosity model of Zhang et al. (2003) and a jump distance of symbols represent data during apatite dissolution; open symbols represent data during monazite dissoluti 2.8×10−10 m.

DEyring is calculated using the viscosity model of Zhang et al. (2003) and a jump distance of 2.8x10-10 m

Zhang, Ni, Chen

354

where viscosity needed to estimate DEyring is from Zhang et al. (2003). However, no universal relation exists between P diffusivity and Eyring diffusivity. Hence, such relations would have to be established at each specific H2O contents, meaning that there is not much advantage for doing so. As diffusion. No As diffusion data in silicate melts are known. Sb diffusion. Two papers reported Sb tracer diffusion data in silicate melts (18 points). Koepke and Behrens (2001) investigated Sb diffusion in HA1 (Table 1) with one datum in dry HA1 and 4 points in wet HA1 (4.5-5.2 wt% H2O). MacKenzie and Canil (2008) studied Sb diffusion in dry CMAS1 and NMAS1 melts. The data are summarized in Figure 28. Sb diffusivity in wet HA1 melt at 1373-1673 K, 0.5 GPa, and 4.5-5.2 wt% H2O can be described by: (16423 ± 1934)   wet HA1 DSb = exp ( -12.56 ± 1.28) TD  T  

(42)

Sb diffusivity in andesite melt is slightly greater than (about 1.5 to 2 times) Ba diffusivity in Zhang et al. (Ch 8) Diffusion data in silicate melts the same melt.

Fig. 28. Sb tracer diffusivities in silicate Figure 28. Sbmelts. tracer

diffusivities in silicate melts.

Bi diffusion. No Bi diffusion data in silicate melts are known.

O, S, Se, Te, Po O diffusion. Oxygen diffusion in silicate melts is reviewed in another chapter (Zhang and Ni 2010). S diffusion. Sulfur in silicate melts can be present in various species controlled mainly by oxygen fugacity (the valence of sulfur can be −2,  + 4, and  + 6, with corresponding species of S2−, SO2 and SO42 + ). Sulfur diffusivity is expected to depend on the sulfur species and hence on oxygen fugacity, which is a main complication. Three papers reported S diffusion data in silicate melts (41 points). In a review paper, Watson (1994) previewed sulfur diffusivity in a variety of melts, including rhyolite, dacite, haplodacite, haploandesite, and a lunar ultramafic melt using (i) the devolatization technique (obtaining tracer diffusivity or FEBD), (ii) the

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diffusion couple technique (obtaining tracer diffusivity or FEBD), and (iii) the thin source technique (obtaining tracer diffusivity). Increasing oxygen fugacity leads to decreasing sulfur SEBD. Winther et al. (1998) characterized S tracer diffusion in dry albite melt at 1573-1773 K, 1 GPa, andZhang oxidized conditions. Freda et al. (2005) investigated S tracer diffusion in Etna and et al. (Ch 8) Diffusion data in silicate melts Stromboli basalt melts. Some diffusion data are summarized in Figure 29.

Figure 29. Fig. S tracer diffusion ininsilicate melts.Data Data sources and conditions: dry andmelts wet(from basalt 29. S tracer diffusion silicate melts. sources and conditions: dry and wet basalt Etnamelts and (from Etna and Stromboli) at reduced conditions of QFM-3 (Freda et al. 2005); dry albite under “oxidized” Stromboli) at reduced conditions of QFM-3 (Freda et al. 2005); dry albite under “oxidized” condition (Winther et al. condition (Winther et al. 1998). 1998).

Due to the complexity of S speciation as well as the limited number of studies, sulfur diffusion in silicate melts is not well understood. Furthermore, errors for S diffusion data are larger than those of other elements, likely also due to the various sulfur species. Zhang et al. (2007) obtained the following equation to describe sulfur tracer diffusivity in Etna and Stromboli basalt melts (ignoring the small compositional difference between the two) at 14981723 K, 0.5-1 GPa (with negligible pressure effect in this range), oxygen fugacity of 3 log units below QFM (hence sulfur is in the form of S2−), and 0-4 wt% H2O: 27692 - 651.6 w   DSbasalts TD = exp  -8.21  T  

(43)

The maximum error of Equation (43) in reproducing the experimental diffusivities is 0.77 in lnD. The activation energy is about 230 kJ/mol in dry basalt melt, decreasing to 209 kJ/mol in wet basalt melt containing 4 wt% H2O. 29 Se diffusion. No Se diffusion data in silicate melts are known.

Te diffusion. Only one experimental study reported Te diffusion data (SEBD). Te can exist in different valences, similar to S. Hence, its diffusivity may depend on the oxygen fugacity. MacKenzie and Canil (2008) carried out devolatization (desorption) experiments of CMAS1 melt (Table 1) at 1523-1623 K and in air, and extracted Te diffusivities. The data are shown in Figure 30. As can be seen, there is considerable scatter in the data (about a factor of 5 difference in diffusivity values at 1573 K), similar to the large scatter in S diffusivity. Because CMAS1 composition is close to a haplobasalt, the Te diffusion data in CMAS1 melt

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Zhang, Ni, Chen

Figure 30.Fig. Te30. tracer diffusivities inCMAS1 CMAS1 1) under oxidized conditions (MacKenzie and Te tracer diffusivities in melt melt (Table (Table 1) under oxidized conditions (MacKenzie and Canil 2008) Canil 2008) compared with S tracer diffusivities in dry Etna and Stromboli basalt melts under reduced compared with S tracer diffusivities in dry Etna and Stromboli basalt melts under reduced conditions (Freda et al. conditions (Freda et al. 2005). 2005).

are compared in Figure 30 with S diffusion data in basalt melt, though the condition was oxidized for Te diffusion and reduced for S diffusion. The two sets of data are comparable. Po diffusion. No Po diffusion data in silicate melts are known. Summary of O, S, Se and Te diffusivities. The diffusivities of O, S, Se and Te are not readily comparable because of limited and scattered data on S and Te and no data on Se, as well as different speciation and oxidation states of O, S and Te.

F, Cl, Br, I, At F diffusion. Five papers reported F diffusion data in silicate melts (84 points). Dingwell and Scarfe (1984, 1985) explored FEBD of F in jadeite and albite melts (containing up to 6.3 wt% F) using the diffusion couple method at 1473-1673 K and 1.0-1.5 GPa by packing F-bearing and F-free jadeite powders into a Pt capsule. Dingwell and Scarfe (1985) reported more FEBD values of F in jadeite, albite and a peraluminous silicate melt (45.5 wt% O; 5.4 30 wt% Na) using devolatization experiments wt% F; 31.58 wt% Si; 10.87 wt% Al; and 6.14 at 1473-1673 K and 0.1 MPa of pure oxygen gas. Gabitov et al. (2005) studied SEBD of F during fluorite dissolution into a haplorhyolite (similar to HR7 in Table 1) melt at 1173-1273 K, 0.1 GPa, and 1.2 to 4.8 wt% H2O, and showed that dissolved H2O enhances F diffusivity significantly. Alletti et al. (2007) characterized F tracer diffusion in dry and wet basalt4 melts at 1523-1723 K and 0.5-1.0 GPa. Balcone-Boissard et al. (2009) investigated F tracer diffusion in Na-phonolite and K-phonolite melts at 1473-1723 K and 0.5-1 GPa and examined the effect of Na/K ratio and H2O content. In addition, Baker and Balcone-Boissard (2009) reviewed halogen diffusion in silicate melts (also Baker et al. 2005). Figure 31 shows F diffusion data in dry jadeite, albite and basalt melts. F diffusivities in jadeite melt at 0.1 MPa are greater than those in albite melt by 3.4 lnD units. In the pressure range of 1.0 to 1.5 GPa, F diffusivities in jadeite melt are almost independent of pressure within 0.27 lnD units, but are significantly (1.8 lnD units) higher than those at 0.1 MPa. The independence of F diffusivity on pressure at high pressure but the dependence on pressure from 0.1 MPa to 1 GPa may imply either a structural change in jadeite melt with pressure, or a

Zhang et al. (Ch 8) Diffusion data in silicate melts

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F diffusivities in in dry jadeite, and basalt4 Data melts. sources: jadeite GPa (Dingwell and Figure 31.Fig.F 31. diffusivities dryalbite, albite, jadeite, andmelts. basalt4 Data 1.0-1.5 sources: jadeite 1.0-1.5 GPa (Dingwell Scarfe and Scarfe 1984); jadeite and albite at 0.1 MPa (Dingwell and Scarfe 1985); basalt4 at 1 GPa 1984); jadeite and albite at 0.1 MPa (Dingwell and Scarfe 1985); basalt4 at 1 GPa (Alletti et al. 2007). (Alletti et al. 2007).

change in the diffusion species from the devolatization method in 0.1 MPa pure O2 gas to the diffusion couple method in Pt capsule (Baker and Balcone-Boissard 2009), or experimental error. Another surprise is that F diffusivities in polymerized jadeite melt are similar to those in less polymerized basalt melt (for comparison, the O self diffusivity in dry basalt melt is greater than that in jadeite melt by about 2 orders of magnitude, Shimizu and Kushiro 1984; Lesher et al. 1996). Taking together, it is possible that F diffusivities in jadeite melt at 1.0-1.5 GPa (Dingwell and Scarfe 1984) are compromised somehow, e.g., because powders (rather than glass cylinders) were packed to make the diffusion couples or because of the presence of unknown amounts of H2O in high-pressure melts (Baker and Balcone-Boissard 2009). F diffusivities in dry basalt4 and jadeite melts can be expressed as (Alletti et al. 2007): (22967 ± 5384)   DFdryTDbasalt4; 1 GPa = exp ( -9.46 ± 3.32)  T  

(44)

31

Equation (44) can reproduce the experimental data to within 0.5 lnD units. The activation energy is 191±45 kJ/mol. A single datum at 0.5 GPa shows that the pressure effect within this range is negligible. Adding 3 wt% H2O in basalt melt increases lnD by 0.54 units. Figure 32 summarizes F diffusion data in natural silicate melts. Experimental F diffusion data at 0.5 and 1 GPa (Balcone-Boissard et al. 2009) show that the pressure effect is negligible. Diffusion data in dry and wet melts suggest that lnD is linear to H2O. The diffusion data in dry and wet Na-phonolite and K-phonolite melts at 1473-1723 K, 1 GPa (likely applicable to 0.5-1.5 GPa), and 0-5 wt% H2O can be fit as follows: (17624 ± 2347) - (312 ± 68)w   1 GPa DFNa-phonolite; = exp ( -13.04 ± 1.47) SEBD  T  

(45)

(21110 ± 4350) - (481 ± 123)w   1 GPa DFK-phonolite; = exp ( -11.43 ± 2.71) SEBD  T  

(46)

Zhang et al. (Ch 8) Diffusion data in silicate melts

Zhang, Ni, Chen

358

Figure 32. Fig. F effective binary diffusivities basalt and phonolite melts. Datadrysources: dry and wet 32. F effective binary diffusivities inin basalt and phonolite melts. Data sources: and wet basalt4 at 0.5 to 1basalt4 at 0.5 to 1 GPa: Alletti et al. (2007); dry and wet Na and K phonolites at 0.5 to 1 GPa: Balcone-Boissard et GPa: Alletti et al. (2007); dry and wet Na and K phonolites at 0.5 to 1 GPa: Balcone-Boissard et al. (2009); and wet al. (2009); and wet HR7 at 0.1 GPa (Gabitov et al. 2005). HR7 at 0.1 GPa (Gabitov et al. 2005).

Equations (45) and (46) can reproduce the experimental data to within 0.34 and 0.47 lnD units, respectively. Cl diffusion. Four papers reported Cl diffusion data in silicate melts (98 points). In addition, there was an early abstract by Watson and Bender (1980) reporting some results, which are not included in this review because no details are available. Bai and Koster van Groos (1994) investigated SEBD (close to tracer diffusion or FEBD) of Cl in rhyolite17 and HR8 melts (Table 1) in contact with NaCl melt or NaCl solution at 0.0001 to 0.46 GPa. Lundstrom (2003) reported a SEBD datum of Cl from basalt7-basanite diffusion couple experiment at 1723 K and 0.9 GPa. Alletti et al. (2007) examined Cl tracer diffusion in dry and wet basalt4 melts at 1523-1723 K and 0.5-1.0 GPa. Balcone-Boissard et al. (2009) determined Cl tracer diffusivities in dry and wet Na-phonolite and K-phonolite melts at 1473-1723 K and 0.5-1 GPa. Furthermore, Baker and Balcone-Boissard (2009) reviewed halogen diffusion in silicate melts (also Baker et al. 2005). 32 Bai and Koster van Groos (1994) inferred that lnDCl in rhyolite17 increases rapidly with H2O from 0 to 2 wt%, and then does not increase much as H2O increases from 2 to 6 wt% (Fig. 33a). However, Balcone-Boissard et al. (2009) deduced that lnDCl in Na- and K-phonolite melts is almost linear to H2O from 0 to 5 wt% H2O (Fig. 33a). Both studies used the method of difference-from-100% in electron microprobe analyses to estimate H2O. Hence, there is some uncertainty in H2O contents in both studies. On the other hand, in some experiments by Bai and Koster van Groos (1994), the pressure is too low to keep H2O content in the rhyolite17 melt (e.g., 6.9 wt% H2O at a pressure of 0.1 GPa), indicating possible problems in the experiments. Tentatively, we suggest that DCl depends linearly on H2O based on BalconeBoissard et al. (2009) and that the inference by Bai and Koster van Groos (1994) is incorrect.

For dry and wet Na-phonolite and K-phonolite melts at 1523-1723 K, 1 GPa and 0-5 wt% H2O, Cl diffusivity can be expressed as follows: (17963 ± 5144) - (605 ± 155)w   1 GPa DClNa-phonolite; = exp ( -14.57 ± 3.22) TD  T  

(47)

Zhang et al. (Ch 8) Diffusion data in silicate melts

Diffusion Data in Siliate Melts

359

Figure 33. Cl diffusivities in various melts as a function of H2O and temperature. Data sources: dry and wet Fig. 33. melt: Cl diffusivities in various melts (1994); as a function of K H2pho O and temperature. Data sources: dry and w rhy17 (rhyolite17) Bai and Koster van Groos Na and (phonolites): Balcone-Boissard et al. (2009); dry basalt4: Alletti et al. (2007). (rhyolite17) melt: Bai and Koster Van Groos (1994); Na and K pho (phonolites): Balcone-Boissard et al basalt4: Alletti et al. (2007)

(15429 ± 5319) - (547 ± 168)w   1 GPa DClK-phonolite; = exp (-15.88 ± 3.32) TD  T  

(48)

Equations (47) and (48) can reproduce the experimental data to within 0.68 and 0.62 lnD units, respectively. When the data on K-phonolite were fit, three outlier points (one is a factor of 10 off, see Fig. 33b, likely due to a typographical error) were excluded. The D values at 0.5 GPa can also be described well by Equations (47) and (48) (for 133 GPa). When different melts are compared, Cl diffusivities in dry basalt4 melt are higher than those in phonolite and rhyolite17 melts by a factor of 2 to 10 at temperature ≥ 1500 K. Those

Zhang, Ni, Chen

360

in dry K-phonolite melt are higher than those in dry rhyolite17 and dry Na-phonolite melts by 0.4 lnD units. Cl diffusivities in dry rhyolite17 melt are similar to those in dry Na-phonolite melt. The similarity in Cl diffusivities between dry Na-phonolite and dry rhyolite melts is unexpected considering (i) the large compositional difference between Na-phonolite and rhyolite17 and (ii) the significant difference in DCl between compositionally similar Naphonolite and K-phonolite (similar composition). Br diffusion. Only one paper (Alletti et al. 2007) reported Br tracer diffusion data (10 points) in basalt4 melt (Table 1) at 1523-1723 K and 1.0 GPa. Figure 34 shows the data, and considerable scatter can be seen. Linearity between lnD and 1/T is not perfect. I and At diffusion. No iodine and astatine diffusion data in silicate melts are known. Comparison of F, Cl, and Br diffusivities. Figure 35 compares diffusivities of F, Cl and Br. The diffusivity increases as the size of halogen ions decreases, similar to the univalent cations. The total increase from Br to F diffusivity is a factor of about 2 (with much scatter) in basalt. In dry Na-phonolite, the increase from Cl to F diffusivity is a factor of about 6. In dry K-phonolite and in wet Na-phonolite, the increase is smaller. The bottom line is that F, Cl and Br diffusivities vary with anion size, but the difference is not large, at least at the temperatures investigated.

He, Ne, Ar, Kr, Xe, Rn Diffusion of noble gas elements in silicate melts is reviewed by another chapter in this volume (Behrens 2010).

Sc, Y, REE Sc diffusion. Only one paper reported Sc diffusion data in silicate melts (13 points). Lowry et al. (1982) investigated Sc tracer diffusion in basalt1 and andesite2 melts at 1570-1676 K and 0.1 MPa. The data are summarized in Figure 36. When compared to REE diffusivities, Sc tracer diffusivity in basalt1 melt is identical to Yb self diffusivity in HB1 melt; and Sc tracer diffusivity in andesite2 melt is slightly higher than La, Nd, Y, Er and Gd diffusivities in a Zhang et al. (Ch 8) Diffusion data in silicate melts haploandesite (HA1) melt (Fig. 36). It appears that Sc diffusivity is similar to Yb diffusivity.

Figure tracer diffusivities basalt melt. Data are(2007). from Alletti Fig. 34. Br 34. tracerBr diffusivities in basalt melt. inData are from Alletti et al.

et al. (2007).

Zhang et al. (Ch 8) Diffusion data in silicate melts

Diffusion Data in Siliate Melts

Figure KK phonolite (pho). Fig.35. 35.Comparison ComparisonofofF,F,ClCland andBrBrdiffusivities diffusivitiesininbasalt basaltand andNaNaand and phonolite (pho). Data sources: basalt4 (Alletti et al. 2007); phonolites (Balcone-Boissard et al. 2009).

361

Data sources

(Alletti et al., 2007); phonolites (Balcone-Boissard et al., 2009)

Y diffusion. Five papers reported Y diffusion data in silicate melts (43 points). Baker and Watson (1988) investigated SEBD of Y (close to FEBD or tracer diffusion) in dry rhyolite1rhyolite8 and HD1-rhyolite8 diffusion couples (Table 1) at 1171-1673 K and 0.01-1 GPa. Nakamura and Kushiro (1998) obtained two tracer diffusivity values of Y in jadeite melt at 1673 K and 1.25 and 1.5 GPa (the two values differ by two orders of magnitude and hence cannot both be correct), as well as two values in diopside melt at 1863 K and 1 and 1.25 GPa. Mungall et al. (1999) studied Y tracer diffusion in dry HR7 and HR7 + Na and wet HR7 containing 3.6 wt% H2O, with D values in wet HR7 (3.6 wt% H2O) and HR7 + Na (20% Na2O) 35 Y tracer diffusivities in wet roughly on the same trend. Koepke and Behrens (2001) reported HA1 melt (4.5-5.2 wt% H2O) and one datum in dry HA1 melt. Behrens and Hahn (2009)

Zhang et al. (Ch 8) Diffusion data in silicate melts

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Zhang, Ni, Chen

Figure 36. Sc tracer diffusivities in dry basalt1 and andesite2 melts (Lowry et al. 1982) compared with REE diffusivities in dry HB1 (LaTourrette et al. 1996) and dry HA1 (Koepke and Behrens 2001).

Fig. 36. Sc tracer diffusivities in dry basalt1 and andesite2 melts (Lowry et al. 1982) compared with REE

characterized Y traceret al. diffusion inHA1 dry(Koepke and wet trachyte1 diffusivities in dry HB1 (LaTourrette 1996) and dry and Behrens 2001).

and phonolite1 melts. The diffusion

data are summarized in Figure 37. SEBD data are limited and not enough to model how they vary with temperature, melt composition and concentration gradients. H2O has a large effect on Y diffusivity. For example, for tracer diffusion in trachyte1, adding 1.7 wt% H2O increases DY by a factor of 28. Assuming that lnD increases linearly with the square root of H2O for each type of melt, Y tracer diffusivities in dry and wet trachyte1 and phonolite1 melts at 1323-1528 K, 0.5 GPa, and ≤ 1.9 wt% H2O can be fit as follows:  (23941 ± 5364) - (4717 ± 568) w  DYtrachyte1 = exp ( -15.22 ± 3.68)  TD T  

(49)

 (31693 ± 5159) - (2132 ± 275) w  DYphonolite1 = exp ( -7.58 ± 3.51)  TD T  

(50)

Equations (49) and (50) can reproduce experimental data to within 0.43 and 0.36 lnD units, 36 respectively. La diffusion. Five papers reported La diffusion data in silicate melts (50 points). Rapp and Watson (1986) examined SEBD of LREE (La, Ce, Nd, and Sm, assumed to have the same diffusivity) during monazite dissolution in wet rhyolite12 melts (containing 1-6 wt% H2O) (Table 1) at 1273-1673 K and 0.8 GPa. Nakamura and Kushiro (1998) studied La tracer diffusion in jadeite and diopside melts at 1673-1863 K and 0.75-2.0 GPa. Koepke and Behrens (2001) investigated La tracer diffusion in a wet haploandesite melt (4.5-5.2 wt% H2O) and reported one datum in dry haploandesite melt at 0.5 GPa. Lundstrom (2003) obtained a SEBD value for La between a basalt7-basanite diffusion couple at 1723 K and 0.9 GPa. Behrens and Hahn (2009) characterized La tracer diffusion in dry and wet trachyte and phonolite melts at 0.5 GPa. The diffusion data are summarized in Figure 38. For SEBD in rhyolite12, from 1 to 4 wt% H2O, lnDLa increases significantly, but from 4 to 6 wt% H2O, there is almost no increase (Rapp and Watson 1986). The data do not conform to a linear relation between lnD and H2O or linear relation between lnD and the square root of

Diffusion Data in Siliate Melts

Zhang et al. (Ch 8) Diffusion data in silicate melts

363

38. La diffusivities in silicatemelts. melts; percentage in the legends means wt% H2O. Data and other Figure 37.Fig. Y diffusivities in silicate Except for dry b-b couple forofwhich the sources: data arejadeite SEBD, data are tracer diffusivities. Data sources: jadeite and diopside melts (Nakamura and Kushiro 1998); dry diopside (Nakamura and Kushiro 1998); SEBD in rhy12 (rhyolite12) during monazite dissolution (1, 2, 4, and 6 HR7, wet HR7 with 3.6 wt% H2O, and HR7 + Na (Mungall et al. 1999); b-b (basalt7-basanite) couple Zhang et al. (Ch 8) Diffusion data in silicate melts (Lundstrom 2003); dry and (4.5-5.2 wt% HA1 (Koepke Behrens and and wet Watson 1986); dry andH wet HA1 (4.5-5.2 wt% H2and O) (Koepke and2001); Behrens dry 2001); drywet and trachyte1 wet wt% H2O) (Rapp 2O) and phonolite1 (Behrens and Hahn 2009). trachyte1 (1.1-1.7 wt% H2O) (Behrens and Hahn 2009); dry and wet phonolite1 (1.8 wt% H2O) (Behrens and Hahn 2009); SEBD in b-b (basalt7-basanite) couple (Lundstrom 2003).

38

Ce diffusivities silicate melts, including FEBD inin drythe jadeite and diopside melts (Nakamura and Data Kushiro Figure 38.Fig. La39. diffusivities ininsilicate melts; percentage legends means wt% of H2O. sources: jadeite and1998), diopside (Nakamura and Kushiro 1998); SEBD in rhy12 (rhyolite12) during monazite TD in dry rhy15 (rhyolite15) melt (Jambon 1982), and SEBD in wet rhy12 (rhyolite12) melts with 1, 2, 4, dissolution (1, 2, 4, and 6 wt% H2O) (Rapp and Watson 1986); dry and wet HA1 (4.5-5.2 wt% H2O) (Rapp anddry Watson and 6Behrens wt% H2O 2001); (Koepke and and1986). wet trachyte1 (1.1-1.7 wt% H2O) (Behrens and Hahn 2009); dry and wet phonolite1 (1.8 wt% H2O) (Behrens and Hahn 2009); SEBD in b-b (basalt7-basanite) couple (Lundstrom 2003).

364

Zhang, Ni, Chen

H2O content. Trying to relate D to viscosity does not lead to a simple relation either. Although the observed lnD versus H2O trends could be real, it is also possible (even likely) that the H2O contents were not known accurately in these early experimental studies, complicating efforts to quantify how diffusivity depends on H2O. The same comment applies to Ce, Nd, Sm and P diffusion in rhyolite12 investigated by Rapp and Watson (1986). Adding H2O increases La tracer diffusivity significantly. For example, for tracer diffusion in phonolite1, adding 1.7 wt% H2O increases DLa by 2.4 lnD units (1.06 logD units). Based on diffusion data in dry and wet trachyte1 melt, it appears that lnD increases linearly with the square root of H2O content. Using such a relation, tracer diffusivities in dry and wet trachyte1 and phonolite1 melts with H2O ≤ 1.9 wt% at 1323-1528 K and 0.5 GPa can be fit as:  (25128 ± 4896) - (4810 ± 519) w  trachyte1 DLa = exp ( -14.13 ± 3.36)  TD T  

(51)

 (33135 ± 4870) - (2941 ± 412) w  phonolite1 DLa = exp ( -6.44 ± 3.34)  TD T  

(52)

Equations (51) and (52) can reproduce experimental data to within 0.38 and 0.40 lnD units, respectively. Ce diffusion. Three papers reported Ce diffusion data in silicate melts (30 points). Jambon (1982) investigated Ce tracer diffusion in dry rhyolite15 (Table 1) at 1148-1573 K and 0.1 MPa. Rapp and Watson (1986) obtained SEBD of Ce in wet rhyolite12 melts during monazite dissolution (assumed to be exactly the same as La diffusivities). Nakamura and Kushiro (1998) studied Ce tracer diffusion in jadeite and diopside melts at 1673-1863 K and 0.75-2.0 GPa. Ce in silicate melts can be present as either trivalent or tetravalent cations, but in all of these studies Ce is likely mainly trivalent, and its diffusivities are similar to La diffusivities. The data are shown in Figure 39. A single Ce tracer diffusivity in dry rhyolite15 at 1573 K does not lie in the same trend as data at 1148-1373 K (red filled circles and solid line in Fig. 39). On the other hand, Ce tracer diffusivities in dry rhyolite15 at 1148-1373 K and 0.1 MPa are coincidentally similar to SEBD of Ce during monazite dissolution in rhyolite12 melt containing 1 wt% H2O at 1273-1673 K and 0.8 GPa; both data sets can be fit by a single equation: (58413 ± 2536)   wet rhyolite12 (1 wt% H 2 O) dry rhyolite15 DCe ≈ DCe = exp (5.78 ± 1.87) TD SEBD  T  

(53)

The maximum error in reproducing the experimental data is 0.55 in terms of lnD. The Ce tracer diffusivity in dry rhyolite15 at 1573 K by Jambon (1982) is a clear outlier, off the trend by 5.9 lnD units (2.6 logD units). Pr diffusion. Only one paper reported Pr diffusion data in silicate melts (8 points). Nakamura and Kushiro (1998) investigated Pr tracer diffusion in jadeite and diopside melts. The data are shown in Figure 40. Pr tracer diffusivities in jadeite melt increases with increasing pressure. At 1573-1723 K and 0.75-2 GPa, the diffusion data can be fit as: (40807 ± 9501) - (1697 ± 704)P   melt DPrjadeite = exp ( -4.57 ± 5.65) TD  T  

(54)

Equation (54) reproduces experimental data to ≤ 0.20 lnD units. The activation energy at 1.5 GPa is 318±71 kJ/mol and the activation volume at 1673 K is −(14±6)×10−6 m3/mol.

Zhang et al. (Ch 8) Diffusion data in silicate melts

Diffusion Data in Siliate Melts

365

Ce diffusivities silicate melts, including FEBDFEBD in dry jadeite andjadeite diopside and meltsdiopside (Nakamura melts and Kushiro Figure 39. Fig. Ce 39. diffusivities ininsilicate melts, including in dry (Nakamura Zhang et al. (Ch 8) Diffusion data in silicate melts and Kushiro 1998), TD in dry rhy15 (rhyolite15) melt (Jambon 1982), and SEBD in wet rhy12 1998), TD in dry rhy15 (rhyolite15) melt (Jambon 1982), and SEBD in wet rhy12 (rhyolite12) melts with 1, 2,(rhyolite12) 4, melts with 1, 2, 4, and 6 wt% H2O (Rapp and Watson 1986). and 6 wt% H2O (Rapp and Watson 1986).

39

Fig. 40. 40. Pr diffusivities in jadeite diopside melts (Nakamura and Kushiro 1998) and Kushiro 1998). Figure Prtracer tracer diffusivities inand jadeite and diopside melts (Nakamura

Nd diffusion. Nine papers reported Nd diffusion data in silicate melts (90 points). Rapp and Watson (1986) examined SEBD of Nd during monazite dissolution in wet rhyolite12 melts (treated as the same as La diffusivities). Lesher (1994) obtained Nd self diffusivities in rhyolite2 and basalt3 melts at 1528-1738 K and 1 GPa. LaTourrette et al. (1996) and LaTourrette and Wasserburg (1997) determined Nd self diffusivities in HB1 melt at 16231773 K and 0.1 MPa (in air and in oxygen fugacity corresponding to the Fe-FeO buffer). Perez and Dunn (1996) measured Nd tracer diffusivities in rhyolite8 melt at 1448-1673 K, 1 GPa and ≤0.8 wt% H2O (but H2O is not always measured). Nakamura and Kushiro (1998) studied

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Nd tracer diffusion in jadeite and diopside melts at 1673-1863 K and 0.75-2.0 GPa. Mungall et al. (1999) investigated Nd tracer diffusion in dry HR7 and HR7 + Na melts at 0.1 MPa, and a wet HR7 melt with 3.6 wt% H2O at 1 GPa. Koepke and Behrens (2001) reported Nd tracer diffusivities in wet HA1 melt (4.5-5.2 wt% H2O) and one datum in dry HA1 melt at 0.5 GPa. Behrens and Hahn (2009) characterized Nd tracer diffusion in dry and wet trachyte1 and phonolite1 melts at 0.5 GPa. Melt compositions can be found in Table 1 except for jadeite and diopside. Zhang Figure 41 shows some diffusion data. et al. (Ch 8) Diffusion data in silicate melts

Figure 41. some melts. In legend, the legend, HB1 under reducing Fig.Nd 41. diffusivities Nd diffusivities in in some melts. In the "HB1-r"“HB1-r” means HB1means under reducing conditions; "rhy" conditions; “rhy” means rhyolite; “pho” means phonolite; “tra” means trachyte. Melt compositions are listed in Table means rhyolite; "pho" means phonolite; "tra" means trachyte. Melt compositions are listed in Table 1. Data 1. Data sources: SD in HB1 and HB1-r (LaTourrette et al. 1996; LaTourrette and Wasserburg 1997); SD in basalt3sources: and rhy2 1994); TD in HA1 and Behrens 2001); TDSD in inrhy8 (Perez and Dunn SD in(Lesher HB1 and HB1-r (LaTourrette et al. (Koepke 1996; LaTourrette and Wasserburg 1997); basalt3 and rhy2 1996); TD in HR7 (Mungall et al. 1999); TD in dry and wet pho1 and tra1 (Behrens and Hahn 2009); TD (Lesher 1994); TD in HA1 (Koepke and Behrens 2001); TD in rhy8 (Perez and Dunn 1996); TD in HR7 (Mungall et in jadeite (Nakamura and Kushiro 1998). al. 1999); TD in dry and wet pho1 and tra1 (Behrens and Hahn 2009); TD in jadeite (Nakamura and Kushiro 1998).

Nd self diffusivities do not depend on oxygen fugacity from air to Fe-FeO buffer (LaTourrette and Wasserburg 1997), which is expected because the valence of Nd is not expected to change. In jadeite melt, Nd tracer diffusivities depend on pressure according to Nakamura and Kushiro (1998). However, Nd self diffusivities in HB1 melt with no Fe, Ti and Na (Table 1) at 0.1 MPa (LaTourrette and Wasserburg 1997) and a single Nd self diffusivity in basalt3 with significant Fe, Ti and Na at 1 GPa (Lesher 1994) are similar, either suggesting negligible pressure dependence and good choice of HB1 melt composition by LaTourrette and Wasserburg (1997), or due to coincidence. Nd self diffusivities in these two melts at 15281773 K and 0.001-1 GPa can be expressed as (Fig. 41): (24001 ± 2803)   dry basalt DNd = exp ( -10.11 ± 1.66) SD  T   41

(55)

Equation (55) reproduces all experimental data to within 0.3 lnD units. In the highly silicic melt HR7 at 1410-1873 K and 0.1 MPa, Nd tracer diffusivities are orders of magnitude lower than self diffusivities in basalt melt, and can be described as (Mungall et al. 1999):

Diffusion Data in Siliate Melts

367

(35847 ± 1488)   dry HR7 DNd TD = exp ( -9.90 ± 0.92)  T  

(56)

Equation (56) reproduces three experimental data to within 0.08 lnD units. In wet haploandesite melt containing 4.5-5.2 wt% H2O, Nd tracer diffusivities at 13731673 K and 0.5 GPa can be expressed as (Koepke and Behrens 2001): (20280 ± 220)   wet HA1 DNd TD = exp ( -11.44 ± 0.15)  T  

(57)

Equation (57) reproduces four experimental data to within 0.01 lnD units. In dry and wet trachyte and phonolite melts with ≤1.9 wt% H2O at 1323-1528 K and 0.5 GPa (Behrens and Hahn 2009), the tracer diffusion data can be fit as follows:  (24381 ± 5125) - (4615 ± 543) w  trachyte1 DNd  TD = exp ( -14.57 ± 3.52) T  

(58)

 (31697 ± 3401) - (2937 ± 288) w  phonolite1 DNd  FEBD = exp ( -7.45 ± 2.33) T  

(59)

Equations (58) and (59) reproduce experimental data to within 0.36 and 0.23 lnD units, respectively. Pm diffusion. No Pm diffusion data in silicate melts are known. Sm diffusion. Four papers reported Sm diffusion data in silicate melts (49 points). Rapp and Watson (1986) studied SEBD of Sm in wet rhyolite12 (treated as the same as La diffusivities). Nakamura and Kushiro (1998) investigated Sm tracer diffusion in jadeite melt at 1573-1723 K and 0.75-2.0 GPa and in diopside melt at 1863 K and 1.0-1.25 GPa. Koepke and Behrens (2001) reported Sm tracer diffusivities in a wet haploandesite (Table 1) melt with 4.5-5.2 wt% H2O at 1373-1673 K an 0.5 GPa, and one datum in dry haploandesite melt at 1673 K and 0.5 GPa. Behrens and Hahn (2009) examined Sm tracer diffusion in dry and wet trachyte and phonolite melts (Table 1) at 0.5 GPa. The diffusion data are shown in Figure 42. Sm diffusivity increases from polymerized melt to depolymerized melt, and increases with increasing H2O. Tracer diffusivities in dry and wet trachyte1 melt with H2O ≤ 1.7 wt% can be fit as:  (24485 ± 4149) - (4790 ± 439) w  trachyte1 DSm  TD = exp ( -14.70 ± 2.83) T  

(60)

Equation (60) can reproduce experimental data to within 0.31 lnD units. Using the same relation to fit tracer diffusion data in dry and wet phonolite1 melt containing ≤ 1.9 wt% H2O, we obtain:  (33207 ± 4055) - (3149 ± 343) w  phonolite1 DSm = exp ( -6.62 ± 2.78)  TD T  

(61)

Equation (61) can reproduce experimental data to within 0.32 lnD units. Eu diffusion. Eight papers reported Eu diffusion data in silicate melts (76 points). Because Eu in natural melts can be present as Eu2 +  and/or Eu3 + , and because Eu2 +  and Eu3 +  have different diffusivities, it is important to control the oxygen fugacity in Eu diffusion studies.

368

Zhang et al. (Ch 8) Diffusion data in silicate melts

Zhang, Ni, Chen

42.tracer Sm tracer diffusivities in melts. DataData sources: dry and wet H 2O at 0.5 GPa Figure 42.Fig. Sm diffusivities insilicate silicate melts. sources: dryHA1 andwith wet4.5-5.2 HA1wt% with 4.5-5.2 wt% H2O at 0.5 GPa (Koepke and Behrens 2001); dry and wet trachyte1 (tra1) and phonolite1 (pho1) (Behrens and (Koepke and Behrens 2001); dry and wet trachyte1 (tra1) and phonolite1 (pho1) (Behrens and Hahn 2009); and Hahn 2009); and jadeite and diopside melts (Nakamura and Kushiro 1998). jadeite and diopside melts (Nakamura and Kushiro 1998).

Magaritz and Hofmann (1978b) explored Eu tracer diffusion in basalt10 and rhyolite12 melts (Table 1) in air. They found that the data (7 points) below the liquidus behaved irregularly, possibly due to convection induced by newly formed crystals. These data are excluded from the compilation and are not used in the discussion here. Jambon (1982) investigated Eu tracer diffusion in dry rhyolite5 melt at 973-1573 K and 0.0001-0.4 GPa. Henderson et al. (1985) studied Eu tracer diffusivities in basalt1, andesite2, dacite2 and pantellerite1 melts at 1475 to 1673 K and in air. LaTourrette and Wasserburg (1997) examined the effect of oxygen fugacity on Eu self diffusivity in dry HB1 melt by conducting experiments in air and in Fe-FeO buffer at 1673-1773 K and 0.1 MPa. Nakamura and Kushiro (1998) investigated Eu tracer diffusion in jadeite melt at 1573-1723 K and 0.75-2.0 GPa and in diopside melt at 1863 K and 1.0-1.25 GPa with uncontrolled oxygen fugacity although attempt was made to change the oxygen fugacity. Koepke and Behrens (2001) reported Eu tracer diffusivities in wet (4.5-5.2 wt% H2O) and dry HA1 melt at 1373-1673 K, 0.5 GPa, and logfO2 of about NNO + 3. Behrens and Hahn (2009) investigated Eu tracer diffusion in dry and wet trachyte and phonolite melts at 0.5 GPa 42 in which the oxygen fugacity was varied. Figure 43 shows Eu diffusion data. The individual Arrhenius relations can be found in Online Supplementary Table 2. LaTourrette and Wasserburg (1997) determined Eu diffusivities in air (where Eu is expected to be mostly Eu3 + ) and in the oxygen fugacity of the Fe-FeO buffer (where Eu is expected to be mostly Eu2 + ). They found that Eu diffusivities at the reduced condition of the Fe-FeO buffer are higher by 42% than those at oxidized conditions in air (the difference in lnD is 0.35), a relatively small difference (compare “HB1” and “HB1-r” in Fig. 43). Based on multi-species diffusion treatment (Zhang et al. 1991a,b), assuming that the fraction of Eu2 +  does not depend on Eu concentration, DEu can be expressed as follows: = DEu X Eu2+ DEu2+ + X Eu3 + DEu3 +

(62)

where DEu2 +  and DEu3 +  are the diffusivities of Eu2 +  and Eu3 + , XEu2 +  = Eu2 + /(Eu2 +   +  Eu3 + ), and XEu3 +  = Eu3 + /(Eu2 +   +  Eu3 + ). Hence DEu lies between DEu3 +  and DEu2 + . DEu3 +  is expected to lie

Zhang et al. (Ch 8) Diffusion data in silicate melts

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369

Figure 43. Eu diffusivities in various melts. (a) All are in air except for HB1-r (reduced at Fe-FeO buffer). Fig. 43. Eu diffusivities in various melts. (a) All are in air except for HB1-r (reduced at Fe-FeO buf Data sources: pantellerite1 (pant1), dacite4, andesite2, and basalt1: Henderson et al. (1985); basalt10: Magaritz and Hofmann (1978b); HB1 in air and in reduced conditions: LaTourrette and Wasserburg (1997). sources: pantellerite1 (pant1), dacite4, andesite2, and basalt1: Henderson et al. (1985); basalt10: Mag (b) Diffusivities at high pressures. Data sources: rhy15 (rhyolite15): Jambon (1982); dry and wet HA1: Koepke and Behrens (2001); dry and wet trachyte1 (tra1) and phonolite1 (pho1): Behrens and Hahn (2009). Hofmann (1978); HB1 in air and in reduced conditions: LaTourrette and Wasserburg (1997). (b) Di high pressures.

Data sources: rhy15 (rhyolite15): Jambon (1982); dry and wet HA1: Koepke and Be

between DSm and DGd, which are similar (Fig. 44a). If the difference between DEu3 +  and DEu2 +  is indeed so small shown by LaTourrette Wasserburg DEu should dryasand wet trachyte1 (tra1) andand phonolite1 (pho1):(1997), Behrensthen and Hahn (2009).not be much greater than DSm and DGd. However, Figure 44a shows that DEu may be greater than DSm and DGd by almost an order of magnitude. Hence, the conclusion that Eu2 +  diffusivity is only slightly greater than Eu3 +  diffusivity by LaTourrette and Wasserburg (1997) is not general.

Behrens and Hahn (2009) also varied oxygen fugacity in determining the tracer diffusivities of Eu in dry and wet trachyte and phonolite melts. Their data did not allow them to infer tracer diffusivities of both Eu2 +  and Eu3 + . Rather, they assumed DEu3 +  is the average of DGd and DSm of Sr2 +  and Eu2 +  are 0.118 and (this should work well), and DEu2 +  is the same as DSr (ionic radii 43 0.117 nm in octahedral sites), and estimated XEu2 +  in each experiment. Figure 44b compares Eu

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Zhang, Ni, Chen

Figure 44. Comparison of Eu diffusivities with those of (a) Sm and Gd, and (b) Sr and Gd in some melts. Fig. 44. Comparison of Eu diffusivities with those of (a) Sm and Gd, and (b) Sr and Gd in some melts. Eu Eu diffusivities are higher by variable amount than both Sm and Gd diffusivities, but are between Sr and Gd diffusivities. Data sources: wet HA1 melt at 0.5 GPa with 4.5-5.2 wt% H2O (Koepke and Behrens 2001); are higher by variable amount than both Sm and Gd diffusivities, but are between Sr and Gd jadeite melt atdiffusivities 1.5 GPa (Nakamura and Kushiro 1998); dry trachyte1 (tra1) and phonolite1 (pho1) at 0.5 GPa (Behrens and Hahn 2009). diffusivities. Data sources: wet HA1 melt at 0.5 GPa with 4.5-5.2 wt% H2O (Koepke and Behrens 2001); jad

melt at 1.5 GPa (Nakamura 1998);indry trachyte1 andwithin phonolite1 (pho1) of at 0.5 GPa (Behren diffusivities with those of Gd and Sr. and Eu Kushiro diffusivities these cases(tra1) do lie the range those of Gd and Sr,2009). consistent with the assumption of Behrens and Hahn (2009). Nonetheless, Hahn it is necessary to directly verify that DEu2 +  is similar to DSr. To estimate Eu diffusivity in natural melts, it is critical to evaluate the Eu2 + /Eu ratio because Eu2 +  diffusivity can be an order of magnitude higher than that of Eu3 +  (Fig. 44).

Gd diffusion. Four papers reported Gd diffusion data in silicate melts (40 points). Magaritz and Hofmann (1978b) investigated Gd (and Eu) tracer diffusion in basalt10 melt in air. They concluded that tracer diffusivities of Gd and those of Eu are identical. The data 44 (7 points) below the liquidus of basalt10 are somewhat problematic and are excluded from the compilation as well as the discussion. Nakamura and Kushiro (1998) investigated Gd

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371

tracer diffusion in jadeite melt at 1573-1723 K and 0.75-2.0 GPa and in diopside melt at 1863 K and 1.0-1.25 GPa. Koepke and Behrens (2001) reported Gd tracer diffusivities in a wet haploandesite (Table 1) melt with 4.5-5.2 wt% H2O at 1373-1673 K and 0.5 GPa, and one datum in dry haploandesite melt at 1673 K and 0.5 GPa. Behrens and Hahn (2009) investigated Gd tracer diffusion in dry and wet trachyte and phonolite melts (Table 1) at 0.5 GPa. The Zhangare et al. shown (Ch 8) Diffusion data in silicate diffusion data in Figure 45. melts

Figure 45.Fig. Gd45.tracer diffusivities insilicate silicate melts. sources: jadeite and diopside melts (Nakamura and Gd tracer diffusivities in melts. DataData sources: jadeite and diopside melts (Nakamura and Kushiro Kushiro 1998), dry and wet HA1 with 4.5-5.2 wt% H2O at 0.5 GPa (Koepke and Behrens 2001); dry and 1998), dry and wet HA1 with 4.5-5.2 wt% H2O at 0.5 GPa (Koepke and Behrens 2001); dry and wet trachyte1 (tra1) wet trachyte1 (tra1) and phonolite1 (pho1) (Behrens and Hahn 2009); dry basalt10 at 0.1 MPa (Magaritz and Hofmann 1978b).(pho1) (Behrens and Hahn 2009); dry basalt10 at 0.1 MPa (Magaritz and Hofmann 1978). and phonolite1

Gd diffusivity increases from polymerized melt to depolymerized melt. Adding H2O increases Gd diffusivity significantly. Gd tracer diffusivities in wet haploandesite melt containing 4.5-5.2 wt% H2O at 1373-1673 K and 0.5 GPa can be expressed as: (19867 ± 1520)   wet HA DGd FEBD = exp ( -11.99 ± 1.00)  T  

(63)

Assuming lnD increases linearly with H2O, Gd tracer diffusivities in dry and wet trachyte1 and phonolite1 melts (≤1.9 wt% H2O) at 1323-1527 K and 0.5 GPa can be fit as:  (26366 ± 3986) - (5021 ± 453) w  trachyte1 DGd = exp ( -13.60 ± 2.73)  TD T  

(64)

45  (33656 ± 5536) - (3294 ± 468) w  phonolite1 DGd = exp ( -6.55 ± 3.79)  TD T  

(65)

Equations (64) and (65) reproduce diffusion data to within 0.31 and 0.51 lnD units, respectively. Tb diffusion. Only one paper reported Tb diffusion data in silicate melts (8 points). Mungall et al. (1999) investigated Tb tracer diffusion in HR7, HR7 + Na, and wet HR7 melts (Table 1). The data are summarized in Figure 46. Tb diffusivities in dry HR7 melt at 1410-

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372

Zhang, Ni, Chen

46.inTb in dry dry and(Mungall wet HR7 Fig. 46. TbFigure diffusivities drydiffusivities HR7, dry HR7+Na, and HR7, wet HR7 (3.6HR7 + Na, wt% H2O) melts et al. 1999). (3.6 wt% H2O) melts (Mungall et al. 1999).

1873 K and 0.1 MPa can be expressed as: (38248 ± 1948)   dry HR7 DTb TD = exp ( -8.95 ± 1.20)  T  

(66)

Equation (66) reproduces experimental data to within 0.10 lnD units. Dy diffusion. Only one paper reported Dy diffusion data in silicate melts (8 points). Nakamura and Kushiro (1998) investigated Dy tracer diffusion in jadeite melt at 1573-1723 K and 0.75-2.0 GPa and in diopside melt at 1863 K and 1.0-1.25 GPa. The data are shown in Figure 47. Assuming constant activation volume, Dy diffusivities in dry jadeite melt at 15731723 K and 0.75-2.0 GPa can be expressed as: 46240 - 2040 P   dry jadeite DDy ≈ exp  -1.93 TD  T  

(67)

Equation (67) reproduces experimental data to within 0.21 lnD units. 46 Ho diffusion. No Ho diffusion data in silicate melts are known. Er diffusion. Two papers reported Er diffusion data in silicate melts (13 points). Nakamura and Kushiro (1998) investigated Er tracer diffusion in jadeite melt at 1573-1723 K and 0.752.0 GPa and in diopside melt at 1863 K and 1.0-1.25 GPa. Koepke and Behrens (2001) studied Er tracer diffusion in dry and wet HA1 melts (Table 1). The data are shown in Figure 48. Assuming constant activation volume, Er diffusivities in dry jadeite melt at 1573-1723 K and 0.75-2.0 GPa can be expressed as: 47186 - 1953P   DErdryTDjadeite ≈ exp  -1.30  T  

(68)

Equation (68) reproduces experimental data to within 0.23 lnD units. For wet haploandesite containing about 4.5-5.2 wt% H2O at 1373-1673 K and 0.5 GPa, Er diffusivities can be expressed as (Koepke and Behrens 2001):

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373

(17895 ± 2876)   DErwetTDHA1 (5 wt%) = exp ( -13.34 ± 1.90)  T  

(69)

Equation (69) reproduces experimental data to within 0.16 lnD units. Tm diffusion. No Tm diffusion data in silicate melts are known. Yb diffusion. Four papersdata reported Yb diffusion data (38 points). LaTourrette et al. (1996) Zhang et al. (Ch 8) Diffusion in silicate melts studied Yb self diffusion in dry HB1 melt (Table 1) at 1623-1773 K and in air. Nakamura and

tracer in dry jadeite diopside (Nakamura and (Nakamura Kushiro 1998). and Kushiro 1998). FigureFig. 47.47.Dy diffusivities dry and jadeite andmelts diopside melts Zhang etDy al.tracer (Ch 8)diffusivities Diffusion data inin silicate melts

47

Figure Er diffusivities in dry (Nakamura and 1998), Fig. 48.48. Er diffusivities in dry jadeite andjadeite diopsideand meltsdiopside (Nakamuramelts and Kushiro 1998), and dryKushiro and wet HA1melts and dry and wet HA1melts (Koepke and Behrens 2001). (Koepke and Behrens 2001).

Zhang, Ni, Chen

374

Kushiro (1998) examined Yb tracer diffusion in jadeite melt at 1573-1723 K and 0.75-2.0 GPa and in diopside melt at 1863 K and 1.0-1.25 GPa. Koepke and Behrens (2001) studied Yb tracer diffusion in a wet HA1 melt (4.5-5.2 wt% H2O) at 0.5 GPa. Behrens and Hahn (2009) investigated Yb tracer diffusion in dry and wet trachyte1 and phonolite1 melts at 0.5 GPa. The diffusion data are shown in Figure 49. Yb diffusivity increases from polymerized melt to depolymerized melt. Adding H2O increases Yb diffusivity significantly. Yb tracer diffusivities in wet haploandesite melt containing 4.5-5.2 wt% H2O at 1373-1673 K and 0.5 GPa can be expressed as: (20310 ± 10329)   wet HA1 DYb TD = exp ( -12.34 ± 6.06)  T  

(70)

In dry and wet trachyte and phonolite melts with ≤1.9 wt% H2O at 1323-1528 K and 0.5 GPa (Behrens and Hahn 2009), the tracer diffusion data can be fit as follows:  (25867 ± 4109) - (4892 ± 467) w  trachyte1 DYb  TD = exp ( -14.19 ± 2.81) T  

(71)

 (33718 ± 5399) - (3570 ± 457) w  phonolite1 DYb  FEBD = exp ( -6.96 ± 3.70) T  

(72)

Equations (71) and (72) reproduce experimental data to within 0.31 and 0.40 lnD units, respectively. Lu diffusion. Two papers reported Lu diffusion data in silicate melts (15 points). Nakamura and Kushiro (1998) characterized Lu tracer diffusion in jadeite melt at 1573-1723 K and 0.75-2.0 GPa and in diopside melt at 1863 K and 1.0-1.25 GPa. Mungall et al. (1999) Zhang et al. (Ch 8) Diffusion data in silicate melts

Fig.Yb 49.diffusivities Yb diffusivities in in silicate melts. DataData sources: self diffusivities in dry HB1 melt 0.1 MPa Figure 49. silicate melts. sources: self diffusivities in atdry HB1(LaTourrette melt at 0.1 MPa (LaTourrette et al. 1996); tracer diffusivities in jadeite and diopside melts (Nakamura and Kushiro 1998), et al. 1996); tracer diffusivities in jadeite and diopside melts (Nakamura and Kushiro 1998), wet HA1 melt with 4.5wet HA1 melt with 4.5-5.2 wt% H2O at 0.5 GPa (Koepke and Behrens 2001); dry and wet trachyte1 (tra1) GPa (Koepke Behrens 2001); dry and wet trachyte1 (tra1) and phonolite1 (pho1) (Behrens 5.2 wt%(pho1) H2O at 0.5 and phonolite1 (Behrens andand Hahn 2009). and Hahn 2009).

Diffusion Data in Siliate Melts

375

investigated Lu tracer diffusion in dry HR7, dry HR7 + Na, and wet HR7 melts (Table 1). The data are shown in Figure 50. Lu diffusivities in HR7 (3 points) are low and can be expressed as: (39949 ± 5206)   dry HR7 DLu TD = exp ( -8.07 ± 3.22)  T  

(73)

Equation (73) reproduces experimental data to within 0.26 lnD units. Adding 3.6 wt% H2O or 20% wt% Na2O increases DLu by roughly 3 orders of magnitude, depending on temperature. Summary of Y and REE diffusion. There are a total of 423 diffusivity values determined for Y and REE. Y and REE diffusion data in selected melts are compared in Figure 51. Because the ionic radius of Y3 +  (0.0900 nm) is similar to that of Ho3 +  (0.0901 nm), and there are no Ho diffusivity data, in the plots, Y is placed in the position of Ho. Under reduced conditions, Eu diffusivity is often significantly higher than the rest of the REE, attributable to the presence of Eu2 + . Other REEs mostly form a smooth trend. Data that are slightly off the trend are attributed to experimental errors. Based on the data in various melts (basalt to rhyolite, and dry to hydrous melts), there is a slight decrease of diffusivity from La to Lu, about 0.5 lnD units. That is, diffusivities of the REE decrease as the size decreases, contrary to the alkali cations. Overall, the rare earth elements as a group behave well in terms of diffusion properties. With the smooth behavior, even though there are no Ho and Tm diffusion data, their diffusivities can be readily predicted from the other REE diffusion data. Eu3 +  tracer diffusivities can also be well predicted. On the other hand, predicting Eu diffusivity at a given temperature, pressure and oxygen fugacity requires knowing the Eu3 + /Eu2 +  ratio and Eu2 +  diffusivity.

Ti, Zr, Hf Ti diffusion. Nine papers reported Ti diffusion data in silicate melts (59 points). Most of the data were SEBD obtained as a byproduct of crystal dissolution experiments. Zhang et al. (1989) reported a single Ti diffusivity (SEBD) in dry andesite melt (Table 1) during rutile dissolution at 1648 K and 1.5 GPa. LaTourrette et al. (1996) investigated Ti self diffusion in dry HB1 melt (Table 1) at 1623-1773 K and in air. Mungall et al. (1999) studied Ti tracer Zhang et al. (Ch 8) Diffusion data in silicate melts diffusion in dry HR7, wet HR7 (3.6 wt% H2O), and dry HR7 + Na melt (Table 1) at 1083-1873

Figure 50. Lu diffusion datamelts. in silicate melts. for jadeite and diopside are Fig. 50. Lu diffusion data in silicate Data for jadeiteData and diopside melts are from Nakamuramelts and Kushiro from Nakamura and Kushiro (1998), and the rest are from Mungall et al. (1999). (1998), and the rest are from Mungall et al. (1999).

Figure 51. Diffusivities of rare earth elements in various silicate melts. Data sources are: jadeite (Jd) and diopside (Di) melts: Nakamura and Kushiro (1998); rhyolite12 (rhy12): Rapp and Watson (1986); dry and wet HR7: Mungall et al. (1999); dry and wet haploandesite (HA1): Koepke and Behrens (2001); dry and wet phonolite1 (pho): Behrens and Hahn (2009).

Fig. 51. Diffusivities of rare earth elements in various silicate melts.

Data sources are: jadeite (Jd) and diopside

376 Zhang, Ni, Chen

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377

K, 0.0001 and 1 GPa. Van Orman and Grove (2000) extracted a Ti diffusivity value (SEBD) in lunar basalt melt (LB1 in Table 1) during clinopyroxene dissolution at 1623 K and 1.3 GPa. Lundstrom (2003) obtained two SEBD values of Ti from dry basalt7-basanite diffusion couple experiment at 1723 K and 0.9 GPa. Morgan et al. (2006) acquired SEBD values of Ti in two lunar picrite melts (LP1 and LP2 in Table 1) during anorthite dissolution at 1673 K and 0.6 GPa. Hayden and Watson (2007) extracted Ti diffusivities (SEBD) in hydrous rhyolite11, trondhjemite, and rhyolite6 (an intermediate mixture between rhyolite11 and trondhjemite) melts during rutile dissolution, with H2O content by difference-from-100% method. Chen and Zhang (2008, 2009) determined Ti diffusivities (SEBD) in basalt11 melt during olivine and clinopyroxene dissolution. As discussed in Zhang et al. (1989) and Zhang (2008, 2010), SEBD values in melts from crystal dissolution experiments are more self-consistent (meaning cross-terms in multicomponent diffusion play a smaller role) if the component is the major component in the crystal that determines the crystal-melt equilibrium. Hence, it is expected that SEBD values of Ti during rutile dissolution (Zhang et al. 1989; Hayden and Watson 2007) are more reliable and consistent than other experimental data. On the other hand, Ti SEBD values in melts obtained from olivine and clinopyroxene dissolution (Van Orman and Grove 2000; Morgan et al. 2006; Chen and Zhang 2008, 2009) are affected or even dominated by cross diffusion terms (sometimes even showing uphill diffusion as during olivine dissolution in andesite melt; Zhang et al. 1989) and are hence less consistent from one condition to another. Nonetheless, as data are compared in Figure 52, the SEBD data in basalt melts are roughly consistent with self diffusivities in HB1 melt. From HB1 to mid-ocean ridge basalt to lunar picrites to basalt7basanite couple during dissolution of various minerals, Ti self diffusivities and SEBD vary by no more than a factor of 5. The scatter in Ti SEBD in hydrous rhyolite to trondhjemite with Zhang (Ch 8) Diffusion silicateis melts 5 todata 12inwt% also a factor of 5 (Hayden and Watson 2007). It is not unspecified H2etOal.between

Figure 52. dryHB1 HB1 0.1(HB, MPa (HB, LaTourrette al. 1996), dry olivine basalt11 melt Fig.Ti52.diffusivities Ti diffusivitiesin in dry meltmelt at 0.1at MPa LaTourrette et al. 1996), dryet basalt11 melt during during olivine dissolution at 0.5-1.4 GPa (Chen and Zhang 2008), dry basalt11 melt during clinopyroxene dissolution at 0.5-1.4 (Chen and Zhang basalt11 melt during at 0.5-1.9 dissolution at 0.5-1.9 GPaGPa (MORB2, Chen2008), and dry Zhang 2009), dry clinopyroxene LB1 melt dissolution during clinopyroxene and anorthite dissolution Orman Grove 2000; Morgan et al. 2006), dry b-b (basalt7-basanite) GPa (MORB2, (Van Chen and Zhang and 2009), dry LB1 melt during clinopyroxene and anorthite dissolution (Van Orman couple (Lundstrom 2003), dry andesite1 melt during rutile dissolution (Zhang et al. 1989), hydrous rhyolite to and Grove al. 2006), b-b (basalt7-basanite) couple (rhy-tron, (Lundstrom 2003), dry andesite1 melt 2007), and trondhjemite melts2000; withMorgan 5-12 etwt% H2Odry during rutile dissolution Hayden and Watson dry HR7, during wet HR7 wt%(Zhang H2O)etand dry HR7+Na melts (Mungall et al.with 1999). rutile (3.6 dissolution al. 1989), hydrous rhyolite to trondhjemite melts 5-12 wt% H2O during rutile dissolution (rhy-tron, Hayden and Watson 2007), and dry HR7, wet HR7 (3.6 wt% H2O) and dry HR7+Na melts (Mungall et al. 1999).

378

Zhang, Ni, Chen

clear whether this is due to insensitivity of Ti diffusivity with respect to H2O at such high H2O, or due to uncertainties in the H2O content. Ti diffusivities in some melts are given below: (24600 ± 3800)   DTidrySDHB1; 0.1 MPa = exp ( -10.08 ± 2.23)  T  

(74)

(39776 ± 4960)   DTidryTDHR7 = exp ( -9.16 ± 2.99)  T  

(75)

Ti SEBD values in basalt11 melt during clinopyroxene dissolution is about 1.6 times, and those during olivine dissolution is about 4 times the self diffusivities in dry HB1 (Eq. 52a). Ti SEBD in dry andesite1 melt during rutile dissolution at 1.5 GPa is 0.6 times the diffusivity from Equation (74). Adding 3.6 wt% H2O or 20 wt% Na2O to HR7 melt increases Ti diffusivity by roughly the same amount. Ti diffusivity increases from HR7 (a haplorhyolite) melt to basalt melt by about 4 orders of magnitude. The effect of 5-12 wt% H2O on Ti duffusivity in silicic melts is greater than that in mafic melts. Zr diffusion. Nine papers reported Zr diffusion data in silicate melts (100 points). Harrison and Watson (1983) extracted SEBD (close to FEBD and tracer diffusivities) of Zr in rhyolite12 melt (Table 1) during zircon dissolution at 1473-1673 K, 0.8 GPa and 0.1-6.3 wt% H2O. Baker and Watson (1988) determined SEBD (close to FEBD) of Zr using the rhyolite1rhyolite7 diffusion couple at 1171-1673 K and 0.01-1 GPa. Baker et al. (2002) studied SEBD (close to FEBD) of Zr in rhyolite14 melt and two other rhyolite melts with slightly different concentrations of FeO (difference of 1.2 wt%), F (difference of 1.2 wt%) and Cl (difference of 0.35 wt%) during zircon dissolution at 1323-1673 K, 1 GPa, and ≤ 5.1 wt% H2O. No difference was found in Zr diffusivity among the three rhyolite melts used by Baker et al. (2002). LaTourrette et al. (1996) investigated Zr self diffusion in dry HB1 melt at 1623-1773 K and in air. Nakamura and Kushiro (1998) characterized Zr tracer diffusion in jadeite melt at 1673 K and 1.25-2 GPa. Mungall et al. (1999) examined Zr tracer diffusion in dry HR7, wet (3.6 wt% H2O) HR7, and dry HR7 + Na melt (Table 1) at 1083-1873 K, 0.0001 and 1 GPa. Koepke and Behrens (2001) measured Zr tracer diffusivities in wet HA1 melt with about 4.55.2 wt% H2O at 1373-1673 K and 1 GPa and one datum in dry HA1 melt. Lundstrom (2003) reported a single SEBD value of Zr from a dry basalt7-basanite diffusion couple experiment at 1723 K and 0.9 GPa. Behrens and Hahn (2009) investigated Zr tracer diffusion in dry and wet trachyte1 and phonolite1 melts at 1323-1528 K and 1 GPa. Zr diffusion in rhyolite melts has been investigated extensively (Harrison and Watson 1983; Baker and Watson 1988; Baker et al. 2002) and hence inter-laboratory comparisons can be made. The data for rhyolite melts are shown in Figure 53. Some data are difficult to reconcile. For example, the huge pressure effect from 0.01 to 1 GPa in the Zr diffusion data of Baker and Watson (1988) (solid squares and open triangles) are difficult to explain. In dry rhyolite melts of similar compositions, Zr diffusivities in samples containing < 0.7 wt% halogens (Baker and Watson 1988) are about 2 orders of magnitude higher than those obtained by Harrison and Watson (1983). Baker and Watson (1988) attributed the difference to the effect of halogens, but a later study (Baker et al. 2002) indicated that adding even 1.2 wt% F has only a minor effect on Zr diffusivities. Assuming that the more recent paper by the same lead author is more likely correct, it appears that the data in Baker and Watson (1988) are erroneous, and the effect of F and Cl at the level of ≤ 1 wt% is minor on Zr diffusion. For zircon dissolution in wet rhyolite melts (solid triangles and circles with center dot), there is no consistency either: Zr diffusivities at 6.0 wt% H2O (Harrison and Watson 1983) are smaller than those at 4.2-4.8 wt% H2O (Baker et al. 2002). Two possible explanations can be advanced although it is difficult to judge which data are more reliable. First, there might be convection in the zircon dissolution experiments (Harrison and Watson 1983; Baker et al. 2002), which could have compromised the diffusion data (see Zhang et al. 1989), although the solubility of zircon

Diffusion Data in Siliate Melts

Zhang et al. (Ch 8) Diffusion data in silicate melts

379

Fig. Zr 53. Zr diffusivities inin rhyolite (rhy) (rhy) and HPG melts. the legend indicates of H2O. Data Figure 53. diffusivities rhyolite and HPGPercentage melts. in Percentage in thewt% legend indicates wt% of H2O. Data sources: HW83 (Harrison and Watson 1983); BW88 (Baker and Watson 1988); B02 (Baker et (Harrison and Watson 1983); BW88 (Baker and Watson 1988); B02 (Baker et al. 2002); M99 al. 2002);sources: M99 HW83 (Mungall et al. 1999). (Mungall et al. 1999).

is small and hence less likely to cause convection. Secondly, in earlier studies (Harrison and Watson 1983), uncertainties in H2O concentration were large. Due to these inconsistencies, at the moment, we cannot confidently assess Zr diffusivities in rhyolite melts even with so many data. More experimental data minimizing convection and with accurate determination of H2O contents are necessary. Tracer diffusivities of Zr in dry and wet (3.6 wt% H2O) haplorhyolite (HR7) melt (Mungall et al. 1999) are also shown in Figure 53 for comparison with natural rhyolite melts. Zr diffusivities in dry HR7 are lower than those in dry natural rhyolite melts. Zr diffusivities in wet HR7 melt containing 3.6 wt% H2O seem to be more consistent with the data of Baker et al. (2002) than with the data of Harrison and Watson (1983), considering that diffusivities are expected to increase with H2O contents. Zr self diffusivities in dry HB1 melt, tracer diffusivities in dry and wet HA1, phonolite1 and trachyte1 melts are shown in Figure 54. Based on Figures 53 and 54 and the original papers, Zr diffusivities in some systems are as53 follows: (42366 ± 6397)   DZrdryTDHR7 = exp ( -7.55 ± 3.87)  T  

(76)

(25865 ± 6299)   DZrdrySDHB1 = exp ( -9.82 ± 3.70)  T  

(77)

(22736 ± 547)   DZrwetTDHA1; 5% = exp ( -11.18 ± 0.36)  T  

(78)

where % in the superscript indicates wt% of H2O. Hf diffusion. Three papers reported Hf diffusion data in silicate melts (31 points), all on tracer diffusion. Mungall et al. (1999) investigated Hf diffusion in dry HR7, wet HR7 (3.6 wt%

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Zhang et al. (Ch 8) Diffusion data in silicate melts

Figure 54.Fig.Zr54.self andand tracer Data sources: HB1 (LaTourrette al.(Koepke 1996);and HA1 (Koepke and Zr self tracer diffusivities. diffusivities. Data sources: HB1 (LaTourrette et al. 1996);et HA1 Behrens Behrens 2001); tra1 (trachyte1) and pho1 (phonolite1) (Behrens and Hahn 2009); jadeite and diopside 2001); tra1 (trachyte1) and pho1 (phonolite1) (Behrens and Hahn 2009); jadeite and diopside melts (Nakamura and melts (Nakamura and Kushiro 1998); dry and wet HR7 and HR7 + Na (Mungall et al. 1999). Kushiro 1998); dry and wet HR7 and HR7+Na (Mungall et al. 1999).

H2O) and dry HR7 + Na melts (Table 1). Koepke and Behrens (2001) examined Hf diffusion in a wet haploandesite melt (4.5-5.2 wt% H2O). Behrens and Hahn (2009) studied Hf in dry and wet trachyte1 and phonolite1 melts. The diffusion data are shown in Figure 55. For dry melts, Hf diffusivities increase from HR7 to trachyte and phonolite to HR7 + Na. Adding 3.6 wt% H2O to HR7 has roughly the same effect as adding 20 wt% Na2O to HR7. Adding ~1.8 wt% H2O to phonolite melt increases diffusivity by a factor of about 50. Hf diffusivities in some systems are as follows: (42614 ± 13496)   dry HR7 DHf TD = exp ( -7.90 ± 8.14)  T  

(79)

(20586 ± 3891)   wet HA1; 5% DHf = exp ( -12.55 ± 2.57) TD  T  

(80)

where % in the superscript indicates wt% of H254 O. Figure 56 compares Ti, Zr, and Hf diffusion data in four melts. In three of the four melts (dry HB1, dry and wet HR7 melts), Ti, Zr and Hf diffusivities (self and tracer diffusivities) are similar (often within a factor of 3). However, in wet rhyolite melts, Ti and Zr diffusivities (SEBD) differ by more than two orders of magnitude (Hayden and Watson 2007). We tentatively attribute the difference to either the unknown difference in H2O content, or the more variable nature of SEBD values, or some experimental difficulties such as convection in the experiments. Hence, it is tentatively concluded that Ti, Zr and Hf diffusivities are similar to each other in silicate melts.

V, Nb, Ta V diffusion. No V diffusion data in silicate melts are known. Nb diffusion. Six papers reported Nb diffusion data in silicate melts (42 points). Baker and Watson (1988) explored SEBD (close to FEBD) of Nb using diffusion couple of dry rhyolite1

Diffusion Data in Siliate Melts

Zhang et al. (Ch 8) Diffusion data in silicate melts

381

Fig.Hf 55. Hf tracer diffusivities diffusivities in dry HR7, wet dry HR7+Na, dry+  and tra1 HA1, (trachyte1) dry wet tra1 Figure 55. tracer inHR7, drywet HR7, HR7, wet dryHA1, HR7  Na,wetwet dryandand Zhang et al. (Ch 8) Diffusion data in silicate melts (trachyte1) and dry and wet pho1 (phonolite1) melts with wt% of H2O indicated for wet melts. Data and wet pho1 (phonolite1) melts with wt% of H2O indicated for wet melts. Data sources: M99 (Mungall et al. sources: M99 (Mungall et al. 1999); KB01 (Koepke and Behrens 2001); BH09 (Behrens and Hahn 2009). 1999); KB01 (Koepke and Behrens 2001); BH09 (Behrens and Hahn 2009).

55

Figure 56. Comparison Zrand and diffusivities inHB1 (i) dry HB1 melt(ii)atdry 0.1HR7 MPa, HR7 Fig. 56. ComparisonofofTi, Ti, Zr HfHf diffusivities in (i) dry melt at 0.1 MPa, melt (ii) at 0.1dry MPa, wet melt at 0.1 MPa, wet HR7 melt with 3.6 wt% H2O at 1 GPa, and (iv) wet rhy-tron (rhyolite-trondhjemite) melt with HR7 melt with 3.6 wt% H2O at 1 GPa, and (iv) wet rhy-tron (rhyolite-trondhjemite) melt with 5-12 wt% H2O at 1 5-12 wt% H2O at 1 GPa and wet rhyolite melt with 6 wt% H2O at 0.8 GPa. GPa and wet rhyolite melt with 6 wt% H2O at 0.8 GPa.

and rhyolite7 (Table 1) melts at 1171-1673 K and 0.01-1 GPa. Nakamura and Kushiro (1998) examined Nb tracer diffusion in jadeite melt at 1673 K and 1.25-2.0 GPa. Mungall et al. (1999) studied Nb tracer diffusion in dry HR7, wet HR7 (3.6 wt% H2O), and dry HR7 + Na melts. Koepke and Behrens (2001) reported Nb tracer diffusivities in a wet haploandesite melt at 1373-1673 K and 0.5 GPa, plus one datum for dry haploandesite melt. Lundstrom (2003) obtained a single datum of SEBD of Nb between basalt7-basanite melts. Behrens and Hahn

382

Zhang, Ni, Chen

(2009) characterized Nb diffusion in dry and wet trachyte1 and phonolite1 melts at 1323-1528 K, 0.5 GPa, and ≤ 1.9 wt% H2O. Figure 57 shows Nb diffusion data in various melts. Nb diffusivities increase from dry HR7 melt at 0.1 MPa, to dry rhyolite, trachyte and phonolite melts, to wet HA1 melt. The effect of adding 3.6 wt% H2O to HR7 is about the same as adding 20 wt% Na2O in terms increasing Nb diffusivity. All tracer diffusion data of Nb in dry silicate melts can be fit as follows: (71534 XSA + 22351X NK - 28958 - 290 P )   all dry melts DNb = exp ( -10.92 TD  T  

(81)

where XSA and XNK are the cation mole fractions of Si + Al and Na + K, respectively. Equation (81) reproduces all data in dry melts (≤0.1 wt% H2O) to within 0.74 lnD units except for one Zhang et al. (Ch 8)melt). DiffusionEquation data in silicate(81) melts cannot reproduce SEBD data of Nb. clear outlier (trachyte

Figure 57.Fig.Nb dryr-rr-r (rhyolite1-rhyolite7) couple 0.01 andand1Watson GPa (Baker 57.diffusivities Nb diffusivities in in dry (rhyolite1-rhyolite7) couple at 0.01 and 1at GPa (Baker 1988), dryand HR7Watson 1988), dry HR7 melt at 0.1 MPa, wet HR7 with 3.6 wt% H2O at 1 GPa, dry HR7 + Na at 0.1 MPa (Mungall melt at 0.1 MPa, wet HR7 with 3.6 wt% H2O at 1 GPa, dry HR7+Na at 0.1 MPa (Mungall et al. 1999), dry jadeite at et al. 1999), dry jadeite at 1.25-2.0 GPa (Nakamura and Kushiro 1998), dry and wet (4.5-5.2 wt% H2O) HA1 at 0.5 GPa GPa (HA, Koepke Behrens 2001), dry b-b wt% (basalt7-basanite) couple (Lundstrom 2003), (HA, Koepke and 1.25-2.0 (Nakamura andand Kushiro 1998), dry and wet (4.5-5.2 H2O) HA1 at 0.5 GPa and dry and wet (1.1-1.7 wt% H2O) tra1 (trachyte1) and dry and wet (1.7-1.9 wt% H2O) pho (phonolite1) Behrens dry b-band (basalt7-basanite) couple (Lundstrom 2003), and dry and wet (1.1-1.7 wt% H2O) tra1 melts at 0.5 GPa2001), (Behrens Hahn 2009). (trachyte1) and dry and wet (1.7-1.9 wt% H2O) pho (phonolite1) melts at 0.5 GPa (Behrens and Hahn 2009).

Ta diffusion. Only one paper (Mungall et al. 1999) reported Ta tracer diffusion data in silicate melts (7 points). Figure 58 shows the data. Ta diffusivities in HR7 melt (Table 1) can be fit as (Mungall et al. 1999): (33792 ± 4683)   dry HR7 DTa TD = exp ( -12.41 ± 2.84)  T  

(82)

Equation (82) reproduces experimental data to within 0.24 lnD units. Adding 3.6 wt% H2O or 20 wt% Na2O increases DTa by roughly the same factor (3 to 5 orders of magnitude, depending on temperature). The limited data indicate that Ta diffusivities are within a factor 2 of Nb diffusivities but can be either larger or smaller than Nb diffusivities (probably within experimental uncertainties). 57

Zhang et al. (Ch 8) Diffusion data in silicate melts

Diffusion Data in Siliate Melts

383

Fig. 58. Ta 58. diffusivities in dry HPG melt at 0.1 MPa,melt dry HR7+Na 0.1 MPa, wet HR7 (3.6 wt%MPa, H2O) at 1 Figure Ta diffusivities in dry HPG at 0.1 at MPa, dryand HR7 + Na at 0.1 GPa.

and wet HR7 (3.6 wt% H2O) at 1 GPa. Data from Mungall et al. (1999).

Data from Mungall et al. (1999).

Cr, Mo, W Cr diffusion. Three papers reported Cr diffusion data in silicate melts (18 points). Zhang et al. (1989) reported a single Cr SEBD value in andesite1 melt during olivine dissolution at 1563 K and 0.55 GPa. Koepke and Behrens (2001) reported Cr tracer diffusivities in wet HA1 containing 4.5-5.2 wt% H2O at 1373-1673 K and 0.5 GPa, and one single datum at 1673 K and 0.5 GPa for dry HA1. Behrens and Hahn (2009) investigated Cr tracer diffusion in dry and wet trachyte1 and phonolite1 melts at 1373-1528 K, 0.5 GPa and ≤ 1.9 wt% H2O. The data are shown in Figure 59. The single Cr SEBD datum in dry andesite1 melt and the single Cr tracer diffusivity in dry HA1 melt are roughly consistent and lie roughly in the same trend as Cr tracer diffusivities in dry phonolite1 melt. Cr tracer diffusivities in dry trachyte melt are lower than those in dry phonolite melt by a factor of 5. Cr tracer diffusivities in wet HA1 melt containing 4.5-5.2 wt% H2O and those in wet phonolite1 melt containing 1.7 wt% H2O are similar. Cr tracer diffusivities in wet HA1 melt containing 4.5-5.2 wt% H2O at 0.5 GPa can be expressed as: 58

wet HA1; 4.5-5.2% Cr TD

D

(27244 ± 6343)   = exp ( -7.00 ± 4.19)  T  

(83)

where % in the superscript indicates wt% of H2O. Mo diffusion. No Mo diffusion data in silicate melts are known. W diffusion. One paper (Mungall et al. 1999) reported W tracer diffusion data in silicate melts (6 points), on dry HR7 and HR7 + Na melts. The data are shown in Figure 60. The three points on dry HR7 do not line up well in an Arrhenius plot. Diffusivities in HR7 + Na are 3 to 4 orders of magnitude higher. W diffusivities in dry HR7 melts are similar to Nd diffusivities (Fig. 60). However, in HR7 + Na melt, this similarity does not hold any more.

Mn, Fe, Co, Ni, Cu, Zn Mn diffusion. Three papers reported Mn diffusion data in silicate melts (28 points). Lowry et al. (1982) investigated Mn tracer diffusivities in dry basalt1 and andesite2 melts at 1567-

384

Zhang, Ni, Chen

Zhang et al. (Ch 8) Diffusion data in silicate melts

Zhang et al. (Ch 8) Diffusionin inand silicate melts Cr tracer diffusivities indata dry wetwet HA1 melts melts (Koepke(Koepke and Behrens 2001), dry andesite1 et Figure 59.Fig. Cr59. tracer diffusivities dry and HA1 and Behrens 2001),melt dry(Zhang andesite1 melt (Zhang etal. al.1989), 1989), and dry and wet trachyte1 and phonolite1 melts (Behrens and Hahn 2009). and dry and wet trachyte1 and phonolite1 melts (Behrens and Hahn 2009).

59

Fig. 60. W tracer diffusivities dry HR7 and HR7+Na in melts. Nd diffusivities are shownmelts. for comparison. Figure 60. Wintracer diffusivities dry HR7 and HR7 + Na

Nd diffusivities are shown for comparison.

1676 K and 0.1 MPa. Henderson et al. (1985) studied Mn tracer diffusivities in dry dacite4 and pantellerite1 melts at 1473-1673 K and 0.1 MPa. Baker and Watson (1988) investigated Mn SEBD (close to FEBD) in dry rhyolite1-rhyolite7 melts at 1218-1672 K and 0.01 and 1.0 GPa. Figure 61 displays all the data. Except for one outlier point (Fig. 61), all TD and SEBD data of Mn at ≤ 0.02 GPa can be fit by: 23993   all dry melts; ≤ 0.02 GPa DMn = exp 1.15 - 15.44 X1 - 39.92 X 2 TD & SEBD T  

(84)

Zhang et al. (Ch 8) Diffusion data in silicate melts

Diffusion Data in Siliate Melts

385

Figure 61.Fig. Mn diffusivities SEBD) in dryand basalt1 and andesite2 (Lowry et al. 61.diffusivities Mn diffusivities(tracer (tracer diffusivities and and SEBD) in dry basalt1 andesite2 melts (Lowry etmelts al. 1982), dry 1982), dry dacite4 and pantellerite1 melts (Henderson et al. 1985), and dry r-r (rhyolite1-rhyolite7) couple and pantellerite1 melts (Henderson et al. 1985), and dry r-r (rhyolite1-rhyolite7) couple (Baker and Watson (Baker anddacite4 Watson 1988). 1988).

where X1 is the sum of cation mole fractions of Si + Ti + Al + P, and X2 = max(Na + K-Al,0) (i.e., X2 = Na + K-Al if Na + K-Al > 0, and X1 = 0 if Na + K-Al < 0, this is because peralkalinity seems to affect diffusivity more than peraluminity). The available data cannot resolve the compositional dependence of the activation energy. The maximum error in reproducing the experimental data by Equation (84) is 0.37 lnD units (Fig. 61). The Mn SEBD data in rhyolite at 1 GPa (Baker and Watson 1988) deviate from Equation (84) (e.g., the data would indicate a negative activation energy), probably due to experimental or analytical problems. Mn and Co diffusivities are similar in all the melts that have been investigated (basalt, andesite, dacite and pantellerite1). Fe diffusion. Eleven papers reported Fe diffusion data in silicate melts (165 points). In silicate melts, Fe can be present as ferrous (Fe2 + ) or ferric (Fe3 + ). Diffusivities of the two different states are likely significantly different. Hence, it is important to characterize Fe oxidation state before the experiment (e.g., whether the two halves of a diffusion couple have 61 the same ferric/ferrous ratio), control oxygen fugacity during the experiments, and/or measure the oxidation state of Fe after the experiment. Lowry et al. (1982) examined Fe radioactive tracer diffusion in basalt1 and andesite2 melts under oxidized condition at 1570-1675 K and in air. Henderson et al. (1985) studied Fe diffusivities in dacite4 and pantellerite1 melts under oxidized condition at 1475-1672 K and in air. Baker and Watson (1988) investigated Fe SEBD in rhyolite1-rhyolite7 and HD2-rhyolite7 couples (Table 1) at 1171-1273 K and 0.01 GPa and 1373-1473 K and 1 GPa (fO2 not characterized). The pressure effect is insignificant compared to data scatter. Dunn and Ratliffe (1990) investigated tracer diffusion of ferrous iron in a peraluminous sodium aluminosilicate (HR1 in Table 1) at 1516-1716 K and 0.1 MPa using a CO-CO2 gas mixture to control oxygen fugacity to be near the FMQ buffer, and at 1523-1723 K and 0.6-2 GPa in graphite capsules. Watson (1991b) examined Fe transport in slightly melted dunite. Because the extracted diffusivity is not for pure liquid, the data will not be used further due to limitations of the scope of this chapter. Baker and Bossanyi (1994) examined the effect of H2O and fluorine on Fe SEBD in dacite1-rhyolite7 diffusion couples at 1373-1673 K and 1 GPa. van der Laan et al. (1994) reported a single Fe diffusivity

386

Zhang, Ni, Chen

(SEBD) in a rhyolite3-rhyolite16 couple at 1523 K and 1 GPa. Koepke and Behrens (2001) obtained Fe tracer diffusivities in hydrous HA1 melt at 1373-1583 K and 0.5 GPa with oxygen fugacity roughly at NNO + 3. Lundstrom (2003) obtained two Fe SEBD values in basalt7basanite couple. Ruessel and Wiedenroth (2004) characterized the compositional effect on Fe tracer diffusivities in Na2O-CaO-MgO-Al2O3-SiO2 melts (Table 1) in air and 1300-1873 K, but diffusivity values were only reported at 1573 K. Morgan et al. (2006) determined Fe SEBD in lunar picrites (LP1 and LP2) during anorthite dissolution at 1673 K and 0.6 GPa. Due to the presence of both Fe2 +  and Fe3 +  in typical geological conditions, with Fe3 +  likely having lower diffusivities (due to higher bond strength than Fe2 + ), Fe diffusivity is expected to lie between that of Fe2 +  and Fe3 + . If oxygen fugacity is uniform in the melt, equilibrium between Fe2 +  and Fe3 +  would lead to uniform ferric to ferrous ratio. Following Zhang et al. (1991b), Fe diffusivities can be expressed as follows: = DFe X Fe2+ DFe2+ + X Fe3 + DFe3 +

(85)

where XFe2 +  = Fe2 + /(Fe2 +  + Fe3 + ). Based on valence and size consideration, it can be argued that Fe2 +  diffusivities lie between that of Mn2 +  and Co2 +  (ionic radii of Mn2 + , Fe2 +  and Co2 +  in octahedral sites at high spin are 0.083, 0.078 and 0.0745 nm, Shannon 1976). If Fe3 +  is in the tetrahedral site, Fe3 +  diffusivity is likely similar to that of Ga3 +  (ionic radii of Fe3 +  and Ga3 +  in tetrahedral sites are 0.049 and 0.047 nm). If Fe3 +  is in the octahedral sites, Fe3 +  diffusivity is likely similar to that of Cr3 + , or V3 + , or Ga3 +  (ionic radii of Fe3 + , Cr3 + , V3 + , and Ga3 +  in octahedral sites are 0.0645, 0.0615, 0.062, and 0.64 nm). In experiments and nature, Mn and Co are divalent, and Ga and Cr are trivalent. Hence for Mn, Co, Cr and Ga, we do not need to worry about the oxidation state. Furthermore, Mn and Co diffusivities are similar (Fig. 62). Hence, Fe diffusivities in melts are expected to lie between those of Mn and Ga, or of Co and Ga; Figure 62 is consistent with this expectation. Furthermore, Figure 62 shows that Fe2 +  diffusivity is about 6 times Fe3 +  diffusivity in dacite melts. Oxygen fugacity was not always controlled in experimental studies. For example, no care was taken to control oxygen fugacity in the experiments of Baker and Watson (1988), Baker and Bossanyi (1994) and van der Laan et al. (1994) because the main goal of their studies was not on Fe diffusion. Some Fe diffusion data are shown in Figure 63. Lowry et al. (1982) and Henderson et al. (1985) conducted the tracer diffusion experiments in air. There might still be some uncertainty about the oxidation state of the diffusion experiments because the initial glass was not equilibrated in air before diffusion experiments. Nonetheless, Fe diffusivities are significantly lower (by a factor of about 3) than both Mn and Co diffusivities, and only slightly higher than Ga diffusivities (Fig. 62), consistent with the expectation that Fe is mostly trivalent in these experiments. Fe diffusivities (TD) in metaluminous basalt1-andesite2-dacite4 melts by Lowry et al. (1982) and Henderson et al. (1985) in air can be expressed as: 29879   basalt-andesite-dacite; air DFe = exp 5.30 - 17.64 X1 TD T  

(86)

where X1 is the sum of cation mole fractions of Si + Ti + Al + P. The available data cannot resolve the compositional dependence of the activation energy (about 248 kJ/mol). The maximum error in reproducing the experimental data by Equation (86) is 0.42 lnD units (Fig. 63). Dunn and Ratliffe (1990) studied Fe2 +  diffusion in HR1 melt (a haplorhyolite melt) by using reducing conditions at 0.0001 and 0.6-2 GPa so that Fe2 +  is the dominant iron species. They found that the pressure effect is insignificant from 0.6 to 2 GPa (diffusivity variation is within 0.3 lnD units). However, Fe tracer diffusivity values at 0.6-2 GPa using graphite capsules is 5 times those at 0.1 MPa at FMQ buffer. Dunn and Ratliffe (1990) attributed the

Zhang et al. (Ch 8) Diffusion data in silicate melts

Diffusion Data in Siliate Melts

387

Zhang et (Ch 8)diffusivities Diffusion data in silicate Fig. 62.al.Tracer Mn and Co inin dacite4 melt melt (Henderson et al. 1985), of Ga inthose dacite1ofmelt Figure 62. Tracer diffusivities ofofMn and Comelts dacite4 (Henderson et those al. 1985), Ga in dacite1 melt (Baker(Baker 1992a) with those Fe in dacite4 in airet(Henderson et al. 1985). 1992)compared compared with those of Feof in dacite4 melt in airmelt (Henderson al. 1985).

62

Figure 63. experimental dataonon Fe diffusion in silicate melts. Data forand drydrybasalt1 and dryareandesite2 Fig.Some 63. Some experimental data Fe diffusion in silicate melts. Data for dry basalt1 andesite2 melts melts are from Lowry et al. (1982); dry dacite4 and pantellerite1 melts are from Henderson et al. (1985); from Lowry et al. (1982); dry dacite4 and pantellerite1 melts are from Henderson et al. (1985); dry HR1 at 0.1 MPa dry HR1 at 0.1 MPa and 0.6-2 GPa are from Dunn and Ratliffe (1990); and wet HA1 are from Koepke and Behrens and (2001). 0.6-2 GPa are from Dunn and Ratliffe (1990); and wet HA1 are from Koepke and Behrens (2001).

difference to a sudden jump from 0.0001 to 0.6 GPa and then no additional pressure effect above 0.6 GPa, a somewhat ad hoc explanation. More likely, the sudden jump is due to experimental difficulties. The oxygen fugacity in the experiments of Koepke and Behrens (2001) using hydrous haploandesite (4.9-5.2 wt% H2O) was estimated to be about NNO + 3, at which the ferric to ferrous ratio is roughly 1. Hence, these data are roughly the average Fe2 +  and Fe3 +  diffusivities

Zhang, Ni, Chen

388 and can be expressed as:

(24421 ± 10053)   wet HA1; 0.5 MPa; NNO+3; 4.9-5.2% DFe = exp ( -7.83 ± 6.84) TD  T  

(87)

where % in the superscript indicates wt% of H2O. The maximum error in reproducing the three experimental data by Equation (87) is 0.28 lnD units. Ruessel and Wiedenroth (2004) produced extensive Fe tracer diffusion data in Na2OCaO-MgO-Al2O3-SiO2 melts at 1300-1873 K. The experiments were implicitly conducted in air, though it was not clearly specified. Hence, the diffusivities are close to Fe3 +  diffusivities. Ruessel and Wiedenroth (2004) reported activation energy and pre-exponential factors for melts with various compositions, but only reported diffusivity values at 1573 K. The activation energy varies from 209 to 284 kJ/mol as the melt composition varies. Due to errors in extracted activation energies and pre-exponential factors, it is easier to treat original diffusion data than to model how activation energies and pre-exponential factors depend on melt composition. The available data of Fe diffusion, though numerous, are not systematic enough to understand both Fe2 +  and Fe3 +  diffusion. In order to better understand Fe diffusion, it is necessary to conduct Fe diffusion studies under a range of oxygen fugacities so that diffusivities of both Fe2 +  and Fe3 +  and their dependence on temperature, pressure and composition can be extracted. Then Fe diffusivity at any given fO2 (or given ferric/ferrous ratio) can be inferred using Equation (85). Co diffusion. Three papers reported Co diffusion data in silicate melts (29 points). Hofmann and Magaritz (1977) studied Co tracer diffusion in dry basalt10 melt (Table 1) at 1553-1713 K and 0.1 MPa. Lowry et al. (1982) investigated Co tracer diffusion in dry basalt1 and andesite2 melts at 1567-1672 K and 0.1 MPa. Henderson et al. (1985) studied Co tracer diffusion in dry dacite4 and pantellerite1 melts at 1473-1672 K and 0.1 MPa. No EBD data of Co are available. Co tracer diffusion data in two different dry basalts (basalt1 by Lowry et al. 1981 and basalt10 by Hofmann and Magaritz 1977) are in good agreement (Fig. 64) and can be fit as follows: (20211 ± 3706)   dry basalts; 0.1 MPa DCo =exp  -(11.05 ± 2.29) TD  T  

(88)

The maximum error in reproducing the experimental data by Equation (88) is 0.16 lnD units. The activation energy is 168±31 kJ/mol. Co tracer diffusivity increases from pantellerite1 to dacite4 to andesite2 to basalt1 melts; the increase from pantellerite to dacite is a little strange because increasing alkalinity is usually thought to cause an increase in diffusivity. All diffusion data are shown in Figure 64 and can be roughly fit by the following equation: (4352 + 25116 X1 )   dry melts; 0.1 MPa DCo =exp  -11.02 - 42.07 X 2 TD  T  

(89)

where X1 is the sum of cation mole fractions of Si + Ti + Al + P, and X2 = max(Na + K-Al,0) (i.e., X2 = Na + K-Al if Na + K-Al>0, and X1 = 0 if Na + K-Al 10. For the alkalis, Mungall (2002) fit the diffusivity empirically as a function of ionic radius, and two compositional parameters (in addition to the temperature dependence): lnDi = -16.16 +

42.6(ri - 1.03)2 + 6.63Y1 - [1239Y2 + 27424(ri - 1.03)2 + 6975] T

(93)

where ri is in Å, Y1 = M/O = [FeO  +  MnO  +  69MgO  +  CaO  +  2(Na2O  +  K2O  +  H2O)]/ [2(SiO2  +  TiO2)  +  3Al2O3  +  FeO  +  MnO  +  MgO  +  CaO  +  Na2O  +  K2O  +  H2O  +  5P2O5], and Y2 = Al/(Na  +  K  +  H), where Al, Na, K, and H are cation mole fractions, and SiO2, TiO2, Al2O3, FeO, MnO, MgO, CaO, Na2O, K2O, H2O and P2O5 are oxide mole fractions. Note that Equation (93) is modified from that in Mungall (2002) in two aspects (i) Mungall (2002) used cm2/s as the diffusivity unit, but we use m2/s; and (ii) Mungall used the base-10 logarithm and we use natural logarithm for consistency with other parts of this chapter. The use of Al/ (Na + K + H) as a parameter means that the melt must contain significant Na, K and H. For IFSE, the diffusivity is empirically related to viscosity η (in Pa·s), temperature, zi2/r, and Y1(M/O) by Mungall (2002):  zi 2   - 12.42  ( 3730Y1 - 2102 ) r 0.564 zi  Di η   -37.8 + + ln  = r T  T  2

(94)

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For HFSE, the diffusivity is empirically related to viscosity, temperature and M/O by Mungall (2002):  Dη -29.2 - 6.61Y1 ln  i  =  T 

(95)

That is, all HFSE have the same diffusivities according to this model. The model of Mungall (2002) has been tested by Behrens and Hahn (2009) using directly measured diffusivity and viscosity values, and the model-predicted diffusivities are 0 to 1.3 logD (0 to 3 lnD) units greater than the experimental values. To further test the model of Mungall et al. (2002), we use our compiled self and tracer diffusivities and FEBD values to compare with calculated diffusivities using the model of Mungall (2002), focusing on silicic melts. The viscosity is calculated from the general model of Hui and Zhang (2007), which has been shown to work well for silicic melts (Wang et al. 2009). We avoid the multi-valent elements so that the field strength can be calculated accurately. To be consistent with Mungall (2002), we used the same radii sent to us by J. Mungall (personal communication), which are crystal radii of Shannon 1976 with coordination number of 6 and high spin, except for the following: coordination number is 8 for Sr2 + , Ba2 + , Nd3 + , and Tb3 + , 4 for B3 + , Si4 + , and Ga3 + , and 3 for Be2 + ; and low spin for Mn2 +  and Co2 + . Figure 70 compares calculated diffusivities with experimental diffusivity values for various silicic melts. It can be seen that the scatter is quite large, as much as 2.8 logD units (6.4 lnD units). The following diffusivity data are off by more than 1.3 log units (a factor of 20): Sr and Ba tracer diffusion data in rhyolite15 melt by Magaritz and Hofmann (1978) and Ce diffusion data in rhyolite15 by Jambon (1982) (partially this is because we adopted the corrected H2O concentration for this rhyolite melt from 0.38 wt% to 0.11 wt% by Jambon et al. 1992); B diffusion data in HR7 melt by Chakraborty et al. (1993), and U, Th and Be diffusion data in HR7 melt by Mungall et al. (1997, 1999). For viscosity of rhyolie15 and HR7, we used the specific viscosity model for rhyolites (Zhang et al. 2003; Hui et al. 2009) and confirmed that Zhang et al. (Ch 8) Diffusion data in silicate melts the large difference between calculated and experimental diffusivities are not due to the choice




Fig. 70.

Figure 70. Comparison of calculated diffusivity based on the model of Comparison of calculated diffusivity based on thediffusivity model of Mungall (2002) andaxis). experimental diffusivity Mungall (2002) and experimental (horizontal

(horizontal axis).

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of the viscosity model of Hui and Zhang (2007). Although some of these discrepancies can be explained away (see Mungall 2002) and some are because Mungall (2002) overlooked a corrected H2O concentration (more than 10 years after the original publications), we concur with Behrens and Hahn (2009) that the empirical model of Mungall (2002) can be used as an order of magnitude estimate for tracer diffusivities, but not accurate enough for practical applications.

Effect of ionic size on diffusivities of isovalent ions For alkali elements or univalent cations (Fig. 8), the diffusivities increase from Cs to Rb to K and to Na in rhyolite and albite melts. That is, the diffusivity increases as the ionic size decreases. But the trend is not perfect: from Na to Li, the diffusivity may increase or decrease or stay the same. For alkali earth elements or divalent cations (Fig. 15), the diffusivities increase from Be to Mg to Ca and to Sr in HR7 and HR7 + Na melts. That is, the diffusivity decreases as the ionic size decreases in these melts. Again, the trend is not perfect: from Sr to Ba, the diffusivities may increase or decrease or stay the same. For trivalent REE species, the diffusivities increase smoothly from La to Lu in melts (Fig. 51). The smoothness makes the REE data excellent references when comparing with diffusivities of other elements. In Figure 71, we examine whether diffusivities continue to decrease to smaller trivalent cations, including B3 +  (the smallest trivalent cation) and Cr3 + . Figure 71 shows that the trend does not continue: at some ionic radius smaller than that of Lu3 + , trivalent cation diffusivities do not decrease any more with decreasing ionic radius. In Figure 72, we compare tetravalent cation diffusivities to examine the size effect for this group in HR7 melt. The ionic radii of Si4 + , Ge4 + , and Ti4 +  in tetrahedral sites increase from 0.026, to 0.039 to 0.042 nm (Shannon 1976). Those of Ti4 + , Hf4 + , Zr4 + , U4 + , and Th4 +  in octahedral sites increase from 0.0605, 0.071, 0.072, 0.089, and 0.094 nm (Shannon 1976). Because Si diffusivity is not available in HR7 melt, the Eyring diffusivity is shown in Figure 72 as the proxy for Si diffusivity. The differences among the diffusivities of these tetravalent cations are small, but there is minor increase from diffusivity of Si (Eyring) to Ge≈Th to Hf (except for an outlier with low diffusivity) to Ti≈Zr≈U. Based on the experimental data discussed above, for alkali elements and halogen elements (Fig. 35), the diffusivities of each isovalent series decrease as the size of the ion increases, and the activation energy increases as the size increases. In a similar fashion, diffusivities of noble gas elements and other neutral molecules also decrease as the size of the neutral atom or molecule increases (Zhang and Xu 1995; Behrens 2010, this volume). On the other hand, for divalent, trivalent and tetravalent cations, the trend is mostly opposite: diffusivities of isovalent ions increase as the size of the ion increases. The observations might be puzzling or even counter-intuitive to some and appear to be unexplained previously. We attribute these opposite trends to the interplay of bond strength and ionic size as follows. Diffusion requires a species to detach from the original site (breaking the bonds between the species and surrounding particles), and then move (or jump) from the old site to a new site, meaning to squeeze through an orifice formed by neighboring ions to arrive at a new site. That is, detachment and going through some orifice are two necessary and sequential steps. Hence, the slowest step determines the overall diffusion rate. The slowest step depends on both the size and ionic valences as follows: (1) If the particle is bonded weakly to other particles, e.g., for neutral atoms and molecules (0-valence “ions”), or for univalent cations and anions, the first step (detachment) is easy and the second step (going through an orifice) is rate limiting. Hence, a smaller particle diffuses more rapidly because it is easier to get through an orifice. (2) If the particle is bonded strongly to other particles, meaning higher valence ions

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Figure 71. Trivalent cation diffusivities. Data sources: HR7 (Mungall et al. 1999); HA1 (Koepke and Fig. 71. Trivalent cation diffusivities. Data sources: HR7 (Mungall et al. 1999); HA1 (Koepke and Behrens 200 Behrens 2001). The Eyring diffusivity is calculated from viscosity data of Hess et al. (1995) on HR7 melt. The Eyring diffusivity is calculated from viscosity data of Hess et al. (1995) on HR7 melt.

(divalent and higher valence), the first step (detaching from other particles) is rate determining. Hence, ions with weaker bonds, meaning low charge to radius ratio, i.e., larger ions in an isovalent series, detach more easily and diffuse more rapidly. The above explanation can be pushed further to explain why Li and Na diffusivities are similar rather than following the trend of increasing diffusivity with decreasing size from Cs to Rb to K to Na. Li +  is small so that the bond strength is fairly high, meaning that from Na +  to Li + , even though the size becomes smaller leading to high diffusivity, the bond strength increases leading to lower diffusivity, with the net effect being 71 similar diffusivity between Li +  and Na + . Similarly, the fact that Ba diffusivity sometimes does not continue the trend of Be to Mg to Ca to Sr for which the diffusivity increases as the bond strength decreases (or size

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Figure cation diffusivities. Diffusivity data are from Mungall al. (1999). The Fig.72. 72. Tetravalent Tetravalent cation diffusivities. Diffusivity data are from Mungall et al. (1999). TheetEyring diffusivity is Eyring diffusivity is calculated from actual viscosity data of Hess et al. (1995) on HR7 melt. calculated from actual viscosity data of Hess et al. (1995) on HR7 melt.

increases) can also be explained: Ba is so large that the bond strength is weak so that the size effect is also significant, leading to Ba diffusivity similar to or even smaller than Sr diffusivity. Exactly when the bond strength effect or the size effect would be more important is expected to depend on the melt composition and structure. In more densely packed melts (mafic or depolymerized melts), there is less free space (low ionic porosity) for particles to move around. Hence, the size effect is expected to be more important than the bond strength effect, so that larger ions would diffuse more slowly. In less densely packed melts (felsic or polymerized melts), more free space is available so that the bond strength effect is expected to be more important, meaning that smaller ions with greater bond strength tend to diffuse more slowly. Hence, the exact ionic size for the trend of diffusivity versus ionic radius to reverse in an isovalent series depends on the melt composition, as shown below in the discussion of diffusivity sequence. Another complexity is that the diffusivities of different species depend on temperature, and above a certain temperature (the compensation temperature), the trend would be reversed. 72 to the activation energy, which is expected Hence, the above explanation may be better applied to increase with the size of the diffusing species for neutral and univalent ions, but to decrease with size (or increase with bond strength) for ions with valence of two or more. Data in Figure 8 support this hypothesis, but the activation energy does not vary much from Mg to Ca to Sr in Figure 15. It is possible that greater temperature range is necessary to resolve the difference in the activation energy.

Dependence of diffusivities on melt composition For noble gas and alkali elements, when the particle size is small, such as He (Jambon and Shelby 1980), Ne (Frank et al. 1961; Perkins and Beagal 1971; Matsuda et al. 1989), Li (Fig. 1), and Na (Fig. 3), the diffusivity (tracer diffusivities and FEBD) increases from depolymerized melts (which are densely packed) to polymerized melts (which are less densely packed), meaning available free space is more important for the diffusion of these elements. The compositional dependence of He, Ne, Li and Na diffusivities means that they increase as melt viscosity increases, opposite to the famous diffusivity-viscosity relations that diffusivity is inversely related to viscosity (e.g., the Einstein relation and Eyring relation). As the particle

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size increases, tracer diffusivity values of K and Rb (Figs. 4 and 6) do not vary strongly with melt composition, and Ar tracer diffusivities (Zhang et al. 2007) increase slightly from polymerized to depolymerized melts. For other elements, especially the divalent and higher valence ions, the diffusivity increases from polymerized to depolymerized melts, suggesting that flexibility (or rigidity) of the structure plays a greater role in controlling diffusion of these elements. The dependence on the melt composition and structure can again be explained by the two necessary steps for diffusion (detachment and squeezing through an orifice). For small and weakly bonded species (such as He, Ne, Li and Na), detaching from the original site is easy. As long as the structure provides large holes (i.e., high ionic porosity), these species can move around easily. However, when the structure is dense (i.e., low ionic porosity), then these species cannot move so rapidly. Hence, for these species, the diffusivity increases as the ionic porosity increases. For silicate melts, ionic porosity increases from depolymerized (also low-viscosity) melts to polymerized (also high-viscosity) melts, leading to the observed positive dependence of diffusivity of these species on melt viscosity. On the other hand, for large and weakly bonded species (such as Ar, Kr, Xe, and Cs), the size of the orifices in all melts, even for the melt with high ionic porosity, are not large enough. Hence, the species must push particles away to expand an orifice significantly so that it can move through the orifice. If the melt structure is more rigid (polymerized melt), it is more difficult to push aside particles, meaning diffusion rate would decrease from depolymerized melt to polymerized melt. For higher valence ions, the bonds between the diffusing species and other particles are strong. It is more difficult to break the bonds if the melt structure is more rigid (polymerized melt). Hence, diffusivity also decreases from depolymerized melt to polymerized melt. Experimental data show that adding water always increases the diffusivity of an element without exception. The magnitude of increase is greater for elements with low diffusivities. Dissolved H2O in a melt results in at least two effects to increase diffusivity: one is to increase the ionic porosity so that the melt is less densely packed (so that diffusivity would increase even for small noble gases and alkalis), and the other is to depolymerize the melt so that the viscosity is decreased and the structure becomes less rigid. Because both effects would increase the diffusivity, adding water always increases the diffusivity.

Diffusivity sequence in various melts For a given melt, we rank the elements by their diffusivity and define the ranking as a diffusivity sequence (by analogy to the incompatible element sequence in presenting data in “spidergrams”). Such a sequence is useful in rough estimation of diffusion behavior in a silicate melt. The concept of diffusivity sequence in a given melt is best applied to self diffusivity, tracer diffusivity and FEBD because they are more closely related to the intrinsic mobility, whereas binary interdiffusivity and SEBD depend on the concentration gradient of other components (e.g., binary interdiffusivity depends on which component is the exchanging component). The sequence may not be universal because diffusing species may change as melt composition or oxygen fugacity or other conditions change. Furthermore, the sequence is for diffusion below the compensation temperature. If there were a single perfect compensation temperature, the diffusivity sequence would become just the opposite above the compensation temperature. Because experimental data often show that there is no perfect compensation temperature, at high temperatures such as above 1700 K, the diffusivity sequence could be irregular. The following sequences do not include multi-valent elements such as Fe and Eu because their diffusivities depend on the oxygen fugacity, which would change their position in the diffusivity sequence. Diffusivity sequence in rhyolite and related melts. Self and tracer diffusivities in dry rhyolite melt and HR7 (haplorhyolite) melt are shown in Figure 73.

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Figure 73. Diffusivities in dry HR7 and rhyolite melts. Data for HR7 melts are from Mungall and Dingwell (1997) andFig. Mungall et al. (1999). in Data rhyolite are melts. from Jambon (1978), (1982), and Ding 73. Diffusivities dryforHR7 and melts rhyolite Data and for Semet HR7 melts areJambon from Mungall Harrison and Watson (1983, 1984), Blank (1993), Lesher et al. (1996), and Bai and Koster van Groos (1994). and Mungall et al. (1999). Data for rhyolite melts are from Jambon and Semet (1978), Jambon (198 Watson and (1983, 1984), Blankdiffusivities (1993), Lesher al. (1996), and Bai Van (1994). In dryand rhyolite5 albite melts, ofetalkali elements canandbeKoster ranked asGroos follows (Fig. 8):

Cs < Rb < K < Na ≈ Li In dry HR7 melt (a haplorhyolite melt) (Mungall et al. 1999), the diffusivities of alkali earth elements can be ranked as follows (Fig. 15a): 73 Be < Mg < Ca < Sr ≈ Ba

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Synthesizing the data in Mungall et al. (1999) and Figure 73 on HR7 and rhyolite5 melts as well as other information, we obtain (Fig. 73): P ≈ Si ≈ Th ≈ Ge ≈ Ti ≈ Hf ≈ U ≈ Zr < REE ≈ Y ≈ Nb ≈ Ta ≈ B ≈ Be < Mg < Ca < Sr ≈ Cs ≈ Ba < F ≈ Sb ≈ Rb ≈ Cl ≈ CO2 ≈ Ar < K < Na ≈ Li where REE means trivalent rare earth elements. Eu2 +  probably diffuses at a rate similar to Sr, and Ce4 +  probably diffuses at a rate similar to U4 + . U in the above sequence could be either U4 +  or U6 +  because the diffusivity difference between the two is small (LaTourrette and Wasserburg 1997). The above sequence is expected to be applicable to dry silicic (highly polymerized) melts. The location of the HFSE Nb and Ta in the above sequence (Fig. 73a) shows that the group of HFSE does not behave the same (Nb and Ta have higher diffusivities than Hf, Ge, Th and U, even though Nb and Ta also have higher field strength), contrary to the suggestion of Mungall (2002). For hydrous rhyolite melts, the main data for multiple elements are on hydrous HR7 with 3.6 wt% H2O by Mungall et al. (1999). By adding H2O, the diffusivity of all components increases, but those with smaller diffusivities increase more, leading to smaller difference in diffusivities of different elements as H2O content is increased. The limited data combined with the alkali element sequence yield the following diffusivity sequence: Hf < Ti < Zr ≈ Nb ≈ Ta ≈ B < REE ≈ Y ≈ Be < Mg < Ba < Cs ≈ Ca < Sr < Rb < K < Na Comparing the diffusivity sequences of alkalis in dry and wet HR7, the largest difference is that Ba diffusivity does not increase as much as the other alkalis as H2O concentration increases, so that its diffusivity in wet HR7 is between that of Mg and Ca, rather than similar to Sr. The positions of Hf, Ti, Nb, Ta, and B also are changed slightly. For HR7 melt plus 20 wt% Na2O, the estimated diffusivity sequence based on the data of Mungall et al. (1999) and other information is: Hf < Zr < Ta ≈ Nb ≈ REE ≈ Y < Ti ≈ W ≈ Be < B < Mg < Ca < Sr ≈ Cs ≈ Ba < Rb < K < Na Diffusivity sequence in jadeite melt. In jadeite melt at 0.1 MPa (Dingwell and Scarfe 1985; Roselieb and Jambon 1997, 2002), the diffusivity sequence is: Cs < Mg < Ba ≈ F < Rb ≈ Sr ≈ Ca < K which differs from the sequence in HR7 in the divalent cations: in HR7, diffusivity increases from Mg to Ca to Sr ≈ Ba, whereas in jadeite, diffusivity increases from Mg to Ca and then decreases from Ca to Sr to Ba. Combining the above sequence with data of Nakamura and Kushiro (1998), the sequence in jadeite melt is: Zr ≈ Th < Nb ≈ U < REE ≈ Y < Cs < Mg < Ba ≈ F < Rb ≈ Sr ≈ Ca < K Diffusivity sequence in trachyte and phonolite melts. For dry trachyte melt, the diffusivity sequence based on the data of Behrens and Hahn (2009) is estimated as: Zr < Hf ≈ Nb ≈ Ta < REE ≈ Y ≈ Sn < Ni ≈ Zn < Ba < Sr < Rb < K < Na For wet trachyte melt containing about 1.5 wt% H2O, the diffusivity sequence based on the data of Behrens and Hahn (2009) is estimated to be: Zr ≈ Hf < Nb ≈ Ta < REE ≈ Y ≈ Cr < Sn < Zn < Ni < Ba < Sr < Rb < K < Na In dry phonolite melt, the diffusivity sequence is estimated to be (Behrens and Hahn 2009): Hf < Zr < Nb < REE ≈ Y < Cr < Ni ≈ Zn ≈ Sn < Ba < Sr < Rb < K < Na

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In wet phonolite melt containing about 1.7 wt% H2O, the diffusivity sequence is roughly (Behrens and Hahn 2009): Hf ≈ Zr < Nb ≈ Ta < REE ≈ Y < Cr ≈ Sn < Ni ≈ Ba ≈ Zn ≈ Sr Diffusivity sequence in dacite melts. In dry dacite melt, data by Cunningham et al. (1983), Henderson et al. (1985), Baker (1992a), and Tinker and Lesher (2001) provide the following constraints: Si ≈ O < Ga ≈ B < Cs < Co ≈ Mn ≈ Ba ≤ Na < Li Diffusivity sequence in andesite melts. In dry andesite2 and haploandesite, the following diffusivity sequence can be obtained based on the data of Lowry et al. (1982) and Koepke and Behrens (2001): Zr < Nb < REE ≈ Y ≈ Sc < Cs < Ba ≈ Zn < Rb ≈ Cr ≈ Co ≈ Sr ≈ Mn ≈ Sb < Na Diffusivity sequence in basalt melts. Self and tracer diffusivity data in dry basalt and haplobasalt melts are shown in Figure 74. The diffusivity sequence can be roughly determined as follows: U ≈ Th ≈ Zr < Si ≈ O ≈ Ti ≈ Te < REE ≈ Y ≈ Sc ≈ Pb ≈ Cs ≈ Tl < Ba < CO2 ≈ Brim < Sr ≈ Ca ≈ Co ≈ Mn ≈ Cl < Cd ≈ F < Na < Li Summary on diffusivity sequences. In summary, there is overall consistency in the diffusivity sequences, with tetravalent cations and P having the smallest diffusivity, then the pentavalent Nb and Ta, then REE and other trivalent ions, and then the divalent and univalent ions having the largest diffusivity. In detail, there are differences among various melts. For example, in dry silicic melts, diffusivity increases from Be to Mg to Ca to Sr ≈ Ba. However, in hydrous silicic melts, diffusivity increases from Be to Mg to Ca ≈ Ba to Sr. In basalt melts, diffusivity increases from Ba to Sr to Ca to Mg. Undoubtedly, the explanation lies in the melt structure (including ionic porosity), and the interplay between bond strength and ionic size as discussed earlier, but it would be very helpful if a quantitative theory can be developed to accurately predict the various sequences. For univalent and divalent ions, the diffusivity depends more strongly on ionic radius or bond strength (related to ionic radius). For ions with valence of 3 or higher, the diffusivity does not depend strongly on ionic radius. For example, diffusivities of REE, Sc, Cr, and B are often within a factor of 2 even though the ionic radius and hence the bond strength change significantly. Similarly, diffusivities of Si, Ge, Ti, Zr, Hf, U, and Th are also not hugely different. One possible explanation is that for the highly charged cations, the diffusing species are different from the simple cations, and the size of the diffusing species does not change significantly from one ion to another. In dry silicic melts at relatively low temperatures such as 1200 K, the diffusivity difference between rapidly and slowly diffusing elements is large, about 8 orders of magnitude from Si to Li and Na. As the melt becomes more mafic, as the temperature increases, and as more H2O is added to the melt, the diffusivity difference becomes smaller, only a few orders of magnitude. For example, Figure 74b shows that the difference between Li and U diffusivities in dry basalt melt at 1640 K is only 2.4 orders of magnitude. Even though the diffusivity sequence is not universal but depends on melt composition and other conditions, a rough sequence for a given type of melt (such as haplorhyolite melt) at some temperature can be useful for at least two purposes: (i) the sequence gives a rough idea of which element diffuses more rapidly, which is useful in forming a semi-quantitative understanding of diffusion rate in natural melts, and in predicting diffusive elemental fractionation; and (ii) the change in the diffusivity sequence may be used to probe the melt structure.

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Figure 74. Self and tracer diffusivities of divalent cations in dry basalt and haplobasalt melts. Fig. 74. Self and tracer diffusivities of divalent cations in dry basalt and haplobasalt melts.

Concluding Remarks An impressive amount of diffusion data on natural melts and analogs of natural melts has been accumulated in the last 30 years. Among the naturally occurring elements, no diffusion data are available for In, N, As, Bi, Se, I, V, Mo, Cu, Ru, Rh, Pd, Ag, Os, Ir, Pt, Au and Hg in natural silicate melts. For some other elements, such as Tl, Te, Ta, W, Zn, Cd, and Re, few studies have been made. On the other hand, for major elements such as O, H, Si, Al, Fe, Mg, and Ca, numerous data are available, and the data can be 74used in quantitative modelings (such as bubble growth, crystal dissolution, etc). However, even for these elements, it is still impossible to accurately predict their diffusivities as a function of temperature, pressure

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and melt composition (including concentration of H2O) due to limited data coverage and/or discrepancies among the data. Much experimental work is still needed in the future. It is more important to carry out well-planned systematic diffusion studies rather than numerous random studies (often with other goals, and extraction of diffusivities being only side products). For example, a dozen data points through well-designed experiments by Tinker and Lesher (2001) are sufficient to pin down the temperature and pressure dependence of Si and O diffusivity at ≤ 4 GPa in a haplodacite melt. Behrens and Zhang (2009) and Wang et al (2009) also showed that with coordination, only a small number of experiments are necessary to quantify the complicated behavior of H2O diffusion as a function of temperature, pressure and H2O content. In general, the compositional effect on diffusivity is much more difficult to constrain compared to temperature and pressure effects because (i) compositional variation of natural melts is complicated and (ii) there is no theory on what compositional parameters are the most important and how the diffusivity should be related to the composition. Furthermore, limited and random experimental data often do not resolve the compositional trends well. Even though in this review we made some ad hoc attempts to derive empirical relations between diffusivity and melt composition, the success is limited. (Trying to use a couple of consistent compositional parameters would not work well, and experimental data are not enough to constrain the fits if many compositional parameters, e.g., fractions of all major cations, are used.) It appears that the compositional dependence of diffusivity for rapidly diffusing (network modifiers) species differs from that for slowly diffusing (network formers) species. Among the effect of various components on diffusivities in silicate melts, dissolved H2O content often has the largest effect, often increasing the diffusivity by orders of magnitude (e.g., Watson 1979a). However, this effect has not been quantified for most elements. Because most natural melts contain significant H2O, it is critical to make major efforts to quantify the effect of H2O for accurate prediction of diffusion rates in natural melts. In the literature, the Eyring diffusivity estimated from viscosity is often associated with oxygen diffusivity. Recent experimental data show that oxygen diffusivity in the presence of H2O is orders of magnitude greater than the Eyring diffusivity (Behrens et al. 2007). Because natural melts often contain a significant amount of H2O, oxygen diffusivity and Eyring diffusivity are rarely similar in nature (Zhang and Ni 2010, this volume). On the other hand, silicon diffusivities are usually similar to or smaller than oxygen diffusivities in silicate melts. Furthermore, the addition of H2O is not expected to affect silicon diffusivity as drastically as oxygen diffusivity because H2O carries oxygen but not silicon. Based on limited data, we hypothesize that the Eyring diffusivity is more likely associated with silicon diffusivity, and the similarity may even apply to hydrous silicate melts. If this hypothesis is verified, the extensive viscosity data for silicate melts may be used to estimated silicon diffusivity, and vice versa. As expected, the SEBD values are less consistent than self and tracer diffusivities. The dependence of SEBD values on the specific conditions (such as a specific mineral dissolving in a specific melt) also means that it is more difficult to quantify SEBD values. Eventually, quantification of chemical diffusion in multicomponent melts will require the use of more complicated diffusion models, either a diffusion matrix approach or more advanced approximate methods (Liang 2010; Zhang 2010, this volume). Almost 30% of all the experimental diffusion data (Online Supplementary Table 1) by geochemists are on silicate melts of simple synthetic systems such as jadeite, albite, anorthite, orthoclase and diopside melts. These data turn out to be less useful in quantitative applications to geological problems. Furthermore, diffusion in these systems is not necessarily simpler in terms of mathematical treatment, or in terms of inferring the diffusion mechanism. We suggest that focusing future diffusion studies on natural melts and their analogs (e.g., well-chosen haplorhyolite and haplobasalt melts) would be more productive.

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Zhang, Ni, Chen Acknowledgments

This research is supported by NSF grants EAR-0711050 and EAR-0838127. H. Ni acknowledges financial support by the visitors program of the Bayerisches Geoinstitut. Charles Lesher and James Mungall provided formal reviews. We thank Daniele Cherniak for editorial handling.

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Reviews in Mineralogy & Geochemistry Vol. 72 pp. 409-446, 2010 Copyright © Mineralogical Society of America

Multicomponent Diffusion in Molten Silicates: Theory, Experiments, and Geological Applications Yan Liang Department of the Geological Sciences Brown University Providence, Rhode Island 02912, U.S.A. [email protected]

Introduction Many silicate melts of geological interests are made of three or more oxide components (e.g., SiO2, Al2O3, TiO2, FeO, MgO, CaO, Na2O, K2O, and H2O). Chemical diffusion in such systems is referred as multicomponent diffusion. Multicomponent diffusion involves mass fluxes driven by chemical potential gradients and is an essential mass transfer mechanism for many transport processes involving melt, solid, and melt-solid mixtures under magmatic conditions. The rate of crystal dissolution in a melt, for example, is determined by the rate of diffusion in the liquid, the thickness of concentration boundary layer surrounding the crystal, and the extent of undersaturation of the melt with respect to the dissolving crystal (e.g., Cooper and Kingery 1964; Watson 1982; Watson and Baker 1991; Zhang et al. 1989; Kerr 1995; Liang 2000; Zhang and Xu 2003). A key to understanding chemical diffusion in multicomponent melt is the diffusive flux. According to Onsager (1945), the diffusive flux of a given component (e.g., SiO2) in a multicomponent fluid or melt is a linear combination of concentration gradients of the independent components in the system. In a mass-fixed or barycentric frame of reference the diffusive flux of component i in an n component melt, Ji, is given by (e.g., Onsager 1945; de Groot and Mazer 1962; Hasse 1969) n -1

J i = -r∑ Dij ∇w j

(1)

j =1

where r is melt density; Dij are elements of an (n-1) by (n-1) diffusion matrix [D] with component n taken as the dependent variable; wj is mass fraction of component j; and ∇ is the gradient operator. In the absence of a sink or source and bulk flow, mass conservation in a fluid volume element leads to the multicomponent diffusion equation ∂ ( rw i ) = ∂t

n -1

∑ ∇ ⋅ (rD ∇w ) ij

j

(2)

j =1

Equations (1) and (2) are generalizations of Fick’s first and second laws for diffusion in isotropic multicomponent systems. When the density and composition variations in a melt of interest are small, Equation (2) can be simplified by neglecting the nonlinear terms associated with the derivatives involving the melt density and the elements of the diffusion matrix, ∂wi = ∂t

n -1

∑D ∇ w j =1

2

ij

j

(3)

The neglected terms are of order DDij / D and Dr / ro when compared with the first order term on the left hand-side of Equation (3) and are small in many practical applications, where DDij and 1529-6466/10/0072-0009$05.00

DOI: 10.2138/rmg.2010.72.9

Liang

410

Dr are magnitudes of diffusivity and density variations in the melt, respectively, D is a characteristic diffusivity and ro is the average melt density (e.g., Richter et al. 1998). For one-dimensional diffusion in a quaternary melt, Equation (3) represents 3 coupled diffusion equations, ∂ 2 w3 ∂w1 ∂2w ∂ 2 w2 = D11 21 + D12 + D 13 ∂t ∂x ∂x 2 ∂x 2 2 2 ∂ 2 w3 ∂w2 ∂w ∂ w2 = D21 21 + D22 + D23 2 ∂t ∂x ∂x ∂x 2 ∂w3 ∂ 2 w3 ∂2w ∂ 2 w2 = D31 21 + D32 + D33 2 ∂t ∂x ∂x ∂x 2

(4a) (4b) (4c)

which can also be written in a matrix form,  w1   D11 ∂   w2 = D21 ∂t     w3   D31

D12 D22 D32

D13   2  w1   ∂  D23   2  w2   ∂x  w3   D33  

(5a)

Let w = [w1 w2 w3]T be a column vector, where T stands for the matrix transpose. Equation (5a) can be written in a compact matrix form, ∂w ∂2w = [ D] 2 ∂t ∂x

(5b)

Concentration of the dependent variable w4 is given by w4 = 1 - w1 - w2 - w3. Equations (3)(5) have been widely used to study multicomponent diffusion in molten silicates of geological interest. The fundamental principles of chemical diffusion in multicomponent fluids were given by Onsager in a series of seminal papers published between 1931-1945 (Onsager 1931a,b, 1945). Due to its importance, the subject of chemical diffusion has been extensively discussed in numerous research articles, review papers, and textbooks, across almost the entire discipline of science and engineering. Examples of general treatment of this subject include, but are not limited to, de Groot and Mazur (1962), Katchalsky and Curran (1967), Haase (1969), Tyrrell and Harris (1984), Miller et al. 1986; Kirkaldy and Young (1987), Eu (1992), Kondepudi and Prigogine (1998). Relevant reviews or summaries in the geological literatures include Anderson (1981), Chakraborty (1995), Lasaga (1998), and Zhang (2008). In this chapter, we focus on some practical aspects of multicomponent diffusion in molten silicates of geological interest. The theoretical background for multicomponent diffusion based on the principles of irreversible thermodynamics is first reviewed. Specifically, we examine the rate of entropy production and show, using a specific example, that it is invariant to the choice of diffusing species. With the help of two simple examples of chemical diffusion in quaternary systems, we then explore the essential features of multicomponent diffusion in molten silicates, including uphill diffusion, quasi steady-state diffusion, transient gradient partitioning, multiple timescales of diffusion, and effective binary diffusion. In the second half of this chapter, we review results of recent experimental studies of multicomponent diffusion in molten silicates, including experimental design and inversion methods. This is followed by an overview of the empirical models for multicomponent diffusion in molten silicates. Selected applications of multicomponent diffusion to crystal growth and dissolution are discussed next. We conclude this chapter with a brief discussion of future research directions.

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Irreversible Thermodynamics and Multicomponent Diffusion The rate of entropy production The diffusive flux in Equation (3) is related to thermodynamic driving forces of the components in the system. This can be shown by considering the rate of entropy production of an n component non-reacting fluid under isothermal and isobaric conditions. The rate of entropy production, s, can be written as a sum of scalar products between the diffusive flux Ji and chemical potential gradient ∇mi of component i in the fluid (e.g., de Groot and Mazur 1962; Haase 1969), n

∑J

= s

i =1

⋅

i

∇mi T

(6)

where T is the temperature. Since the diffusive fluxes and the chemical potential gradients in Equation (6) are not independent, there are a number of ways to write the right hand side of Equation (6). In a volume fixed frame of reference, the diffusive flux obeys the conservation requirement, n

∑V J i

i =1

=0

i

(7a)

where Vi is the partial molar volume of component i in the melt. The chemical potential gradients are related to each other through the Gibbs-Duhem relationship, n

∑ C ∇m i =1

i

i

=0

(7b)

where Ci is the molar concentration of component i in the melt. The diffusive flux and the chemical potential gradient of component n can be eliminated from Equation (6). The entropy production rate then becomes = s

n -1

∑J i =1

 ∇mi Vi ∇m n  = X i = Vn T   T

i

⋅ Xi

n -1



∑ d k =1

(8a)



ik

+

Vi Ck  ∇m k Vn Cn  T

(8b)

where dik = 1 when i = k and dik = 0 when i ≠ k. Xi is called the thermodynamic driving force for chemical diffusion. Expressions for the thermodynamic driving force under other choices of the frame of reference can be found in de Groot and Mazur (1962, pages 240-242). At thermodynamic equilibrium, s = 0, both Ji and Xi vanish in Equation (8a). In an irreversible or non-equilibrium process, the rate of entropy production is always positive. When departures from equilibrium are small, that is, in the linear regime, the diffusive flux can be written as a linear combination of the independent driving forces (Onsager 1945) n -1

J i = -∑ LVik X k

(9)

k =1

where LVik are phenomenological coefficients obeying Onsager’s reciprocal relationship LVik = LVki

(10)

Equation (9) represents the most general relationship between diffusive flux and thermodynamic driving forces for a multicomponent system. It states that the material flux of a component is

Liang

412

not only proportional to its own chemical potential gradient but is also related to chemical potential gradients of all other independent components in the system. At constant temperature and pressure the chemical potential is a function of concentration. Hence the chemical potential gradient in Equation (8b) can be written as a linear combination of the concentration gradients, = ∇m k

 ∂m k   ∇C j j  j =1 

n -1

∑  ∂C

(11)

where the partial derivative ∂mk / ∂Cj is evaluated at a constant temperature and pressure while holding all the components except j constant. Substituting Equation (11) into Equation (9) and exchanging summation signs, we have n -1 n -1 V  L ∂m Ji = -∑∑ ik  k  =j 1 = k 1 T  ∂C j 

 Vk  ∂m n -   V  ∂C n  j 

   ∇C j  

(12)

   

(13)

When chemical diffusion coefficients Dij are defined as DijV =

LVik  ∂m k  k =1  ∂C j

n -1

∑T

 Vk  ∂m n -   Vn  ∂C j  

we have Onsager’s extension of Fick’s first law of diffusion. Equations (12) and (13) can also be written in matrix notation as follows J = -[ DV ] ⋅ ∇c ∂c = [ DV ] ⋅ ∇ 2 c ∂t V [ D= ] [LV ] ⋅ [G]

(14a) (14b) (14c)

where [DV], [LV], and [G] = [∂mk / ∂Cj] are square matrices, J and C are column vectors. [DV] is known as the diffusion matrix and [G] the thermodynamic matrix. Equation (14a) states that the diffusive flux of a component is proportional to the concentration gradients of all the independent components in the system. Although independent of concentration gradients, the chemical diffusion coefficients Dij are, in general, functions of temperature, pressure, as well as concentrations. An important property of the diffusion matrix is that it is positive definite (e.g., de Groot and Mazur 1962; Hasse 1965). Thus all eigenvalues of the diffusion matrix are real and positive. As a consequence, the diffusion equations can be decoupled by a linear transformation (e.g. Toor 1964; Cullinan 1965; Gupta and Cooper 1971). We will make use of this property to solve the coupled diffusion equations below.

Diffusing species and choice of endmember component So far the nature of the diffusing components has not been specified (e.g., molecular or ionic). In many instances, the exact nature of the real diffusing species is unknown (e.g., in most molten silicates). This lack of knowledge about the diffusing species, however, will not affect the use of the chemical diffusion equations so long as the number of the independent variables remains unchanged. Liang (1994) showed that the rate of entropy production of an ionic fluid is independent of the choice of the endmember components. As an example, we consider the rate of entropy production of a simple ionic fluid consisting of n types of cations (Mzi) and one type of anion (OzO). The number of independent variables in such a system is n-1 due to the additional charge neutrality constraint. The rate of entropy production of the ionic fluid is similar to that in Equation (6) with the thermodynamic force being the electrochemical potential gradient due to the presence of a

Multicomponent Diffusion in Molten Silicates

413

local electric field (e.g., Haase 1969). n

∑J

= s

i =0

i

⋅

∇(mi + zi Fy ) T

(15)

where zi is the charge number of component i, F is the Faraday constant, and y is the electric potential. The terms associated with the electrical potential gradient in Equation (15) cancel due to the zero electric current constraint, i.e., ∑in= 0 zi J i = 0. As such, the rate of entropy production for the ionic fluid simplifies to = s

n

∑J i =0

⋅

i

∇mi T

(16)

Similar to that in Equation (8), the flux of the ionic component i can be expressed as a linear combination of concentration (or chemical potential) gradients of the n-1 independent ionic components. Alternatively, the rate of entropy production of the above simple ionic system can be evaluated by taking arbitrary molecules MpiOqi (where pi and qi are stoichiometric coefficients for component i, i = 1, n) as endmember components. Let Ji0 and mi0 be the flux and chemical potential of the molecular component MpiOqi, respectively. Then the relationships between the molecular and the ionic quantities are obtained by considering the following dissociation relation for the molecular component MpiOqi (e.g., Denbigh 1981, pages 304-305), M= pi M zi + qi O z0 pi O qi

(17a)

where the stoichiometric coefficients pi and qi are subject to the charge balance constraint pi zi + qi z0 = 0

(17b)

The molecular flux is related to the ionic flux via Ji 0 =

Ji pi

(18)

With the assumption of local chemical equilibrium, which is true when the rate of dissociation or speciation is much faster than the rate of diffusion, the chemical potential of the molecular component is obtained mi 0 = pimi + qim 0

(19)

Substituting Ji = piJi0, mi = mi0 / pi - qi / pim0, and qi = -zi / z0pi into Equation (16), we have n

∑J

= s

i =1 n

i0

⋅

∇(mi 0 - qim 0 ) ∇m 0 + J0 ⋅ T T

(20)

 z   ∇m 0 ∇m +  J + ∑  i pi J i 0   ⋅ T  z0   T 

n i0 i0 0 =i 1 =i 1

∑J

=

∇m   ∇m 0 +  ∑ zi J i  ⋅ T   Tz0

n i0 i0 =i 1 =i 0

=

n

⋅

∑J

⋅

The second term on the right hand side of the last equation vanishes because of the no net current constraint. The rate of entropy production for the ionic fluid is reduced to = s

n

∑J i =1

i0

⋅

∇mi 0 T

(21)

Liang

414

Hence under the assumption of local thermodynamic equilibrium, the rate of entropy production is independent of the choice of the endmember component. A corollary of this statement is that it is not possible to uniquely determine the diffusing species through chemical diffusion studies alone. The choice of the endmember components for chemical diffusion in multicomponent fluids is simply a matter of convenience. When studying multicomponent diffusion in molten silicates, it is customary and also convenient to choose simple oxides as endmember components, even though the exact nature of the actual diffusing species is not known.

General Features of Multicomponent Diffusion Solutions to multicomponent diffusion equations When elements of the diffusion matrix are constant and uniform, analytical solutions to the coupled diffusion equations can be constructed from solutions to the equivalent binary cases using the method of linear transformation (e.g., Toor 1964; Cullinan 1965; Gupta and Cooper 1971). Exact solutions for multicomponent diffusion in ternary and quaternary systems were first derived by Fujita and Gosting (1956), Kirkaldy (1958), and Kim (1966) for a onedimensional semi-infinite diffusion couple. For more general cases, one can use the linear transformation to decouple the multicomponent diffusion equations (e.g., Toor 1964; Cullinan 1965; Gupta and Cooper 1971), w = [ b] ⋅ u

(22a)

[L = ] [ b]-1 ⋅ [ D] ⋅ [ b]

(22b)

where U is a column vector in the new compositional space; the eigenvectors of [D] form the columns of the [B] matrix; and [L] is a diagonal matrix whose diagonal elements are eigenvalues of [D]. Substituting Equations (22a) and (22b) into Equation (3), we have ∂Ui = l i∇ 2Ui ∂t

(23a)

or in a matrix form  U1    ∂  U2  = ∂t      U n -1 

0   l1 0 0  U1  0 l    0 0 2   ⋅ ∇2  U2              0 0 0 l n -1   U n -1 

(23b)

Equations (23a) or (23b) are now n-1 uncoupled diffusion equations. If the initial and boundary conditions can also be decoupled through the linear transformation w = [B]⋅U, analytical solutions to Equation (23) can be readily constructed using results for diffusion in a binary system (e.g., Carslaw and Jaeger 1959; Crank 1975). Here we consider three examples that are often encountered in diffusion studies. Case 1. Diffusion between two semi-infinite rods initially having uniform but different compositions. This setup has often been used in laboratory studies of multicomponent diffusion in silicate melts (see below). If we choose x = 0 as the position of the original interface between the two semi-infinite rods, solutions to Equation (3) can be written as w + wb -1 w - w a w =a + [ b][ E ][ b] ⋅ b 2 2

(24a)

where wa and wb are column vectors representing initial melt compositions in regions x < 0 and x > 0, respectively; [E] is a diagonal matrix with its diagonal terms given by

Multicomponent Diffusion in Molten Silicates

415

 x  Eii = erf   2 lt  i  

(24b)

and the off-diagonal terms Eij = 0 for i ≠ j; li > 0 are the eigenvalues of the diffusion matrix. Case 2. Diffusion between two rods initially having uniform but different compositions. This is a case of a finite diffusion couple with impermeable walls at the two ends. Let x = 0 be the original interface between the two rods, solutions to Equation (3), in this case, can be written as = w

w a La + w b Lb + [ b][ f ][ b]-1 ⋅ (w b - w a ) La + Lb

(25a)

where wa is the initial composition of rod a (-La < x < 0); wb is the initial composition of rod b (0 < x < Lb); [F] is a diagonal matrix with its diagonal elements given by Fkk ( x, t )

 m 2 π2 l t   mπ ( x + L a )   mπLa  2 ∞ 1 k  sin   cos   exp  ∑ 2 L L π m =1 m  La + Lb  +  L L + a b   b)   ( a 

(25b)

lk > 0 are again the eigenvalues of the diffusion matrix. Equations (25a) and (25b) are identical to the component form given by Trial and Spera (1994). Case 3. Diffusion of a melt pocket (composition wb) initially confined in the region -h  0 are the eigenvalues of the diffusion matrix. The analytical solutions Equations (24)-(26) reduce to the respective binary cases when k = 1 and [B] = [B]-1 = 1. The compact vector solutions (Eqns. 24-26) for chemical diffusion in an n-component system can be easily implemented using computational packages such as MatLab® and Mathematica®. Below we use two simple examples to illustrate the unique and important features of multicomponent diffusion in molten silicates.

Essential features of multicomponent diffusion A key difference between chemical diffusion in a binary melt and a general multicomponent melt is the coupled diffusion in the latter. Diffusive coupling among the various components in the melt gives rise to a number of interesting features that are characteristic of chemical diffusion in multicomponent melts. These include (1) uphill diffusion; (2) quasi steady-state diffusion and transient gradient partitioning; and (3) multiple time scales of diffusion and effective binary diffusion. To highlight these important features of multicomponent diffusion, we consider two simple examples of chemical diffusion in quaternary melts: one is a finite diffusion couple (Eqns. 25a and 25b) and the other a finite melt pocket sandwiched between two semi-infinite melt reservoirs of identical composition (Eqns. 26a and 26b). For purpose of illustration, we choose La = Lb = 2h and use the following hypothetical diffusion matrix to calculate the concentration profiles,

416

Liang 0.5 0.25  D11 D12 D13   2 D =   1.5 -10  × 10 -11 m 2 /s 40 D D 22 23   21   D31 D32 D33   -0.55 -0.45 0.15

(27)

The eigenvalues of the diffusion matrix are [l1 l2 l3] = [1.87 40.13 0.146]×10-11 m2 / s. Given a reference length scale (e.g., h), the eigenvalues of the diffusion matrix can be used to define three diffusion time scales: tk0 = h2 / lk. If h = 1 mm, we have [t10 t20 t30] = [14.8, 0.69, 190] hours. For convenience, we refer to component 2 as the fast component, component 3 as the slow component, and component 1 as the intermediate component. In terms of diffusive behaviors of the major oxide components in natural melts, one might take component 3 as SiO2, component 2, as Na2O + K2O, component 1 as CaO + MgO + FeO, and component 4 as Al2O3 + TiO2, though the connection is loosely defined here. Uphill diffusion. Figures 1a-1c and 1d-1f display calculated diffusion profiles for the three independent components (1, 2, and 3) across the two diffusion setups, respectively, at four selected times. Concentration profiles for the dependent component 4 (not shown) are similar to those for component 2. For comparison between diffusion in the two setups, we use the same starting melt compositions for the two examples and only display data between x / t 1/2 = -2 and x / t 1/2 = 2 in this figure, even though the second setup involves two semi-infinite melt reservoirs. Figure 2 is a zoom out view of Figure 1e showing diffusion in the far field (-20 < x / t 1/2 < 20). Perhaps one of the most striking features of the concentration profiles shown in Figures 1a-1f and 2 is the uphill diffusion of component 2 (curves labeled as t1 and t2 in Fig. 1b and all four curves in Figs. 1e and 2). Uphill diffusion refers to the diffusion of a component against its own concentration gradient. It happens in part of a diffusion couple where the sign of the diffusive flux of the given component of interest is reversed due to the strong coupling of the diffusive flux to the concentration gradients of other diffusing components. This usually occurs around the interface region where the two starting melts are initially in contact (around x / t 1/2 = 0 in Fig. 1b and x / t 1/2 = ± 1 in Fig. 1e). The necessary and sufficient conditions for the uphill diffusion of component 2, in the present case, are D22 Dw2 ≤ D23Dw3

(28a)

∂w3  ∂w2     D22 ∂x  D23 ∂x  < 0   

(28b)

where |∆w2| and |∆w3| are maximum concentration differences across the region of interests (central part of the diffusion couple) for components 2 and 3, respectively, at a given time. Concentration gradients of component 1 are not included in Equations (28a) and (28b) because the diffusive flux of component 2 is only weakly coupled to the concentration gradients of component 1 (i.e., |D22Dw2| >> |D21Dw1|). For the finite diffusion couple shown in Figures 1a-1c, we have |D21Dw1| ≈ 25, |D22Dw2| ≈ 100, and |D23Dw3| ≈ 150 for t < 0.05. For the finite melt pocket diffusion couple (Figs. 1d-1f and 2), the inequalities Equations (28a) and (28b) are always satisfied around the troughs on either side of the original melt pocket (i.e., |x / t 1/2| > 1). However, for diffusion in a finite geometry, Equations (28a) and (28b) are satisfied only at early times (e.g., t < 0.1 for the case shown in Figs. 1a-1c). In general, uphill diffusion is a transient feature of multicomponent diffusion involving a finite diffusion couple. In the presence of impermeable boundaries at the two ends of a finite diffusion couple, uphill diffusion of component 2 driven by the concentration gradients of component 3 eventually reverses the flow direction of component 2 in the finite diffusion couple (cf. points A1A2 and B1B2 in Fig. 1b), reducing the net diffusive flux of component 2. Quasi steady-state diffusion and transient gradient partitioning. At long times (e.g., t > 0.05 in Figs. 1a-1f) when concentration gradients of the fast component are relaxed due to the

1

−2

0

x/sqrt(t)

t2

t3

t4

0

t1

1

t1

1

t4

2

2

5 −2

10

4 −2 15

6

8 B1

10

12

14

A1

−1

−1

0

x/sqrt(t)

t1

t2

t4

0

t3

t4

1

1

(e)

t1

t2

t3

(b)

2

2

B2

A2

−2

50

55

60

65

−2

50

55

60

65

(f)

(c)

−1

−1

0

x/sqrt(t)

t4

t3

t1

0

Figure 1. Plot of calculated diffusion profiles as a function of x / t1 / 2 (in 105 m s-1 / 2) for diffusion in a quaternary melt at four selected times. Panels (a)-(c) are for components 1, 2, and 3 in a finite diffusion couple (Eqns. 25a-25b). Panels (d)-(f) are for a finite melt pocket sandwiched between two semi-infinite melt reservoirs (Eqns. 26a-26b). Diffusion matrix used in the calculation is given in Equation (27) in the text. Color versions of Figures 1-5, 8, 10, 11, and 14 are published online.

10 (d)

15

20

25

−1

−1

−2 30

10

t1 = 0.001 t2 = 0.05 t3 = 1 t4 = 5

t2

15

20

25 t3

w2 w2

w

1

w

3

w 3

w

30 (a)

1

1

t4

t1

2

2

Multicomponent Diffusion in Molten Silicates 417 Figure 1

Liang

418 15

t3

14 13 12

w

2

11

t2

10 9 8 7 6 5 −20

t1 = 0.001 t2 = 0.05 t3 = 1

t1

t4 = 5

−10

0

x/sqrt(t)

10

20

Figure 2. A zoom out view of panel (e) in Figure 1.

diffusion of in a finite melt reservoir, concentration gradients of the slow component become important in driving the diffusion of the fast component such a quasi steady-state is established for the diffusion of the fast component on the slow diffusion time scale. For the fast diffusing component 2, we have ∂w ∂w ∂w - D21 1 - D22 2 - D23 3 ≈ 0 J2 = ∂x ∂x ∂x

(29a)

Since the first term on the right hand side of Equation (29a) is small compared to the last two terms (due to small D21 and small concentration gradients of component 1 at t > t3, see Figs. 1a and 1d), we have an approximate relation between the concentration gradients of components 2 and 3, D22

∂w ∂w2 + D23 3 ≈ 0 ∂x ∂x

(29b)

Hence concentration profiles of the fast component 2 are locked into the concentration profiles of the slow component 3. This is shown in Figures 3a and 4a where concentration profiles of component 2 are linearly correlated with the concentration profiles of the slow component 3 for the two diffusion couples considered (solid and dashed lines, respectively). The slopes of the nearly parallel straight lines, dw2 / dw3, are 0.264 (0.260) at x = 0 and t = 1 (t = 5) for the finite diffusion couple (Fig. 1b) and 0.271 (0.259) at x / t 1/2 = ±1 and t = 1 (t = 5) for the diffusion couple shown in Figure 1e. This is in excellent agreement with the prediction from Equation (29b),  D Dw1  1  D Dw1  D dw2 = - 23 + O  21 + O  21 =  D dw3 D22 D w 4 3   22  D22 Dw3 

(30)

Multicomponent Diffusion in Molten Silicates 16

30

(a)

14

10

w1

2

25

t2

12

w

(b)

t3

t4

8

t4

20

t3

t4

15

t1

t3

6

t1 = 0.001 t2 = 0.05 t3 = 1 t4 = 5

10

4

419

50

55

60

w

65

50

55

3

Figure 4 t2

60

w

65

3

Figure 3. Variations of component 2 (a) and component 1 (b) as a function of the slow diffusing component 3 at the four selected times. The solid lines are diffusion profiles from the finite diffusion couple shown in Figures 1a-1c. The dashed lines are from diffusion profiles shown in Figures 1d-1f.

0

(a)

t3

t4 t2

0.25 0.24 −2

−1

0.25

3

2

dw /dw

3

t3

0.26

−0.2 −0.3

0

1

2

t4

t4 t2

t3 t1

−1

t3

0

x/sqrt(t)

t3

−0.5 −2 0.6

w1/w3

3 2

w /w

−2

t4

−1

0.5

0.2 t2

0.1 (c)

t4

−0.4

t3

0.15

(b)

−0.1

1

0.27

dw /dw

0.28

1

2

t2

1

t4 t3

t4

0.3

0.1 −2

t3

t2

(d)

2

t1

0.4

0.2

t1

0

t1

−1

0

x/sqrt(t)

1

2

Figure 4. Plots of concentration derivatives and concentration ratios calculated from the diffusion profiles shown in Figures 1a-1c (solid lines) and 1d-1f (dashed lines) as a function of x / t1 / 2 (in 105 m s-1 / 2) for the two diffusion couples (see text for details).

Liang

420 where O stands for order of magnitude.

A similar exercise can be carried out for quasi steady-state diffusion of the intermediate component 1 at long times by setting

∂w ∂w ∂w - D11 1 - D12 2 - D13 3 ≈ 0 J1 = ∂x ∂x ∂x

(31a)

In this case both concentration gradients of components 2 and 3 contribute to the diffusive flux of component 1. Since the concentration gradients of component 2 is locked into the concentration gradient of component 3 via Equation (29a), we have ∂w D ∂w ∂w - D11 1 + D12 23 3 - D13 3 ≈ 0 J1 = ∂x ∂x D22 ∂x

(31b)

which can be further simplified to dw1 D12 D23 - D13 D22 = dw3 D11D22

(32)

Substituting the elements of the diffusion matrix from Equation (27) into Equation (32), we find a slope dw1 / dw3 = -0.188, which is in reasonable agreement with the numerical values calculated directly from the diffusion profiles shown in Figure 1a-1f (-0.350 and -0.485, for the two diffusion couples respectively, at t = 1; and -0.247 and -0.211 at t = 5, all at x = 0). Figures 3b and 4b show that concentration profiles of component 1 are approximately linearly correlated with the concentration profiles of component 3 for the two diffusion couples at long time (solid and dashed lines, respectively). The slow convergence rate in this case is due to the moderate diffusion rate of component 1 relative to component 3 such concentration profiles of component 1 were still not relaxed at t = 1 (i.e., |J1| > 0). Diffusive behaviors of the fast, intermediate, and slow diffusing components illustrated in Figures 1-3 are broadly similar to those for diffusion between basalt and molten feldspar or granite (Watson 1982; Watson and Jurewicz 1984; Watson and Baker 1991, see their Fig. 1), if we take the fast component 2 as the sum of Na2O + K2O, the intermediate component 1 as CaO + MgO + FeO, and the slow component 3 as SiO2. Watson and coworkers noted that alkalies are preferentially enriched or partitioned into the felsic melt relative to those in the basalt, by a factor of two to three, after an initial short time, irrespective of their initial abundances in the two starting melts. This “transient equilibrium partitioning” of the fast diffusing alkalies between the high- and low-Si melts in a finite diffusion couple (Watson and Baker 1991) can be understood in terms of quasi steady-state diffusion in a multicomponent melt in which the diffusive fluxes of the alkalies are strongly coupled to the concentration gradient of SiO2. Since the system reaches a quasi steady-state, not a true thermodynamic equilibrium state, concentration variation ratios or slopes in the concentration correlation diagram (e.g., dw2 / dw3 and dw1 / dw3) approach constants (e.g., Eqns. 30 and 32). The concentration ratios (e.g., w2 / w3 and w1 / w3), on the other hand, depend on the boundary conditions (e.g., type of diffusion couple and choice of starting compositions). This can be further demonstrated by the examples below. Figures 4c and 4d display the variations of the concentration ratio w2 / w3 and w1 / w3 as a function of position in the two diffusion couples at the four selected times. In contrast to the nearly constant slopes (dw2 / dw3 and dw1 / dw3 in Figs. 4a and 4b) discussed in the proceeding examples (see also Figs. 3a and 3b), the concentration ratios w2 / w3 and w1 / w3 at the quasi steady-state (t3 or t4) not only vary within a given diffusion couple, but more importantly also depends strongly on the setup or choice of the diffusion couple (cf. the solid and dashed lined in each panel). For example, at t4 = 5, w2 / w3 ≈ 0.158 and w1 / w3 ≈ 0.43 for the finite diffusion couple (Figs. 1a-1c and 4c), whereas w2 / w3 ≈ 0.238 and w1 / w3 ≈ 0.33 for the finite

Multicomponent Diffusion in Molten Silicates

421

melt sandwiched between two semi-infinite melt reservoirs (Figs. 1d-1f and 4d), all at x = 0. Another advantage of using the concentration variation ratios rather than concentration ratios in the interpretation of “transient equilibrium partitioning” is that the former also applies to diffusion in an effectively infinite melt reservoir (see Figs. 1c-d, 2, also solid lines in Figs. 3, 4a-b). Hence it may be more appropriate to use the phrase “transient gradient partitioning” to describe element distributions during quasi steady-state diffusion in a multicomponent melt. Transient gradient partitioning during quasi steady-state diffusion (e.g., Eqns. 30 and 32) may be helpful in designing special chemical diffusion experiments that can be used to better constrain selected elements of the diffusion matrix. Multiple time-scales of diffusion and effective binary diffusion. The contrasting diffusive behaviors between the fast and slow diffusing components in the proceeding examples have already demonstrated the multiple time-scale nature of diffusion in multicomponent melts. The three unique eigenvalues of the hypothetical diffusion matrix Equation (27) imply three independent diffusion time scales for a given length scale: t10, t20, t30. In theory, one can conduct a more rigorous analysis of quasi steady-state diffusion by setting the diffusive flux associated with the fast diffusing eigenvector to zero at long time (e.g., J2 = -l2∂U2 / ∂x = 0 at t > t20). In practice, since the elements of diffusion matrix are usually not known a priori, it is often difficult to define the fast diffusing eigenvector. To further demonstrate the concept of multiple time-scale of diffusion, we take the effective binary approach. Chemical diffusion in multicomponent molten silicates has commonly been modeled in terms of effective binary diffusion (Cooper 1968; see also reviews by Watson and Baker 1991; Chakroborty 1998; Zhang et al. 2010). The multicomponent melt is treated as a pseudo-binary system in which the component of interest is taken as the independent component or variable and all other components are taken together as the second component or dependent variable. The diffusive flux of the component of interest (i) is then assumed to obey a relation of the form, J i = - DiE

∂wi ∂x

(33)

where DiE is the effective binary diffusion coefficient (EBDC) for component i. Since DiE must be positive for diffusion equation to be stable, the effective binary simplification is not capable of modeling uphill diffusion. Cooper (1968) showed how the EBDC defined in Equation (33) could be related to the elements of the diffusion matrix. For example, the EBDCs for component 1 and 3 in the quaternary melt can be calculated using information from diffusion matrix and slopes in concentration correlation diagram, ∂w ∂w2 + D13 3 ∂w1 ∂w1

(34a)

∂w1 ∂w + D32 2 + D33 ∂w3 ∂w3

(34b)

D1E =+ D11 D12 D3E = D31

Since the concentration derivatives, ∂wi / ∂wj, depend on geometry and starting compositions of the diffusion couple, as well as time and spatial coordinates in the diffusion couple (see Figs. 4a and 4b), the EBDCs defined in Equation (34a) and (34b) are in general not constant and uniform in a given diffusion couple even when the elements of the diffusion matrix are constant and uniform. Thus, EBDCs often depend on the direction of diffusion in composition space (Cooper 1968; see also Zhang et al. 2010). For example, the experimentally measured EBDC of SiO2 in molten CaO-Al2O3-SiO2 (1500 °C and 1 GPa) differs by almost an order of magnitude for diffusion along the directions of constant CaO and constant Al2O3 (Liang et al. 1996a). An equally important property of effective binary diffusion is that EBDCs vary as a function of time even at a fixed position in a given diffusion couple. This is illustrated in Figures 5a and 5b where the EBDCs of component 1 and 3 for the quaternary system calculated using

Liang

422 2

2

(a)

1.8

1.8

1.6

1.4

EBDC3

Case 3

1.2 Case 2a

1

Case 2b

EBDC1

Case 3 Case 2a Case 2b

1.6

1.4

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1.2 1 0.8

0.6

0.6

0.4

0.4

0.2 −2 10

(b)

−1

10

0

10

time

1

10

2

10

0.2 −2 10

Case 3 −1

10

0

10

time

1

10

2

10

Figure 5. Plots of calculated efficient binary diffusion coefficients (EBDCs) of component 1 (a) and component 3 (b) as a function of diffusion time for three different diffusion couples (see text for details). The EBDCs were calculated using Equations (34a) and (34b) and concentration derivatives from solutions to the coupled diffusion equations using the hypothetical diffusion matrix Equation (27).

the hypothetical diffusion matrix Equation (27) and Equations (34a) and (34b) are shown as a function of time. Three diffusion couples are considered: a finite melt pocket of length h sandwiched between two semi-infinite melt reservoirs (solid blue lines labeled as Case 3), a finite diffusion couple in which L1 / h = L2 / h = 2 (dashed blue lines labeled as Case 2a), and a short finite diffusion couple in which L1 / h = L2 / h = 1 (solid red lines labeled as Case 2b). Selected concentration profiles of Case 2a and Case 3 are shown in Figures 1a-1c and 1d-1f, respectively. For comparison, starting melt compositions for the three diffusion couples are the same and the EBDCs are calculated at the original interface of the diffusion couples using Equations (34a) and (34b) (x / t 1/2 = ±1 for Case 3, x = 0 for Case 2a and Case 2b). Three interesting observations can be readily made from Figures 5a and 5b. First, there are two time scales of diffusion: the EBDCs of components 1 and 3 at short time when the melt reservoirs are effectively semi-infinite are 2-6 times larger than those at long time. This again can be understood in terms of coupled diffusion. At short time, the concentration gradients of component 1, ∂w1 / ∂x, are significant. The diffusive flux of component 1 is dominated by the diagonal term, as the magnitudes of the off-diagonal terms are relatively small. Hence, D1E ~ D11 = 2×10-11 m2 / s, which is in good agreement with the three cases shown in Figure 5a. Since the sign of ∂w1 / ∂x is opposite to that of ∂w3 / ∂x (Figs. 1a-1f) and D31 < 0, the presence of a significant concentration gradient ∂w1 / ∂x elevates the apparent diffusion rate of component 3 (hence an increase in EBDC of component 3). As the magnitude of ∂w1 / ∂x becomes smaller with increasing time, EBDC of component 3 decreases (Fig. 5b). Contributions of the (relatively) large concentration gradient ∂w3 /∂x to the diffusive flux of component 1 then become significant, reducing the apparent diffusion rate of component 1 at long time, as D13 > 0 (Fig. 5a). At long time, EBDC of component 1 (also component 2) is approximately the same as EBDC of component 3 for the two finite diffusion couples (Case 2a and Case 2b in Figs. 5a and 5b). This is consistent with the observation of Watson and Baker (1991) for diffusion between basalt and molten feldspar or granite: “If the diffusion reservoirs are of limited dimension, the initial rapid transfer of alkalies ceases, and the concentration gradients assumed by all elements become equal in length to (though possibly in the opposite direction of) that of SiO2.”

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Although the relative diffusion information among the independent components appears lost in terms of effective binary diffusion at long time, the nearly constant slopes in the concentration correlation diagrams still offer valuable information regarding the elements of the diffusion matrix (e.g., Eqns. 30 and 32). Also at long time, EBDCs derived from the finite diffusion couples (Case 2a and Case 2b in Figs. 5a and 5b) are different from (lower than) those derived from Case 3 that has two semiinfinite melt reservoirs. This is because the slope, ∂wi / ∂wj, for Case 3 is larger than for Cases 2a and 2b, even though in both cases ∂wi / ∂wj approach their asymptotic values (Figs. 4a and 4b). Finally, Figures 5a and 5b also show that transition from short time to long time diffusive behaviors occurs at different times depending on the size of the diffusion couple (cf. Case 2a and Case 2b). The time- and geometry-dependent natures of EBDC further complicate direct applications of laboratory derived EBDCs for molten silicates to geological problems: one has to match not only the direction of diffusion in composition space, but also the time scale and geometry of diffusion between the laboratory and nature in order to use the EBDCs. That is a lot of requirements. After all, effective binary diffusion is not as simple as it appears.

Experimental studies of multicomponent diffusion Experimental design and strategy The experimental study of chemical diffusion in molten silicates involves juxtaposing two melts of different compositions in an inert container placed in a high temperature (and very often high pressure) furnace. In general, an accurate estimate of the diffusion matrix in an n component system requires at least n-1 different diffusion couples because there are (n-1)2 unknowns in the diffusion matrix and each diffusion couple has only n-1 independent concentration profiles (Gupta and Cooper 1971). Ideally, one should select a set of diffusion experiments such that the n-1 diffusion couples not only cross at the composition point of interest but also are orthogonal to each other in the eigen-space of the diffusion matrix (Liang 1994; Trial and Spera 1994). But, since the diffusion matrix, and thus the eigen-space, is not known a priori, the n-1 directions are often chosen so that one of the components is constant along each of the directions (e.g., Kirkaldy and Young 1987, page 179). Figure 6 shows an example of the starting melt compositions used to form diffusion couples (pairs of open circles joined by thin solid lines) in the ternary CaO-Al2O3-SiO2 (Liang et al. 1996a; Liang and Davis 2002). Three diffusion couples formed by juxtaposing compositions 3 against 5 (designated as 3 / 5), 4 against 2 (4 / 2), and 6 against 1 (6 / 1) were used to determine the diffusion matrix at composition 7, whereas two diffusion couples (16 / 14 + 17 / 10) were used to determined the diffusion matrix at composition 12. The advantage of using multiple diffusion couples to determine the elements of a diffusion matrix can be demonstrated by Monte Carlo simulations in which computer generated diffusion profiles, to which Gaussian noise representing analytical errors has been added, are inverted simultaneously for the elements of the diffusion matrix (Liang 1994; Trial and Spera 1994). Figure 7 compares the relative errors derived from simultaneous inversion of synthetic concentration profiles from single-direction (diffusion couple 6 / 1, open circles), twodirection (3 / 5 + 6 / 1 open triangles, 4 / 2 + 6 / 1 open diamonds, 4 / 2 + 3 / 5 crossed squares), and three-direction diffusion experiments (4 / 2 + 3 / 5 + 6 / 1, open squares) for a range of prescribed analytical uncertainties (Liang 1994). The diffusion couples intercept at composition 7 in composition space (Fig. 6). As shown in Figure 7, for a given analytical uncertainty, uncertainties of the estimated diffusion matrix from simultaneous inversion of concentration profiles from two-direction diffusion experiments are reduced by 220-370% compared to those derived from the single diffusion couple 6 / 1 using the same inversion procedure. Uncertainties in the inverted diffusion matrix are further reduced when concentration profiles from all three diffusion couples are used together in a joint inversion. Overall, uncertainties in the measured diffusion matrix from single-direction experiments are 2-6 times greater than those obtained

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SiO2

SiO2

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CaO

60

Al2O3

3

6

1 2

5 50

20

7

17

16 12

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CaO

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14 40

30

20

Al2O3

Figure 6. Composition space (in wt%) showing the starting compositions (open symbols) used to form diffusion couples for studying multicomponent diffusion in the ternary CaO-Al2O3-SiO2. Tie-lines join starting compositions from which diffusion couples were formed. These diffusion couples intercept at compositions 7 and 12, respectively in composition. Adapted from Liang and Davis (2002) and with permission of Elsevier. http://www.sciencedirect.com/science/journal/00167037.

from two-direction or three-direction diffusion experiments. Another important result of the Monte Carlo simulations shown in Figure 7 is the sensitivity of the inverted diffusion matrix to the quality of the measured concentration profiles. To accurately determine the elements of a diffusion matrix, one needs not only multiple diffusion couple experiments but also high quality concentration profiles. The latter can be achieved by long counting times and high spatial resolution in electron microprobe analysis of the diffusion charge. Several other factors are important in determining the best compositions to juxtapose in a given diffusion experiment. From the point of view of accurately measuring concentration changes along a diffusion profile, one should choose diffusion couples that have large concentration differences, and thus large signal to analytical noise ratios. This, however, is limited by the compositional dependence of the diffusion matrix, which poses real difficulties in terms of determining the diffusion matrix corresponding to a single composition. Hence, the compositions used for diffusion couples are usually a compromise between concentration differences large enough to allow for accurate measurement but small enough to minimize the compositional dependence of the diffusion matrix. Stability with respect to convection is another crucial factor, given the possibility of isothermal double-diffusive convection that can occur even when the lower density melt is placed above the more dense one (McDougall 1983; Turner 1985; Liang et al. 1994; Liang 1995; Richter et al. 1998). The difficulty here is that predicting convectively stable directions requires knowledge of the diffusion matrix, which is unknown at the outset. As a result, some of the initial experiments designed for the purpose of determining the diffusion matrix will prove unsuitable because of convection. Twodimensional X-ray concentration maps of the diffusion couples (e.g., Liang et al. 1994; Liang 1995; Richter et al. 1998) allowed easy identification of convective flow. It is often possible to obtain stable chemical diffusion runs in convectively unstable directions by using short run

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Figure 7. Comparisons of the relative uncertainties of the estimated diffusion matrix using the full nonlinear joint inversion procedure in three types of diffusion experiments: single direction experiments (open circles fitted by long / short dashed lines, couple 6 / 1), two direction experiments (couples 3 / 5+6 / 1, open triangle fitted by dotted lines; couples 4 / 2+6 / 1, open diamonds fitted by dashed lines; and couples 4 / 2+3 / 5, crossed squares fitted by long / short dashed lines), and three direction experiments (couples 4 / 2+3 / 5+6 / 1, open squares fitted by solid lines). The relative error, ∆Dij / Dij, of the estimated chemical diffusion coefficients at a given input data noise level (si) were calculated from Monte Carlo simulations using a normally distributed random number generator. Input diffusion coefficients (in ×10-11 m2 / s) used to generate the perturbed diffusion profiles are D11 = 4.2, D12 = -0.4, D21 = -0.5, and D22 = 1.4. Adapted from Liang (1994).

durations. The use of smaller diameter cylindrical capsules was also helpful in retarding the onset of convection (Liang 1994, 1995; Richter et al. 1998).

Inversion methods Once concentration profiles are determined from a set of chemical diffusion experiments, elements of the diffusion matrix can be calculated by minimizing the c2 defined as m  w - wijk c =∑∑∑  ijk  sijk k 1 =i 1 =j 1  = 2

Nd Nc N p

   

2

(35)

where wijk are predicted concentrations from the exact solution to the diffusion equation for m are measured concentrations of component component i at location j in diffusion couple k; wijk m ; Nd is the number i at location j in diffusion couple k; sijk is the measured uncertainty of wijk of diffusion couples used in the inversion; Nc ( = n) is the number of chemical components;

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and Np is the number of data points analyzed along each diffusion direction. Both Nc and Np can be functions of Nd. Liang (1994) compared five inversion methods through Monte Carlo simulations and recommended the following two for multicomponent diffusion in molten silicates: the Boltzmann-Matano method and the full nonlinear joint inversion method. The Boltzmann-Matano method has been widely used to extract EBDCs for diffusion in molten silicates, especially when diffusion coefficients vary as a function of spatial coordinates or melt composition (see Zhang 2010). To calculate the elements of the diffusion matrix from a one-dimensional diffusion experiment, we first make the substitution h = x / t1 / 2 (e.g., Fujita and Gosting 1956; Kirkaldy 1957) and integrate the resulting one-dimensional diffusion equation from -∞ to h and from h to ∞. Summing the results of the two integrations, we have a set of n-1 linear equations for the (n-1)2 elements of the diffusion matrix, n -1

∑ Dij j =1

dw j dh

w (∞)

= h

w ( h)

1 i 1 i dw h hdwi i 4 wi∫( h) 4 wi (∫-∞ )

(36)

where i = 1, 2, …, n-1; the derivative is evaluated at location h. At least (n-1)2 independent equations are needed to solve for Dij at a given location. The additional m-1 (m ≥ n) equations can be obtained either by evaluating Equation (36) at m-1 different locations along the diffusion profiles in the case of a constant diffusion matrix or, better yet, by conducting m-1 different chemical diffusion experiments all passing wj(h) in the compositional space. As shown in Figure 7, it is advantageous to carry out chemical diffusion experiments along different directions in the compositional space, even in the case of a constant diffusion matrix. Given (n-1)×(m-1) observations, Equation (36) represents a set of linear equations that can be easily inverted for the elements of the diffusion matrix. Perhaps one of the most significant sources of error in the Boltzmann-Matano analysis is the numerical evaluation of the derivatives and integrals from the measured concentration profiles. The derivatives, in particular, are very sensitive to the quality or noise of the measured concentration profiles. To calculate the derivatives and integrals in Equation (36), one usually fits the measured concentration profiles using a polynomial or piece-wise polynomial. Least-squares fitting using a piece-wise polynomial is called spline regression (e.g., de Boor 1978; Seber and Wild 1989). Spline regression is especially useful in fitting uphill diffusion profiles where high-order polynomial regression often introduces artificial oscillation. In the full nonlinear joint inversion, one solves the least squares problem as defined by Equation (35) directly using numerical methods. Here concentration profiles from all the elements analyzed in the diffusion couples around a given composition are inverted jointly for the diffusion matrix. Since solutions to the multicomponent diffusion equations are nonlinear (e.g., Eqns. 24-26), iterative procedures based on the Levenberg-Marquardt method (e.g., Press et al. 1989) are often used in the inversion. Trial and Spera (1994) outlined a least squares procedure in which one first inverts the eigenvalues and eigenvectors of the diffusion matrix from a set of measured diffusion profiles and then computes the diffusion matrix via. Equation (22b), viz., [D] = [B][L][B]-1. Liang (1994) developed a direct inversion procedure in which c2 is minimized with respect to the elements of the diffusion matrix Dij. To use the Levenberg-Marquardt method, one has to provide partial derivatives of c2 with respect to the fitting parameters such as ∂c2 /∂Dij. This is straightforward for a ternary system as explicit analytical expressions of ∂c2 / ∂Dij can be obtained from the exact solutions (Liang 1994). For systems with more than three components, ∂c2 /∂Dij can be calculated numerically using a finite difference method. In general, the full nonlinear joint inversion method is more efficient and reliable than the Boltzmann-Matano method. Figure 8 shows an example of measured concentration profiles from three diffusion couples centered at a haplo-basaltic melt (27.8% CaO - 7.4% MgO

2

SiO

2 3

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14 40

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45 16

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−5

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Figure 8. Concentration profiles (symbols) of three chemical diffusion couples in molten CaO-MgO-Al2O3-SiO2. The three diffusion couples were run together in a three-hole molybdenum capsule at 1500 °C and 1 GPa for 0.7 hours. These diffusion couples intersect at 27.8% CaO, 7.4% MgO, 14.90% Al2O3, and 49.8% SiO2 in composition space. Solid lines are the calculated diffusion profiles using a diffusion matrix (Eqn. 37a) obtained by a simultaneous inversion of the diffusion profiles from the three diffusion couples using the full nonlinear joint inversion methods discussed in the text. Starting compositions are shown as large open circles. The distance (x) is measured in 105× meters and time (t) in seconds.

0

10

20

CaO, MgO

Al O

55

Multicomponent Diffusion in Molten Silicates 427

Figure 8

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- 14.90% Al2O3 - 49.8% SiO2 1500 °C, 1 GPa, 0.7 hrs, Liang, unpublished data). With the Boltzmann-Matano method (Eqn. 36), we had difficulties finding a diffusion matrix that could fit all the measured diffusion profiles, whereas with the nonlinear least squares method, we obtained a diffusion matrix in a matter of seconds. With Al2O3 as the dependent variable, the inverted diffusion matrix for this composition is (in 10-11 m2 / s) Al2 O3  DCaO-CaO  Al O 2 3  DMgO-CaO  Al2 O3  DSiO2 -CaO

Al2 O3 DCaO-MgO Al2 O3 DMgO-MgO Al2 O3 SiO2 -MgO

D

Al2 O3   15.3 ± 1.9 DCaO-SiO -3.40 ± 1.85 -3.75 ± 1.58  2   Al2 O3 DMgO-SiO2  = 0.26 ± 1.16   -2.82 ± 1.48 10.4 ± 1.39  Al2 O3  -3.93 ± 1.22 -0.66 ± 1.14 6.35 ± 0.93  DSiO 2 -SiO2   

(37a)

Similarly, the fits to the measured diffusion profiles reported in Figure 4 of Richter et al (1998) for diffusion around a haplo-dacite melt (8.8% CaO - 7.2% MgO - 20.1% Al2O3 - 63.9% SiO2, 1500 °C and 1 GPa) are improved using the nonlinear least squares method as compared to the Boltzmann-Matano method. With Al2O3 as the dependent variable, the revised diffusion matrix for this melt composition is (in 10-11 m2 / s) Al2 O3  DCaO-CaO  Al O 2 3  DMgO-CaO  Al2 O3  DSiO2 -CaO

Al2 O3 DCaO-MgO Al2 O3 DMgO-MgO Al2 O3 SiO2 -MgO

D

Al2 O3   6.44 ± 0.31 0.89 ± 0.30 1.12 ± 0.24  DCaO-SiO 2    Al2 O3 DMgO-SiO2  =  -1.06 ± 0.21 2.69 ± 0.20 0.46 ± 0.15    -2.69 ± 0.16 -1.64 ± 0.15 -0.52 ± 0.11 Al2 O3 DSiO  2 -SiO2   

(37b)

In summary, the full nonlinear joint inversion procedure is the preferred method for inverting diffusion matrices from multicomponent diffusion experiments involving molten silicates.

Experimental results There are only a limited number of direct experimental measurements of the diffusion matrix for silicate melts and glasses. Diffusion matrices at selected compositions for number of simple molten silicate systems of petrologic interest have been reported: CaO-Al2O3-SiO2 (Sugawara et al. 1977; Oishi et al. 1982; Liang et al. 1996a; Liang and Davis 2002), MgOAl2O3-SiO2 and CaO-MgO-Al2O3-SiO2 (Kress and Ghiorso 1993; Richter et al. 1998), Na2OCaO-SiO2 (Wakabayashi and Oishi 1978, with diffusion matrix given by Trial and Spera 1994), K2O-Al2O3-SiO2 (Chakraborty et al. 1995), K2O-SrO-SiO2 (Varshneya and Cooper 1972), and K2O-NaO-Al2O3-SiO2-H2O (Mungall et al. 1998). However, full diffusion matrices for natural silicate melts have not been obtained to date, though selected elements of diffusion matrices were reported for melts of basaltic compositions (Kress and Ghiorso 1995; Lundstrom 2000, 2003; Morgan et al. 2006). We briefly review the main results of these studies in this section. According to their diffusive behaviors, we group them into alkali-bearing and alkali-free systems. Diffusion in molten CaO-Al2O3-SiO2, MgO-Al2O3-SiO2 and CaO-MgO-Al2O3-SiO2. The ternary CaO-Al2O3-SiO2 (CAS) and MgO-Al2O3-SiO2 (MAS) and quaternary CaO-MgOAl2O3-SiO2 (CMAS) are among the simplest multicomponent systems that bear key components of natural magmas. For the purpose of metallurgical processing, earlier studies of chemical diffusion in molten CAS were focused on the composition 40% CaO, 20% Al2O3, and 40% SiO2 (composition 14 in Fig. 6) (e.g., Sugawara et al. 1977; Oishi et al. 1982). Sugawara et al. (1977) conducted diffusion experiments along three compositional directions around composition 14 at 1450-1550 °C and 1 atm. They used data from two compositional directions and estimated multicomponent diffusion coefficients via the Boltzmann-Matano method (e.g., Fujita and Gosting 1956; Kirkaldy 1957). Oishi et al. (1982) conducted similar diffusion experiments along four compositional directions, three of which are identical to those of Sugawara et al. (1977), at 1370–1550 °C and 1 atm. Although the primary purpose of Oishi et al. (1982) was to test empirical models relating chemical diffusion coefficients to self diffusion coefficients

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and thermodynamic properties of the melt (e.g., Cooper 1965; Oishi 1965), multicomponent diffusion coefficients can be readily extracted from their published diffusion profiles. Liang et al. (1996a) compared the published diffusion profiles between the two studies and noticed a subtle but significant difference in diffusion along the direction of initially constant CaO: the directions of uphill diffusion in CaO with respect to the concentration gradients of SiO2 (or Al2O3) are reversed in the two reports (cf. Fig. 6 of Sugawara et al. 1977 and Fig. 3a of Oishi et al. 1982, all at 1500 °C and 1 atm). The data of Oishi et al. (1982) implies that the diffusive flux Al2 O3 of CaO is negatively coupled to the concentration gradients of SiO2 (DCaO-SiO < 0, where Al2O3 2 is the dependent variable), whereas the data of Sugawara et al. (1977) implies the opposite Al2 O3 (DCaO-SiO > 0). This is indeed confirmed by the inverted diffusion matrix using digitized 2 concentration profiles of Oishi et al. (1982, their Figs. 3a-3c) and the published diffusion matrix of Sugawara et al. (1977) (see Table 3 of Liang et al 1996a). It is not clear how and where this inconsistency arises. The diffusion matrix derived from the digitized concentration profiles of Oishi et al. (1982) is consistent with more recent studies of multicomponent diffusion in molten CAS at 0.5-2 GPa and 1440-1650 °C (Liang et al. 1996a; Liang and Davis 2002, see below). Nevertheless, the diffusion study of Sugawara et al. (1977) has stimulated considerable interests in the geological literature (e.g., Spera and Trial 1993; Kress and Ghiorso 1993; Chakraborty et al. 1995; Liang et al. 1996a). Spera and Trial (1993), for example, tested the Onsager’s reciprocal relations using the diffusion matrices of Sugawara et al. (1977) and the thermodynamic model of Berman and Brown (1984). They concluded that the Onsager’s reciprocal relations are satisfied to within the error of experimental uncertainties for diffusion around composition 14 in molten CAS. Interestingly, the Onsager’s reciprocal relations are better obeyed if the diffusion matrix derived from the digitized concentration profiles of Oishi et al. (1982) is used. Earlier studies of multicomponent diffusion along the join CaMgSi2O6-CaAl2Si2O8 by Kubicki et al. (1990) and in MAS and CMAS by Kress and Ghiorso (1993) have met with only limited success. As pointed out by Zhang (1993) and Trial and Spera (1994), the diffusion matrices reported by Kubicki et al. (1990) have complex eigenvalues and thus do not satisfy the constraint from nonequilibrium thermodynamics that all eigenvalues must be real and positive (e.g., de Groot and Mazur 1962; Haase 1969). The diffusion matrices obtained by Kress and Ghiorso (1993) are problematic because many of their published diffusion profiles were most likely affected by convection (Richter et al. 1998). As part of a broader study of self diffusion and chemical diffusion in molten silicates, Liang and coworkers measured diffusion matrices for selected melt compositions in the systems CAS, MAS, and CMAS over a range of temperatures (1440-1650 °C) and pressures (0.5-2 GPa) using the multiple diffusion couple method (Liang 1994 and unpublished data; Liang et al. 1996a; Richter et al. 1998; Liang and Davis 2002). Similar to Sugawara et al. (1977), they chose simple oxides CaO, MgO, Al2O3, and SiO2 as endmember components and inverted the elements of the diffusion matrix using the full nonlinear joint inversion method. Figure 9 shows a typical example of the measured concentration profiles for diffusion along three directions around composition 7 in molten CAS (filled circle in Fig. 6, 25% CaO - 15% Al2O3 - 60% SiO2, 1500 °C and 1 GPa). General results and conclusions regarding diffusive behaviors of the oxide components in molten CAS, MAS, and CMAS and their implications for chemical diffusion in natural silicate melts are summarized below. First of all, chemical diffusion in molten CAS, MAS, and CMAS show clear evidence of strong diffusive coupling among the oxide components (e.g., the uphill diffusion profiles in Figs. 8 and 9). The extent of diffusive coupling depends strongly on melt composition, moderately on pressure (0.5-2 GPa), and is relatively insensitive to temperature (1440-1650 °C). In general, the diffusive flux of SiO2 is strongly coupled to the concentration gradients of CaO and MgO in melts of more polymerized (or viscous) compositions (cf. Eqns. 37a and 37b, see also Table 4 in Liang et al. 1996a), except when the concentration gradients of CaO and MgO are very small compared to the concentration gradients of SiO2. The diffusive flux

430

Figure 9. Concentration profiles (symbols) of three chemical diffusion couples in molten CaO-Al2O3-SiO2 along three directions around composition 7: the direction of constant CaO (a), constant Al2O3 (b), and constant SiO2 (c) (Liang et al. 1996b). Concentration profiles calculated using the ionic and molecular common-force models are shown as the solid and dotted lines, respectively. The model-derived diffusion profiles were calculated by numerical integration of the coupled diffusion equations with a composition dependent diffusion matrix (Liang 1994). The distance (X) is measured in meters and time (t) in seconds. Adapted from Liang et al. (1997) and with permission of Elsevier. http://www.sciencedirect.com/science/journal/00167037.

Figure 9

Liang

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431

of Al2O3, on the other hand, is strongly coupled to the concentration gradients of CaO and MgO in melts of less polymerized compositions (see Fig. 6 in Liang et al. 1996a). For the range of melt compositions relevant to diffusion in natural molten silicates (45~65% SiO2), 2 O3 the off-diagonal terms of the diffusion matrix D Al j -SiO2 < 0 (j = CaO or MgO, e.g., Eqns. 37a and 37b). The presence of large CaO and / or MgO concentration gradients, therefore, accelerate the diffusion rate of SiO2 when the concentration gradients of SiO2 are opposite in direction to those of CaO and MgO. This is the case for chemical diffusion between molten basalt and more siliceous melt such as dacite or rhyolite (e.g., Watson 1982; Koyaguchi 1989; Baker 1990, 1991; Lesher 1994; van der Laan et al. 1994; Richter et al. 2003; Watkins et al. 2009). Further, the coupling of CaO and MgO with either Al2O3 or SiO2 reverses their signs between melts of more and less polymerized compositions (cf. Eqns. 37a and 37b). The diffusive fluxes of CaO and MgO are positively coupled to the concentration gradients of SiO2 in melts of more siliceous compositions, whereas the diffusive fluxes of CaO and MgO are negatively correlated to the gradients of SiO2 in melts of more mafic compositions. The former retards the diffusion rates of CaO and MgO, whereas the latter speeds up the diffusion rates of CaO and MgO, resulting in the characteristic asymmetric diffusion profiles of CaO and MgO for chemical diffusion between molten basalt and rhyolite. The overall effect of the coupled chemical diffusion between CaO and SiO2, MgO and SiO2, and possibly FeO and SiO2 (see below) is to make their respective diffusion distance very comparable to each other (see the diffusion profiles reported in Lesher 1994; Liang et al. 1996a; van der Laan et al. 1994; Richter et al. 2003; Watkins et al. 2009, among others). This helps to explain the similar EBDCs of the oxide compositions for chemical diffusion between molten basalt and rhyolite (e.g., Richter et al. 2003), despite the fact that the self diffusion coefficient of Si is significantly smaller than the self diffusion coefficients of Ca and Mg in molten CAS and CMAS (Shimizu and Kushiro 1991; Liang et al. 1996b; Liang et al. 2004; see also Fig. 11b below). Secondly, a comparison of the measured diffusion profiles (e.g., Figs. 8 and 9) and the calculated diffusion matrices for molten CAS, MAS, and CMAS show that the diffusive behaviors of CaO are very similar to that of MgO, with the net diffusion rate of CaO being slightly faster than that of MgO. This is in good agreement with the observed diffusion rates of CaO and MgO in basalts and basaltic andesites (e.g., Zhang et al. 1989; Kress and Ghiorso 1995). The diffusive behaviors of FeO in molten silicates are also similar to that of MgO, judging from their similar concentration profiles for diffusion in basaltic to rhyolitic melt compositions (e.g., Watson 1982; Koyaguchi 1989; Zhang et al. 1989; Baker 1990, 1991; Lesher 1994; van der Laan et al. 1994; Kress and Ghiorso 1995; Richter et al. 2003; Morgan et al. 2005; Watkins et al. 2009). In the absence of a complete diffusion matrix, one can simplify the treatment of multicomponent diffusion while retaining essential features of coupled diffusion by treating CaO + MgO + FeO as a single component (e.g., Watson and Baker 1991; see also Eqn. 27). Interdiffusion profiles in natural melts have numerous features that suggest similar coupling between Al2O3 and CaO + MgO + FeO and between SiO2 and CaO + MgO + FeO. Diffusion in Na2O- and K2O-bearing molten silicates. The importance of alkali elements to chemical diffusion in molten silicates of geological interest was recognized in the pioneering work of Watson (1982, see also the summary by Watson and Baker 1991). Chakraborty et al. (1995) conducted chemical diffusion experiments in the ternary K2O-Al2O3-SiO2 (KAS) at 1100-1600 °C and 1 bar. They calculated the diffusion matrices from the measured diffusion profiles in single-direction experiments using the Boltzman-Matano method. As discussed in the previous section, the off-diagonal elements of the diffusion matrix may not be well resolved unless one simultaneously inverts diffusion profiles in different directions in composition space (e.g., Liang 1994; Trial and Spera 1994). Nonetheless, the measured diffusion profiles and diffusion matrices from the experiments of Chakraborty et al. (1995) show strong evidence of coupled diffusion. Mungall et al. (1998) conducted multicomponent diffusion experiments in the system K2O-Na2O-Al2O-SiO2-H2O at 1300 °C and 1600 °C and 1 GPa. They used three

432

Liang

diffusion couples that intersect at a melt composition close to the low-pressure water-saturated eutectic granite composition and estimated the diffusion matrices using a forward iterative least squares method. Lundstrom (2000 and 2003) conducted basanitic melt and tholeiitic melt interdiffusion experiments at 1450 °C and 0.9 GPa and extracted an effective ternary diffusion matrix for the diffusion of Na2O and SiO2. Diffusion profiles reported by Chakraborty et al. (1995), Mungall et al. (1998), and Lundstrom (2000 and 2003) clearly show the contrasting diffusive behaviors between the alkalies and Al2O3 and SiO2, as noted earlier by Watson and coworkers (see also Fig. 1). Together with earlier studies of multicomponent diffusion in molten K2O-SrO-SiO2 (Varshneya and Cooper 1972) and Na2O-CaO-SiO2 (Wakabayashi and Oishi 1978; diffusion matrix given by Trial and Spera 1994), the results of Chakraborty et al. (1995), Mungall et al. (1998), and Lundstrom (2000 and 2003) demonstrate that the diffusive fluxes of Na2O is strongly (and negatively) coupled to the concentration gradients of SiO2, whereas the diffusive flux of K2O is moderately to weakly coupled to the diffusive flux of SiO2, although more work is needed to further confirm the latter observation. The diffusive flux of SiO2 is only weakly coupled to the concentration gradients of Na2O (and possibly K2O) in melts of basaltic composition (Lundstrom 2000), but strongly (and negatively) coupled to the concentration gradients of the alkalies in melts of more siliceous compositions. The diffusion experiments of Mungall et al. (1998) also shed light on the role of water in multicomponent diffusion in granitic melt. Analysis of water abundances in a diffusion charge is considerably more challenging than the analysis of other major and minor elements (see Cherniak et al. 2010). Mungall et al. (1998) analyzed water abundance at selected spots in a diffusion charge using FTIR and calculated the water diffusion profile by interpolating the measured FTIR data. Nonetheless, the reported water concentration profiles by Mungall et al. (1998, their Fig. 2a-2e) are in sufficient detail that inferences about the nature of diffusive coupling involving water can be made. A comparison of the water and silica concentration profiles between the two 1600 °C experiments shown in their Figures 2b and 2c reveals that the diffusive flux of silica is strongly coupled (with a negative sign) to the concentration gradients of water. The net diffusion rate of silica is significantly increased in the presence of an opposing water concentration gradient (see their Fig. 2b). When the concentration gradients of water are very small (their Fig. 2c) or along the same direction as those of silica (their Fig. 2e), the effective or net diffusion rate of silica decreases significantly. Further, the diffusive flux of water is negatively and strongly coupled to the concentration gradients of silica, similar to that of Na2O. Overall, the diffusive behavior of water is similar to that of Na2O, though the diffusive flux of water appears independent of the concentration gradients of the alkalies (their Fig. 2a). Clearly, more work is needed to further understand the role and behavior of water during multicomponent diffusion in molten silicates. Energetics of multicomponent diffusion. The energetics of multicomponent diffusion in molten silicates can be understood by examining the temperature and pressure dependence of the diffusion matrices and their respective eigenvalues and eigenvectors. Figure 10 compares the measured eigenvalues of the diffusion matrices in molten CAS (Sugawara et al. 1977, composition 14; Liang et al. 1996b, compositions 7 and 12 in Fig. 6), for two melt compositions in molten KAS (Chakraborty et al. 1995), and one composition in molten CMAS (Liang and Richter, unpublished data). The eigenvalues of [D] for the selected melt compositions in molten CAS, KAS, and CMAS obey simple Arrhenius relations. The activation energies for the major and minor eigenvalues of [D] at 1 GPa are 215±12 and 240±21 kJ / mol, respectively, for composition 7, and 192±8 and 217±17 kJ / mol, respectively, for composition 12 in molten CAS. These measured values are comparable to the activation energies for Ca and Si self diffusion at the same pressure and melt compositions (219 and 257 kJ / mol for self diffusion of Ca and Si in composition 7, and 211 and 197 kJ / mol for self diffusion of Ca and Si in composition 12, Liang, unpublished data). This is very similar to previous studies of the temperature dependence of self diffusion and chemical diffusion in molten CAS at 1 bar (Towers and Chipman 1957; Sugawara

Multicomponent Diffusion in Molten Silicates 1650

100 1

[7]

2

1 1

2 -11

Eigenvalues (in 10

2 2

1

CAIB

[14]

1

0.1

1300 (°C)

[14]

-1

m s )

1400

[12] 1

10

1500

433

2

[12]

[7]

3

CAIB CAIB

KAS-A 1

KAS-B

0.01

2

0.001

KAS-A

2

KAS-B

0.0001 5

5.2

5.4

5.6

5.8

6

6.2

6.4

-1

10000/T (K ) Figure 10. Plots of eigenvalues of diffusion matrices in selected ternary and quaternary systems as a function of the reciprocal temperature. Three compositions in CaO-Al2O3-SiO2: compositions 7, 12 (Liang and Davis 2002, at 1 GPa), and 14 (40 wt% CaO, 20% Al2O3, 40% SiO2, 1 bar, Sugawara et al. 1977). Two compositions in K2O-Al2O3-SiO2 (Chakraborty et al. 1995, at 1 bar): KAS-A (9% K2O 16% Al2O3, 75% SiO2) and KAS-B (16% K2O, 9% Al2O3, 75% SiO2). One composition in CaO-MgO-Al2O3-SiO2 (labeled as CAIB): 33% CaO, 8% MgO, 25% Al2O3, 34% SiO2. Diffusion matrices in the CAIB melts were calculated at 1 bar using the ionic common-force model of Liang et al. (1997), activity model of Berman (1983) and self diffusivities from the literature. Modified after Liang and Davis (2002) and with permission of Elsevier. http://www.sciencedirect.com/science/journal/00167037.

et al. 1977; Oishi et al. 1982). Figure 10 also shows the importance of melt composition and structure in determining the activation energies and magnitudes of the eigenvalues of diffusion matrices in these simple systems. At a given temperature (e.g., 1500 °C), diffusivities span over 5 orders of magnitude. Eigenvalues of the diffusion matrices in molten CAS, CMAS, as well as MAS (Richter et al. 1998), are 1-4 orders of magnitude higher than those in molten KAS. Kress and Ghiorso (1995) studied multicomponent diffusion in basalts at 1200-1450 °C and 1 bar. They conducted multicomponent diffusion experiments using the diffusion couple method with starting compositions made from a Columbia River basalt that was doped with approximately 5 wt% SiO2, TiO2, Al2O3, FeO, MgO, and CaO. They inverted the 5×5 diffusion matrices using the Boltzmann-Matano method of Kirkaldy (1957) while ignoring Na2O and K2O in the system. Although many of the diffusion profiles and the sense of diffusive coupling reported in their study are consistent with those observed in the simple systems, the eigenvalues of the diffusion matrices do not follow simple Arrhenius relationships. This is probably due to uncertainties associated with their inverted diffusion matrices, judging from the quality of and misfits to their measured diffusion profiles (see their Fig. 2).

Liang

Diffusivity (in 10-11 m2s-1)

434

10

100

(a)

Composition 7

(b)

Composition 12

λ1

λ1 10

DCa

λ2

1

Xbt 0.1

0

0.5

1

1.5

Pressure (GPa)

DAl

λ2

Vδ

-1/2

DSi

1 2

0

0.5

1

1.5

2

2.5

Pressure (GPa)

Figures 11a and 11b display the contrasting pressure dependence of the major and minor eigenvalues of [D] (l1 and l2, respectively) for the haplo-dacite melt (composition 7) and the haplo-basaltic melt (composition 12) at 1500 °C in molten CAS. The activation volumes for the major and minor eigenvalues of [D] at 1500 °C are 0.31±0.44 and 2.3±0.8 cm3 / mol, respectively, for composition 7, and -1.48±0.18 and -0.42±0.24 cm3 / mol, respectively, for composition 12. In spite of the similarities in their dependence on temperature, the major and minor eigenvalues of [D] respond very differently to pressure changes than self diffusivities of Ca and Si for composition 12 (Fig. 11b). Liang and Davis (2002) attributed the latter observation to a difference in the structure and thermodynamic properties of the melt. Finally, the eigenvectors of the diffusion matrices at a given melt composition, which is a measure of the extent of diffusive coupling among the oxide components, are not very sensitive to temperature and pressure variations for the simple ternary and quaternary systems. This is a very interesting observation and may be related to melt structure and diffusion species. Clearly, more work is needed to better understand the relationships among chemical diffusion, self diffusion, and the melt structure.

Empirical Models for Multicomponent Diffusion Empirical models The diffusion matrix [D] is a product of a phenomenological or kinetic matrix [L] and a thermodynamic matrix [G] (e.g., Onsager 1945; de Groot and Mazur 1962; Haase 1969), [D] = [L]⋅[G]. The thermodynamic matrix [G] can be calculated once the activity-composition relations are specified, while the kinetic matrix [L] can be determined by diffusion experiments. Although there is no formal theory that can be used to calculate the kinetic matrix [L] from first principles, a number of empirical models have been developed, relating the elements

Figure 11

Figure 11. Plots of major (l1) and minor (l2) eigenvalues of diffusion matrices at compositions 7 (a) and 12 (b) as a function of pressure (all at 1500 °C). Exponential fits to the measured data (open circles and crossed squares) are shown as solid lines. For comparison, calculated quartz diffusive dissolution (filled red circles, labeled as Xbt-1 / 2) and convective dissolution (open triangles, labeled as Vd) distances (normalized by the square root of time) as a function of pressure are also shown in (a). Composition of the starting melt used in the dissolution calculations is 30% CaO-20% Al2O3-50% SiO2. Clearly the rate of quartz dissolution is dominated by the rate of diffusion of the slower eigen-component in the ternary system. Also for comparison, best fit lines to measured Ca, Al, and Si self diffusion coefficients at the composition 12 and 1500 °C are shown in (b) (dashed lines labeled as DCa, DAl, and DSi, respectively, Liang, unpublished data). Adapted from Liang and Davis (2002) and with permission of Elsevier. http://www.sciencedirect.com/science/journal/00167037.

Multicomponent Diffusion in Molten Silicates

435

of [L] to the kinetic and thermodynamic properties of the components in the melt. These empirical models offer important insights into the nature and origin of the coupled diffusion. There are two types of empirical multicomponent diffusion models, one for a molecular fluid and the other for an ionic fluid (e.g., Darken 1948; Cooper 1965; Oishi 1965; Manning 1970; Lasaga 1979; Richter 1993; Zhang 1993; Liang 1994; Liang et al. 1997). In a binary system, Darken’s formulation is widely accepted for diffusion in a nonelectrolyte fluid, whereas the Nernst-Planck equation is frequently used for diffusion in an electrolyte solution (e.g., Zhang 2010). The difference between the two is the inclusion of Coulombic forces in the ionic models. Extensions of the Darken or the Nernst-Planck equation to a multicomponent fluid have been given by several authors (e.g., Cooper 1965; Oishi 1965; Manning 1970; Lasaga 1979; Liang et al. 1997). A key step in developing the various empirical models is to express the diffusive flux for a component of interest as a sum of two parts: one is due to the mobility or self diffusion of the component of interest and the other arises from interactions between the component of interest and its surrounding environment. Such interactions may include Coulombic forces and other model-dependent force or flow. In a volume fixed frame of reference, the unknown model-dependent force or flow can be constrained by imposing the no net volume flux (Eqn. 7a) constraint and zero electric current constraint, ∑in= 0 zi J i = 0. Since the model-dependent force or flow is the same for all the components in the system, Liang et al. (1997) referred to models derived by specifying the unknown flow as common-velocity models and models derived by choosing the model-dependent force as common-force models. To calculate the elements of the kinetic matrix [L], they chose simple oxides such as Na2O, CaO, Al2O3, and SiO2 as endmember components and mono-atomic cations such as Na+, Ca2+, Mg2+, Al3+, Si4+ and anion O2- as the diffusing species. Self diffusion coefficients of the mono-atomic ions are assumed the same as those for the respective atoms (Na, Ca, Al, Si, O, etc.) measured in the melt, though the actual diffusing species in molten silicates are not known. The use of simple oxides as endmember components for ionic fluid or melt is justified by Equation (21). We briefly summarize the empirical models discussed in Liang et al. (1997) below. In a volume fixed frame of reference, a general expression for the model-derived kinetic matrix [L] can be written as Lij = D j dij + Ci 0 (Vn 0 Dn - V j 0 D j ) Ai + ( pn zn Dn - p j z j D j ) Bi 

(38a)

where component n is the dependent variable; Dj and zj are the self diffusivity and charge number of cation j, respectively; Ci0 and Vj0 are the molar concentration and partial molar volume of the oxide component j, respectively; pj is the stoichiometric coefficient of the cation in oxide component j (for example, p = 2 for Al2O3). The pair of scaling factors Ai and Bi takes the values

(Ai, Bi) = (0, wi), (1, 0), (1, ti), (fi, 0), and (Wi, Qi) (38b)

for the Nernst-Planck equation, the molecular common-velocity model (Cooper 1965), the ionic common-velocity model, the molecular common-force model, and the ionic commonforce model, respectively. Let ck be the molar concentration of cation k. The scaling factors are n

= wi

zi Di = ti n 2 ∑ ck zk Dk

∑(d k =1

- Ck 0Vk 0 ) zk Dk = fi n 2 ∑ ck zk Dk

ik

Vi 0 Di n

∑C

k 0= k 0= k 1 =

= Wi

(38c)

2 k0 k0

V Dk

Di  ziW22 - (Vi 0 pi ) W21  Di (Vi 0 pi ) W11 - ziW12  = Qi W11W22 - W12W21 W11W22 - W12W21

(38d)

Liang

436 W W= = 11 22

n

∑C

k 1 =

2 n n 2 k0 k0 k 12 21 k0 k0 k k k k 0 k 1= k = k

V zD

W =

∑

C V D p

W =

∑c z D

k

(38e)

The Nernst-Planck equation is reduced to the mean field model of Lasaga (1979) when the anion is immobile, that is, D0 = 0. The kinetic matrix Equation (38a) is reduced to the modified effective binary diffusion model of Zhang (1993) when Ai = Bi = 0. Liang et al. (1997) examined the various empirical models in details. Here we highlight two important features of the model-derived kinetic matrix given in Equation (38a). First, the elements of the [L] matrix consist of two parts of different origin. The first part dijDj represents the intrinsic mobility of component i and is therefore retained in the diagonal term Lii of all the models. The second part of the [L] matrix consists of two terms that account for kinetic interactions among different components. The term Ci0(Vn0Dn - Vj0Dj)Ai results from volume relaxation, whereas the term Ci0(pnznDn - pjzjDj)Bi is due to ionic interactions. The overall effect of the kinetic interactions is to slow down larger (volume or charge) and faster diffusing species and to speed up smaller (volume or charge) and slower moving components in order to preserve the condition of no net volume and charge buildup in the melt. Secondly, the off-diagonal element of the [L] matrix, Lij (i ≠ j), is directly proportional to the concentration of oxide component i. Hence off-diagonal terms of [L] are negligible for the diffusive flux of a dilute component in many practical applications. For trace elements obeying Henry’s law, the off-diagonal elements of the diffusion matrix are also negligible. The limit of tracer diffusion is reached.

Experimental tests of the empirical models Specific assumptions were made in the derivation of the empirical multicomponent diffusion models. Whether any of these assumptions are reasonable for characterizing chemical diffusion in a silicate melt can be determined only by comparing model predictions with actual experimental data on molten silicates. Liang (1994) and Liang et al. (1997) conducted rigorous tests of the aforementioned empirical models using data from the ternary system CaO-Al2O3SiO2 at 1500 °C and 1 GPa. This system was chosen because all the relevant diffusion and thermodynamic data needed to for the [L] and [G] matrices are available. Specifically, they computed the [L] matrix corresponding to a given type of empirical diffusion model at several melt compositions using the self diffusion data from Liang et al. (1996b) and partial molar volume data from Lange and Carmichael (1987) and Courtial and Dingwell (1995). They also calculated the thermodynamic matrix [G] at the same melt compositions using a third-order Margules solution model that was parameterized using either experimentally measured activities of Al2O3 and SiO2 (Rein and Chipman 1965; Liang et al. 1997) or phase equilibrium studies in the ternary CAS (Berman and Brown 1984). They then compared the diffusion matrices derived from the empirical models with the diffusion matrices obtained from direct nonlinear joint inversion of the measured chemical diffusion profiles in molten CAS (Liang 1994; Liang et al. 1996a; see Table 7 in Liang et al. 1997). Further, they also compared the modelderived diffusion profiles calculated using composition-dependent [D] with the experimentally measured ones. Of the seven empirical models (including those of Zhang 1993 and Lasaga 1979) tested, they found that the common-force models (molecular and ionic) fit the observed diffusion profiles and diffusion matrices better than other empirical models. Both the kinetic matrix and thermodynamic matrix contribute significantly to the large off-diagonal elements of the diffusion matrix for chemical diffusion in molten CAS. Figure 9 is an example from Liang et al. (1997) in which chemical diffusion profiles calculated using the ionic (solid lines) and molecular (dotted lines) common-force models are compared with experimentally measured concentration profiles (Liang et al. 1996a) around composition 7 along the directions of constant CaO (Fig. 9a), constant Al2O3 (Fig. 9b), and constant SiO2 (Fig. 9c). The model-derived diffusion profiles were calculated by numerical integration of the coupled diffusion equations with composition-dependent [D]. The main source

Multicomponent Diffusion in Molten Silicates

437

of the compositional dependence of the model-derived diffusion matrices comes from measured changes in the self diffusion coefficients with composition (Liang et al. 1996b). With the exception of CaO in Figure 9a and a small shift in the position of the Al2O3 profile (Fig. 9b), the fits to the measured chemical diffusion profiles using the ionic common-force model (solid lines in Fig. 9) are comparable to and in many instances better than the best fits found using a constant [D] derived from a nonlinear least squares fit to the chemical diffusion profiles (see Fig. 3 in Liang et al. 1996a). The ability to include compositional effects in [D] is an important advantage of the model-derived diffusion matrices. The misfits in CaO along the direction of constant CaO in Fig. Al2 O3 9a is the result of off-diagonal term DCaO-SiO < 0 which is the opposite of what was measured 2 directly by Liang et al. (1996a). Liang et al. (1997) attributed the misfits to uncertainties in thermodynamic solution properties that are needed to relate the thermodynamic driving force (chemical potential gradients) to measurable quantities such as concentration gradients (the [G] matrix). Another potential source of error is the assumption of mono-atomic diffusing species, which is convenient but somewhat arbitrary. Oishi et al. (1982) explored several choices of diffusing species for diffusion in molten CAS using a common velocity model, although it is not known how to assign self diffusivity to a complex diffusing species such as SiuAlvOw using measured self diffusivity data, where u, v, w are integers. Nonetheless, such exercises may prove informative in understanding or testing diffusion mechanisms in molten silicates. To date, full diffusion matrices for natural silicate melts have not been measured, due in part to challenges in the direct and accurate measurements of the diffusion matrices for systems having 5 or more major components. With a few exceptions (such as Na), self diffusion coefficients of the major, minor, and trace elements in molten silicates can be routinely measured with accuracy (e.g., Lesher 2010). Thermodynamic models of silicate melts are also improving. The very good match of the model-derived diffusion matrices and concentration profiles with the experimentally measured ones in molten CAS encourages the application of the ionic common-force model to multicomponent diffusion in molten silicates of geological interests.

Geological Applications Applications of multicomponent diffusion in molten silicates to petrologic and geochemical problems are mainly in two areas: (1) to understand the nature and origin of coupled chemical diffusion; and (2) to model mass transfer processes involving silicate melts. Examples of (1) include, but are not limited to, uphill diffusion, transient gradient or equilibrium partitioning, effective binary diffusion, and have already been given in the previous sections. Here we will use diffusive mixing of isotopes, crystal dissolution, and crystal growth as examples of (2) and show how knowledge of multicomponent diffusion and thermodynamic phase equilibrium can be used to model mass transfer processes involving multicomponent crystals and melts in a self-consistent way.

Modeling isotopic ratios during chemical diffusion in multicomponent melts Isotopic ratios have often been used to fingerprinting geochemical mass transfer processes involving heterogeneous sources (e.g., magma mixing and assimilation). It has been demonstrated experimentally that the time scale of diffusive re-equilibration for isotopic ratios in silicate melts is in general different from (often shorter than) the time scale of re-equilibration for the respective element during chemical diffusion in molten silicates (e.g., Lesher 1990, 1994; Liang 1994; van der Lann et al. 1994; Richter et al. 2003, 2009; Watkins et al. 2009). To model isotope fractionation during chemical diffusion in multicomponent melt, one has to consider both self diffusion and chemical diffusion. In a volume fixed frame of reference, the diffusive flux of isotope k of element i can be written as (Liang 1994; Richter et al. 1999)

438

Liang n -1

(

)

J ik = - Dik ∇Cik - ∑ cik DijV - dij Dik ∇C j j =1

(39)

where Dik is the self diffusion coefficient of isotope k; Cik is the molar concentration of isotope k; Cj is the molar concentration of element j; DijV is defined by Equation (13); cik is the mole fraction of k in element i; and dij = 1 if i =j and dij = 1 when i ≠ j. Equation (39) shows that the diffusive flux of individual isotopes depends not only on self diffusion of the respective isotope but also chemical diffusion of the independent components in the multicomponent melt. Figures 12a-12c show an experimental validation of Equation (39) in the case of Si and Ca isotopic fractionation during chemical diffusion in molten CAS. In this case the diffusion matrix DijV was calculated using the ionic common-force model and experimentally determined self diffusion coefficients and thermodynamic properties of the melt in the ternary CAS (Liang 1994). Equation (39) has been used to in several recent studies of isotope fractionation during chemical diffusion in molten silicates (e.g., Richter et al. 1999, 2003, 2009; Watkins et al. 2009). It can also be used to study crystal growth and dissolution in multicomponent melts. When only EBDCs are available, which is the case for chemical diffusion in natural silicates, we can rewrite Equation (39) in terms of the EBDC with the help of Equations (33)(34),

(

)

J ik = - Dik ∇Cik - cik DiE - Dik ∇Ci n -1

∂C j

j =1

∂Ci

DiE = ∑ DijV

(40a) (40b)

where DiE is the EBDC of (the oxide) component i in the melt while holding all other components as the dependent variable (Cooper 1968). Equation (40a) contains two time scales, l 2 / Dik and l 2 / [cik ( DiE - Dik )], where l is a reference length scale, and hence provides a simple explanation to the decoupling of chemical concentration and isotopic ratio during diffusive reequilibration in a multicomponent melt (e.g., Lesher 1990, 1994).

Convective crystal dissolution in a multicomponent melt There is a large body of experimental and theoretical works on crystal dissolution in molten silicates in the geological literature. Most early studies of crystal dissolution in molten silicates treated the crystal + melt as an effective binary system to simplify the mathematical treatment (e.g., Cooper and Kingery 1964; Zhang et al. 1989). One of the main drawbacks of the effective binary approach is that one is not able to predict interface melt and crystal compositions even when the liquidus and solidus of the dissolving crystal are known. This is because for a binary system there are no degrees of freedom at the crystal-melt interface at constant temperature and pressure. For systems involving three or more components, the degrees of freedom at the crystal-melt interface are greater than one at a constant temperature and pressure. The interface melt composition is on the liquidus line or surface in composition space (see Fig. 13 below). In a series of diffusive crystal dissolution studies, Liang (1999, 2000, 2003) showed how the knowledge of multicomponent diffusion can be used to determine compositions of the crystal and melt at the crystal-melt interface. Given the interface melt and solid compositions, dissolution rate can be readily calculated. Figure 13 shows the relative positions of a ternary crystal (point S) and starting melt (point M) with respect to the liquidus of the dissolving crystal (solid line) in composition space (modified after Liang 1999). For simplicity, we neglect coupled diffusion by assuming the off-diagonal elements of the diffusion matrix D12 = D21 = 0. If the diagonal elements of the diffusion matrix are the same, D11 = D22, kinetically the ternary system is reduced to a binary system along the tie-line MS in Figure 13. The interface melt composition is uniquely

Multicomponent Diffusion in Molten Silicates

439

Figure 13 Figure 12. Isotope fractionation during chemical diffusion in molten CaO-Al2O3-SiO2. Application of the ionic common-force model to the measured chemical diffusion profiles (plus symbol) from the isotopically spiked chemical diffusion couple 3 / 7 (Fig. 6, 1500 °C, 1 GPa, 0.5 hrs). Fits (solid lines) to the measured concentration profiles (a), isotopic ratios of 42Ca / 40Ca (b) and 30Si / 28Si (c) across the diffusion couple. The diffusion species used in the calculation are Ca2+, Al3+. Si4+, and O2-. Adapted from Liang (1994).

Liquidus p1w10 + p2w20 = s L

(w10, w20) L2

w2 Melt

M2

L0

M (w11, w21)

w1

Crystal S (ws1, ws2)

Figure 13. Schematic diagram showing relative positions of the liquidus (solid line), dissolving crystal (S), and starting melt (M) in composition space. The dashed tie-line MS joins the starting melt and dissolving crystal. The melt composition at the crystalmelt interface is at L0 if D11 = D22, but at L if D11 > D22. If the starting melt composition is at M2, the interface melt composition is at L2. Modified after Liang (1999) and with permission of Elsevier. http:// www.sciencedirect.com/ science/journal/00167037.

Liang

440

determined by the intercept of the tie-line and the liquidus line (point L0 in Fig. 13). If D11 > D22, more dissolved component 1 diffuses away from the crystal-melt interface when the abundances of component 1 and 2 are higher in the crystal than those in the melt (ws1 > wf 1 and ws3 > wf 2). Given the orientation of the liquidus with respect to the tie-line MS, concentration of component 2 in the interface melt (w20) increases along the liquidus (point L in Fig. 13). (Likewise, the interface melt composition is at L2 if the starting melt composition is at M2 in Fig. 13.) This is an important feature of crystal dissolution in multicomponent melt. It arises when different components in a melt diffuse at different rates. Quantitatively, we can determine the interface crystal and melt compositions (and crystal dissolution rate) by considering mass balance across the crystal-melt interface (Liang 1999, 2000, 2003). In the case of convective dissolution of a pure crystal (i.e., fixed composition ws1 and ws2) in a large melt reservoir (composition wf 1 and wf 2), we can use the following approximate flux balance equations for the two independent components to calculate the interface melt composition (w10 and w20) and dissolution rate (V), D11 D22

w f 1 - w10 d w f 2 - w20 d

+ w10V ≈ ws1V

(41a)

+ w20V ≈ ws 2V

(41b)

where d is the thickness of the compositional boundary layer, which is a function of the vigor and style of convection around the dissolving crystal. Expressions for the compositional boundary layer thickness can be found in the literature (e.g., Kerr 1995; Zhang and Xu 2003). The first term on the left-hand side of Equation (41a) or (41b) is an approximation to the diffusive flux for the respective component in the interface melt. Equations (41a) and (41b) have three unknown: w10, w20 and V. The third equation needed to solve this problem is the liquidus of the dissolving crystal (i.e., the phase equilibrium constraint). For purpose of demonstration, we consider a linearized liquidus, p1w10 + p2 w20 = s

(41c)

where p1, p2, and s are constants at a given temperature and pressure. From Equations (41a)(41c), we obtain simple analytical solutions for the rate of convective dissolution and the interface melt composition in a ternary system, dV ≈

s - ( p1w f 1 + p2 w f 2 )

p1 ( ws1 - w f 1 ) D11

+

p2 ( ws 2 - w f 2 )

(42a)

D22

= w10

Pe11ws1 - w f 1 Vd = Pe11 Pe11 - 1 D11

(42b)

= w20

Pe22 ws 2 - w f 2 Vd = Pe11 Pe22 - 1 D11

(42c)

where Pe11 and Pe22 may be referred to as dimensionless dissolution Péclet numbers for components 1 and 2, respectively. Equations (42a)-(42c) can be readily generalized to systems with more than three components. Hence given the hydrodynamic (d), thermodynamic (p1, p2, and s), and kinetic (D11 and D22) properties of a multicomponent crystal-melt system, the crystal dissolution rate is completely determined for a given starting melt composition, provided the far field melt composition remains constant during the course of crystal-melt interaction (see

Multicomponent Diffusion in Molten Silicates

441

below). There is no need to provide the interface melt composition for crystal dissolution in multicomponent melt. Actually, the interface melt composition depends on the vigor and style of convection around the dissolving crystal (d). Liang and Davis (2002) used Equation (42a) to calculate the convective dissolution rate of quartz in molten CAS and identified the minor eigen-component as the rate-limiting component for quartz dissolution (Fig. 11a). The latter can be understood by rewriting Equation (42a) as, p1 ( ws1 - w f 1 ) p2 ( ws 2 - w f 2 ) 1 ≈ + dV D11  s - ( p1w f 1 + p2 w f 2 )  D22  s - ( p1w f 1 + p2 w f 2 )     

(43)

Hence, all else being equal, the slowest diffusing component determines the rate of crystal dissolution, a conclusion consistent with many previous crystal dissolution studies (e.g., Watson 1982; Harrison and Watson 1983; Zhang et al. 1989; Woods 1992; Liang 1999, 2003). Liang (2003) referred this as the series rule for crystal dissolution.

Crystal growth and dissolution in a multicomponent melt Strictly speaking, Equation (42a) can only be used in simple setups where the melt composition in the far field (wf 1 and wf 2) remains constant during the course of crystal-melt interaction. This may be realized when the volume of dissolving crystal is small compared to the volume of the melt (i.e., dissolution in an effectively infinite melt reservoir). When the volume fraction of the crystal is large, such as a crystal-melt mush, the far field melt composition varies as a function of dissolution time. In this more general case, we can determine the dissolution rate V = dR / dt and the average melt composition (w f 1,w f 2) using the following mass conservation equations (Liang 2003) d

s - ( p1w f 1 + p2 w f 2 ) dR = dt p1 ( ws1 - w f 1 ) p2 ( ws 2 - w f 2 ) + D11 D22

(L - R)

dw f 1

(L - R)

dw f 2

dt dt

(44a)

= ( w f 1 - ws 1 )

dR dt

(44b)

= ( w f 2 - ws 2 )

dR dt

(44c)

where L is the half-length of the crystal + melt dissolution couple, R is the radius of the dissolving crystal. The interface melt composition can be calculated using Equations (42b) and (42c) if one replaces the far field melt composition by the average melt composition in these equations. Hence, the interface melt (and solid) composition varies as a function of time during crystal dissolution in a finite geometry even under the assumption of local thermodynamic equilibrium at the crystal-melt interface. This is shown schematically in Figure 13: as the far-field melt composition changes from M to M2, the interface melt composition shifts from L to L2 along the liquidus. Liang (2003) solved a pair of diffusion equations for diffusive crystal dissolution in ternary melts using a finite difference method and showed how the crystal dissolution rate and the melt and solid compositions at their interface vary as a function of dissolution time in ternary melts of finite size (for additional information, see the movies of crystal dissolution published with that paper). Finally, it is worth noting that Equations (44a)-(44c) can also be used to study crystal growth (and dissolution) under non-isothermal conditions, by allowing diffusivities and the liquidus surface (p1, p2, and s in Eqn. 41c) to vary as a function of temperature and time (e.g., cooling rate). The numerator on the right hand side of Equation (44a) is a measure of the extent of super-saturation or undercooling. Figure 14a shows an example of the variation of

Liang

442 2

10

−10

10

(a)

10

) ,E2B (E 1

−2

10

(E

,E

1

)

2

C

f

−4

10

(E

−12

1

10

D (m2/s)

Growth Rate (µm/hr)

(b)

−11

10

0

(E

−13

1

10

(E

−14

1

10

2

,E 2

)

A

)

B

)

C

D11

−15

10 (E1, E2)A = (190, 210)

−8

10

−10

10

2

,E

−6

10

,E

1000

−16

10

(E1, E2)B = (230, 260)

(E1, E2)C = (280, 350) 1100

1200

1300 o

T ( C)

1400

1500

1600

−17

10

5

D22 5.5

6

4

6.5

−1

10 /T (K )

7

7.5

8

Figure 14. Variation of calculated crystal growth rate as a function of temperature for three choices of the activation energies (cases A, B, C) for chemical diffusion in molten silicates (a). Diffusion coefficients (D11 and D22, solid and dashed lines, respectively) used in the calculations are shown in (b). In this hypothetical example, we used a cooling rate of -4 °C / hr and a temperature-dependent liquidus p1 = 0.2, p2 = 1.0, and s = 0.7 - 0.06time / (T0 - 1000), where T0 is the temperature at the beginning of crystallization; and time is in hours.

crystal growth rate as a function of temperature for three choices of the activation energies for diffusion (Fig. 14b) calculated using Equations (44a)-(44c) and a prescribed cooling rate of -4°C / h. Similar variations of crystal growth rate as a function of temperature were observed for the growth of plagioclase of variable compositions (e.g., Lasaga 1998). Equations (44a)(44c) are in essence a generalization of the continuous growth model that was originally developed for crystal growth from a unary melt (e.g., Flemings 1974; Lasaga 1998). The foregoing examples demonstrate that multicomponent diffusion is important even when the off-diagonal elements of the diffusion matrix are all zero. To use the equations outlined in this section for crystal growth or dissolution in systems with coupled diffusion (i.e., Dij ≠ 0, i ≠ j), one first makes use of the linear transformation, U = [B]-1w (Eqn. 22a), to rotate the concentration vectors (ws, wf, wf , and w0) in compositional space such that the diffusion matrix is diagonalized (Eqn. 22b) in the new compositional space (U1, U2, etc). One then uses the new concentration U in place of w in Equations (42a)-(42c) or (44a)-(44c) to calculate the crystal growth or dissolution rate. The interface melt composition is given by the linear transformation w0 = [B]U0, where U0 is the interface melt composition in the new compositional space.

Future Directions The theory of multicomponent diffusion in molten silicates is firmly established by the application of nonequilibrium thermodynamics. The methodologies for the design and execution of multicomponent diffusion experiments at high temperatures and pressures and for accurately extracting diffusion matrices have matured through systematic studies of multicomponent diffusion in simple molten silicates and Monte Carlo simulations. The nature and origin of diffusive coupling among the major oxide components are relatively well understood, if not quantitatively determined, for natural systems. Applications of empirical models relating the

Multicomponent Diffusion in Molten Silicates

443

elements of diffusion matrices to the kinetic and thermodynamic properties of the components in the multicomponent melts show promising results for simple systems. With ever increasing capabilities in measuring geological materials at higher precision and finer spatial scale, the demands for greater understanding of multicomposition diffusion in molten silicates will likely to increases. We are now in a position to expand multicomponent diffusion studies to geologically more relevant synthetic and natural melt systems. In terms of future research directions, the followings topics or problems come to mind: (1) There is an immediate need for more complete, if not full, diffusion matrices for studying diffusive and advective mass transfer in natural magmas such as basalt, andesite, dacite, and rhyolite. (2) Given the roles of water in controlling transport properties in silicate melts and crystals, it is important to understand how water (as well as other volatile species) behaves and interacts diffusively during multicomponent diffusion in molten silicates. The experimental results of Mungall et al. (1998) are intriguing, begging for more works. (3) Multicomponent diffusion in partially molten silicates is another frontier. Very little work has been done on this subject to date, yet a majority of igneous systems involve crystal-melt two phase or multi-phase aggregates. (4) Applications of multicomponent diffusion to petrologic and geochemical problems remain top priority. This will continue to guide and motivate laboratory, theoretical, and computational studies of multicomponent diffusion in molten and partially molten silicates. The simple examples given in the proceeding section highlight the advantage of multicomponent diffusion over effective binary diffusion. What is needed are more case studies involving geological field observations in which knowledge of multicomponent diffusion in molten silicates plays a crucial role.

acknowledgments I wish to thank Frank M. Richter and E. Bruce Watson for many useful advice and suggestions during various stages of my studies of diffusion and crystal dissolution in molten and partially molten silicates, Youxue Zhang for the opportunity to contribute this chapter, and Chip Lesher, Youxue Zhang, and Daniele Cherniak for their careful reviews. I acknowledge the supports of NSF and NASA on my various diffusion related works, most recently through NSF grant EAR-0911501 and NASA grant NNG06GF75G.

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Reviews in Mineralogy & Geochemistry Vol. 72 pp. 447-507, 2010 Copyright © Mineralogical Society of America

Oxygen and Hydrogen Diffusion in Minerals John R. Farver Department of Geology Bowling Green State University Bowling Green, Ohio 43403, U.S.A. [email protected]

INTRODUCTION This chapter provides a summary and review of experimentally determined oxygen and hydrogen volume diffusion in minerals. A very extensive and detailed review of stable isotope exchange processes including oxygen and hydrogen volume diffusion can be found in Cole and Chakraborty (2001) and a detailed review of hydrogen diffusion in minerals, especially nominally anhydrous minerals, can be found in Ingrin and Blanchard (2006). In addition, a detailed review of oxygen and hydrogen diffusion in silicate melts is provided by Zhang and Ni (2010), and oxygen and hydrogen diffusion rates can also be found in the chapters, in this volume, on specific minerals and mineral groups (e.g., Van Orman and Crispin 2010; Cherniak 2010). Interested readers are encouraged to refer to these reviews for additional information on oxygen and hydrogen diffusion in geological materials. In this chapter only volume diffusion through the crystal lattice will be considered. The majority of the experiments employed isotope tracers (e.g., 18O, 2H). The isotope diffusion of hydrogen in the nominally anhydrous minerals is regarded as an impurity tracer diffusion mechanism, and the effective diffusion coefficients obtained from the hydrogen uptake or extraction experiments correspond to chemical interdiffusion of different species. The relationship between the measured effective diffusivity and the hydrogen diffusivity is governed by the specific reaction involved (e.g., see Kohlstedt and Mackwell 1998).

EXPERIMENTAL METHODS Elsewhere in this volume, a detailed outline and discussion of the experimental methods employed in diffusion studies is presented by Watson and Dohmen (2010). Therefore, only a brief overview of methods commonly employed for oxygen and hydrogen diffusion studies is presented below.

Bulk exchange experiments Hydrothermal bulk exchange. Carefully sized powders of either natural or synthetic crystals of the mineral of interest are loaded into a sealed tube with a known amount of 18O and/or 2 H- enriched water. Most hydrothermal experiments are conducted at elevated confining pressures using cold-seal reaction vessels. After the diffusion anneal, the fractional approach to equilibrium is calculated by comparing the 18O/16O or 2H/H ratio of the sample material before and after the experiment as determined by conventional mass spectrometry. For oxygen isotope analysis, the samples are typically prepared by conventional BrF5 extraction of total oxygen, converted to CO2 (Clayton and Mayeda 1963), and analyzed using a gas source mass spectrometer.

1529-6466/10/0072-0010$10.00

DOI: 10.2138/rmg.2010.72.10

Farver

448

Values for the fractional approach to equilibrium, f, are calculated for the run products using the appropriate equilibrium fractionation equations (e.g., Chacko et al. 2001). The fractional approach to equilibrium is used to determine the value of (Dt/a2)1/2 by using the solution to the diffusion equation for uptake from a well-stirred solution of limited volume by a sphere (Crank 1975, p. 95, Fig. 6.4) or by an infinite cylinder (Crank 1975, p. 79, Fig. 5.7), depending upon the symmetry of the sample. D is the diffusion coefficient, t is the run duration, and a is the radius of the sample grains. The diffusion coefficient can also be obtained from uptake by a sample with a plane sheet symmetry using the equation (Dt/l2)1/2 where l is the half thickness of the sheet (Crank 1975, p. 59, Fig. 4.6). Dry bulk exchange. Dry (water absent) oxygen bulk exchange experiments using carefully sized powders of natural or synthetic minerals have also been performed. These experiments are typically run at ambient pressures (0.1 MPa) by exposing the sample to flowing CO2 or O2 gas that has a measurably different 18O/16O ratio than the sample. By using a flowing stream the isotope composition of the gas remains constant throughout the diffusion anneal and only the isotope change in the mineral grains need be considered. The diffusion coefficients can be calculated from these so-called partial exchange experiments (Muehlenbachs and Kushiro 1974; Connolly and Muehlenbachs 1988) using the diffusion equation for the boundary conditions of a gas of isotopically constant composition and a spherical grain geometry:

δ18Of − δ18Oeq 6 n =∞ 1 = ∑ exp  −n2t /τ δ18O i − δ18Oeq π2 n =1 n 2

(

)

where δ18O refers to the final, f, initial, i, or equilibrium, eq, value of the oxygen isotope ratio of the sample, t is the duration of the experiment, and τ = (r2/π2D), where r is the grain radius and D is the diffusion coefficient (Muehlenbachs and Kushiro 1974). The oxygen isotope analysis of the run products is typically obtained using the BrF5 method (Clayton and Mayeda 1963). This equation assumes a spherical symmetry for the grains, and the validity of the equation can be assessed using a range of run durations and/or grain sizes.

Single crystal experiments The use of single crystals and spatially resolved analysis of diffusion gradients overcomes most of the potential problems associated with the bulk exchange methods (Zhang 2008, p. 288-292). Details of the single crystal method can be found in several previous papers and reviews (e.g., Brady 1995; Cole and Chakraborty 2001; Watson and Dohmen 2010) and only a brief overview of the method will be presented here. In the single crystal method, samples consist of cut and polished or cleaved pieces of single crystals, typically with a specific crystallographic orientation. It is common to thermally anneal the samples (“pre-anneal”) to remove any surface defects acquired during the mechanical polishing and/or to equilibrate the point defect chemistry with the conditions of the diffusion anneal (e.g., at specific buffered pO2 or fH2O conditions, see Ryerson et al. 1989). The diffusion tracer sources may be in a surrounding solution or gas phase, or as a thin-film coating the surface or fine-grained source surrounding the sample. The experiments can be done at ambient pressure (0.1 MPa) in a range of gas compositions, or at elevated pressures using cold-seal pressure vessels, internallyheated vessels, or solid-medium, such as a piston cylinder or multianvil apparatus. The pO2 and fH2O of the experiments can be buffered using gas-mixing (at 0.1 MPa) or a variety of solid oxide pairs or mineral breakdown reactions (e.g., see Huebner 1971). After the diffusion anneal, the tracer concentration profile (e.g., 18O, 2H) is measured as a function of distance from the surface exposed to the tracer. The concentration profile is typically measured by Secondary Ion Mass Spectrometry (SIMS), Nuclear Reaction Analysis (NRA), or Fourier Transform Infrared Spectrometry (FTIR)—a brief description of these methods is provided below.

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The single crystal experiments are typically designed such that the measured concentration profile can be modeled by the constant surface concentration (infinite, well-stirred reservoir) solution to the diffusion equation for a semi-infinite medium: C( x , t ) − C 0 C1 − C0

 x  = erfc    2 Dt 

where Cx,t is the concentration at some distance x from the surface, C0 is the initial concentration in the sample, C1 is the concentration at the sample surface, D is the diffusion coefficient, t is the duration of the anneal, and erfc is the complementary error function. The value of D is often obtained by taking the inverse error function of the concentration ratio and plotting it against the distance from the source. The slope of a straight line fit to the erf−1 values versus distances plot is 1/[2(Dt)1/2], from which D can be evaluated knowing the duration, t, of the experiment. In the case where the sample geometry corresponds to a solid with homogeneous initial concentration bounded by two infinite parallel planes of thickness 2L in contact with an infinite reservoir, the solution to the diffusion equation takes the form of:

C(x,t) − C0 C1 − C0

= 1−

 − D(2n + 1)2 π2t   (2n + 1)πx  4 ∞ ( − 1)n exp   cos  ∑  π n = 0 2n + 1 4L2 2L    

where the origin is located at the mid-point of the slab. Appropriate solutions to the diffusion equation for other geometries and boundary conditions can be derived as well (see Crank 1975; Ingrin and Blanchard 2006; Zhang 2010). The use of single crystals followed by microbeam (e.g., SIMS) or macrobeam (e.g., NRA) analysis for diffusion experiments has the decided advantage over bulk experiments in that the effects of surface reactions like dissolution-precipitation on the tracer uptake can be easily evaluated. In addition, anisotropy of the diffusion can be directly measured using oriented crystals.

ANALYTICAL METHODS A detailed outline and discussion of the analytical methods employed in diffusion studies is presented by Cherniak et al. (2010) and only a brief overview of analytical methods commonly employed for oxygen and hydrogen diffusion studies is presented below.

Mass Spectrometry The majority of the run products from bulk exchange diffusion experiments have been analyzed using conventional gas source mass spectrometers. Secondary ion mass spectrometry (SIMS) has been used extensively for the analysis of single crystal oxygen diffusion experiments (see Giletti et al. 1978; Valley et al. 1998) and occasionally for H-2H exchange experiments (Vennemann et al. 1996; Suman et al. 2000). SIMS analysis involves bombarding the sample surface with an ion beam (typically O−, Cs+, or Ar+) and the secondary ions emitted from the sample are measured with a mass spectrometer. As the primary beam sputters away the sample it provides an average concentration as a function of distance from the sample surface in the form of a depth profile that is the diffusion profile. By controlling the sputter rate, diffusion profiles 10 μm), diffusion profiles, the sample can be sliced in half in a direction normal to the diffusion interface and a step-scan profile can be collected with sample spot sizes down to ~10 μm (e.g., see Farver and Yund 1999). SIMS analysis provides a direct determination of the isotopic concentration versus depth and allows simultaneous determination of multiple isotopes/elements during depth profiling to ensure chemical homogeneity.

450

Farver

Nuclear Reaction Analysis Nuclear Reaction Analysis (NRA) has also been widely employed for determining diffusion profiles for oxygen (see Ryerson et al. 1989; Watson and Cherniak 1997) and hydrogen or deuterium (Dersch et al. 1997). The technique involves using a particle accelerator to direct a beam of monoenergetic particles at the surface of the sample. For 18O analysis, the sample is irradiated with monoenergetic protons inducing the reaction 18O(p,α)15N which produces α particles. The energy of the α particles as they exit the sample is dependent upon the depth within the sample at which they were generated, and the intensity of the α particle signal at a given energy is dependent on the 18O concentration at a particular depth. As the α particles leave the sample surface their energy is solely a function of their depth of origin. The α particle energy spectrum observed is a convolution of the 18O depth profile and spreading factors due to beam energy spread and straggling of protons and α particles as they travel through the sample. Hence the 18O depth profile is obtained by deconvolution of these effects (see Robin et al. 1973; Ryerson et al. 1989). To determine a hydrogen diffusion profile the sample surface is irradiated with 15N with variable energies producing the nuclear reaction 15N + 1H → 12C + α + γ. Depth profiling is performed by varying the incident beam energy to sample H at depth, with depth in the sample determined by the energy loss rate of the 15N in the sample. The number of γ-rays emitted at a given incident energy is proportional to the hydrogen concentration at the respective depth and the hydrogen diffusion profile is obtained by measuring the yield of the characteristic reaction γ-rays as a function of beam energy (Lanford 1992, 1995; Dersch et al. 1997). To determine deuterium diffusion profiles a 3He beam of fixed energy is used. The reaction is 3 He + 2H → 4He + p and in like fashion to collecting the 18O diffusion profile, the deuterium concentration as a function of depth is obtained from the proton energy spectrum (Dersch and Rauch 1999). The primary advantage of these techniques is the very high depth resolution of a few nanometers to tens of nanometers.

Fourier Transform Infrared Spectroscopy The most frequently used method for measuring hydrogen (and water) diffusion is Fourier Transform Infrared (FTIR) spectroscopy. The vibrational modes of the OH dipoles have strong distinct bands in the IR region. The wavenumber of the absorption band is dependent on the strength of the hydrogen (or deuterium) bond, bond geometry, and neighbor interactions. As such, polarized IR spectra can provide information about the structure of the OH as well as its concentration (see Libowitzky and Beran 2006). FTIR provides high sensitivity ( 1125 °C and a lower slope (activation energy of ca. 125 kJ/mol) at temperatures below. Buening and Buseck (1973) themselves provided two alternative explanations for this observation—either a change of diffusion mechanism from intrinsic to extrinsic, or a change of diffusion from volume

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diffusion control to grain boundary diffusion control—and left the issue open. But because they had measured concentration profiles only in the single crystal part of their diffusion couples, and perhaps because Misener (1974) had shown that there was little difference in diffusion rates obtained from crystal-crystal vs. crystal-powder diffusion couples, in the subsequent literature this break in slope was widely interpreted as an example of change of diffusion mechanism in * (Sockel and Hallwig 1977; Morioka a mineralogical system. Some early measurements of DMg 1981 – see above for discussion) and Co-Mg diffusion (Morioka 1980) were combined to further bolster this interpretation. It ultimately gained the status of demonstrated fact and found its place in influential review articles on diffusion in minerals (e.g., Lasaga 1981) and mineralogy textbooks (e.g., Putnis 1992). This profoundly influenced the application of diffusion data in the Earth sciences (Can diffusion coefficients measured at high temperatures be extrapolated to model geological processes that occur at lower temperatures, or does a change of Arrhenius slope due to change of mechanism intervene?). The situation was actually even more complicated when the two data sets are considered together. First, data obtained by Misener (1974) at temperatures below 1125 °C indicated the activation energy of diffusion to be higher, comparable to the ~ 240 kJ/mol found by Buening and Buseck (1973) at higher temperatures. Indeed, the higher temperature data of Buening and Buseck (1973) could be almost smoothly extrapolated to the lower temperature data of Misener (1974) without any break in slope. In numerous subsequent studies that made use of diffusion data in olivine, there were discussions of substantial length as to which of these two data sets were more reliable. Secondly, Misener (1974) had found two sets of diffusion coefficients as a function of composition—one for the compositional range of 0.1 < XMg < 0.8, and another, slower, set of diffusion coefficients for more magnesian olivines (e.g., XMg = 0.9, corresponding to typical mantle rocks). An equation for calculation of activation energy of the first set of diffusion coefficients was provided in the abstract of the paper, and this was used indiscriminately in the subsequent literature for calculation of diffusion coefficients of olivines of all compositions, including mantle olivines. This was the diffusion coefficient that was used largely in the discussion comparing the studies of Buening and Buseck (1973) and Misener (1974) as well. The discussion on the quality of data of the two studies was not straightforward because each study had its strengths and weaknesses. Buening and Buseck (1973) had controlled oxygen fugacity in their experiments, but they had used diffusion couples of single crystals with powders, and only measured concentration gradients in the single crystal part of the couple. They assumed the powder matrix to have behaved as an infinite medium. Misener (1974) had used single crystals and measured concentration profiles on both sides of the diffusion couple, but the oxygen fugacity in his experiments were not explicitly controlled. There was also widespread use of Pt in his experiments which could have led to Fe loss or at least, modification of the point defect structure of the olivines. Because his experiments were carried out in silica capsules with fayalite as part of the diffusion couple, some later workers interpreted his diffusion data to be valid for oxygen fugacities corresponding to the quartz-fayalite-magnetite buffer. There were even attempts to reconcile the two datasets and come up with new expressions describing diffusion rates in olivine where the fO2 dependence was taken from the study of Buening and Buseck (1973) and the pressure dependence from the study of Misener (1974), assuming his data were measured at the QFM buffer (e.g., see Weinbruch et al. 1993). In spite of all these attempts, the fact remained that below 1125 °C there was an unresolved discrepancy between the data sets of Buening and Buseck (1973) and Misener (1974). Even at temperatures above this, there was considerable difference between the results from the two studies, the exact difference depending on exactly which data sets were compared at which composition. A third study on diffusion of Fe-Mg in olivines appeared in the material science literature (Nakamura and Schmalzried 1984). These authors measured diffusion coefficients between

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1050-1280 °C at oxygen fugacities between 10−8 and 10−11.2 bars using olivines with XFe between 0.9 and 0.1. They used synthetic polycrystalline pellets as diffusion couples, but they argued that, because of the coarse grain size (30-50 μm) of the pellets, the contribution from grain boundary transport was minimal, and the diffusion they observed was essentially volume diffusion. Obviously, anisotropy of diffusion could not be resolved and the diffusion coefficient that was obtained was an average of diffusion rates along the different crystallographic directions. Not surprisingly then, these rates were somewhat slower than those obtained by Buening and Buseck (1973) and Misener (1974) in some of their experiments along the [001] direction. However, the activation energy (200 kJ/mol) did not correspond to that obtained from either of the previous studies. It was intermediate between the high and the low temperature values of Buening and Buseck (1973). Thus, this study did not resolve the issue about which diffusion coefficients to use for practical modeling purposes. However, these data did provide the basis for a very successful point defect thermodynamic model for olivines. Indeed, that work (Nakamura and Schmalzried 1983) laid out the formalism of how point defects in silicates can be modeled. But in spite of these advances and the wide use of Fe-Mg diffusion coefficients in olivine for various applications, there remained considerable uncertainty as to the value of diffusion coefficients at different temperatures and compositions. The next set of measurements came from Jurewicz and Watson (1988). They placed olivine crystals in contact (by wrapping with Fe-doped Pt wire) with melt that was in equilibrium with the olivine to study the diffusion of a number of major and trace elements over a restricted range of temperature. They fitted the concentration profiles of Fe and Mg separately to obtain different diffusion coefficients for the two elements. The difference in profile lengths of Fe and Mg may have resulted from significant concentrations of the minor elements accounting for charge balance, in which case the problem was one of multicomponent diffusion. Because of the nature of the experiments involving melts, it was not possible to carry out the experiments over a large temperature range and the question of change of diffusion mechanism and Buening and Buseck (1973) vs. Misener (1974) remained open after this study. Recently, Spandler and O’Neill (2009) have determined Fe-Mg diffusion rates in olivine coexisting with melt and found rates identical to those of Jurewicz and Watson (1988). Chakraborty (1997) set up a set of experiments to specifically test the controversy related to the Buening and Buseck (1973) and Misener (1974) data sets. He used single crystal diffusion couples held together in a container-less set up by spring-loaded alumina rods. The crystals were annealed under controlled oxygen fugacity for long enough durations that diffusion profiles obtained could be analyzed using an electron microprobe without further corrections for convolution or other effects. The diffusion profiles were analyzed in terms of a compositionally dependent diffusion process. Notably, two different kinds of diffusion couples were used. These required slightly different methods for data analysis, but both spanned the important compositional range of mantle olivines—Fo88-Fo92. The rationale was that if all relevant parameters were properly controlled, then both of these couples should yield the same diffusion coefficients for the same compositions. This was found to be the case and therefore it could be further demonstrated that once the relevant set of parameters were controlled (P, T, fO2, XFe of olivine) then variations in contents of other trace elements did not matter and a consistent set of diffusion coefficients can be measured. These data can be then used for modeling processes in natural olivines with other trace element contents. Strictly speaking, another parameter, e.g., the activity of SiO2, needs to be controlled for completely defining the diffusion process. However, it was shown in this and later works that for the transport of divalent cations the influence of this parameter is smaller than the uncertainties of measurement. Note also that subsequent work has shown (Dohmen and Chakraborty 2007) that at low temperatures ( < 800 °C) heterovalent impurities (= trace elements) may begin to control diffusion rates in olivine. The aspect of using two entirely different diffusion couples is particularly important for resolving the discrepancy between the data of Buening and Buseck (1973) and Misener (1974)—if two different diffusion

Diffusion in Olivine, Wadsleytie, Ringwoodite

615

couples and two different methods of analysis of data yielded the same results over a range of experimental conditions (mainly temperature, but also run duration, fO2 etc.), then the data are likely to be more robust. The surprising result from this study was that the diffusion coefficients obtained here were slower than the data of both Buening and Buseck (1973) and Misener (1974) for the Ferich olivines (XMg < 0.1). However, the data of Misener (1974) for more Mg rich crystals was compatible with the results from this study if the data were normalized assuming that Misener’s experiments had been carried out at fO2 equal to that of the QFM buffer. As noted above, it is very difficult to infer what aspects of these older experiments caused the differences. It can be speculated that the lack of explicit fO2 control in the experiments of Misener (1974) affected the more Fe-rich crystals more strongly, leading to enhanced diffusion in those experiments. The discrepancy with the data of Buening and Buseck (1974) proved to be more difficult to rationalize. Subsequently, Dohmen et al. (2007) carried out a detailed analysis of the profile shapes observed by Buening and Buseck (1973) to conclude that an unusually large spatial averaging effect (convolution effect) in the electron microprobe measurements of Buening and Buseck (1973) may have been responsible for yielding the faster diffusion coefficient. This is plausible, but impossible to verify now anymore. Further, it was found (Chakraborty, unpublished results) that in the experimental setup used by Buening and Buseck (1973), the powdered matrix does not function as an infinite medium and concentration gradients develop in that region too. Not considering this may have additionally distorted the analysis of Buening and Buseck (1973). There were additional checks that Chakraborty (1997) could carry out to test the compatibility of different data sets. He used Equation (1) in conjunction with the measured * DMg for the Fe-bearing crystals (see above, Chakraborty et al. 1994) to test whether the DFe-Mg obtained in the 1997 study were consistent. Similar experiments carried out later by Petry et al. (2004) yielded practically identical results. Meissner et al. (1998) could analyze diffusion profiles from similar experiments using analytical TEM on very different spatial scales to obtain the same results. More importantly, the use of TEM established that there were no structural imperfections / formation of new phases in these experiments. Having established the veracity of the data, a number of important inferences could be drawn from these: (i) There was no break in the Arrhenius slope between 980 and 1300 °C, indicating once again that a change of diffusion mechanism does not occur within this range. (ii) An analysis of the data indicated that a simple classification of diffusion mechanisms into intrinsic and extrinsic is inadequate to describe diffusion processes in Fe-bearing silicates; there are three different mechanisms that operate: Intrinsic at highest temperatures close to melting points of crystals, a mechanism that incorporates features of both intrinsic as well as extrinsic diffusion [termed TaMED in this work] that operates at intermediate temperatures, and a truly extrinsic diffusion mechanism (PED = Pure Extrinsic Diffusion) that operates at lower temperatures. (iii) Diffusion and defect formation in Fe bearing systems are fundamentally different from those in Fe-free systems. The processes and defect concentrations can be better controlled, via the control of fO2, in Fe-bearing systems, as already found in Chakraborty et al. (1994). The TaMED regime, where defects can be better controlled, is expected to dominate diffusion behavior observed in nature at magmatic temperatures. Although the Chakraborty (1997) work provided some resolution to the Buening and Buseck (1973) vs. Misener (1974) controversy, it raised other issues. For example, if a transition of diffusion mechanism from TaMED to pure extrinsic was to be expected, at what temperatures does it occur and what does it mean for extrapolation of diffusion data to lower temperatures? The resolution of this issue had to await the development of thin film tools that allowed

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Chakraborty

concentration gradients to be produced, manipulated and measured on nanometer scales. Jaoul et al. (1995) had produced nanometer scale thin films of olivines of restricted composition and measured diffusion coefficients at high pressures at temperatures as low as 600 °C. They also recognized that a simple twofold classification of diffusion mechanisms is not adequate for olivine, although their threefold classification scheme was somewhat different from the one noted above. As they could not yet vary the composition of their thin films and oxygen fugacity of their experiments in a controlled manner, it was not possible to analyze diffusion mechanisms in detail in their work. They had to analyze their data with several assumptions and much of the analysis was carried out in the framework of conventional ideas of enthalpies of defect formation and migration. Although they compared their experimental results to those of computer calculations, the fact that a dependence of experimentally observed diffusion rates on oxygen fugacity was postulated and the computer calculations were carried out on systems without Fe resulted in some inconsistency. A detailed analysis of defect mechanisms through a range of experiments and theoretical analysis became possible with the availability of pulsed laser deposition (PLD) for the production of silicate thin films (Dohmen et al. 2007; Dohmen and Chakraborty 2007). Dohmen et al. (2007) used this study to explore some methodological aspects of diffusion coefficient measurements using thin films and they measured diffusion coefficients down to 700 °C. Measurements at overlapping conditions (e.g., 1000-1200 °C) confirmed that data from Chakraborty (1997) and Petry et al. (2004) were compatible with these results obtained using very different setups and spatial scales of measurement. To obtain a comprehensive expression for the calculation of diffusion coefficients in natural olivines, it is necessary to constrain the pressure dependence of diffusion coefficients as well. Bertran-Alvarez et al. (1992), Jaoul et al. (1995) and Holzapfel et al. (2007) determined the pressure dependence of DFe-Mg by carrying out experiments in the multianvil press. Holzapfel et * * and DMn as well, and the study was carried al. (2007) determined the pressure dependence of DNi out over a wide pressure range. This study also clarified one aspect about the determination of pressure dependence of diffusion rates—high pressure diffusion experiments are usually carried out along an oxygen fugacity buffer (e.g., NNO) and not at a constant fO2. Consequently, the pressure and temperature dependence of the buffering reaction is also embedded in the raw diffusion coefficient data as a function of P and T (Jaoul et al. 1995 mention the problem as well but provide a different analysis). This effect needs to be filtered out to obtain the true activation volume (i.e., pressure dependence of diffusion coefficient). Holzapfel et al. (2007) tried to determine the pressure dependence of diffusion rates using two sets of experiments in order to test internal consistency. Activation volumes could be determined using an isobaric, polythermal set of data at high pressures, to be combined with similar data from atmospheric pressure. Alternately, activation volume could be determined from an isothermal, polybaric data set. Different fitting protocols yielded activation volumes between 4 and 7 cm3/mol for all * * and DMn ), but for diffusion along the NNO buffer, a value three kinds of diffusion (DFe-Mg, DNi of 5.3 cm3/mol was found to be the most optimized result. This is in agreement with other determinations (e.g., Misener 1974 and Farber et al. 2000). However, at constant fO2, correcting for the effects noted above, the activation volume is 7 cm3/mol and this is the value to be used in Arrhenius equations for calculations of diffusion coefficients at high pressures. A consequence of this activation volume is that at depth within the Earth, a maximum in diffusion rates is attained near the base of the lithosphere. Once the geothermal gradient becomes adiabatic in the convecting part of the mantle the pressure dependence of diffusivity dominates and diffusion rates decrease with depth even though temperature increases. Accounting for these pressure effects and considering the dependence of DFe-Mg on temperature, crystallographic direction, composition of olivine (i.e., XFe) and oxygen fugacity in the framework of a quantitative point defect model, Dohmen and Chakraborty (2007) obtained the following expressions for calculation of DFe-Mg:

Diffusion in Olivine, Wadsleytie, Ringwoodite

617

At oxygen fugacities greater than 10−10 Pa a fO2 dependent expression for diffusion along [001], Dc, is 1

DFe-Mg

 fO  6  201000 + ( P − 105 )(7 × 10 −6 )  = 10 −9.21  −27  103( XFe − 0.1) exp  −  RT    10  1 Da ~Db ~ ⋅ Dc 6

At lower oxygen fugacities the diffusion occurs by pure extrinsic mechanism (PED) and is independent of fO2, given by log[ DFe-Mg (m 2 /s)] = −8.91 −

220000 + ( P − 10 5 )(7 × 10 −6 ) + 3( X Fe − 0.1) 2.303 RT

The relationship between Da, Db and Dc remains the same. Considering the stability field of olivine as a function of fO2 (see Fig. 2), these conditions can only be attained at temperatures below about 900 °C. For less accurate work or where fO2 is unknown, the second expression describes diffusion coefficient of olivine at any T-fO2 condition, with a larger error. This is a consequence of the fact that the change in Arrhenius slope at the transition between TaMED and pure extrinsic (PED) mechanisms is very subtle. The main change at this transition is that in the pure extrinsic domain diffusion is independent of fO2. These expressions are validated by the point defect thermodynamic analysis of Dohmen and Chakraborty (2007) as well and they are able to reproduce 113 experimental data points from different studies mentioned above to within half an order of magnitude. Diffusion of Fe-Mg is enhanced by the presence of water in the olivine crystal and this effect has been studied by Hier-Majumder et al. (2005). They find an fH2O exponent, r, of 0.9 ± 0.3, and suggest the expression r

DFe-Mg

 fH O   −(Q + PV * )  2 = D0  02  exp   m /s  fH 0  RT    2 

with logD0 = −14.8±2.7, r = 0.9±0.3, Q = 220±60 kJ/mol and V* = (16±6)×10−6 m3/mol. The dry to wet transition occurs at about 11 ppm water (corresponding to a water fugacity of about log fH2O = 8.5, where fugacity is in Pa, under equilibrium conditions) i.e., the wet equation should be used for calculating diffusivity of olivines containing more water than this amount (Costa and Chakraborty 2008). Moreover, this equation does not account for the effect of oxygen fugacity on diffusion rates and is valid for diffusion along the NNO buffer. The effect of fO2 needs to be accounted for and once this is done (i.e., dry and wet data are compared at the same oxygen fugacities), the enhancement of DFe-Mg under wet conditions is found to be by a factor of two (Costa and Chakraborty 2008). The most interesting effect of water appears to be to enhance the pressure dependence of diffusion rate—the activation volume under wet conditions (16 cm3/mol) is more than twice that under dry conditions (~ 7 cm3/mol) so that at depth in the earth with an appropriate set of parameters it may even be possible to have slower diffusion rates under wet conditions than under dry conditions (i.e., diffusion would occur by the more efficient dry mechanism in these circumstances)! Ni, Mn, Co. Clark and Long (1971) used a novel approach, even by more recent standards, to produce a diffusion couple to study Ni diffusion in olivine. First, they coated an oriented and polished olivine crystal with a thin film of metallic Ni by thermal evaporation in a vacuum chamber. Next, they used this Ni coated surface as the cathode for electroplating the same surface with a much thicker film of metallic Ni from a Ni-sulfate solution. This thick

Chakraborty

618

metallic Ni film coated olivine was then annealed at high temperatures, taking care to prevent oxidation. Concentration profiles were measured with an electron microprobe and fit to an error function solution to determine diffusion coefficients for Ni transport along the three principal crystallographic axes. Thus, this very early study already documented the anisotropy of diffusion in olivine crystals that has been verified in many subsequent studies. Diffusion of Ni, Mn and Co in forsterite has been studied by Morioka (1981) and Ito et al. (1999). Diffusion of Ni and Mn in Fe-bearing San Carlos olivine and olivine from Pakistan has been studied by Petry et al. (2004). Jurewicz and Watson (1988) measured Mn diffusion in Fe-bearing olivine coexisting with melt and Spandler and O’Neill (2009) have studied the diffusion of Ni, Co and Cr in San Carlos olivine coexisting with melt over limited temperature intervals (all data summarized in Figure 5a and b). As with the major elements, it was found that diffusion in a Fe-free system behaved differently and the activation energy was higher (~ 400 kJ/ mol), but a single activation energy could describe the diffusion behavior between temperatures of 800-1400 °C. For the Fe-bearing system, Petry et al. (2004) carried out experiments with several different kinds of diffusion couples (olivine-olivine single crystal diffusion couples, and olivine-metallic alloy diffusion couples, each kind with several different compositions as exchange partners) to verify the internal consistency of data. Their results were in excellent agreement with the pioneering study of Clark and Long (1971) and the main findings were that the diffusion of these trace elements were very similar in rates and activation energies. * * DNi is slightly smaller than DMn , and the latter is practically identical to DFe-Mg. Diffusion rates are anisotropic, they increase with Fe-content of olivine and are dependent on fO2, although the quantitative parameters describing the compositional and fO2 dependences are slightly different * increases by only one and a half order of magnitude across the olivine from those for DFe-Mg. DNi solid solution series whereas for DFe-Mg the change is of three orders of magnitude. Similarly, * is 1/4.25, which is a stronger dependence than the exponent of 1/5.5-1/6 the fO2 exponent for DNi * * and DMn measured by Holzapfel et al. (2007) found for DFe-Mg. The pressure dependence of DNi * for modeling natural samples, based is similar to that of DFe-Mg. An expression to calculate DNi on the results of Petry et al. (2004) and Holzapfel et al. (2007) is: 1

 fo  4.25  220000 + ( P − 10 5 )(7 × 10 −6 )  Dc , Ni (m /s) = 3.84 × 10  −2 6  101.5( XFe − 0.1) exp  −  RT    10  2

−9

1 Da ~Db ~ × Dc 6

For Mn, the expression given for DFe-Mg can be used. Taken together the data indicate that diffusion occurs by a single mechanism in the “Fefree” as well as in the Fe-bearing olivines, but that the mechanisms are different in these two kinds of crystals, as indicated by the large differences in activation energies. In the Fe-bearing olivines diffusion is in the TaMED regime. An important implication of these results is that, at least for olivine, a compensation law kind of behavior is not valid for the diffusion of the divalent cations. Ca, Sr. Inasmuch as Ca occupies primarily the M2 site of olivine, it behaves differently from the other divalent cations. Its diffusion behavior in olivine is also somewhat different and is considerably slower than the diffusion rates of other trace elements such as Ni and Mn. General overall features found above for diffusion of other elements are however retained. In other words, diffusion in “Fe-free” olivine occurs with higher activation energy, and diffusion rates depend on fO2 in both kinds of olivine, but the dependence is stronger in Fe-bearing olivine. Absolute diffusion rates of Ca in Fe-bearing olivine are slower than that of the other trace elements and the dependence of this diffusion rate on fO2 is stronger. Diffusion of Ca

1

7

6

(a)

3 7

1

5

-14

8 9 4 105/T [1/K]

3

10

5

-20

-18

6

(b)

1

4

6 7 10 /T [1/K]

-20

-18

2

8

6 4

T [°C]

1

3

9

6: Mn: Spandler & O'Neill 2009

1: Cr: Ito & Ganguly 2006 Cr: Ito2009 & Ganguly 2006 2: Cr: Spandler &1:O'Neill 2: Cr:(Mn-Mg, SpandlerFo100) & O'Neill 2009 3: Mn: Morioka 1981 Mn: Morioka 1981 (Mn-Mg, Fo100) 4: Mn: Petry et al.3:2004 Mn: Petry et al. 2004 5: Mn: Jurewicz &4:Watson 1989 5: Mn: Jurewicz 6: Mn: Spandler & O'Neill 2009 & Watson 1989

8

1000

7 9 4 10 /T [1/K]

5

1200

indicated in the legend, along with compositions of olivine and nature of diffusion couple, where relevant. (b) Arrhenius plot showing best fits to Cr and Mn diffusion coefficients in olivine from different studies. Data sources are indicated in the legend, along with nature of diffusion couple where relevant. Please note that comparison of results from different studies may additionally require taking into account effects of different compositions and oxygen fugacities at which the data were obtained.

8 9 10 4 10 /T [1/K] Figure 5. (a) Arrhenius plot showing best fits to Ni and Co diffusion coefficients in olivine from different studies. Data sources are

-22

-20

-18

-16

6 1: Ni: Petry et al. 2004

2

1000

1400

T [°C]

2: Ni: Morioka 1981 (Ni-Mg, Fo100) 1: Ni: Petry et al. 2004 Ni: Ito et Fo100) al.1999 (Fo100) 2: Ni: Morioka 19813: (Ni-Mg, Ni: Spandler & O'Neill42009 3: Ni: Ito et al.19994:(Fo100) 5: Co: et al. 1999 (Fo100) -16Ito 4: Ni: Spandler & O'Neill 2009 6: Co: Morioka 1981 (Co-Mg,-16 Fo100) 3 5: Co: Ito et al. 1999 (Fo100) 7: Co: Spandler & O'Neill 2009 6: Co: Morioka 1981 (Co-Mg, Fo100) 7: Co: Spandler & O'Neill 2009 2

7 4

6 7 2 4

-14

log (D(m /s))

-14

800

2

0 1200

2

1200

log (D(m /s))

T [°C] T [°C] 1400 1200 1000 1000 800

log (D(m /s))

1400

Diffusion in Olivine, Wadsleytie, Ringwoodite

619

Chakraborty

620

in olivine has been measured by Morioka (1981) in fosterite, Jurewicz and Watson (1988) in Fe-bearing olivine and Coogan et al. (2005) in forsterite as well as Fe-bearing olivine. In addition, Petry et al. (2004) obtained some Ca diffusion data from their experiments designed primarily to determine Ni diffusion coefficients, and Miyamoto and Mikouchi (1998) carried out a single experiment to determine a Ca diffusion coefficient in Fe-bearing olivine. The data are summarized in Figure 6. All of these results are discussed in Coogan et al. (2005a), who carried out experiments using different setups (e.g., crystal embedded in a powder, crystal coated with a thin film) to test the internal consistency of their results. In considering data from Fe-bearing olivines they found that the data of Jurewicz and Watson (1988) were consistently higher than the diffusion coefficients they obtained, even though the activation energies obtained in the two studies were similar. They postulated that the presence of melt in the experiments of Jurewicz and Watson (1988) may have had something to do with this. This assumption is further strengthened by the fact that the results of Spandler and O’Neill (2009), determined from experiments where olivine coexisted with melt, are very close to those of Jurewicz and Watson (1988). One more data set is available in abstract form— Gaetani et al. (2002). The activation energy reported there is much higher than that found by Coogan et al. (2005a). However, given the temperature range of their experiments, Coogan et al. (2005a) inferred that the data points must lie relatively close to the best fit Arrhenius lines of Coogan et al. (2005a); the higher activation energy would have resulted from fitting scattered data over a small temperature interval. Petry et al. (2004) obtained their Ca diffusion coefficients as a byproduct of their study on Ni and Mn diffusion. These diffusion coefficients were determined from noisy concentration profiles measured with the electron microprobe in olivines containing low concentrations of Ca. As a result, these data scatter considerably. We repeat here the expressions derived by Coogan et al. (2005a) after analysis of all the results for calculating Ca diffusion rates in Fe-bearing olivine: 1

Da,Ca = 16.59 × 10

−12

 fO2  3.2  −19300 ± 11000   −7  exp   RT    10  1

= D 34.67 × 10 b ,Ca

−12

 fO2  3.2  −201000± 10000   −7  exp   RT 10     1

D = 95.49 × 10 c ,Ca

−12

 fO2  3.2  −207000± 8000   −7  exp   RT 10    

All diffusion coefficients are in units of m2/s, fO2 is in Pa and R, the gas constant, is 8.314 J/ mol/K. Sr diffusion has been measured in olivine at a single temperature (Remmert et al. 2008 and in prep) and is found to be 2×10−19 m2/sec at 1000 °C at a fO2 of 10−10 bar. This is considerably slower than the diffusion rates of Ni, Mn and Ca in olivines of the same composition.

Diffusion of Si and oxygen Si and O. Measurement of Si diffusion coefficients is difficult for two reasons. First, the diffusion rates are very slow. Second, one cannot just use two olivine crystals to exchange Si; there is a problem with defining the source. The problem with slow diffusion coefficients was overcome by using nuclear methods to measure very short concentration gradients. The use of Rutherford Backscattering Spectroscopy (RBS) and Nuclear Reaction Analysis (NRA) was pioneered by the group of Olivier Jaoul (e.g., Jaoul et al. 1980). The second problem is addressed by using isotopically enriched material. Isotopically enriched SiO2 was used to produce forsterite with an excess of MgO, and this material was deposited by RF sputtering

Diffusion in Olivine, Wadsleytie, Ringwoodite T1600 [°C] 1400 1200 1000

1600 1400

log (D(m /s))

2

-17

-16

2

2

-17

-18

-18

-19

-19 4

-20

-20 6

1: Ca: Coogan et al. 2005 2: Ca: Morioka 1981 (Fo100) 2: Ca: Morioka 1981 (Fo100) 3: Ca: Jurewicz & Watson 19 3: Ca: Jurewicz & Watson 1989 4: Sr: Remmert 4: Sr: Remmert et al. 2008 and in prep. et al. 2008 an

3

3

-16

1000

1: Ca: Coogan et al. 2005

-15

-15 2

T [°C]

-14

-14

log (D(m /s))

1200

621

7 4 10 /T [1/K]

1

1

4

68

9 7 10 /T [1/K] 4

8

9

Figure 6. Arrhenius plot showing best fits to Ca and Sr diffusion coefficients in olivine from different studies. Data sources are indicated in the legend.

to produce thin films on single crystals in the early studies. These thin films acted as sources of Si for the diffusion process. Oxygen diffusion coefficients were measured by the same groups using similar methods. The problem was that the activation energy of diffusion for Si as well as O was found to be lower than the activation energy for creep of olivine. This behavior was different from that known from studies in oxides, where activation energies of oxygen diffusion often corresponded to the activation energy of creep, identifying diffusive transport of oxygen in these materials as the rate determining step for creep. For olivine, the situation remained unclear, leading to considerable speculation and theoretical attempts to explain how activation energies of diffusion for all concerned species (Fe, Mg, Si and O) could be substantially lower than the activation energy for creep (e.g., see a discussion in the textbook by Poirier 1985). Given these problems with measurement of Si and O diffusion rates in minerals it is not surprising that progress has been slow. Efforts to determine these diffusion coefficients were pioneered by the group of Olivier Jaoul (Jaoul et al. 1980; Jaoul et al. 1981; Jaoul et al. 1983; Houlier et al. 1988; Gerard and Jaoul 1989; Houlier et al. 1990 and a review in Bejina and Jaoul 1997). Other measurements on oxygen diffusion were carried out by Reddy et al. (1980), Ando et al. (1981), Andersson (1989) and Ryerson et al. (1989); limited results are also available in Sockel and Hallwig (1977) and Sockel et al. (1980). The measurements in these studies were carried out on forsterite as well as San Carlos olivine using a variety of experimental methods. Oxygen diffusion is considered in detail in the chapter by Farver (2010, chapter 10 this volume) and a review is also available in Cole and Chakraborty (2001); a summary of Arrhenius relations for Si as well as O is presented in Figure 7 (a and b). As noted above, the main challenges were to define a proper source for the diffusant and to find a proper analytical technique to measure the extremely short concentration profiles. Nuclear Reaction Analysis (NRA), Proton activation analysis (PA), Rutherford Backscattering Spectroscopy (RBS) and Secondary ion mass spectrometry (SIMS) were some of the analytical methods that were tried.

Chakraborty

622 T [°C] 1300

1500

1100

-18

Houlier et al. 1989

Figure 7. (a) Arrhenius plot showing best fits to Si diffusion coefficients in olivine from different studies. Data sources are indicated beside the lines. (b) Arrhenius plot showing best fits to O diffusion coefficients in olivine from different studies. Data sources are indicated in the legend. All data are for Fo90 when not otherwise stated. Please note that comparison of results from different studies may additionally require taking into account effects of different oxygen fugacities at which the data were obtained.

2

log (D(m /s))

-20

-22 Dohmen et al. 2002 -24

-26

(a) 5.5

6.0

6.5

7.0

7.5

4

10 /T [1/K] T [°C] 1300 T [°C] 1300

1500 1500

-17

2

-18

2 4

-18

1100 1100

1= Dohmen et al. 2002 2= Gerard et al. 1989 1= Ryerson Dohmen et et al. al. 1989 2002 3= 2= Ando Gerard 1989 4= et et al.al. 1980(Fo100) 3= Ryerson et al. 1989 4= Ando et al. 1980(Fo100)

4

2

2 log (D(m log (D(m /s)) /s))

-17

-19 -19 -20

3

-20

3

-21 -21

1

(b) 5,5 5,5

6,0

6,5 4

[1/K] 6,0 10 /T6,5

7,0 7,0

1

7,5 7,5

4

10 /T [1/K]

The uniform conclusions from all of these studies were that the diffusion rates of Si and O are much slower than the diffusion rates of the divalent cations and that the diffusion rate of Si is slower than that of O. Beyond that, there were considerable uncertainties and discrepancies in the values of diffusion coefficients and activation energies. The values evolved over time as experimental methods improved with technical advances. For oxygen diffusion, the studies of Ryerson et al. (1989) on San Carlos olivine, Gerard and Jaoul (1989) on San Carlos olivine and Dohmen et al. (2002) on San Carlos olivine as well as olivine from Pakistan yielded similar diffusion coefficients as well as activation energies when data are considered at a given oxygen fugacity. Anisotropy of diffusion was found to

Diffusion in Olivine, Wadsleytie, Ringwoodite

623

be very weak and a positive dependence of diffusion rates on fO2 was observed. The study of Dohmen et al. (2002) measured diffusion of O and Si simultaneously, corrected for possible convolution effects and was carried out at a constant oxygen fugacity (10−9 bars) over the widest temperature range (1100-1500 °C) to yield the following relationship for diffusion along [001]:  ( −338 ± 14 kJ/mol)  DO = (10 −8.34 ± 0.47 m 2 /s)exp   RT  

For diffusion of Si, it turned out that the quality of earlier data were substantially compromised by the inability to produce thick enough films as a source of diffusant and by a large convolution effect that affected the very short diffusion profiles. These were discussed by Dohmen et al. (2002) and after accounting for these effects, diffusion along [001] was found to be well described by  ( −529 ± 41 kJ/mol)  DSi = (10 −4.2 ±1.2 m 2 /s)exp   RT  

The activation energy of diffusion in this expression is practically identical to that found for creep of polycrystalline olivine (e.g., Mei and Kohlstedt 2000; Karato and Jung 2003). The effect of pressure on Si diffusion in San Carlos olivine was studied by Bejina et al. (1997, 1999) in experiments carried out at pressures up to 9 GPa. They found the activation volume to be close to zero: (0.7±2.3)×10−6 m3/mol. This is in sharp contrast to the relatively strong pressure dependence of creep rate of olivine found in recent studies: (9.5±7)×10−6 m3/ mol (Durham et al. 2009) or 14×10−6 m3/mol (Karato and Jung 2003). This leaves the question open as to what atomic mechanism is responsible for the activation volume of creep, or how diffusion relates to deformation under high pressures. The effect of water on Si and O diffusion rate in Fe-bearing olivine were studied by Costa and Chakraborty (2008). They found that at hydrous conditions, DSi ≈ DO and anisotropy of diffusion was weak to non-existent. At 2 GPa and fH2O = 0.93 GPa, fits to D = D0 exp(−Q/RT) yielded:

Si: D0 = (1.68±3.52)×10−7 m2/s; Q = 358±28 kJ/mol O: D0 = (1.43±1.80)×10−4 m2/s; Q = 437±17 kJ/mol

Water enhances the diffusivity of Si much more than that of O (1000 fold vs. 10 fold). Indeed, there were indications that the mechanism of O diffusion remained unchanged under water-present and dry conditions. Analysis of the point defect structure indicated that at hydrous conditions, oxygen diffused by an interstitial mechanism whereas Si diffused by a mechanism involving vacancy complexes. The transition from a dry diffusion mechanism to a wet diffusion mechanism appears to occur after incorporation of very small amounts of H in the olivine lattice (< 10 ppm). As this is smaller than the lowest known water contents of mantle olivines (which are lower limits themselves, because there is expected to have been some water loss during transport to the surface), it is felt that wet diffusion laws should be used to model Si and O diffusion and deformation processes even for nominally “dry” mantle. If it is assumed that dislocation climb and glide contribute equally to the creep process, then absolute deformation rates can be calculated from the diffusion data.

Diffusion of ions that enter olivine via heterovalent substitutions Hydrogen. Hydrogen diffusion in olivine is discussed by Farver (2010, chapter 10 this volume) and only a brief summary is provided here. Hydrogen or water diffusion in olivine has been measured by a number of workers (e.g., Mackwell and Kohlstedt 1990; Kohlstedt and Mackwell 1998; Demouchy and Mackwell 2003, 2006; Ingrin and Kohn 2008) and

Chakraborty

624

Ingrin and Blanchard (2006) provide a review of data available until that time. Diffusion of hydrogen is one of the fastest transport processes in olivine (~ 10−10 m2/s at 1000 °C). Along with Li (see below), it is also one of the most complex transport processes where the species is incorporated and diffuses by more than one mechanism simultaneously. H is incorporated and diffuses in olivine by exchanging with either a electron/polaron or a metal vacancy in the octahedral site. These two processes occur at different rates. Demouchy and Mackwell (2006) have analyzed the overall process and, subject to some assumptions (see details in that work and in the chapter on hydrogen diffusion, Farver 2010, chapter 10 this volume), provide the following expressions for calculation of H diffusivity in Fe-bearing olivines:  −204000 ± 94000  DVMe [100 ][ 010 ] = 10 − ( 4.5 ± 4.1) exp   RT    −258000 ± 11000  DVMe [ 001] = 10 − (1.4 ± 0.5) exp   RT  

For Fe-free olivines, the rates are slightly different (Demouchy and Mackwell 2003), but not as different as for the other elements:  −211000 ± 18000  DVMe [ 001] = 10 − ( 3.3 ± 0.7 ) exp   RT    −211000 ± 18000  DVMe [ 010 ] = 10 − ( 3.9 ± 0.7 ) exp   RT    −211000 ± 18000  DVMe [100 ] = 10 − ( 4.4 ± 0.7 ) exp   RT  

All diffusivities are in m2/s and activation energies are in Joules/mol; the Arrhenius lines are shown in Figure 8.

1400

T [°C] 1200

1000

-9 Demouchy & Mackwell 2006

2

log (D(m /s))

-10 Figure 8. Arrhenius plot showing best fits to H diffusion coefficients in olivine (Fo and Fo90).

-11 -12 Demouchy & Mackwell 2003

-13 5,5

6,0

6,5

7,0 7,5 104/T [1/K]

8,0

8,5

Diffusion in Olivine, Wadsleytie, Ringwoodite

625

Note that one consequence of this diffusion by two mechanisms is that a fraction of the H incorporated in an olivine crystal diffuses faster than the rate of diffusion of vacancies in the crystal—i.e., some fraction of H may be gained or lost by an olivine at rates that are much faster than the expressions given above. Lithium. Dohmen et al. (2010) studied the transport of Li along [001] in San Carlos olivine crystals at temperatures between 800 and 1200 °C. They found a complex diffusion behavior which could be explained by the simultaneous operation of two diffusion mechanisms coupled with a homogeneous reaction within the crystal. Li is thought to be incorporated in interstitial sites as well as in vacant M-sites and transport occurs by hopping within and between these sites. The homogeneous reaction LiMe = VMe + Lii, involving a metal vacancy as well for charge balance is thought to maintain equilibrium distribution of Li between the sites. It occurs instantaneously compared to the rates of diffusion within the sites. The interstitial diffusion occurs much faster than diffusion in the metal sites (see Fig. 9 for a comparison), but the concentration of Li in the metal sites is much higher. One consequence of this behavior is that the diffusion rate of Li in a crystal of olivine and the shapes of profiles that develop depend on the boundary conditions (ambient conditions such as concentration of Li in the surrounding medium and oxygen fugacity, in addition to temperature, pressure etc.). Therefore, a single Arrhenius type equation for calculation of DLi as a function of temperature is not available. It is necessary to simultaneously solve two diffusive transport equations for interstitial and vacancy diffusion, respectively, coupled with the homogeneous reaction that accounts for exchange of Li ions between M-sites and interstitial sites. Dohmen et al. (2010) state the equations explicitly and discuss how such systems of equations can be solved numerically for different

1200 1200 -10

log (D(m /s))

-10

1000

T [°C]

T [°C]

2

log (D(m /s))

-14 -16

800

1 = Lii: Dohmen et al. 2010

800

2 = VMe: Kohlstedt & Mackwell 1998

-12 -14

1 = Lii: Dohmen et al. 2010

3 = Li: For 1-10 ppm Li, based on

1

2

-12

1000

2 = VMe: Kohlstedt & Mackwell 1998 Dohmen et al. 2010

1

4

3 = Li: For 1-10 ppm Li, based 4 = Li: Spandler & on O'Neill 2009 Dohmen al. 2010Dohmen & Chakraborty 5 et = Fe-Mg: 4 = Li: Spandler & O'Neill 2009 (Fo90) 5 = Fe-Mg: Dohmen & Chakraborty 2007 (Fo90)

2 2

4

-16

3 -18

3

-18

5

-20 -20

7

8

5

9

10

8 10 104/T 9[1/K] 104/T [1/K] Figure 9. Arrhenius plot showing different relevant parameters for diffusion of Li in olivine. Line 1 is the best fit to diffusion coefficients of Li interstitials from Dohmen et al. (2010), line 2 shows best fit to diffusion coefficients of metal vacancies (VMe) from Kohlstedt and Mackwell (1998). The grey band represents the region where diffusion coefficients of Li in metal vacancy sites (Dohmen et al. 2010) for different Li concentrations in olivine plot. This also marks the region where overall diffusion rates of Li is expected to lie for concentrations of Li commonly expected in natural olivines (1-10 ppm). The symbol denotes the Li diffusion coefficient measured by Spandler and O’Neill (2009), and the dashed line shows an Arrhenius relation for DFe-Mg from Dohmen and Chakraborty (2007) for reference. 7

626

Chakraborty

assumptions about the rates of the homogeneous reaction. Under typical circumstances to be expected in nature (e.g., concentration of Li in olivine, fO2), diffusion of Li is expected to be about an order of magnitude faster than the diffusion rates of Fe-Mg and other divalent cations at the same conditions (Fig. 9). However, under certain special situations (e.g., highly oxidizing conditions), the fast interstitial diffusion mechanism may dominate. The two isotopes of Li, 6Li and 7Li, diffuse at slightly different rates (the lighter isotope is about 5% faster) so that isotopic fractionation may be expected to occur by diffusion in some situations. When the fast diffusion mechanism dominates, this can lead to a decoupling of the time scales of isotopic equilibration from the time scales of elemental Li equilibration. Diffusive isotopic fractionation is expected to be a transient phenomenon that should be recorded only in crystals that show zoning of elemental Li. The diffusion of Li in olivine is slower than the rate of transport of this element in minerals such as plagioclase (Giletti and Shanahan 1997) and clinopyroxene (Coogan et al. 2005a). It is unusual in that the diffusivities of other elements in these minerals at a given temperature are generally slower than in olivine. Spandler and O’Neill (2006, 2009) studied the diffusion of Li in olivine as part of a multi trace element study on element partitioning and diffusion. They found somewhat slower Li diffusion rates, comparable to diffusion rates of divalent cations in olivine. This may not necessarily be a discrepancy and could be simply a result of lower, natural Li concentrations in their experiments (see Fig. 9). Alternately, the reason for this discrepancy with the experiments described above may be related to the fact that (b) a complex boundary condition is imposed on the olivine in this case that involves gradients of multiple elements and (b) the simultaneous diffusion of several trace elements may set up complex charge balance conditions. Further experiments are necessary to clarify these aspects. Chromium. Cr diffusion in olivine was measured by Ito and Ganguly (2006) at temperatures between 900 and 1100 °C under controlled oxygen fugacity conditions. They used a thin film of Cr2O3 deposited on the surface of an oriented, polished single crystal of San Carlos olivine as the source of Cr (Fig. 5b). They found significant anisotropy of diffusion, described by the following parameters in D = D0 exp(−Q/RT): logD0(a) = −5.31±0.39 [D0 in units of m2/s], Q(a) = 352±10 kJ/mol logD0(c) = −6.65±0.59 [D0 in units of m2/s], Q(c) = 299±14 kJ/mol The rates of Cr diffusion are considerably slower than the diffusion rates of DFe-Mg. REE and Be, V, Ti, Na, Zr. Diffusion data for REE in olivine are necessary to model processes such as the time scales of re-equilibration of melt inclusions trapped in olvine (e.g., Cottrell et al. 2002; Spandler et al. 2007). Various available data are shown in Figure 10. Spandler and O’Neill (2009) found that rates of REE (e.g., Eu) diffusion through olivine crystals are fast enough to re-equilibrate included melt inclusions completely to near surface conditions in shallow magma chambers. In contrast, Cherniak (2010) studied diffusion of REE such as La, Dy and Yb in polished single crystals of synthetic forsterite (~Fo100) or San Carlos olivine (~Fo90) embedded in REE enriched powders and found that diffusion of REE is relatively slow, being described by Arrhenius relations such as  −289 ± 21 kJmol −1  2 DDy = 8 × 10 −10 exp   m /s RT  

Anisotropy was found to be relatively weak, diffusivity for La, Dy and Yb were similar and there was little, if any, difference in diffusion rates of these elements between synthetic forsterite and Fe-bearing San Carlos olivine. Remmert et al. (2008, and in prep) also studied the diffusion of REE in San Carlos olivine using REE enriched thin films deposited on oriented, polished single crystals. At the high temperatures of their experiments, the data were more similar to those of Cherniak (2010) than of Spandler and O’Neill (2009), but the activation

Diffusion in Olivine, Wadsleytie, Ringwoodite T [°C] 1100 1000

1300 1200 -14

T [°C] 1100 1000

1300 1200

-16

2

log (D(m /s))

Be V Ti Na Li Zr Eu2

Eu1 Nd Sm Ce

-18

627

900

Dy: Cherniak 2009 (Fo100) Eu1, Sm, Nd, Ce: Remmert 2008 Be, V, Ti, Li, Na, Eu2, Zr: Spandle

Be V Ti Li Na 900 Eu2 Zr

Dy: Cherniak 2009 (Fo100) Eu1, Sm, Nd, Ce: Remmert 2008 and in prep Be, V, Ti, Li, Na, Eu2, Zr: Spandler & O'Neill 2009

Eu1 Nd

-20

Sm Ce

Dy

-22 -24

Dy

6,5

7,0

7,5

8,0

8,5

9,0

4

10 /T [1/K] 6,5

plot 7,0 Figure 7,510. Arrhenius 8,0 8,5showing 9,0best fits to diffusion coefficients of REE in olivine from different studies. Data sources and elements for which data were obtained are indicated in the legend. Additionally, data for a set of other elements for which diffusion coefficients were measured by Spandler and 104points /T [1/K] O’Neill (2009) are shown. Please note that comparison of results from different studies may additionally require taking into account effects of different oxygen fugacities at which the data were obtained.

energies found by these authors are somewhat higher (Fig. 10). Spandler and O’Neill (2009) also measured diffusivities of Be, V, Ti, Na and Zr and the data are shown in Figure 10. The reasons for the discrepancy between the diffusion coefficients measured by Spandler and O’Neill (2009) and those of Cherniak (2010) and Remmert et al. (2008, in prep) may be the aspects already described in the section on Li diffusion. As all experiments being considered here appear to be carefully carried out, it is important to clarify the reasons of this discrepancy for proper application of these data to natural systems.

Information from olivines other than Fe-Mg binary solid solutions Morioka (1980, 1981, 1983) carried out a series of experiments in Mg2SiO4, Ni2SiO4, Mn2SiO4 olivines, either by coupling two of these crystals to each other to allow exchange of cations, or by depositing an isotopic tracer (e.g., 26Mg) on a crystal (usually forsterite). The rates of Ni-Mg exchange and Mn-Mg exchange, measured in Ni2SiO4-Mg2SiO4 and Mn2SiO4Mg2SiO4 couples, respectively, cannot be compared to each other, and to tracer experiments (for Mg and Ca) on forsterite, to obtain an idea of relative diffusion rates of Ni, Mn, Mg, and Ca. This is because of the distinction between tracer and chemical diffusion coefficients and the kind of compositional dependence of diffusion rates that is expected (see the discussion at the beginning of this chapter on different kinds of diffusion coefficients). In contrast, Morioka (1983) measured the diffusion rates of several elements (Ni, Co, Mn, Ca, Sr and Ba) in the same matrix—crystals of Mn2SiO4. Plotted against ionic radius of the diffusing ion, there is a systematic parabolic behavior. Diffusion rates are maximum for the ionic size of Fe or Mn, and

628

Chakraborty

decreases progressively when the ionic radius becomes smaller (e.g., Ni) or larger (Ca, Sr, Ba in that sequence).

Spectroscopic measurements Nuclear spectroscopic methods such as Mössbauer spectroscopy (for Fe) and NMR (for Mg) can be used to observe the diffusion process on an atomic scale by in situ high temperature measurements. The principle is based on the fact that if the nuclei of the atoms being observed move during the observation time scale of the spectroscopic method, this results in the broadening of the relevant peak in the spectrum. This is a consequence of the fact that the immediate environment of the atom, the parameter that determines the location of peaks in such spectra, changes if the atom moves. More recently, even optical spectroscopy has been used to measure cation dynamics and diffusion in materials with olivine structure. The experiments are demanding because not only do the spectra have to be collected in situ at high temperatures, but the diffusion time scale (jump frequency) must correspond to the observation time scale of the method. Notwithstanding these difficulties, diffusive processes in olivines have been observed using both methods. The group of K.D. Becker has succeeded in observing atomic motion in fayalite and Co-olivine (Co2SiO4) using Mössbauer spectroscopy and relating it to diffusion * measured by Hermeling and Schmalzried 1984). In one of these studies parameters (e.g., DFe (Niemeier et al. 1996), they could delineate the relative frequencies of M1-M1, M1-M2 and M2-M2 jumps. They found that M1-M1 jumps are by far the most dominant. This is consistent with results of computer simulations (see below). The measurements are restricted to a very narrow temperature range, limited by the melting temperature of fayalite (1200 °C) on the one hand and the necessity of having a high enough jump frequency within the observational time scale on the other hand. Further constraints are imposed by the need to have enough Fe atoms to observe the signals, which precludes similar experiments with Mg-rich olivines. More recently, the same group (Shi et al. 2008, 2009) has been able to measure the cation dynamics on and between individual sites (M1, M2) in (Co,Mg)-olivines using in situ optical spectroscopy at temperatures between 600 and 800 °C. They have been able to relate these data to macroscopic diffusion rates, thereby opening the possibility of measuring diffusion coefficients at these relatively low temperatures. The other extremity of the Fe-Mg solid solution series can be explored with NMR spectroscopy. Stebbins (1996) has carried out in situ experiments at 1400 °C on pure forsterite to characterize the jump frequencies of Mg. The results, when translated to diffusion coefficients using very simple models, are compatible with the Mg-tracer diffusion data of Chakraborty et al. (1994). This method sees the strongest signals from the M2-M2 jumps, which are however likely to be the less dominant modes of motion for the overall bulk transport (see above and the results of computer simulations below). Accordingly, the calculated diffusivities are somewhat slower than those measured macroscopically. Further, consideration of the effect of the correlation factor, f, would improve the agreement. In fact, these in situ methods offer an excellent means of determining the correlation factors and the detailed diffusion mechanisms of Fe and Mg. This should be the focus of much future research. In particular, such observations offer a useful bridge between macroscopic measurements of diffusion coefficients and atomistic computer simulations.

Computer calculations Computer models to compute the energetics of defect formation, migration and diffusion in olivines has been carried out for a long time now, a general review of computer calculation

Diffusion in Olivine, Wadsleytie, Ringwoodite

629

methods may be found in the chapter by De Koker and Stixrude (2010, this volume). Beginning with the early static calculations of Lasaga (1980) and Ottonello (1987) based on assumed interatomic potentials, numerous calculations using different methods and programs have tracked the evolution of computational capabilities with time. Now, calculations based on most up to date ab initio quantum mechanical methods are available (Brodholt 1997; Brodholt and Refson 2000; Walker et al. 2009). However, all of these calculations are in idealized Mg2SiO4 systems. Walker et al. 2003 did a very preliminary evaluation of Fe-bearing systems in the framework of an empirical potential model; full ab initio calculations that consider the effect of Fe have not yet been carried out. Considering the kinds of distortion of structure expected in the vicinity of point defects in largely ionic solids such as olivine (see, for example, Fig. 7 of Chakraborty 2008 and the related discussion there), particularly when transition metals such as Fe are involved, ab initio calculations are necessary for properly evaluating defect energetics and diffusion behavior of such solids. We recall from our discussion of experimental data and diffusion mechanisms above that Fe, even when present in very minor trace quantities, determines the diffusion mechanism and energetic of transport in olivine. Therefore, in spite of the number of computer simulations that have been carried out, and the ability to compute at temperatures higher than 0 K including entropic contributions (Vocadlo et al. 1995), there is little that can be directly and quantitatively compared to experimental diffusion data. Nevertheless, one aspect of defect and diffusion energetic that depends more on the structure of olivine than on the detailed energetic contributions of different chemical species is the assessment of diffusion pathways. There is some valuable information that has emerged from the computer simulations of this aspect. Most notably, and in excellent agreement with direct spectroscopic observations described above, it is found that M1-M1 jumps are energetically far more effective, and hence more frequent, than either M1-M2 or M2-M2 jumps. This may result from a combined effect of the brevity of the jump length (Brodholt 1997) as well as the ease of defect formation on the M1 site (e.g., Lasaga 1980; Ottonello 1987; Brodholt 1997; bearing in mind the caveat about energetic noted above). For the diffusion of Si and O, the following aspects that are consistent with available experimental data at this point have emerged from computer simulations: (a) diffusion of Si in wet olivine occurs by the means of different vacancy complexes because incorporation of H significantly reduces the energy cost of creating a Si-vacancy (Brodholt and Refson 2000) and (b) diffusion of oxygen in wet and dry olivine appears to occur by the same mechanism (Walker et al. 2003). The mechanism of diffusion of oxygen is most likely by an interstitial mechanism. We reiterate that all of these conclusions are subject to revision once calculations in Fe-bearing systems become available. With current advances in computational methods, it seems we are very close to the point where this will be the case.

Wadsleyite and Ringwoodite Wadsleyite and ringwoodite (previously named β-spinel and γ-spinel) are high temperature polymorphs of olivine that are found in shocked meteorites and are thought to be the minerals that replace olivine, with increasing depth, in the transition zone (ca. 400-600 km depth) of the Earth’s mantle. Wadsleyite is orthorhombic in structure whereas ringwoodite is cubic. Wadsleyite has the interesting property that one of the oxygen atoms is not or is weakly bonded to Si (essentially, there are Si2O7 groups in wadsleyite although it is not a ring silicate), providing an excellent site for the incorporation of H atoms as (OH) groups. As a result, the solubility of “water” in wadsleyite is many times larger than in olivine and this has important implications for diffusion, transport and rheology in the transition zone. Diffusion rates in wadsleyite and ringwoodite are difficult to measure because crystals large enough in size to measure diffusion coefficients are difficult to obtain. As a result, diffusion coefficients that are known for these materials come from measurements in coarse polycrystals and in rare single crystals. There are additional difficulties related to high pressure

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experiments including the issues of oxygen fugacity control. Much of the discussion of these aspects in the context of diffusion data in olivine is relevant for the design of diffusion experiments and discussion of data in wadsleyite and ringwoodite as well. Controlling the water contents of these phases is a further concern because, unlike olivine, these phases can take up significant amounts of H. Finally, one more problem that is encountered in the study of wadsleyite and ringwoodite is the limited stability ranges of these phases. It is not possible to carry out experiments over large ranges of pressures and temperatures to constrain parameters such as activation volume and activation energy well. There is little information on diffusion anisotropy (although it is expected to be less significant as the structure approaches cubic) because much of the diffusion data comes from polycrystals. On the plus side, however, information on both grain boundary- as well as volume- diffusion rates is often extracted from these experiments. We will primarily discuss the volume diffusion data in this work (see Dohmen and Milke 2010, chapter 21 of this volume on grain boundary diffusion). Diffusion experiments in these materials are typically two stage processes where in a first set of experiments the single crystal/polycrystalline material is synthesized and, after characterization, in a second experiment actual diffusion profiles are induced. These are measured and modeled following more standard procedures after quenching.

Diffusion of divalent cations Fe-Mg. After two large (~ 200-250 μm) single crystals were fortuitously synthesized, the first diffusion data on Fe-Mg diffusion rates in wadsleyite were obtained using single crystalpolycrystal diffusion couples (Chakraborty et al. 1999). Subsequently, three other studies have measured Fe-Mg diffusion rates in Fe-Mg wadsleyites (Farber et al. 2000; Kubo et al. 2004; and Holzapfel et al. 2009) and the results have been analyzed together and discussed by Holzapfel et al. (2009). All data obtained in wadsleyite, along with Fe-Mg diffusion rates in olivine shown for comparison, are illustrated in Figure 11. The combined analysis of Holzapfel et al. (2009) suggests the following expression for calculating diffusion coefficients in wadsleyite as a function of pressure, temperature and composition:  (229000 + ( P − 15)(13.9 × 103 )) J/mol  D(m 2 /s) = 1.24 × 10 −6 exp[11.8(0.86 − X Mg )] exp  −  RT  

Here XMg is the mole fraction of Mg in wadsleyite, T is in Kelvins and P is in Gigapascals. The equation is valid for oxygen fugacity at the NNO buffer and for dry conditions. “Dry” for wadsleyite may include crystals with several tens of ppm of water (Kubo et al. 2004). If the effect of water found by Kubo et al. (2004) at 1230 °C is independent of temperature and water content, then diffusion in wet wadsleyite (several 100 ppm or more water) would be an order of magnitude faster than that calculated using the above expression. The effect of fO2 on diffusion in wadsleyite has not yet been determined; Holzapfel et al. (2009) recommend that for calculations at much reducing conditions compared to NNO, one may take the known fO2 dependence of diffusion in olivine as a first approximation for wadsleyites. Fe-Mg diffusion coefficients in ringwoodite have not yet been measured. Considering the similarity in structure and other transport properties (e.g., see below), a good approximation may be to use the above equation for calculation of diffusion rates in ringwoodite as well until direct data becomes available. The activation energy in the above equation at 15 GPa, where wadsleyite is stable, is of the same order (~ 200kJ/mol) as the activation energy for Fe-Mg diffusion found in spinels at atmospheric pressures (e.g., Liermann and Ganguly 2002). The activation volume of diffusion is relatively high, with important implications for transport and mixing processes in the earth’s mantle. Considering the combined effects of P, T, composition, fO2 and fH2O that are expected along a mantle geotherm, Holzapfel et al. (2009) concluded that diffusion rates would jump

Diffusion in Olivine, Wadsleytie, Ringwoodite 1600

1400

T [°C]

1200

-12 1 2 3 4

2

log (D(m /s))

-14

-16

5 -18

-20 6 5.5

6.0

631

6.5 7.0 4 10 /T [1/K]

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1: Wadsleyite: 13 GPa (Holzapfel et al. 2009) 2: Wadsleyite: 15 GPa (Holzapfel et al. 2009) 3: Wadsleyite: 16 GPa (Holzapfel et al. 2009) 4: Wadsleyite: 17 GPa (Holzapfel et al. 2009) 5: Olivine: 1 atm (Dohmen et al. 2007) 6: Olivine: 15 GPa (Dohmen et al. 2007; Holzapfel et al. 2009) Kubo et al. 2004 Chakraborty et al. 1999 Holzapfel et al. 2009 Farber et al. 2000

Figure 11. Arrhenius plot showing best fits to DFe-Mg in wadsleyite as well as the experimentally measured data points from different studies. Sources of data or best fit lines are indicated in the legend, along with pressures at which these were obtained. Best fit lines for DFe-Mg for olivine at two different pressures (calculated following Dohmen and Chakraborty 2007) are shown for reference.

by 6-7 orders of magnitude (depending primarily on water content and distribution of water in the mantle) at the top of the transition zone when olivine converts to wadsleyite (diffusion in wadsleyite is faster). At greater depths diffusion rates would drop again because of the high, positive activation volume. Therefore, in combination with the inference made above for olivine, it is found that the base of the lithosphere and the top of the transition zone are the two regions in the Earth’s mantle that are particularly efficient sites for chemical mixing, homogenization and transport.

Diffusion of silicon and oxygen Shimojuku et al. (2009) measured the volume- as well as grain boundary diffusion rates of Si and O in polycrystalline samples of wadsleyite in a multianvil press. The samples were coated with isotopically doped thin films of the same compositions as the substrate using pulsed laser deposition (PLD). After the experiments, concentration gradients were measured using the depth profiling mode in SIMS and the surfaces were characterized using a number of tools such as AFM, water contents were determined at different stages using IR spectra. As in Dohmen et al. (2002) for olivine, these experiments had the great advantage that diffusion of Si and O were simultaneously measured in the same samples, so that the data were directly comparable and many sources of uncertainty (e.g., P and T calibration in different experiments) were eliminated. The data analysis had some novel aspects. Along with convolution analysis for the spatial averaging effect of the ion probe, a complete 2-D numerical simulation was used to retrieve the volume and grain boundary diffusion rates. This avoids some of the approximations that need to be made in order to use idealized analytical solutions that have more conventionally been used. The diffusion rates found in this study (Fig. 12) are described by the following parameters in the Arrhenius equation, D = D0 exp (−Q/RT). We report the complete data set for volume

Chakraborty

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-18

-18

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1100 1: Si,Dohmen Olivine: Dohmen et al. 2002 1: Si, Olivine: et al. 2002 2: Si, Ringwoodite: Shimojuku 2: Si, Ringwoodite: Shimojuku et al. 2009et al. 2009 3: Si, Wadsleyite: Shimojuku 3: Si, Wadsleyite: Shimojuku et al. 2009et al. 2009 4: O, Ringwoodite: Shimojuku 4: O, Ringwoodite: Shimojuku et al. 2009et al. 2009 5: O, Wadsleyite: Shimojuku 5: O, Wadsleyite: Shimojuku et al. 2009et al. 2009 6: O,Dohmen Olivine: Dohmen et al. 2002 6: O, Olivine: et al. 2002

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T [°C] T [°C] 1500 1400 1500 1400 1300 1300 1200 1200 1100

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7,5

Figure 12. Arrhenius plot showing best fits to diffusion coefficients for Si and O in wadsleyite and ringwoodite, with fits to 1 atmosphere data for Si and O diffusion in olivine shown for reference. The sources of data are indicated in the legend.

and grain boundary diffusion, although the latter are not discussed in this chapter. The units of D0 are m2/s and the errors represent 1σ, δ is the width of grain boundary (see Dohmen and Milke 2010, chapter 21 this volume for more explanation of the significance of this parameter): logD0

Q (kJ/mol)

Wadsleyite Si volume diffusion: Si grain boundary diffusion (δDGB): O volume diffusion O grain boundary diffusion (δDGB)

−7.6±3.0 −14.9±3.0 −10.5±2.3 −16.8±2.5

409±103 327±101 291±79 244±86

Ringwoodite Si volume diffusion Si grain boundary diffusion (δDGB) O volume diffusion O grain boundary diffusion (δDGB)

−5.5±2.8 −13.2±2.6 −8.5±2.5 −17.1±2.1

483±94 402±88 367±83 246±70

Si was found to be the slowest diffusing species in both wadsleyite and ringwoodite. Comparison with diffusion data from olivines and perovskites indicate that diffusion rates of Si as well as O would increase with depth at both the 410 km as well as the 660 km seismic discontinuities. This suggests that the viscosity of the mantle should decrease with depth across these discontinuities. However, this trend is opposite to the inference drawn from a global inversion study (Mitrovica and Forte 2004). Consideration of additional effects such as the effect of water on diffusion and the water content of the mantle at different depths is necessary to resolve this issue. A deformation map calculated based on the diffusion data suggests that dislocation creep would dominate in most of the transition zone and this may possibly account for the global seismic anisotropy. In very limited regions, such as the base of a cold slab where

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metastable olivine converts to very fine grained (10-100 μm) wadsleyite or ringwoodite, grain size sensitive diffusion creep may occur.

Diffusion of ions that are incorporated by heterovalent substitutions Hydrogen. Hydrogen diffusion in polycrystalline wadsleyite was measured by Hae et al. (2006). For combined grain boundary + volume diffusion in a matrix with average grain size of 9 μm, they provide the following expression for calculation of H transport rates at 15 GPa:  −123 ± 32 kJ/mol  D= 9.6 × 10 −6 exp  H  RT  

These rates are similar to H transport rates in olivine and therefore for H diffusion, there is not a large jump in transport rates at the top of the transition zone, as has been found for DFe-Mg, DSi and DO.

A SUMMARY, AND Applications of diffusion data in olivine, wadsleyite and ringwoodite Based on the foregoing discussion, the relative magnitude of diffusion coefficients that can be used for modeling processes in olivine, wadsleyite and ringwoodite in natural systems are shown in Figure 13. All data discussed in this chapter are available in an Excel spreadsheet as supplemental data associated with this chapter. In addition to the consistency checks for data discussed above, modeling concentration gradients observed in natural systems provide an additional means of evaluating diffusion data. Some checks that may be used for evaluating the quality of diffusion data in the course of modeling natural systems have been outlined by Chakraborty (2006) and an example of such a test for olivine is provided by Hewins et al. (2009). Costa and Dungan (2005) demonstrate how the modeling of concentration gradients of multiple elements with different diffusion rates in the same crystals can be a useful internal check on whether diffusion modeling of natural crystals is yielding meaningful results. Chakraborty (2006) and Costa et al. (2008) provide checklists of factors (with particular focus on olivine) that need to be considered when diffusion data are used for modeling natural systems; Chakraborty (2008) provides some examples of general applications of diffusion data in a number of minerals including olivine. The total citation of papers reporting diffusion coefficients in Fe-Mg olivines alone is in the vicinity of 500, and this does not include the highly cited Misener (1974) because it is not a part of citation databases. Therefore, in a review such as this it is not possible to summarize all possible uses of these diffusion data and I will merely outline some areas of application. Diffusion coefficients in olivine and its polymorphs have recently been used to (i) calculate parameters related to the duration of thermal events (e.g., cooling rates) experienced by volcanic and plutonic igneous rocks, mantle rocks, meteorites and metamorphic rocks (e.g., Fagan et al. 2002; Imai and Yurimoto 2003; Costa and Chakraborty 2004; Frost et al. 2007), (ii) evaluate the closure temperature of geothermometers and infer the geological events that temperatures calculated from these geothermometers relate to (e.g., Liermann and Ganguly 2003), (iii) understand the redox state of the mantle (e.g., Williams et al. 2005; Canil et al. 2006), (iv) understand the distribution of water in the mantle (e.g., Ruedas 2006), (v) evaluate whether melt and fluid inclusions in olivines could retain their primary

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8 9 10 4 10 /T [1/K]

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1600 1400

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T [°C] 1200

H

1000

Figure 13. Comparison of Arrhenius relations for recommended diffusion coefficients (sources shown in legend) in (a) olivine and (b) wadsleyite of composition XMg = 0.9.

O: Dohmen et al. (2002) Ca: Coogan et al. (2005) Ni: Petry et al.(2004) H: Demouchy & Mackwell (2006) Cr: Ito & Ganguly (2006)

Fe-Mg: Dohmen & Chakraborty (2007) Si: Dohmen et al. (2002) O: Dohmen et al. (2002) Ca: Coogan et al. (2005) Ni: Petry et al.(2004) Fe-Mg: Dohmen & Chakraborty (2007) H: Demouchy & Mackwell (2006) Si: Dohmen Cr: Ito & Ganguly (2006)et al. (2002)

Ca

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-8

5

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(b)

6

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1000

8

Fe-Mg, olivine

Fe-Mg

T [°C] 1200

7 4 10 /T [1/K]

O

1600 1400

Fe-Mg: Holzapfel et al. 2009, P = 15 GPa Si: Shimojuku et al. 2009, P = 16 GPa O: Shimojuku et al. 2009, P = 16 GPa H: Hae et al. 2006, P = 15 GPa Fe-Mg, olivine: Dohmen & Chakraborty 2007, P = 15 GPa

2

Fe-Mg: Holzapfe Si: Shimojuku et O: Shimojuku et H: Hae et al. 20 Fe-Mg, olivine:

634 Chakraborty

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signatures through their subsequent thermal history (e.g., Cottrell et al. 2002; Portnyagin et al. 2008), (vi) evaluate the kinetics of order-disorder and other crystal dynamic processes (e.g., Janney and Banfield 1998; Heinemann et al. 2006), (vii) understand the nature of shock processes in meteorites (e.g., Miyahara et al. 2008), (viii) understand the electrical conductivity profile of the Earth’s mantle (e.g., Misener 1974; Karato 1990), (ix) understand the rheological (creep) behavior of the upper mantle (olivine) as well as transition zones (wadsleyite, ringwoodite) (e.g., Mei and Kohlstedt 2000; Kohlstedt 2006; Burgmann and Dresen 2008, Karato 2010), (x) understand the kinetics of evaporation and condensation in the solar nebula (e.g., Ozawa and Nagahara 2000; Wooden 2008; Petaev 2009), as well as compositional evolution (e.g., exsolution) during other extra-terrestrial processes (e.g., Petaev and Brearley 1994), (xi) understand the reaction behavior of A2BO4 type of solids in materials science (e.g., Kimizuka et al. 1990; Devi et al. 2005).

Acknowledgments Much of the research presented here has been carried out with generous funding from the German Science Foundation (DFG). Most recently my research has been supported by grants under the SFB 526 Program of the same organization. Christof Petry, Laurence Coogan, Fidel Costa, Christian Holzapfel, and most significantly, Ralf Dohmen, have played crucial roles in generating much of the data on olivines and in navigating me through the random walk of drunken sailors that is otherwise known as diffusion. Elke Meissner, Achim Hain and Stephan Stahl participated in the early stages of work on olivine, and Tomoaki Kubo and Akira Shimojuku have been collaborators on the most recent work on wadsleyite and ringwoodite. Graduate students Patrick Remmert, Franka Richelmann and Kathrin Faak helped with figures, formatting and references. I thank all of them and the DFG, this chapter is an outcome of what I have learned because of them. Finally, I thank the editors Daniele Cherniak and Youxue Zhang for their patience and help.

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Abstracts, Goldschmidt Conference 72:A409-A409 Ito M, Ganguly J (2006) Diffusion kinetics of Cr in olivine and 53Mn-53Cr thermo-chronology of early solar system objects. Geochim Cosmochim Acta 70:799-809 Ito M, Yurimoto H, Morioka M, Nagasawa H (1999) Co2+ and Ni2+ diffusion in olivine determined by secondary ion mass spectrometry. Phys Chem Miner 26:425-431 Janney DE, Banfield JF (1998) Distribution of cations and vacancies and the structure of defects in oxidized intermediate olivine by atomic-resolution TEM and image simulation. Am Mineral 83:799-810 Jaoul O, Bertranalvarez Y, Liebermann R (1995) Fe-Mg interdiffusion in olivine up to 9 GPa at T = 600-900°C, Experimental data and comparison with defect calculations. Phys Earth Planet Inter 89:199-218 Jaoul O, Froidevaux C, Durham WB, Michaut M (1980) Oxygen self-diffusion in forsterite. Implications for the high-temperature creep mechanism. Earth Planet Sci Lett 47:391-397 Jaoul O, Houlier B, Abel F (1983) Study of O-18-labeled diffusion in magnesium orthosilicate by nuclear microanalysis. J Geophys Res 88:613-624 Jaoul O, Poumellec M, Froidevaux C, Havette A (1981) Silicon diffusion in forsterite: a new constraint for understanding mantle deformation. In: Anelasticity in the Earth. Geodynamics Series Vol. 4. Stacey FD, Paterson MS, Nicolas A (eds) American Geophysical Union, Washington, p 95-100 Jurewicz AJG, Watson EB (1988) Cations in olivine. 2. Diffusion in olivine xenocrysts, with applications to petrology and mineral physics. Contrib Mineral Petrol 99:86-201 Karato S (1990) The role of hydrogen in the electrical conductivity of the upper mantle. Nature 347:272-273

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Portnyagin M, Almeev R, Mateev S, Holtz F (2008) Experimental evidence for rapid water exchange between melt inclusions in olivine and host magma. Earth Planet Sci Lett 272:541-552. Putnis A (1992) An Introduction to Mineral Sciences. Cambridge University Press, New York Reddy KPR, Oh SM, Major LD, Cooper AR (1980) Oxygen diffusion in forsterite. J Geophys Res 85:322-326 Remmert P, Dohmen R, Chakraborty S (2008) Diffusion of REE, Hf and Sr in Olivine. Eos Trans. AGU 89:MR33A-1844 Ruedas T (2006) Dynamics, crustal thicknesses, seismic anomalies, and electrical conductivities in dry and hydrous ridge-centered plumes. Phys Earth Planet Inter 155:16-41 Ryerson FJ, Durham WB, Cherniak DJ, Lanford WA (1989) Oxygen diffusion in olivine – Effect of oxygen fugacity and implications for creep. J Geophys Res [Solid Earth] 94:4105-4118 Shi J, Becker KD, Ebbinghaus SG (2008) Temperature-jump induced cation exchange kinetics in (Co0.1Mg0.9)2SiO4 olivine: an in situ optical spectroscopic study. Phys Chem Mineral 35:1-9 Shi JM, Ganschow S, Klimm D, Simon K, Bertram R, Becker KD (2009) Octahedral cation exchange in (Co0.21Mg0.79)2SiO4 olivine at high temperatures: kinetics, point defect chemistry, and cation diffusion. J Phys Chem C 113:6267-6274 Shimojuku A, Kubo T, Ohtani E, Nakamura T, Okazaki R, Dohmen R, Chakraborty S (2009) Si and O diffusion in (Mg,Fe)2SiO4 wadsleyite and ringwoodite and its implications for the rheology of the mantle transition zone. Earth Planet Sci Lett 284:103-112 Sockel HG, Hallwig D (1977) Ermittlung kleiner Diffusionkoeffizienten mittels SIMS in oxydisehen Verbindungen. Mikrochim Acta (Wien) 7:95-107 Sockel HG, Hallwig D, Schachtner R (1980) Investigations of slow exchange processes at metal and oxide surfaces and interfaces using Secondary Ion Mass-Spectrometry. Mater Sci Eng 42:59-64 Spandler C, O’Neill H, St C, Kamenetsky V S (2007) Survival times of anomalous melt inclusions from element diffusion in olivine and chromite. Nature 447:303-306 Spandler C, O’Neill HS (2006) Trace element diffusion coefficients in olivine. EOS Trans, Abstract in Fall Meeting, AGU 87:V53E-02 Spandler C, O’Neill HS (2009) Diffusion and partition coefficients of minor and trace elements in San Carlos olivine at 1300 °C with some geochemical implications. Contrib Mineral Petrol 159:791-818 Stebbins JF (1996) Magnesium site exchange in forsterite: A direct measurement by high-temperature Mg-25 NMR spectroscopy. Am Mineral 81:1315-1320 Tsai TL, Dieckmann R (2002) Variation of the oxygen content and point defects in olivines, (FexMg1‑x)2SiO4, 0.2 < x < 1.0. Phys Chem Miner 29:680-694 Vocadlo L, Wall A, Parker SC, Price GD (1995) Absolute ionic diffusion in MgO – computer calculations via lattice dynamics. Phys Earth Planet Int 88:193-210 Walker AM, Woodley SM, Slater B, Wright K (2009) A computational study of magnesium point defects and diffusion in forsterite. Phys Earth Planet Inter 172:20-27 Walker AM, Wright K, Slater B (2003) A computational study of oxygen diffusion in olivine. Phys Chem Mineral 30:536-545 Watson EB, Dohmen R (2010) Non-traditional and emerging methods for characterizing diffusion in minerals and mineral aggregates. Rev Mineral Geochem 72:61-105 Weinbruch S, Armstrong J, Palme H (1993) Constraints on the thermal history of the Allende parent body as derived from olivine-spinel thermometry and Fe/Mg interdiffusion in olivine. Geochim Cosmochim Acta 58:1019-1030 Williams HM, Peslier AH, McCammon C, Halliday AN, Levasseur S, Teutsch N, Burg JP (2005) Systematic iron isotope variations in mantle rocks and minerals: The effects of partial melting and oxygen fugacity. Earth Planet Sci Lett 235:435-452 Wooden DH (2008) Cometary refractory grains: Interstellar and nebular sources. Space Science Rev 138:75-108 Zhang Y (2010) Diffusion in minerals and melts: theoretical background. Rev Mineral Geochem 72:5-59

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Reviews in Mineralogy & Geochemistry Vol. 72 pp. 641-690, 2010 Copyright © Mineralogical Society of America

Diffusion in Pyroxene, Mica and Amphibole D.J. Cherniak Department of Earth & Environmental Sciences Rensselaer Polytechnic Institute Troy, New York 12180, U.S.A. [email protected]

A. Dimanov Laboratoire de Mécanique des Solides, UMR C 7649 Ecole Polytechnique Route de Saclay, 91128 Palaiseau, France [email protected]

INTRODUCTION This chapter presents an overview of diffusion data for pyroxenes, amphiboles and micas. These minerals are grouped together since amphiboles and micas are closely related in structure to pyroxenes, with amphiboles essentially constructed of alternating layers with structures of mica and pyroxene. We begin with discussion of diffusion in pyroxenes, for which an extensive literature exists, with diffusion studies of major, minor and trace elements. We consider diffusion mechanisms in light of present understanding of defect chemistry, and discuss various crystal-chemical factors that may affect cation diffusion. The last section of the chapter presents a review of diffusion data for amphiboles and micas. Selected Arrhenius relations for these all these mineral phases are summarized in the Appendix Tables A1, A2, A3 and A4. This chapter focuses primarily on cation diffusion, since oxygen, hydrogen and noble gas diffusion are discussed in other chapters; readers interested in more detailed discussion of diffusion of these species in pyroxene, amphibole and mica are directed to Chapters 10 (Farver 2010, this volume) and 11 (Baxter 2010, this volume).

CATION DIFFUSION IN PYROXENES Since the early 1970s, solid state diffusion of cations in pyroxenes has been recognized to play a fundamental role in numerous geodynamical, petrological and geochemical processes involving mass transport. Knowledge of diffusion coefficients allows constraints to be placed upon time scales and thermodynamic conditions in many natural contexts where mass transport is dominated by solid state diffusive processes. Pyroxenes are major mineral phases at depth (in the lower crust and upper mantle) in both the earth and extraterrestrial bodies. Pyroxene composition evolves with temperature and pressure due to diffusive exchanges of major cations (i.e., Ca, Mg, Fe, Mn, Al) with surrounding mineral phases. The crystalline structure of pyroxenes is also influenced by diffusionally-induced compositional changes, including order-disorder, spinodal decomposition, exsolution and coarsening of lamellae. Equilibrium compositions of pyroxenes and their state of order can provide geothermobarometric constraints. One example is the exchange of Al between pyroxenes and the aluminous phases garnet, spinel, and plagioclase (Gasparik and Lindsey 1980; Gasparik 1984). Chemical zoning within individual crystals is frequently observed (Fraser and Lawless 1978; Sautter and Harte 1988, 1990), because slow intracrystalline 1529-6466/10/0072-0014$05.00

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diffusion kinetics may contribute to the formation of zoning (e.g., Watson and Liang 1995) and its preservation (e.g., Cherniak and Liang 2007). Compositional zoning and isotopic disequilibria may in some cases be interpreted in terms of cooling rates, as in applications to geospeedometry (Dodson 1973, 1986; Lasaga et al. 1977; Hofmann and Hart 1978; Lasaga 1983; Ganguly and Tazzoli 1994; Jaoul and Sautter 1999; Ganguly et al. 2000; Coogan et al. 2005). Cation diffusion in pyroxenes also controls the kinetics of solid state reactions and corona growth (Brady 1983; Joesten 1991; Yund and Tullis 1991); the length scales of such growth allow for estimations of residence times, providing constraints for geothermochronology (Freer 1979). Diffusion-related mechanisms are also controlling processes in microstructural evolution, including grain coarsening, and ductile deformation mechanisms such as dislocation climb (Raterron and Jaoul 1991; Jaoul and Raterron 1994) and grain sliding (Dimanov et al. 2003; Dimanov and Dresen 2005; Dimanov et al. 2007). Therefore, the knowledge of diffusion kinetics is clearly of fundamental importance in enhancing our understanding of geological and planetary processes. Over roughly the past three decades, numerous studies have been undertaken to estimate and directly measure diffusion in pyroxene and apply these data. In this chapter we review the extant diffusion data for major cations, trace and minor elements in pyroxenes, obtained either by direct experimental measurements or indirect estimates from natural and experimental observations. Direct measurements are based on analysis of chemical or isotopic profiles as a function of depth, or with step scans across an interface by means of electron microprobe, electron microscopy (coupled to Energy Dispersive X-ray (EDX)), accelerator-based ion beam techniques (RBS, NRA), or ion probe (SIMS) analysis. The utility of these various techniques for analysis will depend on the experimental configuration and magnitude of diffusion coefficients. Depth profiling techniques (RBS, NRA, SIMS) are characterized by the highest spatial resolution, and permit analysis of the chemical or isotopic composition beneath the sample surface parallel to the diffusion direction. Hence, they are well adapted to slow diffusing species, but the experimental design must allow the retrieval of an intact sample surface (or interface) perpendicular to the diffusion direction. The usual geometries for diffusion experiments involve diffusive exchanges between a substrate and a surface thin film, or a surrounding medium (powders, gaseous environments). Conversely, electron microprobe, SEM and TEM-EDX and transverse mode SIMS are used for experimental geometries where step by step scans are made on a cross-section perpendicular to the diffusion interface. With the exception of TEM-EDX, these are best suited for fast diffusing species, due to limitations of spatial resolution related to the scan step size and spot size of the analyzing beam. Indirect estimates are based on experimentally derived kinetics of homogenization of exsolution lamellae, order-disorder on crystal sites, and solid state reactions with moving interfaces, such as exsolution coarsening, intergrowths, grain growth and spinodal decomposition. While these can be evaluated through controlled laboratory studies, similar estimates may also be inferred from interpretation of chemical and structural inhomogeneities preserved in natural assemblages, but with larger uncertainties given the limitations in precisely knowing time-temperature conditions, pressure, and other critical parameters. Most of the indirect estimates are also subject to the assumption that the observed changes are solely a result of diffusion-controlled kinetics. However, for solid state reactions which involve moving interfaces the kinetics of the process may also be influenced by interface related phenomena (e.g., solute drag mismatch strain). Also, there may be several species which could control the kinetics of solid state reactions for complex silicates. Estimates of diffusion from natural zoning are further dependent on estimates of thermal histories. Hence, indirect estimates should be evaluated with caution, and under most circumstances will provide constraints on the order of magnitude of the diffusion process rather than yielding precise diffusion coefficients. In addition to experimental and empirical determinations of diffusion, we also consider the results of numerical modeling, which has recently been applied to better understand the mechanisms of cation diffusion in pyroxene.

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The diffusion data for major elements will be presented chronologically, and by mineral group, in order to follow the technological evolution and improvement of the spatial resolutions of analytical techniques, which have permitted more and more precise data to be collected and an extension of measurement capabilities to lower diffusion coefficients. The details of reliable diffusion data (parameters of diffusion coefficients, diffusion type and mechanisms, measurement techniques, experimental conditions) are presented in Table A1. This will be followed by sections on pyroxene defect chemistry, and the phenomenon of early partial melting (EPM) and its potential effects on diffusion. Diffusion of trace and minor elements will be grouped by elemental species, and by pyroxene composition for diffusion of a particular element. These data are presented in Table A2. We also consider relative diffusivities of cations in diopside and enstatite, and various parameters that might affect diffusion.

Pioneering approaches In slowly cooled basic igneous rocks, augite and subcalcic clinopyroxene solid solutions commonly experience spinodal decomposition and exsolution of orthopyroxene or pigeonite lamellae (parallel to (001) planes); growth kinetics of these features are controlled by Ca(Fe,Mg) interdiffusion (Huebner et al. 1975; McCallister and Nord 1981; Jantzen 1984; Miyamoto and Takeda 1977, 1994; Weinbruch and Müller 1995; McCallum and O’Brien 1996; Weinbruch et al. 2001, 2003). The lamellar thickness and compositional zoning of adjacent lamellae that result from lamellar coarsening kinetics may be modeled to retrieve cooling rates (e.g., Schwartz and McCallum 2005), provided the interdiffusion coefficients are known. The earliest attempts to estimate interdiffusion rates of major divalent octahedral cations (Fe, Mg, Ca) in pyroxenes were motivated by the interest in constraining cooling rates and annealing durations of extraterrestrial assemblages. Huebner (1976) and Stanford and Huebner (1979) applied comparative analysis of compositionally zoned olivine and pyroxene xenocrysts from lunar basalts (Huebner et al. 1975; Huebner 1976; Stanford and Huebner 1979; Huebner and Nord 1981), on the basis of the known Fe-Mg interdiffusion coefficients for olivine (Buening and Buseck 1973). Assuming as a first approximation isothermal annealing and growth of pigeonite rims by Ca-(Fe,Mg) exchange between augite cores and the surrounding matrix, they estimated DCa-FeMg ~ 4 × 10−15 m2/s at 1050 °C. Huebner et al. (1975) also first attempted to directly measure Ca-Mg interdiffusion rates through cation exchange between orthopyroxene powder and augite single crystals, but electron microprobe analysis failed to detect any chemical profiles even after 628 hours at 1266 °C. Considering the microprobe spatial resolution (~  3 μm), the authors estimated a maximum value for DCa-Mg ~  4 × 10−20 m2/s, which they assumed to be unrealistically low. Later Huebner and Nord (1981) attributed this failure to the experimental difficulty in producing adequate diffusion couples and argued in favor of experimental studies of lamellar growth (McCallister 1978) coupled with electron microprobe and analytical TEM to determine diffusion coefficients. McCallister et al. (1979) attempted direct measurements of diffusion of radioactive 45Ca and Fe tracers. Isotopically-enriched thin films were deposited onto (001) polished diopside faces by evaporation of chloride solutions. The diffusion couples were annealed for up to 6 months, then sectioned perpendicular to (001) in order to obtain radiotracer concentration profiles by exposing samples to β-particle sensitive emulsions. Self-diffusion rates were obtained, but Brady and McCallister (1983) later recognized that the results were not fully consistent with diffusion-controlled processes. Freer et al. (1982) made efforts to measure diffusion of major element cations in diopside single crystals with electron and ion microprobe analyses by attempting to induce Al and Fe interdiffusion between Al- and Fe-rich sintered powders and diopside, and tracer diffusion of 26Mg and 43Ca between isotopically enriched synthetic glass and diopside. Transverse mode analysis of samples failed to show any measurable diffusion profiles, but permitted the setting of new upper limits on diffusivities of ~ 4 × 10−19 m2/s at 1200 °C for Al and Fe, and less than ~ 7 × 10−19 m2/s at 1250 °C for Ca and Mg. These upper limits 57

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for diffusion rates were significantly lower than all previously published tracer-diffusion data for pyroxenes (McCallister et al. 1979; Sneeringer and Hart 1978; Seitz 1973; Lindner 1955). By evaluating marginal and discontinuous zoning in Ca-poor and Ca-rich clinopyroxenes from natural assemblages, Rietmeijer (1983) estimated that Ca-(Fe,Mg) interdiffusion rates along the c-axis in iron-rich clinopyroxenes may be as low as ~ 6 × 10−24 to ~ 2 × 10−21 m2/s at 900 °C. In the early eighties, the most reliable cation interdiffusion data were reported by Brady and McCallister (1983). The authors performed lamellar homogenization experiments at 2500 MPa (in piston cylinder apparatus), between 1100-1250 °C. The homogenization kinetics were analyzed in terms of Ca-(Mg,Fe) interdiffusion normal to (001) between pigeonite lamellae and a sub-calcic diopside host. The average pseudo-binary interdiffusion coefficient obtained by Brady and McCallister (1983) at 1150-1250 °C confirmed the observations of Freer et al. (1982) of the sluggishness of divalent cation diffusion in clinopyroxenes. However, the interdiffusion rates reported by Brady and McCallister (1983) possessed quite large uncertainties (e.g., an activation energy Ea = 361 ± 190 kJ/mol). The limited data did not permit any interpretation in terms of diffusion mechanisms and point defect chemistry, or any quantification of dependence of diffusion rates on pyroxene composition. Moreover, the very few extant diffusion coefficients in clinopyroxenes were all obtained under very different experimental conditions. Most experiments were performed at ambient pressure, in air (e.g., McCallister et al. 1979; Freer et al. 1982), i.e., at extremely oxidizing conditions (pO2 = 0.021 MPa), while the data of Brady and McCallister (1983) were obtained for experiments under high confining pressure, very low oxygen fugacity (graphite-oxygen buffer), and probably the presence of traces of water. At the time theoretically and experimentally derived frameworks of point defect chemistry of enstatite (Stocker 1978) and olivine (Nakamura and Schmalzried 1983) were available, and indicated that point defect concentrations (and hence, diffusion rates) would depend on pO2 for both of these iron-bearing silicates. The lack of rigorously controlled thermodynamic conditions in terms of component activities and volatile fugacities added to the difficulty in comparing the few extant diffusion data.

More recent investigations of major element diffusion From the middle 1980s to date, experimental investigations of diffusion have progressively increased in numerous minerals, including pyroxenes. Among the primary areas of concentration for experimental diffusion studies of clinopyroxenes have been self-diffusion or interdiffusion, and data have been obtained by the depth profiling techniques SIMS, RBS and NRA, all with sufficient spatial resolution (~ 15-30 nm) to resolve short diffusion profiles and access small diffusivities. Some of these studies also considered the effects of sample composition, confining pressure, diffusional anisotropy, and control of point defect chemistry. Considerable effort has also been directed toward the refinement of reliable techniques for diffusion couple preparation. For instance, while self-diffusion in pyroxenes is sometimes evaluated by direct isotopic exchange between simple oxide sources and samples (Schwandt et al. 1998; Ganguly et al. 2007; Zhang et al. 2010), some studies have been designed so that there is isotopic exchange between source materials and samples with the same bulk chemical composition, which may avoid interfacial chemical reactions and changes in chemical composition during diffusion experiments, ensuring that self-diffusion processes can be exclusively quantified (Dimanov and Ingrin 1995; Béjina and Jaoul 1996; Pacaud 1999). In the following section we chronologically present the published diffusion data for major alkaline earth (Ca, Mg), transition metal (Fe, Mn, Cr) and metalloid (Al, Si) cations in pyroxenes obtained by the state of the art techniques. When they have been determined, we will report the dependences on temperature, oxygen fugacity, crystallographic direction, and composition in terms of Fe content. In these cases, the diffusion coefficients will be presented in the form: = D D0 ( pO2 )m exp( n ⋅ X Fe )exp(− Ea RT )

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where, D0 is the pre-exponential constant, m is the oxygen fugacity exponent, n is the coefficient of dependence on the iron content XFe of the pyroxene, Ea is the activation energy, R is the universal gas constant, and T the absolute temperature.

Diffusion of major element cations in clinopyroxenes Aluminium. Sneeringer et al. (1984) and Sautter et al. (1989) were the first to apply accelerator-based ion beam analysis techniques to study cation diffusion in diopside, with Sneeringer et al. (1984) using Rutherford Backscattering Spectrometry (RBS) and Sautter et al. (1989) using Nuclear Reaction Analysis (NRA). Sautter et al. (1989) measured Al diffusion in diopside in order to constrain conditions for use of geothermobarometers based on aluminum exchange. Thin film (~  35 nm) diffusion couples were prepared by radio-frequency (RF) sputtering of Ca-Tschermak (CaAl2SiO6) onto oriented, polished and chemically cleaned (by HF etching) natural diopside substrates with XFe = [Fe]/([Fe]+ [Mg]+ [Ca]) ~ 0.025. Al diffusion profiles were measured along the c-axis by NRA (using the nuclear reaction 27Al(p,γ)28Si). Two experiments were performed at 0.1 MPa, 1180 °C and pO2 = 10−14 MPa (controlled by a flowing Ar-H2-H2O gas mixture), for 3.8 and 16.8 days, respectively. The two penetration profiles yielded the same concentration-independent Al diffusion coefficient (~ 3.7 × 10−21 m2/s and ~  2.7 × 10−21 m2/s). Jaoul et al. (1991) reported additional Al diffusion data obtained at 1000 °C and 1100 °C, constraining the activation energy to Ea = 273 kJ/mol. However, due to the use of an Ar-H2-H2O mixture of constant composition, the dataset was obtained at oxygen fugacity that varied with temperature, equivalent to about an order of magnitude lower than that of the quartz-fayalite-iron (QFI) buffer. This pioneering work confirmed the extremely low diffusivities in clinopyroxenes, but did not clarify whether Al exchanged with Mg, Si or both, nor the atomistic nature of the diffusion process. Silicon self-diffusion. RBS and NRA (using the 30Si(p,γ)31P reaction) techniques were applied to investigate Si self-diffusion in diopside (Béjina and Jaoul 1996). The experimental approach was similar to that for Al. Thin films of synthetic diopside enriched in 30Si were deposited by RF-sputtering onto oriented, polished and chemically cleaned natural diopside single crystals with XFe ~ 0.018. Experiments were performed at T = 1040 °C, 1200 °C and 1250 °C, and at pO2 = 4 × 10−14, 1.3 × 10−14 and 8 × 10−17 MPa, respectively. Oxygen fugacity changed with temperature due to the use of a single-composition Ar-H2-H2O gas mixture, with oxygen fugacities about two orders of magnitude lower that of the QFI buffer. Si self-diffusion along the c-axis proved to be about an order of magnitude slower than Al. Due to the extremely slow diffusivities, only limited data were obtained and the activation energy was constrained with a large uncertainty: Ea = 211 ± 110 kJ/mol. The authors argued that this value may only be an apparent activation energy because it was not obtained at a fixed pO2 value. They speculated that if Si self-diffusion in diopside operates by an interstitial mechanism, as it does in olivine (Houlier et al. 1990), the diffusion coefficient might have been enhanced at lower oxygen fugacity, which would result in a lower apparent activation energy. In order to hypothetically illustrate this effect, the authors corrected their data using the theoretically derived dependence on oxygen fugacity for the concentration of Si interstitials, which is defined by the power law exponent m = − 3/16 (e.g., Jaoul and Raterron 1994). Correspondingly, they calculated a possible activation energy of 280 kJ/mol (see Fig. 1), which coincides with the previously reported value for Si self-diffusion in iron-bearing olivine (Houlier et al. 1990). Calcium self-diffusion. Ca self-diffusion in diopside was first investigated by Dimanov and Ingrin (1995), Dimanov et al. (1996), Dimanov and Jaoul (1998), and recently by Zhang et al. (2010). All studies measured 44Ca tracer diffusion at 0.1 MPa total pressure. The first of these studies (Dimanov and Ingrin 1995; Dimanov et al. 1996; Dimanov and Jaoul 1998) considered Ca self-diffusion in various synthetic and natural iron-bearing diopside single crystals as a function of temperature, pO2, iron content and crystallographic orientation. Thin-film diffusion couples were prepared by RF sputtering of isotopically enriched synthetic diopside onto

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Cherniak & Dimanov

chemically etched diopside substrates; profiles were measured with RBS. These studies covered temperature ranges from 1000 °C up to near the melting point of diopside. Oxygen fugacity was controlled from ambient down to 10−18 MPa by flowing gas mixtures of Ar-H2/Ar-H2O. Depending on the experimental conditions, depth penetration profiles ranged between several tens to hundreds of nm. Diffusion rates were found to depend on iron content, but exhibited little if any anisotropy along the c- and b-axes over the investigated temperature range. The authors observed several diffusion regimes, depending on temperature and pO2 (Fig. 1c, Fig. 2). The most recent study (Zhang et al. 2010) was performed using tracer enriched thin films of CaO, which were thermally evaporated onto natural, iron-bearing diopside single crystals. Experiments were performed at 950-1150 °C, with oxygen fugacity controlled by CO/CO2 gas mixtures to attain conditions equivalent to the iron-wüstite (IW) buffer. Diffusion profiles were analyzed by SIMS. Zhang et al. (2010) determined slight anisotropy in diffusion (Fig. 1c), with the lowest activation energy and the fastest diffusion rates along the c-axis, while diffusivities along the b- and a*-axes were found to be slower than along the c-axis (by about half an order of magnitude at 950 °C) but nearly identical to each other. For 44Ca diffusion between 1000-1380 °C, along the b-axis in synthetic iron-free diopside, Dimanov and Ingrin (1995) observed two regimes characterized by very different activation energies: Ea = 280 ± 26 kJ/mol and Ea = 951 ± 87 kJ/mol below and above 1242 °C, respectively (Fig. 1c). In spite of the low impurity content of the synthetic diopside, it was suggested that the lower temperature regime was extrinsic. The higher temperature regime was considered intrinsic diffusion, with the process proceeding via Ca interstitials related to Ca Frenkel-type Figure 1. (see next two pages) Arrhenius plot of diffusion coefficients for major cations in iron-bearing natural and synthetic pyroxenes (with low Fe). When possible (depending on the experimental results), the data are recalculated to pO2 = 10−12 MPa with the appropriate oxygen fugacity exponent m. When the dependence on oxygen fugacity was not investigated, the reported data correspond to the original buffering conditions (given in brackets). a) Diffusion along the c-axis in clinopyroxene (diopside): Ca-(Mg,Fe) – Brady and McCallister (1983); Al – Sautter et al. (1989); Ca1 – Dimanov et al. (1996); Si – Béjina and Jaoul (1996); Mg1 – Pacaud (1999); (Fe,Mn)-Mg1 – Dimanov and Sautter (2000); Fe – Azough and Freer (2000); (Fe,Mn)-Mg2 – Dimanov and Wiedenbeck (2006); Mg2 – Gasc et al. (2006); Mg3 – Zhang et al. (2010); Ca2 – Zhang et al. (2010); Fe-Mg – Chakraborty et al. (2008). Although oxygen fugacity dependence was not explicitly investigated for Si (Béjina and Jaoul 1996), we have used their empirically suggested dependence (m = −3/16), in order to show an example of how data obtained at varying oxygen fugacities (solid line, buffering conditions along QFI – 2 log units) may result in an apparent activation energy substantially different from the activation energy obtained at fixed pO2 (dotted line, data recalculated to pO2 = 10−12 MPa with m = −3/16). The extrapolation of the data for Mg2 (Gasc et al. 2006) to lower temperatures is performed with the activation energy Ea = 214 kJ/mol proposed by Azough and Freer (2000). b) Diffusion along the c-axis in orthopyroxene (enstatite): Fe-Mg1 – Ganguly and Tazzoli 1994; Mg – Schwandt et al. 1998; Cr – Ganguly et al. 2007 (m = −0.1); Fe-Mg (open square) – Klügel (2001); Fe-Mg (stars) – ter Heege et al. (2006) (m = 0.15). Although the oxygen fugacity dependence was not explicitly investigated for Fe-Mg (Ganguly and Tazzoli 1994), we have used the dependence (m = 0.17) reported by Stimpfl et al. (2005). The regression line for the data of ter Heege et al. (2006) must only be considered as indicative because of the very few available data. It suggests, however, similar activation energy to that for Mg from Schwandt et al. (1998) and for Fe-Mg from Ganguly and Tazzoli (1994). c) Anisotropy of diffusion in clinopyroxenes (crystallographic axes are indicated in brackets): Ca1 – Dimanov et al. (1996), Dimanov and Jaoul (1998); Ca – Dimanov and Ingrin (1995); Fe – Azough and Freer (2000); Mg2 – Gasc et al. (2006); Mg4 and Ca2 – Zhang et al. (2010). For the data showing anisotropy, diffusion is fastest along the c-axis. Extrinsic, intrinsic and transitional diffusion regimes (denoted by CaExtr, Ca-Intr, Ca-Trans) are reported for Ca (Dimanov et al. 1996; Dimanov and Jaoul 1998; Dimanov and Ingrin 1995). Diffusion of Ca and Fe in synthetic crystals (labeled Syn) is shown to be slower than in natural materials. d) Anisotropy of diffusion in orthopyroxenes (crystallographic axes are indicated in brackets): Mg – Schwandt et al. (1998); Cr – Ganguly et al. (2007. Diffusion along the c-axis is the fastest. The regressions of the original data of Ganguly et al. (2007) are indicated by dotted-dashed lines, but the authors recommended the regressions indicated by solid- and dotted lines.

Diffusion in Pyroxene, Mica, Amphibole

1300

-17

1100

1200 Mg

1, 2

1000

647

900

800°C C lin op yro xe ne s

(-6)

c-a xis

-18

pO 2 = 10

C a-M gF e (> L′ C → B2 → B2′→ A′ “Ultrafine-grained” L′ >> d >> sδ/2 C → C′→ B2′→ A′ “Special ultrafine-grained” d d2/Dgb) ∧ (t < d2/Dl < t′) 6/5 B2′ (Lgb* > d) ∧ (t < d2/Dl) (sdDgb / q Dl / t )− p /2 ** A t ≥ 150 d2/Dl 1/2 Dl 1/2 bsδ/dDgb A′ t ≥ 150 d2/Dl A0 t ≥ 150 d2/Dl 1/2 Dgb Migrating grain boundary (Fig. 10b) Mishin and Razumovskii (1992) Migration velocity, v Regime sequence v < v1 C → B2 → B4 v1 < v < v2 C → B2 → MB2 → B4 v > v2 C → MC → MB2 → B4 MC (t > Dl/v2 < t′) ∧ (t < ½ δ/v < t′) 1 (v/(dDgb))1/2 MB2 (t > δ/2v < t′) ∧ (t < δDgb/Dlv < t″) 1 (v/(dDgb))1/2 ∧ (t > Dl/v2 < t″) Dislocations present in the lattice (Fig. 10c) Klinger and Rabkin (1999) Volume fraction of dislocations, fd Regime sequence Dl > fdDd C → B2 → B4 fd2Dd < Dl < fdDd C → B2 → DB2 → DB3 → DB4? Dl < fd 2Dd C → DB2 → DB3 → DB4? DB1 td′ < t < 1/m 6/5 (sdDgb / q fd 2 Dd / t )− p /2 DB2 1/m < t < 1/(fdm) 6/5 ~constant, no relation given DB3 1/(fdm) < t < td″ 6/5 (sdDgb / q fd Dd / t )− p /2

Only valid for influx of material into the aggregate If p ≠ 0.5 as in all B-regimes this empirical fit is only valid for a certain range of the sectioning profile (see text, Eqn. 21) $$ If p = 0.5 the parameter a is equivalent to Dbulk ** The value for parameter q depends on the exact boundary conditions (see text and forward, Table 2) §

Diffusion in Polycrystalline Materials

931

essentially as long as Ll < 2ds, equivalent to t < t′ (Type C regime). Only during the Type C regime is diffusion within the lattice insignificant and transport occurs solely through the grain boundary. With increasing time, the flux normal to the grain boundary becomes more important (e.g., Fig. 3a → 3b) and thus Lgb fails to predict the penetration distance, but instead Lgb* has to be used. The latter does not evolve with the square root of time, but to the power of 1/4 and becomes smaller than Lgb, if it reaches the value L′ (Fig. 4). This evolution is characteristic of the B2 regime where for distances x with Lgb* > x > Ll, the flux parallel to the grain boundary (∂Cl /∂x ≈ 0 ) within the lattice is negligible as long as t < t″. During the B2 regime, the main contribution for redistribution of material into the lattice comes from the diffusion flux normal to the grain boundary (Fig. 3b). Finally, for times approaching t″ and longer, Ll becomes larger than Lgb* (when it becomes larger than L″, Fig. 4) and here the diffusion flux within the lattice is more and more dominated by the diffusion flux normal to the surface, which defines the transition to the B4 regime (e.g., Fig. 3c → 3d). During this regime, the net flux perpendicular to the grain boundary is negligible (∂Cl /∂y ≈ 0) although it cannot be ignored in principle for Equation (4). During this final stage the grain boundary can be basically ignored with respect to diffusion within the lattice and the penetration distance within the grain boundary is almost equal to Ll (see Fig. 4). In an overly simplified view of a grain boundary diffusion problem, one would assume that material transport is much faster within the grain boundary compared with the lattice since Dgb >> Dl. However, as illustrated in Figure 3c,d this appears not to be true for long time scales, which is simply due to the leakage of the diffusing material into the lattice normal to the grain boundary. This is also illustrated in Figure 5, where the concentration profile within the grain boundary is shown for the four cases in Figure 3 in direct comparison by normalizing the distance with the square root of the time. Without leakage into the lattice for a fixed surface concentration, the concentration profile within the grain boundary could be described by a simple error function as given in Figure 5 with Dgb as the effective diffusion coefficient. This is 0 -1

Ln(CGB)

-2

� � x � erfc� � 2� D �t � gb � �

� � x � erfc� � 2� D �t � l � �

B 4 B3

-3

B2

B1

C

-4 -5 1E-5

1E-4

1E-3

0.01

1/2

x/t

Figure 5. Concentration profiles within the grain boundary corresponding to the four different cases shown in Figure 3. The distance is normalized to the square root of t. For comparison two profiles are shown as solid lines, which were calculated from Figure 5 the 1D solution of the diffusion equation with a fixed surface composition, but with either Dgb or Dl as the diffusion coefficient. The formula used for these profiles is shown next to each. In general, the concentration profile within the grain boundary cannot be described by a constant diffusion coefficient using an error function as in the C regime. Once the system has reached the B4 regime, the grain boundary effectively behaves as if the diffusion coefficient is equal to Dl.

Dohmen & Milke

932

only correct for very small time scales. With increasing time the concentration profile develops and cannot appropriately be described by a constant diffusion coefficient. Finally, the leakage into the lattice becomes increasingly relevant until the diffusion within the grain boundary can be almost approximated by an error function that has Dl as the effective diffusion coefficient. The latter behavior is clearly not apparent from simple intuition. Here we have illustrated the different kinetic regimes of a given system as a time sequence. However, a given system will change its behavior after equivalent time if an intensive thermodynamic parameter that affects the diffusion coefficients Dl and Dgb is changed. For example, temperature is well known to strongly affect diffusion coefficients (see below) and different activation energies of grain boundary and lattice diffusion would change the relative diffusion coefficients as a function of temperature. Generally, one would expect lower activation energies for grain boundary diffusion, and hence a decrease in temperature would significantly decrease Dl and increase the ratio of Dgb/Dl. As a consequence both t′ and t″ become larger (Eqn. 6, 7) and the transitions from C to B2 regime and B2 to B4 regime would be shifted to longer times.

The MONoPhase Polycrystalline Aggregate Models and kinetic regimes The Fisher model was used here to illustrate the characteristic features of diffusion problems considering grain boundaries—the interplay between the various diffusion fluxes and the resulting kinetic regimes during evolution with time. The parallel slab model of Harrison (1961), as an extension of Fisher’s (1951) isolated grain boundary model, served as a first basis for the interpretation of diffusion in real rocks (e.g., Joesten 1991). In the classical model of Harrison (1961), he considered equally spaced slabs with thickness 2d, separated by parallel grain boundaries. The temporal evolution of the concentration contours in this case was divided into three different stages and these correspond to three kinetic regimes, C, B, and A, where the C- and B-regime and the corresponding criteria completely coincide with the C and B2-regime from the isolated grain boundary model in which B1 and B3 are only transitional regions (for a comprehensive illustration and description of Harrison’s classification, see also e.g., Joesten 1991; Philibert 1991 p. 251ff; Kaur et al. 1995, p. 62ff; Mishin and Herzig 1999). The main difference between the isolated grain boundary and a set of parallel grain boundaries occurs when the characteristic diffusion distance within the lattice Ll is larger than d and the profiles normal to the grain boundary significantly overlap. This is called the Type A regime, and hence the time constraint for this regime is roughly given by t >> d2/Dl. After this time a parallel diffusion front is propagating away from the source (as in the B4-regime in Fig. 3d) following a square root relationship where the concentration distribution with time can be described by one effective/bulk diffusion coefficient, Dbulk = d/d·Dgb + (1−d/d)·Dl. The parallel slab model with the kinetic regimes A, B, and C, is only a good approximation for a polycrystalline aggregate if during the Type B regime the effective diffusion distance Lgb* within the grain boundary is smaller than the grain size. A more general classification has been introduced by Mishin and Herzig (1995) (Fig. 6, see also Kaur et al. 1995 p. 199ff) where the kinetic regimes of Mishin and Razumovskii (1992a) for the isolated grain boundary model were extended and are also sensitive to the grain size, d, relative to the two characteristic distances L′ and L″ as defined previously (Table 1). In addition to the A-, B2-, B4, and C-regimes, four new kinetic regimes were introduced: the C′-, B2′-, A′-, and A0-regimes (note that the intermediate regimes B1 and B3 have been ignored here for simplification) and the corresponding concentration contours for these are schematically shown in Figure 6b-d. For values of d > L″ the grains are classified as coarse-grained and the sequence of the kinetic regimes with time is basically equivalent to the Harrison model: C → B2 → B4 → A. However, if d