Enthalpies in Alloys

Table of contents :
Enthalpies in Alloys......Page 2
In Memoryof Andries Miedema......Page 4
FOREWORD......Page 5
PREFACE......Page 6
CONTENTS......Page 8
Table of Contents......Page 10
1. The Macroscopic Atom Model; Three Fundamentalquantities......Page 12
2. Estimateof the Chemical Part of the Formation Enthalpy of a Dilute Random Solidsolution......Page 14
3. Estimate of the Formation Enthalpy of Anintermetallic Compound......Page 16
4. Volumeeffects upon Alloying......Page 18
5. Solid Solubility......Page 22
6. Some Thermodynamics......Page 29
7. The Importance of the Structural Term......Page 33
8. Surfaceenergy and Enthalpy of Atomisation......Page 34
1. Application to the Formation and Cohesiveenthalpies of Intermetallic Compounds......Page 37
2. Application to Solid Solubility......Page 38
3. Formation Enthalpy of Concentrated Solidsolutions......Page 39
4. Formationenthalpy of Amorphous Alloys......Page 40
5. Comparison of the Formation Enthalpy of Theamorphous Alloy with the Formation Enthalpy of the Solid Solution......Page 42
6. Formationenthalpy of Ternary Alloys......Page 49
7. Application to Volume Effects upon Alloying......Page 51
8. Applicationto the Heat of Atomisation. 9. Vacancyformation Enthalpies in Pure Metals and Alloys.......Page 52
10. Anti-Site Disorder in Ordered Compounds......Page 54
11. Vacancy Formation in Ordered Intermetallics......Page 60
12. Anti-Site Disorder versus Triple-Defect Disorderand Schottky Defects F.......Page 63
13. Crystallisation Temperature of Amorphous Alloys......Page 66
14. Metal Surfaces......Page 69
ACKNOWLEDGEMENTS. REFERENCES.......Page 74
APPENDIX 1.BACKGROUND OF MIEDEMA’S MODEL......Page 75
2. TABLES......Page 76
Tablea1. Model Parameters in the Metallicstate......Page 78
Table A2. Theinterfacial Enthalpy of Two Transition Metals......Page 79
Table A3. The Interfacial Enthalpy of Anon-Transition Element in a Transition Metal......Page 82
Table A4. The Interfacial Enthalpy of a Transitionmetal in a Non- Transitionelement......Page 85
Table A5. Bulk and Shear Moduli of Theelements......Page 88
Table A6. Surfaceenergies and Vacancy Formation Enthalpies of the Elements......Page 89

Citation preview

Enthalpies in Alloys

In memory of Andries Miedema

FOREWORD Materials are highly diverse, and yet many of the principles, phenomena and processes which are involved in preparing and forming alloys, ceramics, electronic materials, plastics, composites and porous solids are strikingly similar. The very rapid progress of the past two decades, which has transformed not only the silicon-based economy but also the global market, has understandably led to gaps in understanding among busy industrial and academic personnel alike. We are therefore very pleased to announce the publication of a new series of monographs, entitled Materials Science Foundations. Each monograph is treating its chosen topic exhaustively, and seeks in particular to draw together the strands of current theory and industrial practice. The series has been especially tailored to young (and not-so-young) researchers, and has the reading level of a scientific review. But as the target audience is mainly the young researcher, each monograph contains greater detail (theoretical, experimental, and - where applicable - industrial) than is usually to be found in an academic review. For example, in such reviews critically important equations are usually presented without further ado; even though their derivation may no longer be familiar or understandable to the reader. In the new series, all derivations which are necessary to a complete understanding of the topic are provided. Similar criteria are applied to the description of experimental details, and their relationship to the use of materials in the real world are be carefully pointed out. Five monographs are scheduled to be published annually - in both traditional and electronic form; cumulative CD-ROMs will also be available - and, with the passage of time, the collection is growing to become an encyclopedia of permanent value: a basic reference source covering all of the fundamental concepts and phenomena which are the very foundation of materials science research and technology. Our hope is that we are anticipating a real emerging demand in the Materials Science field, but whether we have in fact done so can be answered only by the scientific community itself. The scientific editor Mauro Magini

PREFACE Andries Miedema started developing his rather unconventional model in the sixties as a professor at the University of Amsterdam and continued this work from 1971 at Philips Research Laboratories. At first he encountered scepticism in these environments, where scientists were expecting all solutions from quantum mechanics. Of course, Miedema did not deny that eventually quantum mechanics could produce (almost) exact solutions of problems in solid-state physics and materials science, but the all-day reality he experienced was, - and still is -, that most practical problems that are encountered by researchers are too complex to allow a solution by application of the Schrödinger equation. Besides, he believed in simplicity of nature in that sense that elements could be characterised by only a few fundamental quantities, whereby it should be possible to describe alloying behaviour. On the basis of a comparison of his estimates of the heat of formation of intermetallic compounds with ample experimental data, he was gradually able to overcome the scepticism of his colleagues and succeeded in convincing the scientific world that his model made sense. This recognition became distinct by the award of the Hewlett Packard prize in 1980 and the Hume-Rothery decoration in 1981. At that time, Miedema himself and many others were successfully using and extending the model to various questions, of course realising that the numbers obtained by the model have the character of estimates, but estimates that could not be obtained in a different fast way. It opened new perspectives in materials science. Maybe the power of the model is best illustrated by a discussion, the present author had with Jim Davenport at Brookhaven National Laboratory in 1987. Dr Davenport showed him a result that was obtained by a one-month run on a Gray 1 supercomputer. This numerical result turned out to be only a few kilo Joules different from the outcome by Miedema’s model, obtained within a few minutes by use of a pocket calculator. Andries Miedema passed away in 1992 at a much too young age of 59 years. His death came as a bad shock not only for the scientific community in The Netherlands, but for scientists all over the world. However, for his family, friends and all the people, who knew him well, it is at least some consolation that he lives on by his model, which is being used widely and which is now part of many student programmes in physics, chemistry and materials science. It is the aim of the present review to make the reader familiar with the application of the model, rather than to go in-depth into the details of the underlying concepts. After studying the text, the reader should be able, with the aid of a number of tables from Miedema’s papers and his book, given in the appendix, to make estimates of desired quantities and maybe even to extend the model to problems that, so far, were not handled by others. Beside, the reader

should be aware of the fact that not all applications could be given in this review. For obvious reasons only part of all existing applications are reported and the reader will find further results of using the model in Miedema’s and coworker’s papers, book and in literature. Dr Hans Bakker is an associate professor at the Van der Waals-Zeeman Institute of the University of Amsterdam. In 1970 Andries Miedema was the second supervisor of his thesis on tracer diffusion. He inspired him to subsequent work on diffusion and defects in VIII-IIIA compounds at a time, when, - as Bakker’s talk was recently introduced at a conference -, ‘these materials were not yet relevant’ as they are now. Was it again a foreseeing view of Andries Miedema? Was it not Miedema’s standpoint that intermetallic compounds would become important in the future? Rather sceptic about the model as Bakker was at first like his colleagues, his scepticism passed into enthusiasm, when he began using the model with success in many research problems. In 1984 Bakker’s group started as one of the first in the world ball milling experiments. Surprisingly this seemingly crude technique turned out to be able to induce well-defined, non-equilibrium states and transformations in solids. Miedema’s model appeared again to be very helpful in understanding and predicting those phenomena. Hans Bakker Mauro Magini

CONTENTS

PART I : INTRODUCTION TO THE MODEL 1.

The macroscopic atom model; three fundamental quantities .............. 1

2.

Estimate of the chemical part of the formation enthalpy of a dilute random solid solution ..................................................................... 3

3.

Estimate of the formation enthalpy of an intermetallic compound ................................................................................................... 6

4.

Volume effects upon alloying ................................................................... 8

5.

Solid solubility ......................................................................................... 11 5.1. The mismatch or elastic enthalpy .................................................... 11 5.2. The structural enthalpy .................................................................... 15

6.

Some thermodynamics ............................................................................ 18 6.1. Energy and enthalpy; Helmholtz free energy and Gibbs free energy ....................................................................................... 18 6.2. Solid solubility in case the enthalpy is positive .............................. 20

7.

The importance of the structural term ................................................. 22

8.

Surface energy and enthalpy of atomisation ......................................... 23

PART II : APPLICATION OF THE MODEL 1.

Application to the formation and cohesive enthalpies of intermetallic compounds ........................................................................ 26

2.

Application to solid solubility ................................................................ 27

3.

Formation enthalpy of concentrated solid solutions ........................... 28

4.

Formation enthalpy of amorphous alloys ............................................. 30

5.

Comparison of the formation enthalpy of the amorphous alloy with the formation enthalpy of the solid solution ....................... 31

6.

Formation enthalpy of ternary alloys ................................................... 38

7.

Application to volume effects upon alloying ........................................ 40

8.

Application to the heat of atomisation .................................................. 41

9.

Vacancy formation enthalpies in pure metals and alloys ................... 41

10. Anti-site disorder in ordered compounds ............................................. 44 10.1. The chemical part of the enthalpy ................................................. 44 10.2. The elastic part of the enthalpy ...................................................... 47 11. Vacancy formation in ordered intermetallics ...................................... 49 12. Anti-site disorder versus triple-defect disorder and Schottky defects F .................................................................................................... 53 13. Crystallisation temperature of amorphous alloys ............................... 56 14. Metal surfaces .......................................................................................... 14.1. Surface enthalpy of pure metals ................................................... 14.2. Surface segregation in dilute alloys ............................................. 14.3. Enthalpy of impurity adsorption ..................................................

58 58 59 60

ACKNOWLEDGEMENTS ............................................................................ 63 REFERENCES ................................................................................................ 64 APPENDIX ...................................................................................................... 64 1. Background of Miedema’s model .............................................................. 64 2. Tables .......................................................................................................... 65 Table A1. Model parameters in the metallic state Table A2. The interfacial enthalpy of two transition metals Table A3. The interfacial enthalpy of a non-transition element in a transition metal Table A4. The interfacial enthalpy of a transition metal in a nontransition element Table A5. Bulk and shear moduli of the elements Table A6. Surface energies and vacancy formation enthalpies of the elements

Table of Contents In Memoryof Andries Miedema FOREWORD PREFACE CONTENTS

PART I : INTRODUCTION TO THE MODEL 1. The Macroscopic Atom Model; Three Fundamentalquantities 2. Estimateof the Chemical Part of the Formation Enthalpy of a Dilute Random Solidsolution 3. Estimate of the Formation Enthalpy of Anintermetallic Compound 4. Volumeeffects upon Alloying 5. Solid Solubility 6. Some Thermodynamics 7. The Importance of the Structural Term 8. Surfaceenergy and Enthalpy of Atomisation

1 3 5 7 11 18 22 23

PART II : APPLICATION OF THE MODEL 1. Application to the Formation and Cohesiveenthalpies of Intermetallic Compounds 2. Application to Solid Solubility 3. Formation Enthalpy of Concentrated Solidsolutions 4. Formationenthalpy of Amorphous Alloys 5. Comparison of the Formation Enthalpy of Theamorphous Alloy with the Formation Enthalpy of the Solid Solution 6. Formationenthalpy of Ternary Alloys 7. Application to Volume Effects upon Alloying 8. Applicationto the Heat of Atomisation. 9. Vacancyformation Enthalpies in Pure Metals and Alloys. 10. Anti-Site Disorder in Ordered Compounds 11. Vacancy Formation in Ordered Intermetallics 12. Anti-Site Disorder versus Triple-Defect Disorderand Schottky Defects F. 13. Crystallisation Temperature of Amorphous Alloys 14. Metal Surfaces

ACKNOWLEDGEMENTS. REFERENCES. APPENDIX 1.BACKGROUND OF MIEDEMA’S MODEL 2. TABLES Tablea1. Model Parameters in the Metallicstate

26 27 28 29 31 38 40 41 43 49 52 55 58

b

Enthalpies in Alloys

Table A2. Theinterfacial Enthalpy of Two Transition Metals Table A3. The Interfacial Enthalpy of Anon-Transition Element in a Transition Metal Table A4. The Interfacial Enthalpy of a Transitionmetal in a NonTransitionelement Table A5. Bulk and Shear Moduli of Theelements Table A6. Surfaceenergies and Vacancy Formation Enthalpies of the Elements

Materials Science Foundations ISSN: 2297-7589, Vol. 1, pp 1-2 © 1998 Trans Tech Publications, Switzerland

PART I INTRODUCTION TO THE MODEL 1. THE MACROSCOPIC ATOM MODEL; THREE FUNDAMENTAL QUANTITIES In Miedema’s ‘semi-empirical’ or ‘macroscopic-atom’ model [1] atoms are conceived as ‘blocks’ of the element. In this picture, when bringing dissimilar atoms into contact, enthalpy (energy) effects occur at the interface, where the two different blocks are in contact. Let us consider as an example an A atom (of element A) that is solved in infinite dilution in an excess of B atoms (of element B). Then the enthalpy effect takes place at the A-B interface and will correspondingly be proportional to the area of this interface, i.e. to the surface area of the A atom, because there the A atom is in contact with its B host. A two-dimensional schematic is presented in Fig.1a, where so-called Wigner-Seitz cells are drawn. Fig.1b illustrates the case of A and B atoms of different sizes. Since we will express the enthalpy effect in kJ per mole of A, this area is proportional to V A2 / 3 , a measure of the surface area of A, where VA is the molar volume of A, i.e. the volume of a piece of the element in the solid state consisting of a number of atoms equal to Avogadro’s constant = 6.022  1023.

Fig.1. Two-dimensional schematic of an A atom in excess of B. The contact of A with B is at the boundary of the drawn Wigner-Seitz cells. (a): atoms of equal size; (b): A atoms smaller than B atoms.

A second quantity that plays a role in the enthalpy change upon alloying is similar to the so-called work function and is denoted by . It is a sort of potential that is felt by the outer electrons of the atom and resembles electronegativity [2]. The work function  gives the enthalpy -e that is needed for bringing such an electron with negative charge e to infinity, so has a positive sign and is expressed in Volt. The actual values of the  ’s, used in Miedema’s model are, within experimental error, slight modifications of measured values in order to obtain a set of parameters that adequately describe alloying behaviour.

2

Enthalpies in Alloys

When  A   B electronic charge will be transferred from B to A, giving rise to a negative contribution to the enthalpy, when A is solved in B. This enthalpy is



proportional to -  A   B

 , because an amount of electronic charge Z   2

 A   B is transferred over this ‘potential’ difference with a corresponding enthalpy gain of -Z. We call this negative part of the enthalpy upon alloying Hinter (A in B, negative part), where ‘inter’ stands for ‘interfacial’. The



square -  A   B



2

is also clear from the fact that the enthalpy effect is the

same, irrespective whether  A   B or  A   B . In both cases the same amount of electronic charge is transferred and the only difference is whether the electronic charge will be transferred from A to B or the other way around. A second term in the enthalpy is always positive. It is again and for a similar reason as outlined in the foregoing paragraph, a squared difference. It has to do with the average electron density at the boundary of the Wigner-Seitz cell of both elements. We will refer to this quantity as ‘the density at the boundary of the Wigner-Seitz cell’ and we will denote this positive quantity by nws (Fig.1). The origin of the positive term in the enthalpy lies in the fact that, when solving an A atom in a B host a discontinuity is created in nws , which is not allowed so that the discontinuity should be smoothed at the boundary by bringing electrons to higher energy levels, which explains the positive sign of A 1/ 3 B 1/ 3 2 this contribution. The enthalpy change is proportional to (nws  nws ) . The unity used in the model for this density is the (arbitrary) density unit and the units of the constants occurring in the equations are chosen in such a way that the numbers, given by Miedema have just to be inserted into the equations throughout this review. We call this term Hinter (A in B, positive part) and the total chemical enthalpy is the sum of the negative and the positive part of the interfacial enthalpy. This sum may be either positive or negative in sign, depending on the relative absolute values of both parts. Summarising we can state that there are in principle as few as only three quantities, attached to each element, that determine enthalpy changes upon alloying V,  and nws. Numerical values for the three constants are tabulated in Table A1 of the appendix for (almost) all elements. We will discuss the background of the model in more detail in Appendix 1, but here we give already the equation for the interfacial (chemical) enthalpy for solving one mole of transition metal A in an excess of transition metal B as it was used to obtain the values of Table A2.

Materials Science Foundations ISSN: 2297-7589, Vol. 1, pp 3-4 © 1998 Trans Tech Publications, Switzerland

H

inter

(A in B) =

V A2 / 3 1 1  2  n 1ws/ 3A



 P   2  Q n 1/ 3  ws 1   1/ 3  n ws B 



2

 

(1)

where P and Q are constants to be discussed in the appendix. The power of the model lies in its simple approach by the use of only three fundamental quantities for every element. It makes fast estimates of enthalpies and values for other quantities possible. (We will see in the appendix that a further term has to be taken into consideration, when a transition metal is alloyed with a non-transition metal. In this way the numerical values in Tables A3 and A4 were obtained). The above rather vague information should not give the reader the impression that from this he has to understand the full model. The treatment was only meant to define the basic quantities and to give a feeling for what they mean. In Appendix 1 an outline is given of the background of Miedema’s model. However, it is the aim of the present review to learn how to use the numbers given in the various tables, which we will take for granted. We will see how to make a fast estimate of the value of, for example, a desired unknown enthalpy effect, and we will not present an in-depth and precise content of the model. We will not give an extensive overview of its background and the way the numbers for the above quantities were obtained, we will not discuss a possible fundamental justification of the model, if any would exist. The main point that in this context should be borne in mind is that the semi-empirical model enables us to make fast predictions of values of several effects in alloys within its limitations and that an important justification of the model is that there are many examples where Miedema’s approach ‘works’. Particularly the results for systems involving at least one transition metal are satisfactory and we restrict ourselves to those systems in the solid state. We will present several applications and encounter on our way extensions to various problems. 2. ESTIMATE OF THE CHEMICAL PART OF THE FORMATION ENTHALPY OF A DILUTE RANDOM SOLID SOLUTION It was outlined in the previous section that by use of the three fundamental quantities the chemical part of the enthalpy change, when an A atom is brought into contact with B atoms can be estimated. By ‘chemical’ we mean the effects due to electron transfer and smoothing of the electron density at the boundary of the Wigner-Seitz cell. For the above case of the solution of A in an excess of B, where the A atom is in contact with B atoms only, the chemical part of the enthalpy is given by the interfacial enthalpy H inter (A in B) , which was obtained by Miedema and co-workers using the values of , nws and V defined in the previous section. Values of this quantity for many alloy systems,

4

Enthalpies in Alloys

presented per mole of A are found in the Tables A2-A4. From the tables it can be observed that those values may have either a positive or a negative sign. A negative sign means, that from the point of view of the chemical effect, A atoms ‘like’ a contact with B atoms.

Fig.2. Two-dimensional schematic of Wigner-Seitz cells of A and B atoms of equal size in a random alloy.

Let us as a first application consider a random, dilute solid solution of A in B. Random solid solution means that all atoms are randomly distributed over the lattice sites (see Fig.2). Let us consider such a solution with a fraction cA of A atoms and cB of B atoms (cA < cB) and obviously cA + cB = 1. Furthermore, let us assume that both elements have equal molar volumes. Then, since the average contact of A atoms with B atoms is given in this case by cB (see Fig.2) the chemical enthalpy effect upon the formation of this alloy is simply H chem (1 mole of A) = cB H inter (A in B)

(2)

Of course, for infinite dilution (cB = 1) we recognise the definition of the interfacial enthalpy. Usually we will refer to enthalpy effects per mole of atoms, i.e. A plus B atoms, so that then, because the fraction of A is given by cA H chem (1 mole of atoms) = c AcB H inter (A in B)

(3)

By consulting the tables we are now already able to predict by Eq.(3) the chemical part of the formation enthalpy of a dilute AB alloy. If the volume of A atoms and so their surface area is smaller than that of B atoms, then of course the degree to which A atoms are in contact with B atoms (and not with A atoms) is larger than in the above case, because the A-to-A contact is small. Miedema introduces for this reason the concept of surface fraction (also called ‘surface concentration’) as

Materials Science Foundations ISSN: 2297-7589, Vol. 1, pp 5-6 © 1998 Trans Tech Publications, Switzerland

c sA 

c AV A2 / 3

(4)

c AV A2 / 3  cBVB2 / 3

and a similar equation for cBs . Again obviously c sA  cBs  1 . Since the degree to which A atoms are in contact with B atoms is given in the above case by cBs (>cB), we obtain similar to Eq.(2) H chem (1 mole of A) = cBs H inter (A in B)

(5)

and consequently per mole of atoms, similar to Eq.(3) H chem (1 mole of atoms) = c AcBs H inter (A in B)

(6)

Example Estimate of the chemical part of the formation enthalpy of an 8 at.% Al - 92 at.% Fe random alloy. H inter (Al in Fe)  -91 kJ (from Table A3); 0.92  3.69  0.90 (volumes from Table A1); 0.08  4.64  0.92  3.69 H chem (1 mole of Al) = 0.90  (-91)  -82 kJ. H chem (1 mole of atoms) = 0.08  (-82)  -7 kJ. s cFe 

We already mention that the chemical enthalpy is not the only contribution to the formation enthalpy of a solid solution. In further sections we will see that there is also an elastic part because of mismatch of the solute in the host lattice and, in case of a solid solution of two transition metals, a structural contribution. We proceed now to alloys where the latter two contributions are absent, namely intermetallic compounds, where all atoms fit in the crystal structure and where this structure is different from the original ones of both components. For those intermetallic compounds chemical effects are the only ones. 3. ESTIMATE OF THE FORMATION INTERMETALLIC COMPOUND

ENTHALPY

OF

AN

Intermediate phases in the phase diagram (intermetallic compounds) crystallise in ordered structures. These compounds are characterised by a chemical formula, such as AlFe or AlFe3. Two or even more sublattices can be distinguished in the structure. In stoichiometry, i.e. in the composition given by the formula, one sublattice is occupied by the one atomic species, the other

6

Enthalpies in Alloys

sublattice by the other atomic species. Many of these phases exist just at one composition, but there are intermetallic compounds with a wider range of existence.

Fig.3. Two-dimensional schematic of Wigner-Seitz cell of an intermetallic compound AB with unequal sizes of the constituents.

Fig.3 presents a two-dimensional analogue of an ordered structure of a compound AB. It is clear that in such an ordered compound the surface contact between A atoms and B atoms is larger than in a random (completely disordered) alloy. A atoms are surrounded by B atoms and vice versa. In the schematic picture of Fig.3 the situation is visualised. Even from such a simple schematic it is clear that B atoms are not only in contact with A atoms, but also with B atoms. The surface contact of an A atom with its dissimilar neighbours is denoted by f BA , the degree to which A atoms are in contact with B atoms. Clearly for the random alloy f BA  cBs , but for an intermetallic compound f BA  cBs . Still the contact will not be full, in other words f BA  1 . (In Fig.3 the

situation is given, where this quantity is almost equal to unity, in contrast f AB is considerably less than unity). Characteristic of the empirical character of Miedema’s model is the method for obtaining an estimate of the degree of contact. A number of intermetallic compounds of different compositions were selected for which formation enthalpies were measured. The experimental values were compared to interfacial enthalpies, calculated by the model and the factor by which these differed just gave the values of the f’s. It turned out that these values could be well described by



s s f BA  cBs  1  8  c A cB 

  2

(7)

Subsequently, it was assumed that the relation is generally valid. Note that the term in braces is a correction factor in comparison with the random alloy. Note, furthermore, that the assumption is also that the contact surface is independent of the specific crystal structure of the compound. Of course when B is by far the majority component in the intermetallic compound or when B atoms are much

Materials Science Foundations ISSN: 2297-7589, Vol. 1, pp 7-10 © 1998 Trans Tech Publications, Switzerland

bigger than A atoms f BA  1. Herewith the equations for the formation enthalpy of the compound become H form (1 mole of A) = f BA H inter (A in B)

(8)

H form (1 mole of atoms) = c A f BA H inter (A in B)

(9)

and

Eq.(9) gives the total formation enthalpy of an intermetallic compound, because in such a compound the chemical enthalpy is the only enthalpy that plays a role. In contrast, in a solid solution more terms have to be considered as will become clear in section 5. As an important application, based on this equation, formation enthalpies were calculated for compounds of transition metals with almost all elements. In [1] the outcomes are compared with experimental data and turn out to give a good agreement in the great majority of cases. In the actual calculations a refinement was applied, i.e. molar volumes were corrected for volume changes due to alloying. This is the subject of the next section. Example Estimate of the formation enthalpy of an AlFe3 intermetallic compound. H inter (Al in Fe)  -91 kJ (from Table A3); 0.75  3.69 s cFe   0.70 ; 0.25  4.64  0.75  3.69 s c sAl  1  cFe  0.30 ;





s  s s 2 f FeAl  cFe 1  8 c  Fe c Al   0.95 .   form H (per mole of atoms)= 0.25  0.95  (-91)  -22 kJ. For the moment volume corrections due to alloying were neglected.

4. VOLUME EFFECTS UPON ALLOYING As we saw already in section 1, when we solve A atoms in an excess of B atoms dissimilar atoms are brought into contact and electron charge Z is transferred from the one species to the other species, which gives rise to a negative term in the interfacial enthalpy equal to -Z. We can write

8

Enthalpies in Alloys

H inter A in B, negative part    Z  

(10)

Let us, for example, assume that  A   B , then charge will be transferred from B to the A atom and the size of the A atom will increase, whereas the neighbouring B atoms will shrink. The size increase of the A atom is easily found, because the additional charge will be accommodated at the boundary of A the Wigner-Seitz cell, where the electron density is nws . Correspondingly, the volume increase of the A atom is Z (11) V A  A nws The proportionality sign instead of the equal sign is used because nws is in arbitrary density units and not in electron charge per unit of volume. With the aid of Eq.(10) we obtain for the volume increase in the above example V A 

Z A nws



H inter A in B, negative part  A nws  

(12)

From Eq.(1) we obtain the full expression for the negative enthalpy term and substituting this into Eq.(12) we find (volume change either positive or negative) V A ( 1 mole of A)  

 A   B  A nws

(13)

The quantity  is thus given by

  15 .

1

V A2 / 3

A 1/ 3 nws



(14)

1 1/ 3

B nws where 1.5 (in cm (density units)2/3 Volt-1) is an empirical constant to be discussed later. For A > B we find a volume expansion of the A atom indeed. As already mentioned the B atoms around the A atom will shrink by the electron transfer from B to A (when A > B) and this volume decrease is

VB (per 1 mole of A)  

 A   B  B nws

(15)

Materials Science Foundations Vol. 1

9

so that solving 1 mole of A in B in infinite dilution leads to a total volume change of  1 1  (16) V (1 mole A)    A   B  A  B   nws nws  when A is the dilute component. In a non-dilute alloy charge will be transferred in so far A atoms are in contact with dissimilar atoms, so that  1 1  V alloy (1 mole A)  f BA  A   B  A  B   nws nws 

(17)

where f BA  cBs for a random alloy and where this quantity is given by Eq.(7) for an intermetallic compound. It was on the basis of this equation that the constant 1.5 in Eq.(14) was found in a way, characteristic of the empirical character of the model. In first instance one could assume that the molar volume of an alloy is the weighted average of the molar volumes of the components (c.f. Vegard’s law). A more careful inspection reveals, however, that there are deviations from this simple law. These are due to the electron transfer. By comparing these deviations for a large number of alloys with Eq.(17) the empirical constant 1.5 was derived. When the volumes of the atoms in an alloy are different from the atomic volumes of the pure components, this has also influence on the formation enthalpy of the alloy, because the surface fractions will change. Example Estimate of the volume change of Al and Fe atoms in an AlFe 3 intermetallic compound.     . cm3 ; Per mole of Al atoms V Al  f FeAl Al Al Fe  140 nws Consequently per mole of Al atoms: VFe  

Al nws Fe nws

.   0.68 cm3, but the  14

charge has to be accommodated over three times as many Fe atoms as Al atoms and so the molar volume of Fe (the volume of an Fe atom) increases by 0.68/3= 0.23 cm3 per mole of Fe atoms. alloy alloy V Al  10.00  14 .  8.6 cm3 ; VFe  7.09  0.23  7.32 cm3 ; Estimate the formation enthalpy of an AlFe 3 compound using the above corrected volumes. H inter (Al in Fe)  -91 kJ (from Table A3);

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Enthalpies in Alloys

Using the ‘new’ volumes, we obtain s c sAl  0.28; cFe  0.72; f FeAl  0.95; H form (per mole of atoms)= 0.25  0.95  (-91) = -22 kJ. It is seen that in the above example, there is scarcely any influence on the value of the formation enthalpy, although the volume changes are considerable. There are, however, compounds with volume changes, that are so large that the formation enthalpy is influenced. Example Estimate the formation enthalpy of the intermetallic compound CoGa. We calculate: s s Co Ga cCo  0.41; cGa  0.59; f Ga  0.87; f Co  0.60. We obtain 510 .  4.10 355 . VCo   with   15 .  3.99. 5.36 175 .  131 . This gives VCo = 0.744 cm3; VCo = 6.70 + f GaCo 0.325 = 7.35 cm3. Co n ws 5.36 VGa   Ga VCo    0.744  177 . cm 3 . 2.25 n ws Ga VGa = 11.82 - f Co 1.77 = 10.76 cm3. s s Co cCo  0.437; cGa  0.563; f Ga  0.84. form H = 0.50.84(- 74) = - 31 kJ per mole of atoms. Summarising sections 3 and 4, we conclude that we are now able to estimate formation enthalpies of intermetallic compounds. If the formation enthalpy of a compound is positive, such a compound will not be formed, if it is negative it may exist as an equilibrium intermediate phase, depending on the formation enthalpies (or in fact formation free energies) of possible neighbouring compounds. The discussion of this problem is beyond the scope of the present paper and the reader is referred to text books on phase diagrams. Some more information on the problem is postponed to section 6. Note, by the way, that in Miedema’s model for estimating formation enthalpies the specific crystal structure of the compound is not taken into account. It does not seem to play an important role in these problems.

Materials Science Foundations ISSN: 2297-7589, Vol. 1, pp 11-17 © 1998 Trans Tech Publications, Switzerland

5. SOLID SOLUBILITY 5.1. THE MISMATCH OR ELASTIC ENTHALPY Let us inspect the Mn-Y alloy system. It turns out that there exist three intermetallic compounds [1]: Mn12Y , Mn23Y6 and Mn2Y. Apparently Mn atoms ‘like’ to be surrounded by Y atoms and vice versa. This is reflected by negative interfacial enthalpy H inter (Mn in Y)  -5 kJ. However, the solid solubility of Mn in pure Y is negligible, i.e. Mn is not solvable in crystals of pure Y. (By solid solubility the maximum content of solute is meant that can be solved in the solvent, while the solution is still single phase with the structure of the pure solvent). Is this a paradox? The answer is: not at all. An intermetallic compound ‘chooses’ a crystal structure where, in this case, the big Y atoms and the small Mn atoms match. There are no strains with corresponding mismatch enthalpies. In contrast, if one tries to accommodate a small Mn atom on an Y site in pure Y, the solute atom will not fit and by size mismatch strain will develop and obviously in the case of Mn in Y the mismatch enthalpy is not compensated by the negative interfacial enthalpy. We already mention here that this mismatch enthalpy for the solution of one mole of Mn in Y is equal to 155 kJ. This value is much higher than the absolute value of the corresponding interfacial enthalpy and reduces the solubility to almost nil (for reasons of simplicity we did not take into account here the small structural term of -3 kJ per mole for this case, to be discussed in section 5.2).

Fig.4. ‘Rubbery’ sphere A with volume WA accommodated in a hole of size WB in a rubber medium B. The empty space is filled by expansion of the A atom and by shear in the matrix B until a final size W is attained. The pressure p is continuous across the interface.

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Enthalpies in Alloys

In order to make estimates of the mismatch enthalpies we use continuum elastic theory following Eshelby [3]. In fact this is also a macroscopic approach. In the inside of a rubber medium B a spherical hole with the size of an atom of the solvent is created and in this hole a non-fitting rubbery sphere with the size of the solute atom A is accommodated. Let us call the volume of the A atom WA and that of the hole WB , then when, for example, WA < WB the hole is partly filled by the A atom (Fig.4) and the empty part of the remaining volume (WBWA) will disappear by elastic deformation of inclusion A and matrix B. The A atom will expand and the resistance against expansion is determined by its bulk modulus KA , while the host material accommodates itself to the stresses by means of shear strain. It can be shown that the hole can be handled as having an effective bulk modulus equal to 4/3 times the shear modulus GB of the matrix [3]. (We will accept the latter result without giving the proof). Therefore, the bulk modulus of the solute and the shear modulus of the host material determine, together with the sizes of solute and solvent atoms the mismatch enthalpy. Let us recall the definition of the bulk modulus K. When a hydrostatic pressure p is applied to a material and the relative volume change thereby is V/V the bulk modulus K is defined by V p  V K

(18)

Fig.5. Definition of the shear stress  and the shear angle . When  is the shear angle due to a shear stress  (see Fig.5), then the shear modulus G is defined by

 

 G

(19)

The elastic energy when a material is compressed or expanded by a pressure p is

Materials Science Foundations Vol. 1

H

elastic

V



V

 pdV   0

0

13

V 1  V  K dV  K V 2 V

2

(20)

so that in our case (with a ‘bulk’ modulus of B of 4GB/3 as mentioned above) K  WA  2GB  WB  H  A  (21) 2WA 3WB where WA and WB are the volume changes of the sphere (A) and the hole (B) due to internal stress, respectively. If we call the ‘new’ volume indicated in Fig.4 W, then evidently 2

2

elastic

WB  WA  WB  W  WA  W  WB  WA

(22)

Moreover, the pressure is continuous across the interface, i.e. the pressure on A is equal, but opposite to the pressure in B (Fig.4), so that WA WB 4 (23) KA   GB WA 3 WB or WA 4GBWA   WB 3K AWB

(24)

WA 4GBWA  WA  WB 4GBWA  3K AWB

(25)

or

With the aid of Eq.(22) WA 

4GBWA WA  WB  4GBWA  3K AWB

(26)

WB 

3K AWB WB  WA  4GBWA  3K AWB

(27)

In a similar way

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Enthalpies in Alloys

and substitution of these results into Eq.(21) yields after some algebra H

elastic

2 K A GB WA  WB  (A in B)  4GBWA  3K AWB

2

(28)

This quantity is per mole of A solved in excess of B. Bulk and shear moduli of pure elements are tabulated in Table A5 [4]. Here we used the letter W instead of V for volume to indicate that corrected volumes following section 4 should be used. When we solve A in B, the A volume will be following section 4 WA  V A  V A  V A  

 A   B  A nws

(29)

When A > B the size of the solved A atom grows because of electron transfer from B to A. On the other hand by this very transfer from the wall of the B hole to the A atom, also the hole size grows by (VB is negative!), i.e. in this case     (30) WB  VB  VB  VB   A B B nws Note the difference between atoms and holes. When the B atoms shrink, a hole in B grows, when the A atom is accommodated in it. The conclusion is that first we have to apply the volume corrections due to electron transfer to obtain the above quantities WA and WB and subsequently Eq.(28) accounts for filling of the empty space (WA-WB) by adjusting atom size and hole size by expansion (or compression) and shear, respectively. Example We solve one mole of Al in an excess of Fe (rather visualise this as solving one Al atom in pure Fe). The  value for Fe is larger than the  value for Al (4.93 V and 4.20 V, respectively), so electron charge will be transferred from Al to Fe and the Al atom will shrink following Eq.(29) by an amount -1.47 cm3 per mole (compare the example of section 4, but now the Al atom is completely surrounded by Fe atoms). This electron charge is supplied to the wall of the hole and accommodated there in the electron density of Fe at the boundary of the Wigner-Seitz cell. Since the electron density there is much higher than that of Al, the hole will not shrink so much and Eq.(30) tells us that the hole in Fe shrinks only by -0.72 cm3 per mole of holes. We obtain: WAl = 10.00 - 1.47 = 8.53 cm3 per mole and WFe = 7.09 - 0.72 = 6.37 cm3 per mole. From Table A5 we find KAl = 7.2181010 Pa; GFe = 8.1521010 Pa;

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WAl = 8.5310-6 m3 per mole; WFe = 6.3710-6 m3 per mole (here we have to use m3!). Helastic (1 mole Al in Fe) = 13 kJ, just by using Eq.(28). Of course mismatch (elastic) enthalpies are always positive. Since Hinter (1 mole Al in Fe) = -91 kJ, the sum of both enthalpies is negative and it has to be expected that Al is solvable in Fe and it really is. 5.2. THE STRUCTURAL ENTHALPY So far we saw that the enthalpy of solution of A in B is determined by the interfacial enthalpy that is due to a rearrangement of electronic charge upon alloying and by the mismatch enthalpy, when an A atom has a different size from B atoms. Miedema argues that in case of the solution of a transition metal in an other transition metal there is a third term that has to be taken into account. When we inspect the periodic table of the elements it becomes clear that the crystal structure of a pure transition element depends on the number of valence electrons Z of the metal. Transition metals with 3 and 4 valence electrons prefer the face-centred-cubic (f.c.c.) structure or hexagonal-closepacked (h.c.p.) structure. Examples are Sc with 3 valence electrons and Ti with 4 valence electrons, the number of valence electrons just being equal to the column number of the metal in the periodic table. On the other hand metals with 5 or 6 valence electrons crystallise in the body-centred-cubic (b.c.c.) structure. Examples are V with Z = 5 and Cr with Z = 6, whereas metals with 7-10 valence electrons prefer again the f.c.c. or h.c.p. structure.

16

Enthalpies in Alloys

Fig.6. Structural enthalpies of transition metals as a function of number of valence electrons [1].

Miedema constructed, partly on the basis of band-structure calculations, partly on empirical grounds the curves of Fig.6. The points in this figure give the structural enthalpies for metals with 3,4,....,10 valence electrons. The solid line represents the b.c.c. structure, the dashed and dotted lines the f.c.c. structure and h.c.p. structure, respectively. It is observed that the latter structures have almost an identical structural enthalpy. The metals with Z = 5 and 6 have the lower enthalpy for the b.c.c. structure indeed and naturally crystallise in this structure. From the figure it can be read that, for example, to transform b.c.c. V with 5 valence electrons to the f.c.c. structure takes about 20 kJ per mole of atoms. Of course it should be emphasised that the values given by Fig.6 have again the meaning of estimates only. When we now solve one mole of A atoms in excess of B, then in fact we transform A to B from a structural point of view. This means that we loose the structural enthalpy of A and gain the structural enthalpy of B. The net effect is Estruct(B) - Estruct(A). But there is more. These A atoms, which are virtually transformed into B have a different number of electrons than B itself. And from Fig.6 it is seen, how the structural enthalpy depends on the number of valence electrons. This is given by the slope of the curve in the point corresponding to the B metal E struct (B) Z

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17

Then for a difference of number of valence electrons of ZA - ZB the structural enthalpy will change by an amount

E struct (B) Z A  Z B   Z Thereby the total structural enthalpy change will become per mole of solved A H

struct

E struct (B) (A in B) =  Z A  Z B    E Bstruct  E Astruct Z





(31)

Based on Fig.6 values for this quantity are given in Table 1. Example Solving Fe (Z = 8) in Cr (Z = 6) takes following Table 1 a structural enthalpy of +30 kJ per mole Fe solved. Solving Co (Z = 9) in Mn (Z = 7) yields a structural enthalpy of -12 kJ per mole Co solved.

Materials Science Foundations ISSN: 2297-7589, Vol. 1, pp 18-21 © 1998 Trans Tech Publications, Switzerland

Table 1. The values of Hstruct(A in B) in kJ per mole of solute Zsolute A   Zsolvent

3

4

5

6

7

8

9

10

0 -1 +6 +34 -21 +70 +2 -19 -55

-1 0 +5 +12 -17 +52 0 -17 -47

+7 +16 +7 0 -4 +43 +8 -5 -30

+5 +32 0 -4 0 +25 +6 -3 -22

-3 +21 -14 -15 +9 0 -3 -8 -21

0 +32 -16 -14 +30 -3 0 -1 -8

-2 +37 -23 -18 +46 -12 -1 0 0

-11 +36 -38 -30 +54 -28 -8 0 0

B

3 4 (hcp) 4 (bcc) 5 6 7 8 9 10

Summarising: The total enthalpy associated with solving one mole of A in excess B is given by H sol (A in B)  H chem (A in B)  H elastic (A in B) + H struct (A in B) (32)

where for an infinitely dilute solution the chemical enthalpy per mole of solute is identical to the interfacial enthalpy and is obtained from the Tables A2-A4. For somewhat more concentrated, but still dilute solid solutions the chemical term per mole of solute is calculated by Eq.(5). The elastic enthalpy per mole of solute is calculated by means of Eq.(28), while only in the case of two transition metals the third term is different from zero and should per mole of solute be obtained from Table 1. Concentrated solid solutions are treated in Part II section 3. 6. SOME THERMODYNAMICS 6.1. ENERGY AND ENTHALPY, HELMHOLTZ FREE ENERGY AND GIBBS FREE ENERGY The internal energy of a system is usually indicated by the letter U (sometimes by E), while the enthalpy of the system is denoted by H. The relation between both quantities is H = U + pV

(33)

where p is pressure and V is volume. Often the energy U (or E) and the enthalpy H are used without distinction. This is acceptable at ambient pressure. The

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19

difference between the two quantities is the term pV and this term is small compared with the internal energy U. Atmospheric pressure means p  98 kPa and a typical molar volume is about 20 cm3, so that pV is about 2 J per mole, which is negligible relative to the energies of kiloJoules, that are characteristic of the effects we are discussing. Moreover pV is practically constant as a function of temperature. Consequently, the same applies to the Helmholtz free energy F = U - TS

(34)

(where S is the entropy of the system) compared with the Gibbs free energy G = H - TS

(35)

and in many publications also F and G are used without distinction.

Fig.7. Free energy as a function of composition at a certain temperature, illustrating equilibrium between an intermetallic compound AB and the primary and terminal solid solution of B in A and A in B, respectively. l: liquid state; SA: solid solution of B in A; SB: solid solution of A in B; SAB: solid intermetallic; straight lines: common tangents.

However, in the systems under consideration, it is more correct to use G, because we are considering systems under constant pressure and temperature and not under constant volume and temperature. Under those circumstances G should be a minimum in thermodynamic equilibrium. Of course, when one is studying a system under high pressure, only the Gibbs free energy should be applied. From Eq.(35) we conclude that at zero absolute temperature the

20

Enthalpies in Alloys

enthalpy is the quantity that determines the state of the system. Under this condition H should be a minimum. At higher temperature there will be a competition between enthalpy and entropy. The consequence is that, even when Hsol(A in B) from Eq.(32) is positive, there is some solid solubility as we will demonstrate in the following section. In case Hsol(A in B) is negative, the extent of solid solubility, i.e. the phase field of the primary solid solution is determined by the free energy curve of the solid solution as a function of solute concentration compared with the one of the first intermetallic compound. The phase field is then determined by a common tangent construction to both curves as is explained in detail in text books on phase diagrams. This is beyond the scope of this review paper, but the principle will be outlined on the basis of Fig.7. In this free-energy versus composition diagram at a certain temperature, we observe the curve for the liquid state, labelled l, the curve for the solid solution of B in A, labelled SA, that of the solid solution of A in B (S B) and the free energy curve for an intermetallic compound AB (S AB). Up to a concentration, indicated by point h the solid solution of B in A has the lowest free energy and is single phase up to this concentration. Between h and i the lowest energy is obtained for a two-phase mixture of the solid solution and the compound AB, which, in contrast, exists as a single-phase compound between i and j. Similar considerations apply to the right-hand side of the diagram. This is an example where further compounds have been omitted, because it was assumed for reasons of simplicity that their free energy is higher than that indicated by the common tangents of the two-phase fields. 6.2. SOLID SOLUBILITY IN CASE THE ENTHALPY IS POSITIVE. As an application of Miedema’s estimates of the enthalpy of the solid solution, we consider the consequence of a positive value of the enthalpy on solid solubility. When we consider a very dilute solution of A in B we can write G(P,T,NA) = G0 (P, T) + NA gA - kT ln 

(36)

Here G0(P,T) is the free enthalpy of pure B, gA the free enthalpy increase per solute atom added, NA the number of solute atoms, k Boltzmann’s constant and  the number of ways in which the solute and solvent atoms can be distributed over the number of lattice sites N = NA + NB 

N! N A ! N  N A  !

(37)

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21

Using Stirling’s equation for large numbers and defining cA and cB as the fractions of the total number of lattice sites occupied by A and B atoms, respectively, we obtain the so-called ideal entropy of mixing as Smix  k ln  = -Nk cA ln cA  1  cA  ln 1  cA  

(38)

When this result is substituted into Eq.(36), we obtain G P ,T , N A   G0  P ,T   Nc A g A  NkT c A ln c A  1  c A  ln 1  c A  (39)

Realising that G should be a minimum and so minimising Eq.(39) with respect to cA we obtain c A  exp

 gA s  hA  exp A exp kT k kT

(40)

Switching to corresponding molar quantities c A  exp

S A  H A exp R RT

(41)

The origin of the entropy SA is mainly formed by changes of the frequencies in the phonon spectrum by introducing the A impurities. It is mainly a vibrational entropy, which usually is of the order of the gas constant R. Eq.(41) gives the maximum possible content of the solute at a certain temperature, where the solution is still single phase and has the crystal structure of the solvent. Example Solubility of Ni in f.c.c.() La at 700 K. Hchem (Ni in La) = -81 kJ per mole Ni; Helastic (Ni in La) = +129 kJ per mole Ni; Hstruct(Ni in La) = -11 kJ. Thus HNi = +37 kJ per mole solved Ni. These numbers substituted into Eq.(41) and assuming the value of the entropy to be equal to R yield an estimated solubility of 0.4 at.%, which is a small solubility indeed. As a final remark on thermodynamics let us once more consider Eq.(38). We take the derivative with respect to cB , realising that cA = 1 - cB and considering one mole of atoms A plus B

Materials Science Foundations ISSN: 2297-7589, Vol. 1, pp 22-22 © 1998 Trans Tech Publications, Switzerland

S mix 

c dS mix d cB   Rc A  lnc A  cB  lncB cB   R  ln B cB  dcB dcB 1  cB (42)

We conclude that for cB  1 (very dilute solution) Smix  , so that the mixing entropy increase due to the introduction of the first impurities is enormous. These considerations have an interesting consequence for the purification of pure metals, namely that always some impurities will be present in a pure metal and that in fact it is extremely difficult to remove all impurities. 7. THE IMPORTANCE OF THE STRUCTURAL TERM Let us inspect the Zr-rich side of the Zr-Pd phase diagram as given by Fig.8. At lower temperature the solid solubility is virtually nil, at higher temperature (dashed region) it is high. How can this be explained? Zr (Z = 4) crystallises in the h.c.p. structure below 863 C and above this temperature the structure is b.c.c.

Fig.8. The Zr-rich side of the phase diagram of the Zr-Pd system [1]. From Table 1 it turns out that the structural enthalpy for solving Pd (Z = 10) in h.c.p. Zr is positive and high (+36 kJ per mole Pd), whereas in the b.c.c. structure it is low (-38 kJ per mole Pd). As we saw before there is a competition between intermetallic compounds and the solid solution. The first Zr-rich intermetallic compound in the phase diagram is Zr 2Pd. Since VZr > VPd , in the compound as well in the solid solution Pd atoms are nearly completely surrounded by Zr atoms ( f ZrPd  1 ). This means that per mole of Pd the interfacial (chemical) enthalpy of the solid solution and the formation enthalpy

Materials Science Foundations ISSN: 2297-7589, Vol. 1, pp 23-25 © 1998 Trans Tech Publications, Switzerland

of the compound are nearly equal and only mismatch and structural enthalpies determine the relative positions of the curves of solid solution and intermetallic compound in a plot such as Fig.7. These relative positions determine the solid solubility of Pd in Zr. The mismatch enthalpy is always positive. In contrast the structural enthalpy is positive for the h.c.p. solution, but negative for the b.c.c. solid solution. This means that the free-energy curve for the h.c.p. phase (low temperature) is located relatively at a much higher position than that of the b.c.c. phase (high temperature), what results in a negligble solid solubitity at lower temperature and a high solid solubility at higher temperature in agreement with the phase diagram. Similar phase diagrams are found for Rh, Ru, Mo, Pt, Ir, Os, Re and W in Zr and also for Mo, Ru, Pd, W, Re and Ir in Hf and Ti. Also by the sign of the structural term the large solubility of, for example, Y, Zr, Nb, Mo, Hf, Ta and W in Pd can be understood. 8. SURFACE ENERGY AND ENTHALPY OF ATOMISATION How do we have to conceive atomisation of a solid in the picture of the macroscopic-atom model? Should we attribute the loss of energy to breaking of atomic bonds, such as for example mainly Coulombic bonds in ionic materials or to a quite different electronic structure of single atoms from atoms in a metal in the case of metallic bonding? What should we think, if in fact as in Miedema’s model we consider even single free atoms as elementary blocks of an element, to what is then the loss of energy to be attributed upon atomisation? What is then the essential difference between a solid and an assembly of single, non-interacting atoms? The answer is simple: the difference in this picture is the enormous surface contact of atomised single atoms with vacuum as compared with this contact in the solid, where almost all atoms are surrounded by other atoms. In this way following Miedema’s model there should be a simple relation between the surface energy of a metal A and its heat (energy, enthalpy) of atomisation H Aatomisation  c0V 2 / 3

(43)

where c0 is a constant to be determined,  the surface energy of the metal in units of Jm-2 and V2/3 in m2 a measure for the surface area of an atom. Note that the words enthalpy and energy are sometimes used without distinction. Mostly this is justified as we demonstrated before. Furthermore, note the notation of the heat of atomisation without the symbol ‘’, because here the reference state is that of isolated single atoms, which we will conceive for the present purpose as the ‘real’ zero-enthalpy state of a system, whereas, for example, the heat of alloy formation from two solid metals refers to a non-zero enthalpy, namely to that of

24

Enthalpies in Alloys

the pure solid metals. This is the reason, why we use the symbol ‘’ in such a case. Subsequently in order to estimate the value of the constant c0 a simple argumentation follows now that is characteristic of Miedema’s approach throughout his whole treatment of the problems under consideration in this review. Suppose that atoms were cubes with a cube edge of l, then the surface area O of one mole of isolated atoms would be O = N6l2

(44)

Here N is Avogadro’s number. The molar volume is then V = Nl3

(45)

In contrast, when atoms would have to be conceived as spheres with radius r we would obtain O= N 4r2

(46)

4 V  N r 3 3

(47)

O = c0V2/3

(48)

while

If in both cases we write

we arrive at a constant of 5.1108 for cubes and 4.1108 for spheres. Quite characteristically Miedema concludes that the real value should be somewhere in between and finds by fitting Eq.(43) to available experimental data that the recommended value for the dimensionless constant should be c0 = 4.5108

(49)

Preferred values of  are given in Table A6, obtained from various experimental sources [1], such as the heat of atomisation. The heat of atomisation is of course positive: we have to supply heat to the system in order to atomise it. In contrast the cohesive energy of an element is negative, but apart from this we will not make a distinction between both quantities.

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By this we conclude the first part of this review. In Part II we will encounter several applications, also of this section. Admittedly, we presented already a number of applications in the previous sections in Part I in order to make the treatment more vivid. We summarise these applications also in the following to make Part II more complete. Actually Part I was intended to make the reader familiar with the somewhat unconventional approach inherent to Miedema’s model and to enable him to use the tables in order to obtain numerical values for a number of quantities that can not be evaluated in a different fast and simple way.

Materials Science Foundations ISSN: 2297-7589, Vol. 1, pp 26-26 © 1998 Trans Tech Publications, Switzerland

PART II APPLICATION OF THE MODEL In this second part of the present paper a number of applications of Miedema’s model will be discussed without pretending at all to give a full account of the work of the many investigators, who used the model for various purposes. Those applications are emphasised, where Andries Miedema himself was involved and where the work was inspired by his model at the University of Amsterdam. First we summarise also those applications that we already encountered in Part I. 1. APPLICATION TO THE FORMATION AND ENTHALPIES OF INTERMETALLIC COMPOUNDS

COHESIVE

In Part I the enthalpy of alloy formation was defined as the enthalpy gain (or loss) when two metals are alloyed. The effect is measured as a heat effect and therefore it is also called ‘heat of formation’. For an alloy with composition cA, cB with cA + cB = 1, this heat or enthalpy of formation is defined by H form (1 mole of atoms)  H coh (alloy)  c A H coh (A)  cB H coh (B)

(50)

where ‘coh’ stands for cohesive. These cohesive enthalpies are in absolute values equal to the enthalpies of atomisation, but in contrast these are negative quantities. It is obvious that the definition in the form of Eq.(50) is the same as the one in words given in Part I. Cohesive energies are of an absolute magnitude of hundreds of kiloJoules, while formation heats are normally in the order of tens of kiloJoules. The heat of formation can be measured, for example, by use of a moltentin calorimeter. One mole of the alloy is solved in an excess of molten tin and the heat effect is measured. Subsequently, cA moles of A and cB moles of B are solved and again the corresponding heat effects are evaluated. The difference in the heat effects yields the formation enthalpy. Measurements of formation enthalpies are not very accurate and the experimental values given in literature usually contain large experimental errors, even that large that sometimes it is stated that the values calculated by the Miedema model are more accurate. In many phase diagrams we observe the existence of alloys as intermediate phases (or intermetallic compounds) with a special crystal structure different from the structure of both constituents. In this case there is no atomic size mismatch and also the structural term is absent. After volume corrections following section 4 of Part I we evaluate their formation enthalpy by Eq.(9). Thus Miedema’s model gives us a tool to estimate in a fast way formation enthalpies and thus by Eq.(50) cohesive enthalpies of intermetallic compounds.

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Example The cohesive enthalpy of the intermetallic compound CoGa. Following Eqs (43) and (49) and by use of Tables A6 and A1: atomisation HCo  4.5  108  2.55  355 .  104 J per mole  407 kJ per mole . atomisation HGa  4.5  108  110 .  519 .  104 J per mole  257 kJ per mole . By use of Eq.(50): coh form HCoGa  0.5  407  0.5  257  HCoGa

form and since as we found in the last example of Part I section 4 H CoGa = -31 kJ per mole, we obtain: coh form HCoGa  0.5  407  0.5  257  HCoGa  363 kJ per mole of atoms. Note the opposite sign of heats of atomisation and cohesive enthalpies. Note furthermore that the normal way in which a chemist would give the cohesive enthalpy would be -726 kJ mol -1, i.e per mole of formula units. However, we prefer following Miedema values per mole of atoms of Co plus Ga, in order to avoid difficulties for off-stoichiometric compounds, which can be denoted by chemical formulas such as Co0.56Ga0.44. As a consequence of such a notation the ‘chemical’ value of the cohesive enthalpy for such a composition would be equal to about one half of that of CoGa, which would be confusing. The correct ‘chemical’ notation in this case would then be Co 1.12Ga0.88, a notation that is rather unusual in metal physics.

2. APPLICATION TO SOLID SOLUBILITY From section 5 of Part I it became clear that the heat of solution of a dilute concentration of impurities in an excess of an other metal is far more complicated to be evaluated than the heat of formation of intermetallics, discussed in the previous section 1. Here, following Eq.(32) we are dealing with two or three terms that determine the heat of solution of one mole of A in an excess of B. There is first the chemical contribution, due to a redistribution of electronic charge at the boundary of the Wigner-Seitz cell Hinter (A in B), then the mismatch enthalpy, because the impurity with a atomic size different from the host atoms has to be accommodated in a lattice in which it does not fit (section 5.1 from Part I). Moreover, in case of two transition metals we are dealing with the structural term given by Eq.(31), which is easily obtained from Table 1. In case of a positive heat of solution still some impurities can be solved as a result of mixing entropy following Eq.(41) and Eq.(42). When we are dealing with a negative heat of solution, the phase field of the solid solution is determined by its free energy and that of intermediate phases as briefly discussed on the basis of Fig.7 in section 6.1 from Part I.

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The importance of the structural term was emphasised by a lack of solubility of, for example, Pd in Zr in low-temperature h.c.p. Zr and an extended solubility in b.c.c. Zr at high temperatures. 3. FORMATION SOLUTIONS

ENTHALPY

OF

CONCENTRATED

SOLID

In the previous sections we summarised the method to estimate the formation enthalpy of one type of concentrated alloy: the intermetallic compound. Moreover, we found the heat of solution of very dilute solid solutions. In this section we treat the other type of alloy: the concentrated random solid solution (ss), i.e. the one component randomly solved in the crystal structure of the other component. Similar to the dilute solid solution we are dealing with three terms, where the last term is only non-zero for a solid solution of two transition metals. H form (ss)  H chem (ss)  H elastic (ss) + H struct (ss)

(51)

For concentrated solid solutions there is a little difficulty, which we demonstrate on the basis of the chemical term, although similar problems apply to the other terms. If one considers a concentrated solid solution, for example, a 50 at.% A 50 at.% B alloy, from what chemical enthalpy should one then start: from A in B or from B in A? Miedema solved this problem in a pragmatic way by just averaging. Per mole of atoms (A plus B atoms) one obtains [1]



H chem (ss)  c AcB cBs H inter (A in B)  c sA H inter (B in A)



(52)

Here c AcBs H inter (A in B) is the chemical enthalpy per mole of atoms. The same applies to cB c sA H inter (B in A) , but in practical use of the model it turns out that both quantities are not necessarily strictly equal. The first term is now multiplied by cB, whereby a corresponding fraction of the enthalpy is obtained and by adding the second term multiplied by cA the rest of the enthalpy is taken into account. Moreover the chemical enthalpy is equal to zero if one of the fractions is zero as it should be. In a similar way



H elastic (ss)  c AcB cB H elastic (A in B)  c A H elastic (B in A)



(53)

where the first term in parentheses cB H elastic (A in B) accounts for the fact that the material in which A is solved is not completely B, but only partly.

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Furthermore, for the structural enthalpy Fig. 6 shows the empirical curve. Assuming that in the solid solution atoms form a common d-band, the lattice stability of this solution will be the same function of the average number of valence electrons per atom . For an alloy we calculate and take the lattice stability Hstruct() to be the most negative of the three energies Hstruct(hcp), Hstruct(fcc) and Hstruct(bcc). To calculate the structural contribution to the formation enthalpy we have to subtract from Hstruct() a reference enthalpy. This reference enthalpy is a linear extrapolation between the lattice stabilities of the two relevant metals in their equilibrium state, Hstruct(zA) and Hstruct(zB). As a result we have



H struct (ss)  H struct (< z >) - H ref (< z >)



(54)

For alloys in which, apart from the transition metal noble metals, Cu, Ag and Au are involved (filled d-shell), the effect of structure is almost negligible and was estimated by Miedema as +1 kJ per mole for b.c.c and -1 kJ per mole for f.c.c. and h.c.p. By Eqs (51), (52), (53) and (54) the formation enthalpy for concentrated solid solutions is obtained. One could think that the problem is not very important, because metals that form solid solutions over wide concentration ranges are usually rather similar in atomic size, chemical behaviour and structure. A well-known example is the Ag - Au system, where a solid solution is found over the whole concentration range. However, these considerations apply to equilibrium. In contrast, in metastability the solid-solution ranges of rather dissimilar metals can be largely extended and techniques to attain this goal, such as ultra rapid quenching, co-sputtering and ball milling have been developed in the last decades, so that the number of concentrated solid solutions that can be prepared is nowadays larger than only those in thermodynamic equilibrium. Example The formation enthalpy of a solid solution of 50 at.% Ni (A) and 50 at.% Zr (B). Eq.(52): H chem(ss) = - 0.25(0.62165 + 0.38237) = -48 kJ Eq.(53): H elastic(ss) = 0.25(0.592 + 0.589) = +23 kJ Eq.(54) and Fig. 6 (Ni, Z = 10; Zr, Z=4): H struct(ss) = -3 kJ Eq.(51): H form(ss) = -48 + 23 -3 = -28 kJ per mole of atoms. 4. FORMATION ENTHALPY OF AMORPHOUS ALLOYS Metallic systems in equilibrium always exhibit the crystalline state. However, by rapid cooling from the melt, whereby the nucleation and growth of the crystalline phase can be kinetically by-passed, a configuration can be attained of a frozen-in liquid, a metallic glass. In such a material the regular

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Enthalpies in Alloys

crystal structure, i.e. the translational symmetry is lost. In contrast to crystals there is a distribution of nearest-neighbour distances as well as of the number of nearest neighbours (co-ordination). Characteristic is the x-ray diffraction pattern of such a glassy alloy: instead of a number of sharp diffraction lines of the crystalline state, corresponding to reflections by well defined lattice planes, a broad diffraction peak is observed due to the distribution of interatomic distances. Sometimes a much lower second peak is also visible. Another characteristic of amorphisity is observed, when an amorphous alloy is heated in a differential scanning calorimeter (DSC). When the temperature becomes high enough for the atoms to become sufficiently mobile, the metastable structure transforms to the crystalline equilibrium state and since the latter state has a lower enthalpy than the metastable amorphous alloy, such a crystallisation is accompanied by heat release (as is solidification of a liquid). An exothermic peak is then measured in the DSC scan. High cooling rates can be achieved, for example, by the melt-spinning method, that is applied even on industrial scale. In this process a jet of hot molten metal is propelled against the surface of a rapidly rotating copper cylinder, which is cooled. This yields thin ribbons of the amorphous alloy of about 50 m thickness. Other techniques for obtaining amorphous alloys are vapour condensation, electro-deposition, chemical deposition, sputtering and ball milling. The latter technique is the most recent and moreover the simplest. In this process one or more balls are brought into motion and hit the metal powder. Such a ball mill can be as simple as just a steel vial with a hard bottom containing, apart from a few grams of sample, one hardened steel ball with a diameter of, for example, six centimetres. The vial is mounted on a vibrating frame that makes the ball hop up and down and in this way the powdered sample is ground between the ball and the bottom of the vial. It turns out that many intermetallic compounds amorphise under mechanical milling. Since the amorphous state is a metastable state, its formation enthalpy is higher than that of the intermetallic compound with the same composition. For estimating the formation enthalpy of the amorphous alloy it should be realised that in contrast to the solid solution there is no size mismatch, because there is no crystal structure. For the same reason the structural term is absent, so that we can write [5] H form (amorphous)  H chem (amorphous)  H topological (amorphous) (55) For liquids the topological enthalpy, accounting for the difference between the crystalline state and the liquid state is the heat of fusion with a magnitude of about RTm , where the gas constant R = 8.3 J per mole K-1 and where Tm is the average of the two melting temperatures. In amorphous alloys a certain degree of relaxation towards the solid state exists, so that the enthalpy contribution will be lower. As an estimate we use in J per mole of atoms [5]

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H form (amorphous)  H chem (amorphous)  35 .  c ATm, A  cB Tm, B 

(56)

and as a first approximation we will assume the chemical term of the amorphous state and of the solid solution to be equal. Example Formation enthalpy of amorphous 50 at.% Ni - 50 at.% Zr. H chem(amorphous) = -48 kJ (see the previous example). H topological(amorphous) = 3.5(0.51726 + 0.52125)/1000 = 7 kJ. Eq.(56): H form(amorphous) = -48 + 7 = -41 kJ per mole of atoms (as compared with the compound of -72 kJ). 5. COMPARISON OF THE FORMATION ENTHALPY OF THE AMORPHOUS ALLOY WITH THE FORMATION ENTHALPY OF THE SOLID SOLUTION In this section we will deal with metastable states, i.e. atomic configurations that are different from the equilibrium state. Examples of ball milling experiments will be given and we will show how Miedema’s model can successfully be applied to these problems. First we discuss, how an alloy can be forced into a metastable state by ball milling.

Fig.9. Two-dimensional schematic of an ordered structure (a) and of some disorder (b).

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Enthalpies in Alloys

Fig.10. Results of ball milling of  phase NiV2: x-ray intensity versus twice the diffraction angle. The patterns are labelled by the milling period in hours; 0 h: starting compound [5].

Fig.11. DSC scan at a heating rate of 10 K min-1 of ball-milled amorphous NiV2 [6].

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In many intermetallic compounds anti-site disorder of both components is introduced in the early state of ball milling. Atomic disorder is illustrated by Fig.9, where Fig.9a represents a two-dimensional analogue of an ordered intermetallic compound. Two sublattices can be distinguished, the one sublattice occupied by the one atomic species, the other sublattice by the other atomic species. Fig.9b shows some anti-site disorder, atoms have been transferred to the ‘wrong’ sublattices. Disorder may also be created in thermodynamic equilibrium by temperature. Admittedly the somewhat disordered state has a higher enthalpy than the completely ordered arrangement of atoms, but at higher temperatures this is compensated by the entropy increase due to disordering. There are even compounds, such as CuZn, that are completely disordered above a certain orderdisorder or critical temperature, but these are exceptions. Usually intermetallic compounds preserve their ordered structure up to the melting temperature, although at high temperature some atoms may have been transferred to the wrong sublattice. It turns out that ball milling is far more effective for disordering than high temperature and eventually ball milling can lead to a phase transformation to an inherently disordered structure: amorphous or the crystalline solid solution. An example of amorphisation upon mechanical milling is given by Fig.10. Here the x-ray-diffraction patterns were taken after various milling periods indicated in the figure. The pattern labelled as 0 h is the starting material, the intermetallic compound NiV2. This material crystallises in a complex crystal structure with as many as thirty atoms per unit cell, the so-called  phase. The complexity of this phase is reflected by many diffraction peaks. These peaks broaden for longer milling times and finally disappear after 240 hours of milling (pattern labelled 240 h). This is the picture of an amorphous material. Fig.11 presents a DSC scan of the amorphous alloy at a heating rate of 10 K min -1. The peak at 815 K is the crystallisation peak. Apparently this is the temperature, where the atoms become mobile enough to restore the crystalline state. The second smaller peak is probably due to grain growth of the small crystallites formed at the crystallisation temperature. An illustration of a transformation to the solid solution is Fig.12, again the x-ray intensity versus twice the diffraction angle at various milling periods. This compound, CrFe is also a  phase and transforms to the b.c.c. solid solution of Cr in  Fe. This phase is also present in the equilibrium phase diagram and could be frozen-in by rapid quenching of the solid starting alloy into water from a state represented by a point in the high-temperature phase field. The x-ray diffraction pattern is included in the figure, labelled by HTP (high-temperature phase). It is concluded that the diffraction peaks are at the same angles, clearly showing that by milling the high-temperature b.c.c. phase was achieved, although milling was performed at ambient temperature.

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Enthalpies in Alloys

Fig.12. X-ray diffraction patterns of  phase CrFe after various milling periods. For

comparison the x-ray diffraction pattern of a CrFe sample quenched from 880  C is also included (HTP) [5].

Fig.13. Formation enthalpy for the solid solution (solid line) and the amorphous state (dashed line) of the Ni-Zr system as a function of composition [5].

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Experiments as well as theoretical considerations show that during milling the material is locally heated in the mill to temperatures only up to 100 - 200 C, so that a temperature rise is not the cause of the transformation. It is really the increasing atomic disorder that eventually leads to the transformation. That the x-ray peaks in the ball milled material are broader than in the quenched material is due to a second effect caused by milling. By milling the material becomes nanocrystalline, i.e. the powder particles, usually of a size of 10 - 20 m in diameter consist of crystallites in the order of typically 10 nm, so that a powder particle contains as many as 109 crystallites! Besides, strain is introduced by milling, a second cause of the broadening. An important question is now: what will happen to an intermetallic compound upon mechanical milling? Will it amorphise or transform to the solid solution? Again Miedema’s model is of great help in answering this question. First one has to realise that, although both the solid solution and the glassy state have, inherent to metastability, enthalpies higher than the starting intermetallic, it is plausible that nature will deal in an economic manner with enthalpy, i.e. most probably that state will be formed that has the lower enthalpy of both. And with the aid of sections 3 and 4 of Part II we are now able to construct enthalpy curves for the glass as well as for the solid solution over the whole concentration range. Fig.13 displays both enthalpy curves for the Ni-Zr system. In the early days of ball-milling experiments it was the compound NiZr that was intensively studied. In those days it was far from generally accepted that milling really would lead to a genuine glass. Therefore several compositions of this system were milled and their properties were compared with ‘real’ glasses of the same composition, which were obtained by rapid quenching from the melt. These investigations demonstrated beyond any doubt that alloys, prepared following the mechanical method exhibited the same characteristics as melt-spun ribbons. On the basis of the enthalpy diagram of Fig.13 we can understand, why, for example, the compound NiZr amorphises upon milling. The formation enthalpy of the compound is -72 kJ per mole of atoms, so below the enthalpies of the corresponding amorphous curve and that of the solid solution. By mechanical impact more and more atomic disorder is induced and thus the enthalpy of the compound is increasing and we are moving upwards in the enthalpy diagram. The first curve we then cross is that for the amorphous state and the material amorphises. By further milling it is not possible to increase the enthalpy, because the amorphous state is largely disordered and so it is not possible to store more enthalpy in the form of disorder and the material has attained its final metastable state. Let us now inspect the diagram for Nb-Au (Fig.14). Here we encounter the opposite situation: for any composition the curve for the solid solution is lower than that for the glass. Therefore, we expect Au-Nb intermetallics to

36

Enthalpies in Alloys

Fig.14. Formation enthalpy for the solid solution (solid line) and the amorphous state (dashed line) of the Au-Nb system as a function of composition [5].

Fig.15. X-ray diffraction patterns of Au2Nb (AlB2 structure) after various milling periods [5].

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transform to the solid solution. On the basis of x-ray diffractograms of Fig.15 for AlB2-structure Au2Nb and Fig.16 for A15 AuNb3 it is observed that Au2Nb transforms to the f.c.c. solid solution of Nb in Au, whereas AuNb3 transforms to the b.c.c. solid solution of Au in Nb. A number of different compositions showed similar behaviour and none of the compounds amorphises. Interesting is to mention that around the composition 42 at.% Au and 58 at.% Nb we found a small f.c.c. - b.c.c. two-phase field.

Fig.16. X-ray diffraction patterns of AuNb3 (A15- structure) after various milling periods [5].

A last example is the V-Ni system. The enthalpy diagram is given by Fig.17. On the basis of this figure only around the composition NiV2 amorphisation is to be expected and as shown in Fig.10 that is what really was found experimentally and indeed different compositions did not amorphise, completely in agreement with the diagram [5]. Of course, realising the approximating character of the Miedema model, this result could be somewhat

Materials Science Foundations ISSN: 2297-7589, Vol. 1, pp 38-39 © 1998 Trans Tech Publications, Switzerland

Fig.17. Formation enthalpy for the solid solution (solid line) and the amorphous state (dashed line) of the V-Ni system as a function of composition [5].

fortuitous, but altogether the model seems to work also well for these problems. Studying the behaviour of many compounds, it was concluded that the predictions have a reliability of about 90 % [5]. 6. FORMATION ENTHALPY OF TERNARY ALLOYS Let us consider a ternary compound with a fraction cA of component A, cB of component B and cC of component C with cA + cB + cC = 1. As before we define f BA as the degree to which A is in contact with B and since cA is the fraction of A atoms this contact contributes c A f BA H inter (A in B) to the contact (formation) enthalpy. In this way taking all contacts into account, we obtain [1] H form (1 mole of atoms) = c A f BA H inter (A in B) + c A f CA H inter (A in C) + + c B f CB H inter (B in C)

(57)

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Example Estimate the formation enthalpy of a Heusler alloy AB2C.

Fig.18. Structure of the Heusler alloy. (Remark: crystal structures are sometimes indicated by the name of a person, such as in this case, often by a prototype material, which is AlNi 2Ti in this case, following the rather old system of the Strukturbericht by L2 1 or by a more modern nomenclature as cF16). This cubic structure is presented in Fig.18 and it is observed that it is derived from the b.c.c. structure, where the ordering over the sublattices is made clear by drawing eight b.c.c. cells. Let us for reasons of simplicity assume that all atoms are of equal size in the alloy and let us consider the central (black) B atom, then following Eq.(7) 2 2  f ABand C  c sA  cCs 1  8  c sA  cCs  c Bs   0.5  1  8  0.5  0.5  0.75. a   nd since A and C are equal in size and occupy equivalent positions relative to the B atom: f AB  f CB  0.375 . Since A atoms are surrounded by B atoms as in







 





the B2 structure (Fig.18) f BA  0.75 and since C atoms are next-nearest neighbours of the A atoms, whereas A atoms are third nearest neighbours of A atoms, A atoms don’t ‘see’ A atoms: f CA  1  f BA  0.25 . From Eq.(57): Hform (1 mole of atoms) = 0.250.75 Hinter (A in B) + + 0.250.25 Hinter (A in C) + 0.50.375 Hinter (B in C). The interfacial enthalpies are again obtained from the tables in the Appendix.

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7. APPLICATION TO VOLUME EFFECTS UPON ALLOYING In section 4 of Part I we discussed how the change in volume upon alloying can be estimated. If A is the dilute component in terms of surface fractions (Eq.(4)), the volume change of A per mole of A is given by Eq.(13), that of B per mole of A by Eq.(15), while the total volume change of the alloy per mole of A is obtained by Eq.(16). A spectacular example of volume change upon solving one metal in the other metal is the so-called fast impurity diffusion in metals. Atomic diffusion usually occurs by jumping of atoms into vacant lattice sites (vacancies). This makes diffusion in the solid state slow, because only few lattice sites are unoccupied. In contrast, the diffusion of, for example, carbon in iron is fast and the reason is that the small carbon atoms are able to occupy sites between the normal lattice positions, so-called interstitial sites. Since all those sites (very dilute C concentration) are empty, atomic migration via those positions is orders of magnitude faster than ‘normal’ diffusion via vacancies. An anomalously fast diffusion was also found for certain metal impurities in metals such as in lead, uranium, tin and indium and soon it became clear that this diffusion process had at least an interstitial component, which means that a part of the impurities would be solved interstitially and the more so when the volume of the impurity was small. However, it was found that gold as an impurity was always a faster diffuser than silver, although these metals are from a chemical point of view rather similar and more importantly have equal volumes. How could this be explained? Let us first consider the diffusion problem in greater depth. The speed of diffusion is characterised by the diffusion coefficient D, the higher the value of D the faster the diffusion. Above we explained that diffusion of interstitial impurities is faster than of substitutional impurities, which means that Di , the diffusion coefficient of interstitials is orders of magnitude higher than that of substitutional impurities Ds. If the fraction ci of the impurities that are solved in interstitial positions is not negligible relative to those in substitutional positions cs , then the diffusion may be fast, because we can write for the observed diffusion coefficient D, which can be measured, for example, by the diffusion of radioactive tracers of the impurity D

cs ci Ds  Di cs  ci cs  ci

(58)

Knowing that Di >> Ds , then even if ci < cs the diffusion will be fast. It is very plausible that the smaller the size of the impurity the more impurities will be solved interstitially and the faster the diffusion. We return to the fast diffusion of Ag and Au with equal atomic volumes, where it was found that Au is always faster in diffusion than Ag. Therefore, one would expect a smaller effective

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volume for Au than for Ag. Let us use Eq.(17) with f BA  1 (i.e. Eq.(16) for the infinitely dilute solid solution), then we observe that for Cu, Ag and Au in Pb, U, Pr, Tl and Sn the value of V obtained from the equation is always negative. Closer inspection shows that the noble metal atoms attract electronic charge and become bigger, the matrix atoms become smaller around the impurity. So the interstitial hole in the host metal, where the impurity is accommodated, expands more by the charge transfer to the noble metal than the noble-metal atom itself, so that effectively more space is created to accommodate the impurity in an interstitial position. In other words the mismatch enthalpy becomes lower than without volume change and the number of interstitial impurities is enhanced. The effect is always larger for Au than for Ag, so that Au has always a smaller effective interstitial volume than Ag. In this way the corrected, say effective volumes following Eq.(16) of Cu, Au and Ag in Pb become 6.69, 8.23 and 9.97 cm3 per mole, respectively, for a solution in Pr: 5.90, 7.18 and 9.31 cm 3 per mole, for a solution in Sn: 6.89, 8.89 and 10.13 cm3 per mole, respectively. These effective volumes were taken as a measure of the effect and experimentally always DCu > DAu > DAg holds, completely in agreement with the corrected volumes. This solved a twenty years old problem! It even turned out that diffusion parameters scale with these corrected volumes [7]. 8. APPLICATION TO THE HEAT OF ATOMISATION So far we considered in this second part only contacts between atoms of two different elements. A new aspect, that was introduced later in the model was the consequence of a contact of an atom with vacuum, i.e. the introduction of the concept of surface energy. Hereby it became possible to estimate heats of atomisation of an element following Eq.(43). In this second part of the paper we will encounter a number of further applications of surface energies. These surface energies are tabulated in Table A6. 9. VACANCY FORMATION ENTHALPIES IN PURE METALS AND ALLOYS In any metal vacancies (unoccupied lattice sites) will be present due to the gain of mixing entropy of those vacancies as a sort of ‘quasi particles’, with the atoms, although their formation enthalpy is considerable. Applying thermodynamics in a very similar way as in Part I section 6.2 an equation for the vacancy fraction cV, similar to Eq.(41) is obtained cV  exp

SV  HV exp R RT

(59)

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Enthalpies in Alloys

SV is the formation entropy per mole of vacancies, which is largely due to changes in the vibrational spectrum and has a magnitude in the order of R (a value of 0.6R is recommended [1]). HV is the vacancy formation enthalpy. As the consequence of Miedema’s concept of atoms as being macroscopic blocks of the element, vacancies have to be conceived as macroscopic holes of the size of an atom in the element. Then the formation enthalpy of vacancies should be due to the surface enthalpy of the hole and therefore [1] HV  CV 2 / 3

(60)

where the proportionality constant C still has to be determined. This was performed in an empirical way by fitting Eq.(60) to available experimental data. The result is HV  162 .  108 V 2 / 3

(61)

The proportionality constant is about 1/3 times the constant c0 from Eq.(49). This is not surprising. The hole is not really empty, but there is some electron density inside. Moreover, neighbouring atoms relax somewhat in the direction of the vacancy. This all makes the formation enthalpy of a vacancy much smaller than the enthalpy for atomisation. Vacancy formation enthalpies in pure metals are tabulated in Table A6, although these values can directly be obtained from Eq.(61). Let us now consider an intermetallic compound composed of A and B atoms with an excess of B and let us form a vacancy on an A site. When A atoms are bigger than B atoms the hole that remains after removing the A atom will be bigger than a hole in pure B so that p

V  HV (vacancy on an A site in AB n )   A  HV (in pure B)  VB 

(62)

where p is a constant and HV is given by Eq.(60). Since in a big A hole there will be a smaller electron density than in a hole of size VB, the formation enthalpy will be larger than the value of HV we would obtain if p would be equal to 2/3. On the other hand it is well-known that a divacancy (two neighbouring vacancies) has a formation enthalpy that is less than twice the formation enthalpy of a single vacancy. Therefore, a hole with twice the hole size of a single vacancy does not lead to the double formation enthalpy, so that the constant p must be less than unity. Quite characteristically Miedema decides that a good estimate of p, - in value between 2/3 and 1 -, is 5/6.

Materials Science Foundations ISSN: 2297-7589, Vol. 1, pp 43-48 © 1998 Trans Tech Publications, Switzerland

The extension to vacancy formation enthalpies in concentrated alloys is straight-forward. In such an alloy a vacancy on the A sublattice is in contact with B atoms to a degree of f BA and with A atoms to a degree of f AA , leading for an intermetallic compound to [1] HV (A vacancy) 

V  f BA  A   VB 

5/ 6

HV (B)  f AA HV (A)

(63)

where f AA  1  f BA . For a random alloy f BA  c Bs and f AA  c sA . In many systems, for example, in pure metals the vacancy content is low and only amounts to a fraction of 10 -4 even at the melting point, so that one could get the impression that vacancies are unimportant in the properties of solids. However, a number of properties are related to these very vacancies. As an example we already mentioned vacancy mediated diffusion in metals (Part II, section 7). Example a) Vacancy formation enthalpy on the Co sublattice in CoGa. Co Co f Ga  0.87, so that f Co  013 . . By Eq.(63) and the vacancy formation enthalpies from Table A6 and volumes from Table A1 (here possible volume corrections are neglected by Miedema [1]): 5/ 6  6.70  HV (Co vacancy)  0.87   .  135  44 kJ   48  013  1182 .  per mole of vacancies. b) Vacancy formation enthalpy on the Ga sublattice in CoGa. Ga Ga f Co  0.60, so that f Ga  0.40 . 5/ 6

.   1182 HV (Ga vacancy)  0.60     135  0.40  48  149 kJ  6.70  per mole of vacancies. Note the large difference between formation enthalpies of vacancies on the two sublattices. The reason is that the Co vacancy is mainly surrounded by Ga atoms and thus is similar to a vacancy in pure Ga with a low formation enthalpy, whereas the Ga vacancy resembles a vacancy in Co. Consequences of these numbers will be discussed later.

10. ANTI-SITE DISORDER IN ORDERED COMPOUNDS [8] 10.1 THE CHEMICAL PART OF THE ENTHALPY The simplest model for describing atomic order in alloys is the Ising model, where it is assumed that there are (chemical) bonds between nearest

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Enthalpies in Alloys

neighbouring atoms only. Let us as an example consider a two-dimensional square lattice of a compound AB with complete order (see Fig. 19 upper part). One of the sublattices is occupied by A atoms, the other sublattice by B atoms. We call those sublattices the  sublattice and the  sublattice, respectively.

Fig.19. Two-dimensional schematic of an ordered structure (upper part) and one with two ‘wrong’ atoms (lower part). Atoms are represented by circles. In the lower part interaction energies following the Ising model are indicated.

In the lower part of Fig.19 an A and a B atom have been exchanged, in other words some disorder has been created. In the completely ordered configuration, we are dealing with bonds between A and neighbouring B atoms only in this far oversimplified model. We assign a negative interaction energy of VAB to such bonds. When there are N atoms in the alloy, half of these will be A atoms and the total energy, which is the cohesive energy of the compound is then obtained by summing the energies of all bonds H coh (AB) =

N 4V AB 2

(64)

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where 4 is the number of nearest-neighbour bonds of an atom and where the factor ½ avoids double counting. In the compound with some disorder we have to take into account two more types of bonds, namely AA and BB bonds as indicated in the lower part of the figure with negative interaction energies VAA and VBB, respectively. Inspection of only the A atom makes clear that by transferring this atom to the wrong sublattice four AB bonds were broken and four AA bonds were made. In the same way by the transfer of the B atom four AB bonds were broken and four BB bonds were made. We lost a total energy of - 8VAB and we gained an energy of 4VAA + 4VBB . Since the cohesive energy per atom is following Eq.(64) 2VAB we decomposed by this action, as it were, four moles of AB into necessarily two moles of ‘pure’ A with a cohesive enthalpy of 2VAA per mole each and two moles of ‘pure’ B with a cohesive enthalpy of 2VBB each. Therefore, we can write for the enthalpy difference between the situation of the upper and lower part of the figure, i.e. - 8VAB + 4VAA + 4VBB H chem (disorder)  4 H coh (AB) + 2 H coh (A)  2 H coh (B)

(65)

per mole of ‘wrong’ pairs. The right-hand side of Eq.(65) is following Eq.(50) four times the formation enthalpy of the compound. In Miedema’s model, in contrast, even in a completely ordered compound A atoms are not only in (surface) contact with B atoms, but partly also with A atoms. The degree to which an A atom is in contact with dissimilar atoms (B atoms) determines the chemical enthalpy. Extending this concept we can state, that by disordering the degree to which contacts change determines the chemical enthalpy. Let us again inspect Fig.19 and let us only evaluate the enthalpy of the A atom in both situations. As we saw the A atom looses -4VAB and gains 4VAA in enthalpy. In Miedema’s terminology this would be the enthalpy difference in transferring an atom to the wrong sublattice if on the right sublattice the atom would be fully in contact with only B atoms and on the wrong sublattice fully with A atoms. However, following Miedema’s model, the A-B contact when A is located on the right sublattice is only f BA  1 , where f BA is the degree to which A atoms on the right sublattice are in contact with B atoms. Extending this concept: on the wrong sublattice the degree to which an A atom is in contact with B atoms is f BA  0 . Here the superscript A means the degree to which the A atom on the

 sublattice is in contact with B atoms. In line with these ideas similar

arguments apply to the contact of the A atom with A atoms in both positions. In other words by disordering the atom does not loose completely its character of an atom in the compound, but only in so far as it looses its AB character, i.e. in so far as it looses contact with B atoms. Therefore, the corresponding enthalpy





coh change is  f BA  f BA 2 H AB . On the other hand the A atom increases by the

transfer its pure A character. In the same way the accompanying enthalpy

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Enthalpies in Alloys





change is  f AA  f AA 2 H Acoh . Assuming by approximation that an A atom on the  sublattice has a surface contact that is the same as that of a B atom on the  sublattice, we can write f BA  f BB and for the same reason f AA  f AB . Since evidently f BB  1  f AB and f AA  1  f BA , the chemical enthalpy change the A atom experiences is obtained as

   1  f

  H



coh H chem (A to the wrong sublattice)   1  f BA  f AB  2 H AB  2 H Acoh  A B

 f AB

coh A

 H Bcoh  2 H form



(66) where the definition of the formation enthalpy Eq.(50) was applied. Note that if the cohesive enthalpy of pure A is more negative than the sum of the cohesive enthalpy of pure B plus twice the formation enthalpy (and that can well be so), that then it is favourable to transfer an A atom to the wrong sublattice. However, such a transfer never can take place without a combined action, e.g. the transfer of a B atom to the wrong sublattice and the combination will always take enthalpy. An expression similar to Eq.(66) is obtained for a B atom being accommodated on the  sublattice. Summing the two expressions, the chemical part of the enthalpy for one mole of pairs of wrong atoms is obtained as





H chem (1 mole of pairs)   1  f BA  f AB  4H form



(67)

and indeed for a compound, where the formation enthalpy is always negative, anti-site disorders takes enthalpy. For f BA  1 and f AB  1, i.e. A atoms only in contact with B atoms and vice versa we recognise the result following the Ising model. It is simple and straight-forward to show that Eq.(67) is the same for the B2 (or CsCl) structure (see Fig.20) as for its two-dimensional analogue of Fig.19. Example What would be the chemical enthalpy for creating anti-site defects in the compound CoGa, i.e. exchanging one mole of Ga and one mole of Co? Since the formation enthalpy of CoGa is equal to -31 kJ per mole as was demonstrated in the last example of Part I section 4 and the f’s are equal to 0.87 and 0.60 (here neglecting the effect of volume changes on the values of f, which is justified because it introduces negligible errors compared to the over-all

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accuracy: the one f increases somewhat, whereas the other f decreases a little due to volume changes, so that the sum remains almost the same), the chemical part of the disordering enthalpy becomes: 40.4731 = 58 kJ per mole of pairs. 10.2. THE ELASTIC PART OF THE ENTHALPY In a completely ordered intermetallic compound there is no strain as can be concluded, for example, from sharp diffraction lines in x-ray diffraction. From the variety of possible crystal structures the two components apparently ‘choose’ a structure where the atoms, although of different sizes, do match. However, when atoms are accommodated on wrong sublattices, where they do not fit, strain will develop, accompanied by the storage of elastic enthalpy in the compound. In Part I section 4 we discussed volume effects upon alloying and in Part I section 5.1 the mismatch enthalpy. The extension to disorder in intermetallic compounds is straight-forward, although there are differences. We restrict ourselves to equiatomic compounds AB and first volume effects will be discussed. If, for example, A > B the A atoms will increase in size and the ‘new’ volume will be per one mole of A atoms  A  V A   c (A)

 A  B A nws

(68a)

where c is the constant for the compound. (Of course the same equation holds if A < B). The constant c(A) is, however, different from the constant  from Eq.(14), because charge transfer will take place only as far as the A atom is in contact with B atoms and so

 c (A)  15 .

f BAV A2 / 3

A 1/ 3 nws

B 1/ 3  nws

(69a)

In the same way we obtain per mole of A and so per mole of B in the equiatomic compound (B atoms will shrink in our example)  B  VB   c (B)

and

 A  B B nws

(68b)

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Enthalpies in Alloys

 c (B)  15 .

f ABV B2 / 3

A 1/ 3 nws

(69b)

B 1/ 3  nws

Note that a combination of Eq.(68a) and Eq.(69a) in comparison with Eq.(68b) and Eq.(69b) gives the same absolute electronic charge transfer from A to B as from B to A, since f BAV A2/ 3  f ABVB2/ 3 in the equiatomic compound. The elastic enthalpy for transferring one mole of A to the  sublattice can then be written H

elastic

(A on  ) 

2 K A G AB V A   B  4G ABV A  3K A  B

2

(70)

There is a difference between this equation and Eq.(28). The A atom is transferred to the ‘wrong’ sublattice and is there in an almost fully ordered intermetallic compound mainly in contact with A atoms and therefore the atom takes approximately a size as in pure A. Furthermore, it has to be accommodated in a hole of the size of a B atom on the  sublattice, so approximately into a hole size B. Note the rather subtle difference with the arguments that led to Eq.(28). The A atom has to be compressed or expanded in order to fill the hole. This explains the quantity KA in the equation. On the other hand the lattice as a whole accommodates size differences by shear and that is the reason that the shear modulus of the compound GAB has to be inserted. Since scarcely any shear moduli of intermetallics are known, we use as an estimate of this quantity G AB  125 . c A G A  cB GB 

(71)

because from the few experimental values it becomes clear that the shear modulus of an intermetallic compound is larger than the average of the shear moduli of the pure elements and the factor 1.25 accounts for this in a reasonable way, when known shear moduli of intermetallics are compared with those of pure elements. Of course, for an equiatomic compound cA = cB = 0.5. Finally we obtain by Eqs (67) and Eq.(70) and a similar equation as the latter for B on  per mole of anti-site pairs H disorder (pair)  H chem (pair)  H elastic (A on  )  H elastic (B on  ) (72)

Materials Science Foundations ISSN: 2297-7589, Vol. 1, pp 49-51 © 1998 Trans Tech Publications, Switzerland

Example What would be the enthalpy for exchanging one mole of Ga and one mole of Co in the compound CoGa? In the third example in Part I section 4 we calculated: Co Ga f Ga  0.87 and f Co  0.60 . By Eqs.(68) and (69) and reading the molar volumes from Table A1, we obtain: Co = 7.42 cm3 per mole Co = 7.4210-6 m3 per mole Co; Ga = 10.2610-6 m3 per mole Ga. By Eqs (70) and (71) and Tables A1 and A5 we obtain: Helastic (Co on Ga subl.) = 44 kJ per mole ‘wrong’ Co; By an equation similar to Eq.(70): Helastic (Ga on Co subl.) = 34 kJ per mole ‘wrong’ Ga. The previous example yielded 58 kJ per mole pairs for the chemical enthalpy for this disordering. By summing up following Eq. (72) the chemical enthalpy and both elastic contributions we find a total enthalpy of 136 kJ per mole wrong pairs in CoGa. 11. VACANCY FORMATION IN ORDERED INTERMETALLICS As we already mentioned, vacancy formation in ordered structures is not straight-forward, because there are two constraints: 1) the initial composition should not be violated; 2) the correct ratio of sites on the (two) sublattices should be preserved. Let us as an example again keep in mind the two-dimensional structure of the upper part of Fig.19. Creating a vacancy in this ordered structure by just taking out an A atom from the  sublattice and bringing this atom to infinity would mean a change in composition. On the other hand, taking out this atom and putting it in an  position on the surface (the correct way of visualising vacancy formation) would add a site to the  sublattice and would be in conflict with the requirement of an equal number of sites (in the present structure) on the two sublattices. A way out of this dilemma is the formation of a so-called Schottky defect, i.e. a vacancy on the  sublattice and simultaneously a vacancy on the  sublattice. Without interactions between vacancies, which means that the fractions of vacancies are low, the mass-action law for the formation of a Schottky defect is 

SV  SV ln cV cV  R  



HV  HV  RT

(73)

 where HV  HV is the sum of the two formation enthalpies. These can be calculated following Eq.(63) and an equation similar to Eq.(63) for the other

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Enthalpies in Alloys

sublattice (note the somewhat different notation by using the symbols  and  to indicate the respective sublattices). The term SV  SV is the sum of the two formation entropies (c.f. Eq.(59) for vacancy formation in a pure metal). As mentioned before such entropies are mainly the result of changes in the phonon spectrum due to the introduction of defects and so are vibrational entropies, which are not to be confused with the configurational (mixing) entropy. The quantities cV and cV are here the fractions of unoccupied respective sublattice sites. These quantities are necessarily equal in value for the equiatomic compound in order to preserve an equal number of lattice sites of both sublattices. Example The formation enthalpy of a Schottky defect in CoGa. We obtained in the Example of Part II section 9: HVCo  44 kJ per mole of vacancies. By a similar equation: HVGa  149 kJ per mole of vacancies. Therefore, the Schottky defect has a formation enthalpy of 44 + 149 = 193 kJ per mole of Schottky defects. The large difference in the formation enthalpies can be understood from the fact that a vacancy on the Co sublattice is to be considered mainly as a vacancy in pure Ga with a corresponding low formation enthalpy, whereas a vacancy on the Ga sublattice is essentially a vacancy in pure Co with correspondingly a high formation enthalpy.

Fig.20. The B2 or CsCl structure.

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There is an other possibility of vacancy formation. This formation is closely related to disordering and in fact represents a special type of disordering, that could be called ‘vacancy-type of disordering’. It is experimentally found in some group VIII - group IIIA compounds such as CoGa. Again we take in mind the two-dimensional structure of the upper part of Fig.19 as an analogue of the B2 (CsCl) structure in which these compounds crystallise. The B2 structure is given by Fig.20. Upon disordering the CoGa intermetallic compound, Co atoms substitute on the Ga sublattice, but, in contrast, it turns out that Ga atoms can not be accommodated on the Co sublattice. By transfer of a Co atom to the wrong sublattice evidently a vacancy is left behind on the Co sublattice, while the Ga sublattice grows necessarily by one site. This violates the ratio 1:1 of sublattice sites, so that the number of Co sublattice sites has to grow by one site as well. Since no atoms are available for that (constant composition) an additional empty site on the Co sublattice should be created. The over-all result is the formation of two vacancies on the Co sublattice, combined with one anti-site Co atom. Such a defect consisting of three point defects is called a triple defect. It is in fact a type of disorder and consequently CoGa is termed to disorder in tripledefect disorder. This in contrast to ‘normal’ disorder, whereby both atomic species substitute on the wrong sublattices, which we call anti-site disorder (of both components).

Fig.21. Two-dimensional schematic of an ordered structure (a) and one with triple-defect disorder (b).

The two-dimensional schematic of this disorder is presented by Fig.21. It is assumed that the point defects of the triple defect are randomly distributed over the corresponding sublattices. In contrast to pure metals with a vacancy fraction of 10-3 to 10-4 even at the melting point, the vacancy fraction in CoGa amounts to some per cent at high temperature! It is clear that such a high triple-defect fraction has influence on the physical properties of the material.

Materials Science Foundations ISSN: 2297-7589, Vol. 1, pp 52-54 © 1998 Trans Tech Publications, Switzerland

Similar to Eq.(73) we write the mass-action law for triple-defect formation in a completely ordered B2 compound AB as [5]

 

 ln   cV 





2 SV  S A 2 HV  H A    R RT

2  cA 

(74)

Fractions are again relative to the number of corresponding sublattice sites. The quantities in the right-hand side of the equation, in a somewhat different notation than before, are self-explanatory. The vacancy formation enthalpy on the  sublattice is obtained by Eq.(63) and the enthalpy of the wrong A atom by Eqs.(66) and (70). Example Estimate the formation enthalpy of a triple defect in CoGa. We estimated before HV (Co vacancy )  44 kJ per mole vacancies. Eq.(66): H chem (A to the wrong sublattice)   1  f BA  f AB H Acoh  H Bcoh  2 H form 







 0.47    407  257  62  41 kJ per mole wrong atoms. Here the values of the cohesive enthalpies of the pure metals were estimated by Miedema’s model as explained in section 1 of Part II. Note that by the chemical effect enthalpy is gained, because the Co atom in a Co environment gains cohesive enthalpy compared to a Ga environment. We already found in a previous example: Helastic (Co on Ga subl.) = 44 kJ per mole ‘ wrong’ Co. This gives us the total formation enthalpy of a triple defect in CoGa as 244 - 41 + 43 = 90 kJ. This value agrees excellently with the measured value of 92 kJ per mole [5].

12. ANTI-SITE DISORDER VERSUS TRIPLE-DEFECT DISORDER AND SCHOTTKY DEFECTS [8] For CoGa we found for the anti-site defect of both components a formation enthalpy of 136 kJ, whereas the triple defect has a formation enthalpy of 90 kJ. Of course the defect with the lower enthalpy will be predominant, but the situation is a little more complicated than that. One has to compare the fraction of anti-site Co atoms in both defects. Let us write the mass-action law for anti-site disorder in the completely ordered compound

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53





S B  S A H B  H A ln cB c A   R RT  

(75)

Realising that the number of wrong A atoms is necessarily equal to the number of wrong B atoms we obtain

 

 2

ln c A





S B  S A H B  H A   R RT

(76a)

or 



S B  S A H B  H A ln c A   2R 2 RT 

(76b)

and thereby we conclude that the ‘effective’ enthalpy of formation of wrong A atoms is one half of the above calculated value. Similarly, for anti-site atoms in the triple defect, where necessarily by the very nature of the triple defect cV  2c A , we obtain from the mass-action law Eq.(74)

 

 ln   2c A 





2 SV  S A 2 HV  H A   R RT 

2  cA 

(77)

so that for the same reason the ‘effective’ formation enthalpy for wrong A atoms in the triple defect only is one third of the above calculated value. For the Schottky defect it is on the basis of Eq.(73) one half. Example On the basis of the numbers we estimated in the various examples for CoGa, the effective formation enthalpy of anti-site atoms on the Co sublattice in the antisite defect is 136/2 = 68 kJ per mole of anti-site Co atoms, in the triple defect 90/3 = 30 kJ per mole of wrong atoms and we conclude that the difference is still larger than one would think at first sight. The same applies also to vacancies on one sublattice in the triple defect compared to those in the Schottky defect: 30 kJ versus (149 + 44)/2 = 97 kJ. This makes clear that triple defects are predominant at higher temperatures. What about the other possible triple defect with a wrong Ga atom? The effective formation enthalpy is very high as expected, namely 429/3 kJ.

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Enthalpies in Alloys

It should be mentioned that such a low effective formation enthalpy as in CoGa is extreme. Most intermetallics disorder hardly at higher temperature. One of many compounds is B2 CoZr, which disorders in anti-site pairs, but only to a low degree, because the effective enthalpy for this defect is 261/2 kJ. Example Make on the basis of Eq.(77) a rough estimate of the fraction of vacancies in CoGa at 831 K. Assuming the entropies to be equal to R each, we obtain: ln4 30000 ln c A   + 1 0.022. This means that 2.2% of the Ga sites are 3 8.32  831 occupied by Co atoms, corresponding to a fraction of vacancies of 4.4% on the Co sublattice. These vacancy fractions are enormous compared with pure metals, realising that the melting temperature of CoGa is close to 1500 K and so that 831 K is relatively a low temperature. The agreement of the fraction of anti-site atoms with the measured value of 2.1% is excellent. At higher temperatures there are additional effects that are beyond the scope of this review. The simple mass-action law as given above does not hold there any longer, because a second enthalpy term due to the high concentration of defects has then to be taken into account. Very recently it turned out that the vacancy-type defect as the triple defect is not so exceptional as it may seem. In some C15 compounds, such as GdPt 2 the disorder is also of a vacancy type, but here we are necessarily dealing with a four-defect (quadruple defect): three vacancies on the Pt sublattice and one Pt anti-site atom. The extension to such a structure of the above given equations is not straight-forward, but it is feasible. From thermodynamics it is clear that it is the defect with the lowest effective formation enthalpy that will be formed, while the number of defects at a certain temperature depends on the numerical value of the effective enthalpy. Similarly, as we mentioned before, even in metastable materials nature seems to behave economically. There are strong indications that in a ball milling experiment the defect with the lower effective enthalpy will be formed. This was clearly demonstrated, for example, in the case of CoGa. Here triple defects are formed indeed under mechanical action, but in amounts that are scarcely attainable by high temperature treatment. Moreover, it was found by various investigators that CoGa does not transform to the amorphous state as NiZr does or to the solid solution as AuNb 3 does. A plausible reason for that is that during triple-defect disordering the Ga atoms remain on their own sublattice and frame, as it were, in this way the original structure. On the other hand, for a transformation to an inherently disordered structure such as the amorphous one or the solid solution disordering should be in the form of anti-site atoms of both constituents.

Materials Science Foundations ISSN: 2297-7589, Vol. 1, pp 55-57 © 1998 Trans Tech Publications, Switzerland

We are now at the point to predict what would happen to a compound, when it is subjected to ballistic action. We have to follow the following routes: 1) Estimate the effective enthalpy for the various possible defects. 1a) If the vacancy-type defect has the lower value, then this defect will form in the early state of milling and for prolonged milling times we will find no transformation. 1b) If the anti-site type of disorder has the lower enthalpy, this type of defect will be induced and most probably the material will transform to an inherently disordered structure for long milling times. 2a) It will transform to the amorphous state, if this state has a lower enthalpy than the solid solution. 2b) It will transform to the solid solution, if the solid solution has a lower enthalpy than the amorphous state. It turned out experimentally that by following such a path Miedema’s extended model was successful in yielding the correct prediction in about 90% of the cases [5]. Sometimes the amorphous state was found in cases where the solid solution was expected. Usually these were cases, where the difference between both metastable states was small and it might be that the energy of the grain boundaries in the nanocrystalline material drives the transformation to the amorphous state, because thereby the grain-boundary energy is gained. Also there will be an effect of the energy that can be induced by a certain type of mill. This is already clear from simple considerations. If the ball mill is not energetic enough no disorder and transformation will be induced at all. On the basis of Miedema’s model the Schottky defect in the B2 structure will usually have an effective formation enthalpy that is about the average of the two vacancy formation enthalpies of the pure metals, which usually will be a rather high value compared to the melting temperature of the compound. Therefore, Schottky defects are scarcely expected to be present in appreciable amounts. In order to appreciate the importance of the above considerations, one has to realise that by ball milling new metastable states are induced and that, for example, the state of disorder can be tuned by milling time. It is obvious that any metastable state exhibits properties different from the starting material, so that by the simple milling process, that, by the way, can be scaled up to industrial dimensions, always a new material is produced with possibly advantageous properties. 13. CRYSTALLISATION TEMPERATURE OF AMORPHOUS ALLOYS When a glassy alloy is heated in a calorimeter, heat release is measured at a certain temperature. This exothermic heat effect is due to the transformation of the glass to a state of lower enthalpy, i.e. the crystalline one. Would it be possible to predict such temperature? Buschow and Beekman [9] handled this

56

Enthalpies in Alloys

problem. Also in amorphous materials there are some kind of holes, although the holes are ill-defined and there is a distribution in sizes. The total volume of these holes is called excess free volume. The underlying idea is that the glass will crystallise when enough holes are being formed with the size of the smaller atom, so that the smaller atom can migrate and start the crystallisation. Therefore, a relation is expected between the formation enthalpy of a hole Hh with the size of the smaller atom and the crystallisation temperature Tx. The material will crystallise, when the quantity Hh / Tx in a Boltzmann exponential surpasses some critical value or, in other words, when the thermal energy is high enough to supply energy for big enough holes. Assume now that in an amorphous alloy composed of A and B atoms the A atoms are the smaller ones, then the formation enthalpy of the hole of size VA is given by Eq.(63) has to be adopted. In first approximation the f’ s were taken as the corresponding surface concentrations (Eq.(4)), but Weeber [10] argued that in an amorphous alloy some, so-called ‘chemical short-range order’ exists, i.e. A atoms are somewhat more surrounded by B atoms than by similar atoms and vice versa. It was proposed to use an equation similar to Eq.(7), but for this case with a ‘5’ instead of an ‘8’ in parentheses. In this way values for Hh were obtained for a large amount of amorphous alloys. Fitting these Hh values to measured Tx values led to the equation Tx  5  Hh  275

(78)

and although, strictly speaking the crystallisation temperature is not an intrinsic quantity, but depends somewhat on the number of pre-nuclei for crystallisation, Eq.(78) represents the experimental data rather well as is clear from Fig.22.

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57

Fig.22. Tx versus Hh for 214 amorphous alloys. In the rectangle are 87 data points. The straight line represents Eq.(78)[10].

By this section we terminate applications of Miedema’s (extended) model to bulk properties of solids and proceed to some applications to surface properties.

Materials Science Foundations ISSN: 2297-7589, Vol. 1, pp 58-62 © 1998 Trans Tech Publications, Switzerland

14. METAL SURFACES [1] 14.1. SURFACE ENTHALPY OF A PURE METAL In Part I section 8, we encountered the concept of the surface energy  as the energy in Joules per m2 surface area of an element. When a piece of metal consisting of one mole of atoms is atomised, the surface of all atoms

Fig.23. Two-dimensional schematic of Wigner-Seitz cells of a pure metal with a surface. (elementary blocks) are brought into contact with vacuum, where, in contrast, initially they were in contact with other atoms. We saw that the difference in enthalpy between this piece of material in the solid state and in the atomised state is the heat of atomisation in Miedema’s concept. Therefore, the heat of atomisation per mole of atomised atoms is proportional to the surface energy of the element and is given by Eq.(43). Let us study a metal surface in more detail. Fig.23 shows a two-dimensional schematic of a metal with a surface layer of atoms (top layer). The surface atoms are not completely surrounded by vacuum such as atomised atoms, but they are only partly in contact with vacuum and partly with the atoms in the layer below. Therefore, one mole of surface atoms has a lower surface energy than one mole of atomised atoms. It has to be assessed to which degree surface atoms are in contact with vacuum and to which degree they are in contact with other atoms. Although the surface enthalpy per mole of surface atoms may differ somewhat between different crystal planes due to the degree of roughness of a specific surface, a factor of 1/3 for the contact of surface atoms with vacuum was proposed by Miedema [1], which corresponds in a somewhat suggestive way to the drawing of Fig.23. This leads to the following estimate of the surface enthalpy per mole of surface atoms

Materials Science Foundations Vol. 1

H surface 

1 c0V 2/ 3 3

59

(79)

Note the difference between the surface energy of an element  in Joules per m2 and the surface energy (or enthalpy) of an element Hsurface per mole of surface atoms. We can ask the question: how much energy do we have to supply in order to atomise one mole of surface atoms? The answer is simple. By atomising these atoms we break the contact with the underlying atomic layer and since the surface atoms are for 2/3 in contact with these underlying atoms, we have to 2 supply a total energy of c0V 2 / 3 per mole of surface atoms. Moreover, the 3 layer below consisting of one mole of atoms is now brought into contact with 1 vacuum and this takes an enthalpy of another c0V 2 / 3 , so that the sum is just 3 the heat of atomisation. Consequently, when we consider a piece of an element consisting of N stacked molar layers and when we atomise this piece completely layer by layer, N times the heat of atomisation is to be supplied. In other words, when we atomise a piece of an element consisting of one mole of atoms, we have to supply just the heat of atomisation, what visualises atomisation and makes the picture consistent. Example An estimate of the surface enthalpy of gallium per mole of surface atoms is, with  = 1.1 Jm-2 (Table A6) and V 2/3 = 5.19 cm2 per mole of surface atoms: Hsurface = 1/34.51081.15.1910-4 = 8.5104 J = 85 kJ, which is for obvious reasons one third of the heat of atomisation.

60

Enthalpies in Alloys

14.2 SURFACE SEGREGATION IN DILUTE ALLOYS

Fig.24. Two-dimensional schematic of Wigner-Seitz cells of equal size of a dilute solid solution of A in B, where A atoms are segregated in the surface.

Surface segregation is the phenomenon that in an alloy the surface is in comparison with the over-all (bulk) composition usually enriched in one of the components. In Miedema’s model the driving force for that is the difference in surface energy between the components, whereby the component with the lower surface energy will tend to segregate in the surface. A two-dimensional schematic of a dilute alloy AB, poor in A, with A atoms segregated in the surface is presented by Fig.24. Again it is suggested that an atom in the surface is for 1/3 of its surface area in contact with vacuum as we assumed in the previous section. When an A atom is brought from the bulk, where it was completely in contact with B atoms (dilute alloy!) to the surface, where the fraction of contact with B atoms is only 2/3, one third of the interfacial enthalpy is lost. Moreover, in the surface the contact of a B atom with vacuum is lost, because the B surface is now covered by an A atom and a surface contact of an A atom with vacuum is created. This leads per mole of segregated A atoms in the surface to [11]. 1 1 1 H surf.seg. (A in B)   H inter (A in B)  c0 BVB2 / 3  c0 AV A2 / 3 3 3 3

(80)

Miedema introduces a factor f = 0.71 in the above equation, which is due to relaxation in the surface, so that [11] 1 1  1  H surf.seg. (A in B)  0.71   H inter (A in B)  c0 BVB2 / 3  c0 AV A2 / 3  3 3  3  (80a)

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61

Of course, the above equation gives only a rough estimate of the segregation enthalpy. For example, mismatch effects were not included. From simple thermodynamics the fraction of A atoms in the surface relative to that in the bulk at a temperature T is, assuming that the change in (vibrational) entropy between surface and bulk is negligible c surface A c bulk A

 H surf.seg. (A in B)   exp   RT  

(81)

Example The surface segregation of Au in Ni at 1000 K. By Eq.(80a) we obtain: H surf.seg. (Au in Ni)  1 1  1  0.71    33   4.5  2.45  352 .   4.5  15 .  4.70  104 J  25 kJ. 3 3  3  And following Eq.(81) the surface segregation is a factor of 20. And indeed large factors are found in the measurements, which show, by the way, a large scatter. Thus the result is good, although we are able to make rough estimates only and we can not expect too much accuracy in the case of surface segregation.

14.3. ENTHALPY OF IMPURITY ADSORPTION Let us inspect Fig.25, a schematic of an A atom adsorbed on B. The enthalpy for desorption is easily found. By taking the A atom from the B surface we have to supply one third of the surface enthalpy of A, the surface enthalpy of B in so far the B surface was covered by the A atom and moreover, one third of the interfacial enthalpy of A in B. By adsorption we gain these enthalpies, so that 1 1 1 H adsorption (A on B) =  c0 AV A2 / 3  c0 BV A2 / 3  H inter (A in B) 3 3 3

as usual per mole of adsorbed A atoms.

(82)

62

Enthalpies in Alloys

Fig.25. Two-dimensional schematic of Wigner-Seitz cells of B, where an A atom of different size is adsorbed on the surface.

This terminates this section on surface effects. Some of these applications can be refined for mismatch effects and/or elaborated without too much effort for more concentrated and ordered systems, but it was the aim of the section to demonstrate how Miedema’s model is also applicable to surface effects.

ACKNOWLEDGEMENTS I thank Mr A.J.Riemersma for drawing the figures, Mr A.J.Riemersma and Mr H.Schlatter for help with the layout of the paper and helpful advises. Dr C.Tuijn and Prof H. Takano are gratefully acknowledged for a critical reading of the manuscript, for many suggestions that led to improvements, for correcting errors and for checking the numerical results of the examples.

REFERENCES [1] F.R.de Boer, R.Boom, W.C.M.Mattens, A.R.Miedema and A.K.Niessen, Cohesion in Metals, Transition Metals Alloys (North- Holland, Amsterdam, (1988) [2] L.Pauling, The Nature of the Chemical Bond, 2nd Ed. (Cornell University Press, Ithaca, NY, 1952) [3] D.J.Eshelby, Solid State Physics (F.Seitz and D.Turnbull eds, Academic Press, New York, 1956), Vol.3, 193 [4] Recalculated values from K.A.Gschneidner, Solid State Physics (F.Seitz and D.Turnbull eds, Academic Press, New York, 1964), Vol.16, 276 [5] H.Bakker, G.F.Zhou and H.Yang, Progr. Materials Science, 39, 159 (1995) [6] H.Yang and H.Bakker, Mechanical Alloying for Structural Applications (D.D.deBarbadillo, F.H.Froes and R.Schwarz eds. ASM, Materials Park, OH, 1993), 401 [7] H.Bakker, Journ. Less-Comm. Metals, 105, 129 (1985) [8] H.Bakker, I.W.Modder and M.J.Kuin, Intermetallics, in press [9] K.H.J.Buschow and N.M.Beekmans, Solid State Comm., 35, 233 (1980) [10] A.W.Weeber, Journal of Physics F: Metal Physics, 17, 809 (1987) [11] A.R.Miedema, Z.Metallkd., 69, 455 (1978)

APPENDIX 1. BACKGROUND OF MIEDEMA’S MODEL It is the aim of this part of the appendix to give some more information about the underlying concepts of Miedema’s equations for obtaining interfacial enthalpies. However, this information is only very rough and incomplete and for full details the reader is referred to [1]. In Part I section 1 it was argued that for alloying of two transition metals three fundamental quantities play a role and that there is a negative contribution due to the quadratic difference of the work functions  of the two metals and a positive contribution due to the quadratic difference in the 1/3 power of the densities at the boundary of the Wigner-Seitz cell nws. Furthermore, we saw in Part I section 3 that intermetallics can be formed if the sum of both quantities is negative and that certainly no intermetallics will occur in the phase diagram of the two metals if the positive chemical contribution is larger than the negative chemical contribution in absolute value. Therefore, the existence of intermetallic compounds in a binary phase diagram indicates such a negative interfacial enthalpy. The same applies to a considerable mutual solute solubility. A negative interfacial enthalpy is expected, if one or more intermetallic compounds occur in the phase diagram or if there is a appreciable solid solubility, a positive interfacial one, in the opposite case. We recall the equation for the interfacial enthalpy (Eq.(1)) in the form:





2 1/ 3 2  H inter (A in B)    P    Q nws   

(A1)

If such an equation makes sense, then when we put



 P    Q n1ws/ 3 2



2

0

(A2)

a straight line in a   versus nws1/3  plot should separate regions with positive and negative interfacial enthalpies. Fig.26 gives such a plot for a large number of transition-metal - transition-metal alloys and indeed the drawn straight line separates two regions: one for binary systems with negative interfacial enthalpy (filled circles) and one for systems with a positive interfacial enthalpy (crosses). The straight line gives the empirical value of Q/P = 9.4 V2/(density units)2/3. As we already discussed in Part I section 1 the interfacial enthalpy is proportional to the surface contact area V A2 / 3 and the full equation is now obtained as Eq.(1) from Part I, where the denominator enters the expression as a measure of electrostatic screening length, which determines the width of the dipole layer caused by the electron transfer. On analysing available data of

formation enthalpies of binary transition-transition metal alloys we have for P in corresponding units P =14.1. A similar plot as Fig.26 for alloying a transition-metal with a nontransition-element reveals that for those systems an additional term called -R should be inserted between the brackets of Eq.(1), while on empirical grounds also the value of P gets a somewhat different value. For this and also for the method for obtaining numerical values for the quantities in Table A1 the reader is referred to [1]. As already argued we accept the values for the interfacial enthalpies from Tables A2 through A4 as granted and the aim of the present paper is to be able to use these values for estimating enthalpy effects. 2. TABLES In Table A1 the three fundamental quantities that are used as the basis of the semi-empirical model are given. These values were obtained from [1]. The same applies to the interfacial enthalpies from the Tables A2-A4, where we restricted ourselves to systems with at least one transition metal. In Table A5 values for bulk and shear moduli of the elements were recalculated in proper units from values collected by Gschneidner [4], whereas values in Table A6 are again taken from Miedema’s book [1], partly in a somewhat different unit.

Fig.26.   versus nws1/3  for binary solid alloys of two transition metals. Filled circles: in the binary system one or more compounds exist; crosses: no intermetallic exist or both solid solubilities are smaller than 10 at.%. The straight line clearly separates both regions [1].

Table A1. Model parameters in the metallic state,  in Volt, nws in (arbitrary) density units, V in 10-6 m3 per mole [1]. Elem.



nws

V

Elem.



nws

V

Ag Al As Au B Ba Be Bi C Ca Cd Ce 4+ Ce 3+ Co Cr Cs Cu Dy Er Eu 2+ Eu 3+ Fe Ga Gd Ge H Hf Hg Ho In Ir K La Li Lu Mg Mn Mo

4.35 4.20 4.80 5.15 5.30 2.32 5.05 4.15 6.24 2.55 4.05 3.25 3.18 5.10 4.65 1.95 4.45 3.21 3.22 2.50 3.20 4.93 4.10 3.20 4.55 5.20 3.60 4.20 3.22 3.90 5.55 2.25 3.17 2.85 3.22 3.45 4.45 4.65

2.52 2.70 3.00 3.87 5.36 0.53 4.66 1.56 5.55 0.75 1.91 2.41 1.69 5.36 5.18 0.17 3.18 1.82 1.86 0.68 1.77 5.55 2.25 1.77 2.57 3.38 3.05 1.91 1.82 1.60 6.13 0.27 1.64 0.94 1.91 1.60 4.17 5.55

10.25 10.00 11.85 10.20 4.70 38.10 4.90 19.32 3.25 26.20 13.00 16.04 21.62 6.70 7.23 69.23 7.12 19.00 18.45 28.95 19.97 7.09 11.82 19.90 9.87 1.70 13.45 14.08 18.76 15.75 8.52 45.63 22.55 13.00 17.77 14.00 7.35 9.40

N Na Nb Nd Ni Os P Pb Pd Pm Pr Pt Pu Rb Re Rh Ru Sb Sc Si Sm Sn Sr Ta Tb Tc Th Ti Tl Tm U V W Y Yb 2+ Yb 3+ Zn Zr

6.86 2.70 4.05 3.19 5.20 5.40 5.55 4.10 5.45 3.19 3.19 5.65 3.80 2.10 5.20 5.40 5.40 4.40 3.25 4.70 3.20 4.15 2.40 4.05 3.21 5.30 3.30 3.80 3.90 3.22 3.90 4.25 4.80 3.20 2.58 3.22 4.10 3.45

4.49 0.55 4.41 1.73 5.36 6.33 4.49 1.52 4.66 1.77 1.73 5.64 2.99 0.22 6.33 5.45 6.13 2.00 2.05 3.38 1.77 1.90 0.59 4.33 1.82 5.93 2.10 3.51 1.40 1.86 3.44 4.41 5.93 1.77 0.78 1.86 2.30 2.80

4.10 23.78 10.80 20.58 6.60 8.45 8.60 18.28 8.90 20.25 20.79 9.10 12.06 56.07 8.85 8.30 8.20 16.95 15.03 8.60 20.01 16.30 33.93 10.81 19.32 8.64 19.80 10.58 17.23 18.12 13.15 8.36 9.55 19.90 24.87 17.97 9.17 14.00

Table A2. Predicted values for the interfacial enthalpy for transition element A (in kJ per mole of A), solved in infinite dilution in transition element B. A hyphen (-) means: no values given in the original tables [1]. HAintin B Transition - Transition Element

 Solvent B (matrix element)  A  Ag Au Co Cr Cu Fe Hf Ir La Mn Mo Nb Ni Os Pd Pt Pu Re Rh Ru Sc Ta Tc Th Ti U V W Y Zr

Ag

Au

Co

Cr

Cu

Fe

Hf

Ir

La

Mn

0 +69 +98 +103 -53 +62 -143 +46 +147 +66 +56 +111 -29 -3 +151 +37 +91 -119 +61 +96 -6 +63 +172 -136 -87

0 +25 -1 +28 -257 +48 -328 -39 +13 -124 +25 +68 0 +17 +76 +27 +56 -291 -125 +52 -180 -69 +44 -318 -303

+84 +33 0 -18 +25 -2 -168 -15 -93 -21 -22 -111 -1 0 -7 -33 -88 +9 -9 -3 -139 -109 -1 -166 -126 -110 -58 -6 -112 -197

+120 -1 -18 0 +49 -6 -44 -78 +93 +8 +2 -32 -27 -46 -65 -110 +8 -19 -58 -49 +3 -30 -39 +12 -33 -13 -8 +4 +59 -58

+26 +51 0 +53 -81 +1 -115 +15 +82 +12 +14 +46 -62 -56 +82 -11 +31 -115 +9 +37 -40 +21 +101 -117 -110

+123 +37 -2 -6 +50 0 -98 -38 +25 +1 -9 -70 -6 -17 -19 -59 -29 -1 -23 -20 -53 -67 -13 -58 -74 -53 -29 0 -6 -118

-48 -253 -119 -32 -58 -71 0 -253 +72 -41 -15 +15 -145 -178 -301 -340 -4 -111 -232 -188 +21 +11 -172 +25 +1 -7 -8 -24 +51 -1

+61 +52 -13 -65 +1 -32 -293 0 -230 -66 -85 -216 -5 -2 +24 +2 -183 -13 +2 -2 -255 -211 -6 -316 -228 -215 -130 -63 -242 -331

-100 -255 -51 +52 -64 +14 +53 -158 0 +8 +103 +120 -81 -70 -270 -267 +24 +9 -162 -90 +7 +111 -75 +10 +64 +53 +69 +104 +1 +45

+57 -53 -21 +9 +15 +1 -57 -80 +14 0 +22 -17 -33 -38 -100 -128 -17 -2 -70 -46 -40 -17 -37 -42 -36 -25 -3 +28 -8 -74

Table A2. (continued) Predicted values for the interfacial enthalpy for transition element A (in kJ per mole of A), solved in infinite dilution in transition element B. A hyphen (-) means: no values given in the original tables. HAintin B Transition - Transition Element

 Solvent B (matrix element)  A  Ag Au Co Cr Cu Fe Hf Ir La Mn Mo Nb Ni Os Pd Pt Pu Re Rh Ru Sc Ta Tc Th Ti U V W Y Zr

Mo

Nb

Ni

Os

Pd

Pt

Pu

Re

Rh

Ru

+150 +14 -18 +1 +67 -7 -17 -86 +156 +18 0 -23 -27 -55 -59 -114 +35 -27 -59 -56 +45 -20 -45 +63 -15 +9 0 -1 +114 -27

+66 -134 -88 -26 +9 -57 +17 -207 +177 -13 -22 0 -107 -152 -208 -267 +36 -100 -176 -156 +76 0 -139 +95 +8 +17 -4 -33 +139 +17

+68 +33 -1 -27 +14 -6 -204 -7 -146 -33 -32 -136 0 +6 0 -22 -118 +10 -4 +2 -180 -133 +3 -218 -154 -140 -75 -14 -162 -237

+112 +75 0 -39 +37 -15 -210 -2 -104 -32 -56 -161 +5 0 +32 -2 -111 -4 +8 0 -162 -156 +1 -195 -164 -144 -88 -39 -128 -241

-28 0 -5 -53 -48 -16 -342 +24 -388 -82 -57 -212 0 +31 0 +8 -240 +26 +7 +24 -352 -210 +17 -440 -256 -254 -132 -26 -379 -391

-3 +18 -26 -87 -42 -47 -374 +1 -368 -101 -108 -264 -17 -2 +8 0 -257 -17 -6 -4 -354 -260 -12 -445 -290 -286 -167 -79 -367 -421

-66 +6 -22 -4 -167 +34 -13 +31 +34 -88 -100 -224 -247 0 -39 -157 -112 -3 +30 -98 +6 +16 +31 +18 -13

+152 +83 +8 -16 +65 -1 -129 -13 +14 -2 -27 -105 +8 -4 +26 -18 -43 0 +4 -4 -72 -100 +1 -81 -101 -75 -48 -17 -20 -153

+38 +30 -8 -50 -9 -20 -277 +3 -245 -60 -61 -188 -3 +8 +8 -7 -177 +4 0 +5 -257 -185 +1 -317 -211 -201 -113 -37 -253 -316

+94 +64 -3 -43 +25 -17 -227 -2 -137 -40 -58 -169 +2 0 +25 -4 -127 -4 +5 0 -185 -164 -1 -225 -176 -158 -95 -40 -158 -260

Table A2. (continued) Predicted values for the interfacial enthalpy for transition element A (in kJ per mole of A), solved in infinite dilution in transition element B. A hyphen (-) means: no values given in the original tables. HAintin B Transition - Transition Element

 Solvent B (matrix element)  A  Ag Au Co Cr Cu Fe Hf Ir La Mn Mo Nb Ni Os Pd Pt Pu Re Rh Ru Sc Ta Tc Th Ti U V W Y Zr

Sc

Ta

Tc

Th

Ti

U

V

W

Y

Zr

-108 -293 -100 +2 -83 -39 +20 -227 +10 -28 +38 +66 -130 -141 -319 -333 -3 -63 -221 -157 0 +59 -141 0 +28 +13 +25 +34 +4 +16

+60 -135 -86 -24 +7 -54 +12 -203 +164 -14 -19 0 -105 -147 -206 -263 +31 -96 -173 -152 +67 0 -135 +85 +5 +14 -4 -29 +127 +12

+97 +58 -1 -33 +29 -11 -201 -6 -111 -31 -45 -147 +2 +1 +17 -13 -108 +1 +1 -1 -162 -142 0 -195 -155 -137 -80 -28 -132 -232

-93 +7 -33 +20 -216 +11 -24 +43 +67 -122 -131 -306 -320 -55 -210 -147 0 +60 -132 0 +28 +28 +40 +5 +15

-6 -203 -103 -27 -33 -62 +1 -227 +99 -30 -14 +8 -126 -161 -260 -304 +7 -100 -204 -169 +33 +6 -153 +41 0 -1 -6 -23 +73 -1

-77 -9 -38 -7 -184 +70 -18 +8 +15 -98 -121 -222 -257 -64 -167 -130 +14 +12 -116 -1 0 +3 +4 +47 -13

+73 -87 -54 -8 +19 -28 -10 -146 +118 -3 0 -5 -69 -97 -151 -198 +20 -54 -123 -103 +33 -4 -89 +45 -7 +4 0 -3 +85 -17

+172 +48 -5 +4 +80 0 -27 -62 +154 +23 -1 -34 -11 -38 -26 -82 +33 -16 -36 -38 +39 -30 -28 +57 -23 +5 -3 0 +111 -39

-103 -268 -67 +35 -70 -4 +41 -180 +1 -5 +81 +102 -97 -93 -286 -289 +14 -14 -181 -112 +3 +93 -96 +5 +51 +39 +54 +81 0 +34

-78 -294 -137 -41 -78 -85 -1 -281 +60 -52 -23 +15 -165 -202 -339 -377 -12 -129 -261 -212 +16 +10 -195 +19 -1 -13 -13 -34 +42 0

Table A3. Predicted values for the interfacial enthalpy for non - transition element A (in kJ per mole of A), solved in infinite dilution in transition metal B. A hyphen (-) means: no values given in the original tables [1]. HAintin B Non-Transition - Transition Element

 Solvent B (matrix element)  A  Al As B Ba Be Bi C Ca Cd Cs Ga Ge Hg In K Li Mg N Na P Pb Rb Sb Si Sn Sr Tl Zn

Ag

Au

Co

Cr

Cu

Fe

Hf

Ir

La

Mn

-

-

-124 -149 -119 +65 -21 +8 -170 +13 -5 +353 -96 -89 +22 -18 +293 +30 +3 -320 +180 -213 +32 +323 -62 -138 -62 +53 +40 -49

-87 -156 -146 +299 -33 +61 -233 +196 +46 +628 -50 -77 +65 +50 +516 +131 +98 -439 +329 -274 +88 +571 -34 -134 -10 +271 +110 -9

-

-91 -131 -127 +213 -20 +71 -193 +129 +42 +522 -55 -64 +69 +43 +433 +94 +68 -361 +277 -229 +95 +477 -16 -122 -8 +191 +106 -14

-189 -364 -242 +280 -129 -212 -388 +179 -113 +625 -179 -244 -137 -128 +482 +108 +29 -752 +273 -488 -169 +551 -295 -262 -216 +249 -103 -117

-156 -144 -116 -70 -24 -5 -128 -97 -32 +214 -128 -95 +5 -52 +179 -27 -58 -222 +102 -177 +16 +197 -66 -149 -87 -73 +12 -75

-155 -322 -158 +69 -91 -277 -329 +32 -155 +291 -173 -231 -193 -178 +207 +19 -31 -720 +92 -401 -241 +247 -319 -207 -239 +58 -182 -122

-127 -213 -150 +176 -45 -58 -249 +102 -25 +480 -106 -132 -17 -39 +387 +76 +37 -487 +235 -305 -26 +432 -132 -167 -99 +156 +6 -55

Table A3. (continued) Predicted values for the interfacial enthalpy for non transition element A (in kJ per mole of A), solved in infinite dilution in transition element B. A hyphen (-) means: no values given in the original tables. HAintin B Non-Transition - Transition Element

 Solvent B (matrix element)  A  Al As B Ba Be Bi C Ca Cd Cs Ga Ge Hg In K Li Mg N Na P Pb Rb Sb Si Sn Sr Tl Zn

Mo

Nb

Ni

Os

Pd

Pt

Pu

Re

Rh

Ru

-64 -133 -148 +386 -30 +116 -237 +263 +82 +730 -20 -53 +104 +95 +601 +169 +137 -439 +390 -270 +142 +665 +10 -120 +33 +351 +162 +16

-116 -252 -213 +437 -90 -17 -339 +299 +10 +820 -82 -144 +8 +16 +657 +185 +121 -634 +408 -407 +18 736 -125 -195 -68 +396 +67 -35

-139 -155 -116 +1 -22 -15 -158 -37 -24 +284 -114 -99 +4 -42 +235 +3 -25 -297 +140 -206 +8 +259 -79 -145 -84 -6 +14 -63

-113 -110 -112 +74 -13 +74 -139 +17 +24 +365 -76 -57 +62 +20 +307 +34 +12 -246 +194 -175 +94 +336 -3 -120 -18 +62 +88 -33

-216 -215 -109 -298 -32 -178 -127 -272 -136 -40 -213 -170 -110 -186 -38 -123 -162 -247 -53 -197 -152 -39 -204 -190 -220 -285 -144 -143

-207 -191 -120 -233 -37 -110 -121 -226 -102 +45 -192 -145 -69 -141 +35 -96 -138 -212 -3 -188 -87 +40 -151 -184 -176 -225 -86 -126

-

-77 -90 -114 +210 -7 +130 -160 +125 +70 +514 -33 -31 +106 +76 +431 +94 +75 -288 +281 -186 +151 +472 +39 -101 +33 +188 +148 +2

-167 -169 -115 -108 -26 -57 -139 -124 -59 +168 -148 -118 -27 -86 +139 -43 -76 -256 +72 -194 -33 +154 -109 -160 -123 -108 -30 -90

-126 -124 -114 +32 -17 +43 -140 -16 +4 +320 -93 -72 +40 -5 +268 +16 -9 -250 +166 -181 +64 +294 -29 -130 -44 +22 +60 -47

Table A3. (continued) Predicted values for the interfacial enthalpy for non transition element A (in kJ per mole of A), solved in infinite dilution in transition element B. A hyphen (-) means: no values given in the original tables. HAintin B Non-Transition - Transition Element

 Solvent B (matrix element)  A  Al As B Ba Be Bi C Ca Cd Cs Ga Ge Hg In K Li Mg N Na P Pb Rb Sb Si Sn Sr Tl Zn

Sc

Ta

Tc

Th

Ti

U

V

W

Y

Zr

-170 -344 -196 +141 -122 -258 -363 +78 -149 +416 -178 -242 -185 -164 +305 +45 -20 -755 +150 -445 -220 +359 -316 -232 -234 +122 -158 -128

-120 -255 -211 +417 -89 -28 -337 +284 +3 +794 -88 -148 +1 +7 +635 +176 +113 -632 +393 -405 +7 +713 -133 -197 -76 +377 +57 -40

-121 -128 -115 +57 -17 +42 -150 +5 +8 +346 -88 -72 +42 -1 +290 +27 +2 -272 +180 -190 +64 +318 -31 -129 -41 +46 +63 -42

-

-163 -320 -227 +321 -111 -143 -364 +210 -69 +677 -145 -207 -84 -78 +530 +130 +58 -699 +313 -452 -103 +601 -233 -236 -163 +288 -43 -88

-

-113 -224 -182 +337 -66 -17 -295 +224 +6 +685 -83 -129 +10 +6 +552 +144 +93 -560 +343 -354 +15 +617 -113 -175 -68 +304 +56 -36

-52 -104 -135 +383 -18 +153 -214 +261 +101 +720 -4 -30 +129 +118 +597 +169 +144 -394 +391 -239 +176 +658 +46 -102 +61 +348 +189 +28

-161 -330 -170 +90 -101 -272 -340 +45 -154 +329 -175 -235 -191 -175 +237 +26 -29 -731 +110 -416 -235 +281 -319 -216 -239 +76 -175 -125

-208 -397 -258 +269 -147 -256 -411 +169 -142 +613 -203 -272 -173 -160 +466 +100 +13 -797 +257 -521 -210 +537 -337 -284 -250 +238 -139 -137

Table A4. Predicted values for the interfacial enthalpy for transition element A (in kJ per mole of A), solved in infinite dilution in non - transition element B. A hyphen (-) means: no values given in the original tables [1]. HAintin B Transition - Non-Transition Element

 Solvent B (matrix element)  A  Ag Au Co Cr Cu Fe Hf Ir La Mn Mo Nb Ni Os Pd Pt Pu Re Rh Ru Sc Ta Tc Th Ti U V W Y Zr

Al

As

B

Ba

Be

Bi

C

Ca

Cd

Cs

-105 -74 -79 -215 -163 -231 -106 -65 -121 -118 -115 -230 -229 -79 -169 -126 -195 -124 -124 -162 -101 -54 -221 -240

-104 -111 -94 -359 -122 -411 -151 -113 -223 -109 -92 -187 -172 -77 -139 -102 -339 -226 -107 -274 -170 -90 -389 -398

-148 -182 -161 -407 -177 -333 -184 -218 -325 -145 -168 -170 -194 -172 -170 -167 -319 -322 -173 -333 -238 -205 -332 -440

+31 +141 +104 +169 -43 +57 +81 +216 +244 +1 +44 -185 -152 +124 -63 +19 +89 +233 +34 +170 +163 +222 +68 +163

-26 -41 0 -25 -209 -38 -184 -54 -43 -134 -27 -20 -50 -61 -11 -39 -25 -191 -132 -25 -158 -84 -27 -190 -239

+4 +33 +39 -159 -3 -276 -31 +74 -12 -8 +47 -118 -75 +84 -36 +27 -197 -19 +27 -93 -10 +101 -249 -196

-250 -349 -292 -807 -229 -836 -370 -422 -632 -233 -246 -232 -228 -286 -242 -242 -709 -628 -266 -656 -471 -390 -797 -867

+8 +114 +77 +133 -73 +32 +58 +182 +207 -22 +13 -208 -182 +92 -90 -11 +61 +196 +4 +138 +135 +187 +42 +127

-4 +34 +32 -109 -30 -192 -18 +72 +9 -18 +22 -129 -102 +64 -53 +4 -144 +3 +7 -58 +5 +92 -177 -138

+149 +248 +219 +290 +122 +180 +182 +342 +364 +122 +197 -23 +28 +270 +90 +169 +200 +353 +184 +280 +269 +355 +189 +283

Table A4. (continued) Predicted values for the interfacial enthalpy for transition element A (in kJ per mole of A), solved in infinite dilution in non transition element B. A hyphen (-) means: no values given in the original tables. HAintin B Transition - Non-Transition Element

 Solvent B (matrix element)  A  Ag Au Co Cr Cu Fe Hf Ir La Mn Mo Nb Ni Os Pd Pt Pu Re Rh Ru Sc Ta Tc Th Ti U V W Y Zr

Ga

Ge

Hg

In

K

Li

Mg

N

Na

P

-74 -38 -43 -185 -122 -233 -80 -18 -77 -88 -70 -205 -192 -31 -135 -84 -186 -82 -82 -130 -67 -3 -219 -212

-72 -63 -53 -278 -93 -343 -108 -52 -147 -80 -55 -171 -151 -30 -112 -68 -276 -151 -70 -204 -113 -30 -322 -314

+15 +44 +48 -122 +4 -222 -12 +85 +7 +3 +52 -97 -63 +90 -23 +34 -165 +1 +35 -66 +7 +108 -204 -155

-11 +32 +28 -112 -41 -205 -25 +73 +13 -27 +15 -151 -119 +60 -66 -4 -145 +6 -1 -59 +4 +93 -185 -141

+151 +249 +221 +277 +123 +158 +180 +345 +360 +123 +201 -26 +26 +275 +90 +171 +181 +348 +187 +271 +267 +360 +167 +267

+31 +127 +96 +127 -36 +30 +71 +195 +206 +3 +44 -165 -138 +118 -55 +20 +54 +196 +34 +135 +141 +204 +38 +117

+3 +75 +54 +29 -57 -41 +28 +125 +111 -20 +11 -161 -144 +72 -71 -8 -20 +104 +2 +50 +74 +136 -35 +13

-

+127 +221 +195 +221 +95 +98 +152 +311 +313 +100 +173 -50 -3 +246 +64 +145 +125 +302 +159 +225 +231 +327 +108 +208

-176 -229 -193 -569 -178 -588 -254 -269 -425 -170 -173 -202 -199 -186 -189 -175 -502 -423 -189 -456 -317 -244 -562 -617

Table A4. (continued) Predicted values for the interfacial enthalpy for transition element A (in kJ per mole of A), solved in infinite dilution in non transition element B. A hyphen (-) means: no values given in the original tables. HAintin B Transition - Non-Transition Element

 Solvent B (matrix element)  A  Ag Au Co Cr Cu Fe Hf Ir La Mn Mo Nb Ni Os Pd Pt Pu Re Rh Ru Sc Ta Tc Th Ti U V W Y Zr

Pb

Rb

Sb

Si

Sn

Sr

Tl

Zn

+18 +50 +54 -132 +11 -250 -15 +95 +13 +5 +62 -105 -62 +101 -22 +41 -176 +5 +43 -70 +9 +121 -225 -167

+151 +249 +221 +285 +123 +170 +182 +345 +364 +123 +200 -24 +27 +273 +90 +171 +192 +352 +186 +277 +269 +359 +179 +277

-35 -19 -9 -237 -46 -336 -76 +7 -90 -45 -2 -144 -110 +27 -73 -19 -256 -96 -21 -162 -70 +32 -310 -275

-120 -119 -110 -323 -158 -331 -147 -126 -216 -126 -126 -206 -206 -107 -165 -133 -286 -217 -136 -252 -166 -110 -318 -355

-37 -6 -5 -181 -63 -266 -60 +24 -51 -50 -13 -163 -134 +24 -87 -30 -201 -58 -29 -119 -44 +45 -245 -213

+27 +136 +99 +160 -47 +50 +76 +209 +236 -3 +39 -188 -157 +119 -67 +14 +82 +224 +29 +162 +157 +216 +61 +154

+25 +67 +66 -84 +9 -197 +3 +117 +50 +9 +65 -110 -69 +110 -22 +43 -132 +42 +47 -31 +36 +141 -175 -115

-45 -9 -13 -141 -88 -191 -50 +17 -39 -60 -37 -171 -157 +3 -102 -52 -154 -44 -49 -93 -34 +32 -180 -166

Table A5. Bulk modulus K and shear modulus G [4] Elem.

K in 1010 Pa

G in 1010 Pa

Elem.

K in 1010 Pa

G in 1010 Pa

Ag Al As Au B Ba Be Bi C Ca Cd Ce Co Cr Cs Cu Dy Er Eu Fe Ga Gd Ge Hf Hg Ho In Ir K La Li Lu Mg Mn Mo Na

10.07 7.218 3.942 17.32 17.85 1.030 10.04 3.148 54.54 1.521 4.675 2.395 19.15 19.03 0.2030 13.10 3.844 4.108 1.472 16.83 5.690 3.834 7.724 10.89 2.825 3.968 4.109 35.51 0.3178 2.430 1.158 4.113 3.544 5.967 27.26 0.6817

2.865 2.659 1.462 2.757 20.31 0.4905 14.32 1.285 45.13 0.7358 2.413 1.197 7.642 11.67 0.0647 4.513 2.541 2.963 0.5886 8.152 3.747 2.227 3.924 5.297 1.001 2.668 0.3728 20.99 0.1275 1.491 0.4228 3.384 1.736 7.652 11.58 0.3434

Nb Nd Ni Os P Pb Pd Pm Pr Pt Pu Rb Re Rh Ru Sb Sc Si Sm Sn Sr Ta Tb Tc Th Ti Tl Tm U V W Y Yb Zn Zr

17.03 3.268 18.64 41.79 1.919 4.299 18.09 3.532 3.058 27.84 5.356 0.3145 37.18 27.06 32.09 3.829 5.729 9.888 2.941 11.09 1.162 20.01 3.993 29.72 5.429 10.52 3.593 3.970 9.879 16.20 32.33 3.662 1.327 5.985 8.335

3.747 1.452 7.505 20.99 0.7161 0.5396 5.111 1.668 1.354 6.102 4.375 0.1001 17.85 14.72 15.99 2.001 3.129 3.973 1.265 1.844 0.5229 6.867 2.286 14.22 2.786 3.934 0.2747 3.041 7.358 4.650 15.30 2.580 0.6965 3.718 3.414

Table A6. Surface energy  per m2 and vacancy formation enthalpy HV [1] Elem.



-2

Jm Ag Al As Au B Ba Be Bi C Ca Cd Ce Co Cr Cs Cu Dy Er Eu Fe Ga Gd Ge Hf Hg Ho In Ir K La Li Lu Mg Mn Mo Na

1.25 1.16 0.85 1.50 3.05 0.37 2.70 0.49 0.49 0.74 1.04 2.55 2.30 0.095 1.825 1.14 1.17 0.45 2.475 1.10 1.11 1.00 2.15 0.575 1.15 0.675 3.00 0.13 1.02 0.525 1.225 0.76 1.60 3.00 0.26

HV

kJ mol 90 63 50 100 95 62 90 40 65 41 120 135 120 23 100 120 120 63 140 48 120 55 180 40 120 46 190 24 120 40 125 50 90 200 32

-1

Elem.



-2

Jm Nb Nd Ni Os P Pb Pd Pm Pr Pt Pu Rb Re Rh Ru Sb Sc Si Sm Sn Sr Ta Tb Tc Th Ti Tl Tm U V W Y Yb Zn Zr

2.70 1.08 2.45 3.45 1.10 0.60 2.05 1.10 1.08 2.475