Defect Structure and Properties of Nanomaterials. Second and Extended Edition [2nd Edition] 9780081019184, 9780081019177

Table of contents :
Content:
Related titles,Front Matter,Copyright,List of Figures,List of Tables,PrefaceEntitled to full textChapter 1 - Processing Methods of Nanomaterials, Pages 1-25
Chapter 2 - Characterization Methods of Lattice Defects, Pages 27-57
Chapter 3 - Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation, Pages 59-93
Chapter 4 - Defect Structure in Low Stacking Fault Energy Nanomaterials, Pages 95-119
Chapter 5 - Lattice Defects in Nanoparticles and Nanomaterials Sintered From Nanopowders, Pages 121-153
Chapter 6 - Lattice Defects in Nanocrystalline Films and Multilayers, Pages 155-173
Chapter 7 - Correlation Between Defect Structure and Mechanical Properties of Nanocrystalline Materials, Pages 175-223
Chapter 8 - Defect Structure and Properties of Metal Matrix–Carbon Nanotube Composites, Pages 225-246
Chapter 9 - Effect of Lattice Imperfections on Electrical Resistivity of Nanomaterials, Pages 247-269
Chapter 10 - Lattice Defects and Diffusion in Nanomaterials, Pages 271-295
Chapter 11 - Relationship Between Microstructure and Hydrogen Storage Properties of Nanomaterials, Pages 297-315
Chapter 12 - Thermal Stability of Defect Structures in Nanomaterials, Pages 317-371
Index, Pages 373-376

Citation preview

Related titles Nanomaterials (ISBN: 978-0-08-044964-7) Nanomaterials, Nanotechnologies and Design (ISBN: 978-0-7506-8149-0) Modeling, Characterization and Production of Nanomaterials (ISBN: 978-1-78242-228-0)

DEFECT STRUCTURE AND PROPERTIES OF NANOMATERIALS Second and Extended Edition

} GUBICZA JENO Department of Materials Physics, €tvo €s Lorand University, Institute of Physics Eo Budapest, Hungary

Woodhead Publishing is an imprint of Elsevier The Officers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom Copyright © 2017 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/ permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).

Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-08-101917-7 (print) ISBN: 978-0-08-101918-4 (online) For information on all Woodhead Publishing publications visit our website at https://www.elsevier.com/books-and-journals

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LIST OF FIGURES

Figure 1.1 Figure 1.2 Figure 1.3 Figure 1.4 Figure 1.5

Figure 1.6 Figure 1.7 Figure 1.8 Figure 1.9 Figure 1.10 Figure 1.11 Figure 1.12 Figure 1.13 Figure 1.14 Figure 1.15 Figure 1.16 Figure 1.17 Figure 1.18 Figure 1.19 Figure 1.20 Figure 1.21 Figure 1.22 Figure 1.23 Figure 1.24 Figure 1.25 Figure 2.1 Figure 2.2 Figure 2.3

Classification of processing routes of bulk nanomaterials. (a) The schematic depiction of equal-channel angular pressing (ECAP) processing and (b) the four fundamental processing routes in ECAP. A schematic of ECAP-Conform process [1]. The principle of high-pressure torsion. Schematic illustration of the steps of multidirectional forging procedure. The axes of the reference system attached to the sample are denoted as x, y, and z. The loading directions are indicated by black arrows. The letters a, b, and c denote the sample dimensions. The principle of twist extrusion processing. The principle of accumulative roll bonding method. The principle of repetitive corrugation and straightening method. Schematic picture of an inert gas condensation facility. Schematic depiction of the apparatus of cryogenic melting. Schematic diagram of the setup for electro-explosion of wire: (1) thin metal wire electrode, (2) metal plate electrode, (3) batteries, and (4) glass vessel. The flowchart of the sonochemical method, resulting in Fe2O3 nanoparticles [20]. Schematic of the main steps of hydrothermal synthesis of titanate nanotubes from anatase TiO2 powder. Jar or drum mill. Szegvari attritor. Motion of balls and jars in a planetary mill. Illustration of a vibratory ball mill. Motion of balls in a magnetic mill. Schematic depiction of shock wave consolidation process showing the experimental setup. Illustration of the fusion of particles during pressureless sintering. Schematic depiction of hot pressing. Schematic depiction of the spark plasma sintering apparatus. Schematic illustration of Ni electrodeposition. Illustration of melt spinning. Schematic depiction of copper mold casting. Ranges of application for different experimental methods in the determination of vacancy concentration, dislocation density, and stacking or twin fault probability. The division of reflecting crystallites into scattering columns according to Bertaut theorem. Characterization of lattice distortions parallel to the diffraction vector g [or normal to lattice planes (hkl )].

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List of Figures

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Schematic of some arrangements of dislocations yielding weak or strong screening of the strain fields that correspond to a larger or smaller value of the dislocation arrangement parameter M, respectively. Full width at half maximum of diffraction peaks as a function of magnitude of diffraction vector (g) for the cases where the broadening is caused only by (a) crystallite size in Cu, (b) both size and dislocations in Cu, and (c) both size and twin faults in SiC. The datum points corresponding to harmonic reflection pairs are connected by dotted lines. The convolutional multiple whole profile evaluation of the X-ray diffraction pattern for ultrafine-grained Au processed by severe plastic deformation. The open circles and the solid line represent the measured and the fitted X-ray diffraction patterns, respectively. A magnified part of the pattern is presented in the inset. The difference between the measured and the fitted patterns is also shown at the bottom of the figure. Log-normal crystallite size distribution density function, f(x), and the arithmetic (hxiarit), the area- (hxiarea), and the volume-weighted (hxivol) mean crystallite sizes obtained from m and s. The 11 possible dislocation slip systems in materials with hexagonal crystal structure. The arrows indicate the three different Burgers vector types: < a >, < c >, and < c þ a >. The slip planes are denoted by gray color. The Burgers vectors and the slip planes are listed in Table 2.3. Grain boundary misorientation distribution for an ultrafine-grained Al-1%Mg alloy processed by high pressure torsion at room temperature under the pressure of 6 GPa at a rate of 1 rpm for 10 turns [24]. The columns and the solid curve represent the experimental result and the statistical prediction for a set of random orientations, respectively. Schematic of the coordinate system attached to the sample and the pixels arranged in a square grid on electron backscatter diffraction image. The misorientations between the studied gray pixel and the two neighboring white pixels in x1 and x2 directions are characterized by quaternions Dq1 and Dq2, respectively. Electron backscatter diffraction image (a) and the corresponding dislocation density map (b) obtained with the step size of 35 nm. In (b) the grain boundaries with misorientation angles larger than 5 degrees are indicated by thin blue lines, and the higher the dislocation density, the darker the gray contrast. The dislocation density determined by electron backscatter diffraction as a function of the scan step size. The dislocation density obtained by X-ray line profile analysis is also indicated in the figure. Schematic showing the difference between the grain and crystallite sizes determined by transmission electron microscopy and X-ray line profile analysis, respectively, in nanocrystalline and ultrafine-grained metallic materials processed by severe plastic deformation.

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List of Figures

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Figure 2.16 Figure 2.17 Figure 3.1 Figure 3.2

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Figure 3.6

Figure 3.7

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Figure 3.9

The correlation between the mean twin fault spacing values determined by transmission electron microscopy and X-ray line profile analysis for Ag and SiC samples. The intrinsic resistivity in logarithmic scale as a function of temperature for 99.999% purity Cu. The resistivity for the vacancy concentration of cv ¼104 and dislocation density of r ¼ 1015 m2 are also indicated in the figure. Schematic of poslston lifetime spectrum: the photon counts (N) versus time (t). The poslston lifetime as a function of number of vacancies in clusters for Al. The dislocation density and the crystallite size as a function of the number of equal channel angular pressing (ECAP) passes for 99.98% purity Cu. (a) A transmission electron microscopy micrograph of a subgrain in 99.99% purity Cu processed by repetitive corrugation and straightening. The inset in (a) is a highresolution transmission electron microscopy (HRTEM) image showing that the subgrain boundaries are almost parallel to two sets of {111} planes. (b) A Fourierfiltered HRTEM image from the boundary as pointed out by a black arrowhead in (a). The white arrow in (b) points out the grain boundary (GB) orientation. The black and white circles mark interstitial and vacancy loops, respectively. (a) A boundary in a 99.99% purity Cu sample processed by 14 cycles of repetitive corrugation and straightening. The grain boundary plane is curved and changes from the (5 5 12) plane to the (002) plane. (b) The corresponding electron diffraction pattern. (c) and (e) high-resolution transmission electron microscopy images from the upper-left and lower-right part of the boundary in (a) (see the framed areas). (d) A structural model of the boundary segment in (c). The two types of dislocations in the boundary are marked by black and white symbols. Transmission electron microscopy images of the microstructure of Cu processed by equal channel angular pressing for the 5th (a) and 25th (b) passes. The variation of (a) dislocation density, (b) fractions of hai-type basal and nonbasal edge dislocations, and (c) screw dislocations in the center of an AZ31 disk as a function of high-pressure torsion (HPT) revolutions [35]. Schematic depiction of the grain refinement in hexagonal close-packed metals along the preexisting grain boundaries when the initial grain size is larger than a critical value. Transmission electron microscopy images showing the microstructure of Cu specimens immediately after 20 cycles of multidirectional forging (a), 15 passes of twist extrusion (b), 25 passes of equal channel angular pressing (c) and 25 revolutions of high-pressure torsion (d). Comparison of dislocation densities (a) and grain sizes (b) in different materials processed by equal channel angular pressing (ECAP) or high-pressure torsion (HPT) [55]. The saturation grain and crystallite size values determined by transmission electron microscopy and X-ray line profile analysis, respectively, for severe plastic deformation-processed ultrafine-grained materials as a function of the saturation dislocation density.

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List of Figures

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The saturation crystallite size and dislocation density as a function of melting point for different pure face-centered cubic metals processed by equal channel angular pressing at room temperature. The minimum grain size and the maximum dislocation density as a function of Mg content in Al solid solutions. Transmission electron microscopy images taken on pure Al (a) and Ale3% Mg (b) processed by eight equal channel angular pressing passes at room temperature. Schematic drawings of two edge dislocations climbing in opposite directions by increasing or decreasing the extension of the extra half plane that leads to a production or annihilation of vacancies, respectively. The direction of atomic migration is indicated by arrows. The produced and the annihilated vacancies are denoted by solid and dashed squares, respectively. The vacancy cluster concentration in high-pressure torsion (HPT)-processed Cu as a function of the number of revolutions. The individual datum points given in Ref. [87] are not shown here, but all of them follow this line irrespective of the location in the HPT disk and the applied pressure (2 or 4 GPa). The scatter of datum points is illustrated by the error bar. The concentration of vacancies in the center and at the periphery of Cu disks processed by different revolutions of high-pressure torsion (HPT). The individual datum points given in Ref. [88] are not shown here, but all of them follow this line irrespective of the applied pressure (2 or 4 GPa). The scatter of datum points is illustrated by the error bar. The equilibrium Ag solute concentration in the Cu matrix as a function of the size of Ag nanoparticles at 623 K for two interface energies 0.38 and 1 J/m2. The horizontal dotted line represents the equilibrium Ag concentration in Cu matrix containing Ag particles with very large radius (cN). Reflection (220) of the Cue3 at.% Ag sample in cryorolled state and after annealing at 623 K for 20 min. The symbols and the solid lines represent the measured data and the fitted curves, respectively. The diffraction peak in the annealed condition is a sum of two reflections related to Regions 1 and 2 having different average lattice parameters (for details see the text). In this figure the integrated intensity (the area under the peak after background subtraction) is normalized to unity for both cryorolled and annealed states. Schematic of the development of heterogeneous microstructure in cryorolled ultrafine-grained Cue3 at.% Ag alloy during annealing at 623 K up to 120 min. The darker the gray in the matrix, the higher the solute Ag content. The variation of (a) the solute Ag concentration, (b) the X-ray intensity fraction, and (c) the dislocation density for Regions 1 and 2 in the Cu matrix as a function of annealing time. The solid curves serve only as guide to eyes.

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List of Figures

Figure 4.1

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Figure 4.9

Figure 4.10

Figure 4.11

High-resolution transmission electron microscopy image showing a dissociated screw dislocation bounded by partials in Ag processed by eight equal-channel angular pressing passes (at right) and a picture illustrating the corresponding crystallographic directions and lattice planes (at left). The forces f1 and f2 acting on partials in dissociated screw dislocations due to the shear stress, s, in the glide plane. The glide plane is lying in the sheet, and the stacking fault between partials is marked by light gray color. The Burgers vectors of partials are denoted by b1 and b2. The components of these vectors lying parallel and perpendicular to the dislocation line vector, l, are also presented. (a) and (b) show that if the stress is parallel to the Burgers vector of the undissociated dislocation, the forces acting on partials have the same directions; therefore they do not change the plitting distance between partials. If the stress is perpendicular to the Burgers vector of the undissociated dislocation, it may yield a decrease (c) or an increase (d) in the splitting distance between partials. A schematic illustration of a model for dissociated dislocation with wide stacking fault (SF) ribbon in a nanograin. Two Shockley partial dislocations, XEFY and XCDY, are emitted consecutively from the grain boundary XY, with their ends pinned at X and Y. The effective equilibrium splitting distance between partials, dp,eff, in nanocrystalline Al as a function of grain size (solid line). The dashed line corresponds to the equilibrium splitting distance characteristic for coarse-grained counterparts. Cross-slip of a dissociated screw dislocation. s0 and ss are the shear stresses pushing the partials toward each other on the initial glide plane (S1) and pulling the partials in opposite directions on the cross-slip plane (S2), respectively. Transmission electron microscopy images taken after (a) 1, (b) 4, (c) 8, and (d) 16 passes of equal-channel angular pressing. Examples of twin boundaries are indicated by white arrows. The dislocation density and the twin boundary frequency as a function of number of equal-channel angular pressing (ECAP) passes for 4N5 purity Ag. The saturation twin boundary frequency achieved by equal-channel angular pressing at room temperature versus the twin-fault energy, gT, for pure face-centered cubic metals. The formation of twins at LomereCottrell locks (a) and grain boundaries (b) by dissociation of lattice dislocations into twinning partials and their slip on successive {111} planes. The Schockley partials are denoted by “L.” The untwinning process. The twin boundaries are indicated by “TB.” The partial dislocations are denoted as it is usual in a standard Thompson tetrahedron (see Fig. 4.11). When a dissociated dislocation meets a twin lamella, the leading partial (aB) splits into a sessile stair-rod dislocation (ad) and a glissile Shockley partial (dB) that slips on a layer of the twin lamella resulting in untwinning on that layer (a). If dislocations on successive slip planes meet twin lamella, a complete disappearance of twin segment can occur (b). (a) Thompson tetrahedron ABCD and (b) its two dimensional representation illustrating the possible slip planes and the Burgers vectors of dislocations in a face-centered cubic crystal. The four faces of the tetrahedron corresponding to the slip planes are denoted by a, b, c, and d while the centers of the faces are indicated as a, b, g, and d.

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Figure 5.1

Figure 5.2 Figure 5.3

Figure 5.4 Figure 5.5

Figure 5.6 Figure 5.7

Figure 5.8 Figure 5.9

Figure 5.10

The dislocation density and the twin boundary frequency as a function of number of equal-channel angular pressing (ECAP) passes for 4N purity Ag. Transmission electron microscopy image of the microstructure in 4N Ag processed by 20 high pressure torsion revolutions [34]. The twin boundary frequency as a function of stacking fault energy (SFE). The grain size is also given at some data points. Schematic illustration of the grain-refinement mechanism for the Cue30 wt.% Zn alloy processed by high pressure torsion. A high-resolution transmission electron microscopy image of a bent twin boundary (TB) in high pressure torsion-processed Cue30% Zn alloy showing a high density of dislocations, which are indicated using white and black “T,” accumulated at the TB. Two white solid lines were drawn parallel to {111} at each side of the TB, respectively, and one dash white line was also drawn parallel to (111)M to indicate the misorientation between the {111} planes across the TB. The mean crystallite size, the dislocation density, and the Mg concentration as a function of the milling period for the powder blend with the nominal composition of Ale6 wt.% Mg. Schematic illustration of the particle/grain structure in milled metallic powders. The parameters q and M describing the edge/screw character and the screening of dislocations, respectively, as a function of milling time for a powder blend with the nominal composition of Ale6 wt.% Mg. The mean crystallite size, the dislocation density, and the Mg concentration in Al as a function of the nominal Mg content in AleMg powder blends milled for 3 h. The parameters q and M describing the edge/screw character and the screening of dislocations, respectively, as a function of the nominal Mg content in AleMg powder blends milled for 3 h. The smallest crystallite size in pure metal powders achieved by milling at room temperature as a function of the melting point. (a) Bright-field transmission electron microscopy (TEM) image of Ni powder processed by electro-explosion of Ni wire. The particle size distribution obtained from TEM images is shown in (b) [47]. The Volterra constructions of (a) wedge and (b) twist disclinations. Decahedral (a) and icosahedral (b) nanoparticles. The twin faults in the decahedron are indicated by gray color. The disclinations are illustrated by the arrows, and the symbol u in both decahedron and icosahedron. (c) shows that when five tetrahedra assembled with a common edge an angle gap of 7.35 degrees remains and after joining the tetrahedra a disclination is formed in the resulted decahedron. The schematic of stepwise growth of twinned decahedral nanoparticles from tetrahedral units.

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Figure 5.11

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Figure 6.2

Figure 6.3

Figure 6.4 Figure 6.5

Figure 6.6

Figure 6.7 Figure 6.8

Energy-filtered transmission electron microscopy images on Ni nanopowder showing the element maps for (a) nickel and (b) oxygen, respectively. The lighter the color in (a) and (b), the higher the local nickel and oxygen contents, respectively. The corresponding X-ray diffraction pattern is presented in (c) [47]. Transmission electron microscope images of the sample processed by (a) Hot-isostatic pressing (HIP) and (b) Spark-plasma sintering (SPS). Grain size distributions for the (c) HIP- and (d) SPS-processed bulk samples determined from transmission electron microscopy images. The distributions of twinned grains are also shown by streaked columns [47]. Scanning electron microscopy picture showing clusters in coarse-grained powder with an individual particle size of about 5 mm. Scanning electron microscopy images of the microstructure for samples CG (a), A (b), and B (c). Dislocation density and the crystallite size in nanodiamond samples as a function of consolidation pressure. (a) Transmission electron microscopy images of specimens sintered at 1800  C and 2 GPa, (b) 5.5 GPa, and (c) 8 GPa. Please note the different magnification for (b) [67]. (a) The crystallite size, (b) the dislocation density, and (c) the twin density for 10 sintered specimens as a function of the sintering pressure and temperature. The wire grids in the figures are to guide the eye [67]. Contour maps showing the influence of organic additive concentration and current density on crystallite size, dislocation density, and twin fault probability in textured Ni film processed by electrodeposition. Schematic cross-sectional view of twin fault structures in two epitaxially grown sputtered Ag films where the foil surface is parallel to (a) plane (111) or (b) plane (110) [12]. These planes are perpendicular to the sheet. Schematic of the structure for a pair of Cu and Nb layers in the Cu-Nb multilayer processed by direct current magnetron sputtering at room temperature.     The arrows on the top surface indicate crystallographic directions 101 and 111 in Cu and Nb grains, respectively [13]. Schematic of the dislocation structure in Cu and Nb layers of the Cu-Nb multilayer processed by direct current magnetron sputtering at room temperature. (a) Schematic of the formation of dislocations with Burgers vector [001] with the
 interaction of gliding dislocations with Burgers vectors 1=2ð111Þ and 1 2 111 in Nb layers during rolling of a Cu-Nb multilayer. Figure (b) shows a Bravais cell of Nb with the crystallographic planes and directions which play important role in the dislocation reaction. Schematic illustration of the confined glide of hairpin dislocation loops on two glide planes in a Nb layer of a nanoscale Cu-Nb multilayer, depositing misfit dislocations at the interfaces [21]. Extension and retraction (detwinning) of coherent twin boundaries by moving of groups of three twinning partials at incoherent twin boundaries. Schematic of the formation of interstitial dislocation loops in the vicinity of a pressurized He bubble (loop punching).

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Figure 7.1

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Figure 7.7 Figure 7.8 Figure 7.9

Figure 7.10

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A schematic illustration of the model of a perfect screw dislocation emitted from grain boundary XY and dissociated into two partials in the slip plane (111). The stacking fault (SF) ribbon between the partials is indicated by color. A schematic illustration of the dislocation model for the nucleation of deformation twins. The stacking sequence of the (111) planes is indicated by the letters A, B, and C. A schematic illustration of the emission of a trailing partial from grain boundary XY that partially removes the stacking fault formed previously by passing a leading partial. The shear stresses sP, sL, stwin, strail, and sshrink as a function of the grain size, d, for nanocrystalline Al. The angle between the stress direction and the line of slipping dislocations is b ¼ 30 degrees. Schematic illustration of the AshbyeVerrall model for grain boundary sliding. The arrows in the grains indicate the sliding directions along the grain boundaries. At the bottom of the figure, the arrows show the directions of diffusion in the vicinity of grain boundaries that yields the change of the grain shape as a complementary process in addition to grain boundary sliding. Two grains A and B with the size d in a polycrystalline material loaded by a tensile stress s. The dashed lines represent the slip planes. In grain A, the plastic deformation has already been initiated by the external stress and a FrankeRead (FR) source emits dislocations that are gliding due to the shear stress s. The stress field of the dislocations accumulated at the boundary in grain A assists the activation of FR sources in the unfavorable oriented grain B. The HallePetch plot of the yield strength (sY) versus the grain size (d) for Cu. The stress (s) required for the operation of a FrankeRead source versus the length (L) of the edge dislocation segment between the pinning points of the source in Cu. The shear stresses sP, sL, stwin, strail, and sshrink calculated from Eqs. (7.1)e(7.5) as a function of the grain size, d for nanocrystalline Cu. The angle between the stress direction and the line of slipping dislocations is b ¼ 30 degrees. Hardness of coarse-grained and ultrafine-grained (UFG) Pb, Sn, and In versus the homologous temperature, T/Tm (Tm is the melting point). The arrows indicate softening at RT due to grain refinement achieved by high-pressure torsion (HPT) processing. The yield strength at room temperature (RT ) reduced by the friction stress (sY  s0) versus the product of Gbr1/2 for different fcc metals and alloys processed by SPD. The errors on the individual datum points are indicated by solid horizontal and vertical lines at the symbol diamond. The value of a in the Taylor equation as a function of the equilibrium splitting distance between partials in Burgers vector unit (dp/b) for pure fcc metals processed by equal-channel angular pressing at room temperature till saturation of the dislocation density. The value of a in the Taylor equation as a function of the number of equal-channel angular pressing (ECAP) passes for pure Al and Cu.

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Figure 7.24

Schematic illustration of decreasing ductility (εmax) with decreasing grain size according to Considere criterion. ultrafine-grained material. Grain size dependence of uniform and total elongations at room temperature for interstitial free steel with various mean grain sizes. The strain rate sensitivity, m, measured at room temperature as a function of grain size for (a) face-centered cubic Ni [64e66] and (b) body-centered cubic Fe processed by powder metallurgy methods. Room temperature (RT) tensile engineering stressestrain curves for Cu with different microstructures. Ultrafine-grained (UFG): processed by eight passes of equal-channel angular pressing at RT and then rolled at liquid nlstogen temperature for a reduction of the cross-sectional area of 93%. Bimodal: sample UFG annealed at 200  C for 3 min. Coarse-grained: conventional sample. The strain rate for all the tests is 104 s1. The dislocation density and the grain size (a) and the ultimate tensile strength and the elongation to failure (b) as a function of annealing temperature for Ale1% Mg processed by 10 turns of high-pressure torsion at room temperature [73]. Engineering stressestrain curves for Cue3 at.% Ag alloy rolled at liquid nlstogen temperature (LNT) and room temperature (RT), and annealed for a short time at 375 and 400  C, respectively [74]. Compression stressestrain curves for Ni samples consolidated from powders with the grain size of 50 or 100 nm, in Ar or air and by hot isostatic pressing (HIP) or spark plasma sintering (SPS). HallePetch relation between the yield strength (sY) and the grain size (d ) for Ni samples processed by electrodeposition and powder metallurgy. The solid line represents a linear fit on the data obtained for electrodeposited Ni specimens. The strength contribution of NiO dispersoids in sintered Ni samples as a function of the intensity ratio (INiO/INi) of the X-ray diffraction peaks for NiO and Ni at 2Q ¼ 37.4 and 44.6 degrees, respectively. (a) The dislocation density and (b) the twin boundary frequency for the hot isostatic pressing (HIP)- and spark plasma sintering (SPS)-processed samples before and after compression test as determined by X-ray line profile analysis. The distribution of the angle of misorientation between the grains in the (c) HIP-processed sample and after compression up to 10% strain and (d) in the SPS-processed sample and after compression up to 10% strain as determined by electron backscatter diffraction. (a) Transmission electron microscopy image showing twins in a large grain after compression test of Ni sample processed from a powder with particle size of 100 nm by spark plasma sintering in air. The arrows show steps on twin boundaries as a result of untwinning process. The untwinning mechanism is illustrated in (b) showing the Burgers vectors of dislocations interacting with a twin boundary (TB).

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Figure 8.1 Figure 8.2 Figure 8.3

Figure 8.4 Figure 8.5 Figure 8.6 Figure 8.7

Scanning electron microscopy images showing the surface of the sample processed by spark plasma sintering (SPS) in air deformed by compression to the strain values of (a) 7%, (b) 18%, and (c) 22% and (d) the surface of the hot isostatic pressing (HIP)-processed specimen at a strain of 25%. The loading axis is vertical in the figures. Some cracks are indicated by black arrows. (a) True stresselogarithmic plastic strain curves obtained by compression test at room temperature for samples CG, A, B, and ultrafine-grained (UFG) consolidated from blends of nano- and coarse-grained (CG) Ni powders with different volume fractions. The notations of the samples are explained in the text. The volume fraction of the CG component is also indicated. (b) The yield strength values obtained from the stressestrain curves in (a). The nanohardness distributions for (a) coarse-grained (CG) specimen, (b) sample A, (c) sample B, and (d) ultrafine-grained (UFG) specimen. The nanohardness of the coarse-grained (CG) and ultrafine-grained (UFG) fractions and their volume-weighted average as a function of the volume fraction of the UFG component for the as-processed specimen CG, sample A, sample B, and specimen UFG. The dashed line represents the linear interpolation between the values characteristic for the fully CG and UFG samples. The yield strength, ultimate tensile strength, and strain to failure as a function of coarse-grained powder fraction for bimodal Ale7.5% Mg and 5083 Al alloys consolidated from blends of cryomilled NPs and unmilled coarse particles. The flow stress at the strain of ε ¼ 0.05 versus the logarithm of the strain rate for both CG-Zn and UFG-Zn. Evolution of crystallite size and dislocation density in ultrafine-grained (UFG)-Zn sintered from nanopowder due to compression as a function of strain rate. The dashed lines indicate the crystallite size and the dislocation density in the as-sintered material. Schematic of plasma spray forming of a blend of AleSi powder and CNTs. SEM image of fracture surface of plasma spray formed Al-CNT nanocomposite showing intergranular fracture and cluster of CNTs. Schematic figure depicting the attachment of Cu ions to the functional groups on the surface of a CNT for the “molecular level mixing” process of Cu-CNT composite powders. Schematic depiction of the Cu-CNT composite powder processed by “molecular level mixing”. Schematic of aligned Cu-CNT composite samples obtained by (a) die-stretching and (b) electroplating. CNT, carbon nanotube. High-resolution transmission electron microscopy image taken at the half-radius of the CueCNT-RT disk. The arrows indicate fragments of CNTs. The area-weighted mean crystallite size, < x>area (a), the dislocation density, r (b), and the twin boundary frequency, b (c) at the center, half-radius, and periphery of the HPT-processed Cu, CueCNT-RT, and CueCNT-373 disks.

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List of Figures

Figure 8.8 Figure 8.9 Figure 8.10 Figure 8.11

Figure 8.12

Figure 8.13

Figure 9.1 Figure 9.2

Figure 9.3 Figure 9.4

Figure 9.5 Figure 9.6 Figure 9.7

Figure 9.8

Dark-field TEM images for samples (a) Cu, (b) CueCNT-RT, and (c) CueCNT-373. Some twin boundaries in sample CueCNT-RT are indicated by white arrows in (d). The Young’s modulus and the yield strength as a function of the volume fraction of MWNTs in Al-CNT composites. The ratio of the Young’s moduli, the yield and tensile strengths and the elongations to failure in tension obtained for CNT composites and their pure matrices. The microhardness (HV) as a function of the distance from the center of the HPT-processed Cu, CueCNT-RT, and CueCNT-373 disks (a). The hardness values determined at the half-radius of the disks at RT and 350 K (b). The calculated yield strength versus the measured values obtained as one-third of the hardness at the center, half-radius, and periphery of the HPT-processed Cu, CueCNT-RT, and CueCNT-373 disks. The yield strength was calculated from the grain size obtained by TEM using the HallePetch formula and also from the dislocation density measured by X-ray line profile analysis using the Taylor equation. The ratio of the conductivities and the strength-to-resistivity values obtained for laminar Cu-CNT composites and the pure Cu matrix processed by electroplating [24]. The electrical resistivity of coarse-grained pure Cu as a function of temperature between 1 and 1356 K (the melting point of Cu) [1]. Typical electrical resistivity ranges for different lattice defects and Ni solute atoms in nanostructured Cu. The intrinsic resistivity is also indicated for different temperatures. Variation of the porosity factor of electrical resistivity as a function of volume fraction of pores according to Eqs. (9.11) and (9.12). Strength versus resistivity for ultrafine-grained pure copper and its alloys. The different processing routes are indicated by various symbols. The strength-toresistivity ratio values are reflected by the slopes of the lines connecting the data points and the origin of the coordinate system. The materials with the lowest and the highest strength-to-resistivity ratios are indicated by dashed lines. The variation of strength and conductivity due to high-pressure torsion (HPT) and subsequent annealing in Ale5.4 wt.% Cee3.1 wt.% La alloy and pure Al. Schematic of nanotwinned ultrafine-grained microstructure in electrodeposited polycrystalline Cu possessing high strength and good conductivity. Three-dimensional schematic view of highly twinned, epitaxially grown Cu film processed by magnetron sputtering. The domains are separated by S3{112} twin interfaces, and they contain many S3{111} twin faults. These films have high strength and low resistivity. Strength, resistivity, and strength-to-resistivity ratio as a function of layer thickness for Cu/Cr multilayers processed by sputtering.

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List of Figures

Figure 10.1

Figure 10.2

Figure 10.3

Figure 10.4

Figure 10.5

Figure 10.6

Figure 10.7

Figure 10.8 Figure 10.9

Figure 10.10

Schematic of diffusion coefficient (D) versus reciprocal of temperature (T ) for bulk lattice, general grain boundaries (GBs), and free surface. The diffusion coefficient is plotted in logarithmic scale. DS, DGB, and DL stand for the diffusion coefficients for surface, grain boundary and lattice diffusion. (a) Schematic showing the basic diffusion processes and the tracer atom diffusion profile (indicated by light gray color) developed in the vicinity of a grain boundary due to simultaneous lattice and grain boundary (GB) diffusion. d is the thickness of the grain boundary. Plot of lnc versus x6/5 gives a straight tail part for large x values, which is characteristic for grain boundary diffusion, as shown in (b). This straight line can be used for the determination of grain boundary diffusion coefficient (see the text for details). Schematic of the three basic tracer spatial distribution types: A, B, and C. The tracer atoms diffuse from the surface layer at the right side. The volumes containing larger amount of tracer atoms than a given concentration are indicated by gray color (tracer profile) [6]. (a) Schematic of the tracer penetration profiles before and after a boundary moving parallel to the surface layer. (b) The improved Fisher’s model with secondary short-circuit diffusion paths (dislocations, subgrain, and interface boundaries). This diffusion regime is also referred to as type D kinetics. Schematic showing the variation of the logarithm of grain boundary diffusion coefficient in Cu as a function of misorientation angle for [001] symmetric tilt boundaries. The plot was constructed according to the data presented in [11] and [13]. The dashed line in the inset shows the ideal misorientation angle for S5 coincidence site lattice boundary. Dependence of self-diffusion activation energy on high-angle grain boundary energy for symmetric tilt grain boundaries in Cu [15]. The error bar at the straight line reflects the difference between the data measured parallel and perpendicular to the tilt axis. Schematic of diffusion profile obtained in ultrafine-grained materials in which both slow and fast grain boundary diffusion occur. c is the average tracer concentration determined by serial sectioning method while x is the tracer penetration depth. Model of the hierarchical microstructure with nonequilibrium and relaxed grain boundaries, acting as fast and slow diffusion paths, respectively [23]. Schematic illustrating the difference between the diffusivities of Ni for slow and fast grain boundaries in Cue0.17 wt.% Zr alloy processed by equal channel angular pressing at room temperature. The diffusivity of grain boundaries in coarsegrained pure Cu is also shown [23]. Heterogeneous grain boundary structures with slow and fast diffusion pathways in nanomaterials processed by (a) crystallization of amorphous materials or (b) sintering from nanopowders.

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List of Figures

Figure 10.11

Figure 10.12

Figure 10.13

Figure 11.1 Figure 11.2

Figure 11.3

Figure 11.4

Figure 11.5 Figure 11.6

Figure 11.7 Figure 11.8 Figure 11.9

Figure 11.10

Figure 11.11

Comparison of grain boundary diffusion coefficients for nanomaterials processed by different techniques: severe plastic deformationeprocessing, sintering from powders, and crystallization of amorphous materials. The diffusivity levels for conventional lattices, grain boundaries, and surfaces are also indicated by dashed horizontal lines. Variation of the logarithm of bulk lattice self-diffusion coefficient normalized by the frequency factor as a function of grain size for Cu at room temperature, if the influence of grain size on melting point is taken into account. Illustration of dissolution of Mo in V for Mo/V multilayer while the interface between the two materials remains atomically sharp. (a) Schematic of a pair of Mo and V layers. (b) Change of V atom distribution perpendicular to the layers as a function of time of diffusion experiment. Schematic diagram of pressureecomposition isotherm. Schematic depiction of phase transformation according to (a) JohnsoneMehle AvramieKolmogorov and (b) contracting volume models. The dark areas represent the growing new phase. Absorption (left side) and desorption curves (right side) of (a) conventional coarse-grained MgH2; (b) nanocrystalline MgH2 processed by ball milling for 20 h; (c) MgH2 ball-milled for 700 h, and (d) Nb2O5-catalyzed ball-milled MgH2. The first and the second desorption curves measured in vacuum at 300  C for MgH2 ball-milled for 10 h. The first desorption corresponds to the activation of the powder. Variation of the average crystallite size of MgH2 as a function of the number of dehydrogenationehydrogenation cycles. Schematic depiction of the crystallite structure in MgH2 (a) immediately after milling, after (b) the first and (c) the second dehydrogenationehydrogenation cycles. Variation of the average crystallite size of MgH2 as a function of the fraction of Mg and MgH2 during (a) desorption and (b) absorption, respectively. Variation of the normalized crystallite size of MgH2 as a function of the transformed fraction of Mg during desorption according to Eq. (11.7). Front walls of one and the same single-walled nanotubes just after ion impact (a) and after annealing (b). During annealing the divacancy (D) transformed to an agglomeration of nonhexagonal rings. The numbers in the center of the rings indicate the numbers of atoms that constitute the rings. The single vacancy (S) and the nearby carbon adatom (A) in the upper right-hand corner in (A) transformed to a StoneeWales 5e7 defect in (B). TEM images of (a) defective multiwalled carbon nanotubes (MWCNTs) processed by oxidation, (b) defective MWCNTs doped with PdeNi catalyst nanoparticles (appeared as black dots). Schematic representation of hydrogen spillover in (a) defect free and (b) defective multiwalled nanotubes decorated by catalyst particles. H, atomic hydrogen, H2, hydrogen molecule.

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List of Figures

Figure 12.1

Figure 12.2

Figure 12.3

Figure 12.4 Figure 12.5 Figure 12.6

Figure 12.7

Figure 12.8

Figure 12.9

Figure 12.10 Figure 12.11

Figure 12.12

Differential scanning calorimetry thermograms taken at the heating rate of 10 K/min on 99.995% (4N5) purity Ag processed by 1, 4, 8, and 16 passes of equal-channel angular pressing (ECAP). (a) The temperature of the maximum of the differential scanning calorimetry (DSC) peaks presented in Fig. 12.1 for 4N5 Ag processed by different number of passes of equal-channel angular pressing (ECAP). (b) The heat released in the DSC peaks and the activation energies determined by the Kissinger method as a function of number of ECAP passes. The mean crystallite size and the dislocation density as a function of annealing temperature for Ti processed by eight equal-channel angular pressing passes. The corresponding differential scanning calorimetry (DSC) thermogram is also presented. The relative fractions of hai-, hci-, and hc þ ai-type dislocations as a function of annealing temperature. The released heat obtained in differential scanning calorimetry experiments as a function of grain size for pure face-centered cubic metals. The microstructure of 4N5 purity Ag (a) immediately after equal-channel angular pressing processing and (b) after the differential scanning calorimetry peak as determined by transmission electron microscope and electron backscatter diffraction. Transmission electron microscope image of the interior of a recrystallized grain in equal-channel angular pressingeprocessed Ag heat-treated up to the end of the exothermic differential scanning calorimetry peak. The black spots indicate small dislocation loops. Differential scanning calorimetry thermograms for high-pressure torsion (HPT)eprocessed bulk-Cu and the counterpart consolidated by HPT from a microcrystalline Cu powder. Evolution of crystallite size (a) and dislocation density (b) in high-pressure torsion (HPT)eprocessed bulk-Cu and the counterpart consolidated by HPT from a microcrystalline Cu powder, as determined by X-ray diffraction line profile analysis (XLPA). The notation “recryst.” indicates the occurrence of recrystallization, which made the XLPA evaluation impossible. Differential scanning calorimetry thermograms measured at 40 K/min for consolidated-Cu and Cuecarbon nanotube (CNT) samples. The average crystallite size (a), the dislocation density (b), and the twin-fault probability (c) in consolidated-Cu and Cuecarbon nanotube samples obtained by X-ray diffraction line profile analysis (XLPA) as a function of the temperature in differential scanning calorimetry annealing at a heating rate of 40 K/min. “Recryst.” indicates the occurrence of recrystallization, which yielded higher crystallite size and lower dislocation density than the detection limits of XLPA. Dark field transmission electron microscopy micrographs taken on consolidatedCu specimen (a) immediately after high-pressure torsion (HPT) and after subsequent heating up to (b) 750 K and (c) 1000 K, as well as on Cuecarbon nanotube composite (d) immediately after HPT and after subsequent heating up to (e) 750 K and (f) 1000 K.

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List of Figures

Figure 12.13

Figure 12.14

Figure 12.15

Figure 12.16 Figure 12.17

Figure 12.18

Figure 12.19

Figure 12.20

Figure 12.21 Figure 12.22

Figure 12.23

Figure 12.24

Figure 12.25

Figure 12.26

Scanning electron microscopy images of the polished cross sections at the half-radius of the high-pressure torsion processed disks after heating up to 1000 K: (a) consolidated-Cu and (b) Cuecarbon nanotube composite. The hardness as a function of temperature for consolidated-Cu and Cuecarbon nanotube (CNT) composite specimens in differential scanning calorimetry annealing at a heating rate of 40 K/min. Differential scanning calorimetry thermogram obtained at a heating rate of 10 K/min for 4N purity Ag sample processed by 10 revolutions of high-pressure torsion. The temperatures of heat treatments are indicated by solid circles. Nanoindentation layout on the cross section of the high-pressure torsion (HPT)-processed Ag disk. Nanohardness distributions as a function of the distance from the bottom of the high-pressure torsion (HPT)eprocessed Ag disk measured on the cross section in the axial direction. The lines serve only as guides to the distributions. The typical error bar is illustrated on the left side of the figure. Electron backscatter diffraction micrographs showing the ultrafine-grained (UFG) microstructure before the exothermic differential scanning calorimetry (DSC) peaks at 400 K (a) and after the second peak at 497 K (b). The microstructure after the first DSC peak at 440 K is shown in (c) where the transition layer between the recrystallized interior and the UFG surface layer is indicated by a dashed line. The inset in (c) shows a part of the recrystallized grain in a higher magnification, illustrating that the large grain contains smaller twinned subgrains. A part of the UFG microstructure in the surface region is shown in a higher magnification in (d). Differential scanning calorimetry thermograms obtained immediately after equal-channel angular pressing (ECAP) and storage at room temperature for 4 years in the case of Cu samples processed by 1 and 10 passes. The reduction in vacancy concentration determined from the decrease of stored energy in Cu as a function of number of equal-channel angular pressing (ECAP) passes. Transmission electron microscopy images taken immediately after high-pressure torsion processing (a) and 4 years of storage (b). The microhardness of samples processed by different numbers of equal-channel angular pressing (ECAP) passes as a function of the time of storage at room temperature. DebyeeScherrer rings for the 220 reflections of X-rays (a) immediately after eight equal-channel angular pressing (ECAP) passes and (b) after eight ECAP passes and storage at room temperature for 4 months. Bright field transmission electron microscopy images from a sample processed through eight equal-channel angular pressing passes (a) and after storage at room temperature for 4 months (b). (a) The dislocation density and (b) the twin boundary frequency in the recovered volumes of samples stored at room temperature up to 4 months after processing by equal-channel angular pressing (ECAP) through 1, 4, 8, and 16 passes. Values of the microhardness after processing by equal-channel angular pressing (ECAP) for 1, 4, 8, and 16 passes as a function of the storage time at room temperature for 4N purity Ag.

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List of Figures

Figure 12.27

Figure 12.28

Figure 12.29

Figure 12.30

Figure 12.31 Figure 12.32 Figure 12.33

Bright field transmission electron microscopy images showing the microstructures of (a) the 4N5 purity sample processed through eight equal-channel angular pressing (ECAP) passes and stored at room temperature for 4 months and (b) the 4N purity specimen after eight ECAP passes and storage for 1 year. Schematic graph of the hardness distribution along the diameter of the Zne22% Al disk processed by one turn of high-pressure torsion (HPT). The hardness before HPT and the evolution of the hardness distribution due to storage at room temperature are also shown [83]. Transmission electron microscopy images obtained on Auecetyltrimethylammonium bromide nanoparticles. (a) Immediately after production the size of the spherical Au nanoparticles was 2e5 nm, and (b) after 1 year of storage the gold particles grew to about 25 nm and exhibited different shapes. Transmission electron microscopy images showing the different types nanoparticles with regular morphology after 1 year of storage. (a) Decahedron (D) and triangular plate (TP). (b) Deca-tetrahedron (DT) and rod (R). The inset shows a schematic drawing of the three-dimensional morphology of a DT. (c) Bipyramid (BP). (d) The arrow indicates that the twin boundaries in a rod are lying parallel to its longitudinal axis. Transmission electron microscopy image of fused Au nanoparticles after storing them for 1 year. The arrow indicates the joint surface of two fused nanoparticles. The factors influencing the shape of face-centered cubic nanoparticles and the resulted crystal morphologies. WilliamsoneHall plot of the full width at half maximum (FWHM) of the X-ray diffraction peaks as a function of the length of the diffraction vector (g) for Au nanoparticles after 1 year of storage.

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LIST OF TABLES Table 2.1 Table 2.2 Table 2.3 Table 2.4 Table 2.5 Table 3.1

Table 3.2 Table 4.1

Table 4.2 Table 5.1

Table 5.2

Table 5.3

Table 5.4 Table 6.1 Table 6.2

Comparison of the features of experimental methods used for the study of defect structure in nanomaterials The values of C h00 and q for edge and screw dislocations in Al, Cu, and Fe The notations, the Burgers vectors, and the slip planes of the 11 hexagonal slip systems [16] Specific electrical resistivity for vacancies, dislocations, and stacking faults in some pure fcc metals Positron lifetimes for defect-free lattice and different lattice defects in Al [57] The maximum dislocation density and the minimum crystallite size determined by X-ray line profile analysis, and the minimum grain size obtained by transmission electron microscopy for metallic materials processed by severe plastic deformation The concentration and the cluster size of vacancies obtained for metals processed by severe plastic deformation The equilibrium splitting distance (dp) for screw and edge dislocations, the constriction energy (W0), and the waiting time for cross-slip of screw dislocations (tcs) in various pure face-centered cubic metals The maximum values of the twin boundary frequency achieved by severe plastic deformation for some nanostructured low SFE metals and alloys The maximum dislocation density and the minimum crystallite size determined by X-ray line profile analysis, and the minimum grain obtained by TEM for powder materials processed by long time milling The grain size, the dislocation density, and the twin fault probability determined by X-ray line profile analysis for powder particles processed by bottom-up processing methods The intensity ratio (INiO/INi) of the X-ray peaks for NiO and Ni at 2Q ¼ 37.4 degrees and 44.6 degrees, respectively, the mean grain size obtained from TEM images, the mean crystallite size, the dislocation density, and the twin fault probability for the samples processed by hot-isostatic pressing and spark-plasma sintering [47] The parameters of the microstructure for nanocrystalline SiC specimens sintered at different pressures and temperatures Parameters of the microstructure for electrodeposited nanocrystalline Ni films prepared with different additives Parameters of the microstructure for textured Ni films prepared under different conditions (concentration of organic additive 2-butyne-1,4-diol and current density). The crystallite size is the arithmetically averaged size of crystallites obtained by XLPA (see Chapter 2). The twin fault spacing obtained by XLPA method was calculated from the measured twin fault probability using Eq. (4.15) in Chapter 4

28 35 40 50 53

71 80

96 114

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133

140 147 156

159

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List of Tables

Table 6.3 Table 7.1

Table 7.2

Table 7.3

Table 8.1

Table 8.2

Table 9.1 Table 9.2

Table 9.3

Table 10.1 Table 10.2 Table 12.1

Table 12.2

Table 12.3

The area-weighted mean crystallite size and the dislocation density for Cu-Nb multilayers with different layer and film thicknesses [16,17] The dislocation density (r) determined by X-ray line profile analysis and the yield strength (sY) at room temperature (RT) for face-centered cubic (fcc) metals and alloys processed by severe plastic deformation The average grain size (d), the mean twin spacing (t), the activation volume (V*), the strain rate sensitivity (m), the yield strength (sY), the uniform and total elongations for various Cu samples at room temperature The relative density, the intensity ratio (INiO/INi) of the X-ray peaks for NiO and Ni at 2Q ¼ 37.4 and 44.6 degrees, respectively, the mean grain size (d) obtained from transmission electron microscopy images, the yield strength, maximum strength, and the strain to failure determined by compression for Ni samples processed from nanopowders by hot isostatic pressing (HIP) or spark plasma sintering (SPS) The features of the matrix and CNTs, the processing method, the Young’ modulus, the yield and tensile strengths, and the elongation to failure for different metal matrixeCNT composites Resistivity, conductivity, and strength-to-resistivity ratio measured at RT for different values of CNT volume fraction in laminar Cu-CNT composite films processed by electroplating [24] Specific electrical resistivities for vacancies, dislocations, stacking faults, twin faults, and general high-angle grain boundaries (HAGBs) in Cu Grain size, mechanical strength, conductivity in IACS, resistivity, and strength-to-resistivity ratio measured at RT for different UFG and nanocrystalline materials Grain size, mechanical strength, conductivity in International Annealed Copper Standard (IACS), resistivity, and strength-to-resistivity ratio measured at room temperature (RT) for Cu/Cr multilayers with different layer thicknesses. The strength is calculated as one-third of the hardness [41,43] The grain boundary diffusion kinetics types, the related experimental conditions, and the evaluated function of concentration ðcÞ versus penetration depth (x) The frequency factor (D0), the activation energy (Q) of diffusion for different UFG and nanocrystalline materials The average grain size determined by transmission electron microscopy, the mean crystallite size, the dislocation density, and the twin boundary frequency obtained from X-ray line profile analysis and the onset temperature of recovery/ recrystallization (Tonset) measured by differential scanning calorimetry at a heating rate of 40 K/min for Cu processed by different severe plastic deformation methods The average grain size determined by transmission electron microscopy, the mean crystallite size, and the dislocation density obtained from X-ray line profile analysis and the peak temperature of recovery/recrystallization (Tpeak) measured by differential scanning calorimetry at a heating rate of 40 K/min for Ni processed by different severe plastic deformation methods and electrodeposition The processing method, the grain size, the heat released in the exothermic peak, and the activation energy of recrystallization determined for various ultrafine-grained and nanomaterials

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323

List of Tables

Table 12.4

Table 12.5

The balance table for stored energies calculated for the different lattice defects (dislocations, high-angle grain boundaries, twin faults, small dislocation loops, and vacancies) in equal-channel angular pressingeprocessed 4N5 purity Ag before and after the differential scanning calorimetry (DSC) peak The crystallite size, the dislocation density (r), and the parameter q describing the edge/screw character of dislocations obtained by X-ray line profile analysis

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PREFACE

Nanostructured materials (or nanomaterials) are built up from microstructural units (e.g., grains) having smaller size than 100 nm at least in one dimension of space. This class of materials includes thin layers that have thicknesses smaller than 100 nm, while their lateral dimensions are usually several centimeters, which means that the criterion for nanostructured materials fulfills only in one dimension. Nanotubes also pertain to nanomaterials as their diameter is several nanometers, i.e., they satisfy the criterion in two dimensions. Crystalline nanoparticles and bulk materials having grain sizes smaller than 100 nm are also members of this class; however, these materials are also called as nanocrystalline materials since their microstructural unit (the crystalline grain or particle) is smaller than 100 nm in all directions. It is noted that in some cases the size of microstructural unit depends on the method used for studying the microstructure. For instance, in severely deformed metallic materials the grain size measured by electron microscopy is usually several hundreds of nanometers while the crystallite size obtained from the broadening of X-ray diffraction peak profiles is smaller than 100 nm. This apparent dichotomy is attributed to the fact that in these materials the crystallite size corresponds rather to the subgrain size. Therefore, these specimens are often classified as ultrafine-grained (UFG) materials for which the average grain size is between 100 and 1000 nm but the term of nanomaterials is also used for these samples as the subgrain size is smaller than 100 nm. In 1990’s, nanomaterials have become a focal point of materials science due to their unique physical, chemical, and mechanical properties that destine these materials to novel and promising applications. The small dimension of the grains or particles in nanomaterials and their specific processing methods affect their defect structure (vacancies, dislocations, disclinations, stacking and twin faults as well as grain boundaries) that has a significant influence on the properties of these materials. The knowledge of the relationships between the production methods, the lattice defects, and the physical properties of nanomaterials is very important not only to understand the specific phenomena occurring when the grain size is very small but also from the point of view of practical applications of these materials. Knowing these correlations, the functional properties of nanomaterials can be tailored by tuning the defect structure via an appropriate selection of the processing conditions. This process can be referred to as “defect structure engineering”. This book aims to synthesize the existing knowledge of lattice defects formed in processing and subsequent plastic deformation of nanomaterials and their effect on functional properties. The first edition of this work was published in 2012 under the title “Defect

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structure in nanomaterials”. The great success of that book triggered the publication of this second and extended edition. The eight chapters in the first edition were improved and completed with four additional chapters, resulting in a doubled volume of the book. In the second edition a separate chapter is devoted to the characterization methods of lattice defects. In addition, four chapters deal with the influence of defect structure on functional properties of nanomaterials. These include the electrical resistivity, the diffusivity as well as the mechanical and hydrogen storage properties. Accordingly, the title of the second edition was changed to “Defect structure and properties of nanomaterials” which reflects these improvements. The information in this book is organized and presented in the form that is hopefully beneficial for a wide audience: materials scientists, engineers as well as lecturers, and students at universities. This book contains 12 chapters. In the following a brief description of each chapter is given. Chapter 1 reviews the processing methods of bulk nanomaterials and nanocrystalline particles. These processing methods include “top-down” procedures that produce nanomaterials by severe plastic deformation (SPD), and “bottom-up” synthesis methods of nanoparticles where the particles are built up from individual atoms or from their clusters. Another route for production of nanostructured powders is milling, which is a “topdown” nanopowder processing technique. The powders obtained by either “top-down” or “bottom-up” methods can be sintered into bulk nanomaterials. The most frequently used consolidation techniques, e.g., hot isostatic pressing and spark plasma sintering, are described in this chapter. Finally, the details of electrodeposition of nanostructured thin films and nanocrystallization of bulk amorphous materials are presented. In Chapter 2, the characterization methods of crystal lattice defects, such as vacancies, dislocations, planar faults, and grain boundaries are overviewed. These include direct methods, such as transmission electron microscopy and electron backscatter diffraction imaging, as well as indirect methods, such as X-ray line profile analysis (XLPA), electrical resistometry or positron annihilation spectroscopy. In the latter methods, first only a fingerprint of the defect structureda data seriesdis obtained and then the parameters of the lattice defects are extracted by analyzing these data. Special attention is paid to XLPA since the majority of the results presented in this book are obtained by this technique. The most important advantages of this method are (1) the easy sample preparation, (2) its nondestructivity, and (3) the good statistics of the resulted microstructural parameters compared with microscopic methods. A comparison of the different methods is also presented. Chapter 3 presents the evolution of defect structure during processing of nanostructured materials by SPD. It is shown that at high imposed strain in SPD-processing, the grain size reaches its minimum value and the dislocation density gets saturated. Although there is no strict correlation between the grain size and the dislocation density, the higher dislocation density is usually associated with smaller grain size. The saturation values of the dislocation density and the grain size are strongly influenced by (1) the homologous

Preface

temperature, the pressure, and the strain rate applied in SPD, (2) the solid solution alloying, (3) the second phase particles, and (4) the degree of dislocation dissociation (i.e., the stacking fault energy). It is revealed that the vacancy concentration in metals processed by SPD at room temperature is as high as the equilibrium value at the melting point and the majority of vacancies are clustered. The lattice defects formed during SPD (e.g., dislocations) facilitate precipitation, thereby influencing the phase composition of the asprocessed UFG alloys. In addition, due to the GibbseThomson effect the volume fraction of secondary phase nanoparticles is influenced by the energy of interfaces and the size of particles. In Chapter 4, the effect of low stacking fault energy (SFE) on defect structure in nanomaterials is summarized. The low value of SFE leads to a large degree of dislocation dissociation into partials that hinder strongly the cross-slip and climb of dislocations. As a consequence, a relatively large dislocation density develops during SPD of low SFE metallic materials. Additionally, the low SFE is accompanied by a small value of twin boundary energy resulting in a significant twinning activity during plastic deformation that alters grain refinement mechanisms. Among the pure face-centered cubic (fcc) metals, silver has the lowest SFE. Therefore, the effect of processing conditions and impurity content on dislocation density and twin-fault probability in UFG and nanocrystalline Ag is revealed and discussed in detail. In addition, the lattice defect structure in UFG alloys with low SFE is also studied. It is shown that the reduction of grain size increases the splitting distance between partials and the probability of occurrence of twinning in UFG and nanocrystalline materials. Chapter 5 reviews the type and densities of lattice defects formed during processing of nanoparticles and consolidation of bulk nanomaterials from nanopowders. The evolutions of arrangement, edge/screw character and density of dislocations, and grain size during milling of metallic powders are presented. The minimum grain size achievable by milling at room temperature is correlated to the melting point of metals. The formation of dislocations, disclinations, and twin faults in nanoparticles produced by bottom-up methods are discussed. The effects of the initial powder particle size and the consolidation conditions on the defect structure in sintered metals, diamond, and ceramics are discussed. When blends of nano- and coarse-grained powders are sintered, the defect structure in the nanocrystalline fraction of the consolidated material is strongly influenced by the amount of the coarse-grained powder component and vice versa. In Chapter 6, the lattice defects in nanocrystalline thin films and multilayers are overviewed. For electrodeposited Ni the organic additives increase the dislocation density and twin fault probability as well as reduces the grain size. For magnetron sputtered Cu foil relatively small dislocation density and large twin boundary probability were revealed. Subsequent rolling at room temperature yielded an increment in the dislocation density and a reduction of the twin fault probability due to untwinning. It was found that the substrate orientation has a deterministic effect on the average twin fault spacing in

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Preface

sputtered Ag film. In CueNb multilayer processed by magnetron sputtering at room temperature, the stresses induced by the lattice mismatch between the Cu and Nb phases were relaxed by formation of misfit dislocations with high density and Burgers vector parallel to the interface of the layers. The influence of He ion implantation on the lattice defect structure in different multilayers is also discussed. Chapter 7 overviews the defect-related mechanical properties of nanomaterials. The influence of small grain size on plastic deformation mechanisms, strength, and ductility is discussed. The smaller the grains size in fcc metals, the higher the activity of twinning at the expense of dislocation glide, while hexagonal close-packed materials behave contrarily. The relationship between the dislocation structure and the yield strength is studied in details. Above the grain size of w20 nm, the decrease of grain size is accompanied by an increase of yield strength and a decrease of ductility. The loss of ductility can be moderated by the incorporation of coarse-grains into the nanocrystalline matrix. A combination of large strength with good ductility can be achieved with nanograins containing high density of twin boundaries. Below the grain size of w20 nm, the yield strength is found to decrease with the reduction of grain size or remains unchanged within the experimental error. The possible explanations of this inverse Hall-Petch behavior are discussed in details. In Chapter 8, the processing methods, the defect structure and the mechanical properties of metal matrixecarbon nanotube (CNT) composites are overviewed. It is revealed that the high dispersity of CNTs in the matrix is the most important criterion for processing composites with high strength and good ductility. The correlation between the flow stress and the dislocation density for Cu-CNT composites suggests that the CNT fragments strengthen the composite rather indirectly via the increase of the dislocation density. Chapter 9 reviews the influence of lattice defects (vacancies, vacancy clusters, dislocations, planar faults, and grain boundaries) on the electrical resistivity of UFG and nanocrystalline materials. It turned out that the resistivities of vacancies, dislocations, and twin faults are much smaller than that for high-angle grain boundaries (HAGBs), solute atoms, and the intrinsic resistivity at room temperature. For pure metals and equilibrium solid solutions, nanocrystallization by SPD methods yields only a few percentage increase in resistivity while the strength is improved considerably, thereby improving the strength-to-resistivity ratio. The best combination of high strength and good conductivity in alloys is obtained if the strengthening is achieved by grain boundaries and nanosized secondary phase particles while the grain interiors are purified from solute elements. In pure metallic materials an improvement in strength-to-resistivity ratio can be obtained if the majority of HAGBs are substituted by coherent twin faults as the latter interfaces have very low specific electrical resistivity. In Chapter 10, the most important models, formulas, and methods for the evaluation of diffusion along grain boundaries and dislocations in nanomaterials are overviewed. It

Preface

was found that the larger the free volume and the energy of grain boundaries, the faster the grain boundary diffusion. In UFG metallic materials processed by SPD, a hierarchical microstructure develops with nonequilibrium and relaxed grain boundaries, which are paths for fast and slow grain boundary diffusion, respectively. As the fraction of boundaries exhibiting fast diffusion is very small, the diffusion rate in nanomaterials is determined by the relaxed boundaries. The diffusivity for these boundaries is very close to that observed for grain boundaries in coarse-grained materials. Therefore, the faster diffusion in nanomaterials compared to coarse-grained counterparts is caused basically by the larger amount of grain boundaries and not by their different quality. In nanomaterials processed by powder metallurgy, the diffusivity for interagglomerate interfaces between particles and pores is several orders of magnitude larger than that for intraagglomerate boundaries. Similar bimodal diffusivity is observed for nanomaterials obtained by crystallization of amorphous materials. Here, the slow and fast diffusion pathways are the amorphous and conventional crystalline interfaces. The diffusion along amorphous boundaries is slower than that for crystalline interfaces. Chapter 11 reviews the influence of particle size, crystallite size, and lattice defects on hydrogen storage capacity and absorptionedesorption kinetics of nanostructured materials. It is shown that the particle and crystallite sizes, and catalyst play two different roles in the sorption process. By decreasing the particle and crystallite sizes, the required diffusion path of hydrogen is drastically reduced that enhances the sorption kinetics significantly. At the same time, catalysts have a promoting effect on dissociation or recombination of hydrogen molecules on the particles’ surfaces and they also act as chemisorption sites. The lattice defects (dislocations, stacking faults, and twin boundaries) facilitate the diffusion of hydrogen and the nucleation of hydride phases. It is revealed that large amounts of dislocations and/or planar faults are formed due to stresses induced by phase transformations in hydrogenation and dehydrogenation processes. The effect of defects on hydrogen storage capacity of carbon nanotubes is also reviewed. Finally, in Chapter 12 the thermal stability of the defect structure in nanomaterials is overviewed. At high temperatures, the activation energy of recovery/recrystallization in pure fcc nanomaterials is close to the activation energy of grain-boundary diffusion. The onset temperature of recovery/recrystallization and the released heat depend on the grain size and the defect density contrary to the activation energy. Some nanomaterials with low SFE and/or low melting point (e.g., Ag or PbeSn alloy) tend to recover/recrystallize even during storage at the processing temperature (e.g., at room temperature) that is referred to as self-annealing. The low SFE promotes self-annealing as the recrystallized grains may be easily separated from the matrix by low energy twin boundaries. In addition, the thermal stability of UFG Cu samples processed by SPD and powder metallurgy is compared. The effect of CNT additive on recovery and recrystallization of the UFG Cu matrix is also investigated. Finally, the coarsening of Au nanoparticles and the development of their defect structure during their storage at room temperature is studied.

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CHAPTER 1

Processing Methods of Nanomaterials 1.1 PROCESSING OF BULK NANOMATERIALS BY SEVERE PLASTIC DEFORMATION The processing methods of bulk nanomaterials are usually classified into two groups as depicted in Fig. 1.1. In the course of “bottom-up” methods the materials are built up from individual atoms, molecules, or their clusters (particles), such in electrodeposition or inert gas condensation. In the case of “top-down” methods, the nanosized microstructural units (grains or crystallites) are achieved by refinement of coarse-grained materials. The grain refinement usually occurs by severe plastic deformation (SPD) that can be carried out either on bulk materials as in the case of equal-channel angular pressing or on powder samples, e.g., by milling. In the case of nanomaterials processed by powder metallurgy, the classification may be more complex. The consolidation procedures of nanopowders are usually declared as “bottom-up” methods, but the nanopowders used for sintering can be produced by milling that is a “top-down” procedure. “Bottom-up” processing oft bulk nanomaterials Bulk nanomaterials are assembled from atoms or powder particles

e.g. Electrodeposition Physical vapor deposition Crystallization from amorphous

“Top-down” processing of bulk nanomaterials Grain refinement by severe plastic deformation (SPD) of bulk materials

e.g. Equal-channel angular Pressing high-pressure torsion Twist extrusion

Powder metallurgy Nanopowder production + Conslidation e.g. Pressureless sintering Hot isostatic pressing Spark plasma sintering Shock wave consolidation

“Bottom-up’’ powder production Building-up of particles from atoms

e.g. Inert gas condensation Plasma synthesis Electro-explosion of wire Laser ablation Cryomelting

“Top-down” powder production Refinement of coarse particles by SPD

Milling

Figure 1.1 Classification of processing routes of bulk nanomaterials. Defect Structure and Properties of Nanomaterials ISBN 978-0-08-101917-7, http://dx.doi.org/10.1016/B978-0-08-101917-7.00001-3

© 2017 Elsevier Ltd. All rights reserved.

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Defect Structure and Properties of Nanomaterials

Bulk ultrafine-grained (UFG) or nanomaterials can be produced by SPD of their coarse-grained counterparts. Formally, processing by SPD may be defined as those metal forming procedures in which a very high strain is imposed on a bulk solid without the introduction of any significant change in the overall dimensions of the solid [1]. As a result of SPD, dislocations are formed and arranged into low energy configurations (e.g., low-angle grain boundaries) that transform into high-angle grain boundaries with increasing strain [1,2]. SPD processing can be performed on both bulk materials and powder samples as can be seen in the classification scheme of the processing methods of nanomaterials in Fig. 1.1. In this section, the SPD methods applied on bulk materials are overviewed. A solid is considered as a bulk if its size is in the millimeter range or larger in any direction. The most important advantage of bulk SPD techniques compared to other processing methods (e.g., powder metallurgy) of nanomaterials is that they avoid undesired contamination and porosity in the final microstructure. The initial material is usually a bulk coarse-grained workpiece that is deformed plastically at an equivalent strain higher than one. The equivalent strain in a deformation corresponds to the strain in a uniaxial tension that yields the same plastic work as produced in the given deformation. The most frequently applied SPD procedure is equal-channel angular pressing (ECAP) [1,3e5], known also as equal-channel angular extrusion (ECAE). The principle of ECAP is illustrated in Fig. 1.2a [5]. A circular rod or a square bar is pressed through a die containing two intersecting channels. The cross sections of the channels and the billet match; therefore deformation takes place only at the intersection of the two lubricated channels. Since the cross-sectional dimensions of the billet remain unchanged, the pressings may be repeated to attain exceptionally high strains. It is noted that after removal of the billet from the exit channel, its diameter increases slightly due to an elastic relaxation. Therefore, if the entry and the exit channels have the same cross sections, then a surface layer of the billet should be removed between the consecutive passes. However, this problem may be overcome if the exit channel exhibits a slightly smaller diameter (by about Plunger

(a)

(b) Route BA

Route A

90°

Die Sample

Φ Route C

Route BC 90°

180°

ψ

Figure 1.2 (a) The schematic depiction of equal-channel angular pressing (ECAP) processing and (b) the four fundamental processing routes in ECAP. (Reprinted from K. Nakashima, Z. Horita, M. Nemoto, T.G. Langdon, Development of a multi-pass facility for equal-channel angular pressing to high total strains, Materials Science and Engineering A 281 (2000) 82e87 with permission from Elsevier.)

Processing Methods of Nanomaterials

0.1 mm) than for the entry channel. In the latter case, the billet is extruded when it passes through the intersection of the two channels. The equivalent strain, ε, introduced in ECAP is determined by a relationship incorporating the angle between the two parts of the channel, F, and the angle representing the outer arc of curvature where the two parts of the channel intersect, J. The relationship is given by the formula [1]



  F J F J 1 2ctg þ þ J sin þ 2 2 2 2 pffiffiffi ε ¼ N ; 3

(1.1)

where N is the number of passes. In practice, the channel angle F ranges from 45 to 180 degrees and the arc of curvature varies from 0 to 90 degrees [1]. The usual values of F and J are 90 and 20 degrees, respectively, and in this configuration, one pass corresponds to an equivalent strain of w1 [6]. The imposed strain increases proportionally with increasing number of passes. Between the consecutive passes, the billet can be rotated about its longitudinal axis. According to the various manners of rotation, four different basic routes of ECAP can be distinguished as it is depicted in Fig. 1.2b. In the case of route A, there is no rotation of the billet, while in routes BA or BC, rotations by 90 degrees in alternate directions or the same direction are applied, respectively. In route C the billet is rotated by 180 degrees about the longitudinal axis. The number and orientations of shear planes are different for various routes [1,7], which affects the evolution of the microstructure during ECAP. When using a die with a channel angle of F ¼ 90 degrees, route BC is generally the most expeditious way to develop a UFG microstructure consisting of homogeneous and equiaxed grains with grain boundaries having high angles of misorientation [1]. The advantages of ECAP are that the as-processed workpiece (1) has relatively large dimensions (several centimeters) in all directions, and after several passes the UFG microstructure exhibits (2) a high degree of homogeneity and (3) a large fraction of equilibrium high-angle grain boundaries. The latter feature gives improved fatigue behavior and a good corrosion resistance [8]. The classical ECAP technique was modified to provide a solution for the continuous production of ultrafine-grained materials. This method is referred to as ECAP-Conform process that is shown schematically in Fig. 1.3 [1]. The setup consists of a rotating shaft and a stationary constraint die. The rotating shaft in the center contains a groove, and the workpiece is fed into this groove. Then, the workpiece rotates together with the shaft due to the friction forces between them; however, it is constrained within the groove by a stationary constraint die. The material is subjected to strong shear deformation at the intersection of the two ECAP channels at the end of the groove (see Fig. 1.3). Another frequently studied SPD method is high-pressure torsion (HPT). HPT refers to processing in which the sample, generally in the form of a thin disk, is subjected to

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Defect Structure and Properties of Nanomaterials

Stationary constraint die Workpiece

Rotating shaft

Figure 1.3 A schematic of ECAP-Conform process [1]. ECAP, equal-channel angular pressing.

torsional straining under a high hydrostatic pressure: the principle of HPT is illustrated schematically in Fig. 1.4 [2,5]. The disk is located within a cavity, a hydrostatic pressure is applied, and plastic torsional straining is performed by rotation of one of the anvils. The torsion of the disk is achieved due to the friction between the sample and the anvils. The unique feature of this method is the extremely large applied pressure, 1e9 GPa [2], that hinders the annihilation processes of lattice defects thereby yielding to very large Load

Upper anvil

Specimen (disk)

Lower anvil

Torsion

Figure 1.4 The principle of high-pressure torsion.

Processing Methods of Nanomaterials

defect densities and small grain size. Usually, the thickness and the diameter of the HPT-processed disks are about 1 and 10 mm, respectively, that limits their practical applications. The equivalent strain (ε) depends on the distance from the center (r) according to the equation 2prN ε ¼ pffiffiffi ; 3h

(1.2)

where h is the height of the disk and N is the number of revolutions [2]. It is noted that usually there is some outward flow of material between the two anvils during HPT and the value of h is reduced accordingly, which should be taken into account in the calculation of the strain. The SPD processing of circular rods or rectangular bars by multidirectional forging (MDF) includes multiple repeats of forging operations with changes of the axes of the applied load [9,10]. The principle of MDF is illustrated in Fig. 1.5 for the case of a rectangular workpiece. After one cycle of MDF, the original shape of the billet is regained and its longitudinal axis has the same orientation in the reference system attached to the sample as before deformation. In the case of a circular rod, usually many forging steps (about 20) are needed to restore the rod shape of the sample after one cycle. As in one forging step, the imposed z σ

z

y

y

x

b

x

a σ

c

c

σ

σ

a b

z y

σ a c

x b

σ

Figure 1.5 Schematic illustration of the steps of multidirectional forging procedure. The axes of the reference system attached to the sample are denoted as x, y, and z. The loading directions are indicated by black arrows. The letters a, b, and c denote the sample dimensions.

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strain is about 0.2; therefore one cycle corresponds to an equivalent strain of w4. The choice of the appropriate temperatureestrain rate regimes of MDF leads to the desired grain refinement. The operation is usually realized over the temperature interval of 0.1e0.5 Tm, where Tm is the absolute melting temperature. MDF is useful for producing large-sized billets with nanocrystalline or UFG microstructures [5]. During twist extrusion (TE), a workpiece is pushed through a twisted die that yields torsion of the sample with a designated angle around its longitudinal axis (see Fig. 1.6). The workpiece regains its shape and size after each TE pass; therefore it is possible to repetitively process a sample for stronger grain refinement [5,11]. By analogy to HPT, the plastic strain is not uniform across the cross section but rather increases with the distance from the twist axis; therefore the more distant regions have a finer grain size. This microstructural heterogeneity leads to inhomogeneous mechanical properties: the hardness increases with increasing the distance from the center in the cross section. The homogeneity of the microstructure can be improved by increasing the number of TE passes. During accumulative roll bonding (ARB), first a sheet is rolled so that the thickness is reduced to one-half of the initial value (see Fig. 1.7). The rolled sheet is cut into two halves that are stacked together. The stacked sheets are then rolled again to one-half thickness. The repetition of rolling, cutting, and stacking operations yields to a large accumulated strain in the sheet. To achieve good bonding during the rolling step, the surfaces of the sheets are degreased and wire-brushed before placing them in contact [5]. In practice, the UFG grains produced by ARB are not equiaxed but rather they have a pancake-like shape, which is elongated in the rolling direction. The most important advantage of this method is its high productivity and that it can be performed by a conventional rolling facility. It is noted that beside ECAP, HPT, MDF, TE, and ARB methods, numerous other SPD procedures exist for processing bulk UFG or nanocrystalline materials. In the process of repetitive corrugation and straightening (RCS), the workpiece is initially deformed to a corrugated shape and then straightened that may be repeated many times as depicted in Fig. 1.8. An advantage of RCS is that it can be adapted easily to current industrial rolling facilities and therefore it has the potential of producing nanostructured materials in a continuous and economical way [5]. Initial workpiece

After twist extrusion Twist extrusion die

Figure 1.6 The principle of twist extrusion processing.

Processing Methods of Nanomaterials

Rolling

Stacking

Cutting

Surface treatment

Figure 1.7 The principle of accumulative roll bonding method.

Figure 1.8 The principle of repetitive corrugation and straightening method.

1.2 PRODUCTION OF NANOPOWDERS AND NANOPARTICLES The processing of bulk nanomaterials by powder metallurgy includes the following basic steps: (1) nanopowder production, (2) compaction of the powder into a high-porosity specimen having the same shape as the final product (forming step), and (3) consolidation

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of the sample to high density usually at high pressure and temperature (sintering step). The second step is especially important in the case of brittle materials, such as ceramics, since after sintering the manufacturing of the very hard and rigid workpiece to the desired form is difficult without fracture. Although, during the forming step the sample receives the final shape, its size is still larger than that of the final product due to the large porosity. This specimen is called as “green body.” In the sintering step, the shape of the sample does not change, only its size is reduced as a consequence of the decrease of porosity. For tough metallic materials, the forming step is often missing, since the workpiece can be manufactured even after sintering. The most important advantages of powder metallurgy methods compared to SPD procedures are that (1) usually smaller grain size can be achieved, (2) the thermal stability is better, and (3) the lack of texture gives an isotropic behavior. At the same time, the drawbacks are (1) the small productivity, (2) the remaining porosity, and (3) the undesired contamination. In the following, first some nanopowder production procedures are presented and in the next section the consolidation methods are reviewed. The processing methods of nanopowders can be classified as “bottom-up” and “top-down” procedures. In the first case, the powder particles are built up from individual atoms or atomic clusters. In the second case, coarse-grained initial powder particles are subjected to SPD, usually with the application of milling. As a result, UFG or nanocrystalline microstructure is formed inside the metallic particles, while the particle size remains in the micrometer range [12]. In the case of brittle materials (e.g., ceramics), the fracture of particles into smaller parts also operates during milling. In the following, seven “bottom-up” nanopowder processing methods are presented: inert gas condensation, laser ablation, cryogenic melting, radio frequency plasma synthesis, electro-explosion of wire, sonochemical synthesis, and hydrothermal reaction. In the process of inert gas condensation, first atoms are thermally evaporated from a metallic source inside an ultrahigh vacuum chamber filled with inert gas, typically helium (see Fig. 1.9) [13,14]. The vaporized species then lose their energy via collisions with helium molecules. As collisions limit the mean free path, supersaturation can be achieved above the vapor source that yields the formation of atomic clusters [14,15]. The clusters are transported to a liquid nitrogenefilled cold finger by a convection flow, and then they condensed as nanoparticles on the surface of the finger. The particles are removed from the cold finger by means of a scraper assembly. They are collected and transported to an in situ compaction device. Consolidation is performed first in the low-pressure compaction unit and then in the high-pressure compaction unit [13]. It is noted that when atoms are ejected from the target surface by the impact of energetic ions, the process is called as sputtering. Sputtering is capable of depositing high-melting point materials such as Mo, Ta, W, and ceramics, which are difficult to fabricate using evaporation. In the method of laser ablation, an intense pulsed laser beam irradiates the target of interest, thereby vaporizing atoms and clusters from the target [13].

Processing Methods of Nanomaterials

Liquid N2 filled cold finger Scraper Deposited nanoparticles Evaporation source

Gas inlet

Collection funnel

To vaccum pumps

To consolidation system

Figure 1.9 Schematic picture of an inert gas condensation facility.

In the process of cryogenic melting (or cryomelting), a metal rod melts by a radio frequency inductor as depicted in Fig. 1.10 [13,16]. The molten metal droplets fall into a cool gas flux evaporated from a cryogenic medium, such as liquid Ar, at a temperature of 77 K. Rapid overheating of the metal via radio frequency technique produces a substantial evaporation rate of atoms from the hot surface of the droplets into the cool gas. Then, nanoparticles are formed by rapid condensation of the supersaturated metal vapor. The condensation region, where the particles are formed by nucleation, growth, and coalescence processes, is featured by a high temperature gradient, i.e., typically from 2200 K at the metallic surface but drop to 77 K in the cryogenic medium. The low temperature of the surrounding AI feeding bar Powder collector

AI liquid drop

Ar flux + AI nanoparticles

Induction coil Liquid Ar R.F. generator

Figure 1.10 Schematic depiction of the apparatus of cryogenic melting.

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medium produces a high rate of nucleation and a rapid cooling of the as-formed particles, which yields their small size. The upward cryogenic gas flux transports the nanoparticles into the powder collector. The method of electro-explosion of wire is basically used to prepare metallic nanoparticles [17e19]. The facility for electro-explosion of wire process is illustrated in Fig. 1.11. A very thin wire (about 0.5 mm) and a plate electrode are made of a metal and connected to a battery (tension z10 V). The electrical circuit remains open until the contact is made by the wire onto the plate. The very high current density (104e106 A/mm2) in the thin wire results in an explosion of the wire in a very short time: the material boils up in a burst, a bright light flashes, and a mixture of superheated vapor and boiling droplets scatter into the ambient atmosphere [17]. The scattered material is condensated into nanoparticles in the surrounding medium (e.g., in Ar gas). After explosion, the circuit opens again, and the feeding of the wire causes further explosion processes. The frequency of explosions is about 1 Hz. Sonochemical synthesis is an effective tool for producing organic and inorganic nanoparticles. In this method, ultrasonic waves with frequency between 20 kHz and 2 MHz propagate in the precursor liquid and generate very extreme reaction conditions, such as high-pressure oscillations (maximum pressure is about 100 MPa), elevated local temperature (w5000 K), and cavitation. The latter phenomenon is caused by the stress gradient, which overcomes the cohesion forces in the liquid. As a result, vacuum bubbles are created, which first start to grow but after reaching a critical size they collapse. The cavitational collapse yields intense local heating and a very high pressure, which accelerate the chemical reactions. This method was successfully applied for the production of Fe2O3 [20,21] and ZnO [22,23] nanoparticles. In the case of Fe2O3, the flowchart of the sonochemical method is illustrated in Fig. 1.12. First, NaOH is dissolved in deionized water and the obtained solution is added dropwise to an aqueous FeCl3$6H2O solution. The chemical reactions occurring during sonochemical synthesis of Fe2O3 are also

(3)

(4) (1)

(2)

Figure 1.11 Schematic diagram of the setup for electro-explosion of wire: (1) thin metal wire electrode, (2) metal plate electrode, (3) batteries, and (4) glass vessel.

Processing Methods of Nanomaterials

Mixing of FeCI3 • 6H2O and NaOH

FeCI3 + 3NaOH Reactions with the help of ultrasonic waves

Fe(OH)3

2FeOOH

Drying at RT

Fe(OH)3 + 3NaCl

FeOOH + H2O

Fe2O3 + H2O

amorphous Fe2O3 nanoparticles

Calcination at 773 K for 1 h

nanocrystalline Fe2O3

Figure 1.12 The flowchart of the sonochemical method, resulting in Fe2O3 nanoparticles [20].

shown in Fig. 1.12. Finally, the by-products are removed by washing with methanol and distilled water [20]. Then, the sample is dried in air at room temperature and then heat-treated at 773 K for 1 h. This calcination step is required for crystallizing the as-synthesized amorphous iron oxide nanoparticles. The process parameters such as temperature, time, and power of ultrasonication strongly influence the size and morphology of the final products [20]. For instance, the higher the intensity of ultrasound wave (power/area) and the lower the reaction temperature, the smaller the size of iron oxide nanoparticles [21]. The effectivity of the sonochemical method in processing of nanoparticles was also demonstrated on other metal oxide materials, such as ZnO and Pr-doped ZnO [23]. In the latter case, Pr(NO3)3$6H2O was added to ZnCl2 solution that was then mixed with NaOH solution until the pH ¼ 10 was reached. Then, the solution was irradiated by ultrasonic waves for 3 h to facilitate the precipitation of Pr-doped ZnO. In the final step, the crystallization was carried out by heat treatment. For the production of Er-doped ZnO nanoparticles by sonochemical synthesis, Er(CH3COO)3$H2O was added to ZnCl2 solution [22]. Hydrothermal synthesis is a successfully used method for the synthesis of different nanostructures, such as nanoparticles and nanotubes [24e27]. In this technique, the nanomaterials are synthesized from high-temperature aqueous solutions at high vapor pressures. As an example, Fig. 1.13 shows the schematic of the steps of hydrothermal synthesis of titanate nanotubes from TiO2 precursor. First, a titania powder is mixed with NaOH and the mixture is placed into an autoclave at a temperature between 373 and 433 K for 24 h [27]. In this step, anatase crystals decompose into TiO6 octahedra and these octahedral units form titanate sheets. These sheets are rolled up, forming titanate nanotubes,

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TiO6 octahedral units

Anatase particle NaOH at ~400 K

Rolling up

Titanate nanotube

Titanate sheet from TiO6 units

Figure 1.13 Schematic of the main steps of hydrothermal synthesis of titanate nanotubes from anatase TiO2 powder.

as shown in Fig. 1.13. The synthesized samples are filtered and washed by HCl aqueous solution (pH ¼ 1.6) several times. Finally, the sample is washed in distilled water until the pH value becomes 7. The last step is a calcination at 673 K. Besides nanotubes, different nanoparticles can also be produced by hydrothermal synthesis. For instance, CdSe and ZnS nanoparticles with variable Pr fractions as well as Nd-doped PbSe nanoparticles were successfully synthesized using this method [24e26]. Ball milling of powders is a “top-down” procedure of nanopowder processing. The essence of the method is that coarse-grained powder together with balls made of hard materials (e.g., steel or ceramics) are filled into a mill. The collisions of the powder particles with the balls and the internal wall of the mill yield SPD that leads to grain refinement inside the particles in a similar manner as in SPD processing of bulk metallic materials. It must be noted that nanopowders do not necessarily consist of nanoparticles. The powder particles may be actually of micrometric sizes, yet the particle interior may be divided into many nanograins [12]. The advantages of ball milling techniques are the simplicity and the cost-effectiveness compared to other nanopowder production procedures. A disadvantage of the method is that the powder can be contaminated by the material of the balls, e.g., Fe contamination can occur due to steel balls. Additionally, the milling is often carried out in inert gas (e.g., in Ar) to minimize oxidation of particles. There are many different designs of ball mills, which can be used for processing of nanopowders. Some often applied equipments are jar (drum) mill, Szegvari attritor, planetary mill, vibratory (shaker) mill, and magnetic ball mill. The milling balls usually have a diameter between 5 and 10 mm, and the ball-to-powder weight ratio can vary from 1 to 100. In the case of the laboratory jar mill (or industrial drum mill), large numbers of grinding balls are placed inside a cylindrical container that is rotated by rollers around a horizontal axis (see Fig. 1.14). The balls may roll down the inside wall surface, which produces

Processing Methods of Nanomaterials

Jar Balls ω

Drive rollers

Figure 1.14 Jar or drum mill.

shear forces on powder trapped between the wall and the ball, but mostly they fall freely accelerated only by gravitational force, then impacting the powders (and other balls) beneath them [12]. The jar mills are low-energy mills, although they can provide higher energy milling if sufficiently large diameter in order of meters and many balls are used. Larger acceleration than gravitational force, hence higher velocity and higher kinetic energy of balls can be achieved in Szegvari attritor. In such mills the milling is conducted in a cylinder filled with balls that are stirred by rotating impellers (see Fig. 1.15). The impact of impellers causes differential velocities between the balls and the powder; therefore the powder particles are deformed mainly by shear. An attritor-type ball mill delivers 10 times more energy than a jar mill with comparable size, and attritors can be considered

Drive shaft

Rotating impeller

Balls

Figure 1.15 Szegvari attritor.

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as medium-energy ball mills. The efficiency of vertical attritor mills is limited by the tendency of the powder to fall by gravity to the bottom of the cylinder [12]. In a planetary ball mill the jars containing the powder and the balls are rotating around their own axes with an angular velocity of uv, and simultaneously they orbit around a common axis with an angular velocity of up, similarly as the movement of the planets around the Sun. The motion of the jars and the balls in a planetary mill is shown in the schematic Fig. 1.16. The impact forces achieved in planetary mills are usually higher than 50 times the gravitational force acting on the balls [12]. Planetary ball mills can be considered as medium- to high-energy ball mills; however, milling times needed to process submicron-sized and nanostructured powders may be long. In vibratory or shaker ball mills, an eccentric motion is imparted to a cylindrical vial containing the powder and the balls at frequencies ranging from 10 to 2000 Hz and at small amplitudes of vibrations [12]. The balls oscillate in three mutually perpendicular, or more, planes within a small vial as illustrated in Fig. 1.17. Most of energy transfer in vibratory mills is conducted in the mechanical impact mode although substantial shear is also present on the powder particles trapped between the balls and the internal wall of the vial. This is a high-energy milling process as the balls impact the powder at high speeds and at high frequencies. To prevent the powder against oxidation, the milling is usually performed in inert gas. Other way to apply force to a milling ball, besides the gravitational and the centrifugal means discussed earlier, is to drive ball motion by magnets. In the case of the magnetic ball mill, the magnetic balls are attracted to the inside surface of the nonmagnetic jar by strong permanent magnets placed outside the jar (see Fig. 1.18). The pull imparted by the external magnet on the magnetic balls is so strong that the centrifugal force acting on the balls becomes a secondary factor in milling. The point where each ball is detached from the wall is well determined by the position of the magnet; hence each ball imparts the same energy impact on the milled powder. In magnetic ball mill, the high-energy

ωp

ωv

Figure 1.16 Motion of balls and jars in a planetary mill.

Processing Methods of Nanomaterials

Vial motion

Vial

Balls

Powder

Figure 1.17 Illustration of a vibratory ball mill.

Jar Magnet

ω

Figure 1.18 Motion of balls in a magnetic mill.

milling can be achieved at low rotational speeds (30e200 rpm); therefore the Fe contamination can be reduced. The shear and impact forces can be controlled by the position of the external magnets [12].

1.3 CONSOLIDATION TECHNIQUES OF NANOPOWDERS The driving force of nanopowder consolidation during sintering is the reduction of the large surface area of nanoparticles. Additionally, the high sintering pressure also assists the consolidation by inducing large shear stresses at the contact points between particles that yield plastic deformation thereby contributing to the filling of pores. This mechanism is especially important in the case of metallic powders having good deformability. Although the high temperature during sintering facilitates the consolidation by increasing the atomic mobility, it may also cause grain coarsening that is an unwanted phenomenon in processing of nanomaterials. To minimize grain growth, usually the time and temperature of sintering are reduced together with a simultaneous increase of pressure for maintaining the low porosity level in the final product. In this section, some nanopowder consolidation procedures are overviewed: shock wave consolidation, pressureless sintering,

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Defect Structure and Properties of Nanomaterials

hot pressing, hot isostatic pressing, and spark plasma sintering. The reason of selection of these methods for review is that mainly they were used for the consolidation of the samples studied in the next chapters. In shock wave consolidation, the high pressure and rapid loading rates applied on the powder result in interparticle bonding due to localized melting at the interfaces between the particles [28,29]. In practice, the powder particles are enclosed in a steel block, which is covered by a plate on the top that drives the shock wave caused by the explosion toward the sample (see Fig. 1.19). The driver plate is usually made of a highly conductive, highly ductile material, such as copper. Explosives and the detonator are packed on the top of the driver plate. Of late, gas guns are also widely used in performing shock wave consolidation. In the latter case, hydrogen gas activates the projectile, which travels all the way through the barrel and hits the powder sample. The duration and the pressure of the shock wave, which yields consolidation of the powder, are w106 s and 20e600 GPa, respectively. Shock wave consolidation technique is mainly used for metallic powders. As only the surfaces of the particles are heated up, the grain growth is suppressed. During pressureless sintering, the precompacted sample (green body) is kept at high temperature for long time that yields fusion of the particles thereby decreasing the porosity in the specimen. The driving force for consolidation is the high surface energy in the precompacted sample compared to the dense material. The surface area is reduced by diffusion on the particle surfaces resulting in their fusion as it is illustrated in Fig. 1.20. Fusion occurs well below the melting point of the material, but at a temperature sufficiently high enough to allow an acceptable rate of diffusion to occur, usually at greater than one-half of the melting point on a Kelvin scale [28]. The high temperature and the relatively long time (hours) of pressureless sintering usually yields grain growth in the consolidated sample.

Detonator

Explosive

Driver plate Powder for consolidation

Figure 1.19 Schematic depiction of shock wave consolidation process showing the experimental setup.

Processing Methods of Nanomaterials

Figure 1.20 Illustration of the fusion of particles during pressureless sintering. (Reprinted from V. Viswanathan, T. Laha, K. Balani, A. Agarwal, S. Seal, Challenges and advances in nanocomposite processing techniques, Materials Science and Engineering R 54 (2006) 121e285 with permission from Elsevier.)

In hot pressing process, a uniaxial loading up to 25 tonne assists the consolidation at high temperature [28]. The schematic depiction of the equipment is presented in Fig. 1.21. The application of high pressure enables the reduction of temperature and time of consolidation to decrease the grain growth compared to pressureless sintering process. The external pressure induces internal stresses between the particles that facilitates diffusion and also yields plastic deformation of the particles in the case of metallic powders, thereby speeding up the consolidation. A vacuum or inert atmosphere (e.g., Ar) can be applied to prevent oxidation. In hot isostatic pressing (HIP), high hydrostatic pressure and high temperature are applied to consolidate fine particles [28,30,31]. If only high hydrostatic pressure is used and no heating is performed, then the process is called cold isostatic pressing. During the manufacturing process, the powder is placed in a container, typically steel can. The container is subjected to a very high vacuum to remove air and moisture from

Powder

Figure 1.21 Schematic depiction of hot pressing.

17

Defect Structure and Properties of Nanomaterials

the powder. This processing step usually takes long time (even 100 h). The container is then sealed and subjected to HIP. The application of high inert gas pressures and elevated temperatures yields the removal of internal voids and creates a strong bond throughout the material. The result is a clean homogeneous material with a uniform fine grain size and a near 100% density. The reduced porosity of HIP-processed materials leads to improved mechanical properties and increased workability. One of the primary advantages of the HIP process is its ability to create near-net shapes that require little machining. In spark plasma sintering (SPS), the consolidation is assisted by high-current density pulses that enable to decrease the time and temperature of sintering thereby reducing grain growth. Schematic depiction of typical SPS apparatus is shown in Fig. 1.22, consisting of a graphite die where powder is loaded and is subjected to high electric current [28,32,33]. The high current density results in spark discharge (high temperature plasma) between the gaps of particles that activates the surface by removing surface oxide. This leads to faster diffusion on the surfaces of the purified particles resulting in easier consolidation. Additionally, the high current density at the contact points of the particles yields a local melting of the particles’ surfaces that also facilitates the mass transport. Typical SPS processing parameters include (1) an applied pressure between 50 and 100 MPa, (2) a pulse duration of w10 ms with oneoff cycle of 2e2.5 ms, and (3) maximum pulse current and tension of 10,000 A and 10 V, respectively.

1.4 PRODUCTION OF THIN FILMS BY ELECTRODEPOSITION Electrodeposition is a successful technique for producing thin films on a surface of a substrate by the action of electric current. The substrate material is immersed into an electrolyte solution containing a salt of the metal to be deposited and attached to the negative pole of a Pressure

Current pulse Powder Pyrometer

Graphite die

DC pulse generator

18

Vacuum chamber

Figure 1.22 Schematic depiction of the spark plasma sintering apparatus.

Processing Methods of Nanomaterials

power supply, i.e., it acts as a cathode in the electrolytic cell [28]. The anode is also immersed into the electrolyte and connected to the positive pole of the power supply. Usually, the anode and the electrolyte are the metal to be deposited and the aqueous solution of its salt, respectively. For example, in the case of Ni the electrolyte may be nickel chloride salt that dissociates in water to positively charged nickel cations and negatively charged chloride anions. As the object to be plated is negatively charged, it attracts the positively charged nickel cations, and electrons flow from the object to the cations to neutralize them to metallic form (see Fig. 1.23). Meanwhile, the negatively charged chloride anions are attracted to the positively charged nickel rod. At the anode, electrons are removed from the nickel metal, oxidizing it to the nickel cations. Thus, nickel dissolves as ions into the solution. The deposit formed on the cathode surface may be nanocrystalline, if the electrodeposition parameters, e.g., bath composition, temperature, pH, etc., are properly controlled [14]. Pulse plating is particularly attractive because it can yield finer grain structures than that achievable by direct current plating. In pulse plating, the current is imposed in a repetitive square wave with the following controlling parameters: peak current density, pulse-on time, and pulse-off time. When the pulse is switched off, the grain growth stops and the next pulse induces nucleation of new grains with other crystallographic orientations, thereby reducing the grain size. The pulse electrodeposition technique permits the application of a very high current density (several orders of magnitude higher than for direct current electrolysis) because the pulse-on time (several tens of milliseconds) is much shorter than the pulse-off time (few seconds). The metal ion concentration in the vicinity of the cathode, which is greatly decreased during pulse-on period, can be effectively recovered by ion migration during the relatively long off-time period. In the case of Ni deposition, the bath solution consists of nickel sulfate, nickel chloride, boric acid, and saccharin inhibitor (C7H4NO3S). The deposits generally have an equiaxed grain structure with fairly narrow grain size distribution. To achieve a significant grain refinement, Power supply + – Ni anode

Ni → e– + Ni+

Ni deposit

Electrolyte

Ni+ + e– → Ni

Cathode (substrate)

Figure 1.23 Schematic illustration of Ni electrodeposition.

19

20

Defect Structure and Properties of Nanomaterials

the pulse-off times must be longer than pulse-on times and a grain refiner such as saccharin is needed to retard the grain growth of Ni deposits. In the absence of saccharin, large crystals in the micrometer range are obtained. However, sulfur and carbon impurities content tends to increase with increasing saccharin content in the bath until saturation occurs. These impurities originate from the chemicals and inhibitor used for the bath [34] and may degradate the mechanical properties of electrodeposits. The electrodeposits are also found to exhibit a texture structure, depending on the bath chemistry. For instance, the preferred orientation of Ni deposits progressively changed from a strong (200) fiber texture for a saccharin-free bath to a (111) texture for a bath containing saccharin [35].

1.5 NANOCRYSTALLIZATION OF BULK AMORPHOUS ALLOYS When a conventional metal or alloy cools from the liquid melt, equilibrium is reached when it solidifies into the lowest energy state structure, i.e., a crystalline lattice. It was discovered that if a molten metal having several components is undercooled uniformly and rapidly enough, the different atoms have no enough time at the high temperature regime to rearrange for crystal nucleation [36]. The liquid reaches the glass transition temperature, Tg, and solidifies as an amorphous material (metallic glass). Under the glass transition temperature the atomic transport slows down very much that is also indicated by the large increase of viscosity. At room temperature, the amorphous state remains for very long time due to the low atomic mobility since crystallization needs the development of heterogeneous distribution of elements by diffusion. The alloys processable into metallic glasses must have three features: (1) multicomponent systems, (2) significant atomic size ratios above 12%, and (3) negative heats of mixing [37]. Different rapid cooling processing methods of metallic glasses have been elaborated, e.g., melt spinning or copper mold casting. In melt spinning a wheel is cooled internally, usually by water or liquid nitrogen, and rotated as depicted in Fig. 1.24. A thin stream of a molten alloy is Gas pressure Quartz tube Molten alloy

Heating coil Ribbon of metallic glass Cold wheel

Figure 1.24 Illustration of melt spinning.

Processing Methods of Nanomaterials

then dripped onto the wheel and cooled, causing rapid solidification. The metallic glasses produced by melt spinning have ribbon shape with small thickness (several tens of micrometers); therefore their practical applications are limited. This technique is used for alloys that require extremely high cooling rates to form, such as metallic glasses. The cooling rates achievable by melt spinning are in the order of 104e107 K/s. In the case of the alloys that can be solidified into amorphous state even at low cooling rates (1e10 K/s), copper mold casting technique can be applied for processing metallic glasses with the dimensions of several centimeters. Fig. 1.25 shows schematically the copper mold casting facility. The melt is poured into a water-cooled copper mold under inert atmosphere. The fast flow of the melt into the mold is assisted by a vacuum pump. The melt is stopped in the cooled mold by a copper mesh. The diameter and the length of the as-processed glassy rods are about 10 and 100 mm, respectively. It is noted that bulk amorphous alloys are also produced by hot pressing and warm extrusion of atomized amorphous powders at temperatures fall into the supercooled liquid region. Careful heat treatment of bulk metallic glasses (BMGs) results in their nanocrystallization. For multicomponent alloys, crystallization usually occurs in two to three successive steps when the time increases at a fixed temperature or the temperature increases at a constant rate during annealing. These steps manifest as distinct exothermic peaks on the thermograms detected by differential scanning calorimetry. In the first step, the metallic glass only partially crystallizes and often nanosized, metastable quasicrystalline grains are formed in the amorphous matrix with the volume fraction of about 20e50% (see, e.g., Ref. [38]). It is noted that the redistribution of elements by diffusion is a prerequisite of partial crystallization or quasicrystallization; therefore the distribution of nuclei is rather homogeneous. It was suggested [39,40] that the ability of metals to form amorphous structure by fast cooling is enhanced by the dominance of icosahedral short-range order (ISRO) in melts that is incompatible with translational periodicity of crystallographic structures. An icosahedral packing of 20 slightly distorted tetrahedra is denser than either face-centered Molten alloy To vacuum pump Quartz tube

Copper mold Copper mesh Cooling liquid

Figure 1.25 Schematic depiction of copper mold casting.

21

22

Defect Structure and Properties of Nanomaterials

cubic or hexagonal close-packed structures; therefore although it is incompatible with translational periodicity, it might be a natural choice for liquid and amorphous structures [41,42]. The existence of ISRO in the supercooled liquid state brings about an extremely small interfacial free energy between an icosahedral quasicrystals phase (i-phase) and a metallic glass of the same composition [43]. Consequently, the nucleation of the i-phase during annealing of BMGs is easier than the formation of the more stable crystalline phases. In support of this, the i-phase is frequently reported as the primary devitrification phase, particularly for Zr- and Hf-based bulk metallic glasses [44e46]. It was found that the chemical composition of metallic glasses has a deterministic influence on the local atomic order in the glassy state and therefore on the crystallization sequence during annealing [47]. Comparing Zr70Ni30 and Zr70Cu30 metallic glasses, the former shows a tetragonal atomic order while the latter exhibits an icosahedral local atomic configuration. As a consequence, in Zr70Cu30 as a first step quasicrystalline i-phase forms while Zr70Ni30 crystallizes into tetragonal Zr2Ni phase during heat treatment. It is suggested that structural differences in the glassy phase is caused by a strong chemical affinity of a ZreNi pair compared with that of a ZreCu pair. The sensitivity of local atomic order to the chemical composition was demonstrated by adding 1 at.% Pd into a Zr70Al10Ni20 metallic glass [47]. Without the Pd addition, tetragonal Zr2Ni was observed in the initial stage of transformation; however, the primary crystallization process changes markedly into single i-phase formation by the addition of 1 at.% Pd. Addition elements, such as Ag, Pd, Au, Pt, Ti, or even oxygen, to Zr-based bulk metallic glasses are believed to generate inhomogeneous atomic configuration regions including ISRO configurations in the supercooled liquid [47e51], which then promote the precipitation of an icosahedral phase. In the subsequent crystallization steps occurring with increasing the time and/or temperature of annealing, stable crystalline phases form from both the quasicrystalline phase and the remaining amorphous matrix. This phase transformation usually occurs at the interfaces of quasicrystalline and amorphous phases (peritectic-type phase transformation) [51]. As an alternative to the annealing of amorphous precursor samples, SPD by cold rolling [52] or HPT [53e55] can also induce nanocrystals at room temperature. As distinct from the uniformly dispersed nanocrystals or nanoquasicrystals formed during annealing, the crystallites induced by cold rolling are mostly localized in shear bands formed during deformation of metallic glasses [52]. The heterogeneous nucleation of nanocrystals can be attributed to the increase of the free volume at shear bands that facilitates diffusion necessary for crystallization in these alloys. Additionally, the temperature rises locally at shear bands [56], which also increases atomic mobility. A rather homogeneous dispersion of nanocrystals at an extremely high number density is observed after HPT. For instance, in melt-spun Al88Y7Fe5 metallic glass processed by five revolutions of HPT, the crystalline particle volume density is 1022 m3 that is larger than the values of 4  1021 or 1021 m3 obtained after annealing at 245  C for 30 min or by cold rolling to an equivalent strain of 12, respectively [53]. The maximum size of deformation-induced nanocrystals after cold rolling [52]

Processing Methods of Nanomaterials

and HPT straining [53] does not exceed 15e18 nm, although the amount of strain varied from about 12 for cold rolling to about 300 for HPT [53]. The weak dependence of the nanocrystal size on the strain at higher strain levels is explained by a dislocation-mediated fragmentation of the deformation-induced nanocrystals exceeding the critical size range [52]. Annealing of the HPT-processed samples yields full nanocrystallization with a grain size of w100 nm [53]. In this combined procedure, the HPT treatment gives a very high dispersity of crystalline nuclei; therefore the final grain size is smaller and the size distribution is narrower than that obtained by thermal annealing alone [53].

REFERENCES [1] R.Z. Valiev, T.G. Langdon, Principles of equal channel angular pressing as a processing tool for grain refinement, Progress in Materials Science 51 (2006) 881e981. [2] A.P. Zhilyaev, T.G. Langdon, Using high-pressure torsion for metal processing: fundamentals and applications, Progress in Materials Science 53 (2008) 893e979. [3] V.M. Segal, Materials processing by simple shear, Materials Science and Engineering A 197 (1995) 157e164. [4] R.Z. Valiev, R.K. Islamgaliev, I.V. Alexandrov, Bulk nanostructured materials from severe plastic deformation, Progress in Materials Science 45 (2000) 103e189. [5] R.Z. Valiev, Y. Estrin, Z. Horita, T.G. Langdon, M. Zehetbauer, Y.T. Zhu, Producing bulk ultrafine-grained materials by severe plastic deformation, Journal of the Minerals, Metals and Materials Society 58 (4) (2006) 33e39. [6] Y. Iwahashi, J. Wang, Z. Horita, M. Nemoto, T.G. Langdon, Principle of equal-channel angular pressing for the processing of ultra-fine grained materials, Scripta Materialia 35 (1996) 143e146. [7] K. Nakashima, Z. Horita, M. Nemoto, T.G. Langdon, Development of a multi-pass facility for equalchannel angular pressing to high total strains, Materials Science and Engineering A 281 (2000) 82e87. [8] T. Watanabe, H. Fujii, H. Oikawa, K.I. Arai, Grain boundaries in rapidly solidified and annealed Fe-6.5mass% Si polycrystalline ribbons with high ductility, Acta Metallurgica 37 (1989) 941e952. [9] G.A. Salischev, S.V. Zherebtsov, R.M. Galeyev, Evolution of microstructure and mechanical behavior of titanium during warm multiple deformation, in: Y.T. Zhu, T.G. Langdon, R.S. Mishra, S.L. Semiatin, M.J. Saran, T.C. Lowe (Eds.), Ultrafine Grained Materials II, TMS (The Minerals, Metals and Materials Society), 2002, pp. 123e131. [10] Y. Nakao, H. Miura, Nano-grain evolution in austenitic stainless steel during multi-directional forging, Materials Science and Engineering A 528 (2011) 1310e1317. [11] Y. Beygelzimer, D. Orlov, V. Varyukhin, A new severe plastic deformation method: twist extrusion, in: Y.T. Zhu, T.G. Langdon, R.S. Mishra, S.L. Semiatin, M.J. Saran, T.C. Lowe (Eds.), Ultrafine Grained Materials II, TMS (The Minerals, Metals and Materials Society), 2002, pp. 297e304. [12] R.A. Varin, T. Czujko, Z.S. Wronski, Nanomaterials for Solid State Hydrogen Storage, Springer ScienceþBusiness Media, New York, 2009. [13] C.G. Granqvist, R.A. Buhrman, Ultrafine metal particles, Journal of Applied Physics 47 (1976) 2200e2219. [14] S.C. Tjong, H. Chen, Nanocrystalline materials and coatings, Materials Science and Engineering R 45 (2004) 1e88. [15] H. Gleiter, Nanocrystalline materials, Progress in Materials Science 33 (1989) 223e315. [16] Y. Champion, J. Bigot, Synthesis and structural analysis of aluminum nanocrystalline powders, Nanostructured Materials 10 (1998) 1097e1110. [17] Y.A. Kotov, Electric explosion of wires as a method for preparation of nanopowders, Journal of Nanoparticle Research 5 (2003) 539e550. [18] W. Kim, J.-S. Park, C.-Y. Suh, H. Chang, J.-C. Lee, Fabrication of alloy nanopowders by the electrical explosion of electrodeposited wires, Materials Letters 61 (2007) 4259e4261.

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Defect Structure and Properties of Nanomaterials

[19] A. Alqudami, S. Annapoorni, Govind, S.M. Shivaprasad, AgeAu alloy nanoparticles prepared by electro-exploding wire technique, Journal of Nanoparticle Research 10 (2008) 1027e1036. [20] A. Hassanjani-Roshana, M. Reza Vaezi, A. Shokuhfar, Z. Rajabali, Synthesis of iron oxide nanoparticles via sonochemical method and their characterization, Particuology 9 (2011) 95e99. [21] G.Q. Zhang, H.P. Wu, M.Y. Ge, Q.K. Jiang, L.Y. Chen, J.M. Yao, Ultrasonic-assisted preparation of monodisperse iron oxide nanoparticles, Materials Letters 61 (2007) 2204e2207. [22] A. Khataee, S. Saadi, M. Safarpour, S. Woo Joo, Sonocatalytic performance of Er-doped ZnO for degradation of a textile dye, Ultrasonics Sonochemistry 27 (2015) 379e388. [23] A. Khataee, A. Karimi, S. Arefi-Oskoui, R. Darvishi Cheshmeh Soltani, Y. Hanifehpour, B. Soltani, S. Woo Joo, Sonochemical synthesis of Pr-doped ZnO nanoparticles for sonocatalytic degradation of Acid Red 17, Ultrasonics Sonochemistry 22 (2015) 371e381. [24] A. Khataee, S. Arefi-Oskoui, A. Karimi, M. Fathinia, Y. Hanifehpour, S. Woo Joo, Sonocatalysis of a sulfa drug using neodymium-doped lead selenide nanoparticles, Ultrasonics Sonochemistry 27 (2015) 345e358. [25] A. Khataee, A. Khataee, M. Fathinia, Y. Hanifehpour, S. Woo Joo, Kinetics and mechanism of enhanced photocatalytic activity under visible light using synthesized PrxCd1xSe nanoparticles, Industrial and Engineering Chemistry Research 52 (2013) 13357e13369. [26] Y. Hanifehpour, B. Soltani, A. Reza Amani-Ghadim, B. Hedayati, B. Khomami, S. Woo Joo, Praseodymium-doped ZnS nanomaterials: Hydrothermal synthesis and characterization with enhanced visible light photocatalytic activity, Journal of Industrial and Engineering Chemistry 34 (2016) 41e50. [27] H.-H. Ou, S.-L. Lo, Review of titania nanotubes synthesized via the hydrothermal treatment: fabrication, modification, and application, Separation and Purification Technology 58 (2007) 179e191. [28] V. Viswanathan, T. Laha, K. Balani, A. Agarwal, S. Seal, Challenges and advances in nanocomposite processing techniques, Materials Science and Engineering R 54 (2006) 121e285. [29] S. Ando, Y. Mine, K. Takashima, S. Itoh, H. Tonda, Explosive compaction of Nd-Fe-B powder, Journal of Materials Processing Technology 85 (1999) 142e147. [30] H.V. Atkinson, S. Davies, Fundamental aspects of hot isostatic pressing: an overview, Metallurgical and Materials Transactions 31 (2000) 2981e3000. [31] S. Billard, J.P. Fondere, B. Bacroix, G.F. Dirras, Macroscopic and microscopic aspects of the deformation and fracture mechanisms of ultrafine-grained aluminum processed by hot isostatic pressing, Acta Materialia 54 (2006) 411e421. [32] H.C. Kim, I.J. Shon, J.E. Garay, Z.A. Munir, Consolidation and properties of binderless sub-micron tungsten carbide by field-activated sintering, International Journal of Refractory Metals and Hard Materials 22 (2004) 257e264. [33] G.D. Zhan, J. Kuntz, J. Wan, J. Garay, A.K. Mukherjee, A novel processing route to develop a dense nanocrystalline alumina matrix ( x, and

) ( ½lnðx=mÞ2 1 f ðxÞ ¼ pffiffiffiffiffiffi exp
. 2s2 2psx

(2.5)

where m and s are the median and the square root of the log-normal variance of the crystallite size distribution, respectively. Substituting Eqs. (2.4) and (2.5) into Eq. (2.3) and then Eq. (2.3) into Eq. (2.2), the size Fourier transform is written as [8,9]:     pffiffiffi 1 lnðjLj=mÞ 3 lnðjLj=mÞ pffiffiffi pffiffiffi pffiffiffi A ðLÞ ¼ erfc
1:5 2 s

2s jLjerfc 2 4m expð8:125s2 Þ 2s 2s   3 lnðjLj=mÞ 3 p ffiffi ffi ; þ 3 jLj erfc 4m expð10:125s2 Þ 2s (2.6) S

where erfc is the complementary error function given as: 2 erfcðxÞ ¼ pffiffiffiffi p

Z

N

  exp
t 2 dt.

(2.7)

x

Thus, both the size Fourier transform, AS(L) and the size peak profile, which can be obtained as the inverse Fourier transform of AS(L), depend only on two parameters (m and s) of the size distribution of spherical crystallites. It is emphasized that the peak broadening of the size profile with hkl indices is determined by the size of the reflecting crystallites measured parallel to the hkl diffraction vector (i.e., normal to the (hkl) lattice planes). The smaller the crystallite size the broader the diffraction peak.

Characterization Methods of Lattice Defects

The Fourier transform of strain or distortion peak profile can be approximated by the following formula [10]:  D E AD ðg; LÞ ¼ exp
2p2 L 2 g2 ε2g;L ; (2.8) 2 sin Q where Q is the Bragg-angle l D E X-rays) and ε2g;L is the mean-square

where g is the length of the hkl diffraction vector (g ¼ of the diffraction peak and l is the wavelength of

strain normal to (hkl) lattice planes that depends on both g and L. For understanding the D E meaning of ε2g;L , first let us imagine a distortion-free crystal lattice where the section between points A and B is lying normal to lattice planes (hkl) and has a length of L (see Fig. 2.3). Due to lattice distortions caused by defects (e.g., dislocations) in a real crystal, the positions of points A and B are shifted to A0 and B0 . The projection of section A0 B0 normal to lattice planes (hkl) is L0 . The ratio (L0
L)/L gives the local strain normal to planes (hkl). Averaging the square of this quantity for the reflecting volume yields the D E value of ε2g;L . It is emphasized that the strain broadening of peak profile hkl is determined by the lattice distortions normal to lattice planes (hkl) in the reflecting crystallites. The larger the lattice distortion the broader the diffraction peak. The lattice distortions are mainly caused by the strain fields of lattice defects. The strain field of a point defect decreases with r
3, where r is the distance from the defect, therefore it decays very quickly with increasing r. At the same time, the strain field of an individual dislocation is of long-range character as it decays with r
1. Due to the reciprocity between crystal and reciprocal spaces, the scattering related to point defects (referred to as Huang scattering) is extended far from the fundamental Bragg reflections Perfect crystal

B' B g

L' L A' A

hkl planes

Figure 2.3 Characterization of lattice distortions parallel to the diffraction vector g [or normal to lattice planes (hkl)].

33

34

Defect Structure and Properties of Nanomaterials

and therefore not involved in the evaluation of line profiles. Due to the long-range character of strain field of dislocations, these defects yield considerable broadening of line profiles. For dislocations, the mean-square strain can be given as:

D E b2 2 * L εg;L ¼ ; (2.9) rCf Re 4p where r and b are the dislocation density and the magnitude of Burgers vector, respectively, C is the contrast factor of dislocations, and Re is the effective outer   cut-off radius of dislocations with length dimension. The function f * RLe is given in Ref. [11]. The parameter Re reflects that the strain field of a dislocation structure depends not only on the dislocation density but also on the arrangement of dislocations. If the strain fields of the individual dislocations are screening each other, the total distortion in the lattice is lower that yields a smaller value of Re. Strong screening of strain fields of dislocations occurs, e.g., when the dislocations are arranged into dipoles or low-angle grain boundaries (LAGBs) as depicted in Fig. 2.4. Instead of Re, the screening of strain fields of dislocations is rather described by the dimensionless dislocation arrangement parameter M that can be calculated as: M ¼ Re r1=2 .

(2.10)

The stronger the screening of the strain fields of dislocations, the smaller the value of M and longer the tails in the diffraction profiles. Due to the anisotropic strain field of a dislocation, the mean-square strain is different in the various crystallographic directions for given values of r and Re that yields a strong dependence of peak broadening on hkl indices of reflections. This effect is referred to as strain anisotropy [12,13] and taken into account by the contrast factor C in Eq. (2.9). For an individual dislocation, the value of C depends on the orientation of the Burgers and line vectors of dislocation, and the diffraction vector. Due to the latter effect, factor C usually depends on the indices of reflection and therefore it is denoted as Chkl. In a texture-free polycrystal or if the different slip systems are equally populated by Dislocation dipole

M is large

M is small

Low angle grain boundary

M is small

Figure 2.4 Schematic of some arrangements of dislocations yielding weak or strong screening of the strain fields that correspond to a larger or smaller value of the dislocation arrangement parameter M, respectively.

Characterization Methods of Lattice Defects

dislocations, the factors Chkl can be averaged over the permutations of indices hkl [14] and the average contrast factor, C hkl , should be inserted into Eq. (2.9). For materials with cubic crystal structure, the average contrast factor is given as [14]:   C hkl ¼ C h00 1
qH 2 ; (2.11) where C h00 is the average contrast factor for reflection h00 and H2 ¼

h2 k2 þ h2 l 2 þ k2 l 2 ðh2 þ k2 þ l 2 Þ2

;

(2.12)

where hkl are the indices of reflection. The values of parameters C h00 and q depend on the anisotropic elastic constants of the crystal and the type of prevailing dislocations (e.g., edge or screw). The contrast factors can be determined by software Anizc that can be reached on the Website http://metal.elte.hu/anizc [15]. Assuming 12 h110if111g and 12 h111if110g slip systems for face-centered cubic (fcc) Al and Cu and body-centered cubic Fe, respectively, the values of C h00 and q for edge and screw dislocations are listed in Table 2.2. In hexagonal close-packed (hcp) crystals the average contrast factor, C hkl can be given as [16]:   C hkl ¼ C hk0 1 þ q1 z þ q2 z2 ; (2.13) where q1 and q2 are two parameters depending on the anisotropic elastic constants of the crystal and the type of slip system. z ¼ (2/3)(l/ga)2, where a is the lattice constant in the basal plane. C hk0 is the average dislocation contrast factor of hk0 type reflections. The values of C hk0 , q1, and q2 are listed for the 11 most common slip systems in a large number of hexagonal materials in Ref. [16]. The peak profile caused by planar defects depends on the type of defect (intrinsic or extrinsic stacking fault or twin boundary) and their frequencies. The stacking or twin fault probability is defined as the fraction of faulted planes and usually given in percentage. For instance, in the case of fcc crystals the habit planes of stacking and twin faults are planes {111}. Therefore, the stacking or twin fault probability is the relative frequency of faulted planes among planes {111}. The twin fault probability is denoted as b. In the evaluation of twin faults from peak profiles, it is usually assumed that the appearance of a fault in a lattice plane is stochastic with the probability of b, i.e., it does not depend on the occurrence of twin boundaries in the neighboring (111) planes (referred to as Table 2.2 The values of C h00 and q for edge and screw dislocations in Al, Cu, and Fe C h00 (edge/screw) q (edge/screw)

Al Cu Fe

0.2011/0.1839 0.3076/0.3029 0.2548/0.3013

0.36/1.33 1.67/2.33 1.28/2.67

35

Defect Structure and Properties of Nanomaterials

random twinning). Hereafter, the crystal between two adjacent twin boundaries will be referred to as twin lamella. A twin lamella with the thickness corresponding to n lattice planes is formed if n
1 unfaulted planes are followed by a twin fault. If we consider that n can be expressed as the ratio of the lamella thickness (dtwin) and the spacing of (111) planes (d111), the probability of the formation of a lamella with the thickness dtwin, W(dtwin), can be obtained as [17]. dtwin

W ðdtwin Þ ¼ ð1
bÞ d111


1

b.

(2.14)

Therefore, in this model the variation of twin-fault spacing can be described by a geometric distribution density function. Detailed description of peak profiles corresponding to planar faults in fcc and hcp crystals is given in Refs. [17] and [18], respectively. The nature of line broadening caused by crystallite size, dislocations, and planar faults are essentially different that enables the separation of their contributions and therefore the determination of the parameters characterizing the different lattice defects. This is illustrated in Fig. 2.5 where the full width at half maximum (FWHM) values for different (b) 0.05

0.006

111 200

FWHM (1/nm)

(a) 0.008 FWHM (1/nm)

220 222 311 400

0.004 0.002

Cu, crystallite size = 166 nm Dislocation density = 17 x 1014 m–2

0.04 0.03 0.02 0.01

Cu, crystallite size = 166 nm 0.000 0

2

4

6 8 g (1/nm)

10

(c) 0.20 FWHM (1/nm)

36

12

0.00

14

111 0

2

311 400 200 220 222

4

6 8 g (1/nm)

10

12

14

SiC, crystallite size = 13 nm Twin boundary frequency = 7%

0.15

200 111

0.10

400 222 220 311

0.05 0.00 0

2

4

6 8 g (1/nm)

10

12

Figure 2.5 Full width at half maximum of diffraction peaks as a function of magnitude of diffraction vector (g) for the cases where the broadening is caused only by (a) crystallite size in Cu, (b) both size and dislocations in Cu, and (c) both size and twin faults in SiC. The datum points corresponding to harmonic reflection pairs are connected by dotted lines.

Characterization Methods of Lattice Defects

diffraction peaks are plotted as a function of the magnitude of the diffraction vector, g (WilliamsoneHall plot) for the cases where the broadening is caused only by crystallite size (Fig. 2.5a), both size and dislocations (Fig. 2.5b) and both size and twin faults (Fig. 2.5c). Fig. 2.5a shows that for defect-free spherical crystallites, FWHM is independent of the indices of reflections. When dislocations also contribute to broadening, the breadth of profiles depends on the indices of reflections and harmonic reflection pairs (e.g., 111 and 222) have different FWHM values (see Fig. 2.5b). Twin faults yield same breadths for harmonic reflections, although the widths of the peaks reflected from planes with various crystallographic orientations (e.g., for 111 and 200 reflections) are different as illustrated in Fig. 2.5c. Reliable characterization of defect structure can be achieved by fitting the whole X-ray diffraction profiles by theoretical functions calculated on the basis of a model of microstructure. Such powder pattern fitting methods have been elaborated recently [9,19,20]. The results presented in this book are obtained mostly by the convolutional multiple whole profile (CMWP) fitting method [20]. In this procedure, the diffraction pattern is fitted by the sum of a background spline and the convolution of the instrumental pattern and the theoretical line profiles related to the crystallite size, dislocations, and stacking or twin faults. In the calculation of the theoretical size profile, the crystallites are modeled by spheres with log-normal size distribution. Fig. 2.6 shows a diffraction pattern for a severely deformed UFG Au sample fitted by the CMWP method. The details of the CMWP procedure are available in Refs. [17,20]. This method gives the following parameters of the microstructure: 1. m, the median of the crystallite size distribution; 2. s, the square root of the log-normal variance of the crystallite size distribution; 3. r, the dislocation density; 50

20

311

15

Intensity

111

Intensity

40 30

10

222

5 0 75

20 10

220

311

79 2Θ (degree)

222

200

400 0 30

60

90

83

331 420 120

2Θ (degree)

Figure 2.6 The convolutional multiple whole profile evaluation of the X-ray diffraction pattern for ultrafine-grained Au processed by severe plastic deformation. The open circles and the solid line represent the measured and the fitted X-ray diffraction patterns, respectively. A magnified part of the pattern is presented in the inset. The difference between the measured and the fitted patterns is also shown at the bottom of the figure.

37

38

Defect Structure and Properties of Nanomaterials

4. Re, the outer cutoff radius of dislocations. As it was mentioned, instead of Re, often the dimensionless parameter M is used for the characterization of arrangement of dislocations 5. q, or q1 and q2 parameters of the dislocation contrast factors for cubic or hexagonal crystals, respectively. These parameters can be used for the determination of the prevailing dislocation slip systems; and 6. b, the twin (or stacking) fault probability. From the two parameters, m and s, of log-normal size distribution function, f(x), three different mean crystallite sizes can be determined. The arithmetic average of crystallite diameters (hxiarit) is obtained as: Z N   x$f ðxÞdx ¼ m$exp 0:5s2 ; (2.15) hxiarit ¼ 0

where x denotes the diameter of crystallites. The mean size of crystallites weighted by their surface areas (referred to as area-weighted mean crystallite size, hxiarea) is given by the following formula: RN   x$x2 p$f ðxÞdx ¼ m$exp 2:5s2 . (2.16) hxiarea ¼ 0R N 2 0 x p$f ðxÞdx The mean size of crystallites weighted by their volumes (referred to as volumeweighted mean crystallite size, hxivol) can be obtained as: RN

hxivol

x3 p x$ $f ðxÞdx   6 ¼ m$exp 3:5s2 . ¼ 3 RN x p $f ðxÞdx 0 6 0

(2.17)

As an example, Fig. 2.7 shows a log-normal crystallite size distribution function for m ¼ 14 nm and s ¼ 0.45 where the three different mean crystallite size values are also indicated. It is noted that in this book the crystallite size is usually characterized by hxiarea. XLPA enables the determination of the prevailing dislocation slip systems in hexagonal crystals. Parameters q1 and q2 of dislocation contrast factors obtained by this method depend on the type of dislocations [9]. The 11 possible dislocation slip systems are depicted in Fig. 2.8. The notations of the slip systems used in Fig. 2.8 are listed in Table 2.3. The 11 dislocation slip systems can be classified into three groups based on 1 2110 (< a > type), b ¼ h0001i (< c > type), and b ¼ their Burgers vectors: b ¼ 1 2 3 3  1 2113 (< c þ a > type). There are four, two, and five slip systems in the < a >, 3 < c >, and < c þ a > Burgers vector groups, respectively. The theoretical q1 and q2 values for the 11 possible slip systems in hexagonal crystals can be calculated according to the formulas presented in Refs. [15,21] and by using the software Anizc on the Website http://metal.elte.hu/anizc. The theoretical values of q1 and q2 for some hexagonal materials

Characterization Methods of Lattice Defects

0.08

m = 14 nm σ = 0.45

f(x)

0.06

vol = 28 nm

0.04 area = 23 nm 0.02

arit = 15 nm

0.00 0

10

20 30 Crystallite size (nm)

40

50

Figure 2.7 Log-normal crystallite size distribution density function, f(x), and the arithmetic (hxiarit), the area- (hxiarea), and the volume-weighted (hxivol) mean crystallite sizes obtained from m and s. BE

PrE

PyE

S1 + Screw

:

Pr2E :

S3 + Screw

Pr3E

Py2E

Py3E

Py4E

:

S2 + Screw

Figure 2.8 The 11 possible dislocation slip systems in materials with hexagonal crystal structure. The arrows indicate the three different Burgers vector types: < a >, < c >, and < c þ a >. The slip planes are denoted by gray color. The Burgers vectors and the slip planes are listed in Table 2.3. ðmÞ

ðmÞ

are listed in Ref. [16]. Comparing the measured values of q1 and q2 with the theoretical values calculated for the 11 slip systems, the prevailing slip systems in the studied material can be obtained. A computer program was elaborated to facilitate the determination of Burgers vector population from dislocation contrast factors [22]. First, the software selects some slip systems from one of the three Burgers vector groups. In the second step, the theoretical C hk0 q1 and C hk0 q2 values of the selected slip systems are averaged with equal weights. This procedure is carried out for each Burgers vector group. The relative fractions of the three groups, hi (i ¼ 1,2,3), are calculated by solving the following three equations: ðmÞ

q1

¼

3   1X hi b2i C hk0 q1 i; P i¼1

(2.18)

39

40

Defect Structure and Properties of Nanomaterials

Table 2.3 The notations, the Burgers vectors, and the slip planes of the 11 hexagonal slip systems [16] Slip plane and edge/screw Burgers Slip Burgers vector Notation character vector plane type

BE PrE

Basal edge Prismatic edge

PyE

Pyramidal edge

S1

Screw

Pr2E

Prismatic edge

S3

Screw

Pr3E

Prismatic edge

Py2E

Pyramidal edge

Py3E

Pyramidal edge

Py4E

Pyramidal edge

S2

Screw





1 2110 3 1 2110 3  1 2110 3  1 2110 3

h0001i h0001i  1 2113 3  1 2113 3  1 2113 3  1 2113 3  1 2113 3

ðmÞ q2

3   1X ¼ hi b2i C hk0 q2 i; P i¼1

{0001}

 0110

 1011 e 

0110 e

 0110

 2112

 1121

 1011 e









(2.19)

and 3 X

hi ¼ 1;

(2.20)

i¼1

    where C hk0 q1 i and C hk0 q2 i are obtained by averaging for the ith Burgers vector   P group and P is given as P ¼ 3i ¼ 1 hi b2i C hk0 i. If the three hi weights have positive values, the program stores them as one of the possible solutions. After examining all the possible combinations of the slip systems, ranges of the three weights are obtained as the final solution. A detailed description of fundamentals, methodology, and applications of XLPA is given in Ref. [23].

2.3 ELECTRON BACKSCATTER DIFFRACTION EBSD is generally used for the determination of grain boundary misorientation distribution. The grain orientation in a reference system attached to the sample is determined from the Kikuchi bands scattered from the crystal planes. Then, the misorientation between the neighboring grains is calculated by suitable software. As an example, the grain boundary misorientation distribution for an UFG Al-1%Mg alloy determined by

Characterization Methods of Lattice Defects

Number fraction (%)

20 Al-1% Mg HPT: RT, 6.0 GPa, 1 rpm N = 10 turns

15

High-angle grain boundaries: 80% 10

5

0 0

10

20 30 40 50 Misorientation angle (deg)

60

70

Figure 2.9 Grain boundary misorientation distribution for an ultrafine-grained Al-1%Mg alloy processed by high pressure torsion at room temperature under the pressure of 6 GPa at a rate of 1 rpm for 10 turns [24]. The columns and the solid curve represent the experimental result and the statistical prediction for a set of random orientations, respectively. (Reproduced from O. Andreau, J. Gubicza, N.X. Zhang, Y. Huang, P. Jenei, T.G. Langdon, Effect of short-term annealing on the microstructures and flow properties of an Ale1% Mg alloy processed by high-pressure torsion, Materials Science and Engineering, A 615 (2014) 231e239 with permission from Elsevier.)

EBSD is shown in Fig. 2.9. The sample was processed by high pressure torsion (HPT) at room temperature under the pressure of 6 GPa at a rate of 1 rpm for 10 turns [24]. The columns and the solid curve represent the experimental result and the statistical prediction for a set of random orientations, respectively. According to the measured misorientations, the grain boundaries with misorientations higher than 15 degrees are usually referred to as high-angle grain boundaries (HAGBs) while the rest is called as LAGBs. The reason of this limit is that except the special coincidence site lattice (CSL) boundaries the interface energy only slightly changes with the misorientation angle above 15 degrees. Below this value the grain boundary energy increases monotonously with increasing the angle of misorientation. According to the ReadeShockley equation the grain boundary energy for LAGBs varies with the misorientation angle as:

q q E ¼ Emax 1
ln (2.21) qmax qmax where Emax is the maximum grain boundary energy, which is achieved at qmax z 15 degrees. The value of Emax is w0.5 J/m2. As an example, Fig. 2.9 shows that the fraction of HAGBs is about 80% in UFG Al-1%Mg processed by HPT for 10 turns. Among HAGBs there are specific CSL boundaries with much lower energy than the general HAGB energy of w0.5 J/m2. In the misorientation distribution obtained by EBSD, peaks at the angles of CSL boundaries are often observed. It should be noted that the misorientation solely is not enough for a full characterization

41

42

Defect Structure and Properties of Nanomaterials

of grain boundaries. The energy and other features of boundaries (e.g., the corrosion resistance) depend not only on the misorientation between the neighboring grains but also on the boundary plane orientation; however, the latter information cannot be extracted from the EBSD data. EBSD is also capable to determine the density of geometrically necessary dislocations from the misorientations detected between the neighboring pixels [25]. A software application was written for evaluation of dislocation density from misorientations determined by EBSD. This program is based on the calculation proposed in [26,27]. Since this evaluation procedure has been developed recently, its fundamentals will be presented briefly in the following paragraphs. In the EBSD experiments the pixels in the images are arranged into a square point grid. The coordinate system attached to the sample and the arrangement of pixels are shown schematically in Fig. 2.10. The steps between the pixels in directions x1 and x2 are denoted by Dx1 and Dx2, respectively. Let us estimate the dislocation density in the gray pixel in Fig. 2.10 from its misorientations with the two neighbors in directions x1 and x2. The EBSD measurement gives the Euler angles for each pixel. The orientation of a pixel is described by the Bunge orientation tensor that can be determined from the Euler angles according to Ref. [28]. The Bunge tensors for the gray pixel and its b G b 1 , and G b 2 , respectively. Then, the neighbors in directions x1 and x2 are denoted as G, b ) for a pixel and one of its neighbors is obtained from the Bunge misorientation matrix ( M orientation tensors of the two pixels and the symmetry matrices of the crystal ( b S ) [25]. For instance, in direction x1 the misorientation tensor is given as: 
1  bb b 1b b ¼ G S1 ¼ bg bg
1 (2.22) M S G 1 ; b 1b bb where bg ¼ G S and bg 1 ¼ G S 1 . In the case of a cubic crystal there are 24 symmetry matrices, yielding 576 misorientation matrices for two neighboring pixels. From these

x3 Plane of EBSD image

x2

Δx2

Δq2 Δq1 Δx1 x1

Figure 2.10 Schematic of the coordinate system attached to the sample and the pixels arranged in a square grid on electron backscatter diffraction image. The misorientations between the studied gray pixel and the two neighboring white pixels in x1 and x2 directions are characterized by quaternions Dq1 and Dq2, respectively.

Characterization Methods of Lattice Defects

matrices only that one is selected in the analysis for which the misorientation angle is minimal. With this condition, b S and b S 1 are selected. Then, the orientation of the three pixels in Fig. 2.10 are described by tensors bg , bg 1 , and bg 2 or their quaternionic representation ! q, ! q 1 , and ! q 2 , respectively [26]. Quaternions are mathematical objects consisting of one scalar and a three dimensional vector part. The misorientations between the gray pixel and its neighbors in directions x1 and x2, ! ! respectively, are expressed with the quaternions Dq 1 and Dq 2 as:  
1  
1 ! ! Dq 1 ¼ ! q ! q1 and Dq 2 ¼ ! q ! q2 . (2.23) The components of these two quaternions are denoted as ! ! Dq 1 ¼ ðDq01 ; Dq11 ; Dq21 ; Dq31 Þ and Dq 2 ¼ ðDq02 ; Dq12 ; Dq22 ; Dq32 Þ. These quaternions can be used for the calculation of the curvature tensor defined as [26]: kkl ¼

Dqk ; Dxl

(2.24)

where Dqk ¼ Dq$rk . In the latter formula Dq is the disorientation angle and rk is the kth coordinate of the rotation axis. The component kkl of the curvature tensor can be expressed with the quaternion components as [27]: 2 arccosðDq0k Þ 1 kkl ¼ Dqkl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 1
Dq0k 2 Dxl

(2.25)

As local lattice orientations are determined only in the plane of EBSD image (along directions x1 and x2) but not perpendicular to it (along x3), only those components kkl of the curvature tensor can be obtained for which l ¼ 1 or 2. From these six available curvature tensor components, five components of the Nye’s dislocation density tensor (akl) can be determined: a12 ¼ k21 ; a13 ¼ k31 ; a21 ¼ k12 ; a23 ¼ k32 ; a33 ¼
k11
k22 .

(2.26)

Thus, Nye’s dislocation tensor is an incomplete matrix, and therefore only a local apparent dislocation density in the gray pixel (see Fig. 2.10) can be calculated from the available tensor components as: r ¼

1 ðja12 j þ ja13 j þ ja21 j þ ja23 j þ ja33 jÞ; b

(2.27)

where b is the magnitude of the Burgers vector. In the evaluation procedure for each pixel only its misorientations with the neighbors in directions x1 and x2 are used [25]. Therefore, only those pixels are evaluated that have neighbors in both directions x1 and x2 in the square grid. For calculating the total apparent dislocation density for the investigated area, the values obtained for the individual pixels are averaged. In addition, the dislocation density map in the studied area can be generated.

43

44

Defect Structure and Properties of Nanomaterials

The effectiveness of this procedure was illustrated on a low carbon low alloyed steel specimen [25]. The specimen was annealed at 1100 C for 15 min to achieve stress-free austenitic structure. Then, the sample was quenched in ice cold (0 C) water to obtain full lath martensite structure. The EBSD image obtained on this material is shown in Fig. 2.11a while the corresponding dislocation density map is presented in Fig. 2.11b. It is noted that there is an uncertainty in the determination of the Euler angles from the Kikuchi patterns obtained by EBSD that results in an error in the calculation of the dislocation density from the misorientations. To estimate the error in the misorientation angles, EBSD experiments were performed on an undeformed single crystalline (a)

2 μm

(b)

2 μm

Figure 2.11 Electron backscatter diffraction image (a) and the corresponding dislocation density map (b) obtained with the step size of 35 nm. In (b) the grain boundaries with misorientation angles larger than 5 degrees are indicated by thin blue lines, and the higher the dislocation density, the darker the
r
e, J. La
ba
r, J. Gubicza, P.J. Szabo
, Determination gray contrast. (Reproduced from T. Berecz, P. Jenei, A. Cso of dislocation density by electron backscatter diffraction and X-ray line profile analysis in ferrous lath martensite, Materials Characterization 113 (2016) 117e124 with permission from Elsevier.)

Characterization Methods of Lattice Defects

silicon wafer. For this material the difference between the crystallographic orientations of the neighboring EBSD image pixels was expected to be practically zero. At the same time, a considerable misorientation distribution was obtained for Si single crystal with the step size of 35 nm. The distribution had a maximum misorientation angle of w0.6 degrees and a median of w0.25 degrees. This indicates that a small but considerable misorientation can be observed even if the material is practically defect free. The same misorientation distributions were obtained for Si with other EBSD step sizes. Since this misorientation angle distribution practically ends at 0.5 degrees, this value was selected as a lower limit of the misorientations used for the calculation of the dislocation density in lath martensite. This means that if both misorientations between a pixel and its two neighbors in directions x1 and x2 were smaller than 0.5 degrees, the dislocation density was taken as zero in that pixel. If any of the two misorientations was higher than 5 degrees, the pixel was omitted from the dislocation density calculation [25]. For all other pixels the dislocation density was calculated from the misorientations, as described earlier. Fig. 2.12 shows the average dislocation density determined for the whole studied area as a function of the EBSD step size. The error was calculated from the uncertainty of the misorientation determination, which was estimated by the median of the misorientation distribution (0.25 degrees) obtained on single crystalline Si [25]. The error in the dislocation density was obtained by shifting the measured misorientation angles by 0.25 degrees, cutting the two ends of the distribution at 0.5 and 5 degrees, and then recalculating the dislocation density. Fig. 2.12 reveals that the smaller the pixel size, the higher the

Dislocation density (1014 m–2)

40 X-ray line profile analysis

35 30 25 20

EBSD

15 10 5 0 0

100

200 300 Step size (nm)

400

500

Figure 2.12 The dislocation density determined by electron backscatter diffraction as a function of the scan step size. The dislocation density obtained by X-ray line profile analysis is also indicated in
r
e, J. La
ba
r, J. Gubicza, P.J. Szabo
, Determinathe figure. (Reproduced from T. Berecz, P. Jenei, A. Cso tion of dislocation density by electron backscatter diffraction and X-ray line profile analysis in ferrous lath martensite, Materials Characterization 113 (2016) 117e124 with permission from Elsevier.)

45

46

Defect Structure and Properties of Nanomaterials

calculated dislocation density. This trend suggests that the microstructure is finer than the largest three step sizes used in EBSD (80, 200, and 500 nm). To reveal the subgrain structure, TEM was performed that indicated that the laths contain subgrains with the size varying between 50 and 100 nm. Therefore, in the EBSD experiments the smallest achievable step size (35 nm) provided the most reliable values of the geometrically necessary dislocation density, which was w30  1014 m
2 [25]. The total dislocation density in this lath martensite was determined by XLPA using the CMWP fitting procedure [25]. XLPA gave w35  1014 m
2 for the total dislocation density that agrees well with the value obtained from EBSD. It should be noted that the latter procedure yielded only the density of the geometrically necessary dislocations, while XLPA provides the total dislocation density (including both geometrically necessary and statistically stored dislocations). Therefore, it can be concluded that in this as-quenched lath martensite the majority of dislocations are geometrically necessary. It is also noted that the dislocation density obtained for the present material by XLPA is in good agreement with the value determined for another as-quenched lath martensite by other authors using X-ray diffraction broadening analysis [29]. It should be noted that other methods were also elaborated for the determination of density of geometrically necessary dislocations from EBSD experiments [26,30]. The rotation gradients obtained from the difference between the Kikuchi patterns of the neighboring image pixels are used for the calculation of dislocation density according to the Nye’s theory [26]. The effectiveness of this method is limited by the fact that the misorientation angle can be resolved only above a limit, which is usually between 0.5 and 1.5 degrees [26,31]. This is caused by the uncertainty of the band detection in Kikuchi patterns, which also results in an error in the obtained dislocation density values. A better angular resolution (0.03e0.1 degree) can be achieved by cross-correlationebased EBSD measurements [30,32,33]. The latter method has been successfully applied for the determination of GNDs in Cu [30] and Ti alloys [32,34]. If the method based on the Nye’s theory is applied for severely deformed materials with high dislocation density, the small required EBSD pixel size, and the relatively large uncertainty in the disorientation angle may yield a large error in the dislocation density [26]. The uncertainty of the misorientation angle has a strong effect on the error of the dislocation density. This uncertainty can be reduced by increasing the resolution of the EBSD detector as for a higher camera resolution the EBSD evaluation software can fit the theoretical Kikuchi lines to a higher number of experimental data points [25]. It should be noted, however, that the better resolution yields longer EBSD measuring time.

2.4 TRANSMISSION ELECTRON MICROSCOPY Lattice defects, such as dislocations and stacking faults, can be observed in TEM images due to the contrast caused by their strain fields. The fundamentals of strain contrast of

Characterization Methods of Lattice Defects

lattice defects in TEM can be found in numerous papers and textbooks (e.g., in Ref. [35,36]). Tilting the sample and using the extinction rules, not only the density but also the Burgers vector of dislocations can be determined from the TEM images. The details of this evaluation procedure are not presented here, since they are well known and have already published in many textbooks (e.g., Ref. [37]). It should be noted, however, that conventional TEM can determine a reliable average dislocation density only if its value is smaller than w1015 m
2 because above this limit the contrasts of individual dislocations strongly overlap. In nanocrystalline and UFG materials the subgrain and grain boundaries often comprise dislocations, especially in the samples processed by severe plastic deformation (SPD). In these boundaries the spacing between dislocations may be as low as 1e2 nm that corresponds to a local dislocation density of 1017e1018 m
2. In these materials the individual dislocations can be resolved and the dislocation density can be determined only by HRTEM. The theory and practice of HRTEM were summarized in Ref. [38]. When the dislocation density or twin fault probability obtained by HRTEM is interpreted or compared with the values determined by other methods, it must be kept in mind that these quantities characterize the defect structure locally, and they may deviate from the values averaged for a larger volume in the specimen. In the next paragraph the grain size, dislocation density, and twin fault probability obtained by TEM and XLPA methods are compared. The grain size obtained by TEM is usually larger with a factor of 2e10 than the crystallite size measured by XLPA for bulk UFG and nanocrystalline metals processed by SPD methods (e.g., by equal-channel angular pressing or HPT). This phenomenon can be attributed to the fact that the crystallites are equivalent to the domains in the microstructure that scatter X-rays coherently. As the coherency of X-rays breaks even if they are scattered from volumes having quite small misorientations (1e2 degrees), the crystallite size corresponds rather to the subgrain size in the severely deformed microstructures, as shown schematically in Fig. 2.13 [4]. At the same time, for UFG or nanocrystalline materials processed by condensation or deposition methods the crystallite size obtained by XLPA agrees well with the grain size determined by TEM due to the lack of subgrain structure. For milled nanopowders, the crystallite and grain sizes are in good agreement, although they are about two orders of magnitude smaller than the particle size. Only few experimental works can be found in the literature that compare the defect densities obtained locally by TEM and in a larger volume by XLPA. The dislocation density was determined by both TEM and XLPA in a refractory high entropy alloy (HEA) with equiatomic composition (Ti20Zr20Hf20Nb20Ta20) after compression up to the plastic strain of w10% [39]. Both methods gave similar value of w1015 m
2 for the dislocation density, although the volume studied in TEM was 107 times smaller than that investigated by XLPA. This agreement between the dislocation densities obtained by the two methods suggests a high degree of homogeneity of the dislocation structure in the studied HEA material.

47

48

Defect Structure and Properties of Nanomaterials High-angle grain boundaries Dislocation

Grain size from TEM

Low-angle grain boundaries

Dislocation wall

θ

Crystallite size from XLPA

Figure 2.13 Schematic showing the difference between the grain and crystallite sizes determined by transmission electron microscopy and X-ray line profile analysis, respectively, in nanocrystalline and ultrafine-grained metallic materials processed by severe plastic deformation.

To compare the planar fault probability determined by XLPA with the direct observation performed by TEM, the planar fault probability obtained by XLPA can be transformed into mean fault-spacing. For instance, in fcc crystals the mean twin-fault spacing (dtwin) can be obtained from the twin fault probability (b) as: dtwin ¼

100$d111 b

(2.28)

where d111 is the distance between the neighboring {111} planes. The mean twin-fault spacing values obtained by XLPA and TEM show good agreement within the experimental error as illustrated in Fig. 2.14. Not only the average value of twin-fault spacing but also its distribution was determined by TEM in nanocrystalline Ni films processed by electrodeposition [40]. It was found that the twin-fault spacing histograms obtained by TEM follow the distributions determined by XLPA, except for spacings smaller than 5 nm. In this region, the relative frequency obtained by TEM was much smaller than that determined by XLPA. This deviation can be attributed to nanotwins that have a considerable contribution to XLPA; however, they can be hardly observed by TEM. Indeed, HRTEM investigations proved the presence of nanotwins with the thickness varying between 2 and 6 nm. The grain boundary characterization in TEM can be improved by using Automated Crystal Orientation Mapping (ACOM-TEM). This technique yields an EBSD-like image in which the phase and misorientation analyses can be performed with a space resolution of 1 nm [41e43]. The method uses a precessed electron beam that increases the number of detected diffraction spots, thereby improving the precision of orientation and/or phase

Characterization Methods of Lattice Defects

Twin-spacing from X-ray line profiles (nm)

300 250

Ag

200 SiC

150 100

1

Ag 50 SiC

1

0 0

50 100 150 200 250 Twin-spacing from transmission electron microscope (nm)

300

Figure 2.14 The correlation between the mean twin fault spacing values determined by transmission electron microscopy and X-ray line profile analysis for Ag and SiC samples. (The data were taken from J.
ba
r, T.W. Zerda, T. Unga
r, Influence of sintering temperature and Gubicza, S. Nauyoks, L. Balogh, J. La pressure on crystallite size and lattice defect structure in nanocrystalline SiC, Journal of Materials Research,
ba
r, Z. Heged} 22 (2007) 1314e1321 and J. Gubicza, N.Q. Chinh, J.L. La us, T.G. Langdon, Principles of selfannealing in silver processed by equal-channel angular pressing: the significance of a very low stacking fault energy, Materials and Science Engineering, A 527 (2010) 752e760.)

identification. The electron beam scans the sample area of interest. The experimentally determined local diffraction patterns are compared with calculated data using crosscorrelation matching techniques [41e43]. The geometrically necessary dislocation density mapping can also be performed by ACOM-TEM technique using the Nye tensor with a much better resolution than in EBSD (see Section 2.3). The effectiveness of this method was demonstrated on a Cu foil with a dislocation density of w1016 m
2 [44]. The characteristic features of twin-faults can also be studied with a high precision using ACOMTEM method [45].

2.5 ELECTRICAL RESISTIVITY MEASUREMENT The electrical resistivity of a crystalline metallic material is due to scattering of conducting electrons by phonons and lattice defects such as vacancies, interstitials, dislocations, stacking faults, grain boundaries, and solute atoms. The resistivity of a chemically pure metal without lattice defects is referred to as intrinsic resistivity, denoted here as (ri). According to the Matthiessen’s rule, the electrical resistivity (r) is the sum of the intrinsic resistivity and the contributions of the different lattice defects (rl): X rl ; (2.29) rðT Þ ¼ ri ðT Þ þ l

where T is the temperature and index l stands for the different lattice defects (including impurities and alloying elements). The electrical resistivity caused by defects is

49

50

Defect Structure and Properties of Nanomaterials

temperature independent, which is also referred to as residual resistivity. The contributions of lattice defects increase with increasing their concentrations or densities. For vacancies, dislocations and stacking faults the resistivity contributions can be approximated as rv ¼ cv hv ; (2.30) rd ¼ rhd ;

(2.31)

rsf ¼ rsf hsf ;

(2.32)

respectively, where hv, hd, and hsf are the resistivity increments for unit vacancy concentration, dislocation density (given in m
2), and stacking fault density (rsf given in m
1). The quantity rsf gives the area of stacking faults in a unit volume of material. For parallel stacking faults the value of rsf is equivalent to the reciprocal of stacking fault spacing. For fcc materials rsf can be related to the stacking fault probability as: b ¼ 100$d111 rsf .

(2.33)

Table 2.4 shows the specific electrical resistivities hv, hd, and hsf of vacancies, dislocations, and stacking faults, respectively, in some pure metals. It is noted that the resistivity of interstitials in fcc metals is about three times larger than that for vacancies. The specific resistivity of twin faults was estimated as about one half of the resistivity of stacking faults. In addition, Eq. (2.33) also holds for twin faults. Fig. 2.15 compares the intrinsic resistivity and the resistivity caused by vacancies and dislocations in a strongly deformed high purity Cu. The intrinsic resistivity is plotted as a function of temperature between 20 and 300 K. It can be seen that the resistivity caused by the lattice defects is about two orders of magnitude smaller than the instrinsic resistivity even for very high defect densities, therefore the resistivity measurement is suggested to carry out at low temperatures. For instance, at liquid nitrogen temperature (77 K) the resistivity of lattice defects is lower only by about one order of magnitude than the intrinsic resistivity. It should also be noted that the resistivity values of vacancies, dislocations, and stacking faults (hv, hd, and hsf) are available in the literature only for the most common metallic materials that limits the application of this method for the determination of defect densities in specific materials. Table 2.4 Specific electrical resistivity for vacancies, dislocations, and stacking faults in some pure fcc metals Metal Vacancy, hv (U m) Dislocation, hd (U m3) Stacking fault, hsf (U m2)

Cu Au Ag Ni Al

1.9  10
6 2.4  10
6 1.4  10
6 2.9  10
6 2.2  10
6

The data were taken from Refs. [48e55].

1.3  10
25 2.6  10
25 1.9  10
25 2.8  10
25 1.5  10
25

3.4  10
17 1.8  10
17 e e e

Characterization Methods of Lattice Defects 10–7

99.999% purity Cu

Resistivity (Ωm)

10–8 Intrinsic resistivity 10–9 cv = 10–4

10 –10

ρ = 1015 m–2

10–11 0

50

100 150 200 Temperature (K)

250

300

Figure 2.15 The intrinsic resistivity in logarithmic scale as a function of temperature for 99.999% purity Cu. The resistivity for the vacancy concentration of cv ¼ 10
4 and dislocation density of r ¼ 1015 m
2 are also indicated in the figure. (The data were taken from R.A. Matula, Electrical resistivity of copper, gold, palladium and silver, Journal of Physical and Chemical Reference Data, 8 (1979) 1147e1298.)

2.6 POSITRON ANNIHILATION SPECTROSCOPY The method of PAS enables the determination of vacancy concentration as well as type (edge/screw character) and density of dislocations. PAS technique includes different evaluation methods such as positron lifetime spectroscopy, angular correlation of annihilation radiation, Doppler broadening spectroscopy, and ageemomentum correlation [56]. In the experimental setup the conventional positron source is a 22Na isotope. The radioactive decay of this isotope yields the emission of a positron and a photon with the energy of 1274 keV. The birth of positron is detected by this photon. The emitted positrons irradiate the studied material and after the decrease of their energy (referred to as thermalization) they annihilate via interaction with electrons in the lattice. The annihilation results in two photons with energy of 511 keV, which move nearly in opposite directions. The death of positron is detected capturing these photons. Thus, the positron lifetime can be determined as the difference between the birth and death times, which is usually between 100 and 400 ps for crystalline solids but the exact value depends on the studied material. As the positive atom cores in a metallic material repel the positive positrons, they prefer to stay at interstitial sites and open-volume defects such as vacancies, vacancy clusters, and dislocations. In a vacancy the Coulomb repulsive force is lower, therefore the positron has lower energy that results in an attractive force between the positron and the vacancy (the binding energy is about 1 eV). The lifetime of the positrons trapped at vacancies is larger by a factor of 1.6 than in the defect-free lattice due to the lower electron density (one atom with its electrons is missing from

51

Defect Structure and Properties of Nanomaterials

the lattice at a vacancy) [56]. Therefore, the investigation of lifetime spectrum of positrons can be used for the determination of vacancy concentration. In the dislocation cores positrons are also trapped with a longer lifetime than in the bulk lattice, therefore the dislocation density can also be obtained by positron lifetime spectroscopy. A schematic of positron lifetime spectrum is shown in Fig. 2.16 where the counts (N) is plotted as a function of time (t). Usually at least 105e107 positron annihilation events are collected in each lifetime spectrum. The analysis of the spectrum is carried out by a fitting to the decaying right side using the following function [57]:

X

k Ib t Ii t NðtÞ ¼ exp
þ (2.34) exp
; sb si sb s i¼1 i where Ib and sb, respectively, are the fraction and lifetime for positrons annihilated in the bulk (defect free) material, k is the number of defect types, and Ii and si are the positron fraction and lifetime, respectively, for the ith defect. The sum of fractions for bulk material and defects equals one. In practice, only a maximum of three terms in Eq. (2.34) are used in an evaluation procedure that does not apply any constraint on fitting parameters. Before fitting, the source contribution of the lifetime spectrum caused by positron annihilation in the source spot and the covering mylar foils is measured on a reference sample and subtracted from the spectrum. The trapping rate for the ith defect type (ki) can be determined as:

Ii 1 1 ki ¼ . (2.35)
Ib sb si 106

105 N, counts

52

104

103

0

1

2 3 t, time (ns)

4

5

Figure 2.16 Schematic of positron lifetime spectrum: the photon counts (N) versus time (t).

Characterization Methods of Lattice Defects

Then, the concentration of the ith defect type (ci) can be obtained as: ci ¼

ki ; mi

(2.36)

where mi is the specific trapping rate that must be determined by an independent method. For dislocations ci represents the dislocation density. For Cu the specific trapping rates of dislocations and monovacancies are w7  10
5 m2/s and w1.2  1014 s
1, respectively [58]. The specific trapping rate for a vacancy cluster can be approximated as the product of the number of vacancies and the trapping rate for a monovacancy. Typical lifetimes for a defect-free lattice and different defects in Al are listed in Table 2.5 [57]. The lifetime for a dislocation is only slightly lower than the value for a monovacancy. This can be explained by the trapping and annihilation processes of positrons at dislocations. Dislocation lines are shallow traps for positrons with binding energy of about 0.1 eV. Positrons trapped at dislocations diffuse rapidly along the dislocation lines to deeper traps, such as vacancies at dislocations, where they are annihilated [58]. Therefore, trapping at dislocations is only a precursor state for trapping and annihilation at deeper traps (e.g., at vacancies). As a vacancy at the compressed region of an edge dislocation is squeezed by compressive stresses, the lifetime of positrons trapped at dislocations is usually slightly smaller than that for positrons trapped at isolated monovacancies [59]. Since the stresses around edge and screw dislocation differ, the positron lifetimes for edge and screw dislocations are also different. For instance, in deformed Fe the lifetime of positrons trapped at edge and screw dislocations are 165 and 142 ps, respectively [60]. Hence, the fraction of screw dislocations (fscrew) can be estimated from the lifetimes determined experimentally for dislocations (sexp) and theoretically for edge (sedge) and screw (sscrew) dislocations [59]: fscrew ¼

sedge
sexp . sedge
sscrew

Table 2.5 Positron lifetimes for defect-free lattice and different lattice defects in Al [57] Positron lifetime (ps)

Bulk (defect-free lattice) Dislocation Monovacancy Divacancy Cluster of 5 vacancies Cluster of 10 vacancies Cluster of 20 vacancies Cluster of 50 vacancies

161 230 243 291 360 406 443 474

(2.37)

53

Defect Structure and Properties of Nanomaterials

It was found for HPT-processed interstitial-free Fe samples that both the dislocation density and the fraction of screw dislocations determined by PAS agree well with the values obtained by XLPA [59]. Fig. 2.17 shows that with increasing the number of vacancies in a cluster the positron lifetime increases; however, the rate of this increment reduces for larger clusters. Therefore, the size of vacancy clusters can be estimated from the positron lifetime. The detection limit of PAS method is reached when all positrons are trapped at defects due to the very high defect density. These limits for vacancies and dislocations are listed in Table 2.1. In addition to lifetime spectroscopy, the angle between the two quanta formed at positron annihilation is also analyzed (angular correlation of annihilation radiation) in PAS method. Furthermore, the energy spectra of these photons is broadened around 511 keV due to the momentum component of the annihilating electron in the direction of the photon emission (Doppler broadening spectroscopy). In the latter method, the observed line shapes are usually characterized with shape parameters S and W [56]. The former and latter parameters describe the relative contributions of the central and tail parts of the line to the total peak area. The central part is defined as 511  0.75 keV, while the tail parts correspond to the energy ranges 505e508 keV and 514e517 keV. Parameter S is higher if the relative contribution of lowermomentum valence electrons to positron annihilation is enhanced (i.e., the annihilating positron is relatively far from the atom cores) while parameter W becomes greater if the contribution of core electrons with higher momenta tend to increase. Therefore, the increase in fraction of positrons trapped at open-volume defects is reflected in the increase of the value of S-parameter [56].

500 Al 450 Positron lifetime (ps)

54

400 350 300 250 200 0

10 20 30 40 Number of vacancies in clusters

50

Figure 2.17 The positron lifetime as a function of number of vacancies in clusters for Al. (The data were taken from Table 2.5.)

Characterization Methods of Lattice Defects

REFERENCES [1] A. Leineweber, E.J. Mittemeijer, Anistropic microstrain broadening due to compositional inhomogeneities and its parametrisation, Zeitschrift f€ ur Kristallographie (Suppl. 23) (2006) 117e122. [2] M. Leoni, P. Scardi, Surface relaxation effects in nanocrystalline powders, in: E.J. Mittemeijer, P. Scardi (Eds.), Diffraction Analysis of the Microstructure of Materials, Springer Verlag, Berlin, 2004, pp. 413e454. [3] T. Ung
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arik, T. Ung
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ar, J. Gubicza, G. Rib
arik, A. Borb
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ar, M. Wilkens, Asymmetric X-ray line broadening of plastically deformed crystals I. Theory, Journal of Applied Crystallography 21 (1988) 47e53. [13] T. Ung
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arik, T. Ung
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ar, Contrast factors of dislocations in the hexagonal crystal system, Journal of Applied Crystallography 35 (2002) 556e564. [17] L. Balogh, G. Rib
arik, T. Ung
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arik, J. Gubicza, T. Ung
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athis, K. Nyilas, A. Axt, I.C. Dragomir, T. Ung
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[24] O. Andreau, J. Gubicza, N.X. Zhang, Y. Huang, P. Jenei, T.G. Langdon, Effect of short-term annealing on the microstructures and flow properties of an Ale1% Mg alloy processed by high-pressure torsion, Materials Science and Engineering, A 615 (2014) 231e239. [25] T. Berecz, P. Jenei, A. Cs
or
e, J. L
ab
ar, J. Gubicza, P.J. Szab
o, Determination of dislocation density by electron backscatter diffraction and X-ray line profile analysis in ferrous lath martensite, Materials Characterization 113 (2016) 117e124. [26] W. He, W. Ma, W. Pantleon, Microstructure of individual grains in cold-rolled aluminium from orientation inhomogeneities resolved by electron backscattering diffraction, Materials and Science Engineering, A 494 (2008) 21e27. [27] W. Pantleon, Resolving the geometrically necessary dislocation content by conventional electron backscattering diffraction, Scripta Materialia 58 (2008) 994e997. [28] H.-J. Bunge, Texture Analysis in Materials Science: Mathematical Methods, Butterworths, London, 1982. [29] F. HajyAkbary, J. Sietsma, A.J. B€ ottger, M.J. Santofimia, An improved X-ray diffraction analysis method to characterize dislocation density in lath martensitic structures, Materials and Science Engineering, A 639 (2015) 208e218. [30] J. Jiang, T.B. Britton, A.J. Wilkinson, Evolution of dislocation density distributions in copper during tensile deformation, Acta Materialia 61 (2013) 7227e7239. [31] B.L. Adams, J. Kacher, EBSD-based microscopy: resolution of dislocation density, CMC-Computers Materials & Continua 14 (3) (2009) 185e196. [32] P.D. Littlewood, T.B. Britton, A.J. Wilkinson, Geometrically necessary dislocation density distributions in Tie6Ale4V deformed in tension, Acta Materialia 59 (2011) 6489e6500. [33] C. Maurice, J.H. Driver, R. Fortunier, On solving the orientation gradient dependency of high angular resolution EBSD, Ultramicroscopy 113 (2012) 171e181. [34] P.D. Littlewood, A.J. Wilkinson, Geometrically necessary dislocation density distributions in cyclically deformed Tie6Ale4V, Acta Materialia 60 (2012) 5516e5525. [35] D.B. Williams, C. Barry Carter, Transmission Electron Microscopy a Textbook for Materials Science, Springer Science þ Business Media, New York, NY, USA, 2009. [36] A. Berghezan, A. Fourdeux, S. Amelinckx, Transmission electron microscopy studies of dislocations and stacking faults in a hexagonal metal: zinc, Acta Metallurgica 9 (1961) 464e490. [37] L. Reimer, Transmission Electron Microscopy: Physics of Image Formation and Microanalysis, fourth ed., Springer-Verlag, Heidelberg, Germany, 1997. [38] P. Buseck, J. Cowley, L. Eyring (Eds.), High-resolution Transmission Electron Microscopy: And Associated Techniques, Oxford University Press, New York, 1988. [39] G. Dirras, J. Gubicza, A. Heczel, L. Lilensten, J.-P. Couzini
e, L. Perriere, I. Guillot, A. Hocini, Microstructural investigation of plastically deformed Ti20Zr20Hf20Nb20Ta20 high entropy alloy by X-ray diffraction and transmission electron microscopy, Materials Characterization 108 (2015) 1e7. [40] T. Kolonits, P. Jenei, B.G. T
oth, Z. Czig
any, J. Gubicza, L. P
eter, I. Bakonyi, Characterization of defect structure in electrodeposited nanocrystalline Ni films, Journal of the Electrochemical Society 163 (2016) D107eD114. [41] E.F. Rauch, M. Veron, Coupled microstructural observations and local texture measurements with an automated crystallographic orientation mapping tool attached to a TEM, Materialwissenschaft und Werkstofftechnik 36 (2005) 552e556. [42] E.F. Rauch, J. Portillo, S. Nicolopoulos, D. Bultreys, S. Rouvimov, P. Moeck, Automated nanocrystal orientation and phase mapping in the transmission electron microscope on the basis of precession electron diffraction, Zeitschrift f€ ur Kristallography 225 (2010) 103e109. [43] J. Portillo, E.F. Rauch, S. Nicolopoulos, M. Gemmi, D. Bultreys, Precession electron diffraction assisted orientation mapping in the transmission electron microscope, Materials Science Forum 644 (2010) 1e7. [44] A.C. Leff, C.R. Weinberger, M.L. Taheri, Estimation of dislocation density from precession electron diffraction data using the Nye tensor, Ultramicroscopy 153 (2015) 9e21. [45] A.C. Leff, M.L. Taheri, Quantitative assessment of the driving force for twin formation utilizing Nye tensor dislocation density mapping, Scripta Materialia 121 (2016) 14e17. [46] J. Gubicza, S. Nauyoks, L. Balogh, J. L
ab
ar, T.W. Zerda, T. Ung
ar, Influence of sintering temperature and pressure on crystallite size and lattice defect structure in nanocrystalline SiC, Journal of Materials Research 22 (2007) 1314e1321.

Characterization Methods of Lattice Defects

[47] J. Gubicza, N.Q. Chinh, J.L. L
ab
ar, Z. Heged} us, T.G. Langdon, Principles of self-annealing in silver processed by equal-channel angular pressing: the significance of a very low stacking fault energy, Materials and Science Engineering, A 527 (2010) 752e760. [48] H.J. Wollenberger, in: R.W. Cahn, P. Haasen (Eds.), Physical Metallurgy, vol. 9, Elsevier, Amsterdam, 1983, pp. 1189e1221. [49] R.A. Brown, Journal of Physics F: Metal Physics 7 (1977) 1283e1295. [50] R.M.J. Cotterill, An experimental determination of the electrical resistivity of stacking faults and lattice vacancies in gold, Philosophical Magazine 6 (1961) 1351e1362. [51] D. Setman, E. Schafler, E. Korznikova, M.J. Zehetbauer, The presence and nature of vacancy type defects in nanometals detained by severe plastic deformation, Materials Science and Engineering, A 493 (2008) 116e122. [52] Y. Fukai, Electrical resistivity due to vacancies and impurities in aluminum: band structure effects in the defect scattering in polyvalent metals, Physics Letters 27A (1968) 416e417. [53] D.R. Smith, F.R. Fickett, Low-temperature properties of silver, Journal of Research of the National Institute of Standards and Technology 100 (1995) 119e171. [54] W. Sch€ ule, Properties of vacancies in copper determined by electrical resistivity techniques, Zeitschrift f€ ur Metallkunde 89 (1998) 672e677. [55] R.A. Matula, Electrical resistivity of copper, gold, palladium and silver, Journal of Physical and Chemical Reference Data 8 (1979) 1147e1298. [56] I. Proch
azka, Positron annihilation spectroscopy, Materials Structure 8 (2001) 55e60. [57] P. Hautojarvi (Ed.), Positrons in Solids, Springer-Verlag, Berlin, Heidelberg, New York, 1979. [58] J. Cizek, I. Proch
azka, M. Cieslar, R. Kuzel, J. Kuriplach, F. Chmelik, I. Stulikov
a, F. Becv
ar, O. Melikhova, Thermal stability of ultrafine grained copper, Physical Review B 65 (2002) 094106. [59] J. Cizek, M. Janecek, T. Krajn
ak, J. Str
ask
a, P. Hruska, J. Gubicza, H.S. Kim, Structural characterization of ultrafine-grained interstitial-free steel prepared by severe plastic deformation, Acta Materialia 105 (2016) 258e272. [60] Y.K. Park, J.T. Waber, M. Meshii, C.L. Snead, C.G. Park, Dislocation studies on deformed single crystals of high-purity iron using positron annihilation: determination of dislocation densities, Physical Review B 34 (1986) 823e836.

57

CHAPTER 3

Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation 3.1 EVOLUTION OF DISLOCATION STRUCTURE AND GRAIN SIZE DURING SEVERE PLASTIC DEFORMATION PROCESSING Severe plastic deformation (SPD) techniques are effective methods for producing bulk, porosity- and contamination-free ultrafine-grained (UFG), or nanostructured materials [1,2]. The high pressure applied during these procedures promotes the evolution of a high dislocation density in two ways (1) by impeding the vacancy migration that hinders the annihilation of dislocations and (2) by suppressing the cracking thereby keeping the integrity of the workpiece even at high strains. The most frequently used SPD procedure is equal channel angular pressing (ECAP) that enables the elaboration of bulk UFG or nanomaterials with dimensions of several centimeters in all directions that is favorable in practical applications [1]. One pass of ECAP corresponds to an equivalent strain value of about 1. The imposed strain increases proportionally by the increase of the number of passes. The evolution of the dislocation density and the crystallite size determined by X-ray line profile analysis as a function of number of ECAP passes is illustrated in Fig. 3.1 for 99.98% purity Cu [3,4], but the tendencies are similar for other metals. The dislocation density increases while the crystallite size decreases with increasing strain in 24

150

Crystallite size (nm)

16 12

90

8 Crystallite size Dislocation density

60 0

5

10 15 20 Number of ECAP passes

25

4

Dislocation density (1014 m–2)

20 120

Figure 3.1 The dislocation density and the crystallite size as a function of the number of equal channel angular pressing (ECAP) passes for 99.98% purity Cu. (Reprinted from J. Gubicza, N.Q.
ba
r, S. Dobatkin, Z. Chinh, J.L. La Heged} us, T.G. Langdon, Correlation between microstructure and mechanical properties of severely deformed metals, Journal of Alloys and Compounds 483 (2009) 271 e274 with permission from Elsevier.)

Defect Structure and Properties of Nanomaterials ISBN 978-0-08-101917-7, http://dx.doi.org/10.1016/B978-0-08-101917-7.00003-7

© 2017 Elsevier Ltd. All rights reserved.

59

60

Defect Structure and Properties of Nanomaterials

ECAP up to three passes in Fig. 3.1. The values obtained after three passes can be regarded as the saturation values achievable by ECAP in pure Cu at room temperature (RT). The saturation of the dislocation density is a consequence of the dynamic equilibrium of the formation and annihilation of dislocations. The maximum dislocation density is reached usually after two to four passes of ECAP for the majority of metallic materials [4e11]. The grain size determined by transmission electron microscopy (TEM) has similar evolution as for the crystallite size. The minimum grain size in pure Cu processed by ECAP at RT is about 200 nm [7,12]. It is noted that the crystallite size measured by X-ray line profile analysis is usually smaller than the grain size obtained by TEM. This phenomenon can be attributed to the fact that the crystallites are the domains in the microstructure which scatter X-rays coherently. As the coherency of X-rays breaks even if they are scattered from volumes having quite small misorientations (1
e2
), the crystallite size corresponds rather to the subgrain size in the severely deformed microstructures [13]. The reduction in the subgrain and grain sizes with increasing imposed strain is attributed to the increase of dislocation density as the grain refinement during SPD usually starts by the arrangement of dislocations into low-energy configurations such as low-angle grain boundaries (the misorientation is smaller than 3
per definition). The clustering of dislocations minimizes the energy stored in the dislocation structure as the strain field of dislocations is screened by other dislocations. The clustering of dislocations was observed directly in TEM images [14,15] and also detected by X-ray line profile analysis [16]. As an example Fig. 3.2 shows a subgrain with a size of about 250 nm in a (a)

(b)

Isolated subgrain

Figure 3.2 (a) A transmission electron microscopy micrograph of a subgrain in 99.99% purity Cu processed by repetitive corrugation and straightening. The inset in (a) is a high-resolution transmission electron microscopy (HRTEM) image showing that the subgrain boundaries are almost parallel to two sets of {111} planes. (b) A Fourier-filtered HRTEM image from the boundary as pointed out by a black arrowhead in (a). The white arrow in (b) points out the grain boundary (GB) orientation. The black and white circles mark interstitial and vacancy loops, respectively. (Reprinted from J.Y. Huang, Y.T. Zhu, H. Jiang, T.C. Lowe, Microstructures and dislocation configurations in nanostructured Cu processed by repetitive corrugation and straightening, Acta Materialia 49 (2001) 1497e1505 with permission from Elsevier.)

Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation

99.99% purity Cu sample processed by 14 cycles of repetitive corrugation and straightening (RCS) [14]. The black arrowhead at the left side of the subgrain in Fig. 3.2a points to a boundary segment whose Fourier-filtered high-resolution transmission electron microscopy (HRTEM) image is presented in Fig. 3.2b. The inset in Fig. 3.2a is an HRTEM image from the subgrain showing that the subgrain boundaries are almost parallel to two sets of {111} planes. The average distance between dislocations in the boundary is about 2 nm as estimated from Fig. 3.2b that yields a local dislocation density of about 3  1017 m2 (the average distance between dislocations can be approximated as the inverse square root of dislocation density). This local dislocation density value is about two orders of magnitude higher than the average dislocation density in the whole sample. In addition to dislocations, other lattice defects such as interstitial and vacancy loops are also observed in the boundary (see Fig. 3.2b). X-ray line profile analysis also enables the investigation of dislocation clustering as this method gives a dimensionless dislocation arrangement parameter, M, that is defined as the product of the outer cutoff radius of dislocations and the square root of the dislocation density. Practically, in this quantity the outer cutoff radius of dislocations is normalized by the average dislocation distance that can be expressed as the inverse of the square root of total dislocation density. The value of M usually decreases with increasing number of ECAP passes [16], which indicates a stronger screening of the strain field of dislocations due to their arrangement into low-energy configurations such as dipolar walls or lowangle grain boundaries. The lattice misorientation across a grain boundary is accommodated by geometrically necessary dislocations. SPD processing results in nonequilibrium grain boundaries that contain high density of extrinsic dislocations in addition to geometrically necessary dislocations. As an example, Fig. 3.3a shows a boundary in a 99.99% purity Cu sample processed by 14 cycles of RCS [14]. The grain boundary plane is curved and changes from the (5 5 12) plane to the (002) plane. The misorientation between the adjacent grains is about 9
as it is revealed by the corresponding electron diffraction pattern in Fig. 3.3b. HRTEM images from the upper-left and lower-right part of the boundary (see the framed areas in Fig. 3.3a) are shown in Fig. 3.3c and d, respectively. Fig. 3.3d is a structural model of the boundary segment in Fig. 3.3c. From this model, it can be seen that two types of dislocations marked by black and white symbols are needed to accommodate the geometrical misorientation. These six dislocations are referred to as geometrically necessary or intrinsic dislocations. However, there are three other dislocations in Fig. 3.3c which are extrinsic (or nongeometrically necessary) dislocations. Therefore, this segment of grain boundary is in nonequilibrium state. At the same time, the segment in Fig. 3.3e contains only geometrically necessary dislocations therefore that is an equilibrium grain boundary. Large density of extrinsic dislocations in grain boundaries was also observed for hexagonal metals processed by SPD [15].

61

62

Defect Structure and Properties of Nanomaterials

(a)

(c)

(b)

(d)

(e)

Figure 3.3 (a) A boundary in a 99.99% purity Cu sample processed by 14 cycles of repetitive corrugation and straightening. The grain boundary plane is curved and changes from the (5 5 12) plane to the (002) plane. (b) The corresponding electron diffraction pattern. (c) and (e) high-resolution transmission electron microscopy images from the upper-left and lower-right part of the boundary in (a) (see the framed areas). (d) A structural model of the boundary segment in (c). The two types of dislocations in the boundary are marked by black and white symbols. (Reprinted from J.Y. Huang, Y.T. Zhu, H. Jiang, T.C. Lowe, Microstructures and dislocation configurations in nanostructured Cu processed by repetitive corrugation and straightening, Acta Materialia 49 (2001) 1497e1505 with permission from Elsevier.)

Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation

With increasing strain in SPD processing, the dislocation density in the newly formed subgrain or grain boundaries increases, thereby enhancing the angle of misorientation that can be approximated as the ratio of the magnitude of Burgers vector and the spacing between geometrically necessary dislocations in low-angle grain boundaries. Electron backscatter diffraction experiments revealed that the fraction of high-angle grain boundaries (the misorientation is larger than 15
per definition) monotonously increased with increasing number of ECAP passes even after the achievement of the minimum grain size (about 200 nm for pure Cu) [12]. At very large equivalent strain values (ε z 10e30) of SPD processing, the fraction of high-angle grain boundaries in various metals (e.g., Cu, Al, Ni, steel) increased up to 70%e90% [12,17e19]. According to the model of grain refinement, nonequilibrium grain boundaries evolve from dislocation cell boundaries and/or low-angle grain boundaries by absorbing dislocations gliding in the lattice during SPD processing. The grain boundary misorientation increases with increasing dislocation density in grain boundary, similar to the ReadeShockley model of low-angle grain boundaries. At the same time, contrary to low-angle grain boundaries, dislocations in high-angle grain boundaries are stored in nonperiodic, disordered arrays, and the nonequilibrium nature of the boundary can be characterized by the excess free volume and/or energy. The values of these parameters increase with increasing the density of extrinsic dislocations. Fig. 3.1 shows that between the 3rd and 10th passes of ECAP, the dislocation density and the crystallite size for Cu remains unchanged within the experimental error. At the same time, during ECAP processing between the 10th and 15th passes the dislocation density decreases and after 25 passes even the crystallite size increases. These changes can be attributed to the structural relaxation of grain/subgrain boundaries. The equilibrium state can be approached by the annihilation of extrinsic dislocations. The decrease of the dislocation density after 15 passes can be explained by this annihilation of extrinsic dislocations. This structural recovery at large strain is accompanied by a decrease in the grain boundary thickness corresponding to an evolution from nonequilibrium boundaries to a more equilibrated structure and by an increase in the misorientation between neighboring grains. This is illustrated in Fig. 3.4 where the TEM images taken on the microstructures processed by the 5th and 25th passes are presented [3,12]. It is noted that the decrease of the dislocation density in SPD processing can be also observed for other materials (e.g., in Al [6] or Ag [10]) if the imposed strain is high enough and the applied hydrostatic component of the stress field does not suppress recovery processes. The grain refinement in metallic materials processed by SPD at high homologous temperatures (0.3e0.4  Tm or above where Tm is the absolute melting point) may occur by continuous dynamic recrystallization (CDRX) [20]. In CDRX process, new fine grains with HAGBs nucleate homogeneously during SPD as a consequence of the large stored energy. This primary recrystallization may contribute to grain refinement in Mg

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Figure 3.4 Transmission electron microscopy images of the microstructure of Cu processed by equal channel angular pressing for the 5th (a) and 25th (b) passes. (Reprinted from S.V. Dobatkin, J.A. Szpunar, A.P. Zhilyaev, J-Y. Cho, A.A. Kuznetsov, Effect of the route and strain of equal-channel angular pressing on structure and properties of oxygen-free copper, Materials Science and Engineering A 462 (2007) 132e138 with permission from Elsevier.)

and Al even at RT due to their low melting points [21]. A secondary recrystallization process should be avoided to maintain the fine grain structure. Beside the density and the arrangement of dislocations, X-ray diffraction line profile analysis also enables the determination of the type of the prevailing dislocations in nanomaterials [22]. In the case of cubic crystal systems, the edge/screw character of dislocations can be determined. It was found that for face-centered cubic (fcc) metals having high stacking fault energies (SFEs), e.g., for Al and its alloys, the character of dislocation structure is rather edge type after RT SPD processing [23]. Due to the high SFE, the dissociation of screw dislocations is marginal that gives a relatively easy annihilation by cross-slip compared to climb of edge dislocation segments. With decreasing SFE, the splitting distance between partials in dissociated dislocations increases that retards both cross-slip and climb [24,25]. As cross-slip is more sensitive to the degree of dislocation dissociation than climb [10], the decrease of SFE leads to a gradual enhancement of the screw character of dislocation structure. In the case of body-centered cubic (bcc) metals, e.g., interstitial-free (IF) steel processed by ECAP at RT, the character of dislocations is more screw [11], which can be explained by the reduced mobility of screw dislocations compared to edge dislocations in bcc structures. This difficulty in motion of screw dislocations is due to the fact that the ground state dislocation core is dissociated into a nonplanar configuration [26]. As a consequence, during ECAP processing the edge dislocation segments can annihilate more easily than the screw ones thereby the remaining dislocations have more screw character. In the case of hexagonal close-packed (hcp) materials, X-ray line profile analysis provides the relative fractions of dislocations having hai, hci, and hc þ ai Burgers vectors [27,28]. Generally, SPD processing of hexagonal materials is carried out at high homologous temperatures (at about 0.4  Tm) due to their rigidity. However, high-pressure torsion (HPT) processing can be carried out on hcp materials even at RT without an

Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation

early failure of the samples, as the high pressure applied in HPT suppresses crack propagation. There is an abundance of hai-type dislocations (usually with 60%e70%) in all hexagonal UFG or nanomaterials (e.g., sintered WC [29,30], SPD-processed Ti [15,31], and Mg alloys [32]) that can be explained by their smallest Burgers vector compared to other types of dislocations. The amount of hci-type dislocations is usually negligible; however, a significant fraction (30%e40%) of hc þ ai dislocations is detected. The relatively high fraction of hc þ ai dislocations in these materials can be attributed to the elevated temperature of SPD processing and/or the high stresses developed at grain boundaries which facilitates the activation of hc þ ai dislocations [33]. At RT, the critical resolved shear stress of pyramidal hc þ ai dislocations is about five times larger than that for basal slip [34], but this value decreases with increasing temperature. The evaluation of X-ray diffraction profiles also enables the determination of the population of the individual slip systems (e.g., basal, prismatic, or pyramidal systems) in UFG or nanocrystalline hcp materials [28]. As an example, Fig. 3.5 describes the (b)

Fractions of basal and non-basal edge dislocations (%)

10 Dislocation density in HPT-processed AZ31 disk center

8 6 4 2 0 0

1

5 2 3 4 Number of turns

(c)

Fraction of screw dislocations (%)

Dislocation density (1014 m–2)

(a)

14

15

70

-type edge dislocations in HPT-processed AZ31 disk center

60

Basal edge Nonbasal edge

50 40 30 20 10 0

5 2 3 4 Number of turns

1

14

15

60 Screw dislocations in HPT-processed AZ31 disk center

50 40 30 20 10 0 0

1

2

3

4

5

14

15

Number of turns

Figure 3.5 The variation of (a) dislocation density, (b) fractions of hai-type basal and nonbasal edge dislocations, and (c) screw dislocations in the center of an AZ31 disk as a function of high-pressure torsion (HPT) revolutions [35].

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Defect Structure and Properties of Nanomaterials

evolution of the dislocation structure as a function of HPT revolutions in the center of an AZ31 alloy (Mge3% Ale1% Zn) disk [35]. The HPT processing was carried on a wellannealed material at RT under the pressure of 2.5 GPa and the zero turn corresponds to a compression in the HPT device without torsional deformation. The parameters of the dislocation structure saturate even after 1/4 HPT turn, and significant changes were not observed with increasing deformation up to 15 revolutions. The maximum dislocation density is about 7  1014 m2. The analysis of the population of the different slip systems shows that the majority of dislocations have hai-type Burgers vector (60%e80%) for the compressed sample (zero turn) and also in the HPT-processed specimens. Fig. 3.5b shows that among hai-type edge slip systems the fraction of nonbasal dislocations (the sum of the prismatic and pyramidal edge dislocations) decreases from about 55% to 30%, while the population of basal edge dislocations slightly increases from 15% to 20%. A similar evolution of the dislocation fractions occurred in the periphery of the disks. This change in the dislocation population can be explained by the fact that in Mg, which has larger c/a lattice parameter ratio than 1.6, the slip occurs most easily in basal plane at low temperatures [36]. Fig. 3.5c reveals that the relative fraction of screw dislocations increases from w10% (only compressed state) to 30%e40% owing to HPT straining. It should be noted that the fraction of screw dislocations saturates already at 1/4 turn and significant difference between the values in the center and the periphery was not observed, in accordance with the evolution of other parameters of the dislocation structure. It is noted that the fraction of screw dislocations during ECAP processing was even larger (50%e70%) than in the case of HPT, since the high pressure in the latter procedure hinders the lattice diffusion, thereby impeding the annihilation of edge dislocations [37]. The grain refinement in hcp metals during SPD processing has different features than in fcc materials. For instance, there is an evidence in different Mg alloys (e.g., AZ31 and ZK60) that above a critical initial coarse grain size, the grain refinement starts with the development of an inhomogeneous grain structure [38e40]. The new grains form along the boundaries of the initial coarse grains in a necklace-like arrangement as depicted in Fig. 3.6. As a consequence, if the initial grain size is large enough, the fine grains at

Figure 3.6 Schematic depiction of the grain refinement in hexagonal close-packed metals along the preexisting grain boundaries when the initial grain size is larger than a critical value. SPD, severe plastic deformation.

Initial grain structure

After SPD

Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation

the grain boundaries and the internal volumes of the initial coarse grains give a bimodal grain structure in the beginning of SPD processing. With increasing strain, the grain refinement spreads into the initial grain interiors, thereby leading to a more homogeneous fine grain structure. In hcp metals and alloys, dislocation mechanism of grain refinement requires the activation of both nonbasal and basal slips [41]. As dislocations on the basal plane have the smallest hai-type Burgers vector, their activation is easy by the external stresses. The glide on nonbasal slip systems is more difficult, and the activation of these dislocations is facilitated by the high stresses and/or the elevated temperature. Therefore, the stress concentrations at the boundaries of the initial grains yield the activation of both basal and nonbasal slip processes leading to the formation of fine grains at the preexisting grain boundaries [42]. If the grain size is smaller than a critical value, the formation of fine grains at the initial grain boundaries does not cause a bimodal grain structure. This critical grain size is between 3 and 9 mm for AZ31 alloy processed by ECAP at RT [40]. In this alloy, the bimodal grain size distribution developed during the first pass disappears after six passes. However, the critical grain size depends not only on the selected alloy but also on the SPD conditions. For instance, the critical grain size becomes larger, when the temperature of ECAP increases as the higher temperature facilitates the activity of nonbasal slip systems.

3.2 COMPARISON OF DEFECT STRUCTURES FORMED BY DIFFERENT ROUTES OF BULK SEVERE PLASTIC DEFORMATION The effect of SPD processing method on grain size and the dislocation structure is illustrated on 99.98% purity Cu [43]. The TEM images of the microstructures of Cu samples produced by 20 cycles of multidirectional forging (MDF), 15 passes of twist extrusion (TE), 25 passes of ECAP, and 25 revolutions of HPT (at the half radius) are shown in Fig. 3.7. In the TEM images the majority of grains are equiaxed, only a few slightly elongated grains can be observed. The average grain size determined by TEM is 220  5 nm for all Cu samples except the HPT-processed material that has a smaller grain size of 160 nm. According to the literature data, these values are around the minimum grain size that can be achieved in bulk pure Cu by SPD at RT. The crystallite size, the dislocation density, and the twin boundary frequency were determined by X-ray line profile analysis. The dislocation density increases in the order of MDF (7  1014 m2), TE (10  1014 m2), ECAP (15  1014 m2), and HPT (37  1014 m2) [43]. The higher the dislocation density, the smaller the crystallite size for the four studied samples: 142, 107, 101, and 75 nm for MDF-, TE-, ECAP-, and HPT-processed samples, respectively [43]. The twin boundary frequency for all samples was relatively low (0.1  0.1%), close to the detection limit of this quantity for the applied experimental setup of X-ray line profile analysis. The extremely high dislocation density after HPT is in good agreement with the values obtained by other authors [44].

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Defect Structure and Properties of Nanomaterials

(a)

(b)

(c)

(d)

Figure 3.7 Transmission electron microscopy images showing the microstructure of Cu specimens immediately after 20 cycles of multidirectional forging (a), 15 passes of twist extrusion (b), 25 passes of equal channel angular pressing (c) and 25 revolutions of high-pressure torsion (d). (Reprinted from J.
ba
r, Microstructural stability of Cu processed by Gubicza, S.V. Dobatkin, E. Khosravi, A.A. Kuznetsov, J.L. La different routes of severe plastic deformation, Materials Science and Engineering A 528 (2011) 1828e1832 with permission from Elsevier.)

The very high dislocation density after HPT can be attributed to the high pressure (p ¼ 4 GPa) applied during HPT as it hinders annihilation of dislocations by retarding climb and cross-slip in the following ways: 1. The climb velocity is proportional to the diffusion coefficient [25] that depends on the concentration and the migration enthalpy of vacancies. Previous experiments have shown that in SPD-processed metals a large amount of vacancies are formed due to forced plasticity, therefore their concentration is much higher than the equilibrium value [8,45e50]. When a high pressure is applied in SPD processing, larger work (by pVVF where VVF is the vacancy formation volume) is needed for vacancy formation but this does not reduce the vacancy concentration as the necessary extra work is supplied by the external forces. Rather an increased vacancy concentration

Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation

was observed in previous studies [45e48] with increasing the pressure of HPT that can be explained by the suppressed vacancy annihilation at dislocations due to the slower migration of vacancies. The reduction in vacancy migration rate at high pressure is caused by the increase of the migration enthalpy by pVVM, where VVM is the vacancy migration volume. The increased difficulty of vacancy migration due to the high pressure applied during HPT hinders effectively the annihilation of dislocations by climb compared to other SPD processing. 2. In fcc metals the larger the splitting distance between partials in dissociated dislocations, the more difficult the climb or cross-slip of edge or screw segments of dislocations, respectively [24,25]. The values of equilibrium splitting distance between partials for edge and screw dislocations in Cu are 2.3 and 0.9 nm, respectively [10]. The very large loading on the anvils during HPT most probably results not only in a high hydrostatic pressure but also in high shear stresses since the idealized constrained conditions are usually not achieved. The shear stresses acting on the slip plane of a dislocation perpendicular to the Burgers vector can decrease or increase the splitting distance by pushing the partials toward each other or pulling the partials in opposite directions, respectively, depending on the stress orientation (see Fig. 4.2 in Chapter 4). Therefore, the higher shear stresses during HPT may result in much larger or smaller splitting distance depending on their directions. As the splitting distance without stresses is not very large in Cu due to the medium value of SFE (data between 45 and 78 mJ/m2 are reported in the literature [51,52]), the stresses pushing the partials toward each other do not accelerate climb and cross-slip very much. At the same time, the high stresses resulting in much larger splitting distance slow down climb and cross-slip effectively. As a result of the effects (1) and (2), both climb and cross-slip of dislocations are retarded due to the high load applied during HPT that explains the extremely large dislocation density measured immediately after HPT processing. This argumentation is also supported by the increase of dislocation density with increasing pressure applied during HPT (e.g., Refs. [45,50]). Fig. 3.8a and b compare the maximum dislocation density and the minimum grain size, respectively, achievable by ECAP and HPT at RT for 2N5 purity Al, Ale1% Mg alloy, IF steel, oxygen-free (99.98% purity) Cu, and 4N purity Ag. The data are taken from Refs. [6,10,11,43,53,54]. For Cu and Ag, the maximum dislocation density after HPT is much larger than after ECAP, while for the other three materials the saturation dislocation densities after ECAP and HPT are practically the same. Most probably, in the latter materials the dislocation density was also higher under the large pressure applied during HPT than in ECAP processing; however, when the pressure was released a large fraction of dislocations annihilated due to the accelerated diffusion, as shown in Ref. [44]. In the case of Cu and Ag, the relatively low SFE (compared to other materials in Fig. 3.8) retarded this annihilation process, as discussed in the previous paragraphs, hereby resulting in a higher dislocation density after HPT than that obtained in ECAP processing.

69

Defect Structure and Properties of Nanomaterials

Dislocation density (1014 m–2)

Figure 3.8 Comparison of dislocation (a) 180 densities (a) and grain sizes (b) in different 160 materials processed by equal channel 140 angular pressing (ECAP) or high-pressure torsion (HPT) [55]. IF, interstitial free. 120

ECAP HPT

100 80 60 40 20 0 u

Ag

C ee

st l

M

Al

%

5

-1

IF

Al

2N

g

(b)

1000 ECAP HPT

800 Grain size (nm)

70

600 400 200 0 u

Ag

C l

ee

st

M

Al

%

5

-1

IF

Al

2N

g

The low SFE also hindered the arrangement of excess dislocations into grain boundaries in Cu and Ag during HPT, therefore the minimum grain size values are similar after processing by the two different SPD methods. At the same time, in Al, AleMg alloy, and IF steel the increased dislocation density during HPT processing yielded a smaller grain size as a part of excess dislocations were arranged into grain boundaries. The grain boundary character depends on both the imposed strain and the route of SPD processing. The evolution of a reasonably equiaxed UFG microstructure with large fraction of high-angle grain boundaries is usually most rapid for route BC among the different routes of ECAP [56,57]; however, the difference in the grain boundary misorientation distribution produced by the various ECAP routes decreases with increasing number of passes [12]. At very high imposed strains (ε z 10e40), the high-angle grain boundary fraction is the largest in the materials processed by ECAP and HPT compared to other SPD methods [12,17].

Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation

3.3 MAXIMUM DISLOCATION DENSITY AND MINIMUM GRAIN SIZE ACHIEVABLE BY SEVERE PLASTIC DEFORMATION OF BULK METALLIC MATERIALS At high imposed strain in SPD processing, the grain size and the crystallite size (subgrain size) reach their minimum values and the dislocation density gets saturated. The saturation dislocation density, crystallite, and grain sizes for different SPD-processed metals are listed in Table 3.1. In the case of HPT-processed materials, the data measured at the half radius of the disks are selected as characteristic values. For Cue10% Zn and Cue30% Zn alloys [58], the grain size values at the half radius are not available, therefore the mean of the data measured in the center and the periphery is presented in Table 3.1. The minimum grain and crystallite size values as a function of the maximum dislocation density are plotted in Fig. 3.9. It can be concluded that, although there is no strict correlation between the grain or crystallite size and the dislocation density of the UFG metals processed by SPD procedures, the higher dislocation density is associated with smaller grain and crystallite sizes. This is a consequence of the fact that the basic mechanism of grain refinement in SPD materials is the arrangement of dislocations into grain and subgrain Table 3.1 The maximum dislocation density and the minimum crystallite size determined by X-ray line profile analysis, and the minimum grain size obtained by transmission electron microscopy for metallic materials processed by severe plastic deformation Grain Dislocation size Crystallite density Material Processing method (nm) size (nm) (1014 mL2) References Pure metals

Al Al Al Cu Cu Au Ni Ni Ag Ag Ti

ECAP at RT HPT at RT Cryogenic rolling at 77 K, reduction ¼ 0.75 ECAP at RT HPT at RT ECAP at RT 8 ECAP þ 5 HPT at RT HPT at RT ECAP at RT HPT at RT ECAP at 700 K

1200 800 200

272 e e

e e

1.8

[6] [59] [60]

200 160 460 140 e 200 200 265

70 75 72 48 e 60 50 40

21 37 17 25 34 46 154 30

[4,7] [43,48] [61] [62] [48] [10] [53] [15]

450 200 300

88 114 65

Solid solution alloys

Ale1% Mg Ale1% Mg Ale3% Mg

ECAP at RT HPT at RT ECAP at RT

3.9 3.8 23

[9] [63] [9] Continued

71

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Defect Structure and Properties of Nanomaterials

Table 3.1 The maximum dislocation density and the minimum crystallite size determined by X-ray line profile analysis, and the minimum grain size obtained by transmission electron microscopy for metallic materials processed by severe plastic deformationdcont'd Dislocation Grain density size Crystallite (1014 mL2) (nm) size (nm) Material Processing method References

Al-7075, solid solution Cue10% Zn Cue30% Zn Cue5% Al Cue8% Al IF steel IF steel 316L steel (Fe-Cr-NiMo-Mn)

HPT at RT HPT at RT HPT at RT ECAP at RT ECAP at RT ECAP at RT HPT at RT HPT at RT

26 145 85 107 82 360 275 45

e 60 34 e e 66 80 21

320 34 81 e e 10 8 133

[64] [58] [58] [65] [65] [11] [54] [66]

ECAP at RT ECAP at 473 K

300 500

76 165

5.4 3.2

[9] [67]

ECAP at 473 K

300

119

3.4

[67]

50

37

HPT at RT ECAP at 543 K

190 1200

170 97

4.2 2

[68,69] [32]

Extrusion at 623 K þ ECAP at 523 K Extrusion at 623 K þ ECAP at 493 K HPT at RT

1500

219

0.9

[70]

800

217

0.7

[71]

200

70

7

[35]

Extrusion at 623 K þ ECAP at 453 K HPT at RT ECAP at RT HPT at RT HPT at RT HPT at RT HPT at RT HPT at RT

500

120

1.7

[37]

280 250 209 143 40 155 108

100 73 58 52 36 50 47

20 39 38 75 163 64 68

[72] [73] [74] [74] [74] [75] [75]

Cryorolling at 77 K

150

30

48

[76]

Dispersion-strengthened alloys

Al-6082 Ale4.8% Zne1.2% Mge0.14% Zr Ale5.7% Zne1.9% Mge0.35% Cu Ale5.9% Mge0.3% Sce0.18% Zr Ale30% Zn Mge9% Ale1% Zn0.25Mn (AZ91) Mge4% Ale1% Ca (AX41) Mge4% Ale1% Ca (AX41) Mge3% Ale1% Zn (AZ31) Mge3% Ale1% Zn (AZ31) Cue0.18% Zr Cue0.18% Zr Cue0.7% Cr Cue9.85% Cr Cue27% Cr Cue0.9% Hf Cue0.7% Cre0.9% Hf Cu-3% Ag

HPT at RT

24

ECAP, equal channel angular pressing; HPT, high-pressure torsion; IF, interstitial free; RT, room temperature.

[13]

Grain size, crystallite size (nm)

Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation

Figure 3.9 The saturation grain and crystallite size values determined by transmission electron microscopy and X-ray line profile analysis, respectively, for severe plastic deformation-processed ultrafine-grained materials as a function of the saturation dislocation density. (The data are taken from Table 3.1 and plotted in double logarithmic scales as the values span about two orders of magnitude. The experimental error corresponds to the size of symbols.)

Grain size Crystallite size

1000

100

10 1

10

100

Dislocation density (1014 m–2)

boundaries. Table 3.1 also shows that the grain size determined by TEM is usually two to six times larger than the crystallite size obtained by X-ray line profile analysis. This is a consequence of the fact that the coherently scattering domains (crystallites) in SPDprocessed materials correspond to the subgrains and/or the dislocation cells inside the grains bounded by high-angle grain boundaries. The saturation dislocation density is determined by the dynamic equilibrium between the formation and annihilation of dislocations. The more difficult the dislocation annihilation during SPD processing, the higher the maximum dislocation density. The annihilation processes are hindered by (1) the low homologous temperature of SPD, (2) the solid solution alloying, (3) the second phase particles, and (4) the high degree of dislocation dissociation due to the low SFE. The homologous temperature of SPD can be kept at low values either by processing at low temperatures (e.g., cryogenic rolling at liquid nitrogen temperature, 77 K [60]) or by applying SPD on metals having high melting points. Fig. 3.10 illustrates that the higher the melting point (Tm) in pure fcc metals

Crystallite size (nm)

40 200 30 20 100 10 Al 0 900

Ag

Au

Cu

1200 1500 Melting point (K)

Ni

0 1800

Dislocation density (1014 m–2)

50

300

Figure 3.10 The saturation crystallite size and dislocation density as a function of melting point for different pure face-centered cubic metals processed by equal channel angular pressing at room temperature.

73

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Defect Structure and Properties of Nanomaterials

having medium or high SFE (Al, Ni, Cu, and Au), the larger the maximum dislocation density achievable by ECAP at RT. As the grain refinement occurs by arranging of dislocations into subgrain and grain boundaries, the higher saturation dislocation density is accompanied by smaller crystallite and grain sizes as shown in Fig. 3.10. The effect of the melting point on the saturation dislocation density can be explained by the thermally activated processes (cross-slip and climb) of dislocation annihilation. For instance, in fcc metals having medium or high SFE, the degree of dislocation dissociation is not very large, therefore cross-slip of screw dislocations occurs more easily than climb of edge segments during SPD processing. As a consequence, the maximum dislocation density at high strains is determined by the diffusion-controlled climb process. The diffusion coefficient is given by D ¼ D0 exp

Qself ; RT

(3.1)

where Qself is the activation energy of self-diffusion, R is the universal gas constant, and T is the absolute temperature. It is known that Qself for fcc metals is proportional to the melting point [77]. Qself ¼ B$Tm ;

(3.2)

where Bz155 J$K1 $mole1 . As the preexponential factor, D0, has similar values for Al, Cu, Au, and Ni, the diffusion coefficient depends primarily on the homologous temperature (T/Tm) due to Eqs. (3.1) and (3.2). At RT, the larger the melting point, the higher the activation energy and therefore the slower the climb process resulting in larger saturation dislocation density. It is noted that climb is usually considered as an important process only at high temperatures due to the slow diffusion at RT. However, UFG materials produced by SPD have a high volume fraction of grain boundaries and a large density of dislocations together with a large concentration of excess vacancies. The activation energy of diffusion along grain boundaries and dislocation cores is about half of that for self-diffusion; therefore the vacancy migration in nanomaterials is faster than in their coarse-grained counterparts by a factor of 1010 to 1020 [78]. Consequently, diffusion may control the annihilation of dislocations even at RT. The high internal stresses developed in SPDprocessed nanomaterials due to the large defect densities also assist climbing considerably. Fig. 3.10 shows that the saturation dislocation density for pure Ag is much higher than the values for other fcc metals having similar melting points (Au, Cu). The very large dislocation density in Ag can be explained by its extremely low SFE (16 mJ/m2 [51]) yielding a high degree of dislocation dissociation that hinders both cross-slip and climb thereby retarding dislocation annihilation. The evolution of the defect structure in low SFE materials will be presented in Chapter 4.

Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation

The temperature and strain rate of SPD processing have a significant effect on the defect structure and grain size in nanocrystalline or UFG metals and alloys. For a given strain and processing method, the higher the strain rate and/or lower the temperature of deformation, the larger the dislocation density and smaller the grain size. The combined effects of strain rate (_ε) and temperature can be expressed by the ZenereHollomon parameter defined as
 Q Z ¼ ε_ exp ; (3.3) RT where Q is the related activation energy for the dominant diffusion process (bulk or grain boundary diffusion). For a constant strain value, the following relationship between the grain size (d) and parameter Z is suggested: ln d ¼ A  B ln Z;

(3.4)

where A and B depend on the studied material. In the case of Cu, assuming a grain boundary diffusion activation energy of 73 kJ/mol, the value of ln Z is 22 for RT deformation at a strain rate of 103 s1 [79]. With increasing ln Z from 22 to 66 in uniaxial deformation of Cu, the grain size decreases from 320 to 66 nm, while the dislocation density increases from 14  1014 m2 to 30  1014 m2 at a fixed equivalent strain of 2 [79]. Deformation twinning occurs in Cu when ln Z exceeds 30, and the twin boundary frequency increases at higher Z. The grain sizes achieved in Cu at different strain rates and temperatures of SPD are close to each other for similar values of parameter Z [80]. Similarly to pure metals, in solid solution alloys the dislocation density, the crystallite, and grain sizes saturate after approximately four to eight passes of ECAP [5e9,81e84]. At the same time, alloying usually leads to an increase of the critical imposed strain that corresponds to saturation. For instance, in 99.99% purity Al and Ale3% Mg alloy processed by ECAP at RT the maximum value of the dislocation density is achieved after the first and fourth passes, respectively. Additionally, the maximum dislocation density increases while the minimum values of the crystallite and grain sizes decrease due to alloying. As an example, Fig. 3.11 shows the saturation values of the dislocation density and the grain size as a function of the Mg concentration in Al processed by ECAP at RT [9]. Both the larger strain needed for saturation and the higher value of the saturation dislocation density in alloys can be explained by the pinning effect of solute atoms on dislocations that hinders their annihilation. As grain refinement in SPD metals occurs usually by the rearrangement of dislocations into subgrain and grain boundaries, the higher dislocation density results in a decrease of grain size for higher Mg concentrations as shown in Fig. 3.11 and demonstrated in the TEM images of Fig. 3.12. In pure Al the saturation grain size is 1200 nm while 3 wt.% Mg alloying leads to a reduction to 300 nm. It is noted that although the minimum grain size in Ale3% Mg alloy is only 4 times smaller than in

75

Defect Structure and Properties of Nanomaterials 1500

20 900 600 10 300 0

Dislocation density (1014 m–2)

30 Grain size Dislocation density

1200 Grain size (nm)

76

0 0

1 2 Mg content (wt.%)

3

Figure 3.11 The minimum grain size and the maximum dislocation density as a function of Mg con
llics, I. Schiller, T. Unga
r, tent in Al solid solutions. (Reprinted from J. Gubicza, N.Q. Chinh, Gy. Kra Microstructure of ultrafine-grained fcc metals produced by severe plastic deformation, Current Applied Physics 6 (2006) 194e199 with permission from Elsevier.)

Figure 3.12 Transmission electron microscopy images taken on pure Al (a) and Ale3% Mg (b) processed by eight equal channel angular pressing passes at room temperature. (Reprinted from J. Gubicza, N.Q. Chinh, Z. Horita, T.G. Langdon, Effect of Mg addition on microstructure and mechanical properties of aluminum, Materials Science and Engineering A 387e389 (2004) 55e59 with permission from Elsevier.)

Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation

pure Al, the maximum dislocation density is approximately 13 times higher (see Table 3.1). The dragging effect of solute atoms on dislocations also hinders the evolution of low-angle grain boundaries into high-angle boundaries during SPD. For instance, although in both 99.99% purity Al and Ale1% Mg alloy the fraction of high-angle grain boundaries increases with increasing the number of ECAP passes, its maximum value in Ale1% Mg (w65%) is lower than that for pure Al (w75%) [85,86]. It is worthwhile to mention that the minimum grain size achievable by SPD is rather affected by the deformation conditions than the initial grain size before deformation. For instance, if RT HPT processing is performed on electrodeposited nanostructured Ni with the grain size of 30 nm, there is a grain growth to 129 nm [87], which is close to the grain size limit for HPT-treated coarse-grained Ni (140e170 nm [18]). Although the dislocation density increases from 120  1014 to 200  1014 m2 during HPT of electrodeposited Ni, this cannot compensate the softening effect of grain growth and the hardness decreases from 7.2 to 6.1 GPa. Similar grain growth was also observed during indentation of nanomaterials, while there is a lack of grain growth during tensile testing. This suggests that only stresses above a critical value induce grain growth in nanocrystalline materials. The large dislocation density in the electrodeposited Ni after HPT may be a consequence of the high impurity content that is usually a characteristic feature of electrodeposited materials.

3.4 EXCESS VACANCY CONCENTRATION DUE TO SEVERE PLASTIC DEFORMATION It is well known that during deformation in addition to dislocations, vacancies are also formed. The most important phenomenon resulting in vacancy production is the climb of edge dislocation segments. However, an edge dislocation may climb to different directions relative to its extra half plane; therefore it may act as either a source or a sink of vacancies. Fig. 3.13a and b shows schematic drawings of two edge dislocations climbing in

Figure 3.13 Schematic drawings of two edge dislocations climbing in opposite directions by increasing or decreasing the extension of the extra half plane that leads to a production or annihilation of vacancies, respectively. The direction of atomic migration is indicated by arrows. The produced and the annihilated vacancies are denoted by solid and dashed squares, respectively.

77

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Defect Structure and Properties of Nanomaterials

opposite directions by increasing or decreasing the extension of their extra half planes that leads to a production or annihilation of vacancies, respectively. The significant difference between the two climb processes is that the second one needs vacancy migration to the core of dislocation contrary to the first one. As a consequence, in the beginning of SPD processing at RT, due to the slow diffusion the climb-producing vacancy occurs more easily than the climb-sinking vacancy. Therefore the vacancy production rate is higher than the rate of annihilation resulting in an increase of excess vacancy concentration with increasing strain. At large strain values, the vacancy migration becomes faster due to the large amounts of dislocations and grain boundaries, thereby the formation and annihilation of vacancies equilibrate yielding to a saturation concentration of excess vacancies at the given conditions of SPD. The concentration of excess vacancies was investigated mainly in Cu and Ni processed by ECAP and HPT [8,48,78e83]. In HPT-deformed 99.95% purity Cu, positron annihilation spectroscopy (PAS) revealed the absence of monovacancies while a large amount of vacancy clusters were identified [88]. Most probably, the monovacancies created during HPT processing either disappear by diffusion to sinks such as grain boundaries or agglomerate into small clusters. The vacancy clusters at the center of the HPTprocessed Cu and Ni disks consist of four to five vacancies, while the clusters at the periphery consist of seven to nine vacancies [88]. Neither the increase of number of HPT revolutions nor the raise of pressure from 2 to 4 GPa results in a change of the size of vacancy clusters. The densities of dislocations and vacancies were so high in the HPT-processed Cu samples that every positron is very quickly trapped at dislocations or vacancy clusters (referred to as saturated positron trapping). As a consequence, the concentration of vacancy clusters could be only evaluated from the PAS signal with the help of the dislocation density determined by X-ray line profile analysis. The ratio, I2/I1, of the intensities of PAS signals corresponding to positrons trapped in vacancy clusters and at dislocations is directly proportional to the ratio of cluster concentration and dislocation density [88]: I2 nV c V ¼ ; I1 nD r

(3.5)

where nV, nD are the specific positron trapping rate to vacancy clusters and dislocations, respectively, cV is the vacancy cluster concentration, and r is the dislocation density. The values of nV, nD for Cu are 1.2  1014 at s1 and 0.6  104 m2 s1, respectively [88]. This methodology most probably overestimates the vacancy cluster concentration as the calculation uses the total dislocation density while the positrons are mainly trapped at edge dislocation segments. The concentration of vacancy clusters increases with increasing of number of HPT revolutions in Cu from about 1.0  104 after 1 turn to about 2.5  104 after 25 revolutions as it is shown in Fig. 3.14 [88]. Multiplying the cluster concentration with the numbers of vacancies in the clusters, the total

Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation

Figure 3.14 The vacancy cluster concentration in high-pressure torsion (HPT)processed Cu as a function of the number of revolutions. The individual datum points given in Ref. [87] are not shown here, but all of them follow this line irrespective of the location in the HPT disk and the applied pressure (2 or 4 GPa). The scatter of datum points is illustrated by the error bar.

Concentration of vacancy clusters (10–4 at.)

3

2

1

0 0

5

10 15 20 Number of HPT revolutions

25

vacancy concentration is determined for the center and the periphery of the disks as a function of the number of revolutions and plotted in Fig. 3.15 [88]. The periphery exhibits enhanced concentration of vacancies due to larger size of vacancy clusters. The concentration of vacancies at both the center and the periphery of the disks increase with increasing number of HPT revolutions due to the increase of the imposed strain. At the center, the vacancy concentration is 5  104 after one turn that increases to 12  104 after 25 revolutions. At the periphery, the corresponding concentrations are 12  104 and 20  104 after 1 and 25 revolutions, respectively. These values have the same order of magnitude as the concentrations observed in other Cu and Ni disks processed by HPT at RT [48]. The vacancy concentration and the cluster size determined for different SPD-processed samples are listed in Table 3.2.

Figure 3.15 The concentration of vacancies in the center and at the periphery of Cu disks processed by different revolutions of high-pressure torsion (HPT). The individual datum points given in Ref. [88] are not shown here, but all of them follow this line irrespective of the applied pressure (2 or 4 GPa). The scatter of datum points is illustrated by the error bar.

Vacancy concentration (10–4 at.)

25

20

Periphery

15

10 Center 5

0

0

5

10 15 20 Number of HPT revolutions

25

79

80

Cu Cu 99.95% Cu 99.95% Cu 99.95% Cu 99.95% Cu 99.99% Cu 99.998% Ni

1 ECAP at RT 4 ECAP at RT 1 HPT at RT, center of disk, p ¼ 2e4 GPa 1 HPT at RT, periphery of disk, p ¼ 2e4 GPa 25 HPT at RT, center of disk, p ¼ 2e4 GPa 25 HPT at RT, periphery of disk, p ¼ 2e4 GPa HPT at RT, p ¼ 4e8 GPa, shear strain ¼ 20 HPT at RT, p ¼ 2 GPa, shear strain ¼ 20

e e 4e5 7e9 4e5 7e9 e 1e2 þ clusters with unknown size

99.998% Ni

HPT at RT, p ¼ 8 GPa, shear strain ¼ 20

1e2 þ clusters with unknown size

Cue0.18% Zr

8 ECAP at RT

4

ECAP, equal channel angular pressing; HPT, high-pressure torsion; RT, room temperature.

1.5  104 3.5  104 5  104 12  104 12  104 20  104 4  104 1.1  104 (single/ double vacancies) 3.5  104 (in clusters) 0.9  104 (single/ double vacancies) 5  104 (in clusters) 5  105

References

[8] [8] [88] [88] [88] [88] [48] [48]

[48]

[73]

Defect Structure and Properties of Nanomaterials

Table 3.2 The concentration and the cluster size of vacancies obtained for metals processed by severe plastic deformation Material Processing method Cluster size Vacancy concentration

Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation

For ECAP-processed Cu, the vacancy concentration was determined by a combination of X-ray line profile analysis and residual electrical resistometry or differential scanning calorimetry (DSC) [8]. The decrease of electrical resistivity during annealing of the ECAP-processed samples is caused by the annihilation of lattice defects such as vacancies (vacancy clusters) and dislocations. The contribution of dislocations to resistivity can be estimated from the dislocation density determined by X-ray line profile analysis and the specific dislocation resistivity of Cu being 0.8  1025 Um3 [89]. Subtracting this contribution from the total decrease of resistivity, the vacancy concentration can determined as the ratio of the remaining resistivity and the resistivity per unit vacancy concentration (0.62  104 Ucm for Cu [90]). The vacancy concentrations obtained by this way are 1.5  104 and 3.5  104 after the first and fourth ECAP passes, respectively. These values are smaller than the concentrations determined in HPT-processed Cu samples. This can be explained by the larger imposed strain in HPT compared to ECAP. The vacancy concentration values determined from the released heat measured in DSC agree well with those obtained by resistometry [8]. In the DSC thermograms for ECAPprocessed Cu samples, one exothermic peak was observed. The released heat is assumed to be a sum of the contributions of vacancies and dislocations. The stored energy in a unit volume corresponding to dislocations having half edgedhalf screw character is determined as [8]: 1 Edisl ¼ 0:2Gb2 rln pffiffiffi ; b r

(3.6)

where G is the shear modulus (47 GPa for Cu), b is the magnitude of Burgers vector (0.25 nm for Cu), and r is the dislocation density. Subtracting Edisl from the released heat and dividing the remaining stored energy by the formation energy of a vacancy, 0.195  1018 J [90], the vacancy concentration can be determined. It is worth to note that the values of cv obtained for Cu after ECAP or HPT at RT has the same order of magnitude as the equilibrium vacancy concentration near the melting point. Taking 1.28 eV as the value of the vacancy formation energy (EF) and 12.2 as the preexponential factor (c0) [91], the Arrhenius-type formula of vacancy concentration cV ¼ c0 exp

EF ; RT

(3.7)

gives 4  1021 and 2  104 for the equilibrium vacancy concentrations at RT and the melting point, respectively. This means that the vacancy concentration after ECAP at RT is 17 orders of magnitude larger than its equilibrium value due to SPD. It is noted that vacancies have a considerable contribution to the diffuse background scattering under X-ray diffraction peaks; therefore the ratio of the integrated background and the integrated peak intensity can be used for the determination of vacancy concentration after a calibration procedure [92].

81

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Defect Structure and Properties of Nanomaterials

In the case of HPT-processed 99.99% and 99.998% purity Ni, two exothermic peaks were observed on the DSC thermograms [48]. The first peak at lower temperature corresponds to the disappearance of single/double vacancies while the second one at higher temperature is related to the annihilation of vacancy clusters and dislocations. The first DSC peak was not observed for 99.99% Cu processed under the same conditions (see earlier) as most probably all the vacancies are clustered due to the smaller values of SFE and melting point of Cu. In Ni with lower purity than 99.99%, single/double vacancy peak is also not detected as impurities facilitate the formation of vacancy clusters. The single/double vacancy concentration in Ni was determined as the ratio of the heat released in the first peak and the formation energy of a vacancy (0.29  1018 J for Ni [90]). The concentration of single/double vacancies increases with increasing shear strain during HPT, and it saturates at about 1.0  0.1  104 after a shear strain of about 20 irrespective of the pressure (2e8 GPa) [48]. The concentration of vacancy clusters also increases with increasing strain and its saturation values are 3.5  1.0  104 and 5.1  1.0  104 for 2 and 8 GPa, respectively [48]. The cluster size in the HPTprocessed Ni samples was not determined. However, if we assume that the clusters consist of four to nine vacancies similarly as in Cu and we take into account the difference between the concentrations of clusters and single/double vacancies, it is evident that more than 90% of vacancies are clustered.

3.5 DEFECTS AND PHASE TRANSFORMATION IN NANOMATERIALS PROCESSED BY SEVERE PLASTIC DEFORMATION When SPD processing is performed on supersaturated solid solution alloys at high temperatures, grain refinement and formation of precipitates occur simultaneously. The precipitation processes are promoted by SPD since lattice defects, such as dislocations and grain boundaries, act as nucleation sites for precipitates. This can be explained by (1) the faster diffusion of alloying elements at lattice defects (kinetic effect) and (2) the decrease of free energy due to annihilation of defects during phase transformation (thermodynamic effect). The influence of SPD processing on phase composition was studied in Ale4.8Zne1.2Mge0.14Zr and Ale5.7Zne1.9Mge0.35Cu (wt.%) alloys [67]. First, these materials were solution heat-treated for 30 min at 743 K and water-quenched to introduce supersaturated solid solutions. Then, the samples were processed by eight passes of ECAP at 473 K which yielded UFG microstructure with the grain sizes of about 300e500 nm. It is noted that in these AleZneMgeZr and AleZneMgeCu alloys ECAP processing for eight passes results in a two times higher dislocation density (3.2e3.4  1014 m2) and a three times smaller grain size than in pure Al, despite the much higher temperature of ECAP in the former case (473 K for the alloy and RT for pure Al) [67]. The higher dislocation density and the smaller grain size can be attributed to the pinning effect of alloying elements on lattice defects.

Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation

The microstructure and the phase composition of the ECAP-processed AleZne MgeZr and AleZneMgeCu alloys were compared with samples artificially aged at the same temperature and for the same time (30 min) as applied during ECAP. In the ECAP-processed specimens, stable incoherent MgZn2 precipitates (h-phase particles) formed while artificial aging yielded only negligible concentration of h precipitates and mainly coherent Guinier-Preston (GP) zones develop [67]. As the precipitation sequence in these alloys is coherent GP zones, semicoherent h0 precipitates and incoherent h-phase particles, the observed difference in the phase compositions of the ECAP-processed and the aged samples indicates the promoting effect of SPD on precipitation. The experimental results show that processing by ECAP at a high temperature influences not only the precipitation kinetics but also there is a significant effect on the shape of the h precipitates. The AleZneMgeZr and AleZneMgeCu alloys aged for long times contain long rod-like precipitates but these are absent and only essentially spherical particles are present in the specimens processed by ECAP [67]. On the other hand, the precipitation during SPD most probably influences the grain refinement since the pinning effect of the second phase particles on dislocations and grain boundaries yields higher dislocation density and smaller grain size than for the pure counterparts. In multiphase nanomaterials, the energy of interfaces strongly influences the phase composition [76,93]. This phenomenon is caused by the GibbseThomson effect which reveals that the solubility limit in the matrix phase depends on the size of secondary phase particles (d) and the interface energy (g) as [94,95]:
 6gVm cd ¼ cN exp ; (3.8) dRT where T is the temperature of annealing, cN and cd are the solute concentrations in the matrix with precipitates having infinitely small curvature (large diameter) and diameter of d, respectively, R is the molar gas constant, Vm is the molar volume. The value of g depends on the interface structure. For instance, the energies of semicoherent (111)/(111) and (100)/(100) Ag/Cu interfaces are 0.23 and 0.53 J/m2, respectively [96]. These values are much lower than that for coherent interfaces (w1 J/m2) [97]. Electron microscopy studies revealed both cube-on-cube and heterotwin orientation relationships between Cu and Ag phases separated by semicoherent interfaces [98e100]. It was shown that the size of Ag precipitates influences the nature of interfaces: below 2 nm misfit dislocationsdwhich are characteristic features of semicoherent boundariesdwere not observed in Cu/Ag interfaces, therefore these boundaries were considered to be coherent [101,102]. The dashed and solid curves in Fig. 3.16 show the Ag solute concentration in the Cu matrix as a function of the size of Ag nanoparticles at 623 K for interface energies of 0.38 and 1 J/m2, respectively. The former energy was obtained as the arithmetic average of the energies of (111)/(111) and (100)/(100) Ag/Cu semicoherent interfaces.

83

Defect Structure and Properties of Nanomaterials

Figure 3.16 The equilibrium Ag solute concentration in the Cu matrix as a function of the size of Ag nanoparticles at 623 K for two interface energies 0.38 and 1 J/m2. The horizontal dotted line represents the equilibrium Ag concentration in Cu matrix containing Ag particles with very large radius (cN).

4

γ = 1 J/m2 (Coherent interface)

3 cd (at.%)

84

T = 623 K

γ = 0.38 J/m2 (Semi-coherent interface)

2

1

0.33%

0 0

4

8 12 Ag particle size, d (nm)

16

20

The value of 1 J/m2 corresponds to coherent interfaces, as discussed above. Fig. 3.16 shows that the GibbseThomson effect becomes significant if the Ag particle size is smaller than 10e20 nm. It can be seen that the smaller the Ag particle size and the higher the interface energy, the larger the solute Ag concentration in the Cu matrix. Therefore, interface engineering by thermomechanical treatments of alloys may be an effective tool for tailoring multiphase microstructures. The influence of the GibbseThomson effect on the microstructure evolution in multiphase materials was demonstrated in a CueAg alloy [76]. Supersaturated solid solution Cue3 at.% Ag alloy was SPD-processed by cryorolling at liquid nitrogen temperature (w77 K). The total thickness reduction during cryorolling was 85% which corresponds to a logarithmic strain of w2. The Ag solute content in the cryorolled Cu matrix was about 1 at.% while w2 at.% Ag was present as secondary Ag-rich phase. This phase was formed from the supersaturated alloy due to the promoting effect of SPD on precipitation (see above). The grain size of the Cu matrix was 100e200 nm. The average size of secondary Ag particles was w20 nm, however, a large number of Ag precipitates with sizes smaller than 8 nm was also detected by TEM. The dislocation density in the cryorolled Cue3 at.% Ag alloy was very high (w48  1014 m2) due to the pinning effect of Ag solute atoms on dislocations. Annealing of the cryorolled UFG Cue3 at.% Ag alloy at 623 K yielded formation of a strongly heterogeneous microstructure, as indicated by the splitting of each Cu diffraction peak into two components [76]. As an example, Fig. 3.17 shows reflection (220) for the sample annealed for 20 min. For comparison, the same reflection for the cryorolled specimen is also presented. The splitting of the peaks is caused by the development of an inhomogeneous solute atom distribution in the Cu matrix during annealing, resulting in a variation of the lattice parameter of the Cu matrix. For the simplicity, each line profile was evaluated by fitting it with the sum of two profile components having

Normalized intensity (a.u.)

Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation

73

Cryorolled Cryorolled + annealed

Reflection 220 Region 1

Region 2

74 2Θ (degree)

75

Figure 3.17 Reflection (220) of the Cue3 at.% Ag sample in cryorolled state and after annealing at 623 K for 20 min. The symbols and the solid lines represent the measured data and the fitted curves, respectively. The diffraction peak in the annealed condition is a sum of two reflections related to Regions 1 and 2 having different average lattice parameters (for details see the text). In this figure the integrated intensity (the area under the peak after background subtraction) is normalized to unity for both cry
ba
r, A. Kauffmann, orolled and annealed states. (Reprinted from J. Gubicza, Z. Heged} us, J.L. La J. Freudenberger, V. Subramanya Sarma, Solute redistribution during annealing of a cold rolled CueAg alloy, Journal of Alloys and Compounds 623 (2015) 96e103 with permission from Elsevier.)

different Bragg angles which correspond to two distinct regions of the matrix with different average lattice parameters. The subprofiles appeared at higher and lower diffraction angles correspond to the matrix volumes with low- and high-solute Ag contents, which are referred to as Regions 1 and 2, respectively [76]. From the positions, intensities and shapes of the diffraction subprofiles, the solute Ag content in the Cu matrix, the fractions of Cu and Ag phases, and the dislocation density in the two matrix regions were determined, respectively. According to the results, the development of the heterogeneous microstructure during annealing at 623 K is depicted schematically in Fig. 3.18. In the cryorolled sample, the average Ag solute concentration is 1%. However, as it is shown in Fig. 3.16, around Ag particles with semicoherent interfaces and sizes smaller than w4 nm, the equilibrium Ag solute concentration in the Cu matrix is higher than 1%. Therefore, these particles tend to dissolve into the matrix during annealing, resulting in an increase of solute Ag content and a corresponding decrease of the volume fraction of Ag particles (see the second picture in Fig. 3.18). The matrix regions around these small Ag particles correspond to Region 2. The particles larger than the critical value (w4 nm for semicoherent interfaces, see Fig. 3.16) grow since the equilibrium solubility limit around them is smaller than the initial 1%. In these volumes, denoted as Region 1, the solute Ag content decreases while the fraction of Ag phase increases (see the second picture in Fig. 3.18). Thus, it can be concluded that the GibbseThomson effect resulted

85

86

Defect Structure and Properties of Nanomaterials Cryorolled + annealed for 20 min

Cryorolled Cu–1% Ag matrix

Cryorolled + annealed for 120 min

Cu–2.3% Ag matrix Ag

Region 2 Ag Annealing at 623 K

Ag particles

Annealing at 623 K Ag

Ag

Ag

Ag Ag

Ag

Ag

Region 1

Ag

Ag

Region 1 Cu–0.3% Ag matrix

Cu–0.3% Ag matrix

Figure 3.18 Schematic of the development of heterogeneous microstructure in cryorolled ultrafinegrained Cue3 at.% Ag alloy during annealing at 623 K up to 120 min. The darker the gray in the matrix, the higher the solute Ag content.

in the development of a heterogeneous two-phase microstructure in Cue3 at.% Ag alloy processed by cryorolling and subsequent annealing. It is noted that for coherent interfaces the critical Ag particle size is larger (w10 nm), as shown by the solid curve in Fig. 3.16. Fig. 3.19 shows the solute Ag content in Regions 1 and 2, the relative fractions of these two matrix regions and their dislocation densities as a function of annealing time at 623 K for cryorolled UFG Cue3 at.% Ag alloy [76]. In Region 2, the solute Ag content increased from w1 at.% to w2.6 at.% within 5 min annealing and it decreased slightly to w1.9 at.% with increasing annealing time up to 75 min (see Fig. 3.19a). For longer durations of heat treatment the solute Ag concentration in Region 2 was not determined due to the low volume fraction of this region (see below). In Region 1, the solute Ag concentration decreased to w0.3 at.% during the first 10 min of annealing and remained unchanged within the experimental error up to 120 min. This value agrees with the equilibrium Ag concentration in Cu at 623 K (0.33%) within the experimental error. The increase and the decrease of solute Ag concentration during annealing suggest the dissolution and the precipitation of Ag in Regions 2 and 1, respectively. In the cryorolled state, the relative X-ray intensity of the Ag phase is w8% which decreased to w5% after 5 min annealing and remained unchanged within the experimental error up to 30 min (Fig. 3.19b). This observation is in accordance with the dissolution of Ag precipitates in Region 2. In this period, the fractions of Regions 1 and 2 are w60% and w35%, respectively [76]. Between 30 and 120 min the fractions of Regions 1 and 2 increased and decreased, respectively. This indicates that precipitation occurred in

Defect Structure in Bulk Nanomaterials Processed by Severe Plastic Deformation

T = 623 K

2.5

Region 2

2.0 1.5 Initial solute Ag concentration

1.0

Region 1

0.5 0.0 0

30 60 90 Annealing time (min)

120

(b) 100 Intensity fractions (%)

Solute Ag concentration (%)

(a) 3.0

T = 623 K Region 1

80 60 40

Region 2 20 0

Ag 0

30 60 90 Annealing time (min)

120

Dislocation density (1014 m–2)

(c) 60 T = 623 K 50 40 30

Region 2

20 10 Region 1 0

0

30 60 90 Annealing time (min)

120

Figure 3.19 The variation of (a) the solute Ag concentration, (b) the X-ray intensity fraction, and (c) the dislocation density for Regions 1 and 2 in the Cu matrix as a function of annealing time. The solid
ba
r, A. Kauffmann, curves serve only as guide to eyes. (Reprinted from J. Gubicza, Z. Heged} us, J.L. La J. Freudenberger, V. Subramanya Sarma, Solute redistribution during annealing of a cold rolled CueAg alloy, Journal of Alloys and Compounds 623 (2015) 96e103 with permission from Elsevier.)

Region 2 which reduced the solute Ag concentration to the equilibrium value (w0.3 at.%). After 120 min annealing, the fraction of Region 2dwhere the dissolution of Ag had occurred in the beginning of the heat treatmentdwas practically zero. Concerning the relative intensity of the Ag phase, it increased gradually to w13% between 30 and 120 min in accordance with the reduction of the fraction of the matrix volume with high solute Ag content (Region 2). The variation in the dislocation density in Regions 1 and 2 of the Cu matrix as a function of the annealing time is shown in Fig. 3.19c. In Region 1, the dislocation density decreased to w1  1014 m2 during the first 30 min of annealing, then it remained unchanged within the experimental error up to 120 min [76]. In Region 2, the dislocation density decreased only to about half of the initial value (w20  1014 m2) due to the stronger pinning effect of the higher solute Ag concentration. Further annealing up to

87

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Defect Structure and Properties of Nanomaterials

75 min resulted in only a slight change in the dislocation density in Region 2. Between 75 and 120 min the dislocation density and the crystallite size in Region 2 were not determined since the fraction of this region was very low (area ¼ 15  1 nm b ¼ 11  1% r < 1013 m2

< x>area ¼ 19  1 nm b ¼ 11  1% r < 1013 m2

1600

< x>area ¼ 13  1 nm b ¼ 11  1% r < 1013 m2

< x>area ¼ 16  1 nm b ¼ 11  1% r < 1013 m2

< x>area ¼ 94  7 nm b ¼ 0.20  0.03% r ¼ 4.0  0.8  1014 m2

1800

< x>area ¼ 13  1 nm b ¼ 11  1% r < 1013 m2

< x>area ¼ 124  10 nm b ¼ 0.10  0.02% r ¼ 3.0  0.8  1014 m2

< x>area ¼ 126  10 nm b ¼ 0.17  0.03% r ¼ 4.0  0.8  1014 m2

< x>area is the area-weighted mean crystallite size, b is the twin fault probability and r is the dislocation density. The samples containing significant amount of dislocations are marked by light gray color [67].

not larger than 5.5 GPa. The crystallite size increases as a result of sintering. The crystallite size is increasing with the temperature for a given pressure and also with the pressure at a constant temperature. The crystallite size may increase due to coalescence of grains similarly as Ostwald ripening. The increase of temperature at constant pressure causes an increase of mobility of atoms resulting in a faster grain growth. Although the higher pressure usually reduces the mobility of atoms, the increased elastic shear strains at the contact surfaces may increase the driving force for reducing the fraction of grain boundaries, i.e., induce grain growth. At low sintering pressure values, twins are formed during consolidation of SiC NP and the dislocation density remains below the detection limit of line profile analysis. At 1600 and 1800  C, and above a certain pressure limit, the planar fault density decreases to a very low level, i.e., to about 0.1  0.02%, and dislocations become the main type of lattice defects. The values of the pressure limit are 5.5 and 4 GPa at 1600 and 1800  C, respectively, i.e., this limit decreases with increasing temperature [67]. The decrease of the planar fault density when the pressure is increasing from 2 to 5.5 GPa at 1800  C is also justified by the comparison of TEM images in Fig. 5.16a and b. The values of twin fault spacing determined from the TEM images are 3e6 nm and 100e200 nm for the specimens sintered at 2 and 5.5, respectively. Taking into account that the distance between the neighboring {111} planes in SiC is d111 ¼ 0.252 nm and using Eq. (4.15), the twin fault probability is obtained to be in the range of 6%e12% and 0.1%e0.2%, for the samples prepared at 2 and 5.5 GPa, respectively. These values are close to the twin fault probabilities determined by XLPA (see Table 5.4).

147

148

Defect Structure and Properties of Nanomaterials

(a)

(b)

100 nm

1000 nm

(c)

100 nm

Figure 5.16 (a) Transmission electron microscopy images of specimens sintered at 1800  C and 2 GPa, (b) 5.5 GPa, and (c) 8 GPa. Please note the different magnification for (b) [67].

For the specimen prepared at 8 GPa and 1800  C the crystallite size, the twin fault probability, and the dislocation density are 73  8 nm, 1.8  0.3%, and 15  2  1014 m2, respectively [67]. The crystallite size decreased while both the dislocation density and the twin fault probability increased compared to the sample processed at 5.5 GPa at the same temperature. The TEM images shown in Fig. 5.16b and c confirm the higher planar fault density and the smaller grain size at 8 GPa compared to the specimen sintered at 5.5 GPa. Although the higher shear strains at 8 GPa would induce higher driving force for the increase of the grain size, the diffusion rate and rearrangement of atoms are impeded with the increased pressure and these effects could slow down grain growth when pressure exceeds a critical value. The distances between the twin boundaries in the TEM images is 10e20 nm for the specimen sintered at 8 GPa that gives 1%e2% for the twin fault probability, which is close to the value determined by XLPA (see Table 5.4). The values of the grain sizes observed in the TEM images of Fig. 5.16 are between 30 and 100 nm, 100 and 300 nm, and 80 and 120 nm for the specimens sintered at 1800  C and 2, 5.5, and 8 GPa, respectively [67]. The larger grain sizes for 5.5, and 8 GPa than for 2 GPa are consistent with the trend observed for the crystallite

Lattice Defects in Nanoparticles and Nanomaterials Sintered From Nanopowders

(b)

(a)

16

120

40 1800 1700 1600 1500 1400

(nm)

10 8 6 4

20

pe

re

re

tu

tu

ra

ra

pe

)

)

2

(K

(K

3

2

m Te

m Te

2

7 5 6 a) P 4 G ( sure Pres

8

1800 1700 1600 1500 1400

0

(c)

6 7 5 ) 4 GPa ( 3 sure Pres

Dislocation de

80 60

12

Crystallite size

100

14

nsity (10 14 m –2 )

size (see Table 5.4). However, the grain sizes obtained by TEM are somewhat larger than the crystallite sizes obtained by XLPA. This difference can be attributed to the fact that XLPA measures the coherently scattering domain size that is generally smaller than the grain size observable by microscopic methods. The coherency of X-rays scattered from a grain can be broken down by special arrangement of dislocations such as dipolar dislocation walls or low-angle boundaries, which gives no or only weak contrast difference in TEM images [69]. The crystallite size, the dislocation density, and the twin fault probability for the 10 sintered specimens as a function of the sintering pressure and temperature are plotted in Fig. 5.17a, b and c, respectively. It seems that at high temperatures and pressures the relatively large crystallite size enables the formation of dislocations during the sintering process. The elimination of planar faults when grain growth takes place during firing at high temperatures has already been observed for SiC [70,71]. This result is also in line

8 0

12

8 6 4 2

m Te

1400 1500 1600 1700 1800

Twin density (%)

10

pe

ra

tu

re

(K

8

7

6

5

4

)

sure

Pres

2

3

0

)

(GPa

Figure 5.17 (a) The crystallite size, (b) the dislocation density, and (c) the twin density for 10 sintered specimens as a function of the sintering pressure and temperature. The wire grids in the figures are to guide the eye [67].

149

150

Defect Structure and Properties of Nanomaterials

with observations for fcc metals, where planar faults and dislocations control plasticity below and above a critical crystallite size, respectively [72]. The value of the critical crystallite size where that transition takes place depends on the properties of materials. For example, for Cu w40 nm is suggested as the critical crystallite size below which deformation proceeds by twinning rather than by dislocation glide [72,73]. The data presented in Table 5.4 do not allow us to precisely identify the critical grain size for which dislocations become prevalent, but it appears to be bounded between 20 and 90 nm. It has been mentioned earlier that the pressure limit, where dislocations become the main lattice defects, decreases with increasing temperature. This phenomenon can be explained by the promoting effect of temperature for the growth of crystallites [71]. It is interesting to note that when the crystallite size decreases, e.g., for sintering at 8 GPa and 1800  C, the planar defects have significant densities, which again supports the observed correlation between the crystallite size and the defect structure. It is noted that the grain growth during consolidation of SiC powders at high temperature can be suppressed by the application of SPS method [71]. For instance, SPS processing of SiC NP with the crystallite size of about 10 nm at 1700  C for 10 min under a pressure of 40 MPa yields only a moderate growth of crystallites to 25 nm [71]. The TEM analysis gives slightly larger grain size of 40 nm. Despite the small grain size, the stacking disorder is lowered during sintering that indicates the role of high temperature in the decrease of planar faults in nanocrystalline SiC. It is also noted that if the dislocation density is very high in the initial powder then it can decrease simultaneously with the grain growth during high temperature sintering. It is manifested during consolidation of WC NP processed by high energy ball milling. The crystallite size and the dislocation density in the ball-milled powder are 8 nm and 1.9  1017 m2, respectively. After sintering at 1420  C and 7 MPa, the crystallite size increases to 113 nm, while the dislocation density decreases to 2  1015 m2 [37].

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agyi, Microstructure and mechanical characteristics of bulk polycrystalline Ni consolidated from blends of powders with different particle size, Materials Science and Engineering, A 527 (2010) 1206e1214. [64] C. Pantea, J. Gubicza, T. Ung
ar, G.A. Voronin, T.W. Zerda, Dislocation density and graphitization of diamond crystals, Physical Review, B 66 (2002) 094106. [65] C. Pantea, J. Gubicza, T. Ung
ar, G.A. Voronin, N.H. Nam, T.W. Zerda, High-pressure effect on dislocation density in nanosize diamond crystals, Diamond and Related Materials 13 (2004) 1753e1756. [66] H. Hirai, K. Kondo, Shock-compacted Si3N4 nanocrystalline ceramics: mechanisms of consolidation and of transition from a-to b-form, Journal of the American Ceramic Society 77 (1994) 487e492. [67] J. Gubicza, S. Nauyoks, L. Balogh, J. L
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ar, Influence of sintering temperature and pressure on crystallite size and lattice defect structure in nanocrystalline SiC, Journal of Materials Research 22 (2007) 1314e1321. [68] X.J. Ning, P. Pirouz, A large angle convergent beam electron diffraction study of the core nature of dislocations in 3C-SiC, Journal of Materials Research 11 (1996) 884e894. [69] T. Ung
ar, G. Tichy, J. Gubicza, R.J. Hellmig, Correlation between subgrains and coherently scattering domains, Powder Diffraction 20 (2005) 366e375. [70] K. Koumoto, S. Takeda, C.H. Pai, T. Sato, H. Yanagida, High-resolution electron microscopy observations of stacking faults in b-SiC, Journal of the American Ceramic Society 72 (1989) 1985e1987. [71] T. Yamamoto, H. Kitaura, Y. Kodera, T. Ishii, M. Ohyanagi, Z.A. Munir, Consolidation of nanostructured b-SiC by spark plasma sintering, Journal of the American Ceramic Society 87 (2004) 1436e1441. [72] Y.T. Zhu, X.Z. Liao, S.G. Srinivasan, E.J. Lavernia, Nucleation of deformation twins in nanocrystalline face-centered-cubic metals processed by severe plastic deformation, Journal of Applied Physics 98 (2005) 034319. [73] L. Balogh, G. Rib
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CHAPTER 6

Lattice Defects in Nanocrystalline Films and Multilayers 6.1 DEFECTS IN NANOCRYSTALLINE FILMS Electrodeposition (or electroplating) is a frequently used method to produce nanocrystalline thin films. Among the electrodeposited nanocrystalline metals nickel is the one of the most extensively studied material. The grain size and the defect structure in electrodeposited nickel can be controlled by several deposition parameters such as bath temperature, solution pH, applied current density, deposition mode (direct current or pulse-plating), and additives. The effect of the different organic additives on the lattice defects in electrodeposited Ni films was studied in [1]. Ni layers with a thickness of about 20 mm were deposited on a thick Ti sheet at room temperature (RT) applying direct current in an electrolyte with the composition of 0.6 mol/L nickel sulfate, 0.30 mol/L sodium sulfate, 0.25 mol/L H3BO3, and 0.15 mol/L H3NO3S. Three different samples were produced. During the deposition of the first film no additive was used and this sample was denoted as Ni-NA. In the case of the second film 1.22 mol/L formic acid was added to the electrolyte and this sample was referred to as Ni-FA. In the deposition process of the third sample 1 g/L saccharin was added to the electrolyte and this film was denoted Ni-SC. The pH of all the three solutions was set to 3.25 using NaOH. The study of the defect structure was performed on both sides of the deposits. In the following, the phrase “electrolyte side” is used to describe the side of the sample toward the solution, while the other specimen side is referred to as “substrate side” and they will be abbreviated as “es” and “ss”, respectively. Table 6.1 lists the average grain size and twin fault spacing obtained by transmission electron microscopy (TEM) and the arithmetically averaged crystallite size, dislocation density, and twin fault probability determined by X-ray line profile analysis (XLPA). It was found that the microstructure differed in the two sides of the film only for the sample deposited without additives (Ni-NA). For other specimens there was no significant difference between the electrolyte and substrate sides. In the case of Ni-NA film, the grains are nearly equiaxed at the substrate side of the film with an average size of w50 nm. At the electrolyte side there is a columnar grain morphology with more than 3 mm long and w120 nm wide columns that are aligned perpendicular to the film surface. In Table 6.1 the column width is given as the grain size. The transition of the structure along the growth direction from disordered small crystals to a columnar system is common for the direct current electrodeposition process carried out in an additive-free bath [1]. The columnar morphology is in accordance with T-zone structural zone models [2] that were Defect Structure and Properties of Nanomaterials ISBN 978-0-08-101917-7, http://dx.doi.org/10.1016/B978-0-08-101917-7.00006-2

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Table 6.1 Parameters of the microstructure for electrodeposited nanocrystalline Ni films prepared with different additives Grain Dislocation Twin fault Twin fault Twin fault size Crystallite density probability spacing from spacing from Sample (nm) size (nm) (1014 mL2) (%) XLPA (nm) TEM (nm)

NiNA (es) NiNA (ss) Ni-FA Ni-SC

124
1

55
6

9
1

0
0.1

n.a.

n.a.

50
1

34
4

16
2

0
0.1

n.a.

n.a.

48
1 24
1

23
3 16
2

46
5 176
18

0.8
0.1 3.3
0.1

25
3 6.1
0.5

25.1
0.1 17.3
0.1

Ni-NA, no additive; Ni-FA, addition of formic acid; Ni-SC, addition of saccharin. Due to the heterogeneity of sample Ni-NA along the growth direction, the substrate (ss) and electrolyte/solution (es) sides were distinguished. At the electrolyte side of sample Ni-NA, there is a columnar grain structure and the grain size given in the table corresponds to the width of the columns (the length of the columns is larger than 3 mm) [1]. The crystallite size is the arithmetically averaged size of crystallites obtained by XLPA (see Chapter 2). The twin fault spacing obtained by XLPA method was calculated from the measured twin fault probability using Eq. (4.15) in Chapter 4.

developed for the growth of films by physical vapor deposition (PVD). In this model, columns develop if the substrate temperature is held in the homologous temperature range of 0.1e0.3, considering the melting point of the deposited material. According to the model, the grain growth is hindered in the bulk of the film due to the low rate of bulk diffusion, and the surface morphology supports competitive growth of grains and selection of grain orientations, resulting in the formation of texture [2]. During electrodeposition, the substrate temperature is similarly low to hinder diffusion process in the bulk, but there is sufficient surface mobility at the deposit electrolyte interface for the development of a competitive growth. In accordance with the model, a crystallographic texture was detected on the electrolyte side of sample Ni-NA by X-ray diffraction where the (220) planes were parallel to the film surface. XLPA showed differences between the two sides of Ni-NA film. Similar to the grain size the crystallite size was also smaller in the substrate size, although the difference was not as high as suggested by the TEM results. This effect can be attributed to the high penetration depth of X-rays (estimated as w14 mm) compared to the thickness of the film (about 25 mm) since the cross-sectional TEM images were taken very close to the deposit surfaces on both sides. In addition, dark field TEM images revealed that the columns at the electrolyte side were fragmented into subgrains that serve as crystallites in XLPA evaluation. XLPA method often yields smaller crystallite size than the grain size obtained by TEM as the former method determines the coherently scattering domain size, which may be smaller than the grain size if the misorientations inside the grains break the coherency of diffracted X-rays [3]. The dislocation density was larger in the substrate side than in the electrolyte side, which is in accordance with the smaller crystallite size since the subgrain

Lattice Defects in Nanocrystalline Films and Multilayers

boundaries with low misorientations usually consist of dislocations. In the additive-free sample twin boundaries were not observed inside the grains by either XLPA or TEM. Table 6.1 shows that the organic additives reduced the grain and crystallite sizes and increased the defect density. In this sense saccharin addition is more effective than that of formic acid. In Ni-SC sample the grain size decreased to about 24 nm, while the dislocation density and the twin fault probability reached very high values of w176  1014 m2 and w3.3%, respectively. There are some hypotheses for the influence of additives on the defect structure [1]. The first explanation may be that hydrophilic functional groups of organic additives (e.g., SO2 or carboxyl group) can enclave the additive molecule between water (in the electrolyte) and the previously deposited metal surface [4]. This “organic shield” does not block the electron transfer but hinders the ion transfer. The preferential sites for the adsorption of the surface-active additive molecules are the kink sites and step edges. Since these crystal features are, at the same time, also the most active growth centers, their blocking leads to the necessity of new grain nucleation, which obviously results in grain refinement. This shielding mechanism on the growth surface can inhibit the grain growth and induce secondary nucleation. It is noted that when using formic acid as additive, components from the inhibiting additive could not be detected in the deposit by EDS analysis; however, impurity segregation at grain boundaries below the EDS detection limit cannot be excluded. Such a small impurity level may have significant effect on the grain size as pointed out formerly [2,5]. In addition, in the case of saccharin passivation of the cathode surface by precipitation of nickel hydroxide may lead to the nucleation of new grains [6]. Most probably, this electrochemical process is enhanced by the segregation of sulfur or sulfide phases that can occur both within the grains and at the grain boundaries [1]. Similar explanation for grain size refinement by the segregation of an oxide phase was established for PVD films [2]. Beside the hindrance of the characteristic crystal growth sites, saccharine may decompose during electrodeposition of Ni, resulting in benzamide formation and sulfur inclusion [7]. Although the formation of a well-defined NiS compound may not take place, the inclusion of the sulfur atom in the Ni lattice leads to a lattice deformation that impedes the coherent growth of the crystals and to a higher secondary nucleation probability. Hence, the sulfur inclusion has a contribution to the grain refinement. The formation of twins at the growth surface may also be facilitated by impurity segregation on the close-packed (111) crystal facets of Ni. The segregated species can serve as two-dimensional nucleation centers for the successive (111) plane; however, they may cause a shift of the plane compared to the ideal fcc stacking, resulting in twin boundaries or stacking faults [1]. In addition, the elastic stresses developed inside the grains during deposition can relax by forming lattice defects such as twin boundaries. The dislocation density also increases due to the addition of formic acid and saccharin. Most probably, this effect is associated with the refinement of the microstructure owing to additives. As the interiors of nanograins usually do not store dislocations, their majority

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is assumed to be located in the grain/subgrain boundaries. These dislocations may form to reduce the elastic incompatibility stresses developed between the grains/subgrains at their boundaries [1]. From the twin fault probability determined for the electrodeposited Ni samples by XLPA the twin fault spacing was calculated using Eq. (4.15) (see Chapter 4) and listed in Table 6.1. The value obtained for sample Ni-FA by XLPA agrees well with the average twin fault spacing determined from the TEM images. At the same time, in the case of saccharin addition the twin fault spacing obtained by XLPA was much smaller than the value determined by TEM. This difference can be attributed to nanotwins that have a considerable contribution to XLPA; however, they can be hardly observed by TEM. High-resolution TEM investigations on Ni-SC film revealed the presence of nanotwin lamellae with the thickness of 2e6 nm, which explains the difference between the twin fault spacing values determined by XLPA and TEM [1]. It should be noted, however, that the agreement between the twin boundary spacings measured by the two methods is still satisfactory; especially if we consider that the volume studied by XLPA is at least six orders of magnitude larger than that examined by TEM. The addition of tungsten to nickel in electrodeposited films led to finer grain and crystallite sizes as well as a lower fraction of special Ʃ3n grain boundaries including coherent twin boundaries [8]. For W content smaller than 3 at.% the grain size was about 200 nm; while for Ni‒15 at.% W and Ni‒18 at.% W alloys, the grain size reduced to 8 and 5 nm, respectively. The fraction of Ʃ3n grain boundaries was between 17% and 33% for tungsten concentration lower than 3 at.%. When the W content increased to 15 at.% or higher, the fraction of Ʃ3n boundaries decreased to 1e5%. In addition, a weak (111) texture was formed when the tungsten content increased above 10 at.%. The effect of tungsten additive on the defect structure in Ni, Cu, and Ag thin films processed by magnetron sputtering was studied [9]. The foils with the thickness of 500 nm were grown on Si(001) substrate. The W content was 12, 14, and 13 at.% in Ni, Cu, and Ag films, respectively. All the three samples were single phase fcc materials with columnar grains and strong (111) texture. The diameter and the height of the columns were 40e50 and 300e400 nm, respectively. Coherent twin faults parallel to the film surface were observed for all the three films. The twin boundary spacing values were 0.8, 5, and 2 nm for Ni-12%W, Cu-14%W, and Ag-13%W, respectively, which correspond to very large twin fault probabilities of 25, 4, and 10%. It was revealed that the tungsten distribution in Ni-12%W and Cu-14%W was homogeneous; however, in Ag-13%W the tungsten concentration was found to be higher at twin faults and grain boundaries [9]. The influence of the processing conditions on the defect structure in textured Ni films processed by electrodeposition was studied by XLPA [10]. The concentration of organic additive 2-butyne-1,4-diol varied between 0 and 10 mMol/dm3 while the current density was changed in the range 2e10 A/dm2. The thickness of the

Lattice Defects in Nanocrystalline Films and Multilayers

Table 6.2 Parameters of the microstructure for textured Ni films prepared under different conditions (concentration of organic additive 2-butyne-1,4-diol and current density). The crystallite size is the arithmetically averaged size of crystallites obtained by XLPA (see Chapter 2). The twin fault spacing obtained by XLPA method was calculated from the measured twin fault probability using Eq. (4.15) in Chapter 4 Twin fault Organic spacing additive from content Current Dislocation Twin fault XLPA (mMol/ density density probability Crystallite (nm) dm3) (A/dm2) (1014 mL2) (%) Texture size (nm)

0 0 0 5 5 5 10 10 10

2 5 10 2 5 10 2 5 10

[211] [100] [100] [211] [111] þ [100] [100] [111] [111] þ [511] [111] þ [511]

73
6 58
4 58
3 30
2 8
3 56
6 31
6 13
4 8
3

1.2
0.2 2
0.2 1.3
0.2 6
1 4
1 9
2 4
1 5
1 23
4

0
0.1 0
0.1 0
0.1 0
0.1 0.5
0.1 0
0.1 3.8
0.3 3.1
0.3 0
0.1

n.a. n.a. n.a. n.a. 37
7 n.a. 5
1 6
1 n.a.

electrodeposited Ni films was w15 mm. It was found that both the organic additive content and the current density have a strong influence on texture and lattice defect structure. The results are summarized in Table 6.2. The various Ni films deposited under different conditions exhibited either [111] or [100] or [211] fiber textures or double fiber textures consisting of a main [111] and an additional minor texture component. The [111] texture component was stronger for low current densities and high additive contents [10]. It was found that the dislocation density in the different thin films varied between 1  1015 m2 and 23  1015 m2. The dislocation density in the grains with random orientation was either equal to or smaller than that for the major texture component. The crystallite size, dislocation density, and twin fault probability as a function of organic additive concentration and current density for electrodeposited Ni are plotted in the contour maps of Fig. 6.1, which were created from the measured data listed in Table 6.2. The dislocation density was the smallest in the additive-free samples irrespective of the current density. The dislocation density increased with increasing the additive concentration. For a given additive content, the dislocation density was raised when the current density was set to 10 A/dm2. The higher dislocation density was accompanied by a smaller crystallite size. The crystallite size determined by XLPA was in good agreement with the grain size obtained by TEM [10]. It was revealed that in the films with textures [111] and [100] a considerable fraction of the Burgers vectors are lying parallel to the plane of the films. However, in the [211]-textured films the Burgers vectors are oriented randomly. It was found that the largest current density and the largest amount of additive resulted in the

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Defect Structure and Properties of Nanomaterials

Crystallite size (nm)

10

(b) Current density (A/dm2)

(a) Current density (A/dm2)

8 20

6

13 26 40 33 46

4 2

60 53 66

0

Dislocation density (1014 m –2)

10

19

14

21

16 12

8 7.5 3.0

9.7

5.2

6 4 5.2

2

2 6 8 10 4 Organic additive (mMol dm–3)

0

4 6 8 10 2 Organic additive (mMol dm–3)

Twin fault probability (%)

(c) 10 Current density (A/dm 2)

160

0.11

8 0.93

6

0.11

0.52

1.3 1.8 2.2 2.6 3.0

4 2

3.4

0

4 6 8 2 Organic additive (mMol dm–3)

10

Figure 6.1 Contour maps showing the influence of organic additive concentration and current density on crystallite size, dislocation density, and twin fault probability in textured Ni film processed by electrodeposition. (The maps were created from the data listed in Table 6.2).

highest dislocation density (see Fig. 6.1b and Table 6.2). It is noted that a very high dislocation density of w1015e1016 m2 was also found in thermally deposited [111]-oriented gold films, ranging in thickness from 30 to 190 nm [11]. The dislocations were lying in plane (111) parallel to the film surface. Fig. 6.1c and Table 6.2 show that a considerable twin fault probability was formed for high organic additive content (10 mMol/dm3) and low current densities (2e5 A/dm2). In the case of epitaxially grown films the substrate orientation has a deterministic effect on the twin fault probability. This effect was demonstrated on nanotwinned Ag films deposited by magnetron sputtering on Si substrate [12]. Two substrate surface orientations were used in the experiments: (111) and (110). The epitaxially grown silver films have the same crystallographic orientations as the substrates. For both orientations the film thickness was w2 mm. It was found that the sputtered Ag(111) film consists of columns aligned parallel to the growing direction. The width of these columns was about 50e100 nm and their h111i axis was perpendicular to the film surface. In the columns a very high density of twin faults was observed. These twin faults were formed on plane (111) lying parallel to the film surface (i.e., perpendicular to the growth direction) while

Lattice Defects in Nanocrystalline Films and Multilayers

(a)

(b) Ag(111)

Columns

Ag(110)

Twin faults

Twin faults

Growth direction

Figure 6.2 Schematic cross-sectional view of twin fault structures in two epitaxially grown sputtered Ag films where the foil surface is parallel to (a) plane (111) or (b) plane (110) [12]. These planes are perpendicular to the sheet.

twin faults were not observed on the other three {111} planes, as shown in Fig. 6.2a. This is in accordance with observations on other materials that indicated that in sputterdeposited polycrystalline Cu and 330 stainless steel films, twin boundaries are orientated predominantly perpendicular to the growth direction. At the same time, in the case of electrodeposited thick films twin boundaries are often randomly orientated. The average twin fault spacing in Ag(111) film was w9 nm, which corresponds to a twin fault probability of w2.7% [12]. In Ag(110) film the twin faults are formed on the two planes {111}, which are orientated w55 degrees from the growth direction, while the other two {111} planes which are parallel to the growth direction, did not contain twin faults (see Fig. 6.2b). We can say that in both films the twin faults were formed on {111} planes, which have orientations closest to the film surface. In Al(110) film the average twin fault spacing was larger (w42 nm) than that in Ag(111) film that corresponds to the twin fault probability of w0.6%. The smaller twin spacing in Ag(111) film yielded a two times higher hardness (w2 GPa) than that for Ag(110) film (w1 GPa). It was concluded that by controlling the Ag epitaxial orientation, it is possible to modify the orientation and density of growth twins, thereby tuning the film strength.

6.2 LATTICE DEFECTS IN MULTILAYERS The dislocation structure in a Cu-Nb multilayer was investigated by XLPA [13]. The Cu-Nb multilayer was synthesized by direct current magnetron sputtering at RT. The details of magnetron sputtering can be found in Ref. [14]. The as-deposited sample with the thickness of 7.5 mm consisted of alternating layers of Cu and Nb. The total number of layers was 100, therefore the thickness of each layer was 75 nm. The size

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Defect Structure and Properties of Nanomaterials

Crystallites

– [101]

Cu layer ~75 nm

– [111]

Nb layer

– Plane (111) Plane (110)

~75 nm

Figure 6.3 Schematic of the structure for a pair of Cu and Nb layers in the Cu-Nb multilayer processed by direct current magnetron  sputtering   at room temperature. The arrows on the top surface indicate crystallographic directions 101 and 111 in Cu and Nb grains, respectively [13].

of Cu and Nb crystallites both parallel and perpendicular to the layer was about 75 nm. Although the layers were polycrystalline, there was an orientation relationship
between the adjacent Cu and Nb grains, as shown in Fig. 6.3. The lattice planes 111 and (110) in the adjacent crystallites of face-centered (fcc) Cu and body-centered cubic (bcc) Nb, respectively, were parallel to each  other and also  to the interface of the layers. In addition, in these planes the directions 101 in Cu and 111 in Nb were parallel to each other. As a consequence of this orientation relationship, both Cu and Nb layers had a texture, namely for all Cu and Nb crystallites the directions 111 and [110], respectively, were perpendicular to the surface of the multilayer. The dislocation structure in both Cu and Nb layers was determined by XLPA [13]. Twin boundaries were not incorporated into the evaluation of peak profiles. Both phases in the Cu-Nb multilayer have strong texture and if dislocations were mainly formed to
compensate the misfit between the 111 and (110) planes in Cu and Nb, respectively, then not all the slip systems were populated by dislocations. Therefore, XLPA evaluation method must have been adapted to this special case. The details of the evaluation methodology are described in Ref. [3]. The XLPA evaluation for the Cu layers in the as-deposited film showed that the majority (about 75%) of the dislocations were edge
dislocations with the glide plane 111 and Burgers vector 12 101 [15]. A smaller fraction
of dislocations (21%) were edge dislocations with the glide plane 111 and Burgers 1 ½110, and screw dislocations with the same Burgers vector. Both Burgers vectors vector 2   1 101 and 1 ½110 were lying parallel to the plane of the interface. In the as-deposited 2 2 Nb layers the glide plane of the majority of dislocations (about 75%)   was the interface plane (110) with the Burgers vectors 12 111 (w45%) and 12 111 (w30%). A small fraction (w15%) of dislocations with the glide plane (011) and Burgers vector [100] were also detected [15]. The Burgers vector of these dislocations did not lie in the plane

Lattice Defects in Nanocrystalline Films and Multilayers

Cu layer

Nb layer

Figure 6.4 Schematic of the dislocation structure in Cu and Nb layers of the Cu-Nb multilayer processed by direct current magnetron sputtering at room temperature.

of the interface. The rest of dislocations (w10%) were of screw type. The fact that the majority of the dislocations have Burgers vectors parallel to the interface in both Cu and Nb layers is consistent with the assumption that the dislocations in the asdeposited samples were misfit-compensating defects at interfaces, as shown schematically in Fig. 6.4. The dislocation density was about 170  1014 m2 in both the as-processed Cu and Nb phases [13]. The high initial dislocation density is presumably due to the large misfit between the Cu and Nb layers. The type and density of dislocations in magnetron sputtered Cu-Nb multilayers were investigated by XLPA as a function of the thickness of the individual Cu and Nb layers [16]. Three different layer thicknesses were studied: 20, 50, and 75 nm. In each sample the thicknesses of Cu and Nb layers were the same. For each layer thickness value, two samples were produced with different total film thicknesses, namely for 1 and w7 mm. The multilayers with the total thickness of 1 mm were kept on the substrate while the thicker films (w7 mm) were removed from the substrate and tested as self-supported films. For all layer and film thicknesses Cu and Nb layers have sharp fiber textures with [111] and [110] directions normal to the foil surface, respectively. This texture is similar to the case studied above (see Fig. 6.3). XLPA revealed that 80e90% and 70e80% of dislocations have edge character in Cu and Nb layers, respectively [16]. In both Cu and Nb layers the majority of Burgers vectors of these dislocations are lying in the foil plane. The glide planes of edge dislocations in Cu layers are strictly parallel to the foil plane while in Nb layers all of them are inclined to the foil plane. The Burgers vector population does not depend on the layer thickness or on the total thickness of the foils. In Cu layers the dislocation density was 160
20  1014 m2, which did not change significantly with the variation of the thicknesses of either the layers or the foils. In Nb layers the dislocation density decreased from about 410  1014 m2 to 180  1014 m2 with increasing layer thickness; however, for the same layer thickness its value did not depend on the total thickness of the foil [16]. The latter reduction of the dislocation density in Nb layers can be understood if all the dislocations are misfit dislocations located at the interfaces between Cu and Nb layers. In this case the total interface area per unit volume decreases with increasing the layer thickness, resulting

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Defect Structure and Properties of Nanomaterials

in a reduction of the total dislocation density. For Cu layers the constancy of the dislocation density suggests the existence of nonmisfit dislocations within the layers and their density increases with increasing the layer thickness. The crystallite size and the dislocation density were also studied in Cu-Nb layers with the total thickness of 500 nm and individual layer thicknesses of 2.5, 5, and 50 nm [17]. It was found that similar to thicker Cu-Nb multilayers (layer thicknesses of 20e75 nm) the very thin layers (2.5e5 nm) have the crystallite size close to the layer thickness. At the same time, the multilayer with the smallest layer thickness (2.5 nm) contains the highest density of dislocations, 550  1014 m2 and 350  1014 m2 for Cu and Nb layers, respectively. In the Cu layer with the thickness of 50 nm, two-third of dislocations are edge ones with Burgers vector and slip planes lying parallel to the layer interface [plane (111) in Cu], while one-third of them are also edge dislocations with {111} type slip planes inclined to the interface. With reducing the thickness to w5 nm, the majority of dislocations are edge dislocations lying on these inclined {111} slip planes. For the smallest layer thickness (w2.5 nm) the dislocations have screw character in the Cu phase [17]. In the Nb layers with the highest thickness (w50 nm) the dislocations are mainly edge dislocations with Burgers vector and slip planes lying parallel to the interface. This suggests that for the thick layers the dislocations are misfit dislocations at the Cu/ Nb interfaces. However, in the thinnest Nb layer most of dislocations have screw type, similar to the Cu layer with the same thickness.

6.3 EVOLUTION OF DEFECT STRUCTURE DURING PLASTIC DEFORMATION OF THIN FILMS The effect of cold rolling on the lattice defect structure in magnetron sputtered Cu foils was investigated by XLPA [18]. Cu foils with a total thickness of approximately 40 mm were magnetron sputtered at RT on a Si substrate. The as-processed Cu films have a strong [111] fiber texture normal to the specimen surface. However, the crystallites are randomly rotated around the [111] axis, therefore practically there is no texture in the lateral directions of the thin foil. According to TEM observations the crystallites contain twin boundaries on (111) planes oriented perpendicular to the growth direction, i.e., to the normal vector of the film surface. The average twin boundary spacing in the asdeposited samples was 5 nm. The sputtered Cu foils were rolled to a reduction of 50% at RT. Both the as-deposited and the rolled specimens were investigated by XLPA. In the as-deposited sample the dislocation density was relatively small (4
1  1014 m2) while the twin boundary probability was large (3.8
0.5%). Due to rolling at RT the dislocation density increased by a factor of about 7 to 27
3  1014 m2, while the twin fault probability decreased by about 20%. The decrease of the twin fault probability can be attributed to untwinning, which may occur due to the interaction between the dislocations and twin boundaries (see Chapter 7). The average crystallite size decreased

Lattice Defects in Nanocrystalline Films and Multilayers

during rolling, while the grain size parallel to the Cu foil surface increased by a factor of about three, as revealed by TEM observations [19]. The difference between the evolution of the grain and crystallite sizes can be explained by the hierarchy of the severely deformed microstructures. In plastically deformed metallic materials the grains contain subgrains, which correspond to crystallites measured by XLPA. In the present case, the severe plastic straining flattened the grains along the foil surface; however, the increasing dislocation density reduced the subgrain (crystallite) size simultaneously. The dislocation arrangement parameter decreased during rolling indicating the stronger shielding of the strain fields of dislocations, most probably due to their arrangement into low energy configurations such as low angle grain boundaries and/or dipoles. The important role of dislocation glide in plastic deformation of nanocrystalline Ni film was also proved by postmortem high-resolution TEM (HRTEM) study [20]. The foil with the thickness of 150 mm and the grain size of 24 nm was prepared by electrodeposition. Then, the Ni film was deformed by uniaxial tension up to the strain of 4% or rolling to a thickness reduction of 30%, both at the liquid-nitrogen temperature (LNT). The reason of the very low deformation temperature was to retain at least a portion of dislocations formed during deformation for postmortem HRTEM investigation. The storage of both full and partial dislocations in Ni nanograins was evidenced in both tensile deformed and rolled samples. Closely spaced full dislocations with the Burgers vector of 1=2h110i were often observed near grain boundaries and twin faults. The angle between the dislocation line and the Burgers vector was often 60 degrees. Frank type dislocation loops with a Burgers vector of 1=3h111i and a diameter of w1 nm were also found, which correspond to vacancy clusters. These loops act as obstacles against slip of glissile dislocations. It is noted that these loops cannot be detected by XLPA method due to their strongly shielded strain field. It seems that full lattice dislocations formed during plastic deformation at LNT may be trapped inside nanocrystalline Ni grains due to pinning effect of other defects such as twin faults, grain boundaries, and small Frank dislocation loops. Impurities from the electrolyte bath may also have a pinning force on dislocations. The order of full dislocation density was w1016 m2, which is in accordance with XLPA investigations performed on other plastically deformed films (see later). The dislocation structure in a Cu-Nb multilayer foil with the total thickness of 7.5 mm (layer thickness: w75 nm) was studied in the previous section. This multilayer was deformed by rolling at RT [14]. First, the foil was peeled off from the substrate, cut into strips, and stacked, hereby obtaining a sample with the thickness of about 250 mm for rolling. The rolling strains were determined as the engineering strain corresponding to the foil elongation along the rolling direction. Two samples were processed by rolling at the strain values of 100% and 150%. The thickness of the individual Cu and Nb layers was equivalent to the crystallite size normal to the multilayer surface. This size was reduced from 75 nm to 45 and 30 nm while the lateral dimension of the crystallites increased from 75 nm to 150 and 240 nm for the strains of 100% and 150%, respectively [15]. In the

165

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Defect Structure and Properties of Nanomaterials

sputtered material the dislocation density was w170  1014 m2 in both Cu and Nb layers that did not change significantly during rolling. Therefore, dislocations were formed and annihilated in a dynamic equilibrium during deformation. However, the types of dislocations changed considerably in both Cu and Nb phases. Generally, the fraction of dislocations with Burgers vectors parallel to the interface decreased during rolling while new dislocations were formed for which the Burgers vector did not lie in the interface plane. For instance, after  150%  cold rolling in the Cu layers, about 27% of the dislocations had a Burgers vector 1 2 110 or 1=2½101, which has a component out of the interface plane (this fraction was only 4% in the as-deposited Cu phase). In the Nb layers  rolled  up to the strain of 150% the fraction of dislocations with Burgers vectors 1 2 111 and    1 2 111 became marginal, while the majority of dislocations have Burgers vector [100], [010], or [001]. In the Nb layers the fraction of dislocations with Burgers vector lying in the interface plane was reduced from 75% to 40% due to rolling. This means that there was a significant fraction of Burgers vectors h001i in the Nb layers that were not lying in the interface plane. The dislocations with Burgers vectors h001i can be formed via the reaction of dislocations with Burgers vector type 1=2h111i gliding on planes inclined to the interface, as shown in Fig. 6.5a[21].  The  formation of Burgers vector h001i by merging Burgers vectors 1=2½111 and 1 2 111 is energetically favorable as the sum of the squares of the latter vectors is larger than the square of vector [001]. After this reaction the obtained Burgers vector h001i either may be in the interface plane (110) or point out of this plane, as illustrated in Fig. 6.5b. Vector [001] is in plane
 (110) and it is obtained as the sum of vector 1=2½111 in plane 011 and vector 1 2 111 in plane (101). At the same time, vector   010 does  not lie in plane (110)
and this vector   wasthe result of the reaction between vector 1 2 111 in plane 011 and vector 1 2 111 in plane (101). (a)

(b) Nb layer –– 1/2[111]

1/2[111] ––

(110)

1/2[111]

[001]

(110) 1/2[111]

(101) – (011) (101)

–

[010]

––

1/2[111]

– (011)

[001]

Figure 6.5 (a) Schematic of the formation of dislocations with Burgers 
vector [001] with the interaction of gliding dislocations with Burgers vectors 1=2ð111Þ and 1 2 111 in Nb layers during rolling of a Cu-Nb multilayer. Figure (b) shows a Bravais cell of Nb with the crystallographic planes and directions which play important role in the dislocation reaction.

Lattice Defects in Nanocrystalline Films and Multilayers

It should be noted that the layer thickness strongly influences the arrangement of dislocations in the individual layers during plastic deformation of multilayered foils. TEM investigations on magnetron sputtered Cu-Nb multilayers with the layer thickness of 2 mm showed that rolling yielded dislocation cell structure formation and large lattice rotations inside the individual layers [21]. Due to this rotation the initial fiber texture of the layers was altered. In the Cu layers the h111i fiber texture normal to the foil surface split into two h111i components oriented about 35e40 degrees from the initial interface normal. For the Nb layers the normal of the interface plane changed from h110i to h111i due to rolling. At the same time, for Cu-Nb multilayers with the layer thickness of 75 nm rolling resulted in a uniform reduction of the thickness of both layers and there was neither dislocation cell structure formation nor significant rotations of the interface normal. Therefore, the initial h111i and h110i fiber textures in the Cu and Nb layers, respectively, were preserved. The unique plastic behavior of nanoscale multilayers was interpreted by glide of individual dislocations in the form of hairpin-shaped loops confined within a layer. This glide occurred in slip planes inclined to the interface of Cu and Nb layers, as shown for a Nb layer in Fig. 6.6. The slip of the hairpin loop deposits misfit type dislocations at interfaces. As interfaces act as sources and sinks of dislocations, all dislocations are stored at the interfaces. This mechanism ensures the plastic stability in rolling (i.e., the uniform thickness reduction), the unchanged crystallographic orientation (texture), and the lack of dislocation cell formation [21]. It should be noted that dislocation glide in multilayers may also occur during heating. It has been shown for a Cu-Ni multilayer with the layer thickness of 21 nm that a heating/ cooling cycle between RT and 325  C resulted in a reduction of internal coherency stresses by an increase of the interfacial misfit dislocation density [22]. These coherency stresses were evolved due to the different lattice spacings of the Si(001) substrate and the “cube-on-cube” epitaxially grown Cu and Ni layers. In both layers, direction [001] was perpendicular to the film surface. The total film thickness was 5 mm. Due to the different Nb layer

(110)

– (011) (101)

Figure 6.6 Schematic illustration of the confined glide of hairpin dislocation loops on two glide planes in a Nb layer of a nanoscale Cu-Nb multilayer, depositing misfit dislocations at the interfaces [21].

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Defect Structure and Properties of Nanomaterials

lattice parameters of Cu and Ni, misfit dislocations were formed at the semicoherent interfaces. In addition internal stresses were developed parallel to the film surface. During heating stress relaxation occurred via dislocation glide, which finally resulted in an increase of interfacial dislocation density by w25%. As a consequence, the hardness of the multilayer decreased by w11%. For Cu-Ni layers with the thickness of 11 nm plastic deformation and corresponding internal stress relaxation after the heating/cooling cycle were not observed, although these stresses were much higher than for the thicker layers. The lack of plasticity was most probably due to the almost coherent interfaces which contain negligible density of misfit dislocations [22]. In the previous section it was shown that a high density of growth twin faults can be formed in electrodeposited or sputtered fcc films (e.g., in Cu). During plastic deformation of these foils a decay of both coherent and incoherent twin faults was observed [23]. This phenomenon is referred to as detwinning or untwinning. Detwinning usually occurs by gliding of twinning dislocations along {111} planes, as described in Chapter 7. These dislocations are Shockley partials with Burgers vectors of 1=6h112i, which may be formed by (1) the dissociation of gliding dislocations at twin boundaries, (2) their emission from high-angle grain boundaries, and (3) nucleation of pair of twinning dislocations inside a twin crystal. It is noted that twin lamellas inside grains often end at incoherent twin boundaries and the movement of these boundaries may also cause shortening of twin lamellas, thus finally resulting in detwinning [23]. These incoherent twin boundaries are usually S3{112} boundaries which may migrate during plastic deformation. During migration, discrete steps form with a step height of three or multiples of three {111} planes [24]. The motion of a S3{112} boundary can lead to the extension or retraction (detwinning) of coherent twin boundaries, as shown in Fig. 6.7. It is noted that a S3{112} incoherent twin boundary can be regarded as a series of Shockley partial dislocations on every {111} plane along the incoherent boundary, Incoherent twin boundary

Twin Twin retraction

Group of three partials Twin extension

Coherent twin boundaries on {111} planes Parent grain

Figure 6.7 Extension and retraction (detwinning) of coherent twin boundaries by moving of groups of three twinning partials at incoherent twin boundaries.

Lattice Defects in Nanocrystalline Films and Multilayers

therefore the motion of this boundary can be considered as a glide of these twinning partials. The partial dislocations on three neighboring {111} planes   have  different   Burgers  vectors and their sum equals zero (e.g., they may be 1 6 112 , 1 6 121 , and    1 6 211 ). These three partials can be considered as a group of dislocations that is repeated along the incoherent twin boundary. There is an attractive interaction between the partials in this group that inhibits them from breaking apart [24]. Therefore, a group of three partials has a collective glide during migration of S3{112} incoherent twin boundaries. It was also shown that the transmission of a 1=2f110g type glide dislocation across a S3{112} boundary results in a sessile dislocation in the boundary which locally pins the incoherent twin boundary at the site of the slip transmission [24].

6.4 INFLUENCE OF IRRADIATION ON DEFECT STRUCTURE IN MULTILAYERS The irradiation-induced change of the microstructure is a very important phenomenon in structural materials used in high radiation environments. It is well known that point defects (e.g., vacancies and interstitials) with large concentrations form due to irradiation with subatomic particles or ions (e.g., He2þ). Helium irradiation is an important phenomenon in nuclear power plants. At RT the solubility limit of helium in metals is very low and the implanted helium form bubbles inside the material. As the helium atom per vacancy ratio increases in the bubbles, the internal pressure increases and the surrounding lattice becomes more distorted that may lead to the formation of interstitial loop as shown in Fig. 6.8. This mechanisms is referred to as dislocation loop punching. The lattice distortion and loop punching usually hinder the passage of glide dislocations, thereby hardening the material but reducing the fracture toughness. Recent studies [25,26] have shown that incoherent interfaces are capable of trapping implanted He,

He bubble

Lattice planes

He bubble

Interstitial loops

Figure 6.8 Schematic of the formation of interstitial dislocation loops in the vicinity of a pressurized He bubble (loop punching).

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Defect Structure and Properties of Nanomaterials

thereby lowering the probability of formation of He bubbles inside the material. The higher the density of interfaces, the stronger the He trapping. Therefore, much attention is paid to the effect of He ion implantation on the defect structure in multilayers, which contain a large amount of interfaces. The influence of He ion irradiation on the defect structure in magnetron sputtered CuNb multilayers was studied by XLPA [17]. Three foils were deposited on Si(100) substrates at RT with layer thicknesses of 2.5, 5, and 50 nm. The total thickness of the foils was 500 nm for all samples. Both Cu and Nb layers had sharp textures where h111i Cu and h110i Nb directions were perpendicular to the foil surface. The sharp texture of the layers remained unchanged after He implantation. The concentration of He in the irradiated specimens was 7 at%, which could be found mainly in He bubbles with size of w2 nm inside the material [25]. The average crystallite size and dislocation density in Cu and Nb layers before He ion irradiation were presented in Section 6.2 and listed in Table 6.3. For the layer thicknesses of 50 and 5 nm, the crystallite size decreased due to He ion irradiation, i.e., they became smaller than the layer thickness. In the case of layer thickness 2.5 nm, the crystallite size remained equal to the layer thickness. He implantation did not change the order of magnitude of the dislocation density. In Cu layers with the thickness of 50 nm, both the total dislocation density and the fraction of “in-plane” edge dislocations decreased. The “in-plane” dislocations have slip planes parallel to the Cu/Nb interface. This change in the dislocation structure indicates an irradiation-induced recovery of misfit dislocations at the interfaces. In Nb layers with the thickness of 50 nm, the dislocation density only slightly reduced; however, the fraction of screw dislocations increased at the expense of edge dislocations. Therefore, in the

Table 6.3 The area-weighted mean crystallite size and the dislocation density for Cu-Nb multilayers with different layer and film thicknesses [16,17] Layer/film thickness Phase Crystallite size (nm) Dislocation density (1014 mL2)

75 nm/1 mm 50 nm/1 mm 50 nm/500 nm 20 nm/1 mm 5 nm/500 nm 2.5 nm/500 nm

Cu Nb Cu Nb Cu Nb Cu Nb Cu Nb Cu Nb

75
5 75
5 40
2 30
2 35
4 46
5 22
2 19
2 5.4
1.0 5.7
1.0 2.7
1.0 2.5
1.0

160
50 180
50 150
20 280
30 50
10 130
30 170
20 410
30 150
50 400
80 550
80 350
60

Lattice Defects in Nanocrystalline Films and Multilayers

Nb layers a recovery of the misfit dislocations also occurred due to He implantation. For the sample with the layer thickness of 5 nm, the screw character of dislocations became stronger in both Cu and Nb layers. In the film with the thickness of 2.5 nm, where in-plane misfit dislocations were not observed in the as-processed state, the dislocation density did not change considerably during He ion irradiation. Twin and stacking faults were not observed either before or after implantation in the studied Cu-Nb multilayers. Thus, it can be concluded that the main effects of He implantation on the defect structure in Cu-Nb multilayers are (1) the formation of points defects and He bubbles and (2) the annihilation of in-plane misfit dislocations and the formation of out-of-plane dislocations without changing the order of magnitude of the dislocation density. It is noted that with decreasing layer thickness the volume fraction of He bubbles decreased and He atoms are rather trapped at Cu-Nb interfaces, grain boundaries, or dislocations in the form of Hevacancy clusters [25]. For 5 nm layer thickness approximately 32% of He atoms are in these clusters. Due to He implantation the hardness of Cu-Nb multilayers with different layer thicknesses increased by 20%e50%, mainly owing to the dispersion strengthening effect of He bubbles and vacancy clusters. The effect of He implantation on the defect structure and hardness was also studied in magnetron sputtered Cu-Mo, Cu-Nb, and Cu-V nanolayered composites (multilayers with alternating fcc/bcc structures) [26]. The foil and layer thicknesses were 1.2 mm and 5 nm, respectively. The He2þ ion energies and doses were varied to produce a controlled distribution of bubbles, in which the majority of He bubbles formed at the interfaces. It was shown that the misfit dislocation intersections at the interfaces are referred sites for He bubble nucleation. The density of these intersections was different in the three multilayers. However, the implanted He concentration was tuned to reach an approximately equal average He content in the bubbles for the three materials. The size of He bubbles was w2 nm and they contained about 30e40 He atoms. The order of magnitude of the density of He bubbles was w1023 m3. Micropillar compression tests showed that He implantation increased the maximum value of the flow stress by about 10% for all the three layers [26]. The maximum flow stress values were observed at true strains between 0.1 and 0.2. Thus, it can be concluded that the strengthening effect of interfacial bubbles with a given He content is proportional to the strength of the unimplanted multilayer. This can be explained by a preferred nucleation of He bubbles at interfacial misfit dislocation intersections. The strength of the unimplanted multilayers was mainly determined by the density of these intersections and the shear resistance of interfaces. It was suggested that dislocations at interfaces dissociate, and the larger the degree of dissociation, the more difficult the constriction, i.e., the interface has higher shear resistance [26]. When He bubbles with equal size formed at interfacial misfit dislocation intersections, they simply increased the strength of the interfaces with the same percentages (w10% in this case) for all multilayers.

171

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REFERENCES [1] T. Kolonits, P. Jenei, B.G. T
oth, Z. Czig
any, J. Gubicza, L. P
eter, I. Bakonyi, Characterization of defect structure in electrodeposited nanocrystalline Ni films, Journal of Electrochemical Society 163 (2016) D107eD114. [2] I. Petrov, P.B. Barna, L. Hultman, J.E. Greene, Microstructural evolution during film growth, Journal of Vacuum Science and Technology, A 21 (2003) S117eS128. [3] J. Gubicza, X-ray Line Profile Analysis in Materials Science, IGI-Global, Hershey, PA, USA, 2014. [4] H. Natter, R. Hempelmann, Nanocrystalline copper by pulsed electrodeposition: the effects of organic additives, bath temperature, and pH, Journal of Physical Chemistry 100 (1996) 19525e19532. [5] L.P. Bicelli, B. Bozzini, C. Mele, L. D’Urzo, A review of nanostructural aspects of metal electrodeposition, International Journal of Electrochemical Science 3 (2008) 356e408. [6] P. Vanden Brande, A. Dumont, R. Winand, Nucleation and growth of nickel by electrodeposition under galvanostatic conditions, Journal of Applied Electrochemistry 24 (1994) 201e205. [7] M. Kouncheva, G. Raichevski, St. Vitkova, The effect of sulphur and carbon inclusion on the corrosion resistance of electrodeposited Ni-Fe alloy coatings, Surface and Coating Technology 31 (1987) 137e142. [8] N. Shakibi Nia, J. Creus, X. Feaugas, C. Savall, Influence of metallurgical parameters on the electrochemical behavior of electrodeposited Ni and Ni-W nanocrystalline alloys, Applied Surface Science 370 (2016) 149e159. [9] G. Csisz
ar, S.J.B. Kurz, E.J. Mittemeijer, Stability of nanosized alloy thin films: faulting and phase separation in metastable Ni/Cu/Ag-W films, Acta Materialia 110 (2016) 324e340. [10] G. Csisz
ar, K. Pantleon, H. Alimadadi, G. Rib
arik, T. Ung
ar, Dislocation density and Burgers vector population in fiber-textured Ni thin films determined by high-resolution X-ray line profile analysis, Journal of Applied Crystallography 45 (2012) 61e70. [11] R.W. Cheary, E. Dooryhee, P. Lynch, N. Armstrong, S. Dligatch, X-ray diffraction line broadening from thermally deposited gold films, Journal of Applied Crystallography 33 (2000) 1271e1283. [12] D. Bufford, H. Wang, X. Zhang, High strength, epitaxial nanotwinned Ag films, Acta Materialia 59 (2011) 93e101. [13] K. Nyilas, A. Misra, T. Ung
ar, Micro-strains in cold rolled Cu-Nb nanolayered composites determined by X-ray line profile analysis, Acta Materialia 54 (2006) 751e755. [14] P.M. Anderson, J.F. Bingert, A. Misra, J.P. Hirth, Rolling texture in nanoscale Cu/Nb multilayers, Acta Materialia 51 (2003) 6059e6075. [15] K. Nyilas, Burgers vektor popul
aci
ok r€ ontgen diffrakci
os meghat
aroz
asa egykrist
alyokban
es erTsen text
ur
alt anyagokban (Ph.D. dissertation), Eotvos Lorand University, Budapest, Hungary, 2010. [16] G. Csisz
ar, A. Misra, T. Ung
ar, Burgers vector types and the dislocation structures in sputter-deposited Cu-Nb multilayers, Materials Science and Engineering A 528 (2011) 6887e6895. [17] G. Csisz
ar, Evolution of the Burgers-vector population of Cu-Nb multilayers with 7 at% Heimplantation determined by X-ray diffraction, Materials Science and Engineering A 609 (2014) 185e194. [18] G. Csisz
ar, L. Balogh, A. Misra, X. Zhang, T. Ung
ar, The dislocation density and twin-boundary frequency determined by X-ray peak profile analysis in cold rolled magnetron-sputter deposited nanotwinned copper, Journal of Applied Physics 110 (2011) 043502. [19] O. Anderoglu, A. Misra, J. Wang, R.G. Hoagland, J.P. Hirth, X. Zhang, Plastic flow stability of nanotwinned Cu foils, International Journal of Plasticity 26 (2010) 875e886. [20] X.-L. Wu, E. Ma, Dislocations in nanocrystalline grains, Applied Physics Letters 88 (2006) 231911. [21] A. Misra, J.P. Hirth, R.G. Hoagland, J.D. Embury, H. Kung, Dislocation mechanisms and symmetric slip in rolled nano-scale metallic multilayers, Acta Materialia 52 (2004) 2387e2394. [22] M.D. Gram, J.S. Carpenter, E.A. Payzant, A. Misrad, P.M. Anderson, X-ray diffraction studies of forward and reverse plastic flow in nanoscale layers during thermal cycling, Materials Research Letters 1 (2013) 233e243. [23] J. Wang, N. Li, O. Anderoglu, X. Zhang, A. Misra, J.Y. Huang, J.P. Hirth, Detwinning mechanisms for growth twins in face-centered cubic metals, Acta Materialia 58 (2010) 2262e2270.

Lattice Defects in Nanocrystalline Films and Multilayers

[24] N. Li, J. Wang, J.Y. Huang, A. Misra, X. Zhang, Influence of slip transmission on the migration of incoherent twin boundaries in epitaxial nanotwinned Cu, Scripta Materialia 64 (2011) 149e152. [25] N. Li, M. Nastasi, A. Misra, Defect structures and hardening mechanisms in high dose helium ion implanted Cu and Cu/Nb multilayer thin films, International Journal of Plasticity 32e33 (2012) 1e16. [26] N. Li, M. Demkowicz, N. Mara, Y. Wang, A. Misra, Hardening due to interfacial He bubbles in nanolayered composites, Materials Research Letters 4 (2016) 75e82.

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CHAPTER 7

Correlation Between Defect Structure and Mechanical Properties of Nanocrystalline Materials 7.1 EFFECT OF GRAIN SIZE ON DEFORMATION MECHANISMS IN FCC AND HCP NANOMATERIALS In conventional coarse-grained (CG) metallic materials, the deformation is mainly controlled by the motion of full lattice dislocations in the grain interiors. Grain boundaries act as obstacles against dislocation motion, therefore the dislocations formed at FrankeRead (FR) sources inside the grains are accumulated in pileups, e.g., at grain boundaries. The dislocations in pileups apply repulsive forces to successive dislocations emitted by the sources. As the grain size decreases, the sources are located closer to the grain boundaries; therefore the emission of dislocations from the FR sources becomes more difficult. Thus, for very small grain sizes (10e20 nm) dislocation pileups do not form and the dislocation sources are rather located in the grain boundaries. In this case, the dislocations emitted from a boundary travel throughout the grain and annihilated in the opposite boundary. Experiments and theoretical modeling revealed that when the grain size decreases below w10 nm, rather partial dislocations are emitted from grain boundaries instead of perfect lattice dislocations [1e7]. Nanostructured metals produced by severe plastic deformation (SPD) techniques often have nonequilibrium grain boundaries that contain excess dislocations (referred to as extrinsic or nongeometrically necessary or statistically stored dislocations) in addition to the geometrically necessary ones. Some dislocations may dissociate into pairs of Shockley partials, which could move away from the grain boundary under a stress. Such grain boundaries thus act as partial dislocation sources. However, partial dislocations can also be emitted from grain boundaries by atomic reshuffling [2]. The effect of grain size on the emission of partial dislocations from grain boundaries and the related deformation twinning has been studied for face-centered cubic (fcc) nanostructures on the model depicted in Figs. 7.1e7.3 [8,9]. The model assumes a grain with a square (111) slip plane, as shown in Fig. 7.1. Under an external shear stress s, a leading Shockley partial dislocation with the Burgers vector b1 is emitted from the grain boundary XY. This leading partial dislocation consists of the three segments XE, EF, and FY in Fig. 7.1. s is oriented at an angle b with line EF. The ends of this partial dislocation are pinned at the triple junctions X and Y, so that two segments of Defect Structure and Properties of Nanomaterials ISBN 978-0-08-101917-7, http://dx.doi.org/10.1016/B978-0-08-101917-7.00007-4

© 2017 Elsevier Ltd. All rights reserved.

175

Defect Structure and Properties of Nanomaterials

F

(111)

bo

E

Y

b2

rib

β

n

b1

τ

b

D

SF

176

d

C

b

X

Figure 7.1 A schematic illustration of the model of a perfect screw dislocation emitted from grain boundary XY and dissociated into two partials in the slip plane (111). The stacking fault (SF) ribbon between the partials is indicated by color. (After Y.T. Zhu, X.Z. Liao, X.L. Wu, Deformation twinning in nanocrystalline materials, Progress in Materials Science 57 (2012) 1e62 with permission from Elsevier.)

C B A C D b1

b1 (111)

τ β b1 C

Y

A

Twin boundary

B

SF

C

b1

SF

B

Twin boundary

B

X

A d

Figure 7.2 A schematic illustration of the dislocation model for the nucleation of deformation twins. The stacking sequence of the (111) planes is indicated by the letters A, B, and C. SF, stacking fault. (After Y.T. Zhu, X.Z. Liao, X.L. Wu, Deformation twinning in nanocrystalline materials, Progress in Materials Science 57 (2012) 1e62 with permission from Elsevier.)

partial dislocation lines XE and FY are deposited on grain boundaries. Similarly, a trailing Shockley partial dislocation with the Burgers vector b2 is emitted from the grain boundary XY. This partial consists of the segments XC, CD, and DY. The two partial segments EF and CD are separated by a stacking fault (SF) ribbon. The two partials react to

Correlation Between Defect Structure and Mechanical Properties of Nanocrystalline Materials

b

D b1

SF

τ

β

Y

b2

d

(111) C

b

X

Figure 7.3 A schematic illustration of the emission of a trailing partial from grain boundary XY that partially removes the stacking fault formed previously by passing a leading partial. (After Y.T. Zhu, X.Z. Liao, X.L. Wu, Deformation twinning in nanocrystalline materials, Progress in Materials Science 57 (2012) 1e62 with permission from Elsevier.)

form two perfect edge dislocation segments XC and YD with the Burgers vector b [ b1 D b2 at grain boundaries. The partial segments EF and CD would form a perfect screw dislocation with the Burgers vector b if they reacted. For the formation of a single leading partial, s has to perform the work needed to increase both the area of the SF behind the partial and the length of the dislocation segments XE and FY. Using the formulas for the forces acting on the leading partial dislocation segment (see Chapter 4), the critical stress, sP for moving the leading partial XEFY can be derived as [9] "pffiffiffi pffiffiffi # 1 Ga$ð4  nÞ 6g 2d þ pffiffiffi ; (7.1) sP ¼ ln  cosðb  30 Þ a a 8 6pð1  nÞ$d where G is the shear modulus, a is the lattice parameter, n is the Poisson’s ratio, g is the stacking fault energy (SFE), and d is the grain size. The shear stress, sL, needed to move the leading and trailing partials together is equivalent to the stress for moving a full screw lattice dislocation. sL has to overcome the work needed to lengthen the lattice dislocation segments XC and YD, and can be obtained as [9]: pffiffiffi Ga 2d sL ¼ pffiffiffi . (7.2) ln a 2 2pð1  nÞ$d cos b When sP < sL, the deformation is performed by gliding of partials. For instance, in Al for b ¼ 135 degrees and below the grain size of 17 nm, sP is smaller than sL [8].

177

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Defect Structure and Properties of Nanomaterials

Moreover, the slipping of partial dislocations on adjacent planes may yield twin nucleation. After formation of an SF by slipping of a partial dislocation across the grain, a twin may nucleate via the emission of a second leading partial from the grain boundary on a plane adjacent to the SF (see Fig. 7.2). After slipping of the two partials on adjacent planes, the resulted stacking sequence is shown by letters A, B, and C in Fig. 7.2. The sequence of the (111) planes reveals that a thin twin lamella has been formed whose boundaries are also indicated in Fig. 7.2. The critical twin nucleation stress, stwin can be expressed as [9] pffiffiffi Ga$ð4  nÞ 2d . (7.3) stwin ¼ pffiffiffi ln  a 8 6pð1  nÞ$d cosðb  30 Þ Another possible scenario is that after the slipping of the first leading partial across the grain, a trailing partial is emitted from the boundary XY that slips on the SF plane and erases the SF in its path as can be seen in Fig. 7.3. The trailing partial requires the following stress to move [9]: pffiffiffi pffiffiffi  pffiffiffi 1 6ð8 þ nÞ$Ga 6g 2d strail ¼ ln  . (7.4) cosðb þ 30 Þ 48pð1  nÞ$d a a Once a twin is nucleated, it may grow via the emission of more twinning partials on the adjacent planes under stress stwin. However, the twin may shrink via the emission of a trailing partial with the Burgers vector b2 on a plane adjacent to the twin boundary inside the twin (not in the parent grain). The stress needed to move a shrinking partial can be expressed as [9] pffiffiffi pffiffiffi 2d 1 6$ð8 þ nÞ$Ga sshrink ¼ . (7.5) ln  a cosðb þ 30 Þ 48pð1  nÞ$d As an example, Fig. 7.4 shows the grain size dependence of the stresses sP, sL, stwin, strail, and sshrink for nanocrystalline Al. The stresses are calculated using G ¼ 26 GPa, a ¼ 0.405 nm, n ¼ 0.34, g ¼ 166 mJ/m2, and b ¼ 30 degrees. The point M in Fig. 7.4 represents the critical grain size (dC z 4.7 nm) below which a deformation twin nucleates because stwin < strail. However, when a deformation twin nucleates by partial dislocation slipping on a plane (111), an initial SF is needed on the adjacent plane. Fig. 7.4 shows that at the critical grain size w4.7 nm sL < sP; therefore the stress corresponds to the point N yields the operation of full lattice dislocation but it is not enough for the emission of partial dislocation from grain boundary that is necessary for the formation of the initial SF. At the same time, for the values of sL at point N and dC, the splitting distance between partials in a lattice dislocation (the SF width) reaches the grain size [8]. This means that the SF ribbon of a dissociated full dislocation spreads across the grain and this will serve as an initial SF for the nucleation of a deformation twin. It should

Correlation Between Defect Structure and Mechanical Properties of Nanocrystalline Materials

3000 Al

4.7 nm

Shear stress (MPa)

2500

τ shrink

2000

τtrail

1500

τP N

1000

τtwin

τL M

500 0 0

5

10 d (nm)

15

20

Figure 7.4 The shear stresses sP, sL, stwin, strail, and sshrink as a function of the grain size, d, for nanocrystalline Al. The angle between the stress direction and the line of slipping dislocations is b ¼ 30 degrees.

be emphasized that although the twin nucleation at the critical grain size dC requires a high stress corresponding to point N, the nucleated twin can grow under a much smaller stress corresponding to point M in Fig. 7.4 [9]. It is noted that the grain size dependence of the stresses sP, sL, stwin, strail, and sshrink is strongly influenced by their orientation to the possible slip systems, i.e., by b, therefore the deformation mechanism may change from grain to grain in a nanocrystalline material. Experiments also show that the operation of partial dislocations may produce deformation twins, even in nanostructured aluminum having high SFE [3e5], which in its CG state never deforms by twinning except at crack tips. For materials with lower SFE, the critical grain size and the stress required for twinning are increased and decreased, respectively. For instance, while in the case of Al the critical grain size and the stress for twin nucleation are w5 nm and w1000 MPa, respectively, these values for Cu are 46 nm and 370 MPa, respectively [9]. The probability of twinning in hexagonal close-packed (hcp) structures decreases with decreasing grain size [10] that can be explained by the stress dependence of deformation mechanisms. In an hcp metal, the strain along the basal plane is resulted by the easy glide of dislocations with hai-type Burgers vectors. At the same time, the deformation in the direction of the crystallographic c-axis may be performed by glide of dislocations with hc þ ai-type Burgers vectors or by twinning (both processes operate mainly on pyramidal   planes). The most frequently observed twinning planes and directions are f10:1g 10:2     and f11:2g 11:3 (compressive twins), and f10:2gh10:1i and f11:1g 11:6 (tensile twins). The generation and propagation of hc þ ai dislocations is difficult due to their large Burgers vectors. However, an elevated deformation temperature and/or high stresses

179

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Defect Structure and Properties of Nanomaterials

increase the activation of pyramidal hc þ ai dislocations at the expense of twin formation. In deformed hcp metals with smaller grain size, the stresses are higher mainly in the vicinity of grain boundaries [11] that facilitate the operation of hc þ ai dislocations in addition to hai dislocations, thereby reducing the occurrence of twinning. It is noted that deformation twins are observed in nanocrystalline Mgd10 at.% Ti alloy with an average grain size of 33 nm processed by mechanical attrition. The formation of deformation twins is attributed to the alloying effect, which may change the energy path for twinning [12]. As the grain boundary fraction in nanomaterials increases, the relative importance of deformation mechanisms occurring at grain boundaries increases. One example for these deformation mechanisms is grain boundary sliding which is controlled by grain boundary diffusion. Grain boundary diffusion is much faster than crystal lattice diffusion inside grains, since the activation energy of the former process is about half of that for the latter mechanism. However, in CG materials (d > 10 mm) the contribution of grain boundary diffusion to plastic deformation (Coble creep) is marginal due to the small volume fraction of grain boundaries. For these materials, lattice diffusion inside grains is fast enough for controlling plastic deformation (NabarroeHerring creep) only at temperatures above 0.5  Tm, where Tm is the absolute melting temperature. At the same time, in nanomaterials, grain boundary sliding by Coble creep may operate even at room temperature (RT) as very high fraction of atoms are located in the vicinity of grain boundaries. Fig. 7.5 shows σ

σ

σ

σ

Figure 7.5 Schematic illustration of the AshbyeVerrall model for grain boundary sliding. The arrows in the grains indicate the sliding directions along the grain boundaries. At the bottom of the figure, the arrows show the directions of diffusion in the vicinity of grain boundaries that yields the change of the grain shape as a complementary process in addition to grain boundary sliding.

Correlation Between Defect Structure and Mechanical Properties of Nanocrystalline Materials

the AshbyeVerrall model of grain boundary sliding [13,14]. The grains are sliding at their boundaries due to the external tensile stress. The sliding of the grains without any change of their shape would yield voids between them, therefore the integrity of the material is sustained by a complementary diffusion in the vicinity of the grain boundaries as it is shown at the bottom part of Fig. 7.5. The combination of sliding and diffusion yields a strain of 0.55 in this model. For metallic materials with large melting points grain boundary sliding may contribute to plasticity at RT even if the grain size is only in the ultrafine-grained (UFG) regime, i.e., between 100 nm and 1 mm. For instance, atomic force microscopy (AFM) study shows rumpling on the surface around a Vickers pattern in pure Al processed by equal-channel angular pressing (ECAP) and having a grain size of about 1 mm [15,16]. The quantitative evaluation of the AFM measurements reveals that adjacent grains may slide up to w30 nm over each other and the contribution of grain boundary sliding to the total strain is about w70% during indentation of the ECAP-processed Al sample. Other experimental investigations [17e19] and three-dimensional molecular dynamics computer simulations [20,21] also suggest the occurrence of grain boundary sliding as a viable deformation process in nanocrystalline solids at low temperatures. In addition to grain boundary sliding, other boundary-related deformation mechanism, such as grain rotation is also observed in nanomaterials [22]. During this process, nanosized grains rotate in a manner that brings their orientations closer together. The disappearance of the boundary between these grains provides a path for more extended dislocation motion [23].

7.2 BREAKDOWN OF HALLePETCH BEHAVIOR IN NANOMATERIALS Considering the grain boundary strengthening in polycrystalline CG metallic materials, the yield strength is usually correlated to the grain size by applying the HallePetch equation [24e26]. In a grain of a polycrystal, the plastic deformation sets in when the resolved shear stress in one or more slip systems reaches the critical value necessary for the initiation of dislocation glide. In a polycrystalline material, the grains have different crystallographic orientations relative to the direction of the external load, therefore the resolved shear stresses acting on the various slip systems also change from grain to grain. In the grains with favorable orientations (i.e., having slip systems with large Schmid factors), the resolved shear stresses are high and the plastic deformation starts by the activation of one or more slip systems. In unfavorably oriented grains, the shear stresses are not enough to initiate dislocation glide, therefore these grains deform only elastically. In the plastically deformed grains, dislocations are formed at the (FR) sources, and these dislocations are accumulated in pileups at the grain boundaries as depicted for grain A in Fig. 7.6. The elastic stresses of dislocations in the pileups are added to the shear stresses caused by the external load, thereby assisting the initiation of plastic deformation in the adjacent, elastically deformed grain having unfavorable orientation relative to the applied

181

Defect Structure and Properties of Nanomaterials

σ

B

FR

182

A τ

FR

FR τ

d

σ

Figure 7.6 Two grains A and B with the size d in a polycrystalline material loaded by a tensile stress s. The dashed lines represent the slip planes. In grain A, the plastic deformation has already been initiated by the external stress and a FrankeRead (FR) source emits dislocations that are gliding due to the shear stress s. The stress field of the dislocations accumulated at the boundary in grain A assists the activation of FR sources in the unfavorable oriented grain B.

force (grain B in Fig. 7.6). The macroscopic plastic deformation sets in when all grains start to deform plastically. As the repulsive force between dislocations in pileups is inversely proportional to dislocation spacing, the smaller the grain size, the lower the number of dislocations in the pile-ups. As a consequence, when the grain size is reduced, the contribution of the stress field of dislocations to the initiation of plasticity in the adjacent grain is smaller; therefore larger applied stress is necessary for yielding the material, i.e., the yield strength will be higher. Thus, there is an inverse relationship between grain size and yield strength, as demonstrated by the HallePetch equation [24e26]: sY ¼ s0 þ kY $d1=2 ;

(7.6)

where sY is the yield strength, s0 is the friction stress, kY is the strengthening coefficient (a constant unique to each material), and d is the average grain diameter. For instance, the values of s0 and kY are w25 MPa and w5000 MPa nm1/2, respectively, for Cu [26] as illustrated in Fig. 7.7. When the grain size is reduced to the range 20e100 nm, the yield strength further increases with decreasing grain size but the slope kY continuously decreases (see Fig. 7.7) [23]. This deviation from the conventional HallePetch behavior can be attributed to the grain size dependence of the stress (s) required to operate an FR source [32,33]. If an edge dislocation segment pinned at its ends is considered as an FR source

Correlation Between Defect Structure and Mechanical Properties of Nanocrystalline Materials

1000

1000 100

20

10

5

d (nm)

σ Y = 25 + 5000 x d –1/2

σY (MPa)

800

600

400

Chokshi et al. Hansen et al. Fougere et al. Sanders et al. Lu et al. Gubicza et al.

200

0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

d –1/2 (nm –1/2)

Figure 7.7 The HallePetch plot of the yield strength (sY) versus the grain size (d) for Cu (Chokshi et al. [27], Hansen et al. [26], Fougere et al. [28], Sanders et al. [29], Lu et al. [30], Gubicza et al. [31]).

inside a grain and n ¼ 0.33 is chosen for the Poisson’s ratio, the stress s versus the length of the segment L can be given as     0:12Gb L s ¼ ln  1:653 ; (7.7) L b where b is the magnitude of the Burgers vector and G is the shear modulus [33]. s as a function of L1/2 for Cu is plotted in Fig. 7.8. When the length of the dislocation segment is large, the stress needed for the operation of a source increases more or less linearly with L1/2. However, when the length of an FR source is reduced below 10e15 nm, the slope of s versus L1/2 plot decreases. Moreover, if L is smaller than 4 nm in Cu, the stress required for the operation of an FR source decreases with the reduction of its length. In materials with grain sizes between 20 and 100 nm, a considerable fraction of FR sources may have smaller lengths than 10e15 nm that can explain the deviation from the conventional HallePetch behavior. Additionally, the strong reduction of the numbers of dislocations in pileups with decreasing grain size yields a decrease of the exponent of d in Eq. (7.6) from 1/2 to 1 which may cause the observed reduction of kY in the HallePetch plot [16]. The value of 1 for the HallePetch exponent is also predicted by the model in which two dislocations locate within one grain with the distance of d/3 between them, where d is the grain diameter [34]. As the grain size decreases to 10e20 nm, the number of dislocations in pileups is eventually reduced to one and thus the pileup mechanism breaks down [23]. As a consequence, the assisting effect of the stress field of dislocation pileups on the initiation of

183

Defect Structure and Properties of Nanomaterials

1000 100

20

10

5

L (nm)

2

500

400

τ

L τ (nm)

184

300

200

100

0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

L–1/2 (nm–1/2)

Figure 7.8 The stress (s) required for the operation of a FrankeRead source versus the length (L) of the edge dislocation segment between the pinning points of the source in Cu.

plastic deformation is lost and the HallePetch equation is not valid. In this grain size regime, if dislocations contribute to plasticity, they are emitted from grain boundaries, then travel through the grains with a small chance of interaction with other dislocations, and finally annihilate at the opposing boundaries. Below 10e20 nm, the experimental results do not show a clear trend in the grain size dependence of the yield strength. In some cases, the yield strength is found to decrease with the reduction of grain size while in other cases only a plateau is observed (see Fig. 7.7). When the yield strength decreases with decreasing grain size, the HallePetch plot remains more or less linear even below the grain size of 10e20 nm but with a negative slope; therefore this behavior is termed as negative or inverse HallePetch behavior. This effect is usually attributed to the dominance of deformation mechanisms occurring in the grain boundaries as the reduced grain size is accompanied by the increase of grain boundary fraction. Such a mechanism is grain boundary sliding that may occur by grain boundary diffusion controlled Coble creep [35]. In this case, the stress (s) required for yielding at a strain rate of ε_ can be given by the following formula [23]: s ¼

pkT ε_ d3 ; 150UdDGB

(7.8)

where k is the Boltzmann’s constant, T is the absolute temperature, U is the atomic volume, d is the grain boundary width, and DGB is the grain boundary diffusion coefficient. Assuming U ¼ 1.3  1029 m3, d ¼ 1 nm, DGB ¼ 4.7  1020 m2/s at RT, and ε_ ¼ 104 for Cu with d ¼ 20 nm, the stress required for Coble creep is

Correlation Between Defect Structure and Mechanical Properties of Nanocrystalline Materials

112 MPa as obtained from Eq. (7.8), which is much less than the calculated stress needed for dislocation glide (w800e900 MPa). The easy occurrence of Coble creep suggests its significant contribution to plasticity at RT for grain sizes less than 20 nm. Moreover, Eq. (7.8) predicts that when the grain boundary sliding becomes the main deformation mechanism, the yield strength will decrease with decreasing grain size, which corresponds to a considerable fraction of experimental results. However, Coble creep cannot be the sole deformation mechanism in the inverse HallePetch behavior as the experimental yield strength values do not fall so steeply with decreasing grain size as the trend, s w d3, predicted by Eq. (7.8) [36]. For instance, for inert gasecondensed Cu the inverse HallePetch effect was described as sY ¼ A  k0 $d1=2 ;

(7.9)

where A z 1290 MPa and k0 z 2170 MPa nm1/2 [27]. Most probably, intragranular dislocation motion as a complementary deformation mechanism also occurs to sustain the integrity of the material as the sliding of grains without changing their shape would yield voids between them. It should be also noted that the voids, micro- or nanocracks, and poorly bonded interfaces in nanomaterials processed by powder consolidation (e.g., inert gas condensation) may also yield a similar strength degradation as the inverse HallePetch behavior [23]. Although, the experimental results obtained on different nanomaterials do not prove convincingly the general existence of the inverse HallePetch effect, molecular dynamic simulations predicts this behavior for very small grain sizes [37]. It is noted, however, that the results of molecular dynamic simulations are valid only for very high strain rates (108e109 s1). Moreover, the change of the deformation mechanism inside the grains when their sizes decrease below w20 nm may also contribute to the inverse HallePetch behavior. This is illustrated in Fig. 7.9, where the shear stresses required for the emission of full lattice (sL), leading partial (sP), trailing partial (strail), twinning partial (stwin), and shrinking partial (sshrink) dislocations from grain boundaries are plotted as a function of grain size in Cu. These stresses are calculated according to Eqs. (7.1)e(7.5) with assuming G ¼ 47 GPa, a ¼ 0.361 nm, n ¼ 0.34, g ¼ 62 mJ/m2, and b ¼ 30 degrees. It seems that below w30 nm, twinning becomes a considerable deformation mechanism. For twin nucleation, first an initial SF should be formed at the stress corresponding to point N. Then, twinning occurs at the stress corresponding to point M that is much lower than that at point N. When the grain size falls below w20 nm (point L in Fig. 7.9), partial dislocations are emitted from the grain boundaries instead of full dislocations since the former mechanism requires smaller stress. In nanocrystalline materials, the grains have a size distribution. By decreasing the average grain size, more and more grains in the distribution become so small that their interiors deform rather by twinning which requires smaller stress than for dislocation glide in the large nanograins. Therefore, the macroscopic yield strength decreases with decreasing the average grain size in accordance with the inverse HallePetch behavior.

185

Defect Structure and Properties of Nanomaterials

3000 Cu 2500 Shear stress (MPa)

186

2000 1500 1000

L

τ twin

500

τ trail

τ shrink

τP N

τL

M

0 0

10

20

30 d (nm)

40

50

60

Figure 7.9 The shear stresses sP, sL, stwin, strail, and sshrink calculated from Eqs. (7.1)e(7.5) as a function of the grain size, d for nanocrystalline Cu. The angle between the stress direction and the line of slipping dislocations is b ¼ 30 degrees.

It is worth to note that the inverse HallePetch behavior depends on the strain rate if the reason of this behavior is the enhanced grain boundary sliding. For large strain rates, the role of this mechanism in plasticity is reduced, and dislocation gliding in grain interiors becomes dominant. In this case, softening is not observed with decreasing grain size. It should be also noted that for pure metals with low melting point (e.g., Al, Pb, Sn, or In), grain boundary sliding may yield softening at RT even if the grain size is refined only into the UFG regime [38]. In these materials, grain sizes of several hundreds of nanometers are enough for the dominance of grain boundary sliding at low strain rates. Fig. 7.10 shows the decrease in hardness at RT due to grain refinement in Pb, Sn, and In processed by high-pressure torsion (HPT). If deformation after HPT is carried out at high strain rates, most probably hardening can be observed for these materials. A decrease of hardness at RT due to grain refinement by SPD was also detected in Ale30 wt.% Zn alloy [39]. SPD was carried out by HPT at RT. Before HPT, the material was a supersaturated solid solution where the Zn concentration inside the Al(Zn) grains varied due to spinodal decomposition [40]. This lamellar structure in the coarse grains led to high hardness. HPT destroyed this lamellar structure and yielded an elemental decomposition of supersaturated Ale30 wt.% Zn alloy. Therefore, a refinement of Al(Zn) grains to about 300 nm and a precipitation of Zn particles occurred simultaneously. In addition, w3-nm thick Zn-rich layers were formed in AleAl grain boundaries [41]. The deterioration of the fine lamellar structure and the grain boundary wetting caused softening in the HPT-processed Ale30 wt.% Zn alloy. For Zn concentration smaller than 10 wt.%, this softening was not observed [39].

Correlation Between Defect Structure and Mechanical Properties of Nanocrystalline Materials

80 HPT processing Sn

Hardness (MPa)

60 Pb 40

20 Coarse-grained UFG

0 0.45

0.50

0.55

In 0.60

0.65

0.70

0.75

Homologous temperature, T/Tm

Figure 7.10 Hardness of coarse-grained and ultrafine-grained (UFG) Pb, Sn, and In versus the homologous temperature, T/Tm (Tm is the melting point). The arrows indicate softening at RT due to grain refinement achieved by high-pressure torsion (HPT) processing. (The data were taken from K. Edalati, Z. Horita, Significance of homologous temperature in softening behavior and grain size of pure metals processed by high-pressure torsion, Materials Science and Engineering A 528 (2011) 7514e7523.)

7.3 CORRELATION BETWEEN DISLOCATION STRUCTURE AND YIELD STRENGTH OF ULTRAFINE-GRAINED FCC METALS AND ALLOYS PROCESSED BY SEVERE PLASTIC DEFORMATION For UFG fcc metals processed by ECAP the dislocation density and the yield strength show similar developments as a function of strain and both quantities saturate approximately at the same strain [16,42]. This suggests that it is worthwhile studying the relationship between these quantities. The dislocation density determined by X-ray line profile analysis and the corresponding yield strength measured by mechanical tests for different fcc metals and alloys are listed in Table 7.1. The relationship between the dislocation density (r) and yield strength (sY) for plastically deformed metals is generally characterized by the Taylor equation: 1

sY ¼ s0 þ aM T Gbr2 ;

(7.10)

where s0 is the friction stress, a is a constant depending on the arrangement of dislocations, and MT is the Taylor factor. Since the value of aMT is of the order of 1, it is possible to check the validity of Eq. (7.10) by plotting the value of (sY  s0) against the product of Gbr1/2 for different fcc metals and solid solution alloys processed by SPD, as shown in Fig. 7.11. The good correlation between these two quantities in Fig. 7.11 demonstrates that in UFG fcc metals and solid solutions

187

Al Al Al Al Cu Cu Cu Cu Cu Cu Cu Au Ni Ni Ag

1 ECAP at RT 2 ECAP at RT 4 ECAP at RT 8 ECAP at RT 1 ECAP at RT 3 ECAP at RT 5 ECAP at RT 10 ECAP at RT 15 ECAP at RT 25 ECAP at RT 25 HPT at RT 4 ECAP at RT 6 ECAP 8 ECAP þ 5 HPT at RT 8 ECAP at RT

1.9 2.1 1.9 1.8 15 21 21 20 15 15 37 17 18 25 46

95 110 120 120 294 386 394 412 404 402 563 245 730 1103 330

20 20 20 20 35 35 35 35 35 35 35 27 60 60 29

26 26 26 26 47 47 47 47 47 47 47 27 82 82 30

0.286 0.286 0.286 0.286 0.256 0.256 0.256 0.256 0.256 0.256 0.256 0.288 0.249 0.249 0.289

0.24 0.27 0.32 0.33 0.18 0.21 0.21 0.23 0.26 0.26 0.24 0.22 0.25 0.33 0.17

[44] [44] [44] [44] [31] [31] [31] [31] [31] [31] [45] [46] [46] [44] [46]

8 1 8 5 5

3.9 12 23 34 81

220 250 400 773 987

25 50 50 35 44

26 26 26 46 41

0.286 0.286 0.286 0.256 0.256

0.43 0.25 0.32 0.35 0.33

[44] [44] [44] [47] [47]

4.0 5.0 5.4 3.2 3.4 24 39

193 246 265 290 380 750 416

25 25 25 20 20 20 35

26 26 26 26 26 26 47

0.286 0.286 0.286 0.286 0.286 0.286 0.256

e e e e e e e

[44] [44] [44] [48] [48] [48] [48]

Solid solution alloys

Ale1Mg Ale3Mg Ale3Mg Cue10% Zn Cue30% Zn

ECAP at RT ECAP at RT ECAP at RT HPT at RT HPT at RT

Dispersion-strengthened alloys

Al 6082 Al 6082 Al 6082 Ale4.8Zne1.2Mge0.14Zr Ale5.7Zne1.9Mge0.35Cu Ale5.9Mge0.3Sce0.18Zr Cue0.18% Zr

1 ECAP at RT 4 ECAP at RT 8 ECAP at RT 8 ECAP at 200  C 8 ECAP at 200  C 15 HPT at RT 8 ECAP at RT

The calculated values of a in the Taylor equation Eq. (7.10) for pure metals and solid solutions are also presented. s0, friction stress; b, Burgers vector; ECAP, equal-channel angular pressing; G, shear modulus; HPT, high-pressure torsion.

Defect Structure and Properties of Nanomaterials

Pure metals

188

Table 7.1 The dislocation density (r) determined by X-ray line profile analysis and the yield strength (sY) at room temperature (RT) for face-centered cubic (fcc) metals and alloys processed by severe plastic deformation Material Processing method r (1014 mL2) sY (MPa) s0 (MPa) G (GPa) b (nm) a References

Correlation Between Defect Structure and Mechanical Properties of Nanocrystalline Materials

1200 Al Ag Au Cu Ni

σY – σ0 (MPa)

900

Al-1 Mg Al-3 Mg Cu-10 Zn Cu-30 Zn

600

300

0 0

300

600

900

1200

Gbρ 1/2 (MPa)

Figure 7.11 The yield strength at room temperature (RT) reduced by the friction stress (sY  s0) versus the product of Gbr1/2 for different fcc metals and alloys processed by SPD. The errors on the individual datum points are indicated by solid horizontal and vertical lines at the symbol diamond. r, dislocation density; b, Burgers vector; G, shear modulus.

processed by SPD the yield strength is essentially determined by the interactions between dislocations. This observation can be explained by the fact that in materials processed by SPD at low homologous temperatures, the grain refinement occurs mainly by the arrangement of dislocations into subgrain boundaries and/or dislocation walls. Consequently, large fraction of boundaries consists of dislocations and the effect of subgrain/grain boundaries on dislocation motion can be regarded as the interaction between dislocations [43]. Concerning the solid solution strengthening, the solute atoms may increase the strength in two ways: (1) directly by pinning the dislocations and impeding their motion in a soluteedislocation interaction and (2) indirectly by hindering the annihilation of dislocations during deformation, leading to an increase in the dislocation density in a dislocationedislocation interaction. The direct and indirect effects of solute atoms are taken into account in the friction stress, s0, and the dislocation term in the Taylor equation. For severely deformed metals where the dislocation density is relatively high, the latter term is dominant so that the hardening effect of the solute atoms is manifested basically in the higher dislocation density. The scattering of the datum points in Fig. 7.11 is attributed to the difference between the values of a for the different metals. Therefore, the values of a in Table 7.1 are calculated from Eq. (7.10) by using the experimental values of sY and r. In this calculation, the Taylor factor is estimated by the value characteristic for a random crystallographic orientation (3.06) as weak textures in fcc metals after SPD, if any, yield only a deviation less than 3% from this value of MT [46]. Most of the values of a are between 0.17 and 0.35, and this range is similar than that observed for CG materials. In practice, the value of a

189

190

Defect Structure and Properties of Nanomaterials

depends upon the arrangement of dislocations in the material [49,50]. For example, calculations have shown that the value of a increases from w0.15 to w0.37 when dislocation clustering increases and the dislocation structure evolves from a uniform random distribution through thick cell walls to sharp boundaries [49]. The dissociation of lattice dislocations into partials in fcc metals has a strong effect on the arrangement of dislocations. In the formation of dense dislocation structures, as in walls or subgrain boundaries, an important role is played by cross-slip and climb. Thus, the higher the degree of dissociation, the more difficult the occurrence of cross-slip and climb. The degree of dislocation dissociation is characterized by the equilibrium splitting distance (dp) between the partial dislocations, which can be determined from the shear modulus, the Burgers vector, and the SFE using Eq. (4.3) given in Chapter 4. The values of dp are listed in Table 4.1 of Chapter 4 for edge and screw dislocations in pure fcc metals. The values of a obtained for pure fcc metals processed by ECAP at RT till the saturation of the dislocation density are plotted in Fig. 7.12 as a function of the arithmetic average of splitting distances for edge and screw dislocations in Burgers vector unit, dp/b. The higher the value of dp/b, the higher the degree of dislocation dissociation which impedes the clustering of dislocations inside the grains resulting in a relatively low value of a. It should be noted that, beside the interaction between dislocations, the relatively high twin boundary frequency in Ag (see Chapter 4) is expected to make an additional contribution to the strength. However, if this hardening effect is taken into account, the Taylor-type contribution to the total yield strength will be reduced and this will lead to an even 0.4 Al

0.3

Ni

α

0.2 Au Cu

Ag

0.1 0

4

8

12

16

dp /b

Figure 7.12 The value of a in the Taylor equation as a function of the equilibrium splitting distance between partials in Burgers vector unit (dp/b) for pure fcc metals processed by equal-channel angular pressing at room temperature till saturation of the dislocation density. (Reprinted from J. Gubicza, N.Q.  ba r, S. Dobatkin, Z. Heged} Chinh, J.L. La us, T.G. Langdon, Correlation between microstructure and mechanical properties of severely deformed metals, Journal of Alloys and Compounds 483 (2009) 271e274 with permission from Elsevier.)

Correlation Between Defect Structure and Mechanical Properties of Nanocrystalline Materials

0.36 AI

0.32

Cu 0.28

α 0.24

0.20

0.16

0

5

10

15

20

25

Number of ECAP passes

Figure 7.13 The value of a in the Taylor equation as a function of the number of equal-channel angular pressing (ECAP) passes for pure Al and Cu.

smaller value of a for Ag, providing additional confirmation that the higher splitting distance is accompanied by a smaller value of a. In the cases of Cue10% Zn and Cue30% Zn, the values of a are most probably also overestimated in Table 7.1 due to the large twin boundary hardening. It is noted that a varies with increasing strain in SPD processing of pure fcc metals. Fig. 7.13 plots the value of a as a function of the number of ECAP passes for pure Al and Cu. The increase of the value of a with increasing strain is explained by the thinning of the grain/subgrain boundaries and the evolution to a more equilibrated structure as discussed in Chapter 3. It is worthwhile to note that in the case of Cu, a saturates only after 15 ECAP passes, although the maximum dislocation density has already been achieved after 5 passes indicating that the evolution of the dislocation structure continues even after the saturation of its density. The yield strength values for some dispersion-strengthened UFG alloys are also presented in Table 7.1 but for these materials the value of a is not determined due to the strength contribution of the secondary phase dispersoids (precipitates). Considering nonshearable dispersoids in UFG alloys and assuming the additivity of the different strengthening contributions, the yield strength may be expressed by the following relationship [51]: sy ¼ s0 þ aMGbr1=2 þ 0:85M

Gb lnðx=bÞ ; 2pðL  xÞ

(7.11)

where x is the average size of dispersoids and L is the average distance between them. For instance, in supersaturated Ale4.8Zne1.2Mge0.14Zr and Ale5.7Zne1.9Mge0.35Cu (wt.%) alloys processed by eight passes of ECAP at 200  C, stable incoherent MgZn2

191

192

Defect Structure and Properties of Nanomaterials

precipitates (h-phase particles) form [48]. The different contributions to yield strength are calculated from Eq. (7.11) using the values of L and x obtained from transmission electron microscopy (TEM) images and the dislocation density determined by X-ray line profile analysis. The average size of the precipitates in the Ale4.8Zne 1.2Mge0.14Zr and Ale5.7Zne1.9Mge0.35Cu alloys are 30 and 20 nm, respectively [48]. The average distance, L, between the precipitates is approximately 120 nm in the Ale4.8Zne1.2Mge0.14Zr alloy and 80 nm in the Ale5.7Zne1.9Mge0.35Cu alloy. The value of s0 was taken as 20 MPa measured for pure Al [52] because the lattice parameter of the matrix obtained from the X-ray diffractograms agrees with the value for pure Al within the experimental error. The strength contributions originating from the dislocation density given by the second term in Eq. (7.11) are 133 and 137 MPa for Ale4.8Zne1.2Mge0.14Zr and Ale5.7Zne1.9Mge0.35Cu, respectively, as the dislocation densities in the two matrices are very close to each other. The third term in Eq. (7.11) related to the dislocationeprecipitation interaction gives 156 and 214 MPa for Ale4.8Zne1.2Mge0.14Zr and Ale5.7Zne1.9Mge0.35Cu, respectively, where the difference between the two values is due to the different size and dispersion of precipitates [48]. Thus, the smaller particles with a more dense distribution lead to a higher strength for the Ale5.7Zne1.9Mge0.35Cu alloy. Finally, the sum of the three components leads to estimated total strengths of 309 and 371 MPa, which are in good agreement with the values of 290 and 350 MPa determined by mechanical testing for the Ale4.8Zne1.2Mge0.14Zr alloy and the Ale5.7Zne1.9Mge0.35Cu alloy, respectively (see Table 7.1).

7.4 DEFECT STRUCTURE AND DUCTILITY OF NANOMATERIALS Beside the yield strength, the ductility of UFG and nanomaterials is also an important parameter describing the mechanical behavior. The ductility is usually characterized by the maximum elongation or the maximum true strain (εmax) achieved until inhomogeneous deformation (necking) sets in during tension. After necking, the deformation is localized mainly to the neck, and due to the decrease of the cross section at the neck, the force required for further straining decreases, i.e., the deformation becomes unstable (it does not stop immediately when the force is reduced). As a consequence of inhomogeneous deformation after necking, the true stress (s) and true strain (ε) vary along the longitudinal axis of the specimen; therefore the following usual formulas for the calculation of s and ε can only be applied before necking: l ε ¼ ln ; l0

(7.12a)

Al ¼ A0 l0 ;

(7.12b)

Correlation Between Defect Structure and Mechanical Properties of Nanocrystalline Materials

s ¼

F ; A

(7.12c)

where l and l0 are the actual and initial lengths of the sample, respectively, F is the applied force, A and A0 are the actual and initial cross sections of the sample, respectively. Eq. (7.12b) reflects the unchanged volume during plastic deformation. The deformation is stable only in the homogeneous regime, i.e., when the first derivative of the force over the sample length, dF dl , is larger than zero. Using Eqs. (7.12aec), this criterion can be expressed by the true stress and true strain as ds  s. dε

(7.13)

UFG

σ UFG

⎛dσ ⎝dε

⎛ ⎝

⎛dσ ⎝dε

⎛ ⎝

True stress, σ Work hardening rate, dσ/dε

This formula is referred as Considere criterion. In the case of UFG and nanomaterials, the flow stress at RT is large even in the beginning of tension due to the high initial dislocation density and/or the small grain size, and it tends to saturate at small strain values. As a result, for UFG and nanomaterials the true stress, s becomes equal to the work hardening rate, ds dε at smaller strains than for CG counterparts, i.e., the ductility decreases with decreasing grain size as illustrated in the schematic picture of Fig. 7.14. As an example, Fig. 7.15 shows the grain size dependence of ductility at RT for interstitial-free steel processed by accumulative roll bonding [53]. The grain size varies between 200 nm and 20 mm, and the ductility is described by the uniform and total elongations. The former and the latter quantities are determined when dF dl becomes zero and after the failure of the sample, respectively. Fig. 7.15 reveals that a large loss of ductility occurs when the grain size decreases below 1 mm. Similar effect is also observed for Al alloys processed by SPD [53]. In addition to the small value

CG

σ CG

(εmax)UFG

(εmax)CG

True strain, ε

Figure 7.14 Schematic illustration of decreasing ductility (εmax) with decreasing grain size according to Considere criterion. CG, coarse-grained material; UFG, ultrafine-grained material.

193

Defect Structure and Properties of Nanomaterials

60 Uniform elongation

Uniform and total elongation (%)

194

50

Total elongation

40 30 20 10 0 0.1

1 Grain size (μm)

10

Figure 7.15 Grain size dependence of uniform and total elongations at room temperature for interstitial free steel with various mean grain sizes. (The data are taken from N. Tsuji, Y. Ito, Y. Saito, Y. Minamino, Strength and ductility of ultrafine grained aluminum and iron produced by ARB and annealing, Scripta Materialia 47 (2002) 893e899.)

of work hardening rate, the considerable porosity and amount of oxide phase at the interfaces in UFG and nanomaterials consolidated from nanopowders (NPs) also reduce the ductility. The relative fractions of pores and oxide particles usually increase with decreasing grain size as a result of the larger specific surface area for a smaller initial particle size. The porosity and the oxide phase may yield weaker particle bonding resulting in an easier cracking under external loads. Generally, in UFG and nanomaterials the total elongation is typically less than a few percent and the regime of uniform deformation is even smaller, which limits their practical utility. For materials whose plastic deformation is sensitive to strain rate ð_εÞ, the strain regime of homogeneous deformation is determined by Hart’s stability criterion: ds  sð1  mÞ; dε where m is the strain rate sensitivity parameter defined as   vln s . m ¼ vln_ε T

(7.14)

(7.15)

The value of m is usually between 0 and 1, but it depends on the microstructure, the temperature, and the strain rate of deformation. The higher strain rate sensitivity helps to sustain the homogeneous deformation since the material hardens at the neck when the strain rate increases in the beginning of necking that yields larger ductility. If the value

Correlation Between Defect Structure and Mechanical Properties of Nanocrystalline Materials

of m is larger than 0.3, very large strains (higher than 100%) can be achieved during tension and this behavior is termed as superplasticity. Superplastic deformation in CG metals and alloys is usually observed at low strain rate (105e102 s1), high homologous temperature (>0.5Tm, where Tm is the melting point), and for relatively small grain size (area Þ, the dislocation density (r), and the twin boundary frequency (b) were

231

Defect Structure and Properties of Nanomaterials

(b)

(a)

120

80

100 60 ρ (1014 m–2)

area (nm)

40

20

0

Cu-CNT-RT Cu-CNT-373 Cu

Center

80 60 40 Cu-CNT-RT Cu-CNT-373 Cu

20 Half-radius

0 Periphery

Center

Half-radius

Periphery

(c) 1.5 1.2

β (%)

232

Cu-CNT-RT Cu-CNT-373 Cu

0.9 0.6 0.3 0.0 Center

Half-radius

Periphery

Figure 8.7 The area-weighted mean crystallite size, < x>area (a), the dislocation density, r (b), and the twin boundary frequency, b (c) at the center, half-radius, and periphery of the HPT-processed Cu, CueCNT-RT, and CueCNT-373 disks. CNT, carbon nanotube. (Reprinted from P. Jenei, E.Y. Yoon, J.
ba
r, T. Unga
r, Microstructure and hardness of copper-carbon nanotube Gubicza, H.S. Kim, J.L. La composites consolidated by High Pressure Torsion, Materials Science and Engineering A 528 (2011) 4690e4695 with permission from Elsevier.)

plotted in Fig. 8.7aec, respectively. Usually, the crystallite size is larger while the dislocation density and the twin boundary frequency are smaller in the center than at the halfradius or periphery due to the smaller imposed torsional strain. The curves lying on the measured data points in Fig. 8.7aec are only guides for eyes. They reach the saturation values of the microstructural parameters between the center and the half-radius as suggested by the hardness investigations (see Section 8.4 in this chapter). For comparing the three samples, it is reasonable to use the values of the microstructural parameters determined at the half-radius of the disks. The crystallite size, the dislocation density and the twin boundary frequency measured at the half-radius in sample Cu are 60  6 nm, 43  4  1014 m2, and 0.0%  0.1%, respectively [17]. In sample Cu the mean grain size is 173 nm as determined from dark-field TEM images (see Fig. 8.8a). It is noted that in the case of sample Cu, the parameters of the microstructure only slightly change along the radius of the disk,

Defect Structure and Properties of Metal MatrixeCarbon Nanotube Composites

(a) (b)

100 nm

100 nm

(c) (d)

100 nm

100 nm

Figure 8.8 Dark-field TEM images for samples (a) Cu, (b) CueCNT-RT, and (c) CueCNT-373. Some twin boundaries in sample CueCNT-RT are indicated by white arrows in (d). CNT, carbon nanotube.
ba
r, T. Unga
r, Microstructure and (Reprinted from P. Jenei, E.Y. Yoon, J. Gubicza, H.S. Kim, J.L. La 2hardness of copper-carbon nanotube composites consolidated by High Pressure Torsion, Materials Science and Engineering A 528 (2011) 4690e4695 with permission from Elsevier.)

indicating that the microstructure achieved the near saturation state in all locations of the sample. It is also worth to note that the crystallite size determined by X-ray line profile analysis is smaller than the grain size observed by TEM. This phenomenon has been usually observed for plastically deformed metals [25] and it can be attributed to the fact that the crystallite size determined from X-ray line profiles corresponds essentially to the mean size of cells/subgrains, which is usually smaller than the conventional grain size measured in metals by electron microscopy methods [26,27]. The mean crystallite size of the composite sample CueCNT-RT (36  4 nm) is half as large as that of pure Cu (60  6 nm), while the dislocation density (111  10  1014 m2) is three times higher in the former sample at the half-radius of the disks (see Fig. 8.7a and b). Fig. 8.8b shows a dark-field TEM image of the grain structure for sample CueCNT-RT [17]. The average grain size in this specimen is 74 nm that is about half of the value obtained for pure Cu. The smaller crystallite and grain sizes as well as the much higher dislocation density in sample CueCNT-RT can be explained by the pinning effect of CNT fragments on lattice defects (dislocations and grain boundaries) during HPT that is especially significant where the imposed strain is higher, i.e., at the half-radius and the periphery of the disk. The principle of producing nanostructures

233

234

Defect Structure and Properties of Nanomaterials

by HPT is to form subgrain boundaries via dislocation accumulation and rearrangements under severe shear strain. These subgrain boundaries then develop into high-angle grain boundaries, resulting in grain refinement. At the same time, dynamic recovery and recrystallization reduce the dislocation density and may increase the grain size. The balance of these two opposite processes determines the final grain size. With the incorporated CNTs, the dislocation motion is blocked at CNT-Cu interfaces, therefore the dislocation accumulation is enhanced. As a consequence, the addition of CNTs leads to a decrease in grain size [5]. Fig. 8.7c shows that in contrast with the lack of observable twin boundaries in pure Cu, in sample CueCNT-RT the twin boundary frequency is high (1.1  0.1%). The pinning effect of CNT fragments in sample CueCNT-RT hinders the escape of dislocations from pile-ups during HPT resulting in high stresses at glide obstacles such as LomereCottrel barriers and grain boundaries. If the local stresses at these obstacles exceed the critical stress for twin nucleation, deformation twins are formed (see Chapter 4). Most probably, the small grain size has also contributed to the evolution of the very high twin boundary frequency. As it was shown in previous papers [28], the reduction in grain size of Cu led to an increasing contribution of twinning to plasticity at the expense of dislocation activity (see also Chapter 7). Some twin boundaries in sample CueCNT-RT are shown in Fig. 8.8d. Fig. 8.7aec reveal that when the Cu-CNT composite was processed by HPT at 373 K the crystallite size (49  5 nm) is larger by 36%, while the dislocation density (66  8  1014 m2) and the twin boundary frequency (0.3%  0.1%) are one-half and one-third of the values obtained after processing at RT, respectively. This can be explained by the higher mobility of lattice defects at 373 K, yielding their easier annihilation during HPT-straining. Fig. 8.8c shows a dark-field TEM image of the microstructure in sample CueCNT-373. The average grain size is 83 nm that is larger than the value determined for sample CueCNT-RT [17].

8.4 CORRELATION BETWEEN DEFECT STRUCTURE AND MECHANICAL PROPERTIES OF NANOTUBE-REINFORCED COMPOSITES The extremely high elastic modulus (1 TPa) of CNTs along their axes yields a very large increment of Young’s modulus with their addition to the metal matrix. As an example, Fig. 8.9 shows the measured Young’s modulus of Al-CNT composites as a function of volume fraction of CNTs [9]. The modulus increases linearly with increasing the volume fraction of MWNTs. The experimental results are compared with the calculated Young’s modulus values of the composites that are obtained from the expression [9]. Ec ¼ KCNT ECNT VCNT þ Em ð1  VCNT  Vp Þ;

(8.1)

where Ec, ECNT, and Em are the Young’s moduli of the composite, CNTs and the matrix, and VCNT and Vp are the volume fractions of CNTs and pores, respectively, and KCNT is

120

700

100

600 500

80

400 60 300 40 Young’s modulus Yield strength

20 0 0

1

2 3 4 CNT volume fraction (%)

200 100

Yield strength (MPa)

Young’s modulus (GPa)

Defect Structure and Properties of Metal MatrixeCarbon Nanotube Composites

0 5

Figure 8.9 The Young’s modulus and the yield strength as a function of the volume fraction of MWNTs in Al-CNT composites. CNT, carbon nanotube. (The data are taken from H. Choi, J. Shin, B. Min, J. Park, D. Bae, Reinforcing effects of carbon nanotubes in structural aluminium matrix nanocomposites, Journal of Materials Research 24 (2009) 2610e2616.)

a reinforcement factor. For example, if CNTs are randomly oriented within the matrix or they are aligned along the loading direction, KCNT is about 0.2 or 1, respectively [9]. The dashed line in Fig. 8.9 illustrates the moduli calculated from Eq. (8.1) for the case of KCNT ¼ 1 taking ECNT ¼ 1 TPa and Vp ¼ 0. The experimental and calculated values are very well matched, that can be explained by the alignment of CNTs along the loading direction. These Al-CNT composites are consolidated from powder blends by hot rolling [9]. Most MWNTs in the final composites are aligned along the rolling direction, following the macroscopic variation of the powder container during rolling [9]. As in the tensile test the samples were loaded in the rolling direction, therefore CNTs were lying parallel to the loading direction. The Young’s modulus values for different metal matrixeCNT composites are listed in Table 8.1. For comparison, the moduli of the matrix materials without CNTs are also given in the table. The ratio of the moduli of the CNT composites and their matrices are plotted as a function of volume fraction of CNTs in Fig. 8.10. The higher CNT content yields larger Young’s modulus irrespectively of the matrix composition and the characteristics of CNTs (single- or multiwalled, diameter and length). The pinning effect of CNTs on the lattice defects (dislocations, grain boundaries) causes an increment in yield strength as shown in Fig. 8.9 for Al-CNT composites [9]. Table 8.1 lists the yield strength obtained on different metaleCNT composites together with the values determined on the matrix materials. Fig. 8.10 shows that the higher amount of CNTs usually results in larger increment in yield strength. The correlation between the yield strength and lattice defect structure was examined in details on Cu-CNT composites consolidated by HPT [17]. The processing and the defect structure in these materials are presented in Section 8.3 of this chapter. The samples processed from

235

Al/100 nm

Al/150 nm Al/150 nm Al/150 nm Al/1 mm

2e40 nm/1 mm/2 vol.% 1e2 nm/e/7.3 vol. % (5 wt.%) 10 nm/5 mm/1.5 vol.% 10 nm/5 mm/3 vol.% 10 nm/5 mm/4.5 vol.% 10 nm/5e15 mm/ 0.75 vol.% (0.5 wt.%)

Al/1 mm

10 nm/5e15 mm/ 1.5 vol.% (1 wt.%)

Al/1 mm

10 nm/5e15 mm/3 vol.% (2 wt.%)

Al/e

140 nm/3e4 mm/ 0.75 vol.% (0.5 wt.%)

Al/e

140 nm/3e4 mm/ 1.5 vol.% (1 wt.%)

Al/e

140 nm/3e4 mm/3 vol.% (2 wt.%)

Al/e

40e100 nm/10 mm/ 0.75 vol.% (0.5 wt.%)

Shock-wave consolidation Ultrasonic mixing þ high-pressure torsion Ball milling þ hot rolling Ball milling þ hot rolling Ball milling þ hot rolling Mechanically mixed/ Cold uniaxial pressing/ Free-sintering/Hot extrusion Mechanically mixed/ Cold uniaxial pressing/ Free-sintering/ Hot extrusion Mechanically mixed/ Cold uniaxial pressing/ Free-sintering/Hot extrusion Blending by shaker mixer and planetary mill þ hot rolling Blending by shaker mixer and planetary mill þ hot rolling Blending by shaker mixer and planetary mill þ hot rolling Milling þ pressureless sintering þ hot extrusion

e

130 (133)

e

1.4 (6.5)

[7]

e

150 (100)

210 (150)

0.5 (0.8)

[8]

83 (70)

386 (262)

391 (282)

7 (13)

[9]

95 (70)

483 (262)

491 (282)

4 (13)

[9]

110 (70)

610 (262)

615 (282)

2 (13)

[9]

62

114 (91)

122 (98)

2.2 (2.5)

[10]

66

139 (91)

151 (98)

3.0 (2.5)

[10]

75

176 (91)

184 (98)

2.7 (2.5)

[10]

60 (50)

100 (70)

144 (130)

19 (25)

[11]

40 (50)

68 (70)

105 (130)

7.5 (25)

[11]

47 (50)

44 (70)

62 (130)

2 (25)

[11]

e

140 (160)

180 (105)

12 (20)

[12]

Defect Structure and Properties of Nanomaterials

Al/600 nm

236

Table 8.1 The features of the matrix and CNTs, the processing method, the Young’ modulus, the yield and tensile strengths, and the elongation to failure for different metal matrixeCNT composites Matrix CNT characteristics Young’s Yield Tensile Elongation Material/grain Diameter/length/volume modulus strength strength to failure size fraction Processing method (GPa) (MPa) (MPa) (%) References

40e100 nm/10 mm/ 2.2 vol.% (1.5 wt.%)

Al/e

40e100 nm/10 mm/ 3 vol.% (2 wt.%)

Al/e

10e20 nm/10 mm/1 vol.%

93.5% Al e 4.4% Cu 1.5% Mg e 0.6% Mn (2024Al)/ 200 nm 74% Al e 23% Si e 2% Ni e 1% Cu/ 100 nm Ni/28 nm

20e40 nm/10 mm/ 1.5 vol.% (1 wt.%)

90% Mg e 9% Al e 0.6% Zn e 0.4% Mn (AZ91D)/ 10 mm 90% Mg e 9% Al e 0.6% Zn e 0.4% Mn (AZ91D)/ 10 mm

Milling þ pressureless sintering þ hot extrusion Milling þ pressureless sintering þ hot extrusion Milling þ spark plasma sintering þ hot extrusion Ball milling þ cold isostatic pressing þ extrusion

e

190 (160)

240 (105)

14 (20)

[12]

e

180 (160)

250 (105)

16 (20)

[12]

e

e

170 (127)

23 (20)

[29]

88 (71)

336 (289)

474 (384)

3 (16.5)

[13]

40e70 nm/ 0.5e2.0 mm/ 12 vol.% (10 wt.%)

Blending þ plasma spray forming

120 (68)

e

83 (80)

0.08 (0.2)

[19,20]

10e30 nm/1e2 mm/e

Electrodeposition

e

e

2.09 (2.39)

[18]

100 nm/5 mm/0.5 vol.%

Ball milling þ hot press þ extrusion

43 (40)

281 (232)

1475 (1162) 383 (315)

6 (14)

[14]

100 nm/5 mm/1 vol.%

Ball milling þ hot press þ extrusion

49 (40)

295 (232)

388 (315)

5 (14)

[14]

Continued

Defect Structure and Properties of Metal MatrixeCarbon Nanotube Composites

Al/e

237

Yield strength (MPa)

Tensile strength (MPa)

Elongation to failure (%)

References

100 nm/5 mm/3 vol.%

Ball milling þ hot pressing þ extrusion

51 (40)

284 (232)

361 (315)

3 (14)

[14]

100 nm/5 mm/5 vol.%

Ball milling þ hot pressing þ extrusion

51 (40)

277 (232)

307 (315)

1 (14)

[14]

8e15 nm/50 mm/1 vol.%

Ball milling þ microwave sintering

e

176 (110)

307 (136)

e

[30]

8e15 nm/50 mm/2 vol.%

Ball milling þ microwave sintering

e

134 (110)

211 (136)

e

[30]

8e15 nm/50 mm/3 vol.%

Ball milling þ microwave sintering

e

117 (110)

218 (136)

e

[30]

40e70 nm/4e7 mm/ 1 vol.%

Disintegrated melt deposition þ hot extrusion

e

221 (203)

321 (310)

12 (8.7)

[31]

40 nm/1e2 mm/5 vol.%

Ball milling þ spark plasma sintering þ cold rolling

127 (70)

149 (135)

220 (175)

8 (14)

[6]

Defect Structure and Properties of Nanomaterials

90% Mg e 9% Al e 0.6% Zn e 0.4% Mn (AZ91D)/ 10 mm 90% Mg e 9% Al e 0.6% Zn e 0.4% Mn (AZ91D)/ 10 mm 86% Mg e 5.9% Al e 0.7% Zn e 0.3% Mn e 6.8% O (AZ61)/ 21 mm 86% Mg e 5.9% Al e 0.7% Zn e 0.3% Mn e 6.8% O (AZ61)/ 24 mm 86% Mg e 5.9% Al e 0.7% Zn e 0.3% Mn e 6.8% O (AZ61)/ 35 mm AZ31 e 3 wt.% AA5083 hybrid alloy/ 3.8 mm Cu/250 nm

the yield and tensile strengths, and the elongation to

238

Table 8.1 The features of the matrix and CNTs, the processing method, the Young’ modulus, failure for different metal matrixeCNT compositesdcont'd Matrix CNT characteristics Young’s Material/grain Diameter/length/volume modulus size fraction Processing method (GPa)

Cu/250 nm Cu/100 nm

40 nm/1e2 mm/10 vol.% 40 nm/2 mm/5 vol.% (1 wt.%)

Cu/100 nm

40 nm/2 mm/10 vol.% (2.2 wt.%)

Cu/22 nm

10 nm/1 mm/4.76 vol.% (1 wt.%) 5e20 nm/1e10 mm/ 3 vol.%

Cu/74 nm

2e6 nm/5e30 mm/ 5 vol.%

Cu/e

0.26 vol.%

Cu/e

0.52 vol.%

Cu/e

0.78 vol.%

Cu/e

1.04 vol.%

137 (70)

197 (135)

281 (175)

6.5 (14)

[6]

114 (82)

350 (110)

e

e

[15]

135 (82)

450 (110)

e

e

[15]

117 (91)

1125 (738)

e

e

[5,16]

e

770 (580)

e

e

[17]

e

e

371 (327)

5 (4)

[22]

e

145 (110)

213 (195)

9 (12.5)

[23]

e

180 (110)

254 (195)

6 (12.5)

[23]

e

190 (110)

264 (195)

5 (12.5)

[23]

e

214 (110)

287 (195)

3.5 (12.5)

[23]

The values of Young’s modulus, yield and tensile strengths as well as elongation to failure obtained without CNTs are also given in parenthesis. The CNT volume fractions are taken from the original papers or calculated from the weight percentages using the densities of CNTs and the matrices.

Defect Structure and Properties of Metal MatrixeCarbon Nanotube Composites

Cu/e

Ball milling þ spark plasma sintering þ cold rolling CNT surface functionalization by Cu þ spark plasma sintering CNT surface functionalization by Cu þ spark plasma sintering Ball milling þ highpressure torsion High-energy milling þ cold isostatic pressing þ high-pressure torsion Electroless plating þ ultrasonic dispersion þ sintering þ forging þ die-stretching Aligning of CNTs þ electroplating of Cu Aligning of CNTs þ electroplating of Cu Aligning of CNTs þ electroplating of Cu Aligning of CNTs þ electroplating of Cu

239

Defect Structure and Properties of Nanomaterials

4.5 Ratio of the values for the CNT composite and the pure matrix

240

4.0 Young’s modulus Yield strength Tensile strength Elongation to failure

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0

2

8 10 6 4 12 CNT volume fraction (%)

14

16

Figure 8.10 The ratio of the Young’s moduli, the yield and tensile strengths and the elongations to failure in tension obtained for CNT composites and their pure matrices. CNT, carbon nanotube. (The data are taken from Table 8.1.)

pure Cu, the blend of Cu, and 3 vol.% CNTs at RT and 373 K are denoted as Cu, CueCNT-RT, and CueCNT-373, respectively. The microhardness as a function of the distance from the center of the HPT-processed disks is plotted in Fig. 8.11a. It can be revealed that the hardness increases with increasing the distance from the center and gets saturated already at about 20% of the radius for all the three samples. The smaller hardness in the center can be explained by the smaller strain imposed during HPT compared to the other locations along the radius of the disk. Despite the linear dependence of the strain on the distance from the center, the hardness varies in a nonlinear manner due to decreasing work hardening with strain that is in agreement with previous observations [8,16]. It is noted that although the saturation hardness values of the Cu-CNT composites are higher than for the pure Cu sample, close to the disk center this relation is reversed (see Fig. 8.11a). It was shown in Section 8.3 that in the three samples near the center the dislocation density values are close to each other; therefore they have similar hardening effects. At the same time, the significant remaining porosity in the composite samples (about 3%) results in hardness reduction. Far from the center, the hardening effect of the much larger dislocation density in the composites overwhelms the influence of porosity leading to higher hardness than for the pure Cu sample. The hardness measured at 350 K also saturated already at about 20% of the radius for all the three samples (not shown here). The saturation hardness values determined at the half-radius of the disks for RT and 350 K are plotted in Fig. 8.11b. For all the three specimens the hardness reduced by about 20% when the temperature increased from RT to 350 K. This means that the relative hardening effect of CNTs was maintained at least till 350 K. The smaller grain size as well as the larger dislocation and twin boundary densities may explain the much higher hardness of specimen CueCNT-RT compared to pure

Defect Structure and Properties of Metal MatrixeCarbon Nanotube Composites

(a) 2.4 2.2

HV (GPa)

2.0 1.8 1.6 1.4 1.2

Cu Cu-CNT-RT Cu-CNT-373

1.0 0.8

0

2 4 6 8 Distance from the center (mm)

(b) 2.5

10

RT 350K

HV (GPa)

2.0 1.5 1.0 0.5 0.0

Cu

Cu-CNT-RT

Cu-CNT-373

Figure 8.11 The microhardness (HV) as a function of the distance from the center of the HPT-processed Cu, CueCNT-RT, and CueCNT-373 disks (a). The hardness values determined at the half-radius of the disks at RT and 350 K (b). CNT, carbon nanotube. (Reprinted from P. Jenei, E.Y.
ba
r, T. Unga
r, Microstructure and hardness of copper-carbon nanotube Yoon, J. Gubicza, H.S. Kim, J.L. La composites consolidated by High Pressure Torsion, Materials Science and Engineering A 528 (2011) 4690e4695 with permission from Elsevier.)

Cu. The smaller defect densities and larger grain size in sample CueCNT-373 may yield smaller hardness compared with specimen CueCNT-RT. The parameters of the microstructure and the yield strength are usually correlated using the Taylor and/or HallePetch formulas. The former and the latter equations express the hardening effects caused by dislocations and boundaries, respectively. The yield strength values of the samples Cu, CueCNT-RT, and CueCNT-373 were determined at the center, half-radius, and periphery as one-third of the hardness obtained at RT. These values are compared with the yield strength calculated from the dislocation density using the Taylor equation and corrected for the porosity as [32]:   1 s ¼ s0 þ aM T Gbr2 expð0:05$Vp Þ; (8.2)

241

Defect Structure and Properties of Nanomaterials

where s0 is the friction stress (35 MPa [33]), a is a constant (0.22 [34]), G is the shear modulus (47 GPa [35]), b is the length of the Burgers vector (0.256 nm), MT is the Taylor factor (3.06 was selected as strong texture was not observed in the samples), and Vp is the porosity in percents (0% for Cu and 3% for CueCNT-RT and CueCNT-373). The measured and the calculated yield strength values are in good agreement as shown in Fig. 8.12. The yield strength was also calculated at the halfradius of the disks from the grain size (d) determined by TEM using the HallePetch formula corrected for the porosity:   1 s ¼ s0 þ kd 2 expð0:05$Vp Þ; (8.3) where k is the HallePetch slope (w4100 MPa nm1/2 for UFG Cu [36]). Fig. 8.12 demonstrates that the yield strength determined from the grain size is much less than the experimental values in agreement with previous studies on Cu-CNT composites [16]. It seems that although the Taylor equation takes only the interaction between dislocations into account, it is capable solely to estimate the yield strength of the microstructure in the HPT-consolidated Cu and Cu-CNT composites. This can be explained by the severely deformed microstructures in these samples containing extremely high dislocation densities (in the order of 1016 m2). First, in severely deformed structures the majority of grain and subgrain boundaries are built up from dislocations, therefore their strengthening contributions can be described by a Taylor-type equation as pointed 900 800 σcalculated (MPa)

242

Hall–petch equation Taylor-formula

700 600

1 1

500 400 300 300

400

500 700 600 σmeasured (MPa)

800

900

Figure 8.12 The calculated yield strength versus the measured values obtained as one-third of the hardness at the center, half-radius, and periphery of the HPT-processed Cu, CueCNT-RT, and CueCNT-373 disks. The yield strength was calculated from the grain size obtained by TEM using the HallePetch formula and also from the dislocation density measured by X-ray line profile analysis using the Taylor equation. CNT, carbon nanotube. (Reprinted from P. Jenei, E.Y. Yoon, J. Gubicza, H.S. Kim, J.L.
ba
r, T. Unga
r, Microstructure and hardness of copper-carbon nanotube composites consolidated by High La Pressure Torsion, Materials Science and Engineering A 528 (2011) 4690e4695 with permission from Elsevier.)

Defect Structure and Properties of Metal MatrixeCarbon Nanotube Composites

out theoretically by Hughes and Hansen [37]. Secondly, the stress needed for passing a dislocation through the high density dislocation structure is so large that it is also enough for crossing the twin boundaries, therefore they have no additional strengthening contribution. It is also interesting to note that the graphite-like fragments would be expected to have an additional dispersion strengthening contribution to the flow stress but this effect cannot be detected since the yield strength determined by the Taylor formula agrees well with the measured values. This observation can be explained by the severe plastic deformation during sample production as follows. In HPT-processing, a very large number of dislocations passed through the CNT fragments, resulting in Orowan loops around them. When the consolidated samples are deformed subsequently in hardness testing, the gliding dislocations interact directly with these loops instead of the CNT fragments, therefore this strengthening effect is included in the Taylor equation. Therefore, the CNT fragments in Cu have no direct strengthening effect but rather they harden via the increase of the dislocation density. Although, the incorporation of CNTs into metallic matrices increases the elastic modulus and the yield strength with the nanotube volume fraction, the elongation to failure during tension is usually reduced with the addition of CNTs into the matrix materials. Table 8.1 shows the maximum elongation in tension for different metal matrixeCNT composites. The incomplete bonding between CNTs and the matrix grains, as well as porosity caused by clustering of nanotubes during blending may yield lower fracture toughness and therefore smaller ductility than that for the materials without CNTs. As a consequence, the maximum elongation is very sensitive to the processing conditions and solely the amount of CNTs is not determining the ductility as seen in Fig. 8.10. Similar argumentation can be applied for the tensile strength (see Fig. 8.10). It is noted that for a given processing technique and matrix material, the tendency of agglomeration of CNTs increases with increasing nanotube volume fraction that yields a lower ductility for higher CNT contents (see Table 8.1). The high dispersity of CNTs in the matrix can be achieved by “molecular level mixing” of nanotubes and the matrix material [10] or by decreasing the adhesion between CNTs using surfactants [15] as discussed in Section 8.1 of this chapter. The application of these processing steps increases the tensile strength and the strain to failure of the composites.

8.5 ELECTRICAL CONDUCTIVITY OF METALeCARBON NANOTUBE COMPOSITES The addition of CNTs to metal matrices may overcome the usual incompatibility between high strength and good electrical conductivity. Lattice defects and alloying increase strength, but reduce conductivity. At the same time, the highly conductive CNTs may increase both strength and conductivity of the matrix concomitantly. This effect is clearly demonstrated on laminar Cu-CNT composite film processed by

243

Defect Structure and Properties of Nanomaterials

Table 8.2 Resistivity, conductivity, and strength-to-resistivity ratio measured at RT for different values of CNT volume fraction in laminar Cu-CNT composite films processed by electroplating [24] CNT volume Resistivity Conductivity Strength-to-resistivity fraction (%) (10L8 U m) in IACS (%) ratio (1010 MPa/U m)

0 0.26 0.52 0.78 1.04

2.5 2.4 2.3 2.4 2.1

69 70 74 73 81

0.44 0.59 0.77 0.80 1.00

For the calculation of strength-to-resistivity ratio, the strength values are taken from Table 8.1.

electroplating (see Section 8.1). Table 8.2 lists the conductivity and the strength-toresistivity ratio measured at RT for different values of CNT volume fraction [24]. The conductivity is expressed in terms of percentage IACS (International Annealed Copper Standard). IACS at 20
C corresponds to conductivity of w5.8  107 S/m or resistivity of w1.72  108 U m. The addition of CNTs increases both the conductivity and the strength-to-resistivity ratio of the Cu matrix. Fig. 8.13 shows the conductivity and the strength-to-resistivity ratio of the electroplated laminar Cu-CNT composites normalized with the values measured on pure Cu matrix processed by the same technique. The increase of the two quantities as a function of CNT volume fraction is nearly linear. For a given CNT content, the enhancement of strength-to-resistivity ratio is about eight times larger than the increase of conductivity. This indicates that the influence of CNT addition on strength is higher than its effect on conductivity. It should be noted, however, for these electroplated composites and pure Cu material both conductivity and 2.4 Ratio of the values for the CNT composite and the pure matrix

244

2.2 2.0

Strength-to-resistivity ratio

1.8 1.6 1.4 Electrical conductivity

1.2 1.0 0.0

0.2

0.4 1.0 0.6 0.8 CNT volume fraction (%)

1.2

Figure 8.13 The ratio of the conductivities and the strength-to-resistivity values obtained for laminar Cu-CNT composites and the pure Cu matrix processed by electroplating [24]. CNT, carbon nanotube. (The data are listed in Table 8.2.)

Defect Structure and Properties of Metal MatrixeCarbon Nanotube Composites

strength-to-resistivity ratio have poor values compared to those obtained for other Cu nanomaterials and listed in Table 9.2 (see Chapter 9). Most probably, the applied electroplating technique introduces structural defects (e.g., porosity) and/or contamination into the as-processed films which deteriorate their electrical transport and mechanical performances. It is also noted that reduction of electrical conductivity due to CNT addition was also observed for a Cu matrix composite processed by die-stretching [22]. This technique is described in Section 8.1. Addition of 5 vol.% CNT resulted in a lower conductivity (about 81% IACS) than that for pure Cu matrix manufactured by the same method (102% IACS). It is assumed that the main reason of the poor conductivity of the composite is the large fraction of semiconductive single-walled CNTs in the pristine sample, since chiral separation of CNTs was not performed.

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erez-Bustamante, C.D. G
omez-Esparza, I. Estrada-Guel, M. Miki-Yoshida, L. Licea-Jim
enez, S.A. P
erez-García, R. Martínez-S
anchez, Microstructural and mechanical characterization of AIMWCNT composites produced by mechanical milling, Materials Science and Engineering A 502 (2009) 159e163. [13] C. Deng, X.X. Zhang, D. Wang, Q. Lin, A. Li, Preparation and characterization of carbon nanotubes/ aluminium matrix composites, Materials Letters 61 (2007) 1725e1728. [14] Y. Shimizu, S. Miki, T. Soga, I. Itoh, H. Todoroki, T. Hosono, K. Sakaki, T. Hayashi, Y.A. Kim, M. Endo, S. Morimoto, A. Koide, Multi-walled carbon nanotube-reinforced magnesium alloy composites, Scripta Materialia 58 (2008) 267e270. [15] S.I. Cha, K.T. Kim, S.N. Arshad, C.B. Mo, S.H. Hong, Extraordinary strengthening effect of carbon nanotubes in metal-matrix nanocomposites processed by molecular-level mixing, Advanced Materials 17 (2005) 1377e1381.

245

246

Defect Structure and Properties of Nanomaterials

[16] H. Li, A. Misra, Y. Zhu, Z. Horita, C.C. Koch, T.G. Holesinger, Processing and characterization of nanostructured Cu-carbon nanotube composites, Materials Science and Engineering A 523 (2009) 60e64. [17] P. Jenei, E.Y. Yoon, J. Gubicza, H.S. Kim, J.L. L
ab
ar, T. Ung
ar, Microstructure and hardness of copper-carbon nanotube composites consolidated by High Pressure Torsion, Materials Science and Engineering A 528 (2011) 4690e4695. [18] P.-Q. Dai, W.-C. Xu, Q.-Y. Huang, Mechanical properties and microstructure of nanocrystalline nickel-carbon nanotube composites produced by electrodeposition, Materials Science and Engineering A 483e484 (2008) 172e174. [19] T. Laha, A. Agarwal, T. McKechnie, S. Seal, Synthesis and characterization of plasma spray formed carbon nanotube reinforced aluminium composite, Materials Science and Engineering A 381 (2004) 249e258. [20] T. Laha, Y. Chen, D. Lahiri, A. Agarwal, Tensile properties of carbon nanotube reinforced aluminium nanocomposite fabricated by plasma spray forming, Composites: Part A, Applied Science and Manufacturing 40 (2009) 589e594. [21] S. Salimi, H. Izadi, A.P. Gerlich, Fabrication of an aluminium-carbon nanotube metal matrix composite by accumulative roll-bonding, Journal of Materials Science 46 (2011) 409e415. [22] S. Zhao, Z. Zheng, Z. Huang, S. Dong, P. Luo, Z. Zhang, Y. Wang, Cu matrix composites reinforced with aligned carbon nanotubes: mechanical, electrical and thermal properties, Materials Science and Engineering A 675 (2016) 82e91. [23] Y. Jin, L. Zhu, W.D. Xue, W.Z. Li, Fabrication of superaligned carbon nanotubes reinforced copper matrix laminar composite by electrodeposition, Transactions of Nonferrous Metals Society of China 25 (2015) 2994e3001. [24] J. Shuai, L. Xiong, L. Zhu, W. Li, Enhanced strength and excellent transport properties of a superaligned carbon nanotubes reinforced copper matrix laminar composite, Composites: Part A 88 (2016) 148e155. [25] Y.T. Zhu, J.Y. Huang, J. Gubicza, T. Ungar, Y.M. Wang, E. Ma, R.Z. Valiev, Nanostructures in Ti processed by severe plastic deformation, Journal of Materials Research 18 (2003) 1908e1917. [26] T. Ung
ar, G. Tichy, J. Gubicza, R.J. Hellmig, Correlation between subgrains and coherently scattering domains, Powder Diffraction 20 (2005) 366e375. [27] J. Gubicza, T. Ung
ar, Nanocrystalline materials by powder diffraction line profile analysis, Zeitschrift fur Kristallographie 222 (2007) 567e579. [28] L. Balogh, G. Rib
arik, T. Ung
ar, Stacking faults and twin boundaries in fcc crystals determined by x-ray diffraction profile analysis, Journal of Applied Physics 100 (2006) 023512. [29] K.P. So, D. Chen, A. Kushima, M. Li, S. Kim, Y. Yang, Z. Wang, J.G. Park, Y.H. Lee, R.I. Gonzalez, M. Kiwi, E.M. Bringa, L. Shao, J. Li, Dispersion of carbon nanotubes in aluminum improves radiation resistance, Nano Energy 22 (2016) 319e327. [30] A.D. Akinwekomi, W.-C. Law, C.-Y. Tang, L. Chen, C.-P. Tsui, Rapid microwave sintering of carbon nanotube-filled AZ61 magnesium alloy composites, Composites Part B 93 (2016) 302e309. [31] M. Paramsothy, M. Gupta, J. Chan, R. Kwok, Carbon nanotube addition to simultaneously enhance strength and ductility of hybrid AZ31/AA5083 alloy, Materials Sciences and Applications 2 (2011) 20e29. [32] J. Luo, R. Stevens, Porosity-dependence of elastic moduli and hardness of 3Y-TZP ceramics, Ceramics International 25 (1999) 281e286. [33] N.Q. Chinh, J. Gubicza, T.G. Langdon, Characteristics of face-centered cubic metals processed by equal-channel angular pressing, Journal of Materials Science 42 (2007) 1594e1605. [34] J. Gubicza, N.Q. Chinh, J.L. L
ab
ar, Z. Heged} us, C. Xu, T.G. Langdon, Microstructure and yield strength of severely deformed silver, Scripta Materialia 58 (2008) 775e778. [35] D.H. Chung, W.R. Buessem, in: F.W. Vahldiek, S.A. Mersol (Eds.), Anisotropy of Single Crystal Refractory Compounds, vol. 2, Plenum Press, New York, 1968, pp. 217e245. [36] N. Hansen, HallePetch relation and boundary strengthening, Scripta Materialia 51 (2004) 801e806. [37] D.A. Hughes, N. Hansen, Microstructure and strength of nickel at large strains, Acta Materialia 48 (2000) 2985e3004.

CHAPTER 9

Effect of Lattice Imperfections on Electrical Resistivity of Nanomaterials 9.1 CONTRIBUTION OF LATTICE DEFECTS TO ELECTRICAL RESISTIVITY Crystal lattice has a resistivity against electric current since the conducting electrons are scattered by phonons. This phenomenon is referred to as intrinsic resistivity of crystalline materials (hereafter denoted as ri). The intrinsic resistivity is the electrical resistivity of a defect-free and chemically pure material. With decreasing temperature (T ) lattice vibrations diminish; therefore, intrinsic resistivity strongly decreases. For metallic materials the resistivity at high temperatures (e.g., for Cu between 100 K and the melting point) is a linear function of T [1,2]. As the temperature is reduced, the temperature dependence of resistivity follows. At lower temperatures the resistivity varies with wT 5 (if scattering by phonons is the main contribution), and the resistivity becomes constant at 10e20 K. This residual resistivity is caused by the lattice defects and the impurities, i.e., it is not related to the intrinsic resistivity. As an example, Fig. 9.1 illustrates the temperature dependence of resistivity of Cu in double logarithmic scale. Lattice defects such as vacancies, interstitials, dislocations, stacking faults, grain boundaries, and solute atoms have considerable contributions to electrical resistivity. The 10–7 99.999% purity Cu

Resistivity (Ωm)

10–8

10–9

10–10

10–11 1

10

100

1000

Temperature (K)

Figure 9.1 The electrical resistivity of coarse-grained pure Cu as a function of temperature between 1 and 1356 K (the melting point of Cu) [1]. Defect Structure and Properties of Nanomaterials ISBN 978-0-08-101917-7, http://dx.doi.org/10.1016/B978-0-08-101917-7.00009-8

© 2017 Elsevier Ltd. All rights reserved.

247

248

Defect Structure and Properties of Nanomaterials

intrinsic resistivity of a crystalline material and the contributions of different lattice defects (rl) are summed up in the total electrical resistivity (r) as discussed in Chapter 2 (Matthiessen’s rule): X rðT Þ ¼ ri ðT Þ þ rl ; (9.1) l

where T is the temperature and index l stands for the different lattice defects (including impurities and alloying elements). The contributions of lattice defects are practically independent of the temperature but increase with increasing concentrations or densities of defects. For vacancies, dislocations, stacking faults, twin faults (twin boundaries), and high-angle grain boundaries (HAGBs) the resistivity contributions can be approximated as: rv ¼ cv hv ;

(9.2)

rd ¼ rhd ;

(9.3)

rsf ¼ rsf hsf ;

(9.4)

rTB ¼ rTB hTB ;

(9.5)

rHAGB ¼ rHAGB hHAGB ;

(9.6)

respectively, where hv, hd, hsf, hTB, and hHAGB are the resistivity increments for unit vacancy concentration, dislocation density (given in m2), stacking fault density (rsf given in m1), twin fault density (rTB given in m1), and HAGB density (rHAGB given in m1). The quantity rsf gives the area of stacking faults in a unit volume of material. rTB and rHAGB are defined similarly. It is noted that the planar fault or grain boundary density can be given as the reciprocal of the fault or boundary spacing, respectively. Table 9.1 lists the values of the specific resistivities hv, hd, hsf, hTB, and hHAGB for Cu. It should be noted that the specific resistivity of interstitials is about three times larger than that for vacancies. In addition, the electrical resistivity of a divacancy is about 20% less than the electrical resistivity of two separated vacancies [3]. It was shown that the specific resistivity of a vacancy in Cu is approximately the same in the temperature range between 4 K and the melting point [4]. It should also be Table 9.1 Specific electrical resistivities for vacancies, dislocations, stacking faults, twin faults, and general high-angle grain boundaries (HAGBs) in Cu Stacking fault, hsf Dislocation, Twin fault, Grain boundary, Metal Vacancy, hv (Um) hd (Um3) (Um2) hTB (Um2) hHAGB (Um2)

Cu

1.9  106

1.3  1025

The data were taken from Refs. [5e10].

3.4  1017

1.7  1017

3.4  1016

Effect of Lattice Imperfections on Electrical Resistivity of Nanomaterials

noted that the resistivity of HAGBs depends on the structure of HAGBs. For instance, in Cu the value of hHAGB decreased from 5.5  1016 to 2  1016 Um2 due to grain boundary relaxation during annealing [4]. Instead of rTB the amount of twin faults is often characterized by the quantity called as twin fault probability (b). In face-centered cubic (fcc) structures b is defined as the fraction of faulted planes among {111} crystallographic planes. If only one {111}-type plane is populated with twin faults in each grain, the value of rTB is equivalent to the reciprocal of twin fault spacing, and b can be expressed by rTB as: b ¼ 100$d111 rTB ;

(9.7)

where d111 is the spacing between the neighboring {111} planes and b is given in percentage. Inserting Eq. (9.7) into Eq. (9.5), the resistivity caused by twin faults can be related to twin fault probability as: rTB ¼

b h . 100$d111 TB

(9.8)

Similar considerations hold for stacking faults as well. For a cubic grain shape the resistivity of HAGBs can be expressed with the grain size (d ) as: 3 rHAGB ¼ hHAGB . d

(9.9)

Impurities and alloying elements, except for silver, increase the electrical resistivity of Cu. For a given concentration, the effect of solute atoms on the resistivity of Cu increases in the following order: Pb, Zn, Ni, Al, Sn, Sb, Mn, Cr, Si, Fe, and P. For instance, 1 at.% Zn, Ni, Al, or Sn causes w0.3, 1.2, 1.5, or 3  108 Um increment in resistivity of Cu, respectively [11e13]. It is noted that with increasing oxygen content in Cu, the resistivity first decreases, but above w200 ppm oxygen the resistivity starts to increase similar to effect of other solute elements. For a given alloying element concentration, the resistivity of solid solution state is higher than that for a precipitation microstructure. Simply saying, the purer the matrix, the lower the resistivity. The electrical resistivities caused by different lattice defects and solute atoms in Cu are compared with the intrinsic resistivity in Fig. 9.2. The plotted resistivity ranges correspond to lattice defect densities characteristic to nanostructured Cu materials. It can be seen that the resistivities of vacancies, dislocations, and twin faults are much smaller than the effects of HAGBs, solute atoms, and the intrinsic resistivity at room temperature (RT). Therefore, if the lattice defect densities are intended to be determined by resistivity measurements, the experiments are usually carried out at very low temperatures where the intrinsic resistivity is relatively small (e.g., at liquid nitrogen or helium temperature with the value of 77 or 4 K, respectively) [8].

249

250

Defect Structure and Properties of Nanomaterials

cv = 10–5

10–4

10–3

Vacancies

1016 m–2

ρ = 1014 m–2 β = 0.1%

1%

Dislocations

d = 100 nm cNi = 10–3 wt.% T = 20K 10–11

10–10

Twin faults

10%

0.1 wt.% 80K

300K

10–9

10–8

10 nm

HAGBs

10 wt.%

Solute atoms (e.g., Ni)

1300K

Intrinsic resistivity

10–7

r (Ωm)

Figure 9.2 Typical electrical resistivity ranges for different lattice defects and Ni solute atoms in nanostructured Cu. The intrinsic resistivity is also indicated for different temperatures. HAGBs, high-angle grain boundaries.

In nanomaterials sintered from powders, the porosity also increases resistivity. The total resistivity can be expressed as the product of the resistivity of the porosity-free matrix (rm) and a factor describing the effect of pores ( f(Vp)) [14e16]: r ¼ rm f ðVp Þ;

(9.10)

where Vp is the volume fraction of pores. For Vp < 0.5, the porosity factor, f(Vp), can be expressed as (WiedemanneFranzeLorenz equation) [15]: f ðVp Þ ¼

1 þ 11Vp 2 . 1  Vp

(9.11)

It should be noted, however, that the Schroeder formula elaborated for the description of the effect of precipitates on resistivity [16] can also be adopted for the calculation of the porosity factor [14]: f ðVp Þ ¼

1 þ 0:5Vp . 1  Vp

(9.12)

Fig. 9.3 shows the variation of the porosity factor as a function of volume fraction of pores according to Eqs. (9.11) and (9.12). The electrical conductivity of metallic materials is often expressed in terms of percentage IACS. IACS is the abbreviation of International Annealed Copper Standard, and 100% IACS is equivalent to the conductivity of a commercially pure annealed copper. The standard was established in 1913 by the International Electrotechnical Commission. 100% IACS at 20  C corresponds to conductivity of w5.8  107 S/m or resistivity of

Effect of Lattice Imperfections on Electrical Resistivity of Nanomaterials

Porosity factor of resistivity

8 7 6

Wiedemann–Franz–Lorenz eq.

5 4 3 2 1 0.0

Schroeder eq. 0.2 0.3 0.1 0.4 Volume fraction of pores (%)

0.5

Figure 9.3 Variation of the porosity factor of electrical resistivity as a function of volume fraction of pores according to Eqs. (9.11) and (9.12).

w1.72  108 Um. The application of IACS is especially useful when change of conductivity of copper samples is investigated during annealing since the variation of IACS reflects the influence of lattice defects, chemical, and phase compositions on conductivity (i.e., the change of temperature independent residual resistivity). For instance, 20% IACS at a given temperature is equivalent to 20% of the conductivity of an annealed pure Cu at the same temperature. It is noted that conductivities higher than 100% IACS are often measured nowadays for Cu since the improved processing techniques may produce higher purity Cu with a better conductivity than that obtained in 1913.

9.2 CHANGE OF RESISTIVITY IN NANOMATERIALS PROCESSED BY SEVERE PLASTIC DEFORMATION The electrical resistivity increment caused by monovacancies, vacancy clusters, and dislocations was determined in high purity ultrafine-grained (UFG) Ni and Cu [8]. The resistivity contribution of single and double vacancies in 99.998% purity Ni processed by two turns of high-pressure torsion (HPT) at RT was about 3  1010 Um, which was caused by a vacancy concentration of w104. This value was measured at the periphery of the HPT-processed disk where the strain was the largest. The resistivity increment caused by vacancy agglomerates, dislocations, and HAGBs was w17  1010 Um. It was found that the resistivity increment caused by the lattice defects formed in Ni during HPT saturated at a shear strain of w20, which can be achieved already at the periphery of a disk deformed by one-third turn of HPT. After one turn of HPT, both resistivity and strength in Ni change only slightly with increasing either the distance from the disk center or the number of turns [17]. It is worth to note

251

252

Defect Structure and Properties of Nanomaterials

that the contribution of lattice defects to resistivity in HPT-processed Ni (w20  1010 Um) is two orders of magnitude larger than the intrinsic resistivity measured at 4 K. At the same time, at RT the resistivity of lattice defects is only w10% of the intrinsic electrical resistivity of Ni. In UFG copper processed by five passes of equal-channel angular pressing (ECAP) at RT the lattice defects formed during severe plastic deformation (SPD) yielded a resistivity increment of w4  1010 Um, irrespective of the applied ECAP route (A, BC, and C, as defined in Chapter 3) [18]. The change of resistivity for ECAP-processed Cu is much smaller than that in Ni deformed by HPT, which can be explained by the much lower shear strain applied in the former method (w5). The contribution of lattice defects to resistivity of ECAP-processed pure Cu at RT is only w2%. Similar results were obtained for 99.96% purity UFG copper produced by 16 passes of ECAP via route BC at RT [19]. Although, the grain size was refined from 6 mm to 500 nm when the annealed Cu sample was processed by 16 ECAP passes, the conductivity changed only from 97% IACS to 95% IACS, i.e., the contribution of lattice defects (including vacancies, dislocations, and grain boundaries) to resistivity is only w2%. In Cue3 wt.% Ag alloy 12 passes of ECAP at 343 K yielded reduction of conductivity from 94% to 88% IACS [20]. Generally, it can be concluded that in pure metals and equilibrium solid solutions, SPD results only in a slight increase of resistivity owing to the increase of amount of lattice defects. At the same time, the lattice defects result in a strong increase of mechanical strength. For instance, the yield strength in pure copper increased from w100 to w350 MPa owing to 16 ECAP passes [19]. Therefore, SPD methods usually significantly improve the strength-to-resistivity ratio. In the case of Cu processed by 16 passes of ECAP, the strength-to-resistivity ratio increased from 0.56  1010 to 1.9  1010 MPa/Um. At the same, for low carbon steel four passes of constrained groove pressing (CGP) led to only a 20% increment in strength-to-resistivity ratio [21]. In the CGP process the grain size of the steel was refined from 30 mm to 231 nm. Table 9.2 summarizes the grain size, the mechanical strength, the conductivity in IACS, the electrical resistivity, and the strength-to-resistivity ratio for different UFG and nanocrystalline materials. In the present case, the mechanical strength is characterized by the yield strength or the strength calculated as one-third of the hardness, depending on the availability of these data. It should be noted that the stress determined as one-third of hardness gives the flow stress corresponding to 8% plastic strain since hardness measurement itself yields an additional plastic deformation. For materials exhibiting strong strain hardening after the onset of yielding, one-third of hardness may be considerably larger than the yield strength. At the same time, for SPD-processed samples the strain hardening is usually moderate; therefore, there is only a slight difference between one-third of hardness and yield strength. The addition of alloying elements to coarse-grained metallic materials increases both strength and resistivity; however, the former effect is usually stronger, thereby improving the strength-to-resistivity ratio. For instance, in coarse-grained Cue0.18% Zr alloy the

Table 9.2 Grain size, mechanical strength, conductivity in IACS, resistivity, and strength-to-resistivity ratio measured at RT for different UFG and nanocrystalline materials Strength-toresistivity ratio Grain size Strength Conductivity Resistivity Material and processing method (nm) (MPa) in IACS (%) (10L8 Um) (1010 MPa/Um) References

500 70

350 930*

95 74

1.8 2.3

1.9 4.0

[19] [10,26]

146

700*

100

1.7

4.1

[10,26]

400

900

97

1.75

5.1

[27]

220 200

590 530*

78 59

2.2 2.9

2.7 1.8

[28] [22]

200

570*

51

3.4

1.7

[22]

e

570*

75

2.3

2.5

[22]

e

600*

57

3.0

2.0

[22]

350 37 55

617 700* 670*

88 31 47

1.9 5.5 3.7

3.2 1.3 1.8

[20] [14] [14]

170 170 e

460 620 600*

50 44 47

3.4 3.9 3.7

1.4 1.6 1.6

[24] [24] [25]

370

525

86

2.0

2.6

[23] Continued

Effect of Lattice Imperfections on Electrical Resistivity of Nanomaterials

99.96% purity Cu, 16 ECAP at RT 99.999% purity Cu film, magnetron sputtering, twin spacing: 7 nm 99.999% purity Cu film, magnetron sputtering, twin spacing: 16 nm 99.998% purity Cu, pulsed electrodeposition, twin spacing: 15 nm Cu, SPS þ CD Cue0.18 wt.% Zr with Cu5Zr precipitates, 15 HPT at RT Cue0.18 wt.% Zr solid solution, 15 HPT at RT Cue0.18 wt.% Zr with Cu5Zr precipitates, 15 HPT at RT þ annealing for 1 h at 723 K Cue0.18 wt.% Zr solid solution, 15 HPT at RT þ annealing for 1 h at 723 K Cue3 wt.% Ag, 8 ECAP at 343 K Cue1.1 wt.% Al2O3, cold spray Cue1.1 wt.% Al2O3, cold spray þ annealing at 1223 K Cue1vol.% Al2O3, ball milling þ HIP Cue5vol.% Al2O3, ball milling þ HIP Cue4 wt.% Al2O3, ball milling þ compaction by hot-pressing Cue0.5 wt.% Al2O3, 4 ECAP at RT

253

254

Cue1.1 wt.% Al2O3, 4 ECAP at RT Cue0.3 wt.% Mge0.05 wt.% Ce, 8 ECAP at RT Cue0.3 wt.% Mge0.05 wt.% Ce, 8 ECAP at RT þ annealing for 2 h at 573 K Cue0.2 wt.% Mg, 4 ECAP þ CR þ CD at RT Cue0.2 wt.% Mg, 4 ECAP þ CR þ CD at RT þ annealing for 2 h at 573 K Cue0.4 wt.% Mg, 4 ECAP þ CR þ CD at RT Cue0.4 wt.% Mg, 4 ECAP þ CR þ CD at RT þ annealing for 2 h at 573 K Cue0.75 wt.% Cr, Q þ 5 HPT at RT Cue0.75 wt.% Cr, Q þ 5 HPT at RT þ annealing for 1 h at 523 K Cue0.75 wt.% Cr, SC þ 5 HPT at RT Cue0.75 wt.% Cr, SC þ 5 HPT at RT þ annealing for 1 h at 523 K Cue9.85 wt.% Cr, Q þ 5 HPT at RT Cue9.85 wt.% Cr, Q þ 5 HPT at RT þ annealing for 1 h at 773 K Cue9.85 wt.% Cr, SC þ 5 HPT at RT Cue9.85 wt.% Cr, SC þ 5 HPT at RT þ annealing for 1 h at 773 K Cue27 wt.% Cr, as-cast þ 5 HPT at RT Cue27wt.% Cr, as-cast þ 5 HPT at RT þ annealing for 1 h at 773 K

350 600

560 520

82 72

2.1 2.4

2.7 2.2

[23] [29]

e

510

75

2.3

2.2

[29]

100e200

567

82

2.1

2.7

[30]

e

426

87

2.0

2.1

[30]

100e200

597

72

2.4

2.5

[30]

e

488

80

2.1

2.3

[30]

209 245

580* 610*

34 35

5.0 4.9

1.2 1.2

[31] [31]

e e

537* 478*

61 72

2.8 2.4

1.9 2.0

[31] [31]

143 229

713* 584*

29 67

5.9 2.6

1.2 2.2

[31] [31]

e e

702* 631*

54 76

3.2 2.3

2.2 2.7

[31] [31]

40 96

900* 879*

20 42

8.6 4.1

1.1 2.1

[31] [31]

Defect Structure and Properties of Nanomaterials

Table 9.2 Grain size, mechanical strength, conductivity in IACS, resistivity, and strength-to-resistivity ratio measured at RT for different UFG and nanocrystalline materialsdcont'd Strength-toresistivity ratio Grain size Strength Conductivity Resistivity Material and processing method (nm) (MPa) in IACS (%) (10L8 Um) (1010 MPa/Um) References

155 189

660* 800*

41 71

4.2 2.4

1.6 3.3

[32] [32]

e e

660* 600*

50 80

3.4 2.1

1.9 2.9

[32] [32]

108

800*

23

7.4

1.1

[32]

131

933*

61

2.8

3.3

[32]

e

673*

46

3.7

1.8

[32]

e

770*

67

2.6

3.0

[32]

810 449

200* 167*

57 60

3.0 2.8

0.7 0.6

[33] [34]

539

163*

60

2.8

0.6

[34]

454

167*

59

2.9

0.6

[34]

199

510*

26

6.6

0.8

[34]

215

437*

26

6.6

0.7

[34]

185

493*

26

6.6

0.8

[34]

130

448

48

3.6

1.2

[35] Continued

Effect of Lattice Imperfections on Electrical Resistivity of Nanomaterials

Cue0.9 wt.% Hf, Q þ 5 HPT at RT Cue0.9 wt.% Hf, Q þ 5 HPT at RT þ annealing for 1 h at 723 K Cue0.9 wt.% Hf, SC þ 5 HPT at RT Cue0.9 wt.% Hf, SC þ 5 HPT at RT þ annealing for 1 h at 723 K Cue0.7 wt.% Cre0.9 wt.% Hf, Q þ 5 HPT at RT Cue0.7 wt.% Cre0.9 wt.% Hf, Q þ 5 HPT at RT þ annealing for 1 h at 773 K Cue0.7 wt.% Cre0.9 wt.% Hf, SC þ 5 HPT at RT Cue0.7 wt.% Cre0.9 wt.% Hf, SC þ 5 HPT at RT þ annealing for 1 h at 773 K 99.5 wt.% Al, 10 HPT at RT Ale0.3 wt.% Sie0.4 wt.% Fe (AA 1050), 4 HE at RT Ale0.3 wt.% Sie0.4 wt.% Fe (AA 1050), 4 ECAP at RT Ale0.3 wt.% Sie0.4 wt.% Fe (AA 1050), 2 ECAP þ 4 HE at RT Ale4.6 wt.% Mge0.9 wt.% Mne0.1 wt.% Sie0.1 wt.% Fe (AA 5483), 4 HE at RT Ale4.6 wt.% Mge0.9 wt.% Mne0.1 wt.% Sie0.1 wt.% Fe (AA 5483), 2 ECAP at RT Ale4.6 wt.% Mge0.9 wt.% Mne0.1 wt.% Sie0.1 wt.% Fe (AA 5483), 2 ECAP þ 4 HE at RT Ale0.8 wt.% Mge0.8 wt.% Sie0.1 wt.% Fe alloy (Al 6201), 20 HPT at RT

255

256

Ale0.8 wt.% Mge0.8 wt.% Sie0.1 wt.% Fe alloy (Al 6201), 20 HPT at 403 K Ale0.8 wt.% Mge0.8 wt.% Sie0.1 wt.% Fe alloy (Al 6201), 20 HPT at 453 K Ale0.8 wt.% Mge0.8 wt.% Sie0.1 wt.% Fe alloy (Al 6201), 20 HPT at 503 K Ale0.6 wt.% Mge0.5 wt.% Sie0.1 wt.% Fe alloy (Al 6101), 20 HPT at RT Ale0.6 wt.% Mge0.5 wt.% Sie0.1 wt.% Fe alloy (Al 6101), 20 HPT at 373 K Ale0.6 wt.% Mge0.5 wt.% Sie0.1 wt.% Fe alloy (Al 6101), 20 HPT at 443 K Ale5.4 wt.% Cee3.1 wt.% La, 20 HPT at RT Ale5.4 wt.% Cee3.1 wt.% La, 20 HPT at RT þ annealing for 1 h at 553 K Ale5.4 wt.% Cee3.1 wt.% La, 20 HPT at RT þ annealing for 1 h at 673 K Ti (50% a þ 50% u), 10 HPT at 100 K Ti (20% a þ 80% u), 10 HPT at RT Fee0.2 wt.% Mn (low carbon steel), 4 CGP at RT

280

380

56

3.1

1.2

[35]

440

326

58

2.9

1.1

[35]

960

218

59

2.9

0.8

[35]

180

570*

47

3.7

1.5

[36]

240

440*

52

3.3

1.3

[36]

430

240*

59

2.9

0.8

[36]

136

475

40

4.3

1.1

[37]

203

495

45

3.9

1.3

[37]

385

255

52

3.3

0.8

[37]

54 118 231

1300* 1200* 400

2.1 1.9 1.7

80 90 100

0.16 0.13 0.04

[38] [38] [21]

It is noted that the mechanical strength usually corresponds to the yield strength. However, if this value is not available, the strength is calculated as one-third of the hardness and these values are indicated by asterisks. CD, cold drawing; CGP, constrained groove pressing; CR, cold rolling; ECAP, equal-channel angular pressing; HE, hydrostatic extrusion; HIP, hot isostatic pressing; HPT, high-pressure torsion; IACS, International Annealed Copper Standard; Q, quenching from temperatures between 1173 and 1273 K; RT, room temperature; SC, slowly cooled in air from temperatures between 1173 and 1273 K; SPS, spark plasma sintering; UFG, ultrafine-grained. The numbers before the abbreviations of the processing techniques indicate the numbers of passes or turns.

Defect Structure and Properties of Nanomaterials

Table 9.2 Grain size, mechanical strength, conductivity in IACS, resistivity, and strength-to-resistivity ratio measured at RT for different UFG and nanocrystalline materialsdcont'd Strength-toresistivity ratio Grain size Strength Conductivity Resistivity Material and processing method (nm) (MPa) in IACS (%) (10L8 Um) (1010 MPa/Um) References

Effect of Lattice Imperfections on Electrical Resistivity of Nanomaterials

strength-to-resistivity ratio is 25%e55% higher than in pure Cu [22]. The smaller and larger increments are related to precipitation microstructure and solid solution, respectively. In SPD-processed UFG copper materials, the addition of small amount of solute atoms yields similar relative increase in both strength and electrical resistivity, resulting in practically unchanged strength-to-resistivity ratio, as shown in Table 9.2. At the same time, secondary phase particles (e.g., alumina) in ECAP-processed Cu matrix can improve the strength-to-resistivity ratio by about 50% since the strength increased considerably without significant reduction of conductivity [23]. It should be noted, however, that the conductivity of Cuealumina composites processed by powder metallurgy techniques are very poor (see Table 9.2), most probably due to the remaining porosity [14,24,25]. The best behavior was observed for Cue3wt.% Ag alloy processed by eight passes of ECAP at 343 K, as for this material the strength-to-resistivity ratio is about 50% higher than that for pure ECAP-processed Cu with similar grain size [20]. In this alloy, silver addition increased the strength considerably, while the conductivity was only slightly reduced. It is worth to note that SPD processing may yield not only the increase of the amount of lattice defects but also phase transformation, which influences the electrical resistivity. For instance, HPT under 6 GPa pressure can result in a transformation from hexagonal close-packed (hcp) a structure to hexagonal u phase [38]. The electrical resistivity of the latter phase at RT is about two times larger (w1.0  106 Um) than that for a-Ti (w0.5  106 Um). The u phase transforms back to hcp structure after annealing the HPT-processed samples at w400 K, which results in a decrease of resistivity. It is noted that the strength-to-resistivity ratio is very small for the HPT-processed two-phase a/u titanium materials (0.13e0.16  1010 MPa/Um) due to the high fraction of u phase, as compared to other SPD-processed pure materials (see Table 9.2). In two-phase alloys, SPD often causes dissolution of equilibrium phases as the solubility limit of alloying elements in the matrix can be increased due to their preferred storage at defects, such as dislocations and grain boundaries. This dissolution usually increases the electrical resistivity due to the scattering of conductive electrons on solute atoms. This phenomenon was observed in Cue0.18 wt.% Zr alloys during processing via 15 turns of HPT at RT [22]. HPT deformation yielded a partial dissolution of secondary phase particles of Cu5Zr into Cu matrix besides the formation of lattice defects. Both processes have contributions to the increase of resistivity from w2.3  108 to w2.9  108 Um during HPT. Since this SPD processing resulted in a higher change of yield strength (from w200 to w530 MPa) than the increment in resistivity, HPT led to an increase in strength-to-resistivity ratio from 0.9  1010 to 1.8  1010 MPa/Um. When the initial state was a solid solution with the same composition, both the strength and resistivity changes caused by HPT were higher; however, the strength-to-resistivity ratio remained practically the same [22]. This ratio for Cue0.18 wt.% Zr alloy can be improved by an additional annealing after HPT as it will be discussed in Section 9.4. In Al alloys SPD

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processing by hydrostatic extrusion, ECAP, and their combination yielded a significant improvement in strength while the resistivity increased only by w2%, similar to copper alloys [34]. Therefore, the strength-to-resistivity ratio was improved with a factor of about two for both low and high solute concentrations (AA 1050 and AA 5483 alloys). It is also worth to note that for alloys with high solute contents, the degree of disorder also influences strongly the electrical resistivity. For instance, SPD processing of Cue40 at.% Pd alloy may yield transition from ordered B2 structure to disordered fcc A1 phase, which increases the resistivity from 7  108 to 28  108 Um [39]. Annealing of the disordered alloy at w620 K results in ordering, which leads to reduction of resistivity. The effect of temperature applied in HPT on resistivity and strength was studied for Al 6101 and Al 6201 alloys with the compositions of Ale0.6 wt.% Mge0.5 wt.% Sie0.1 wt.% Fe and Ale0.8 wt.% Mge0.8 wt.% Sie0.1 wt.% Fe, respectively [35,36]. The HPT temperature varies between RT and 443 K for Al 6101 and between RT and 503 K for Al 6201 alloys. The aim of the application of high temperature during HPT was to develop a UFG microstructure with nanosized secondary phase particles. Owing to the elevated temperatures the initial solid solution was decomposed via dynamic aging, resulting in the formation of secondary phase b0 -Mg2Si nanoprecipitates and concomitant solute depletion in grain interiors. The lower concentration of lattice defects and solute elements inside the grains improved the conductivity considerably, while the nanoprecipitates at the grain boundaries moderate the grain growth during high temperature HPT. The size of these spherical b0 -Mg2Si particles varied between 10 and 50 nm, depending on the temperature of HPT. The higher the temperature, the smaller the precipitate size. With increasing HPT temperature, both strength and resistivity decreased in Al 6101 and Al 6201 alloys due to annihilation of lattice defects, recovery of nonequilibrium grain boundaries, and grain growth. As the strength reduction was stronger than the decrease of resistivity, the strength-to-resistivity ratio was reduced with increasing the temperature of HPT, as shown in Table 9.2.

9.3 PROCESSING OF NANOMATERIALS WITH HIGH HARDNESS AND GOOD CONDUCTIVITY A combination of high strength and good conductivity in alloys can be obtained by applying moderate annealing after SPD processing. Large strength-to-resistivity ratio can be achieved by producing very fine grain structure with very low solute content inside the grains. In these microstructures, the majority of alloying elements are at the grain boundaries and/or in nanosized intermetallic compound particles. The SPD step in the thermomechanical treatment provides a UFG or nanocrystalline matrix that is stabilized by the nanoscaled precipitates formed in the annealing step. The good conductivity is guaranteed by a nearly complete purification of the matrix grain interiors from the solute atoms. The high strength is caused by the high amount of grain boundaries

Effect of Lattice Imperfections on Electrical Resistivity of Nanomaterials

and the nanosized secondary phase particles instead of solute hardening in the grain interiors. The effectiveness of this processing strategy in the optimization of strength and electrical conductivity was illustrated for Cu and Al alloys [22,35e37]. For instance, the strength-to-resistivity ratio of Cue9.85 wt.% Cr deformed by five turns of HPT at RT increased with about 20% when a heat treatment was applied at 773 K for 1 h [31]. In this material the secondary phase particles were Cr nanodispersoids with the size of 10e15 nm. It is noted that for similar Cr content without the application of this thermomechanical treatment the strength-to-resistivity ratio is five times smaller mainly due to the much lower conductivity [40]. Fig. 9.4 illustrates the effect of SPD processing and subsequent annealing on the strength, resistivity, and strength-to-resistivity ratio for pure copper and its alloys. It can be concluded that UFG microstructure with fine precipitates and purified grain interiors (from solutes) may yield copper materials with high strength and low electrical resistance, and the strength-to-resistivity ratio of this alloy may be better than that for pure Cu. It should be noted, however, that a very small Mg solute concentration in SPD-processed UFG Cu can also result in a good strength-to-resistivity ratio, similar to dispersion-strengthened UFG alloys with purified grain interiors [30]. In this case, most probably the large solute Mg atoms are segregated at the grain boundaries, thereby reducing their contribution to resistivity. Annealing after SPD processing for this alloy reduced the strength-to-resistivity ratio since the strength increased significantly without a considerable improvement of conductivity. The best strength-to-resistivity ratio (w3.3  1010) was obtained for thermomechanically treated Cue0.9 wt.% Hf and 1000 Nanotwinned Cu films Aging of bulk Cu(Cr,Hf) after SPD

Strength (MPa)

800 600 400

Cu and its alloys SPD at RT SPD + aging or HT-SPD Nanotwinned Cu Powder metallurgy

200 0 0

2

4 6 Resistivity (10–8 Ωm)

8

10

Figure 9.4 Strength versus resistivity for ultrafine-grained pure copper and its alloys. The different processing routes are indicated by various symbols. The strength-to-resistivity ratio values are reflected by the slopes of the lines connecting the data points and the origin of the coordinate system. The materials with the lowest and the highest strength-to-resistivity ratios are indicated by dashed lines. HT-SPD, high-temperature SPD processing; SPD, severe plastic deformation; RT, room temperature. ( The data were taken from Table 9.2.)

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Cue0.7 wt.% Cre0.9 wt.% Hf alloys [32]. Both materials were first annealed at 1173 K for 1 h and quenched to RT. Then, they were deformed by five turns of HPT at RT. Finally, an aging heat treatment was applied for 1 h at 723 and 773 K for Cue0.9 wt.% Hf and Cue0.7 wt.% Cre0.9 wt.% Hf alloys, respectively. During this aging, Cr and Cu5Hf intermetallic compound nanoparticles with the size of 10e20 nm were formed, which hindered the grain growth effectively. Hence, the grain size increased only by about 20% despite the high temperature of aging. The formation of the very fine precipitates yielded an exceptional change of yield strength, as it increased by about 20% during heat treatment. In addition, a two times higher conductivity was detected after aging, i.e., both the strength and the conductivity were improved during the heat treatment. In Al-based immiscible systems, such as AleCeeLa alloys, the very low solubility limits guarantee the formation of secondary phase particles. It was demonstrated for Ale5.4 wt.% Cee3.1 wt.% La alloy that 20 turns of HPT at RT yielded fragmentation and/or partial dissolution of precipitates [37]. These precipitates are intermetallic phase particles with the composition of Al11RE3 where RE stands for rare earth atoms (Ce and La). During HPT at RT, the grain size was refined from 5.2 mm to 136 nm while the precipitate size decreased from w0.15e2 mm to w44 nm. In addition, the elongated shape of secondary phase particles became spherical. The average interparticle spacing was 72 nm. The fragmentation of precipitates was accompanied by their partial dissolution, leading to a supersaturated solid solution in the Al matrix. The increase of solute content and lattice defect density in the Al matrix during HPT induced a significant reduction in the electrical conductivity from 50% to 40% IACS. Simultaneously, the yield strength increased from 73 to 475 MPa due to HPT processing. Subsequent annealing at 553 K for 1 h led to a moderate grain growth from 136 to 203 nm, reduction of dislocation density, segregation of RE atoms at grain boundaries, and formation of nanoscale clusters of RE atoms with the size of w2 nm. These nanoclusters are distributed homogeneously in the matrix. The size of Al11RE3 secondary particles remained unchanged during annealing. Owing to these changes in the microstructure, the conductivity was improved to 45% IACS while the high value of strength was preserved [37]. It seems that the strengthening effect of nanoclusters in the annealed sample was higher than the hardening of solute atoms in the HPT-processed material, and this difference could compensate the softening caused by the grain growth during annealing. The strengthening of RE clusters is most probably similar to that of GuinierePreston zones in age-hardenable Al alloys. The purification of the grain interiors yielded the 10% increase in conductivity. Therefore, the annealing at 553 K led to a slight improvement of strength-to-resistivity ratio. At the same time, heat treatment at much higher temperature (673 K) resulted in a strong reduction of grain size from 136 to 385 nm, which could not be compensated by the change of the precipitated microstructure, thereby causing a significant reduction of yield strength to 255 MPa. Although, the conductivity was further improved to 52% IACS, the strength-to-resistivity ratio decreased below the

Effect of Lattice Imperfections on Electrical Resistivity of Nanomaterials

value obtained for the HPT-processed sample (see Table 9.2). It should be noted that the sample processed by HPT at RT and then annealed at 673 K has slightly better conductivity than the initial coarse-grained material (w50% IACS), and its strength is also much higher (255 MPa) than for the initial unprocessed sample (73 MPa). Fig. 9.5 summarizes the variation of strength and conductivity due to HPT and subsequent annealing in Ale5.4 wt.% Cee3.1 wt.% La alloy and compares these changes with that observed for pure Al. The much larger reduction of conductivity for Ale5.4 wt.% Cee3.1 wt.% La alloy compared to pure Al during HPT was caused by the increase of solute content due to dissolution of precipitates in the former material. In pure metallic materials, the grain boundary relaxation during annealing after SPD processing yields only a minor (about 5%) reduction of resistivity [28]. At the same time, a significant improvement in strength-to-resistivity ratio can be achieved if the majority of HAGBs are substituted by coherent twin faults as the latter interfaces have very low specific electrical resistivity [10,26,27,40]. For instance, in Cu the specific resistivity of coherent twin boundaries is 1.7  1017 Um2, which is 20 times smaller than that for a general HAGB (3.4  1016 Um2), as shown in Table 9.1. At the same time, coherent twin interfaces possess high resistance to the transmission of dislocations, resulting in a high contribution to strength. Therefore, in pure Cu materials the superior strengthto-resistivity ratio can be achieved by inducing many growth twin faults in the UFG grain interiors, as shown schematically in Fig. 9.6. Nanotwinned UFG microstructures in 5N purity Cu were processed by pulsed electrodeposition [27] and magnetron sputtering [10,26] techniques. In the former and latter methods the thicknesses of the films were w100 and w1.5 mm, respectively. The grain size and the twin spacing in these films

65

CG AI HPT at RT

Conductivity (% IACS)

60

UFG AI

55 CG Al-5.4Ce-3.1La 50 45

HPT at RT

40

1 h at 673 K 1h at 553 K UFG Al-5.4Ce-3.1La

35 0

200

400

600

Yield strength (MPa)

Figure 9.5 The variation of strength and conductivity due to high-pressure torsion (HPT) and subsequent annealing in Ale5.4 wt.% Cee3.1 wt.% La alloy and pure Al. CG, coarse-grained; IACS, International Annealed Copper Standard; RT, room temperature; UFG, ultrafine-grained.

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Defect Structure and Properties of Nanomaterials

Grains

Twin lamella

Figure 9.6 Schematic of nanotwinned ultrafine-grained microstructure in electrodeposited polycrystalline Cu possessing high strength and good conductivity.

were 70e400 nm and 7e16 nm, respectively. The achieved yield strength at RT was as high as 700e930 MPa, which is about 20 times larger than that for a coarse-grained Cu with the same chemical composition. At the same time, the conductivity of the nanotwinned films at RT was similar to that for a coarse-grained Cu due to the negligible contribution of twin faults. The strength and resistivity of UFG Cu with nanoscale growth twin lamellae are compared with the values obtained for other UFG copper materials in Fig. 9.4.

9.4 ELECTRICAL RESISTIVITY OF NANOSTRUCTURED FILMS In general, thin films usually have higher resistivity than bulk materials with similar chemical composition. For instance, a sputtered pure polycrystalline Cu film with the thickness of 1 mm exhibited a resistance of w2.2  108 Um at RT, which is about 20% larger than that for pure bulk Cu with similar grain size (w1.8  108 Um) [41]. For Cr film with the thickness of 1 mm, the resistivity (w116  108 Um) was one order of magnitude higher than that for bulk Cr material (w13  108 Um). In addition, the resistivity of Cr film only slightly increased with increasing the temperature from 4K to RT. These observations can be explained by the very small diameter of the columnar grains (w30e35 nm) and the high free volume at the grain boundaries in the film. The latter phenomenon is manifested by the higher interatomic spacing at grain boundaries and the lower density of pure Cr films than that observed for bulk material (the density of a Cr thin film was found to be w96%e97% of the value of bulk Cr). Therefore, the resistivity of sputtered Cr films is determined mainly by the lattice defects and not by the temperature-dependent electronephonon scattering [41]. It was suggested that the large interatomic spacing at the grain boundaries in Cr films is caused by large

Effect of Lattice Imperfections on Electrical Resistivity of Nanomaterials

internal stresses. In Cu films, these stresses can relax by twinning in the grain interiors; therefore, the free volume at the grain boundaries is smaller, resulting in only a slightly larger resistivity than in bulk Cu. With increasing film thickness, the resistivity of thin films decreases and asymptotically converges to the resistivity of bulk materials. Among metallic materials silver has the lowest resistivity (w1.6  108 Um at RT). In thin Ag films, the resistivity decreases with increasing film thickness, however, the optical properties (e.g., transmittance) deteriorate for thicker Ag films [42]. This problem can be solved by an appropriate selection of the underlayer material, since the resistivity of Ag films is reduced when the layer is deposited on ZnO undercoats. It was shown that the sputtering conditions of aluminum-doped zinc oxide (AZO) undercoats have a strong effect on the structural characteristics of AZO (e.g., preferred orientation), thereby influencing the electrical resistivity of the deposited Ag films [42]. It was revealed for polycrystalline 20 nm Ag/15 nm AZO layer structure magnetron sputtered on sodae limeesilicate glass at RT that the most effective processing parameter is the applied power. With increasing the power of sputter deposition from 0.3 to 1.2 kW, the electrical resistivity decreased from w6  108 to w4  108 Um. Simultaneously, plane (002) of the hexagonal AZO undercoat preferably grew parallel to the glass substrate surface, and the overcoated Ag film became highly oriented with its plane (111) parallel to the substrate. The development of this preferred orientation was accompanied by a reduction of Ag film surface roughness. Other sputtering conditions, such as oxygen concentration and pressure in the sputter gas, have only negligible effect on the structure and conductivity of deposited Ag films [42]. Besides the high strength a very low resistivity can be achieved due to the highly twinned grain structure in magnetron-sputtered 5N purity Cu films with the thickness of w1.5 mm [10,26]. The Cu layer was epitaxially grown on (011)-orientated single crystal Si substrate. Planes {111} of Cu layer were lying parallel to the film surface; therefore, the relationship Cu< 111 >kSi< 110 > held between the out-of-plane directions of the Cu film and the Si substrate. The in-plane directions are also correlated as Cu< 110 >kSi< 111 > and Cu< 112 >kSi< 112 > [10]. A schematic of Cu film microstructure is presented in Fig. 9.7. A columnar microstructure with high amount of twin faults was developed during deposition. The longitudinal axes of columns (or domains) are lying perpendicular to the film surface (i.e., parallel to the growing direction) while the twin faults inside the domains are lying parallel to the film surface. The vertical domain boundaries were identified as S3{112} twin interfaces. Planes {111} coincide in the adjacent columns across the domain boundaries with practically zero misalignment (i.e., the domain boundaries are coherent or semicoherent). Therefore, the epitaxially grown Cu layer can be considered as a single crystalline film with high twin fault density in both parallel and perpendicular to the film surface. It is noted that when the Si substrate was oxidized, the adjacent columns were separated by grain boundaries and the domains have random in-plane orientations [polycrystalline Cu layer with a (111) fiber texture].

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Defect Structure and Properties of Nanomaterials – [111]

Σ3 {112}

Domains (columns)

Σ3 {111}

Cu film Si single crystal substrate

[110]

[011]

– [112] Twin lamella – [111] – [211]

Figure 9.7 Three-dimensional schematic view of highly twinned, epitaxially grown Cu film processed by magnetron sputtering. The domains are separated by S3{112} twin interfaces, and they contain many S3{111} twin faults. These films have high strength and low resistivity.

The average twin fault spacing in the epitaxially grown single crystalline Cu layer can be tailored with changing the deposition rate [10]. When the deposition rate increased from 0.9 to 4 nm/s, the domain size in plan view (column width) decreased from 146 to 70 nm while the twin spacing perpendicular to the film surface was reduced from 16 to 7 nm. In addition, for high deposition rates (e.g., 4 nm/s) an offset was observed for {111} planes across the {112} domain interfaces. The resistivity of epitaxial Cu layer deposited at 0.9 nm/s was w1.7  108 Um at RT, which is only slightly larger than that for bulk oxygen-free high conductivity Cu (OFHC-Cu). At very low temperatures (10e20 K), where the intrinsic resistivity becomes very small, the resistivity of epitaxial Cu layers is two to three orders of magnitude larger than that for bulk OFHC-Cu. With increasing the deposition rate to 4 nm/s, the resistivity increased to w2.3  108 Um due to the higher density of S3{111} and S3{112} twin interfaces, as well as the decay of coherency of domain boundaries. It is noted that the latter resistivity is still one order of magnitude smaller than that for polycrystalline nanotwinned Cu films grown on oxidized Si under the same sputtering conditions [10]. The resistivity of a twinned film depicted in Fig. 9.7 can be expressed as:   1 2 rðT Þ ¼ ri ðT Þ þ h þ h ; (9.14) dtwin;111 TB;111 dtwin;112 TB;112 where dtwin,111 and hTB,111 are the average twin fault spacing and specific resistivity for S3{111} twin interfaces, respectively. dtwin,112 and hTB,112 are the average twin fault spacing and specific resistivity for S3{112} twin faults, respectively. It should be noted that experiments suggested the larger specific resistivity for S3{112} interfaces. As a consequence, the experimentally determined value of the average specific twin fault resistivity in sputtered Cu films increased from w1.5  1017 to w3  1017 Um2 with increasing the deposition rate, which can be attributed to the contribution of

Effect of Lattice Imperfections on Electrical Resistivity of Nanomaterials

semicoherent S3{112} twin interfaces developed at higher sputtering rates. The higher deposition rate also caused an increase in hardness from 2100 to 2800 MPa at RT. Therefore, the strength-to-resistivity ratio of sputtered Cu with high density of nanotwins reached a superior value of w4  1010 MPa/Um, which did not change considerably with the variation of deposition rate. Similar high strength-to-resistivity ratio (w5  1010 MPa/Um) was obtained for nanotwinned 99.998% purity Cu film processed by pulsed electrodeposition (see Table 9.2), although the microstructure was very different from the epitaxially grown sputtered film [27]. The as-deposited Cu film was polycrystalline with a thickness of w100 mm and consisted of irregularshaped grains with random orientations. The microstructure resembles to that depicted in Fig. 9.6. The grains had a wide distribution with an average size of 400 nm and contained a high density of twin faults with a mean twin-fault spacing of 15 nm. Similar grain size was achieved in 99.96% purity Cu by 16 passes of ECAP at RT but without large amount of twin faults in the grain interiors [19]. The twin faults inside the grains of the electrodeposited film increased the strength by a factor of 2.6 without significant reduction of conductivity. It was shown in Section 9.3 that secondary phase body-centered cubic Cr nanoparticles in UFG fcc Cu matrix improve the strength-to-resistivity ratio significantly, since they increase the strength much more strongly than the resistivity. This beneficial microstructure in bulk materials can be achieved by sequential SPD and annealing processes. It is noted that Cu and Cr are practically immiscible in solid state. Applying sputtering techniques, Cu/Cr multiphase microstructures with improved strength-to-resistivity ratio can also be produced in thin films [41,43]. For instance, a Cu/Cr multilayer with 1 mm film thickness and 50 nm layer thickness exhibited a very high strength of 2000 MPa and a resistivity of w7  108 Um at RT. The strength-to-resistivity ratio was w2.9  1010 MPa/Um, which is almost as high as the best value obtained for bulk copper alloys (w3.3  1010 MPa/Um detected for Cue0.7wt.% Cre0.9wt.% Hf processed by five turns of HPT at RT and subsequently annealed for 1 h at 773 K, see Table 9.2). With decreasing layer thickness in Cu/Cr multilayers, the resistivity considerably increases due to the enhanced contribution of layer interfaces [41]. For layer thickness of 150 nm the resistivity was w5.5  108 Um at RT, which increased to w27.5  108 Um when the layer thickness was reduced to 1.25 nm. At the temperature of 4 K, the resistivity is smaller than that at RT with a value of 2.5e3.5  108 Um. This difference decreases slightly with the reduction of layer thickness. As the resistivity of Cu/Cr multilayers is much smaller than that for Cr thin films, it can be concluded that the resistivity is determined mainly by the current flow through the Cu layers, which is strongly influenced by the scattering at interfaces, and the Cr layers carry only a small fraction of current [41]. The ratio of the resistivities measured at RT and 4 K is much smaller (varying between 1 and 3) for Cu/Cr multilayers than for Cu thin films (about 8), which suggests an enhanced role of defects (primarily Cu/Cr interfaces and grain boundaries

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inside the layers) in conductive electron scattering. The strength, resistivity, conductivity, and strength-to-resistivity ratio for Cu/Cr multilayers with different layer thickness values are listed in Table 9.3. Fig. 9.8 shows that the strength-to-resistivity ratio strongly decreases with the reduction of layer thickness since interfaces and grain boundaries in Cu/Cr multilayers are more effective in the deterioration of conductivity than in the improvement of strength. Table 9.3 Grain size, mechanical strength, conductivity in International Annealed Copper Standard (IACS), resistivity, and strength-to-resistivity ratio measured at room temperature (RT) for Cu/Cr multilayers with different layer thicknesses. The strength is calculated as one-third of the hardness [41,43] Material and Strength-toprocessing Grain size Strength Conductivity Resistivity resistivity ratio method (nm) (MPa) in IACS (%) (10L8 Um) (1010 MPa/Um)

CueCr multilayer sputtered at RT, layer thickness: 50 nm CueCr multilayer, sputtered at RT, layer thickness: 25 nm CueCr multilayer sputtered at RT, layer thickness: 10 nm CueCr multilayer sputtered at RT, layer thickness: 5 nm CueCr multilayer sputtered at RT, layer thickness: 2.5 nm

32/10 (Cu/Cr)

2000

24

7.0

2.9

e

2070

18

9.5

2.2

9/7.5 (Cu/Cr)

2330

11

15

1.6

5/5 (Cu/Cr)

2330

9

20

1.2

2.5/2.5 (Cu/Cr)

2400

6

27.5

0.9

30

3.0 Strength-to-resistivity ratio

25

2.5 2.0

20

Strength

1.5

15

1.0 0.5

10

Resistivity 0

10

30 20 40 Layer thickness (nm)

50

Resistivity (10–8 Ωm)

Strength-to-resistivity (1010 MPa/Ωm) strength (GPa)

Effect of Lattice Imperfections on Electrical Resistivity of Nanomaterials

5

Figure 9.8 Strength, resistivity, and strength-to-resistivity ratio as a function of layer thickness for Cu/Cr multilayers processed by sputtering.

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[16] L. Guobin, S. Jibing, G. Quanmei, W. Ru, Fabrication of the nanometer Al2O3/Cu composite by internal oxidation, Journal of Materials Processing Technology 170 (2005) 336e340. [17] E.A. Korznikova, S.Y. Mironov, A.V. Korznikov, A.P. Zhilyaev, T.G. Langdon, Microstructural evolution and electro-resistivity in HPT nickel, Materials Science and Engineering A 556 (2012) 437e445. [18] E. Schafler, G. Steiner, E. Korznikova, M. Kerber, M.J. Zehetbauer, Lattice defect investigation of ECAP-Cu by means of X-ray line profile analysis, calorimetry and electrical resistometry, Materials Science and Engineering A 410e411 (2005) 169e173. [19] O.F. Higuera-Cobos, J.M. Cabrera, Mechanical, microstructural and electrical evolution of commercially pure copper processed by equal channel angular extrusion, Materials Science and Engineering A 571 (2013) 103e114. [20] Y.G. Ko, S. Namgung, B.U. Lee, D.H. Shin, Mechanical and electrical responses of nanostructured Cue3wt.%Ag alloy fabricated by ECAP and cold rolling, Journal of Alloys and Compounds 504S (2010) S448eS451. [21] F. Khodabakhshi, M. Kazeminezhad, The effect of constrained groove pressing on grain size, dislocation density and electrical resistivity of low carbon steel, Materials and Design 32 (2011) 3280e3286. [22] S.V. Dobatkin, D.V. Shangina, N.R. Bochvar, M. Janecek, Effect of deformation schedules and initial states on structure and properties of Cue0.18% Zr alloy after high-pressure torsion and heating, Materials Science and Engineering A 598 (2014) 288e292. [23] R.F. Need, D.J. Alexander, R.D. Field, V. Livescu, P. Papin, C.A. Swenson, D.B. Mutnick, The effects of equal channel angular extrusion on the mechanical and electrical properties of alumina dispersion-strengthened copper alloys, Materials Science and Engineering A 565 (2013) 450e458. [24] S. Nachum, N.A. Fleck, M.F. Ashby, A. Colella, P. Matteazzi, The microstructural basis for the mechanical properties and electrical resistivity of nanocrystalline CueAl2O3, Materials Science and Engineering A 527 (2010) 5065e5071. [25] V. Rajkovic, D. Bozic, M.T. Jovanovic, Effects of copper and Al2O3 particles on characteristics of CueAl2O3 composites, Materials and Design 31 (2010) 1962e1970. [26] O. Anderoglu, A. Misra, H. Wang, F. Ronning, M.F. Hundley, X. Zhang, Epitaxial nanotwinned Cu films with high strength and high conductivity, Applied Physics Letters 93 (2008) 083108. [27] L. Lu, Y. Shen, X. Chen, L. Qian, K. Lu, Ultrahigh strength and high electrical conductivity in copper, Science 304 (2004) 422e426. [28] C. Arnaud, F. Lecouturier, D. Mesguich, N. Ferreira, G. Chevallier, C. Estournes, A. Weibel, A. Peigney, C. Laurent, High strengthehigh conductivity nanostructured copper wires prepared by spark plasma sintering and room-temperature severe plastic deformation, Materials Science and Engineering, A 649 (2016) 209e213. [29] G. Yang, Z. Li, Y. Yuan, Q. Lei, Microstructure, mechanical properties and electrical conductivity of Cue0.3Mge0.05Ce alloy processed by equal channel angular pressing and subsequent annealing, Journal of Alloys and Compounds 640 (2015) 347e354. [30] C. Zhu, A. Maa, J. Jiang, X. Li, D. Song, D. Yang, Y. Yuan, J. Chen, Effect of ECAP combined cold working on mechanical properties and electrical conductivity of conform-produced CueMg alloys, Journal of Alloys and Compounds 582 (2014) 135e140. [31] S.V. Dobatkin, J. Gubicza, D.V. Shangina, N.R. Bochvar, N.Y. Tabachkova, High strength and good electrical conductivity in CueCr alloys processed by severe plastic deformation, Materials Letters 153 (2015) 5e9. [32] D.V. Shangina, J. Gubicza, E. Dodony, N.R. Bochvar, P.B. Straumal, N.Y. Tabachkova, S.V. Dobatkin, Improvement of strength and conductivity in Cu-alloys with the application of high pressure torsion and subsequent heat-treatments, Journal of Materials Science 49 (2014) 6674e6681. [33] T.S. Orlova, A.M. Mavlyutov, A.S. Bondarenko, I.A. Kasatkin, M.Y. Murashkin, R.Z. Valiev, Influence of Grain Boundary State on Electrical Resistivity of Ultrafine Grained Aluminium, Philosophical Magazine, 2016, http://dx.doi.org/10.1080/14786435.2016.1204022. [34] M. Lipinska, P. Bazarnik, M. Lewandowska, The influence of severe plastic deformation processes on electrical conductivity of commercially pure aluminium and 5483 aluminium alloy, Archives of Civil and Mechanical Engineering 16 (2016) 717e723.

Effect of Lattice Imperfections on Electrical Resistivity of Nanomaterials

[35] R.Z. Valiev, M. Yu Murashkina, I. Sabirov, A nanostructural design to produce high-strength Al alloys with enhanced electrical conductivity, Scripta Materialia 76 (2014) 13e16. [36] X. Sauvage, E.V. Bobruk, M. Yu Murashkin, Y. Nasedkina, N.A. Enikeev, R.Z. Valiev, Optimization of electrical conductivity and strength combination by structure design at the nanoscale in AleMgeSi alloys, Acta Materialia 98 (2015) 355e366. [37] M.Y. Murashkin, I. Sabirov, A.E. Medvedev, N.A. Enikeev, W. Lefebvre, R.Z. Valiev, X. Sauvage, Mechanical and electrical properties of an ultrafine grained Ale8.5 wt. % RE (RE¼5.4wt.% Ce, 3.1wt.% La) alloy processed by severe plastic deformation, Materials and Design 90 (2016) 433e442. [38] K. Edalati, T. Daio, M. Arita, S. Lee, Z. Horita, A. Togo, I. Tanaka, High-pressure torsion of titanium at cryogenic and room temperatures: grain size effect on allotropic phase transformations, Acta Materialia 68 (2014) 207e213. [39] O.V. Antonova, A. Yu Volkov, Changes of microstructure and electrical resistivity of ordered Cu-40Pd (at.%) alloy under severe plastic deformation, Intermetallics 21 (2012) 1e9. [40] A.O. Olofinjanaa, K.S. Tan, Achieving combined high strength and high conductivity in re-processed CueCr alloy, Journal of Achievements in Materials and Manufacturing Engineering 35 (2009) 14e20. [41] A. Misra, M.F. Hundley, D. Hristova, H. Kung, T.E. Mitchell, M. Nastasi, J.D. Embury, Electrical resistivity of sputtered Cu/Cr multilayered thin films, Journal of Applied Physics 85 (1999) 302e309. [42] Y. Tsuda, H. Omoto, K. Tanaka, H. Ohsaki, The underlayer effects on the electrical resistivity of Ag thin film, Thin Solid Films 502 (2006) 223e227. [43] A. Misra, M. Verdier, Y.C. Lu, H. Kung, T.E. Mitchell, M. Nastasi, J.D. Embury, Structure and mechanical properties of CueX (X ¼ Nb,Cr,Ni) nanolayered composites, Scripta Materialia 39 (1998) 555e560.

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CHAPTER 10

Lattice Defects and Diffusion in Nanomaterials 10.1 EFFECT OF LATTICE DEFECTS ON DIFFUSION In nanostructured materials, the mass transport is faster than in coarse-grained counterparts due to the large amount of lattice defects, such as dislocations, grain boundaries, and interfaces. Along these defects the atomic diffusion is more rapid than in the bulk material. First, let us overview briefly the influence of lattice defects on diffusion. The rate of diffusion is usually characterized by the diffusion coefficient, D (also referred to as diffusivity). If the basic step of the diffusion of an atom is the position exchange with a vacancy, D can be expressed as: D ¼

1 2 dL cv n0 eGVM =RT ; 6

(10.1)

where dL is the spacing between the atomic sites (i.e., the lattice spacing) in the direction of diffusion, cv is the vacancy concentration, n0 is the frequency of thermal vibrations of atoms in the lattice, GVM is the Gibbs free energy for vacancy migration, R is the universal gas constant, and T is the absolute temperature in Kelvin degrees. If the studied material is in thermal equilibrium, the vacancy concentration can be given as: cv ¼ eGVF =RT ;

(10.2)

where GVF is the Gibbs free energy for vacancy formation. In this case the diffusion coefficient can be expressed by the following simple formula: D ¼ D0 e½EVF þEVM þpðVVF þVVM Þ=RT ;

(10.3)

where D0 is the frequency factor, EVF and EVM are the formation and migration energies of vacancies, respectively, and p is the pressure. VVF and VVM are the formation and migration volumes of vacancies, respectively. For atmospheric pressure the term p(VVF þ VVM) is about five orders of magnitude smaller than EVF or EVM, therefore it can be neglected in Eq. (10.3). The sum of EVF or EVM is referred to as the activation energy of diffusion and denoted as Q, therefore Eq. (10.3) can be written as (Arrhenius equation): D ¼ D0 eQ=RT .

(10.4)

Eq. (10.1) indicates that a larger vacancy concentration yields a higher diffusivity. Therefore, excess vacancies (i.e., the vacancy concentration above the equilibrium value) Defect Structure and Properties of Nanomaterials ISBN 978-0-08-101917-7, http://dx.doi.org/10.1016/B978-0-08-101917-7.00010-4

© 2017 Elsevier Ltd. All rights reserved.

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may result in a much higher diffusion coefficient than the value determined from Eq. (10.4). Large concentration of excess vacancies may develop during grain refinement in metallic materials processed by severe plastic deformation (SPD). It is noted that the Gibbs free energy of lattice self-diffusion is about 40% of the cohesive energy for pure face-centered cubic (fcc), body-centered cubic (bcc), and hexagonal close-packed (hcp) materials [1]. For interstitial diffusion, vacancy is not needed for atomic migration, therefore Q ¼ EIM, where EIM is the migration energy of interstitials. The fast diffusion of interstitials is attributed to their low Q values. For self-diffusion and migration of substitutional atoms, the order of magnitude of Q is 1e3 eV (96e288 kJ/mol); however, its value depends on the atomic environment of diffusing element. For instance, the diffusion along lattice defects with free volumes (e.g., dilational sides of edge dislocations or incoherent grain boundaries) is faster than in the bulk material. The variation of the logarithm of self-diffusion coefficient as a function of the reciprocal of temperature for bulk lattice, general grain boundaries, and free surface is illustrated schematically in Fig. 10.1. The lower slope corresponds to smaller activation energy of diffusion. The activation energy of diffusion along grain boundaries is about two times smaller than for lattice diffusion. Grain boundary self-diffusion in metals is about four to six orders of magnitude faster than bulk lattice diffusion, depending on the temperature (the difference is larger for lower temperatures) [2]. The large difference between the bulk and grain boundary diffusion coefficients is mainly caused by the difference in the activation energies, while the preexponential factors DL0 and DGB0 are only slightly different. It is noted that at temperatures close to the melting point the grain boundary diffusion coefficient approximates the diffusion coefficient in the melt. The diffusion activation energy along dislocations is similar to grain boundaries. The easiest

Surface DS = DS0e–QS / RT Grain boundary InD

272

DGB = DGB0e–QGB / RT Bulk lattice QL : QGB : QS ≈ 4 : 2 : 1

DL = DL0e–QL / RT

DL < DGB < DS 1/T

Figure 10.1 Schematic of diffusion coefficient (D) versus reciprocal of temperature (T) for bulk lattice, general grain boundaries (GBs), and free surface. The diffusion coefficient is plotted in logarithmic scale. DS, DGB, and DL stand for the diffusion coefficients for surface, grain boundary and lattice diffusion.

Lattice Defects and Diffusion in Nanomaterials

atomic migration can be observed on the free surface that is indicated by the four times smaller activation energy compared to the value determined for lattice diffusion. For pure fcc metals, the lattice self-diffusion activation energy has a linear relationship with the melting point (Tm): QL ½kJ=mol ¼ CL $Tm ½K;

(10.5)

where CL z 0.155 kJ mol1 K1. In the case of grain boundary self-diffusion in fcc metals, a similar relationship holds (QGB ¼ CGB  Tm) with CGB of about 0.078 kJ mol1 K1. For bcc and hcp metals the values of CGB are equal to 0.092 and 0.087 kJ mol1 K1, respectively [2]. Because of the fast diffusion along grain boundaries, they can be considered as shortcircuit diffusion paths in materials. However, the diffusion usually occurs in both bulk lattice (inside the grains) and grain boundaries concomitantly. When the grain boundary self-diffusion of a material is investigated experimentally, a tracer atom layer is deposited on the surface and the tracer atom distribution is investigated as a function of time at a given temperature. Fig. 10.2a shows the basic processes and the tracer atom diffusion profile developed in the vicinity of a grain boundary during simultaneous lattice and grain boundary diffusion. The following processes operate during the migration of tracer atoms from the deposited layers: direct lattice diffusion from the surface, diffusion along the grain boundaries, partial leakage from the grain boundaries to the grain interiors, and (a)

y

(b) Lattice diffusion

Leakage

lnc–

Lattice diffusion

δ

GB diffusion

Grain boundary diffusion

x

Leakage

Deposited surface tracer layer

Tracer diffusion profile

Grain boundary x6/5 Lattice diffusion

Figure 10.2 (a) Schematic showing the basic diffusion processes and the tracer atom diffusion profile (indicated by light gray color) developed in the vicinity of a grain boundary due to simultaneous lattice and grain boundary (GB) diffusion. d is the thickness of the grain boundary. Plot of lnc versus x6/5 gives a straight tail part for large x values, which is characteristic for grain boundary diffusion, as shown in (b). This straight line can be used for the determination of grain boundary diffusion coefficient (see the text for details).

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the subsequent bulk diffusion near the grain boundaries (Fisher’s model) [2e5]. In the leakage process, a fraction of atoms diffusing along the grain boundary leaves it and migrates in the bulk lattice adjacent to the boundary. As the grain boundary diffusion is faster than the atomic migration in the bulk lattice, therefore the penetration depth of tracer atoms is larger in the grain boundary than in the bulk lattice, as indicated in Fig. 10.2a. It is assumed that the tracer concentration is constant along the thickness of the grain boundary, i.e., it does not depend on the coordinate y (see Fig. 10.2a). The thickness of the grain boundary is denoted as d, and its value can be approximated as 0.5 nm [2]. According to the diffusion processes, the tracer concentrations in the boundary and the bulk lattice [denoted as cGB(x, t) and cL(x, y, t), respectively, where t is the time, and coordinates x and y are shown in Fig. 10.2a] can be obtained by solving the following equations:
2  vcL v cL v2 cL ¼ DL þ 2 ; for jyj > d=2 (10.6) vt vx2 vy and

 vcGB v2 cGB 2DL vcL  . ¼ DGB 2 þ vt vx d vy y¼d=2

(10.7)

The first and the second terms on the right side of Eq. (10.7) correspond to the diffusion along the grain boundary and the leakage from grain boundary to the bulk lattice. During a diffusion experiment, the penetrated tracer concentration is determined by serial sectioning method. In this procedure, thin layers parallel to the diffusion source film are removed from the sample and the tracer concentrations in these sections are measured. This concentration is an average perpendicular to axis x and denoted as c. Model calculations using Eqs. (10.6) and (10.7) revealed that lnc depends linearly from x6/5. Therefore, plotting lnc versus x6/5 may provide the grain boundary diffusion coefficient. In practice, because of the lattice diffusion at the deposited source layer, this evaluation procedure works only for large x values where grain boundary diffusion dominates, as shown in Fig. 10.2b. Thus, the grain boundary diffusion coefficient can be determined from the absolute value of the slope of the straight line fitted to the right tail part of lnc versus x6/5 plot (m) as (WhippleeLe Claire equation):
1 A DL 2 5 m3; (10.8) DGB ¼ d t where t is the time of experiment and A is a constant depending on the type of tracer source. For constant and instantaneous source types the value of A are 1.322 and 1.308, respectively [2]. The first condition means that the tracer concentration is constant at the surface (x ¼ 0), irrespective of time. The second source type is also

Lattice Defects and Diffusion in Nanomaterials

referred to as thin layer condition for which all tracer atoms are in the deposited layer at t ¼ 0, and during the diffusion experiment, they are completely consumed by the specimen. For the constant source condition, Eq. (10.8) applies only if (1) the lattice 1 diffusion length ðDL tÞ2 is at least 10 times larger than the half thickness of the boundary (d/2) and (2) the grain boundary diffusion coefficient is at least 100 times larger than the lattice diffusion coefficient. For the instantaneous source, Eq. (10.8) is strictly valid only if the ratio of the grain boundary and lattice diffusion coefficients is at least 105. For the determination of DGB from Eq. (10.8), the lattice diffusion coefficient and the grain boundary thickness must be known from independent experiments. In general, d ¼ 0.5 nm is a reasonable approximation for the grain boundary thickness [2]. The measurement of DGB at different temperatures yields the grain boundary diffusion activation energy (QGB) in accordance with Eq. (10.4). The values of QGB are 40%e50% of the lattice self-diffusion activation energies. It is noted that for impurity diffusion in the equations and conditions presented above, d must be substituted by sd where s is the equilibrium impurity segregation factor. This factor describes the segregation of impurities at the grain boundaries, and it can be obtained from the following coupling condition at the periphery of the grain boundary (at y ¼ d/2): cGB(x, t) ¼ s  cL(x, d/2,t). The value of s is one for grain boundary self-diffusion. In practice, dDGB or sdDGB is determined from Eq. (10.8), and DGB can be obtained only if s and d are known from other experiments. The kinetics of the combined lattice and grain boundary diffusion are usually classified into three groups [2e4,6]. Fig. 10.3 shows the three basic tracer spatial distribution types Type A

Type B

Type C

Decreasing temperature and/or time of diffusion

δ Grain boundary

Deposited surface tracer layer

Tracer diffusion profile

Figure 10.3 Schematic of the three basic tracer spatial distribution types: A, B, and C. The tracer atoms diffuse from the surface layer at the right side. The volumes containing larger amount of tracer atoms than a given concentration are indicated by gray color (tracer profile) [6].

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A, B, and C that evolve due to the different contributions of bulk and grain boundary diffusion (Harrison’s classification). The diffusion measurements can be performed in different regimes (A, B, or C) by an appropriate selection of experimental conditions, such as temperature and measuring time. For type A diffusion kinetics, the bulk diffusion regions around the neighboring grain boundaries strongly overlap and the faster diffusion along the grain boundaries is not reflected in the concentration profile. This diffusion 1 type develops when the lattice diffusion length ðDL tÞ2 is much larger than the spacing between the grain boundaries (d). The above condition is fulfilled when the temperature is high, the duration of experiment is long, and/or the grain size is small. For type B diffusion kinetics, although there are simultaneous grain boundary and lattice diffusion processes, the lattice diffusion regions at the neighboring grain boundaries do not overlap   1 (see Fig. 10.3). This diffusion type occurs if the lattice diffusion length ðDL tÞ2 is much smaller than the boundary spacing but larger than the grain boundary thickness (or sd for impurity diffusion). In the diffusion regime, referred to as type C, the bulk lattice diffusion is negligible compared to grain boundary diffusion, therefore diffusion takes place only along the grain boundaries without any considerable leakage to the bulk lat  1 tice. This condition is fulfilled if the lattice diffusion length ðDL tÞ2 is much smaller than the grain boundary thickness (at least by a factor of 10), which can be achieved in low temperature and short time experiments. As a consequence, Eq. (10.8) cannot be applied in the evaluation of diffusion profile in regime C. Instead, the classical Gaussian or error function must be used for the characterization of the diffusion profile for the instantaneous or constant source, respectively. In this case, lnc is plotted as a function of x2 that yields directly the value of DGB. Table 10.1 summarizes the three grain boundary diffusion types and the conditions required for their occurrence. If grain boundary diffusion experiments are performed in regime type B, only the product dDGB or sdDGB can be obtained from the profile evaluation for self-diffusion or impurity diffusion, respectively. Although, d ¼ 0.5 nm is a reasonable approximation for the grain boundary thickness, the value of s is usually difficult to determine. Table 10.1 The grain boundary diffusion kinetics types, the related experimental conditions, and the evaluated function of concentration ðcÞ versus penetration depth (x) Diffusion type Condition Evaluated function Obtained parameter

A

1

ðDL tÞ2 [d 1 2

B

sd  ðDL tÞ  d

C

sd[ðDL tÞ2

1

lnc versus x2

fDGB þ (1f)DL

lnc versus x6/5

sdDGB

lnc versus x2

DGB

d is the grain boundary thickness, d is the spacing between grain boundaries, DL is the lattice diffusion coefficient, f is the volume fraction of grain boundaries in the material, s is the impurity segregation factor (s ¼ 1 for pure metals), and t is the time of experiment.

Lattice Defects and Diffusion in Nanomaterials

In addition, s varies with temperature following an Arrhenius law; therefore the apparent grain boundary diffusion activation energy obtained from the profile evaluation is the sum of the real grain boundary diffusion activation energy and the impurity segregation energy [2]. For instance, if we consider Te diffusion in Ag, the values of grain boundary diffusion activation and impurity segregation energies are comparable (87 and 43 kJ/ mol, respectively), due to the very high level of Te segregation at Ag grain boundaries (s ¼ 103e105 owing to the low solubility limit of Te in Ag). At the same time, for Au impurity diffusion in Cu the value of s is only about 10 due to the high solubility of Au in Cu grains. Therefore, the impurity segregation energy is small (only about 10 kJ/mol) compared to the grain boundary diffusion activation energy (w81 kJ/mol). It is also worth to note that grain boundary diffusion of impurities is often hindered by their favored positions in the grain boundary structure. The higher the segregation tendency, the stronger the binding of impurities to the favored grain boundary sites, which reduces the diffusion rate. At the same time, there may be an opposite trend, namely the binding between impurities and vacancies, which can accelerate diffusion in both the bulk lattice and the grain boundaries. It should also be noted that the grain boundary self-diffusion is influenced by the impurities (or alloying elements) segregated to the grain boundaries. This effect depends on the type of alloying element: for instance 0.2 at.% silver in Cu has only a negligible influence on grain boundary self-diffusion while already 1 ppm of sulfur changes the grain boundary diffusion of Cu [7,8]. It is noted that when a grain boundary moves parallel to the surface layer (e.g., due to grain growth at elevated temperatures), the tracer penetration profiles before and after the moving boundary differ from each other, as shown in Fig. 10.4a. It should also be noted that

Deposited surface tracer layer

(a)

Stationary grain boundary

(b)

Grain boundary

δ Subgrain boundary Moving grain boundary Tracer diffusion profile Tracer diffusion profile Dislocations

Figure 10.4 (a) Schematic of the tracer penetration profiles before and after a boundary moving parallel to the surface layer. (b) The improved Fisher’s model with secondary short-circuit diffusion paths (dislocations, subgrain, and interface boundaries). This diffusion regime is also referred to as type D kinetics.

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Defect Structure and Properties of Nanomaterials

although the calculation of Eq. (10.8) is based on a bicrystal-like simple model presented in Fig. 10.2a; it is also valid for randomly oriented grain boundary structures [9], therefore it gives a basis of the quantitative evaluation of grain boundary diffusion in polycrystalline materials [3]. The Fisher’s model was improved by involving secondary short-circuit diffusion paths (e.g., dislocations, subgrain, and interface boundaries) in grain interiors [10]. In the extended model, these diffusion paths are connected to grain boundaries, thereby resulting in a faster leakage of tracer atoms from grain boundaries to grain interiors, as shown in Fig. 10.4a. The dislocation pipe diffusion dominated grain boundary diffusion regime is often denoted as type D kinetics. According to new leakage paths, additional terms are given to Eqs. (10.6) and (10.7), as well as an extra equation for diffusion along dislocations is introduced. If dislocations are the main reason of leakage from grain boundaries, Eq. (10.8) remains valid for the slope of the straight line fitted to lnc versus x6/5 data, only DL must be substituted by g2Dd or gDd for short and long diffusion times, respectively, where DL is the diffusivity for dislocations and g is the volume fraction of dislocation pipes in the material [10]. Grain boundary diffusion is sensitive to the grain boundary misorientation since the different misorientations are usually related to different grain boundary structures. The misorientation dependence of grain boundary diffusion is usually studied in bicrystals. It was revealed that Bi diffusivity in Cu bicrystals strongly correlates to the energy of grain boundary [11]. These studies were carried out on [001] and [011] twist and symmetric tilt boundaries. The higher the degree of disorder (or free volume) in the atomic arrangement of the boundary, the larger the grain boundary energy and the faster the diffusion along the grain boundary. For low-angle grain boundaries (LAGBs) in Ag with misorientations smaller than 15 degrees and tilt axis [001], the diffusivity increases monotonously with increasing misorientation [12]. At the same time, for high-angle grain boundaries (HAGBs) the diffusivity shows a minima at misorientations with low S values since these boundaries have highly ordered and low energy structures. S is used for the characterization of coincidence site lattice (CSL) boundaries and its value is equivalent to the inverse density of coincidence sites in the grain boundary. The diffusivity is very sensitive to misorientation of HAGBs. For instance, the diffusivity along S5 boundary with misorientation of 36.9 degrees is smaller by a factor of 3 and 5 at 780 and 661 K, respectively, than for a boundary with 1 degree smaller or higher misorientation [13]. Fig. 10.5 shows schematically the variation of the logarithm of grain boundary diffusion coefficient in Cu as a function of misorientation for [001] symmetric tilt boundaries. It is noted that the minimum diffusivity was measured at about 0.4 degree smaller angle than the ideal misorientation for S5 CSL boundary (see the inset in Fig. 10.5). This deviation was caused by an accidental and unintentional formation of a dislocation network at S5 CSL boundary that accelerated the diffusion. At the misorientation angle related to the minimum diffusivity, the diffusion activation energy is 30%e40% higher while the prefactor (DGB0) is one order of magnitude larger than the values obtained a few degrees

Lattice Defects and Diffusion in Nanomaterials

Logarithm of grain boundary diffusion coefficient (a.u.)

Σ17

Σ5

Σ5

Σ17

Σ5 35.5° 36.5° 37.5° 0°

10°

20°

30°

40°

50°

60°

70°

80°

90°

Angle of misorientation (degrees)

Figure 10.5 Schematic showing the variation of the logarithm of grain boundary diffusion coefficient in Cu as a function of misorientation angle for [001] symmetric tilt boundaries. The plot was constructed according to the data presented in [11] and [13]. The dashed line in the inset shows the ideal misorientation angle for S5 coincidence site lattice boundary.

away from this angle. The first effect overwhelms the second one, resulting in a slower diffusion in the very close vicinity of CSL misorientation [13]. Similar misorientation dependence was observed for both impurity and self-diffusion along symmetrical tilt boundaries in Ag [14] and Cu [15]. One order of magnitude difference was observed between the largest and smallest grain boundary self-diffusion coefficients measured for [001] symmetric tilt boundaries with different misorientations in Ag at 771 K [14]. Atomistic simulations revealed that the grain boundary self-diffusion activation energy in Cu decreases with increasing energy of HAGBs with S values between 5 and 13 [15]. This trend is shown schematically in Fig. 10.6. It can be seen that the diffusion activation energy for a general grain boundary with the energy of 0.6 J/m2 is about half of the value determined for vacancy diffusion in the bulk lattice. This result of the atomistic simulations is in a reasonable agreement with the experiments (see aforementioned). Large differences between the diffusivities were also found for twist boundaries. For instance, the diffusion of Bi along [011] twist boundaries in Cu is faster than in [001] twist boundaries [11]. It is emphasized that the diffusivity is more sensitive to the boundary structure than the boundary energy. For instance, 20% smaller boundary energy yields similar change in the activation energy due to the linear relationship between them (see Fig. 10.6). However, this variation in the activation energy leads to a three orders of magnitude larger diffusion coefficient owing to the exponential function in the Arrhenius formula of diffusivity. The effect of grain boundary misorientation on diffusivity can be monitored not only in bicrystals, but also in polycrystalline materials. For instance, in Ale3.3% Mge0.1% Sc alloy the grain boundary character was tailored by an appropriate selection of SPD route [16]. Processing by equal channel angular

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Defect Structure and Properties of Nanomaterials

3.0 Cu Activation energy of diffusion (eV)

280

2.5 Vacancy diffusion in lattice

2.0

Grain boundary diffusion

1.5 1.0 0.5 0.0 0.0

0.2

0.4 0.6 0.8 Grain boundary energy (J/m2)

1.0

1.2

Figure 10.6 Dependence of self-diffusion activation energy on high-angle grain boundary energy for symmetric tilt grain boundaries in Cu [15]. The error bar at the straight line reflects the difference between the data measured parallel and perpendicular to the tilt axis.

pressing (ECAP) via route C for two passes and a subsequent diffusion annealing at 523 K for 5 days resulted in an HAGB fraction of 21%. At the same time, eight ECAP passes through route BC and a subsequent diffusion annealing at 523 K for 20 days yielded an HAGB fraction of 60%. The diffusion coefficient of Mg in Al was higher for the sample containing higher fraction of HAGBs, in accordance with Fig. 10.5. Similar trend was observed for Zn diffusion in Ale2.1% Zne0.1% Sc alloy, which was also processed by two different routes of ECAP and subsequent annealing. In these materials, SPDprocessing yielded ultrafine-grained (UFG) microstructure with the grain size of 200e400 nm, and this was coarsened to 1e2 mm during subsequent annealing [16]. It was found that the one order of magnitude difference between the grain boundary diffusion coefficients of the Ale3.3% Mge0.1% Sc samples processed by the two different routes was caused by the larger preexponential factor (DGB0) for the material with higher fraction of HAGBs. At the same time, the activation energies of the two specimens were practically the same. In addition to the misorientation dependence of grain boundary diffusivity, there is also an anisotropy of diffusion in grain boundaries [11,15,17]. In general, for tilt boundaries the grain boundary diffusivity parallel to the tilt axis is much larger than perpendicular to this axis. For instance, in Ag for symmetrical tilt boundaries with tilt axis [001] and misorientation angle of 15 degrees the diffusion at 723 K parallel to the tilt axis is faster by a factor of 15 than perpendicular to the tilt axis [17]. This ratio decreases to two when the misorientation angle increases to 45 degrees. Low-angle tilt boundaries with misorientations smaller than 15 degrees usually consist of dislocations lying parallel to the tilt axis,

Lattice Defects and Diffusion in Nanomaterials

which explains well the higher diffusivity parallel to this axis. The similar diffusion anisotropy for HAGBs (the misorientation is larger than 15 degrees) suggests an ordered anisotropic structure instead of an amorphous-like atomic arrangement in these boundaries. It should be noted that atomistic simulation methods revealed that the dominance of vacancy or interstitial diffusion varies for the different S boundaries in Cu [15]. In addition, for some S boundaries the diffusion may be faster perpendicular to the tilt axis than parallel to that. Finally, it should be noted that diffusivity for triple junctions is orders of magnitude larger than that for grain boundaries, as it was shown for Zn diffusion in Al [18]. Impurity concentration at grain boundaries also influences diffusion along interfaces. A detailed investigation of grain boundary self-diffusion revealed that the purer the matrix, the faster the interface diffusion due to the lower activation energy [19]. The largest variation of diffusivity was observed between 99.999 and 99.99 wt.% purity Ni, while further increment of impurity level to 99.6 wt.% did not yield significant change in grain boundary diffusion coefficient. Impurity segregation at grain boundaries decreases the free volume and the energy of interfaces, thereby lowering diffusivity [20]. Therefore, grain refinement to UFG or nanosized regime increases the diffusivity not only due to the increase of amount of grain boundaries but also the purification of grain boundaries (as the same impurity content is distributed on a larger grain boundary area).

10.2 DIFFUSION IN ULTRAFINE-GRAINED AND NANOCRYSTALLINE MATERIALS PROCESSED BY SEVERE PLASTIC DEFORMATION Processing of UFG and nanocrystalline metals and alloys by SPD techniques yields a development of dislocation walls and LAGBs inside the initial grains. With increasing straining, these boundaries evolve into HAGBs, thereby refining the microstructure into UFG or nanocrystalline regime. When a minimum grain size is achieved, the HAGBs exhibit high energy and large free volume which result in high diffusivity along these boundaries (referred to as nonequilibrium HAGBs) [21]. The large free volume in nonequilibrium HAGBs is associated with the high density of extrinsic dislocations in the close vicinity of these boundaries [22]. These extrinsic dislocations are accumulated at HAGBs during SPD-processing. In contrast to geometrically necessary dislocations at grain boundaries, extrinsic dislocations are not needed for the accommodation of lattice misorientations between the neighboring grains. If SPD-processing continues even after achieving the saturation grain size, the HAGBs tend to transform into a more equilibrated state owing to the annihilation of a significant portion of extrinsic dislocations. This grain boundary relaxation can also be achieved by annealing. The relaxed HAGBs exhibit much slower diffusivity than the nonequilibrium boundaries. The grain boundary diffusion coefficients for Ni along nonequilibrium and relaxed boundaries in 99.98 wt.% purity UFG Cu and Cue0.17 wt.% Zr alloy were determined immediately after

281

Defect Structure and Properties of Nanomaterials

ECAP-processing [21,23]. It is noted that these measurements were carried out in C-type regime of grain boundary diffusion (see Table 10.1 and Fig. 10.3) by an appropriate selection of the time and temperature of diffusion. In the data evaluation, lnc is plotted as a function of x2, and the slope of the straight line fitted to the datum points provides the grain boundary diffusion coefficient. If both “fast” diffusion along nonequilibrium HAGBs and “slow” diffusion along relaxed HAGBs occur, the lnc versus x2 curve may have two distinct linear segments due to the bimodality of grain boundary diffusivity distribution, as shown in Fig. 10.7. The first segment at lower penetration depth and with higher slope corresponds to the slow diffusion path (along relaxed HAGBs), while the second segment at higher penetration depth and with lower slope is related to fast short-circuit diffusion along nonequilibrium boundaries [23]. It should be noted, however, that the two distinct segments can be observed only if the average diffusion distance along slow paths is smaller than the spacing between the fast diffusion boundaries. Otherwise, A-type diffusion kinetics (see Fig. 10.3) can be observed with a single effective diffusion coefficient, and lnc versus x2 plot exhibits a single slope accordingly. The different kinetics observable for materials with bimodal grain boundary diffusivity is summarized in [24]. For determination of diffusion coefficient, the measurements are usually carried out at different temperatures, including elevated temperatures. Therefore, the standard evaluation procedures work only if the UFG or nanocrystalline microstructure is stable during high-temperature diffusion experiments. This stability can be achieved by fine secondary phase precipitates distributed at the grain boundaries. For instance, in Cue0.17 wt.% Zr alloy processed by 4 passes of ECAP at room temperature (RT), the UFG microstructure with the grain size of 300 nm remained stable even at 623 K due to Cu5Zr particles at the grain boundaries [23]. This stable microstructure can be modeled by a skeleton of

Slow HAGB diffusion lnc

282

Fast HAGB diffusion

x2

Figure 10.7 Schematic of diffusion profile obtained in ultrafine-grained materials in which both slow and fast grain boundary diffusion occur. c is the average tracer concentration determined by serial sectioning method while x is the tracer penetration depth. HAGB, high-angle grain boundary.

Lattice Defects and Diffusion in Nanomaterials

Figure 10.8 Model of the hierarchical microstructure with nonequilibrium and relaxed grain boundaries, acting as fast and slow diffusion paths, respectively [23].

Deposited surface tracer layer

d

Slow diffusion path

Fast diffusion path

nonequilibrium grain boundaries showing fast diffusion which is embedded in a network of relaxed grain boundaries exhibiting slow diffusion. This hierarchical microstructure is shown schematically in Fig. 10.8. The domain size in the relaxed grain boundary network is equivalent to the grain size (300 nm) while the characteristic cell size in the nonequilibrium grain boundary skeleton is in the micrometer range for ECAPprocessed Cue0.17 wt.% Zr alloy [23]. During grain boundary diffusion in this hierarchical microstructure, there is a leakage from the fast boundaries into the slow paths (indicated by black arrows in Fig. 10.8), in analogy with the model depicted in Fig. 10.2a. In the model shown in Fig. 10.8 the slow diffusion occurs by a direct migration of tracer atoms from the surface layer to nonequilibrium grain boundaries, therefore the slow diffusivity can be determined from the slope of the first, steep part of the penetration profile (see Fig. 10.7) using Eq. (10.8). At the same time, the diffusion coefficient for fast diffusion (DGBf) can be obtained from the slope of the second, less steeper part of the penetration curve (mII) and the slow diffusivity (DGBs) using the following formula [10]:
1 2 DGBs 2 5 DGBf z mII 3 ; (10.9) d t where d is the spacing between the nonequilibrium grain boundaries lying perpendicular to the surface tracer layer (see Fig. 10.8). The frequency factor and the activation energy values for “fast” and “slow” Ni diffusion along nonequilibrium and relaxed HAGBs in Cue0.17 wt.% Zr alloy, respectively, are listed in Table 10.2. The diffusivity in the nonequilibrium boundaries is about two

283

284

99.98 wt.% Cu/slow GB Cue0.17 wt.% Zr/slow GB Cue0.17 wt.% Zr/fast GB 99.6 wt.% purity Ni/fast GB Ni/GB

Ni

4 ECAP at RT

300

Ni

4 ECAP at RT

300

Ni

4 ECAP at RT

300

Ni

4 ECAP at RT

300

Cu

300

Ni/GB

Ni

g-Fe40 wt.% Ni/slow GB g-Fee40 wt.% Ni/fast GB g-Fee40 wt.% Ni/slow GB g-Fee40 wt.% Ni/fast GB g-Fee40 wt.% Ni/slow GB g-Fee40 wt.% Ni/fast GB Fe90Zr7B3/slow GB

Fe

ECAP at RT þ annealing at 398 K for 1 h Compaction of powder at 773 K and 4.4 GPa Compaction of powder at 1123 K and 1.25 GPa Compaction of powder at 1123 K and 1.25 GPa Compaction of powder at 1123 K and 1.25 GPa Compaction of powder at 1123 K and 1.25 GPa Compaction of powder at 1123 K and 1.25 GPa Compaction of powder at 1123 K and 1.25 GPa Melt spinning þ crystallization at 873 K

Fe Ni Ni Ag Ag Fe

References

1.6  107 (2.3  103) 1.4  107 (2.0  103) 1.6  103 (22.9)

85 (0.38)

[21]

84 (0.37)

[23]

96 (0.43)

[23]

1.2  108 (1.3  104) -

67 (0.24)

[28]

43 (0.17)

[26]

46 (0.17)

[35]

80e100 (XRD)

2.2  1012 (2.4  108) 4.2  103 (4.8)

187 (0.62)

[36]

80e100 (XRD)

3.4  103 (3.9)

148 (0.49)

[36]

80e100 (XRD)

9.3  104 (1.2)

177 (0.58)

[37]

80e100 (XRD)

1.8  103 (2.3)

134 (0.44)

[37]

80e100 (XRD)

4.7  104 (0.39)

173 (0.62)

[38]

80e100 (XRD)

8.1  105 (0.07)

91 (0.33)

[38]

18 (XRD)

2.8  107 (1.4  103)

163 (0.65)

[39]

70

Defect Structure and Properties of Nanomaterials

Table 10.2 The frequency factor (D0), the activation energy (Q) of diffusion for different UFG and nanocrystalline materials Activation energy Host material/diffusion Frequency (kJ/mol) path Diffusant Processing Grain size (nm) factor (m2/s)

18 (XRD)

6.8  105 (0.34)

157 (0.62)

[39]

200

-

140 (0.46)

[40]

Mechanical alloying þ hot extrusion

200

-

76 (0.25)

[40]

22 (XRD)

[41]

48 (0.17)

[42]

UO2/GB

O

Molecular dynamic simulation

10

71 (0.29)

[42]

UO2/GB

Xe

Molecular dynamic simulation

10

48 (0.17)

[42]

W-1 at.% H/L 99.9% purity Pd

H H

Molecular dynamic simulation Cold rolling þ 10 HPT at RT under 1.5 GPa

10 350

2.4  1012 (3.7  108) 1.7  1010 (5.7  105) 5.4  108 (4.3  104) 1.4  109 (6.7  104) 1.1  107 (2.2) 1.0  108 (0.04)

35 (0.26)

U

Mechanical alloying þ hot compaction at 673 K Molecular dynamic simulation

74 (3.9) 15 (0.68)

[43] [44]

Fe90Zr7B3/fast GB

Fe

Fee14Cre3We0.4Ti e0.25Y2O3 in wt.%/slow GB Fee14Cre3We0.4Ti e0.25Y2O3 in wt.%/fast GB Al91.9Ti7.8Fe0.3/GB

Fe

Fe

Cu

UO2/GB

Melt spinning þ crystallization at 873 K Mechanical alloying þ hot extrusion

10

Lattice Defects and Diffusion in Nanomaterials

In the parentheses after the values, the ratios of the present values and those obtained for diffusion in bulk lattices are presented. The processing methods of the materials, the diffusants, and the grain size are also given. The grain size was usually determined by TEM, except the cases indicated by “XRD” where the crystallite size obtained from the breadth of X-ray diffraction peak profiles is presented. The path and rate of diffusion are indicated as: GB (grain boundary), L (lattice), “slow,” and “fast.” ECAP, equal channel angular pressing; RT, room temperature.

285

Defect Structure and Properties of Nanomaterials

10–11 10–12 GB diffusion 10–13 in coarse-grained Cu DGB (m2/S)

286

Fast GB diffusion in UFG Cu-0.17% Zr alloy

10–14 10–15 10–16 10–17

Slow GB diffusion in UFG Cu-0.17%Zr alloy

10–18 1.8

2.0

2.2

2.4

2.6

2.8

3.0

1/RT (10–4 mol/J)

Figure 10.9 Schematic illustrating the difference between the diffusivities of Ni for slow and fast grain boundaries in Cue0.17 wt.% Zr alloy processed by equal channel angular pressing at room temperature. The diffusivity of grain boundaries in coarse-grained pure Cu is also shown [23].

orders of magnitude larger than for relaxed HAGBs at RT, which is caused by the much higher value of frequency (or preexponential) factor (DGB0) for the former boundaries, as illustrated in Fig. 10.9. This figure also shows that the diffusivity for Ni migration along relaxed boundaries in the UFG Cue0.17 wt.% Zr alloy is similar to the grain boundary diffusion coefficient measured for Ni in coarse-grained 5N8 purity polycrystalline Cu for all temperatures [23]. Therefore, it can be concluded that grain size is not a deterministic factor in grain boundary diffusivity. It is noted that in this material Zr alloying atoms are in Cu5Zr precipitates, therefore their segregation at grain boundaries do not influence the diffusivity along HAGBs. It should also be noted that the term “slow HAGB diffusion” serves only to make a distinction between the different rates of grain boundary diffusion along various paths in UFG and nanocrystalline materials. In absolute terms it refers to a quick diffusion similar to the migration along general grain boundaries in coarse-grained materials. The rate of this “slow diffusion” is much higher than that for special CSL boundaries (see Section 10.1). In UFG Cue0.17 wt.% Zr alloy the majority of grain boundaries exhibit diffusivity close to that of HAGBs in high-purity coarse-grained copper, while the minority of the grain boundaries (nonequilibrium HAGBs) have much higher diffusion coefficient. The area fraction of fast diffusion boundaries is only 0.2%e0.5% in UFG Cue0.17 wt.% Zr alloy processed by ECAP at RT, therefore their contribution to the total diffusion flux is marginal. Consequently, it can be concluded that the grain boundary diffusion in SPD-processed UFG materials is very similar to the diffusion along general (not specific CSL) boundaries in coarse-grained counterparts. It is noted that for bulk lattice diffusion of Ni in Cu, the frequency factor (D0 ¼ 7  105 m2/s) is 2-3 orders of magnitude larger than for slow diffusion along

Lattice Defects and Diffusion in Nanomaterials

relaxed grain boundaries [23]. The activation energy for bulk lattice diffusion (225 kJ/ mol) is also higher by a factor of about 2.6 than for slow grain boundary diffusion. The latter effect overwhelms the former one, thereby resulting in a 21 orders of magnitude larger diffusion coefficient for slow grain boundary diffusion, compared to bulk lattice diffusion at RT. As in a nanocrystalline sample interfaces occupy a large fraction of the material, grain boundaries determine the diffusivity of the whole sample. Therefore, the diffusion coefficient of nanostructured materials is usually very close to the grain boundary diffusivity. In bulk nanomaterials compacted from nanopowders, usually there is some remaining porosity. The diffusion on the pore surfaces is as fast as the diffusion on the free surfaces. Therefore, the diffusivity for porous nanomaterials may be much larger than that for grain boundaries. Short time annealing of SPD-processed UFG and nanocrystalline materials at moderate temperatures may yield the relaxation of nonequilibrium boundaries without grain growth [25]. For instance, the grain boundary diffusivity of Cu measured at 423 K in ECAP-processed UFG Ni decreased by three orders of magnitude from 9.6  1015 m2/s to 2.8  1018 m2/s during annealing at 523 K for 1 h after ECAPprocessing [26]. This reduction of the grain boundary diffusion coefficient was not accompanied by grain coarsening (the grain size remained 300 nm), therefore it was attributed to grain boundary relaxation. The grain boundary diffusivity of Cu for relaxed grain boundaries in UFG Ni was still one order of magnitude higher than that for coarsegrained Ni (4.3  1019 m2/s at 423 K). This difference can be explained by an incomplete relaxation at 523 K for 1 h and/or the high diffusivity of triple junctions. The amount of triple junctions in UFG samples is higher than in coarse-grained counterparts, which may yield higher diffusivity in the former materials. The grain boundary diffusivity of Cu in electrodeposited Ni with the grain size of 30 nm was found to be 3.8  1017 m2/s at 523 K [26]. This value is about two orders of magnitude smaller than the diffusivity measured for ECAP-processed UFG Ni, indicating the high degree of relaxation for grain boundaries in electrodeposited Ni. At the same time, the grain boundary diffusion coefficient for electrodeposited Ni is two orders of magnitude larger than those coarse-grained counterparts, which can be attributed again to the effect of triple junctions [26]. It is noted that in SPD-processed UFG materials pores may form at triple junctions and grain boundaries from the excess vacancies. When these pores are percolated, a very high diffusivity component can be detected [27e29]. The diffusivity of percolating pores is similar to that measured on the free surface [30]. However, as the volume fraction of these pores is very small (e.g., around 106 for ECAP-processed Cu with the grain size of w300 nm [27]), their influence on the total diffusivity is very small. The chains of interconnected cavities exhibit high stability during annealing after SPD [31]. The back pressure during ECAP retards the formation of percolating pores. The diffusion in SPD-processed UFG and nanocrystalline materials can be accelerated by the excess vacancies formed during plastic straining. For RT SPD-processing, this

287

288

Defect Structure and Properties of Nanomaterials

excess vacancy concentration may reach w104, which is about 17 orders of magnitude larger than the equilibrium value [32e34]. In SPD-processed materials, besides the very large amount of grain boundaries and dislocations that act as fast diffusion paths, the high vacancy concentration also contributes to the large diffusivity according to Eq. (10.1). It should be noted, however, that during SPD-processing the accelerating effect of excess vacancies on diffusion is reduced by the retarding effect of the applied high pressure on vacancy migration. Especially in the course of HPT processing, the pressure may reach w10 GPa, which increases the vacancy migration enthalpy (HVM ¼ EVM þ pVVM) by about 0.5 eV. This 50% increment in HVM yields nine orders of magnitude smaller exponential factor in Eq. (10.1). Combining the effects of the larger vacancy concentration and the more difficult migration, the diffusion coefficient is still eight orders of magnitude larger during SPD-processing of UFG materials than in coarse-grained counterparts.

10.3 DIFFUSION IN NANOMATERIALS PROCESSED BY BOTTOM-UP METHODS In nanostructured materials processed by crystallization of amorphous alloys, the grain boundary diffusivity often exhibits similar bimodality as for SPD-processed UFG metallic materials [25]. Some boundaries resemble classical polycrystalline interfaces while others behave as amorphous layers between crystalline nanograins. This heterogeneous grain boundary structure is illustrated in Fig. 10.10a. The diffusivity of the “amorphous” boundaries is similar to that observed in bulk amorphous materials, which is lower than the grain boundary diffusion coefficient in crystalline counterparts. Therefore, in Crystalline boundaries

Intraagglomerate boundaries

Amorphous boundaries

Interagglomerate boundaries

(a)

(b)

Figure 10.10 Heterogeneous grain boundary structures with slow and fast diffusion pathways in nanomaterials processed by (a) crystallization of amorphous materials or (b) sintering from nanopowders.

Lattice Defects and Diffusion in Nanomaterials

crystallized nanomaterials the crystalline and amorphous interfaces correspond to fast and slow diffusion paths, respectively. As an example, the preexponential factors and the activation energies of Fe diffusion along slow and fast diffusion pathways in nanocrystalline Fe90Zr7B3 alloy (crystallite size is 18 nm) processed by crystallization of melt spun amorphous precursor at 873 K are shown in Table 10.2. Although the activation energies for diffusion along amorphous and crystalline boundaries are close to each other, the preexponential factor is about two orders of magnitude smaller for amorphous boundaries [39]. The fraction of amorphous-like boundaries is much larger than that for the conventional crystalline interfaces. Therefore, the diffusivity of nanocrystalline materials obtained by crystallization of amorphous alloys is often smaller than that for nanomaterials that have similar grain sizes but are processed by other methods [39,45,46]. In nanomaterials processed by powder metallurgy, the remaining porosity strongly influences the diffusion rate [24,25]. Diffusion on the “particle-pore interfaces” is much faster than along boundaries of coalescenced particles. The initial powders often consist of agglomerates of nanocrystallites, and after sintering the pores are usually formed between these agglomerates. Therefore, the fast and slow grain boundary pathways in sintered nanomaterials are often referred to as interagglomerate and intraagglomerate interfaces, respectively. This heterogeneous interface structure is depicted in Fig. 10.10b. In accordance with the coexistence of the two classes of boundaries, the diffusivity of sintered nanomaterials may show a bimodal nature. This effect was observed, for instance, in nanocrystalline g-Fee40 wt.% Ni alloy processed by powder metallurgy [36,47]. The initial powder was obtained by mechanical alloying, which was then compacted under the pressure of 1.25 GPa at 1123 K for 1 h. The sintered material had a relative density of 98%. The preexponential factor and the activation energy of diffusion of different species (Fe, Ni, and Ag) along interagglomerate and intraagglomerate interfaces are listed in Table 10.2. The diffusion coefficient for interagglomerate boundaries is orders of magnitude larger than that for intraagglomerate interfaces, mainly due to the smaller activation energy [36e38]. The relatively slower intraagglomerate boundary diffusivity was found to be very similar to the value determined for conventional grain boundaries in coarse-grained polycrystals. The diffusivities of interfaces in nanomaterials processed by different techniques are summarized schematically and compared to conventional lattice, boundary, and surface diffusion coefficients in Fig. 10.11. Overviewing the grain boundary diffusion activation energy data listed in Table 10.2 for different materials and diffusants, it can be concluded that their values relative to the bulk diffusion activation energies fall between 0.17 and 0.65. The small activation energy and the large amount of grain boundaries yield fast diffusion in nanomaterials. However, there are some exceptions. For instance, molecular dynamic simulations revealed that hydrogen diffusion in nanocrystalline tungsten is much slower than in coarse-grained counterparts due to the four times higher diffusion activation energy, caused by trapping

289

Defect Structure and Properties of Nanomaterials

SPD-processed nanomaterials

Sintered nanomaterials

Crystalline GB

Amorphous GB

Interagglomerate GB

Intraagglomerate GB

Percolating porosity at GBs

Bulk lattice diffusion

Relaxed GB

Conventional GB diffusion

Nonequilibrium GB

Surface diffusion

Diffusivity

290

Crystallized nanomaterials

Figure 10.11 Comparison of grain boundary diffusion coefficients for nanomaterials processed by different techniques: severe plastic deformationeprocessing, sintering from powders, and crystallization of amorphous materials. The diffusivity levels for conventional lattices, grain boundaries, and surfaces are also indicated by dashed horizontal lines. GB, grain boundary.

of H at vacancies in grain boundaries [43]. Trapping at dislocations also retards hydrogen diffusion in severely deformed nanomaterials. The hydrogen diffusivity in nanocrystalline W increases with H content. This can be explained by the larger occupancy of high energy hydrogen sites in grain boundaries with increasing H concentration. The jumping probability of hydrogen atoms occupying these high energy sites is larger, resulting in faster diffusion. This effect depends on the chemical composition of nanomaterials. For instance, in dispersion strengthened Fee25Nie15Cr alloy (wt.%) the refinement of grains to the size of 100e200 nm yielded an increase of hydrogen diffusivity [48]. The same effect was observed in UFG Pd [44]. Grain boundary diffusivity has an effect on the subsurface pore formation in nanocrystalline materials [49]. Excess vacancies may be produced near the surface by interdiffusion between two dissimilar materials (e.g., in multilayers), oxidation or anodic dissolution. These excess vacancies can migrate along grain boundaries into the bulk material, causing cavitation at grain boundaries and triple junctions in the subsurface region. Then, these pores can grow from triple junctions along grain boundaries toward the free surface. The cavities become more narrow and elongated for faster grain boundary diffusion, and they may yield the disintegration of nanomaterials [49]. For nanoparticulate systems with the particles sizes smaller than 20 nm, the reduction of melting point also contributes to the faster diffusion in nanocrystalline materials, since the self-diffusion activation energy is proportional to the melting point [see Eq. (10.5)]. According to thermodynamical calculations, the melting point is a function of the size of spherical particles (d) [50,51]: h am i Tm ¼ Tm0 1  ; (10.10) d

Lattice Defects and Diffusion in Nanomaterials

Figure 10.12 Variation of the logarithm of bulk lattice self-diffusion coefficient normalized by the frequency factor as a function of grain size for Cu at room temperature, if the influence of grain size on melting point is taken into account.

–20 Cu –30

In(DL/DL0)

–40 –50 –60 –70 –80 2

4

6

8

10

12

14

16

18

20

Grain size (nm)

where am depends on the bulk enthalpy of fusion (Hm) and the surface energies in solid and liquid states (denoted by gS and gL, respectively) as: am ¼

6ðgS  gL Þ . Hm

(10.11)

Since gS is usually larger than gL, the melting temperature decreases with decreasing particle size. Therefore, according to Eqs. (10.5) and (10.10), the diffusion activation energy is reduced when the particle size decreases. This reduction is only 5% for the particle size of 10 nm, while for 3 nm the activation energy decreases by 30% [50]. Therefore, this effect is significant only below w20 nm. The corresponding variation of the lattice selfdiffusion coefficient as a function of grain size for Cu at RT is shown in Fig. 10.12. For copper nanomaterials with grain size of 3 nm, the lattice self-diffusion coefficient is 17 orders of magnitude larger than for coarse-grained counterparts. It should be noted, however, that this calculation is valid only if Eq. (10.5) can be used for such a small grain size. It is also noticed that the thermodynamical conditions applied in the calculation of Eq. (10.10) can only be used for grain sizes larger than 2e3 nm [50]. For diffusion distances shorter than about 10 times the atomic spacing (w3 nm), instead of the continuum model a discrete atomic description of diffusion must be used in the interpretation of experiments [3]. It was shown that if the diffusivity exhibits strong concentration dependence, the validity of continuum approach can be extended even to 50 nm. With the application of discrete models (e.g., the model of Martin [52]) we can explain unique effects observed in nanostructures. It is emphasized that these effects are caused by the very small dimensions of structural units of nanomaterials and not by the large defect densities. As this chapter focuses on the influence of lattice defects on

291

Defect Structure and Properties of Nanomaterials

(b) t2 > t1 Mo layer Position

Figure 10.13 Illustration of dissolu- (a) tion of Mo in V for Mo/V multilayer while the interface between the two materials remains atomically sharp. (a) Schematic of a pair of Mo and V layers. (b) Change of V atom distribution perpendicular to the layers as a function of time of diffusion experiment. 3 nm

292

t1 > t0

t0

V layer

0 1 Atomic fraction of V

diffusivity, therefore only one example will be shown for the effect of low dimensionality on diffusion in nanomaterials. This special phenomenon can be observed in multilayers consisting of two different materials. If the interdiffusion coefficient shows strong concentration dependence and the layer thickness is a few nanometers, the interface between the two different layers remains atomically sharp during dissolution of one component in the other one. This effect is illustrated for Mo/V multilayer in Fig. 10.13. It can be seen that the dissolution is asymmetric as Mo diffuses into V layer while V is not dissolved in Mo layer [53]. Therefore, the interface moves in the direction of Mo layer until this material is not consumed. There is a step by step character of the dissolution of Mo in V, since only the Mo atomic layer at the interface is dissolved while the other Mo layers remain unchanged. The velocity of the interface is constant that can be explained only by discrete models, as the continuum description predicts that the velocity is proportional to the inverse square root of time. Similar dissolution of Ni in Cu was observed in Cu/ Ni(111) multilayers [54].

REFERENCES [1] R.V. Patil, G.P. Tiwari, Correlation of self and solute diffusion data with cohesive energy, Transactions of the Japan Institute of Metals 15 (1974) 432e434. [2] Y. Mishin, C. Herzig, Grain boundary diffusion: recent progress and future research, Materials Science and Engineering A 260 (1999) 55e71. [3] J. Bernardini, D.L. Beke, Diffusion in nanomaterials, in: P. Knauth, J. Schoonman, H.L. Tullor (Eds.), Nanocrystalline Metals and Oxides, Electronic Materials: Science and Technology, vol. 7, Kluwer Academic Publishers, Dordrecht, 2002, pp. 41e79. [4] C. Herzig, S.V. Divinski, Grain boundary diffusion in metals: recent developments, Materials Transactions 44 (2003) 14e27. [5] J.C. Fisher, Calculation of diffusion penetration curves for surface and grain boundary diffusion, Journal of Applied Physics 22 (1951) 74e76. [6] L.G. Harrison, Influence of dislocations on diffusion kinetics in solids with particular reference to the alkali halides, Transactions of the Faraday Society 57 (1961) 1191e1199.

Lattice Defects and Diffusion in Nanomaterials

[7] S. Divinski, M. Lohmann, C. Herzig, Grain boundary diffusion and linear and non-linear segregation of Ag in Cu, Interface Science 11 (2003) 21e31. [8] T. Surholt, C. Herzig, Grain boundary self-diffusion in Cu polycrystals of different purity, Acta Materialia 45 (1997) 3817e3823. [9] H.S. Levine, C.J. MacCallum, Grain boundary and lattice diffusion in polycrystalline bodies, Journal of Applied Physics 31 (1960) 595e599. [10] L. Klinger, E. Rabkin, Beyond the Fisher model of grain boundary diffusion: effect of structural inhomogeneity in the bulk, Acta Materialia 47 (1999) 725e734. [11] R. Monzen, T. Kuze, Misorientation dependence of grain boundary diffusion of Bi in Cu bicrystals, Journal of the Japan Institute of Metals and Materials 63 (1999) 1224e1228. [12] D. Turnbull, R.E. Hoffman, The effect of relative crystal and boundary orientations on grain boundary diffusion rates, Acta Metallurgica 2 (1954) 419e426. [13] E. Budke, T. Surholt, S.I. Prokofjev, L.S. Shvindlerman, C. Herzig, Acta materialia 47 (1999) 385e395. [14] J. Sommer, C. Herzig, S. Mayer, W. Gust, Grain boundary self-diffusion in silver bicrystals, Defect and Diffusion Forum 66e69 (1989) 843e848. [15] A. Suzuki, Y. Mishin, Atomistic modeling of point defects and diffusion in copper grain boundaries, Interface Science 11 (2003) 131e148. [16] T. Fujita, Z. Horita, T.G. Langdon, Using grain boundary engineering to evaluate the diffusion characteristics in ultrafine-grained AleMg and AleZn alloys, Materials Science and Engineering A 371 (2004) 241e250. [17] R.E. Hoffman, Anisotropy of grain boundary self-diffusion, Acta Metallurgica 4 (1956) 97e98. [18] B. Bokstein, V. Ivanov, O. Oreshina, A. Peteline, S. Peteline, Direct experimental observation of accelerated Zn diffusion along triple junctions in Al, Materials Science and Engineering A 302 (2001) 151e153. [19] D. Prokoshkina, V.A. Esin, G. Wilde, S.V. Divinski, Grain boundary width, energy and self-diffusion in nickel: effect of material purity, Acta Materialia 61 (2013) 5188e5197. [20] S.V. Divinski, G. Reglitz, G. Wilde, Grain boundary self-diffusion in polycrystalline nickel of different purity levels, Acta Materialia 58 (2010) 386e395. [21] Y. Amouyal, S.V. Divinski, L. Klinger, E. Rabkin, Grain boundary diffusion and recrystallization in ultrafine grain copper produced by equal channel angular pressing, Acta Materialia 56 (2008) 5500e5513. [22] X. Sauvage, G. Wilde, S.V. Divinski, Z. Horita, R.Z. Valiev, Grain boundaries in ultrafine grained materials processed by severe plastic deformation and related phenomena, Materials Science and Engineering A 540 (2012) 1e12. [23] Y. Amouyal, S.V. Divinski, Y. Estrin, E. Rabkin, Short-circuit diffusion in an ultrafine-grained copperezirconium alloy produced by equal channel angular pressing, Acta Materialia 55 (2007) 5968e5979. [24] S.V. Divinski, J.-S. Lee, C. Herzig, Grain boundary diffusion and segregation in compacted and sintered nanocrystalline alloys, Journal of Metastable and Nanocrystalline Materials 19 (2004) 55e68. [25] W. Sprengel, R. W€ urschum, Diffusion in nanocrystalline metallic materials, in: M.J. Zehetbauer, Y.T. Zhu (Eds.), Bulk Nanostructured Materials, Wiley-VCH, Weinheim, 2009, pp. 501e517. [26] Yu.R. Kolobov, G.P. Grabovetskaya, M.B. Ivanov, A.P. Zhilyaev, R.Z. Valiev, Grain boundary diffusion characteristics of nanostructured nickel, Scripta Materialia 44 (2001) 873e878. [27] J. Ribbe, G. Schmitz, H. Rosner, R. Lapovok, Y. Estrin, G. Wilde, S.V. Divinski, Effect of back pressure during equal-channel angular pressing on deformation-induced porosity in copper, Scripta Materialia 68 (2013) 925e928. [28] S.V. Divinski, G. Reglitz, H. Rosner, Y. Estrin, G. Wilde, Ultra-fast diffusion channels in pure Ni severely deformed by equal-channel angular pressing, Acta Materialia 59 (2011) 1974e1985. [29] S.V. Divinski, J. Ribbe, D. Baither, G. Schmitz, G. Reglitz, H. Rosner, K. Sato, Y. Estrin, G. Wilde, Nano- and micro-scale free volume in ultrafine grained Cue1 wt.%Pb alloy deformed by equal channel angular pressing, Acta Materialia 57 (2009) 5706e5717.

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[30] S.V. Divinski, G. Reglitz, I.S. Golovin, M. Peterlechner, R. Lapovok, Y. Estrin, G. Wilde, Effect of heat treatment on diffusion, internal friction, microstructure and mechanical properties of ultra-fine-grained nickel severely deformed by equal-channel angular pressing, Acta Materialia 82 (2015) 11e21. [31] J. Ribbe, G. Schmitz, D. Gunderov, Y. Estrin, Y. Amouyal, G. Wilde, S.V. Divinski, Effect of annealing on percolating porosity in ultrafine-grained copper produced by equal channel angular pressing, Acta Materialia 61 (2013) 5477e5486. [32] D. Setman, E. Schafler, E. Korznikova, M.J. Zehetbauer, The presence and nature of vacancy type defects in nanometals detained by severe plastic deformation, Materials Science and Engineering A 493 (2008) 116e122. [33] J. Gubicza, S.V. Dobatkin, E. Khosravi, Reduction of vacancy concentration during storage of severely deformed Cu, Materials Science and Engineering A 527 (2010) 6102e6104. [34] G. Reglitz, B. Oberdorfer, N. Fleischmann, J.A. Kotzurek, S.V. Divinski, W. Sprengel, G. Wilde, R. W€ urschum, Combined volumetric, energetic and microstructural defect analysis of ECAPprocessed nickel, Acta Materialia 103 (2016) 396e406. [35] B.S. Bokstein, H.D. Br€ ose, L.I. Trusov, T.P. Khvostantseva, Diffusion in nanocrystalline nickel, Nanostructured Materials 6 (1995) 873e876. [36] S.V. Divinski, F. Hisker, Y.-S. Kang, J.-S. Lee, C. Herzig, 59Fe grain boundary diffusion in nanostructured g-Fe-Ni part II: effect of bimodal microstructure on diffusion behavior in type-C kinetic regime, Zeitschrift fur Metallkunde 93 (2002) 265e272. [37] S.V. Divinski, F. Hisker, Y.-S. Kang, J.-S. Lee, C. Herzig, Tracer diffusion of 63Ni in nano-g-FeNi produced by powder metallurgical method: systematic investigations in the C, B, and A diffusion regimes, Interface Science 11 (2003) 67e80. [38] S.V. Divinski, F. Hisker, Y.-S. Kang, J.-S. Lee, C. Herzig, Ag diffusion and interface segregation in nanocrystalline gamma-FeNi alloy with a two-scale microstructure, Acta Materialia 52 (2004) 631e645. [39] S. Herth, M. Eggersmann, P.D. Eversheim, R. Wurschum, Interface diffusion and amorphous intergranular layers in nanocrystalline Fe90Zr7B3, Journal of Applied Physics 95 (2004) 5075e5080. [40] R. Singh, J.H. Schneibel, S. Divinski, G. Wilde, Grain boundary diffusion of Fe in ultrafine-grained nanocluster-strengthened ferritic steel, Acta Materialia 59 (2011) 1346e1353. [41] Y. Minamino, S. Saji, K. Hirao, K. Ogawa, H. Araki, Y. Miyamoto, T. Yamane, Diffusion of copper in nanocrystalline Al-7.8 at.%Ti-0.3 at.%Fe alloy prepared by mechanical alloying, Materials Transactions JIM 37 (1996) 130e137. [42] K. Govers, M. Verwerft, Classical molecular dynamics investigation of microstructure evolution and grain boundary diffusion in nano-polycrystalline UO2, Journal of Nuclear Materials 438 (2013) 134e143. [43] P.M. Piaggi, E.M. Bringa, R.C. Pasianot, N. Gordillo, M. Panizo-Laiz, J. del Río, C. G omez de Castro, R. Gonzalez-Arrabal, Hydrogen diffusion and trapping in nanocrystalline tungsten, Journal of Nuclear Materials 458 (2015) 233e239. [44] H. Iwaoka, M. Arita, Z. Horita, Hydrogen diffusion in ultrafine-grained palladium: roles of dislocations and grain boundaries, Acta Materialia 107 (2016) 168e177. [45] R. W€ urschum, P. Farber, R. Dittmar, P. Scharwaechter, W. Frank, H.-E. Schaefer, Thermal vacancy formation and self-diffusion in intermetallic Fe3Si nanocrystallites of nanocomposite alloys, Physical Review Letters 79 (1997) 4918e4921. [46] R. W€ urschum, T. Michel, P. Scharwaechter, W. Frank, H.-E. Schaefer, Fe diffusion in nanocrystalline alloys and the influence of amorphous intergranular layers, Nanostructured Materials 12 (1999) 555e558. [47] S.V. Divinski, F. Hisker, Y.-S. Kang, J.-S. Lee, C. Herzig, 59Fe grain boundary diffusion in nanostructured g-Fe-Ni part I: radiotracer experiments and Monte-Carlo simulation in the type-A and B kinetic regimes, Zeitschrift fur Metallkunde 93 (2002) 256e264. [48] Y. Minea, K. Tachibana, Z. Horita, Grain-boundary diffusion and precipitate trapping of hydrogen in ultrafine-grained austenitic stainless steels processed by high-pressure torsion, Materials Science and Engineering A 528 (2011) 8100e8105.

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[49] L. Klinger, I. Gotman, E. Rabkin, A mesoscopic model of dissolution/disintegration of nanocrystalline metals via vacancy diffusion along grain boundaries, Computational Materials Science 76 (2013) 37e42. [50] G. Guisbiers, L. Buchaillot, Size and shape effects on creep and diffusion at the nanoscale, Nanotechnology 19 (2008) 435701. [51] B. Arab, S. Sanjabi, A. Shokuhfar, Size dependency of self-diffusion and creep behavior of nanostructured metals, Materials Letters 65 (2011) 712e715. [52] G. Martin, Atomic mobility in Cahn’s diffusion model, Physical Review B 41 (1990) 2279e2283. [53] Z. Erdelyi, D.L. Beke, P. Nemes, G.A. Langer, On the range of validity of the continuum approach for nonlinear diffusional mixing of multilayers, Philosophical Magazine A 79 (1999) 1757e1768. [54] Z. Erdelyi, C. Girardeaux, Z. TTkei, D. Beke, C. Cserhati, A. Rolland, Investigation of the interplay of nickel dissolution and copper segregation in Ni/Cu(111) system, Surface Science 496 (2002) 129e140.

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CHAPTER 11

Relationship Between Microstructure and Hydrogen Storage Properties of Nanomaterials 11.1 FUNDAMENTALS OF HYDROGEN STORAGE IN SOLID STATE MATERIALS The application of hydrogen as a fuel can help to overcome the challenges arisen from the fossil fuelebased economy, such as the release of growing amounts of greenhouse gas CO2, the poor urban air quality, and the reduction in the world crude oil supply. Hydrogen can be produced directly from sunlight and water by biological organisms and using semiconductor-based systems similar to photovoltaics, or indirectly via thermal processing of biomass or fossil fuels [1]. The electric power necessary for the decomposition of water into hydrogen and oxygen can be produced by wind turbines or nuclear power plants during off-peak periods. In a hydrogen fuel cell, electricity and water are produced in the following manner. Hydrogen molecules diffuse through a porous anode toward a catalyst made of Pt and are stripped of their electrons and become positively charged ions (protons). The anode and the cathode are separated by a protonpermeable polymeric membrane that allows protons to migrate to the cathode [1]. The electrons formed at the anode-side catalyst cannot migrate through the polymeric membrane, but rather travel in the external circuit between the anode and the cathode, thereby creating electric current. At the cathode side, on the surface of a catalyst the hydrogen protons recombine with electrons and oxygen molecules in air to produce water and heat. This waste heat gives the fuel cell an operating temperature of 60e80
C. Hydrogen should be stored for supplying the fuel cells. A possible solution is the storage of hydrogen in chemically bonded state, forming solid-state hydrides in metals or intermetallic compounds. Because of the low pressures involved in metal hydride technologies and the fact that the release of hydrogen takes place via an endothermic process, this method of hydrogen storage is very safe. Additionally, the volumetric hydrogen capacity defined as the mass of hydrogen in a unit volume of the storing device is about 80e160 kg/m3 for solid-state hydrides, which is much higher than that either for compressed hydrogen gas under 80 MPa pressure (w40 kg/m3) or for liquid hydrogen in a tank at 252
C (w71 kg/m3). During the solid-state absorption process, first hydrogen molecules are adsorbed onto the surface and dissociate into atoms by breaking the molecular bonds in favor of new bonds to the surface at chemisorption sites [1]. The energy Defect Structure and Properties of Nanomaterials ISBN 978-0-08-101917-7, http://dx.doi.org/10.1016/B978-0-08-101917-7.00011-6

© 2012 J. Gubicza. Published by Woodhead Publishing Ltd. All rights reserved.

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to overcome the activation potential of dissociation is usually supplied by the vibrational energy. Then, the hydrogen atoms enter the material by diffusion. Following diffusion of hydrogen atoms into the bulk lattice sites, a hydride is formed by nucleation and growth. The formation of hydride starts at the surface of the metal, therefore hydrogen atoms should migrate through the hydride to continue hydrogenation in the metal. In the case of desorption, metal has to be nucleated at the surface, and hydrogen atoms have to diffuse to the surface and recombine to hydrogen molecules, which have to physically desorb [2]. In this case the hydrogen atoms should diffuse through the metal formed at the surface of grains [3]. The absorptionedesorption behavior can be characterized by the pressuree composition isotherm depicted schematically in Fig. 11.1. This curve can be obtained by measuring the hydrogen pressure outside the hydrogen storing material as a function of hydrogen concentration in weight percent in the hydrogen storing material during absorption and desorption at a given temperature. The uptake of hydrogen occurs at higher pressure than desorption, which yields a hysteresis in the curve. The origin of hysteresis is mainly attributed to the excess of energy required to override the lattice strains occurring upon hydrogen absorption or to the production of misfit dislocations between the initial and the hydride phases. The metal initially dissolves only small amount of hydrogen, which creates a solid solution of hydrogen in the metal matrix [1]. As the hydrogen pressure and hydrogen concentration in the metal are increasing, interactions between hydrogen and metal atoms become locally important, and nucleation and growth of a new metal hydride phase are observed. In the plateau region, a mixture of solid solution and metal hydride phases exists. The length of plateau determines the reversible capacity of the material that gives the relative weight fraction of hydrogen that can be stored reversibly with a small pressure variation (see Fig. 11.1). The desorption temperatures for high-capacity hydrides at a pressure of 1 bar are between 110 and 600
C. At this

Hydrogen absorption /n P

298

Hydrogen desorption

Reversible capacity

Hydrogen concentration in wt.%

Figure 11.1 Schematic diagram of pressureecomposition isotherm.

Relationship Between Microstructure and Hydrogen Storage Properties of Nanomaterials

atmospheric pressure, the reversible capacity for these materials varies between 5 and 14 wt.%. With increasing temperature, the plateau pressure increases and beyond a critical temperature, the plateau region disappears. The relation between the pressure P measured at the middle of the absorption plateau and the temperature T is given by the Van’t Hoff equation [1]:   P DH DS ln ¼  ; (11.1) P0 RT R where P0 is the atmospheric pressure, DH and DS are the enthalpy and entropy changes during the hydriding reaction, respectively, T is the absolute temperature, and R is the universal gas constant. For almost all hydrides the enthalpy and entropy changes during hydriding reaction are negative, i.e., the hydriding reaction is exothermic and dehydriding reaction is endothermic. The entropy change has nearly constant value of DS z 130 J/mol K for all the solid state hydrogen systems while DH varies between 20 and 200 kJ/mol for different materials [1]. Basically, there are three models describing absorption/desorption kinetics: (1) surface reaction, (2) JohnsoneMehleAvramieKolmogorov (JMAK), and (3) contracting volume (CV) models of phase transformations [4]. The surface reaction model assumes that the slowest step of reaction is the chemisorption, i.e., the dissociation or recombination of hydrogen molecules on the particles’ surfaces. In this case, the transformed fraction, Vrel, linearly depends on time t as Vrel ¼ kCS t;

(11.2)

where kCS is the reaction constant. In JMAK theory of phase transformations, it is assumed that the nucleation and growth of the new (hydride) phase starts randomly in the bulk and at the surface. The transformed fraction as a function of time can be given as Vrel ¼ 1  exp½  ðkJMAK tÞn ;

(11.3)

where kJMAK is the reaction constant and n depends on the dimensionality of the growth of the new phase nuclei (e.g., n ¼ 2 corresponds to two-dimensional growth and n ¼ 3 to three-dimensional growth). Assuming that the hydrogen diffusion is very fast, the transformation rate is controlled by the (constant) velocity of the metal/hydride interface [4]. If the nucleation of the hydride phase starts at the surface of the metal particle and the growth continues from the surface into the bulk, the CV model is used. The analytical model assumes that a thin layer of transformed phase on the surface of the particle already exists. The kinetics can be described as Vrel ¼ 1  ð1  kCV tÞn ;

(11.4)

299

300

Defect Structure and Properties of Nanomaterials

(a)

Particle

Crystallite(grain)

(b)

Figure 11.2 Schematic depiction of phase transformation according to (a) JohnsoneMehleAvramie Kolmogorov and (b) contracting volume models. The dark areas represent the growing new phase.

where n depends again on the dimensionality of the growth (n ¼ 2 and 3 for two and three dimensional growth processes, respectively). The reaction constant kCV is proportional to the velocity of the metal/hydride interface and inversely proportional to the radius of the metal particle. If the hydrogen diffusion is fast enough, the rate of transformation is determined by the constant velocity of the interface. The difference in growth mechanism for JMAK and CV models is illustrated in Fig. 11.2 [4]. If the diffusion of hydrogen through the transformed phase is the rate-limiting step, the interface velocity decreases with time. In case of CV and three-dimensional growth, the relationship between the transformed volume fraction and the time can be given as   2Vrel 1 (11.5)  ð1  Vrel Þ2=3 ¼ kDCV $t; 3 where kDCV is the reaction constant [4]. The comparison of the experimental Vrel versus t data with Eqs. (11.2)e(11.5) may yield an estimation of the rate-controlling mechanism of hydrogen absorption and desorption as it will be shown in the next section.

11.2 MICROSTRUCTURE AND HYDROGEN STORAGE IN NANOMATERIALS PROCESSED BY SEVERE PLASTIC DEFORMATION During absorption process, hydrogen atoms first migrate along the grain boundary network and then they begin to diffuse into grain interiors. Therefore, the nanostructured state of materials used for hydrogen storage is beneficial from the viewpoint of fast absorption and desorption of hydrogen. The nanocrystalline microstructure is usually achieved by severe plastic deformation, such as ball milling of powders or high-pressure torsion (HPT) of bulk materials. During milling, both the powder particle size and the

Relationship Between Microstructure and Hydrogen Storage Properties of Nanomaterials

grain size inside the particles decrease leading to a faster diffusion of hydrogen along the particle surfaces and the grain boundaries. Additionally, grain boundaries are favorable nucleation sites for the formation and decomposition of the hydride phase. Moreover, high-energy milling (1) breaks the oxide layers and passivation coatings on particles’ surfaces, hence exposure of fresh catalytic sites for dissociation of hydrogen molecules and (2) increases the densities of lattice defects (e.g., dislocations, stacking faults, and twin boundaries) that also serve as paths for fast diffusion of hydrogen [1]. Moreover, addition of catalysts to hydriding material further increases the rate of sorption by assisting the dissociation of hydrogen molecules. Thus, the three dominant factors influencing the sorption properties are particle size, grain size, and catalyst. The effects of grain and particle sizes on hydrogen storage properties were studied extensively on milled MgH2 powders. It was shown in Chapter 4 that the grain size determined by electron microscopy and the crystallite size obtained by X-ray line profile analysis agree well for ball milled materials, therefore the two terms will be used equivalently in this chapter. Magnesium is considered as one of the most attractive hydrogen storage materials, mainly because of high-storage capacity (7.6 wt.%), lightweight, and low cost [1]. During hydrogenation, Mg having hexagonal crystal structure transforms to tetragonal MgH2. Hydrogen atoms bind too strongly with the Mg atoms, i.e., the absolute value of formation enthalpy of MgH2 is high (75 kJ per mol H2) [1], therefore the hydride needs to be heated up to very high desorption temperatures (w350
C) to release hydrogen gas at atmospheric pressure (w1 bar). Additionally, the absorption and desorption are also very slow at temperatures w350
C that impedes the direct use of magnesium as hydrogen storage material. To convert Mg completely to MgH2, it requires more than 50 h at 350
C [5]. The slow sorption kinetics at lower temperatures are mainly due to the low dissociation ability of hydrogen gas molecules on the metallic Mg surface and the slow diffusion of hydrogen through the hydride formed on the surface of Mg particles. Ball milling of Mg in hydrogen atmosphere yields a nanocrystalline material with the crystallite size of w30 nm that can absorb about 6 wt.% hydrogen in much shorter time, 120 min, and at lower temperature, 300
C, than in the case of coarse-grained Mg. Milling MgH2 instead of pure Mg results in even smaller crystallite size of w10 nm that yields very short duration, 10 min, of sorption at 300
C. The addition of suitable catalysts (e.g., Nb2O5 leads to very short hydrogen absorption and desorption times of less than 2 min at 300
C for MgH2 with high capacity of 7 wt.% hydrogen) [6]. The milling of MgH2 is usually carried out in hydrogen atmosphere to prevent the decomposition of MgH2 into Mg and H2. The effects of particle and crystallite sizes, and the catalyst on the absorption and desorption behaviors at 300
C are illustrated in Fig. 11.3 [2]. The absorption and desorption are performed under 8 bar hydrogen and in vacuum, respectively. The conventional MgH2 powder with coarse particles (>10 mm) and crystallites (>1 mm) cannot be loaded or unloaded with hydrogen at 300
C in reasonable times. The powders processed by ceramic ball milling for 20 and

301

Defect Structure and Properties of Nanomaterials

8

8 Hydrogen content (wt. %)

302

(d)

(a) (c)

6

6

(b) (b)

4

4

(c)

2

2 (d)

(a) 0

0

2

4 6 Time (min)

8

10 0

5

10 15 20 Time (min)

25

0 30

Figure 11.3 Absorption (left side) and desorption curves (right side) of (a) conventional coarsegrained MgH2; (b) nanocrystalline MgH2 processed by ball milling for 20 h; (c) MgH2 ball-milled for 700 h, and (d) Nb2O5-catalyzed ball-milled MgH2. (Reprinted from M. Dornheim, S. Doppiu, G. Barkhordarian, U. Boesenberg, T. Klassen, O. Gutfleisch, R. Bormann, Hydrogen storage in magnesiumbased hydride composites, Scripta Materialia 56 (2007) 841e846 with permission from Elsevier.)

700 h have comparably small crystallite sizes in the range of 5e20 nm, while the particle sizes are different, w5 and w0.5 mm, respectively. This difference results in a much faster sorption kinetics for the sample milled for 700 h. It should be noted that the reduction of grain and particle sizes also decrease the hydrogen desorption temperature [7]. The addition of Nb2O5 catalyst to MgH2 powder before milling increases further the rate of absorption and desorption of hydrogen (see Fig. 11.3). However, the particle/ crystallite size and catalyst play two different roles in the sorption process. By decreasing the particle and crystallite sizes, the required diffusion path of hydrogen is drastically reduced that enhances the sorption kinetics significantly. However, the size reduction influences neither the adsorption of hydrogen molecules on the surface, nor the ease of their dissociation. At the same time, the metal oxide catalysts seem to have a promoting effect on dissociation or recombination of hydrogen molecules on the particles’ surfaces and they also act as chemisorption sites [6], resulting in a decrease of the activation energy of hydrogenation and dehydrogenation [4,7]. Moreover, the hard Nb2O5 catalyst particles in the milled powder blend results in an additional slight decrease in MgH2 particle size that also contributes to better sorption kinetics [7]. It seems that above a critical fraction, additional catalyst does not yield further acceleration of absorption or desorption. In the case of Nb2O5 catalyst added to MgH2, this critical amount is w0.5 mol% when hydrogenation and dehydrogenation are performed under hydrogen atmosphere at 8.4 bar and in vacuum, respectively [4]. After 100 h milling, the activation energy of desorption is drastically decreased from values around 120 kJ/mol for the chemisorption of uncatalyzed nanocrystalline magnesium hydride down to about 62 kJ/mol by increasing the catalyst content up to w0.5 mol%. Above this catalyst content, further decrease of activation energy is not observed. As the activation energy of desorption for MgH2 with

Relationship Between Microstructure and Hydrogen Storage Properties of Nanomaterials

coarse particles is w140 kJ/mol, the reduction of particle and crystallite sizes in MgH2 during milling yields only a slight decrease in the activation energy. By comparing the experimental absorption and desorption data with Eqs. (11.2)e(11.5) for MgH2 powder milled with Nb2O5 catalyst, the absorption behavior is best described by the three-dimensional diffusion controlled CV model (Eq. 11.5) [4]. In the dehydrogenation process, generally chemisorption (recombination of hydrogen molecules on the surface) is the rate controlling mechanism (Eq. 11.2) but when both milling time and catalyst content are high, CV model with two-dimensional growth (Eq. 11.4, n ¼ 2) best describes the kinetics. Both the increase of milling timedleading to the reduction of particle and crystallite sizesdand Nb2O5 content increase the desorption reaction constants, kCS and kCV for chemisorption and CV model, respectively [4]. These results indicate that during desorption the hydrogen diffusion through the dehydrided Mg is fast and not rate limiting. On the other hand, the absorption of hydrogen in Mg is mainly controlled by the diffusion of hydrogen through the hydride phase formed at the surface of the particles. This difference between the absorption and desorption behaviors can be explained by the smaller diffusion constant of hydrogen through MgH2 compared to Mg [8]. The increment in the reaction constant for absorption with increasing the catalyst content and the milling time can be explained by an enhanced dissociation rate of hydrogen molecules resulting in a higher gradient of the hydrogen chemical potential and thereby in a faster diffusion of hydrogen through the formed magnesium hydride phase [4]. Longer milling reduces the particle and crystallite sizes yielding a faster diffusion of hydrogen along boundaries. When a very high fraction of catalysts is added and the long milling time yields a nanosized dispersion of catalyst particles, the recombination of hydrogen atoms to molecules at the surface is fast and no longer the rate-limiting step in the desorption process. In this case, the powder particles are covered by a dehydrided magnesium phase and the interface between the dehydrided magnesium and the magnesium hydride moves from the particles’ surfaces along the crystallite boundary areas into the volume of the powder particles because the interface velocity along crystallite boundaries is significantly faster than perpendicular to the crystallite boundary area. As a consequence, the desorption kinetics is governed by the interface mobility of the transformed phase along the two-dimensional network of crystallite boundaries that is reflected by the value n ¼ 2 in CV model [4]. It is noted that taking the size distribution of crystallites into account, the kinetics and the reaction constants of the different transformation models are considerably modified compared to the models for monodisperse crystallites [9]. The severe plastic deformation of bulk Mg and its alloys also improves the hydrogen storage properties [10e16]. For instance, in 3N purity Mg processed by HPT up to 10 revolutions under a pressure of 6 GPa, total hydrogen absorption of 6.9 wt.% is achieved at 150
C under a hydrogen pressure of 3 MPa. The dislocation density reaches its maximum value after 1/4 revolution; however, after 10 turns both the dislocation density

303

304

Defect Structure and Properties of Nanomaterials

and the grain size are reduced due to primary recrystallization. Although the dislocation density is much larger in the sample processed up to 1/4 revolution than after 10 turns, the hydrogen storage capacity of the former material is comparably smaller (