progress in materials scienceweb.mit.edu/ceder/publications/van_der_ven_ceder_2010.pdf · a. van...

Progress in Materials Science 55 (2010) 61–105

Contents lists available at ScienceDirect

Progress in Materials Science

journa l homepage : www.e lsev ie r . com/ loca te /pmatsc i

Vacancy mediated substitutional diffusion in binarycrystalline solids

Anton Van der Ven a,*, Hui-Chia Yu a, Gerbrand Ceder b, Katsuyo Thornton a

a Department of Materials Science and Engineering, University of Michigan, 2300 Hayward Street, Ann Arbor, MI, 48109, United Statesb Department of Materials Science and Engineering, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, MA, 02139, United States

a r t i c l e i n f o

Article history:Received 5 March 2009Received in revised form 23 June 2009Accepted 1 August 2009

0079-6425/$ - see front matter � 2009 Elsevier Ltdoi:10.1016/j.pmatsci.2009.08.001

* Corresponding author. Tel.: +1 734 615 3843.E-mail address: [email protected] (A. Van der V

a b s t r a c t

We describe a formalism to predict diffusion coefficients of substi-tutional alloys from first principles. The focus is restricted tovacancy mediated diffusion in binary substitutional alloys. Theapproach relies on the evaluation of Kubo-Green expressions ofkinetic-transport coefficients and fluctuation expressions of ther-modynamic factors for a perfect crystal using Monte Carlo simula-tions applied to a cluster expansion of the configurational energy.We make a clear distinction between diffusion in a perfect crystal(i.e. no climbing dislocations and grain boundaries that can act asvacancy sources) and diffusion in a solid containing a continuousdistribution of vacancy sources that regulate an equilibriumvacancy concentration throughout. A variety of useful metrics tocharacterize intermixing processes and net vacancy fluxes thatcan result in the Kirkendall effect are described and are analyzedin the context of thermodynamically ideal but kinetically non-idealmodel alloys as well as a realistic thermodynamically non-idealalloy. Based on continuum simulations of diffusion couples usingself-consistent perfect-crystal diffusion coefficients, we show thatthe rate and mechanism of intermixing in kinetically non-idealalloys is very sensitive to the density of discrete vacancy sources.

� 2009 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622. Substitutional solids: perfect crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

d. All rights reserved.

en).

http://dx.doi.org/10.1016/j.pmatsci.2009.08.001

mailto:[email protected]

http://www.sciencedirect.com/science/journal/00796425

http://www.elsevier.com/locate/pmatsci

62 A. Van der Ven et al. / Progress in Materials Science 55 (2010) 61–105

2.1. Phenomenological description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.1.1. Thermodynamics of a substitutional solid with vacancies. . . . . . . . . . . . . . . . . . . . . . . . 662.1.2. Atomic transport in a substitutional solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.2. Atomistic description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.2.1. Thermodynamics from first principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.2.2. Diffusion from first principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.3. Fickian flux expressions in a perfect crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.4. Metrics for substitutional diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.4.1. Generalized flux expressions for a kinetically ideal alloy . . . . . . . . . . . . . . . . . . . . . . . . 772.4.2. Generalized flux expressions for non-ideal binary alloys in the dilute vacancy limit . . . . . 782.4.3. Physical interpretation of � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.5. Classification of substitutional alloys based on kinetic metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.5.1. Thermodynamically ideal substitutional alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.5.2. Thermodynamically non-ideal substitutional alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3. The role of vacancy sources and sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.1. A continuous and dense distribution of sources and sinks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.1.1. The Darken approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.1.2. The Kirkendall effect and interdiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.2. Comparison between the perfect crystal and the uniform source-sink approximation . . . . . . . . 91
3.2.1. Thermodynamically ideal alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.2.2. Thermodynamically non-ideal alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.3. Discrete distribution of sources and sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Appendix B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Appendix C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
C.1. Scaling of �. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101C.2. Scaling of � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101C.3. Scaling of �þ and �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Appendix D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
1. Introduction

Substitutional diffusion in multi-component alloys is a kinetic phenomenon that plays an impor-tant role in solid-state phase transformations [1–11]. It also determines degradation rates of multi-layer structures characterized by large concentration gradients.

Theoretical and computational methods in materials research can provide essential guidance in theinterpretation of solid state phenomena and are increasingly capable of predicting materials proper-ties before experiments are ever performed. First-principles electronic structure methods [12], forexample, have reached a point where thermodynamic and kinetic properties of perfect crystals canbe predicted with reasonable accuracy. Phase diagrams of crystalline solids are now routinely calcu-lated by combining first-principles-parameterized lattice models with Monte Carlo simulations ormeanfield approximations such as the cluster variation method [13–36]. The statistical mechanicaltechniques used to calculate phase diagrams from first principles have also been implemented to pre-dict diffusion coefficients in multi-component solids [37–39]. However, these methods describe prop-erties of perfect crystals, whereas kinetic phenomena such as substitutional diffusion can be sensitiveto the presence of dislocations and grain boundaries.

This paper focuses on substitutional diffusion mediated by vacancies and seeks to (i) describe a for-malism that connects phenomenological transport coefficients in a binary alloy to well-defined quan-tities that can be calculated from first principles for a perfect crystal and (ii) to clarify the link betweensubstitutional diffusion in a perfect crystal (i.e., without dislocations and grain boundaries) with dif-fusion in a real solid containing vacancy sources and sinks. We exclude in our analysis fast diffusionalong grain boundaries and dislocation cores, which can be considered separately.

Nomenclature

A slow diffusera atomic hop distanceB fast diffuserCi concentration of the ith diffusing species (i = A, B or V) defined as the number of atoms per

unit volumebCB fast diffuser concentration normalized by the lattice densitybCpcB non-dimensionalized fast diffuser concentration calculated in a perfect-crystal modelbCcsB non-dimensionalized fast diffuser concentration calculated in a continuous-vacancy-

source approximationDi self-diffusion coefficient of the ith diffusing speciesD�i tracer diffusion coefficient of the ith diffusing speciesDij diffusion coefficient relating the ith flux to the jth concentration gradient in a perfect

crystalD diffusion coefficient matrix of a perfect crystaleD interdiffusion coefficient in the continuous-vacancy-source approximationd dimensionality of the substitutional networkEð~rÞ ground state energy of configuration ~rEs energy of microstate sf correlation factor in a single component solidfi correlation factor of the ith diffusing species (i = A, B or V)Ji flux of the ith diffusing species (i = A, B or V) defined as the number of atoms of species i

crossing a unit area per unit timebJ i intermixing flux (i = A or B) in a perfect crystaleJ i interdiffusion flux (i = A or B) within the continuous-vacancy-source approximationG Gibbs free energy of the systemg Gibbs free energy per crystal siteh Planck’s constantkB Boltzmann’s constantLij kinetic-transport coefficient relating the ith flux to the jth driving forceeLij LijXkBTbL non-dimensionalized length of the planar diffusion coupleM total number of crystal sites of a perfect crystalNi number of the ith diffusing species (i = A, B or V)Ng number of grains in the planar diffusion coupleN

DbC norm measuring the deviation of concentration evolution of a diffusion couple containingexplicit vacancy sources from that of a continuous-vacancy-source approximation

P pressurepð~rÞ probability of a configuration ~rQ ðhN2

Ai � hNAi2ÞðhN2Bi � hNBi2Þ � ðhNANBi � hNAihNBiÞ2

T absolute temperaturet timet non-dimensionalized time ðt ¼ ta2CA=L2ÞV vacancyVo effective cluster interaction for the empty clusterVa effective cluster interaction for a cluster of sites av l crystal frame velocity relative to the laboratory framexi concentration variable defined as the fraction of crystal sites occupied by the ith speciesy non-dimensionalized coordinate variable in a planar diffusion coupleZ partition functionz coordination number of each substitutional siteCi exchange frequency of the ith atomic species with an nearest neighbor vacancy (i = A or B)CI ! F frequency of a hop connecting configuration ~rI to configuration ~rF

A. Van der Ven et al. / Progress in Materials Science 55 (2010) 61–105 63

ci activity coefficient of the ith speciesDEb activation barrier of a diffusive hopD~Ri

f vector linking the end points of the trajectory of atom f of atomic species i after time td metric for the susceptibility to the Kirkendall effect in a perfect crystal ðd ¼ DBB � DAAþ

DAB � DBAÞdij Kronecker deltadlV¼0 metric for the susceptibility to the Kirkendall effect at lV ¼ 0, i.e. within the continuous-

vacancy-source approximation� deviation parameter ð� ¼ /ð1� /Þd2=ðkþk�ÞÞeH Hessian of free energy divided by kBT (thermodynamic factor)kþ large eigenvalue of the perfect-crystal diffusion coefficient matrixk� small eigenvalue of the perfect-crystal diffusion coefficient matrixli chemical potential of the ith diffusing species (i = A, B or V) defined per atom~li chemical potential of the ith atomic species with respect to vacancies (i = A or B)m� vibration prefactormi normal-mode vibrational frequency of a stable configuration~mk non-imaginary normal-mode vibrational frequency at the saddle point in the energy sur-

face of a hopq geometric factor that depends on crystal structureri occupation variable for site i~ri collection of all occupation variables/ metric for the fraction of the flux of the fast diffuser exchanging with vacancy flux in the

absence of intermixing ð/ ¼ DBA=ðDAB þ DBAÞÞ/a cluster expansion basis function associated with cluster aX volume per substitutional siteh i ensemble averagehD2

iji hNiNji � hNiihNji


Connecting the characteristics of correlated atomic hops in non-dilute alloys to the diffusion coef-ficients appearing in macroscopic flux expressions requires a statistical mechanical theory that canaccurately account for short- and long-range ordering tendencies within the solid. Several such theo-ries have been developed relying on meanfield approximations, including the path probability methodof Kikuchi and Sato [40–44] and the self-consistent mean field theory of Nastar et al. [45–47]. An alter-native approach is the use of Monte Carlo simulations to sample atomic trajectories explicitly [48–55,37–39]. Such simulations rigorously capture the short- and long-range order that prevails in mostmulti-component solids as well as correlations between successive hops due to differences in hop fre-quencies among the components of the solid.

The calculation of kinetic-transport coefficients for diffusion is greatly simplified with the fluctua-tion–dissipation theorem of statistical mechanics, which lets us link macroscopic diffusion coefficientsto ensemble averages of the trajectories of diffusing atoms in a solid at equilibrium [56]. Mathematicaltechniques that connect phenomenological kinetic coefficients to fluctuations at equilibrium are com-monly referred to as Kubo-Green methods [57–69], and enable the calculation of diffusion coefficientsusing kinetic Monte Carlo simulations [70,71] to sample representative atomic trajectories in a crystalat equilibrium [52–54,37,38,55,39]. These calculated diffusion coefficients can then be implementedin continuum simulations to study the evolution of diffusion couples or to predict the rates of diffu-sional phase transformations [72–74].

The standard treatment of substitutional diffusion relies on the assumption that there are sufficientcrystalline defects (i.e. dislocations and grain boundaries) uniformly distributed to regulate an equi-librium vacancy concentration throughout the solid [2,75,69,76]. Theoretical treatments of substitu-tional diffusion, however, are often only tractable for perfect crystals. Furthermore, as materials areincreasingly synthesized with precisely controlled microstructures, gross assumptions about a contin-uous and dense distribution of vacancy sources and sinks become questionable. Most materials inmodern technologies are likely somewhere in between the two extremes, with the continuous


source/sink treatment approximating diffusion at high temperatures and the perfect crystal descrip-tion approximating diffusion at low temperatures.

We can distinguish between two modes of diffusion in substitutional binary alloys. One corre-sponds to the process of intermixing between the two components of the alloy while the other corre-sponds to the emergence of unbalanced vacancy fluxes. The intermixing diffusion mode is responsiblefor the homogenization of any non-uniform distribution of the components of the alloy. An unbal-anced vacancy flux can arise from a concentration gradient among the components of the alloy ifone component has a higher exchange frequency with vacancies than the other. Vacancy fluxes inthe presence of active vacancy sources cause the crystal to drift relative to a laboratory frame of ref-erence, as was first observed by Kirkendall [78,77], often leading to swelling and void formation. Thepresence of vacancy sources and sinks can also play an important role in mediating intermixing. Themechanisms of intermixing in a perfect crystal therefore differ significantly from that in a solid con-taining many vacancy sources and sinks, especially in solids exhibiting a strong tendency for the Kir-kendall effect. (While in the strictest sense, the term ’Kirkendall effect’ refers specifically to the drift ofmarkers in a diffusion couple, here we will use the term more broadly to refer to any phenomenacaused by vacancy fluxes that arise from differences in vacancy exchange frequencies among the var-ious components of a substitutional alloy.)

The structure of this paper can be summarized as follows. We start with the development of a for-malism to calculate diffusion coefficients from first principles in a defect-free, binary substitutionalsolid containing a dilute concentration of vacancies. This requires a precise description of thethermodynamics of a binary substitutional alloy containing vacancies along with the kinetics usingKubo-Green expressions, first derived by Allnatt for substitutional solids [67,68], to connect phenom-enological kinetic-transport coefficients to density fluctuations at the atomic scale. Next we describeconvenient metrics, derived from the matrix of diffusion coefficients, that characterize intermixingand net vacancy fluxes in a perfect-crystal substitutional solid. Here, we build on earlier work of Kehret al. [51], and analyze the behavior of thermodynamically ideal model systems as well as a thermo-dynamically non-ideal system. In this context, the analysis of the characteristics of substitutional dif-fusion in thermodynamically ideal but kinetically non-ideal alloys is greatly facilitated by thederivations of Moleko et al. [79] of analytical expressions for phenomenological transport coefficients.After a comprehensive analysis of substitutional diffusion in a perfect crystal, we consider substitu-tional diffusion under the assumption that there is a sufficiently high and uniform density of vacancysources and sinks to regulate an equilibrium vacancy concentration throughout the solid. Within thisapproximation, invoked in most textbook treatments of substitutional diffusion, we link the variousdiffusion coefficients, including self-diffusion coefficients and the interdiffusion coefficient, to well-defined quantities that can be calculated from first principles for perfect crystals. We conclude withan analysis of diffusion in a solid containing discrete vacancy sources and sinks, showing that theunbalanced vacancy fluxes responsible for the Kirkendall effect can play an important role in theprocess of intermixing as the density of vacancy sources and sinks increases.

2. Substitutional solids: perfect crystals

2.1. Phenomenological description

Diffusion in the solid state is a kinetic process that proceeds not too far from thermodynamic equi-librium but rather evolves between states that are in local equilibrium. A solid removed from equilib-rium is in local equilibrium if thermodynamic quantities such as temperature and chemical potentialsare well-defined and constant within any macroscopically small subregion of the solid even when theyvary from subregion to subregion. The theory of irreversible thermodynamics, which describes kineticprocesses within the local equilibrium approximation, then stipulates that the flux of diffusing atomsat constant temperature is driven by a gradient in the chemical potentials of the diffusing species [80].Before formulating the relevant flux equations for substitutional diffusion, it is useful to first consideraspects of the thermodynamics of a substitutional alloy containing a dilute concentration of diffusion-mediating vacancies. In this section, we will restrict ourselves to what will be referred to throughout


as a perfect crystal, defined as a multi-component crystal containing vacancies but no extended defectssuch as dislocations and grain boundaries that can serve as vacancy sources and sinks.

2.1.1. Thermodynamics of a substitutional solid with vacanciesA binary substitutional alloy with vacancies has three components that include A, B and vacancies

V. The number of vacancies is conserved in the absence of vacancy sources and sinks. The sum of thenumber of atoms and vacancies NA þ NB þ NV then equals the total number of crystal sites M of thesubstitional alloy. In terms of concentration variables xi ¼ Ni=M, the conservation of lattice sites is ex-pressed as xA þ xB þ xV ¼ 1. The substitutional alloy has a free energy GðNA;NB;NV Þ from which thechemical potentials of each component can be calculated according to

Fig. 1.composdeterm

li ¼@G@Ni

ð1Þ

where i refers to A, B or V. The partial derivatives are taken at constant temperature, T, pressure, P, andNj with j – i. The chemical potential li corresponds to the change in free energy of the solid as com-ponent i is added to the solid. Since the other components j – i are held fixed, adding more of i to asubstitutional can only be achieved by extending the crystalline network of the solid; that is, byincreasing M [81–83].

Normalizing G by the number of crystal sites M yields a free energy per crystal site gðxA; xBÞ. Interms of this quantity, the chemical potentials li can be written as [84]

li ¼ gðxA; xBÞ þ ðdiA � xAÞ@g@xAþ ðdiB � xBÞ

@g@xB

; ð2Þ

where dij is the Kronecker delta. The chemical potentials li have a convenient graphical interpretationin a gðxA; xBÞ versus xA and xB plot: the li are equal to the intercepts with the axes xi ¼ 1 of the planetangent to g at the overall concentration xA; xB. This is illustrated in Fig. 1a. Free energies of a perfectmulti-component crystal can be calculated from first principles. The approach relies on a clusterexpansion of the configurational energy combined with for example the cluster variation method orMonte Carlo simulations to calculate thermodynamic properties [13,20,29,36,85]. More details onthe first-principles computation of the free energy of a substitutional alloy are provided in Section2.2.1.

Real solids contain dislocations and grain boundaries that act as vacancy sources and sinks. In ther-modynamic equilibrium, the chemical potential of vacancies is then equal to zero as vacancies are no

A

B

VxA

xB

xV

μA

μB

μV

(a) (b)

μV = 0

xA(xB)μV=0

g(xA, xB)

A

B

VxB

μA

μB

g(xA, xB)

(a) Schematic illustration of the Gibbs free energy of a perfect crystalline binary solid containing vacancies. (b) Theition of an alloy with an equilibrium vacancy concentration xAðxBÞlV¼0 (dashed line in the xA—xB composition plane) isined graphically by the plane passing through lV ¼ 0 and simultaneously tangent to gðxA; xBÞ.


longer a conserved species [83]. The equilibrium vacancy concentration xV is determined by the con-dition that lV ðxA; xBÞ ¼ 0. Graphically, this constraint manifests itself as the projection in compositionspace (xA, xB) of the line formed by the contact of the envelope of planes passing through lV ¼ 0 atxV ¼ 1 and simultaneously tangent to the gðxA; xBÞ surface. This is schematically illustrated inFig. 1b, where the dashed line on the gðxA; xBÞ surface corresponds to the free energy of the alloy withan equilibrated vacancy concentration. The trajectory in composition space of the alloy having anequilibrium vacancy concentration must satisfy the constraint that dlV ¼ 0. Written explicitly interms of the independent variables xA and xB,

@lV

@xAdxA þ

@lV

@xBdxB ¼ 0; ð3Þ

and substituting expression (2) for lV into (3) yields

dxA

dxB¼ �

xA@2g

@xB@xA

!þ xB

@2g@x2

B

!

xA@2g@x2

A

!þ xB

@2g@xA@xB

! : ð4Þ

This differential equation in terms of second derivatives of the free energy gðxA; xBÞ determines a func-tional relationship between xA and xB for an alloy having a constant vacancy chemical potential,including alloys having an equilibrium vacancy concentration for which lV ¼ 0. This functional rela-tionship will be denoted by xAðxBÞlV¼0. In using free energies for a perfect crystal to determine theequilibrium vacancy concentration, contributions from grain-boundary free energies and disloca-tion-line free energies are neglected.

The second derivatives of gðxA; xBÞ appearing in Eq. (4) play a central role in the formalism of sub-stitutional diffusion and are elements of the Hessian of the free energy

kBT eH ¼@2g@x2

A

@2g@xB@xA

@2g@xA@xB

@2g@x2

B

0BBB@1CCCA: ð5Þ

The matrix eH will be referred to as the thermodynamic factor for perfect-crystal diffusion.

2.1.2. Atomic transport in a substitutional solidIn writing flux expressions for a substitutional solid, we start with a perfect crystal. The vacancy

concentration in a perfect crystalline network, in which the number of sites is conserved, can locallydeviate from its equilibrium value, leading to a vacancy chemical potential that is locally not zero. Thisalso occurs in crystalline solids with a low density of defects in regions away from vacancy sourcesand sinks. Such gradients in vacancy chemical potential drive fluxes in vacancies as well as A and Batoms. For a conserved crystalline network, the flux equations for substitutional diffusion mediatedby vacancies become [75,51]

JA ¼ �LAArlA � LABrlB � LAVrlV

JB ¼ �LBArlA � LBBrlB � LBVrlV

JV ¼ �LVArlA � LVBrlB � LVVrlV ;

ð6Þ

where the fluxes are here defined as the number of atoms (as opposed to mole of atoms) crossing aunit area per unit time. The chemical potentials in Eq. (6) are defined according to Eqs. (1) and (2).Due to Onsager’s reciprocity theorem [57,58,80], the kinetic-transport coefficients form a symmetricmatrix

LAB ¼ LBA; LAV ¼ LVA; LVB ¼ LBV : ð7Þ


The flux equation (6) can be simplified further by exploiting properties resulting from the conserva-tion of crystal sites in a perfect crystal

JA þ JB þ JV ¼ 0: ð8Þ

Substituting Eq. (6) into Eq. (8) and invoking the Onsager reciprocity relations, Eq. (7), leads to linearrelations between the different kinetic coefficients of the form

LAV ¼ LVA ¼ �ðLAA þ LABÞLBV ¼ LVB ¼ �ðLBA þ LBBÞ;

ð9Þ

which can be used to simplify the flux Eq. (6) to yield

JA ¼ �LAAr~lA � LABr~lB

JB ¼ �LBAr~lA � LBBr~lB;ð10Þ

where the driving forces are now the gradients of differences in the chemical potentials of i and thevacancies

~li ¼ li � lV ; ð11Þ

with i=A or B. This form of the diffusion potential is specific for substitutional diffusion and arises fromthe constraints of a crystalline network [81,82]. (In the formalism of Andersson and Ågren [72], thesedriving forces emerge within the volume-fixed frame if vacancies are taken as the reference elementand assigned a volume equal to that of A and B). The substitution of Eq. (2) into Eq. (11) shows that thediffusion potentials are equal to the derivatives of the free energy gðxA; xBÞ with respect to the concen-tration of the ith component

~li ¼@g@xi

ð12Þ

The emergence of ~li as the driving force for substitutional diffusion in a binary system can be under-stood on physical grounds. To this end, it is useful to view diffusion as the redistribution of atoms be-tween different subregions of the solid. As a result of the conservation of sites in a perfect crystal, theaccommodation of excess atoms diffusing to a region from neighboring subregions can only occurthrough an exchange with vacancies. Thus, one subregion gains vacancies while the other loses them.The change in free energy in each subregion due to diffusion occurs by adding or removing atoms atthe expense of vacancies. Hence, the ~li ¼ li � lV instead of the li are the relevant driving force foratomic transport.

2.2. Atomistic description

2.2.1. Thermodynamics from first principlesThe formal link between thermodynamic properties of a binary crystalline alloy, as encapsulated in

the Gibbs free energy G, and the electronic structure of the solid resides with the partition function Zaccording to

G ¼ �kBT ln Z ð13Þ

where at constant temperature T, pressure P (which we effectively take to be equal to zero) and com-positions, the partition function becomes

Z ¼X

s

exp�Es

kBT

� �: ð14Þ

The sum in Eq. (14) extends over all microstates s having energy Es that are accessible to the alloythrough thermal fluctuations under the externally imposed thermodynamic boundary conditions.The energies Es correspond to the eigenvalues of the Schrödinger equation of the solid. In a binary sub-stitutional alloy, the most important microstates contributing to the partition function arise from


electronic excitations, vibrational excitations and configurational degrees of freedom associated withall possible ways of distributing A, B and V over the M sites of the crystal.

Diffusion at the atomic scale results from configurational rearrangements of the various compo-nents of the solid. Hence any atomic scale description of diffusion in an alloy requires an explicit treat-ment of configurational degrees of freedom. Therefore, it is useful to assign occupation variables, ri, toeach substitutional site i of the solid, which is for example +1 if an A atom resides there, �1 if a B atomresides there and 0 if the site is vacant. The collection of all occupation variables,~r ¼ ðr1; . . . ;ri; . . .rMÞ then uniquely characterizes the arrangement of A, B and V components overthe M sites of the substitutional solid. Between successive diffusive hops leading to configurationalrearrangements, the solid will sample a large number of vibrational and electronic excitations. Recog-nizing the large difference in characteristic time scales of the ‘‘fast” vibrational and electronic transi-tions versus the ‘‘slow” configurational rearrangements, it is convenient to introduce a constrainedfree energy by summing over the vibrational and electronic excitations at fixed configuration ~raccording to [17,85]

Gð~r; TÞ ¼ �kBT lnX

s¼vib;elec

exp�Esð~rÞ

kBT

� �" #: ð15Þ

In terms of this coarse grained free energy, the partition function can be reduced to that of a latticemodel according to [85]

Z ¼X~r

exp�Gð~r; TÞ

kBT

� �; ð16Þ

where now the summation extends over the configurational degrees of freedom only. The probabilitythat the solid has a particular configuration ~r then becomes

pð~rÞ ¼exp

�Gð~r; TÞkBT

� �Z

: ð17Þ

We point out that the introduction of coarse grained free energies Gð~r; TÞ is only meaningful as long asevery configuration of A, B and V over the substitutional sublattice is mechanically stable (i.e., itcorresponds to a local minimum with respect to vibrational degrees of freedom). While many hightemperature phases observed in nature have been shown to be mechanically unstable at zero Kelvin[86–88], forming at finite temperature through anharmonic vibrational degrees of freedom [89–91],here we will restrict the treatment to the large class of substitutional alloy phases that are mechanicallystable at zero Kelvin.

The prediction of thermodynamic properties of binary alloys becomes straightforward with theabove coarse graining procedure using standard statistical mechanical tools such as cluster expansions[13,20] and Monte Carlo simulations [92]. The Metropolis Monte Carlo algorithm [93] samples config-urational microstates with a frequency given by Eq. (17), thereby enabling an explicit calculation ofthermodynamic averages of extensive quantities such as the internal energy and number of atomswhen controlling chemical potentials in grand canonical Monte Carlo simulations. With free energyintegration techniques, it is then possible to calculate the free energy of the alloy as a function of tem-perature and composition [29,36]. An essential input for these simulations, however, is knowledge ofthe functional dependence of the coarse grained free energies Gð~r; TÞ on configuration ~r.

The coarse grained free energies Gð~r; TÞ can be calculated within the (quasi) harmonic approxima-tion by integrating over the phonon densities of state at fixed configuration ~r determined with first-principles electronic structure methods based on density functional theory (DFT) [94–97] as describedfor example in Ref. [98]. The electronic contribution to Gð~r; TÞ can also be determined by integratingover the electronic densities of state calculated with DFT; however, this approach provides at best onlyqualitative predictions as there is no formal justification that DFT should be able to predict excitedelectronic states. A direct first-principles calculation of Gð~r; TÞ for all configurations ~r sampled inMonte Carlo simulations is intractable. Therefore, cluster expansion techniques [13,20] are needed


to extrapolate the values of Gð~r; TÞ calculated from first principles for a small number of configura-tions to all other configurations accessible in Monte Carlo simulations.

The cluster expansion formalism provides a set of basis functions to support any property of thecrystal that uniquely depends on how the various components of the solid are arranged [13]. For a bin-ary A–B alloy, for example, these basis functions /a are simply the products of occupation variables ri

of each site of a cluster a of the crystal, where a corresponds to a pair cluster, triplet cluster, quadru-plet cluster, etc.:

/að~rÞ ¼Yi2a

ri: ð18Þ

Defining a scalar product over configuration space ~r between two functions of configuration, f ð~rÞ andgð~rÞ, according to

hf � gi ¼ 12M

X~r

f ð~rÞgð~rÞ; ð19Þ

it can be shown that the collection of cluster functions /að~rÞ then form a complete and orthonormalbasis [13]. Hence any property that depends on configuration ~r, including the coarse grained free en-ergy Gð~r; TÞ, can be expanded in terms of the polynomial basis functions /aðrÞ according to

Gð~r; TÞ ¼ Vo þX

aVa/að~rÞ ð20Þ

where the expansion coefficients, Vo and Va, that depend only on temperature are formally related toGð~r; TÞ through the scalar product [13,20]

Va ¼ hGð~r; TÞ � /að~rÞi: ð21Þ

While the expansion extends over all possible 2M clusters, Eq. (20) can usually be truncated above amaximal sized cluster, while symmetry considerations drastically reduces the number of independenteffective cluster interaction (ECI) coefficients Va. The truncated expansion is then a convenient tool torapidly and accurately calculate values of Gð~r; TÞ for arbitrary configurations ~r sampled in Monte Car-lo simulations.

A common practice is to neglect vibrational and electronic excitations in first-principles predictionsof thermodynamic properties of binary alloys. Within this approximation, the coarse grained freeenergies Gð~r; TÞ then reduce to the ground state energies, Eð~rÞ, for each configuration ~r (i.e., the fullyrelaxed energy of configuration ~r). Furthermore, the ECI, Va, of the cluster expansion, Eq. (20), arerarely calculated using the scalar product, (21), but rather with a least-square fitting procedure,whereby the coefficients of a truncated cluster expansion are fit to reproduce first-principles energiesof a subset of configurations ~r. Several fitting schemes based on the minimization of a cross-validationscore and the use of genetic algorithms to determine the optimal truncation have been proposed andare in common use [28,33].

A treatment of diffusion in a binary substitutional alloy needs to explicitly account for the presenceof vacancies, which renders the solid a ternary mixture. An accurate description of the interactionsamong A, B and V would in principle require a ternary cluster expansion. However, since vacanciesin binary substitutional alloys are often very dilute, it is possible to neglect interactions betweenvacancies and use a local cluster expansion in combination with a standard binary cluster expansionfor A–B disorder to describe interactions between an isolated vacancy and A and B atoms. This is de-scribed in detail in ref. [31,36].

While cluster expansion and Monte Carlo techniques are required to predict thermodynamic prop-erties from first principles for real binary alloys, for an ideal solution the partition function can beevaluated in closed form, and an analytical expression for the Gibbs free energy can then be derivedusing Eq. (13). The underlying assumption that defines an ideal solution, also commonly referred to asa random alloy, is the absence of an energetic preference for any arrangement of atoms: each arrange-ment of atoms in the solid has the same energy and is therefore equally likely to occur as the solidfluctuates from one configurational microstate to the next. (The cluster expansion of an ideal solution


consists of only point cluster terms with all ECI coefficients for multi-site clusters (i.e., clusters withtwo or more sites) being equal to zero). With this assumption, the partition function, Eq. (16), reducesto a sum of M!=ðNA!NB!NV !Þ identical terms and the free energy of the alloy becomes

gðxA; xB ¼ xVDgV þ kBT½xAln xA þ xBln xB þ xV ln xV � ð22Þ

where xV ¼ 1� xA � xB and the free energies of the pure elements are taken as reference states. Thevacancy formation energy, DgV , is independent of composition and configuration in a random alloy.Using Eq. (2) to derive an expression for the chemical potentials yields

lA ¼ kBT ln xA; lB ¼ kBT ln xB; lV ¼ DgV þ kBT ln xV : ð23Þ

An ideal solution serves as a useful reference alloy with which to compare thermodynamically non-ideal alloys.

As noted in Section 2.1.1, the Hessian of the free energy, gðxA; xBÞ plays a central role in substitu-tional diffusion. Although the elements of the Hessian can be calculated from the free energy by takingthe second derivative of gðxA; xBÞ with respect to the concentration variables xA and xB, it is in generalmore convenient to evaluate them directly in Monte Carlo simulations by tracking fluctuations in thenumber of A and B atoms at constant chemical potentials lA and lB (or equivalently at constant ~lA and~lB). Indeed, it can be shown that (see Appendix A) the thermodynamic factor matrix ~H, defined by Eq.(5), is related to the variances of NA and NB in the grand canonical ensemble within a crystal with fixednumber of crystal sites M according to

~H ¼ MQ

hD2BBi �hD2

ABi�hD2

ABi hD2AAi

!; ð24Þ

where

hD2iji ¼ hNiNji � hNiihNji; i; j ¼ A;B;

and

Q ¼ ðhN2Ai � hNAi2ÞðhN2

Bi � hNBi2Þ � ðhNANBi � hNAihNBiÞ2:

The angular brackets hi denote ensemble averages in the usual statistical mechanical sense [56,99].For an ideal solution, Monte Carlo simulations are not necessary to obtain the thermodynamic fac-

tor matrix. Direct differentiation of the ideal solution free energy expression, Eq. (22), yields

~H ¼

1� xBð ÞxAxV

1xV

1xV

1� xA

xBxV

0BB@1CCA: ð25Þ

For non-ideal binary alloys, eH will deviate from this expression, but will nevertheless be dominated bythe 1=xV factor as xV ! 0.

2.2.2. Diffusion from first principlesDiffusion in crystalline solids can be treated at the atomic scale as arising from a sequence of ele-

mentary stochastic hop events. Atoms of a solid spend most of their time in the vicinity of crystallo-graphically well-defined positions, undergoing only small excursions due to thermal vibrations. Veryoccasionally, however, atoms adjacent to a vacant site follow a trajectory that takes them from theiroriginal site to a neighboring vacant site. This is referred to as a diffusive hop, and is a rare event on thetime scale of typical atomic vibrations. The hop results in a change of the atomic configuration withinthe crystal and can be represented by a reaction equation ~rI ! ~rF , where ~rI corresponds to the con-figuration in the initial state before the hop and ~rF is the configuration in the final state after the hop.The frequency, CI ! F , with which the solid in configuration ~rI transforms into configuration ~rF as aresult of an elementary hop can be estimated in a statistical mechanical sense using transition statetheory [100,101],


CI ! F ¼ kBTh

exp �eGð~rI : ~rF ; TÞ � Gð~rI; TÞ

kBT

!ð26Þ

where h is Planck’s constant and eGð~rI : ~rF ; TÞ is a constrained free energy of the solid obtained by sum-ming over vibrational and electronic degrees of freedom in a hypersurface of phase space that passesthrough the saddle point separating ~rI from ~rF [101]. As before, the free energy Gð~rI; TÞ of Eq. (26) cor-responds to the coarse grained free energy of configuration ~rI defined by Eq. (15). While there remainssome ambiguity as to an exact definition of the free energy, eGð~rI : ~rF ; TÞ, in the activated state [102], amore precise expression for the hop frequency in which all terms are accessible from first principlesemerges within the harmonic approximation in the high temperature limit (i.e., above the Debye tem-perature). In this limit, Eq. (26) reduces to the familiar form derived by Vineyard [103].

CI ! F ¼ m� exp�DEb

kBT

� �ð27Þ

where DEb is an activation barrier for a diffusive hop and corresponds to the difference in energy be-tween the saddle point of the transition from ~rI to ~rF and the fully relaxed energy in configuration ~rI

[103]. The vibrational prefactor is equal to

m� ¼Q3N�3

l¼1 mlQ3N�4k¼1 ~mk

ð28Þ

where N ¼ NA þ NB corresponds to the total number of A and B atoms in the crystal. The frequencies ml

are the 3N � 3 normal-mode vibrational frequencies of configuration ~rI while the ~mk are the 3N � 4non-imaginary normal-mode vibrational frequencies at the saddle point between ~rI and ~rF . In non-dilute alloys, both the activation barriers DEb and the vibrational prefactors m� will be a function ofthe local configuration surrounding the migrating atom. Here again, cluster expansions can be usedfor example to parameterize the dependence of the activation barrier on local order/disorder [37–39].

Macroscopic diffusion arises from a succession of many atomic hops over time scales that are sig-nificantly larger than those characterizing elementary hops. In a substitutional alloy, the atoms of thecrystal are usually redistributed through thermally activated exchanges with a dilute concentration ofvacancies wandering stochastically throughout the solid. The various components of the alloy willtypically have different exchange frequencies with vacancies, leading to complex correlations betweensuccessive hops [104,105] that are further complicated by any thermodynamic tendency for short- orlong-range order [38].

The time evolution of local concentration inhomogeneities and variations in short- or long-rangeorder is conveniently encapsulated in the time-dependent probability distribution, pð~r; tÞ, definedas the probability of observing a configuration ~r at time t. The time dependence of this probability dis-tribution is determined by a master equation

dpð~rI; tÞdt

¼X

~rF – ~rI

CF ! I � pð~rF ; tÞ �X

~rF – ~rI

CI ! F � pð~rI; tÞ ð29Þ

which states that the rate of change of pð~rI; tÞ depends on the fraction of equivalent systems transi-tioning from configuration ~rF into configuration ~rI minus the fraction of equivalent systems leavingconfiguration ~rI . Once the solid reaches thermodynamic equilibrium, the probability distributionpð~r; tÞ becomes time invariant and reduces to Eq. (17) predicted by equilibrium statistical mechanics[69]. While the right hand side of the master equation Eq. (29) is equal to zero in thermodynamic equi-librium, the principle of detailed balance is even more stringent, imposing a constraint on individualtransitions according to

CI ! F � pð~rIÞ ¼ CF ! I � pð~rFÞ; ð30Þ

which states that the rate of transitions from ~rI to ~rF is equal to the rate of the reverse transition.Inspection of Eqs. (17) and (26) shows that detailed balance is satisfied when using transition statetheory to estimate hop frequencies C.


Ultimately we are interested in linking macroscopic metrics of atomic mobility, such as the kinetic-transport coefficients Lij, to hop characteristics as well as short- and long-range order tendencies at theatomic scale. An important property of kinetic processes linking states that are in local equilibrium isthat the kinetic parameters describing non-equilibrium phenomena can be determined by consideringfluctuations at equilibrium [57,58,80]. The master equation (29) and the principle of detailed balance(30) shows that even though the average short- or long-range order is invariant with time in thermo-dynamic equilibrium, the instantaneous arrangements of the atoms in a solid are constantly evolvingas a result of diffusive hops. With what are often referred to as Kubo-Green linear response methods, itis possible to link phenomenological kinetic-transport coefficients with fluctuations in atomicarrangements that occur at equilibrium [57,58,61–66]. In the context of diffusion in crystalline solids,Allnatt [67–69] derived expressions for the kinetic coefficients Lij, showing that

Lij ¼1

XkBTeLij; ð31Þ

where X is the volume per substitutional site and

~Lij ¼P

fD~Ri

fðtÞ� � P

nD~Rj

nðtÞ� �D E

ð2dÞtM : ð32Þ

Here, D~Rif is the vector linking the end points of the trajectory of atom f of atomic species i (i = A or B)

after time t. The parameter d denotes the dimension of the substitutional network (e.g., for a three-dimensional substitutional network, d = 3), and M is equal to the number of substitutional sites inthe crystal. The brackets indicate an ensemble average performed at equilibrium. The kinetic coeffi-cients eLij have the units of a diffusion coefficient (e.g., cm2=s).

As with thermodynamic properties, it is useful to consider the behavior of the kinetic-transportcoefficients for a thermodynamically ideal alloy. Fig. 2 illustrates the variation of the different ki-netic-transport coefficients for three thermodynamically ideal fcc alloys where A and B atoms havedifferent exchange frequencies with a vacancy, Ci. Analytical expressions for eLij have been derivedfor binary random alloys [75,51,79]. For an alloy that is both thermodynamically and kinetically ideal,i.e., CA ¼ CB ¼ C, the analytical expressions for the kinetic coefficients can be expressed as [75,51]:

eLAA ¼ xV xAqa2C 1� xB

1� xVð1� f Þ

� �eLBB ¼ xV xBqa2C 1� xA

1� xVð1� f Þ

� �eLAB ¼

xV xAxB

1� xVqa2Cð1� f Þ:

ð33Þ

Here, a is the hop distance and q is a geometric factor determined by the crystal structure of the solid(for a crystal that is also a Bravias lattice, q ¼ z=2d, where z is the coordination number of each siteand d is the dimensionality of the crystal). The parameter f is the correlation factor for tracer diffusion.In a multi-component solid, each component i can be assigned a correlation factor defined as

fi ¼hðD~RiÞ2i

na2 ; ð34Þ

where D~Ri connects the endpoints of the trajectory of a particle of species i after n hops. The bracketsagain denote an ensemble average in thermodynamic equilibrium. In an ideal alloy in which CA ¼ CB,the kinetic behavior of A and B are identical and fA ¼ fB ¼ f . It is this correlation factor, f, that appears inEq. (33).

For random alloys where CA – CB, the analytical expressions for eLij derived by Moleko et al. [79]become more complicated (see Appendix B). In the most general case, the kinetic coefficients eLij

can also be calculated by numerically evaluating the Kubo-Green expressions, Eq. (32), with kineticMonte Carlo simulations to sample representative trajectories of migrating atoms at equilibrium. Ina kinetic Monte Carlo simulation [70,71], atoms exchange sites with adjacent vacant sites with fre-quencies Ci and the vectors connecting the end points of their trajectory are calculated as a function

10-12

10-10

10-08

10-06

0 0.2 0.4 0.6 0.8 1

cm2 /s

Concentration of B

LAA~

LBB~

LAB~

0 0.2 0.4 0.6 0.8 1

LBB~

LAA~

LBB~

LAA~

LAB

0 0.2 0.4 0.6 0.8 1

(a)

(b)

(c)

10-12

10-10

10-08

10-06

cm2 /s

10-12

10-10

10-08

10-06

cm2 /s

~

LAB~

Concentration of B

Concentration of B

Fig. 2. Kinetic-transport coefficients eLij for a random fcc alloy at 600 K with a vacancy concentration of 0.002, a cubic latticeparameter of 4.0 Å, a vibrational prefactor m� � 4� 1013 Hz and an activation barrier of 600 meV for an A hop into a vacancy for(a) CB ¼ CA , (b) CB ¼ 10� CA , (c) CB ¼ 100� CA . The solid line corresponds to the analytical kinetic transport coefficientsderived by Moleko et al. [79] and the circles correspond to values calculated with kinetic Monte Carlo simulations.


of time to enable the numerical evaluation of eLij according to Eq. (32). As an example, Fig. 2a illustratesthe variation of eLAA; eLBB and eLAB for a random fcc alloy where CB ¼ CA, comparing the analytical


expressions, Eq. (33), with values obtained with kinetic Monte Carlo simulations. Fig. 2b and c showthe kinetic coefficients ~Lij for a random fcc alloy in the dilute vacancy limit evaluated with the analyt-ical expressions derived by Moleko et al. [79] and approximated with kinetic Monte Carlo simulations.Fig. 2b is for a random alloy with CB ¼ 10� CA, and Fig. 2c is for a random alloy with CB ¼ 100� CA. Inthese plots, the cubic fcc lattice parameter is equal to 4.0 Å, the activation barrier, DEb, for componentA is equal to 600 meV while its vibrational prefactor, m�, is equal to 4� 1013 Hz. These values are typ-ical for a metal such as Al [38]. The kinetic coefficients are evaluated at 600 K assuming a vacancy con-centration of 0.002, which is relatively large for typical metallic alloys. The eLij coefficients scalelinearly with the vacancy concentration in the dilute vacancy concentration limit in the absence ofstrong interactions among vacancies.

As is clear from Fig. 2, the diagonal kinetic-transport coefficients eLAA and eLBB are always larger thanthe off-diagonal coefficient eLAB. For a thermodynamically ideal alloy that is also kinetically idealðCA ¼ CBÞ, the coefficients are symmetric with respect to composition. This symmetry is broken, how-ever, when CA – CB, as can be seen in Fig. 2b and c.

2.3. Fickian flux expressions in a perfect crystal

Gradients in chemical potentials are difficult to measure experimentally. It is more convenient toexpress fluxes for substitutional diffusion in terms of directly measurable concentration gradients.Starting from the flux expressions, Eq. (10), and using the chain rule of differentiation, we can writefor a perfect crystal

JA ¼ �DAArCA � DABrCB

JB ¼ �DBArCA � DBBrCB;ð35Þ

with Ci the number of atoms of species i per unit volume (i.e., Ci ¼ xi=X, where X is the volume persubstitutional crystal site). These flux expressions can be viewed as generalized Fick equations for sub-stitutional diffusion in a perfect crystal. The matrix of diffusion coefficients D is a product of the matrixof kinetic-transport coefficients eLij with a matrix of thermodynamic factors, eHij, according to

DAA DAB

DBA DBB

� �¼

eLAAeLABeLBAeLBB

! eHAAeHABeHBAeHBB

!: ð36Þ

The eHij are partial derivatives of the chemical potentials, ~li (normalized by kBT), with respect to atom-ic fractions xj and are related to the Hessian of the free energy gðxA; xBÞ according to

eHij ¼1

kBT@~li

@xj¼ 1

kBT@2g@xi@xj

ð37Þ

The elements of the thermodynamic factor matrix can be calculated with grand canonical Monte Carlosimulations using the fluctuation formulas Eq. (24).

2.4. Metrics for substitutional diffusion

While the flux equations in terms of concentration gradients, Eq. (35), are more practical than Eq.(10), expressed in terms of chemical potential gradients, they still do not provide insights into the rel-evant diffusional processes in a substitutional alloy. In this respect, Kehr et al. [51] demonstrated thatthe eigenvalues of D lend well to intuitive interpretation. They showed that for regular solution modelalloys having a dilute vacancy concentration, the larger eigenvalue of D, denoted by kþ, becomes ameasure for the rate with which density fluctuations due to inhomogeneities in the vacancy concen-tration dissipate while the smaller eigenvalue, k�, becomes a measure of the rate with which A and Batoms intermix. In this limit, kþ can be viewed as a vacancy diffusion coefficient and k� as an intermix-ing coefficient in a perfect crystal.


To provide further insight about binary diffusion in a perfect crystal, it is useful to complement kþ

and k� with two additional parameters expressed in terms of elements of the diffusion coefficient ma-trix D according to

d ¼ DBB � DAA þ DAB � DBA ð38Þ

and

/ ¼ DBA

DAB þ DBA: ð39Þ

The parameters d and / along with kþ and k� can serve as an alternative set of coefficients toDAA;DAB;DBA and DBB for the purpose of describing binary diffusion in a perfect crystal. This is shownin Appendix D, where expressions for Dij in terms of kþ; k�; d and / are derived. While a physical inter-pretation of d and / will become evident later, we here point out that for a thermodynamically idealalloy with identical hop frequencies (i.e. CA ¼ CB),

d ¼ 0; / ¼ xB

xA þ xB: ð40Þ

Eq. (40) can be verified by substituting the analytical expressions for the thermodynamic and kineticfactors of a thermodynamically and kinetically ideal alloy, Eqs. (25) and (33), into Eq. (36) and subse-quently into Eqs. (38) and (39). For non-ideal alloys, d and / deviate from the values given by Eq. (40).The two parameters can therefore serve as measures for the deviation of the alloy from ‘‘kinetic” ide-ality. Appendix C shows that d and k� scale linearly with the vacancy concentration, while kþ and /have a negligible dependence on the vacancy concentration. The scaling behavior of kþ and k� withxV in the dilute vacancy limit was first determined by Kehr et al. [51] for a regular solution model. Thisscaling behavior with xV is important in order to attribute a physical interpretation to kþ; k�; d and /for alloys having a dilute vacancy concentration.

The dominant modes of diffusion in a binary, perfect-crystal alloy can be decomposed into a va-cancy flux and an intermixing flux between A and B atoms. To elucidate these dominant modes, itis convenient to consider a pair of generalized fluxes defined as linear combinations of the fluxes ofJA and JB according to

�ðJA þ JBÞ; /JA � ð1� /ÞJB: ð41Þ

In a perfect crystal, the first generalized flux is simply equal to the vacancy flux, JV (see Eq. (8)). Thesecond generalized flux, which is a linear combination of JA and JB having / and �ð1� /Þ as coeffi-cients, will emerge below to be a useful measure of an intermixing flux in a perfect crystal. Both gen-eralized fluxes can be expressed in terms of the four parameters kþ, k�, d and / according to

JV ¼ �kþrCV � dð/rCA � ð1� /ÞrCBÞ; ð42Þ/JA � ð1� /ÞJB ¼ �/ð1� /ÞdrCV � k�ð/rCA � ð1� /ÞrCBÞ; ð43Þ

which is accurate to first order in the vacancy concentration. These flux expressions were obtained bysubstituting expressions for Dij in terms of kþ; k�; d and / (to first order accuracy in xV ) derived inAppendix D into Eq. (35) and usingrCV ¼ �ðrCA þrCBÞ for a perfect crystal. The symmetry inherentto the above flux expressions can be better appreciated when they are expressed in matrix form

JV

/JA � ð1� /ÞJB

� �¼ � kþ d

/ð1� /Þd k�

� � rCV

/rCA � 1� /ð ÞrCB

� �: ð44Þ

The eigenvalues kþ and k� of D form the diagonal elements of the matrix linking the generalized fluxesto the generalized driving forces, while the two additional metrics, d and /, defined by Eqs. (38) and(39), appear as off-diagonal terms.


2.4.1. Generalized flux expressions for a kinetically ideal alloyA useful starting point to analyze the generalized fluxes, Eqs. (42) and (43), for binary, perfect-crys-

tal diffusion is the kinetically ideal alloy for which CB ¼ CA. For a kinetically ideal alloy, d ¼ 0, therebysimplifying Eqs. (42) and (43) to yield the following flux expressions

Fig. 3.alloy. (order foeach po

JV ¼ �kþrCV ; ð45Þ/JA � ð1� /ÞJB ¼ �k�ð/rCA � ð1� /ÞrCBÞ: ð46Þ

For a kinetically ideal binary alloy, it can be verified using Eqs. (25), (33) and (36) that the eigenvaluesof the diffusion coefficient matrix are independent of alloy concentration [51].

The first flux expression, Eq. (45), clearly shows that kþ for a kinetically ideal binary alloy corre-sponds to a vacancy diffusion coefficient in a perfect crystal. It directly couples the vacancy flux, JV ,to a vacancy concentration gradient,rCV . The second flux expression, Eq. (46), describes an intermix-ing mode between A and B. This can be understood by considering an initial concentration profile witha constant vacancy concentration throughout the solid, as illustrated by an example in Fig. 3a. Thisinitial concentration profile will activate only the second generalized flux expression, Eq. (46), asthe driving force for the first flux expression, rCV , is zero everywhere. Thus the vacancy flux, JV , willalso be zero everywhere and the vacancy concentration profile will remain unchanged during the dif-fusion process. In the absence of a vacancy flux, any flux of A must be equal but opposite to a flux of B,i.e., JA ¼ �JB, due to the conservation of crystal sites in a perfect crystal, Eq. (8). Hence, diffusion in akinetically ideal, perfect-crystal alloy arising from initial conditions that only activate the second fluxexpression corresponds to a pure intermixing mode between A and B atoms.

As the second generalized flux expression, Eq. (46), corresponds to an intermixing mode, we willrefer to its generalized flux as an intermixing flux, bJA, for a perfect crystal
bJA ¼ /JA � ð1� /ÞJB; ð47Þ
where by symmetry bJA ¼ �bJB. If initial conditions are such that JV ¼ 0 and rCV ¼ 0, the intermixingflux bJA reduces to JA as then JA ¼ �JB. Furthermore, any concentration modulation of A must then becompensated by an opposite modulation in that of B such that rCA ¼ �rCB everywhere (e.g.,Fig. 3a). Under these conditions, the second normal mode flux expression, Eq. (46), reduces to

JA ¼ �k�rCA ¼ �JB ¼ �k�ð�rCBÞ; ð48Þ

rxAr

(b)(a)

xB

xV

xA

xB

xV

(a) Schematic illustration of concentration profiles that activate an intermixing mode of diffusion in a kinetically idealb) Schematic illustration of concentration profiles in which interdiffusion is suppressed in a kinetically ideal alloy. Inr interdiffusion fluxes to be suppressed, the concentration gradients of A and B must satisfy /rCA � ð1� /ÞrCB ¼ 0 atint.


showing that in the absence of vacancy fluxes and vacancy concentration gradients, k� for the kinet-ically ideal alloy can be interpreted as an intermixing coefficient of a perfect crystal, measuring therate of mixing between A and B.

We can assign an interpretation to / as well by again considering the first flux expression, Eq. (45),in the absence of an intermixing flux, bJA ¼ 0. Fig. 3b schematically illustrates concentration profiles ina perfect crystal in which intermixing is suppressed: the non-uniform concentration profiles for A andB have gradients in the same direction at each point, which are both opposite to the vacancy concen-tration gradient (due to xV ¼ 1� xA � xB in a perfect crystal). Setting bJA to zero in Eq. (47) and using Eq.(8) for a perfect crystal we can write

JA ¼ �ð1� /ÞJV ; JB ¼ �/JV : ð49Þ

These relations indicate that without intermixing fluxes, / is equal to the fraction of the vacancy fluxexchanging with B atoms, and ð1� /Þ is equal to the fraction of the vacancy flux exchanging with Aatoms. For a kinetically ideal alloy, / ¼ xB=ðxA þ xBÞ, which corresponds to the fraction of non-vacantsites occupied by B atoms (likewise, ð1� /Þ ¼ xA=ðxA þ xBÞ corresponds to the fraction of non-vacantsites occupied by A atoms). At the atomic scale, the vacancies in a kinetically ideal alloy perform unbi-ased exchanges with A and B atoms as CA ¼ CB. Hence the fraction of B (A) atoms exchanging with anet vacancy flux, JV , is equal to the local fraction of B (A) atoms, as it emerges in Eq. (49).

With this interpretation of / and ð1� /Þ, we can better understand the meaning of the intermixingflux bJA during general diffusion processes that simultaneously activate both flux expressions, Eqs. (45)and (46), by rewriting Eq. (47) as
bJA ¼ JA þ ð1� /ÞJV ; ð50Þ
where the conservation of crystal sites, Eq. (8), was used. In this form, the intermixing flux bJA can beunderstood as the total flux of A minus the flux of A stemming from exchanges with vacancies. Thisdifference corresponds to the flux of A caused by exchanges with B and is thus the part of the totalflux of A participating in intermixing with B.

Clearly, for a kinetically ideal alloy, the vacancy flux expression, Eq. (45), and the intermixing fluxexpression, Eq. (46), each in isolation describe a normal mode diffusion process in a binary, perfect-crystal alloy. The decay rates of each normal mode are the eigenvalues kþ and k� [51]. The gradientin the vacancy concentration,rCV , is a driving force for the first normal mode diffusion process result-ing in vacancy fluxes, while /rCA � ð1� /ÞrCB is the driving force for the second normal mode dif-fusion process corresponding to intermixing between A and B. The intermixing flux expressionpossesses a symmetry between A and B where / weights both the flux of A and the concentration gra-dient of A while ð1� /Þ weights the flux of B and the concentration gradient of B.

2.4.2. Generalized flux expressions for non-ideal binary alloys in the dilute vacancy limitFor a kinetically non-ideal alloy, d deviates from zero, and the vacancy flux and intermixing flux

expressions, Eqs. (42) and (43), become more complicated. When expressed in matrix form as inEq. (44), the vacancy and intermixing fluxes, JV and /JA � ð1� /ÞJB, are no longer diagonal in termsof the vacancy concentration gradient, rCV , and intermixing driving force, /rCA � ð1� /ÞrCB. Theinterpretations assigned to kþ and k� for a kinetically ideal alloy are therefore no longer strictly valid.Nevertheless, we can still interpret kþ as a vacancy diffusion coefficient and k� as an intermixing coef-ficient in the asymptotic limit of a dilute vacancy concentration (provided that the eLij scale with xV andthat the alloy is thermodynamically well-behaved as defined in Appendix C). We illustrate this by con-sidering the generalized flux expressions, Eqs. (42) and (43), under conditions where either the va-cancy flux is absent or the intermixing flux is absent.

The absence of a vacancy flux, JV ¼ 0, corresponds to a pure intermixing mode since, due to the con-servation of crystal sites, Eq. (8), JA ¼ �JB. Setting the vacancy flux to zero in Eq. (42) imposes a con-straint on the vacancy concentration gradient according to

rCV ¼ �d

kþð/rCA � ð1� /ÞrCBÞ ð51Þ

which, when substituted into Eq. (43) for the intermixing flux, yields

Fig. 4.concen


/JA � ð1� /ÞJB ¼ �k�ð1� �Þð/rCA � ð1� /ÞrCBÞ ð52Þ

where

� ¼ /ð1� /Þd2

kþk�: ð53Þ

Eq. (52) is only valid in the absence of a vacancy flux and therefore describes a pure intermixing modeðJA ¼ �JBÞ. It relates an intermixing flux, /JA � ð1� /ÞJB, to an intermixing driving force,/rCA � ð1� /ÞrCB, with as diffusion coefficient k�ð1� �Þ. The parameter � scales with the vacancyconcentration and vanishes as the vacancy concentration xV approaches zero. Hence, also for kineti-cally non-ideal alloys, k� can be interpreted as a perfect-crystal intermixing coefficient in the asymp-totic limit of a dilute vacancy concentration xV ! 0.

Since crystalline solids of technological importance often have finite vacancy concentrations, it is ofinterest to estimate the size of the dimensionless parameter � for typical vacancy concentrations ofbinary substitutional alloys. We can calculate � exactly for thermodynamically ideal but kineticallynon-ideal alloys using the analytical expressions for the kinetic-transport coefficients eLij (derived byMoleko et al. [79], see Appendix B) and thermodynamic factors eHij (Eq. (25)). For thermodynamicallyideal alloys, � is a function of the ratio of hop frequencies CB=CA as well as the alloy concentration, xB

and xV (with xA ¼ 1� xB � xV ). For small vacancy concentrations, � scales linearly with the vacancyconcentration xV . At constant CB=CA ratio and vacancy concentration xV , the parameter � exhibits amaximum at intermediate alloy concentration. Fig. 4 illustrates the maximum value of �=xV with re-spect to alloy concentration plotted versus CB=CA ratio for thermodynamically ideal alloys. The ratio�=xV is zero for the kinetically ideal alloy where CB=CA ¼ 1 and steadily increases as the alloy deviatesfrom kinetic ideality (i.e., with increasing CB=CA ratio), reaching a plateau close to 4.5 for very largeCB=CA ratios. Hence an upper bound on � for small vacancy concentrations is � � 5xV . For � to be atmost 0.01 (i.e., a 1% error if � were neglected), the vacancy concentration must be 0.002, which is sig-nificantly larger than typical vacancy concentrations in most metal alloys. For xV ¼ 10�6, which isapproximately the vacancy concentration in Al at 600 K, � is of the order of 5� 10�6. It is clear thateven for kinetically non-ideal alloys having a dilute vacancy concentration, k� can be interpreted asa perfect-crystal intermixing coefficient, describing mixing between A and B in the absence of vacancyfluxes.

To assign a physical interpretation to kþ for a kinetically non-ideal alloy, it is useful to analyze theflux expressions Eqs. (42) and (43) in the absence of an intermixing flux. Setting the intermixing fluxbJA ¼ /JA � ð1� /ÞJB equal to zero in Eq. (43) imposes a constraint on the concentration profiles

0

1

2

3

4

5

1 10 102 103 104 105 106

ε

ΓB / ΓA

xV

The maximum value of �, Eq. (53), normalized by the vacancy concentration xV versus the CB=CB ratio. For small vacancytrations, � scales linearly with the vacancy concentration.


/rCA � ð1� /ÞrCB ¼ �/ð1� /Þd

k�rCV ð54Þ

which when substituted into the vacancy flux expression Eq. (42) yields

JV ¼ �kþð1� �ÞrCV ð55Þ

where � is again given by Eq. (53). In the absence of an intermixing flux, a fraction ð1� /Þ of the va-cancy flux exchanges with A atoms while a fraction / of the vacancy flux exchanges with B atoms. Un-der these conditions, Eq. (55) shows that the vacancy flux, JV , is proportional to a gradient in thevacancy concentration, rCV , with as proportionality constant kþð1� �Þ. This proportionality constantcan be interpreted as a vacancy diffusion coefficient. In alloys having a dilute vacancy concentration, �is negligible compared to 1, and kþð1� �Þ � kþ. Hence, even in kinetically non-ideal binary alloys, kþ

can be interpreted as a vacancy diffusion coefficient.For general non-equilibrium concentration profiles, both modes of perfect-crystal diffusion corre-

sponding to vacancy fluxes and intermixing will participate. Furthermore, a pure intermixing modecannot be sustained over time in a non-ideal alloy as off-diagonal elements in the diffusion matrixof Eq. (44) will activate vacancy fluxes.

2.4.3. Physical interpretation of dAs with kþ; k� and /, it is also possible to assign a physical interpretation to the parameter d to

characterize the transport properties of a non-ideal alloy having a perfect crystal. The generalized fluxexpressions, Eqs. (42) and (43), show that for a kinetically non-ideal alloy, d couples the vacancy flux,JV , to the intermixing driving force, /rCA � ð1� /ÞrCB, while /ð1� /Þd couples an intermixing flux,/JA � ð1� /ÞJB, to a gradient in the vacancy concentration, rCV . In a kinetically ideal alloy, d ¼ 0 andvacancy fluxes therefore cannot emerge due to intermixing driving forces. Hence the Kirkendall effect,which results in vacancy fluxes during intermixing, will not occur in a kinetically ideal alloy. The Kir-kendall effect only occurs when differences in the hop frequencies of A and B exist. In kinetically non-ideal alloys, CA – CB such that d – 0 and vacancy fluxes then can emerge in the presence of intermix-ing driving forces. The parameter d, in addition to measuring the deviation from kinetic ideality, there-fore also serves as a metric for the Kirkendall effect. In fact, by examining Eq. (42) under the conditionof rCV ¼ 0, it is evident that d can be interpreted as a Kirkendall diffusion coefficient.

The second generalized flux expressions, Eq. (43), also shows that intermixing fluxes can be causedby a gradient in the vacancy concentration even in the absence of an intermixing driving force. Thestrength of this coupling is /ð1� /Þd, which is again proportional to the degree with which the alloydeviates from kinetic ideality, d.

2.5. Classification of substitutional alloys based on kinetic metrics

The parameters kþ; k�;/ and d are useful metrics to characterize the transport properties of binaryalloys having a perfect crystal (i.e., in the absence of dislocations and grain boundaries). They providedirect information about the relevant modes of diffusion in a substitutional alloy related to intermix-ing and vacancy fluxes responsible for the Kirkendall effect, which is not as easily discerned frominspection of the matrix of diffusion coefficients Dij. We emphasize, however, that the interpretationsassigned to kþ; k� and / for kinetically non-ideal alloys are based on an analysis of equations that areasymptotically valid in the dilute vacancy limit where xV ! 0. In this limit, kþ is equivalent to a va-cancy diffusion coefficient in the binary alloy, k� is an intermixing coefficient describing mixing be-tween A and B, and / represents the fraction of a vacancy flux that exchanges with B atoms in theabsence of intermixing (by symmetry, ð1� /Þ represents the fraction of a vacancy flux that exchangeswith A atoms). The fourth metric d, which is equal to zero for a thermodynamically and kineticallyideal alloy (i.e., CA ¼ CB), can be interpreted as a Kirkendall diffusion coefficient, linking a vacancy fluxto an intermixing driving force.

While the vacancy concentration in most metallic alloys is very small, it is not infinitesimally small.Nevertheless, the interpretations assigned to kþ; k�;/ and d are still meaningful for vacancy concen-trations as high as 0.002, as discussed in Section 2.4.2. An inspection of these kinetic metrics provides


invaluable insights about the diffusional processes in a perfect-crystal alloy, allowing comparisons andclassifications of alloys having differing thermodynamic and kinetic properties.

In this section, we explore the behavior of the kinetic metrics in thermodynamically ideal butkinetically non-ideal alloys, as well as a thermodynamically non-ideal alloy. As a thermodynamicallynon-ideal alloy, we consider fcc Alð1�xÞLix, which exhibits both a solid solution and L12 ordering forcompositions around Al3Li. To facilitate the comparison between the thermodynamically ideal A–Bmodel alloys and fcc Alð1�xÞLix, we assume that pure A has the same vacancy concentration as that pre-dicted for Al with a first-principles cluster expansion [31] at 600 K (i.e., xV ¼ 1:4� 10�6 correspondingto a vacancy formation energy of almost 700 meV). We also assign the A atoms a vacancy exchangefrequency CA similar to that predicted for Al at 600 K from first principles (i.e., DEb ¼ 600 meV andm� ¼ 4:0� 1013 appearing in Eq. (27)) [38].

2.5.1. Thermodynamically ideal substitutional alloysIn the ideal reference alloy (i.e., thermodynamically as well as kinetically ideal), the eigenvalues are

independent of alloy concentration, d ¼ 0 and / ¼ xB=ðxA þ xBÞ. Once a thermodynamically ideal alloydeparts from kinetic ideality (i.e. CA – CB), the coefficients d and / deviate from 0 and xB=ðxA þ xBÞ,respectively. Furthermore, the kinetic coefficients kþ; k� and d become concentration dependent.

Fig. 5 illustrates the composition dependence of kþ and k� for thermodynamically ideal alloys withCB ¼ CA;CB ¼ 10� CA and CB ¼ 100� CA. These were calculated by diagonalizing the diffusion coef-ficient matrix D, Eq. (36), using the analytical expressions for eLAA; eLAB and eLBB derived by Molekoet al. [79] (see Fig. 2 and Appendix B) along with the thermodynamic factors for a random alloy, Eq.(25). Fig. 5a shows that kþ increases with the concentration of the fast diffuser (B in this case) whenCB > CA. Also plotted in Fig. 5a (circles) are values for the vacancy tracer diffusion coefficient for eachof the three random alloys defined by

D�V ¼hðD~RV Þ2ið2dÞt ð56Þ

where D~RV is a vector connecting the end points of a vacancy trajectory after time t in a crystal withdimension d. The vacancy tracer diffusion coefficients were calculated with kinetic Monte Carlo sim-ulations. As is clear from Fig. 5a, the vacancy tracer diffusion coefficients coincide with kþ, confirmingthe interpretation of kþ as a measure of the vacancy mobility within the binary alloy in the dilute va-cancy limit (the kinetic Monte Carlo simulations to calculated D�V were performed with a vacancy con-centration of 0.002).

While the addition of a fast diffuser to an alloy leads to an increase of kþ, its effect on the perfect-crystal interdiffusion coefficient k� is more complicated. Fig. 5b shows that for the random alloy withCB ¼ 10� CA, k� decreases with the concentration of the fast diffuser B. If the difference in vacancyexchange frequencies between A and B is very large (e.g., CB ¼ 100� CA), Fig. 5b shows that low con-centrations of the fast diffuser (B atoms) cause a slight increase of k�, followed by a decrease of k� asmore of the fast diffuser is added to the alloy. This implies that high concentrations of a fast diffuseractually reduces the efficacy of intermixing in the perfect crystalline state.

Fig. 6a illustrates the concentration dependence of / for the three thermodynamically ideal alloyshaving different CB=CA ratios. For the kinetically ideal alloy ðCB ¼ CAÞ;/ is equal to xB=ðxA þ xBÞ andapproaches xB as xV ! 0 (or equivalently xA þ xB ! 1). Fig. 6a shows that / deviates from its lineardependence on xB (when xV ! 0) as the CB=CA ratio deviates from 1. In systems for which CB > CA;/exceeds xB as illustrated in Fig. 6a. In view of the interpretation of / as the fraction of the vacancy fluxexchanging with B (in the absence of interdiffusion fluxes), Fig. 6a shows that a larger fraction of thefast diffuser exchanges with an unbalanced vacancy flux than in the kinetically ideal alloy.

The Kirkendall effect arises from a net vacancy flux as a result of concentration gradients and canonly occur in alloys in which the components have different hop frequencies with vacancies. Any met-ric for the Kirkendall effect in a binary alloy should therefore increase as the ratio of hop frequencies ofthe fast and slow diffuser increases. The analysis of the previous section suggests that d can serve as ametric for the tendency of an alloy to exhibit the Kirkendall effect. For the ideal reference alloy, d ¼ 0and the Kirkendall effect is absent. However, when CB – CA, the alloy becomes susceptible to the

10 -07

10 -06

10 -05

0 0.2 0.4 0.6 0.8 1

λ+(c

m2 /s

)

Concentration of B

10 -04

ΓB=10xΓA

ΓB=ΓA

Concentration of B

10 -13

10 -12

10 -11

10 -10

0 0.2 0.4 0.6 0.8 1

(a)

(b)

λ−(c

m2 /s

)

ΓB=100xΓA

ΓB=ΓA

ΓB=10xΓA

ΓB=100xΓA

Fig. 5. Concentration dependence of (a) kþ and (b) k� for a random alloy with CB ¼ CA , CB ¼ 10� CA and CB ¼ 100� CA . Both kþ

and k� are independent of concentration in a random alloy that is also kinetically ideal, i.e., CA ¼ CB . The lines are obtainedusing analytical expressions for eLij derived by Moleko et al. [79] and the ideal solution thermodynamic factors to calculate thediffusion coefficient matrix, while the circles in (a) are vacancy tracer diffusion coefficients calculated with kinetic Monte Carlosimulations.


Kirkendall effect and d should deviate from zero. Fig. 6b illustrates the concentration dependence of dfor the two random alloys with CB ¼ 10� CA and CB ¼ 100� CA. As is clear from Fig. 6b, the larger theCB=CA ratio, the larger the Kirkendall diffusion coefficient d becomes. Note also that for a given alloy, dincreases with the concentration of the fast diffuser. This dependence is stronger when the ratio CB=CA

is large and thus the susceptibility for the Kirkendall effect becomes significantly more pronouncedwith an increase in the concentration of the fast diffuser in these cases.

Correlation factors offer valuable information about atomic mechanisms of diffusion [104–107]and provide insight about the concentration dependence of the various kinetic coefficients depictedin Figs. 5 and 6. A correlation factor, Eq. (34), measures the average of the square of the displacementof a particular component relative to that of a random walker having the same hop frequency. Fig. 7illustrates the correlation factors for the A, B and V species (referred to as fA; fB and fV respectively) forrandom fcc alloys with CB=CA ¼ 10 and CB=CA ¼ 100 (calculated with kinetic Monte Carlo havingxV ¼ 0:002). The maximum value that a correlation factor can attain is 1 as this corresponds to a per-fect random walk. In a single component fcc crystal in which diffusion occurs via the vacancy mech-anism, the correlation factor of atomic diffusion is 0.781 [108]. This is the value of fA as xB ! 0 as canbe seen in Fig. 7. The correlation factor for vacancy diffusion in a single component solid (e.g., xB ! 0)containing a dilute concentration of vacancies is 1 since an isolated vacancy performs a random walk.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1Concentration of B


φ=

DA

B+

DB

A

DB

A

ΓB=100xΓA

ΓB=10xΓA

ΓB=ΓA

10-12

10-11

10-10

δ(c

m2 /

s)

10-09

ΓB=10xΓA

ΓB=100xΓA

(a)

(b)

Fig. 6. Concentration dependence of (a) / for a random alloy with CB ¼ CA;CB ¼ 10� CA and CB ¼ 100� CA and (b) of d for arandom alloy with CB ¼ 10� CA and CB ¼ 100� CA . d is zero for all concentrations in a kinetically ideal solution ðCA ¼ CBÞ.


However, as is evident in Fig. 7, the vacancy correlation factor drops rapidly upon addition of a smallconcentration of a fast diffuser B.

The dramatic reduction in fV upon alloying with a fast diffuser indicates that the presence of a verymobile component introduces important correlations between successive vacancy hops. For small Bconcentration, B atoms are typically isolated, i.e., separated by a significant distance from other Batoms. Once a vacancy wanders into the nearest neighbor shell of an isolated B atom, it will on averageundergo many back and forth exchanges with that B atom before departing its nearest neighbor shell.However, while the vacancy and an isolated B atom may perform many hops, these are essentially re-stricted to back and forth movements between a pair of sites as opposed to a more extended trajectorycharacteristic of a random walk. As the concentration of B reaches a critical value leading to intercon-nected networks of B, however, the vacancy can migrate more freely along the percolating network ofB atoms, rather than predominantly hopping back and forth with an isolated B. This causes an increaseof both fV and fB.

The rapid rise in kþ; d and / around xB � 0:2 for the alloy having CB=CA ¼ 100 is correlated with theupswing in the vacancy correlation factor resulting from the establishment of a percolating network ofB atoms throughout the crystalline network. Once a percolating network of the fast diffuser is formed,all diffusional processes involving vacancy fluxes become more effective, as reflected by the kineticmetrics kþ; d and /. On the other hand, a well-connected network of fast diffusers leads to a reduction

0

0.2

0.4

0.6

0.8

1

corre

latio

n fa

ctor

corre

latio

n fa

ctor

0

0.2

0.4

0.6

0.8

1

fA

fB

fV

fA

fV

fB

0.0 0.2 0.4 0.6 0.8 1.0

(a)

(b)

0.0 0.2 0.4 0.6 0.8 1.0Concentration of B

Concentration of B

Fig. 7. Concentration dependence of correlation factors for ideal solutions with (a) CB ¼ 10� CA and (b) CB ¼ 100� CA .


in the perfect-crystal interdiffusion coefficient k�. As more fast diffusing B atoms become availablewith increasing xB, the vacancy has more B atoms to exchange with, making it even less likely to ex-change with A and to rearrange A and B atoms relative to each other.

2.5.2. Thermodynamically non-ideal substitutional alloyMost multi-component solids deviate from thermodynamic ideality. A solid exhibits different de-

grees of short- and long-range order depending on its temperature and composition, as well as thestrength of the interactions between its constituents. Varying degrees of order occur because the en-ergy of a solid depends on the arrangement of the different constituents, with certain arrangementsenergetically more favored than most others. Likewise, the activation barrier, DEb, and vibrational pre-factor, m�, that determine the hop frequency according to Eq. (27) also depend on the local degree oforder. A typical atom migrating within a multi-component solid samples many different local environ-ments along its trajectory and executes hops with frequencies that change with the local environment.

The fcc Al–Li alloy serves as a useful example to analyze diffusion in a binary substitutional solidthat deviates from thermodynamic ideality. While Al1�xLix is predominantly stable in the bcc crystalstructure, it is metastable in fcc when Al-rich and exhibits L12 ordering around x = 0.25. A first-prin-ciples investigation has shown that diffusion in this alloy is strongly affected by its thermodynamicproperties [38]. As demonstrated elsewhere, it is possible to accurately describe the configurationalenergy of the Al1�xLix alloy with a cluster expansion parameterized to reproduce the first-principlesenergies (density functional theory within the local density approximation) of a small set of periodicarrangements of Al and Li over fcc [21,31]. When implemented in Monte Carlo simulations, this cluster


expansion predicts the phase stability of the L12 phase at finite temperature as illustrated in the cal-culated phase diagram in Fig. 8a [31]. The L12 phase is predicted to be stable over a large concentrationinterval, undergoing an order–disorder transition around 730 K. By accounting for the configurationdependence of the vacancy formation energy, it is also possible to calculate the equilibrium vacancyconcentration in Al1�xLix as a function of the bulk alloy concentration as illustrated in Fig. 8b at 600 K[31]. The Monte Carlo simulations also predict significant short-range order around the vacancy with-in the solid solution phase, with the vacancy preferring a nearest neighbor shell rich in Al atoms. In theL12 ordered phase, where Li occupies the corner points of the conventional cubic fcc unit cell, the va-cancy predominantly occupies the Li-sublattice, as this allows the vacancy to be surrounded by Al[31].

First-principles calculations of migration barriers for various Al–Li configurations and concentra-tions showed that the average migration barrier (averaged between forward and backward hops)

500

600

700

800

Tem

pera

ture

(K)

10-07

10-06

10-05

Vaca

ncy

con

cent

ratio

n

solid solution

L12

two phase

region

(a)

(b)

10-16

10-14

10-12

0 0.1 0.2 0.3

cm2 /s

LAlAl

LLiAl

LLiLi~

~

~

L12two phase

region

concentration x in Al(1-x)Lix

0 0.1 0.2 0.3concentration x in Al(1-x)Lix


(c)

Fig. 8. (a) First-principles calculated phase diagram of the metastable fcc Alð1�xÞLix alloy and (b) equilibrium vacancyconcentration as a function of bulk alloy concentration x at T = 600 K. (c) First-principles calculated kinetic-transportcoefficients eLij for the metastable fcc Alð1�xÞLix alloy at T = 600 K.


for a Li atom into an adjacent vacancy is less by almost 200 meV than that for Al [38]. This shows thatin addition to exhibiting non-ideal thermodynamic properties, the alloy deviates from kinetic ideality.In fact, the non-idealities tend to work in opposite directions: while Li is kinetically more mobile, hav-ing a lower migration barrier than Al when next to a vacancy, it is nevertheless thermodynamicallydeprived of diffusion mediating vacancies as the vacancies prefer to be surrounded by Al.

By combining the first-principles calculated activation barriers with the cluster expansion for theenergies of end states in kinetic Monte Carlo simulations as described in Ref. [38], it is possible to eval-uate the Kubo-Green expressions for the eLij coefficients (with i,j = Li or Al). These are illustrated inFig. 8c at 600 K. Although the fcc solid solution of Alð1�xÞLix has kinetic coefficients that are qualitativelysimilar to those of an ideal solution (Fig. 2), within the L12 phase, the kinetic coefficients exhibit morestructure as a function of concentration. In the L12 phase, the off-diagonal kinetic coefficient, eLLiAl,which reflects correlations between the trajectories of the center of mass of Al atoms with that of Liatoms, is significantly smaller than the diagonal kinetic coefficients, eLLiLi and eLAlAl. Its small magnitudeleads to large statistical scatter. The diagonal kinetic coefficients exhibit a minimum at the stoichiom-etric composition (x = 0.25) of L12 where the number of anti-site defects that disrupt ordering is min-imized. Beyond the stoichiometric composition of the L12 phase, the kinetic coefficient eLLiLi increasessignificantly due to an anti-site structural bridge diffusion mechanism [109,110]. For x > 0.25, theexcess Li occupy the Al sublattice of L12 and form a bridge to allow vacancies, which predominantlyoccupy the Li-sublattice, to more easily depart one Li-sublattice site and migrate to the next.

10-14

10-12

10-10

10-08

10-06

0 0.1 0.2 0.3

cm2 /s

λ−

λ+

Two phase

region L12

0

0.2

0.4

0.6

0.8

1

φideal

φAl(1-x)Lix

φ=

DA

lLi+

DL

iAl

DL

iAl



(a)

(b)

Fig. 9. (a) Concentration dependence of the eigenvalues kþ and k� of the diffusion coefficient matrix for the metastable fccAlð1�xÞLix alloy. (b) Concentration dependence of / for the metastable fcc Alð1�xÞLix alloy at T = 600 K.


To obtain the Fickian diffusion coefficients Dij as well as the metrics for substitutional diffusionkþ; k�; d and /, it is also necessary to calculate the components of the thermodynamic factor matrix~H appearing in Eq. (36). This can be done with Eq. (24) in terms of ensemble averages of fluctuationsin the number of A = Al and B = Li atoms, numerically evaluated within grand canonical Monte Carlosimulations. Multiplying the thermodynamic factor matrix with the matrix of kinetic coefficientsaccording to Eq. (36) then yields the matrix of diffusion coefficients D. Fig. 9a illustrates the eigen-values, kþ and k� of this matrix as a function of the Li concentration. In the solid solution ofAlð1�xÞLix, both kþ and k� exhibit a negligible concentration dependence, while in the L12 phase, botheigenvalues drop significantly. The thermodynamic preference of the vacancies for the lithium sublat-tice sites of L12 constricts the trajectory of the vacancies, leading to a drop in overall mobility of Li, Aland vacancies. Nevertheless, kþ, which is a measure for the vacancy diffusivity, rapidly increases abovex ¼ 0:25 in the L12 phase as a result of the anti-site structural bridge mechanism [109,110] that be-comes viable once excess Li occupies the Al sublattice.

Fig. 9b illustrates the concentration dependence of / defined as

/ ¼ DLiAl

DAlLi þ DLiAl; ð57Þ

which, as explained in Sections 2.4.1 and 2.4.2, can be interpreted as the fraction of a vacancy flux thatexchanges with Li atoms (in the absence of intermixing). In the solid solution, / is greater than that forthe kinetically ideal alloy, /ideal (ideal solution and CA ¼ CB). This can be attributed to the lower acti-vation barrier for Li hops into an adjacent vacancy as compared to that of Al. A net flux of vacancies(e.g., after a quench when supersaturated vacancies diffuse to surfaces and extended defects) will pref-erentially exchange with Li atoms (compared to a kinetically ideal alloy). In L12, / is slightly less than/ideal below x ¼ 0:25 but becomes larger than /ideal above x ¼ 0:25.

Numerical errors on the individual Fickian diffusion coefficients Dij due to statistical noise inherentto Monte Carlo techniques prevent us from extracting reliable values for the Kirkendall diffusion coef-ficient, d. The metric d is a small quantity as it scales with the vacancy concentration (which is small inAlð1�xÞLix) and is a difference between much larger quantities of approximately the same magnitude.Thus, a different approach must be developed in order to obtain this quantity from first principles.

3. The role of vacancy sources and sinks

The treatment of substitutional diffusion has so far been restricted to perfect crystals. In this lim-iting case, we can make direct contact with diffusion coefficients that are calculated from first princi-ples using a cluster expansion for the configurational energy and kinetic Monte Carlo simulationswithin a defect-free crystal (i.e., without extended defects that can serve as vacancy sources andsinks). The assumption of a perfect crystal may be valid for very precisely synthesized heterostruc-tures, or for a study of diffusion at low temperatures where vacancy creation or annihilation at sourcesand sinks could be kinetically suppressed. Most solids, however, contain defects that can locally reg-ulate the vacancy concentration, complicating the task of linking kinetic properties calculated for per-fect crystals with non-equilibrium phenomena observed in real solids. As a first approximation forsolids containing vacancy sources and sinks, we can examine the other extreme in which sourcesand sinks are assumed to exist uniformly (at least on a continuum scale) such that the vacancy con-centration is locally always equilibrated. This approximation is conventionally made in textbook treat-ments of substitutional diffusion [2,76].

3.1. A continuous and dense distribution of sources and sinks

In the limit of a continuous distribution of vacancy sources and sinks, the assumption can be madethat the vacancy chemical potential is zero throughout the solid and that therefore dlV ¼ 0. This con-dition implies a sufficiently high density of vacancy sources and sinks to regulate the vacancy concen-tration at its equilibrium value. Despite the high concentration of vacancy sources and sinks, wenevertheless assume (as is done in standard treatments of substitutional diffusion) that diffusion


happens through crystalline regions, thereby neglecting contributions from grain boundary diffusionand diffusion along dislocation cores. The perfect-crystal diffusion coefficients, Dij, then still describetransport in the bulk alloy. The only role of defects is to impose the thermodynamic constraint thatlV ¼ 0 throughout the solid, which, using Eq. (4) and the fact that Ci ¼ xi=X, allows us to expressthe concentration gradient of one atomic species in terms of the other according to

rCA

rCB¼ �

xA@2g

@xB@xA

!þ xB

@2g@x2

B

!

xA@2g@x2

A

!þ xB

@2g@xA@xB

! ð58Þ

where the concentration dependence of the volume per substitutional site, X, has been neglected.With Eq. (58) the Fickian flux equations for a perfect crystal, Eq. (35), can be rewritten in terms ofthe concentration gradient of either A or of B (but not both). Expressing the diffusion coefficients Dij

in terms of the kinetic coefficients eLij and thermodynamic factors eHij and using Eq. (58) we obtain

JA ¼ �DArCA

JB ¼ �DBrCBð59Þ

with the self-diffusion coefficients DA and DB defined as

DA ¼eLAA

xA�eLAB

xB

!xA

kBTdlA

dxA

� �lV¼0

DB ¼eLBB

xB�eLBA

xA

!xB

kBTdlB

dxB

� �lV¼0

:

ð60Þ

The derivatives of the chemical potentials are taken under the constraint that lV ¼ 0 (along the line incomposition space corresponding to an equilibrium vacancy concentration, Fig. 1b). In terms of theHessian elements of the free energy, these derivatives can be written as

dlA

dxA

� �lV¼0

¼ xB

@2g@x2

A

@2g@x2

B

� @2g@xA@xB

!2

xA@2g

@xA@xBþ xB

@2g@x2

B

0BBBBB@

1CCCCCA

dlB

dxB

� �lV¼0

¼ xA

@2g@x2

A

@2g@x2

B

� @2g@xA@xB

!2

xA@2g@x2

A

þ xB@2g

@xA@xB

0BBBBB@

1CCCCCA;ð61Þ

where the Hessian elements of the free energy are to be evaluated at concentrations xA and xB forwhich lV ¼ 0. These expressions can be derived starting with Eq. (2) and differentiating under theconstraint that xA and xB are functionally related to each other according to Eq. (4) when holdinglV ¼ 0. It should be emphasized that the above derivatives of the chemical potentials are not partialderivatives, but rather derivatives with respect to one concentration variable xi performed at constantvacancy chemical potential (i.e., lV ¼ 0). In a perfect binary crystal with vacancies, there are two inde-pendent concentration variables. However, once the constraint of an equilibrium vacancy concentra-tion is imposed (i.e., lV ¼ 0), only one composition variable is independent (either xA or xB, but notboth). For an ideal solution having a vacancy concentration independent of alloy composition, Eq.(61) becomes

dlA

dxA

� �lV¼0

¼ kBTxA

;dlB

dxB

� �lV¼0

¼ kBTxB

: ð62Þ


As is clear from Eq. (59), the assumption of an equilibrium vacancy concentration leads to flux equa-tions for each component A and B that depend only on their own concentration gradient. This does not,however, mean that gradients in the vacancy concentration are absent or that rCA ¼ �rCB. Since theequilibrium vacancy concentration can vary depending on the local alloy concentration, any gradientsin A and/or B will lead to a gradient in the vacancy concentration, even when vacancies are locally inequilibrium.

The above flux equations are identical to those derived in the standard textbook treatment of sub-stitutional diffusion based on the assumption that a continuous and dense distribution of vacancysources exist to locally maintain an equilibrium vacancy concentration throughout the solid[2,75,69,76]. The standard derivation of Eq. (59) starts with Eq. (10) and then uses the constraint thatdlV ¼ 0 along with the Gibbs–Duhem relation. The derivation here clarifies the precise meaning of thederivatives of the chemical potentials, Eq. (61), in terms of quantities that can be calculated with grandcanonical Monte Carlo simulations (i.e., the Hessian of gðxA; xBÞ related to the variances in the numberof atoms at constant chemical potentials, Eq. (24)).

3.1.1. The Darken approximationFig. 2 illustrates that the off-diagonal kinetic coefficients ~LAB are always smaller than the diagonal

kinetic coefficients ~LAA and eLBB. We could therefore neglect the off-diagonal term eLAB to simplify Eq.(60). Inspection of the Kubo-Green expression for eLAB shows that this coefficient measures the degreeto which the trajectories of A atoms are correlated with the trajectories of B atoms. A further simpli-fication to Eq. (60) becomes possible if each diagonal kinetic coefficient, for example eLAA, is written as

eLAA

xA¼

PNAf¼1ðDRA

f ðtÞÞ2

D E2dNAt

þPNA

f¼1

PNAf¼1;n – fDRA

f DRAn

D E2dNAt

; ð63Þ

where we have used the fact that xA ¼ NA=M. The ratio eLBB=xB can be written in a similar manner. Thefirst term in Eq. (63) is equal to the well-known tracer diffusion coefficient defined as

D�i ¼hðD~RiÞ2ið2dÞt ; ð64Þ

where D~Ri is the vector that connects the endpoints of the trajectory of an atom of type i. The secondterm in Eq. (63) is similar in nature to the off-diagonal kinetic coefficient eLAB except that it measuresthe degree with which the trajectories of different atoms of the same type are correlated.

By neglecting all terms that measure correlations between the trajectories of different diffusingparticles (i.e., LAB and the second term in Eq. (63)), the self-diffusion coefficients DA and DB reduce to

DA ¼ D�AxA

kBTdlA

dxA

� �lV¼0

DB ¼ D�BxB

kBTdlB

dxB

� �lV¼0

;

ð65Þ

where D�A and D�B are tracer diffusion coefficients of A and B. These expressions for the self-diffusioncoefficients correspond to the well-known Darken approximation [111].

3.1.2. The Kirkendall effect and interdiffusionThe Kirkendall effect arises from a difference in hop frequencies between A and B atoms. If, for

example, B exchanges more frequently with vacancies than A, then a gradient in the concentrationsof A and B should lead to a net vacancy flux in a direction opposite to the flux of B. In the presenceof active vacancy sources and sinks that accommodate any resultant deviation of the vacancy concen-tration from its equilibrium value, the crystal will locally shrink or expand (e.g., through dislocationclimb). This leads to a local movement of the crystalline frame of reference relative to the laboratoryframe and is referred to as a Kirkendall shift. The local velocity of the crystal frame of reference relativeto the laboratory frame is proportional to the net flux of atoms at that point according to [2,75,69,76]


v l ¼ �XðJA þ JBÞ: ð66Þ

Using Eqs. (59) and (60) to eliminate the fluxes JA and JB and Eq. (58) to relaterCA torCB for an equi-librium vacancy concentration, the local crystal frame velocity, v l, can be written as

v l ¼ �XdlV¼0rCB; ð67Þ

where

dlV¼0 ¼eLAA

xA�eLAB

xB

!�

eLBB

xB�eLAB

xA

!" #xB

kBTdlB

dxB

� �lV¼0

: ð68Þ

For alloys in which the vacancy concentration is independent of the alloy concentration,

dlV¼0 ¼ DA � DB; ð69Þ

as then

xAdlA

dxA

� �lV¼0

¼ xBdlB

dxB

� �lV¼0

ð70Þ

according to the Gibbs–Duhem relation [84] and the fact that dxA ¼ �dxB. Eq. (69) is a result com-monly derived in standard treatments of substitutional diffusion [2,75,69,76] and is strictly valid onlyif the equilibrium vacancy concentration is independent of the bulk alloy concentration, which is areasonable approximation for most alloys but not for intermetallic compounds such as B2 NiAl [36].In view of the relationship between ml and Jv implicit in Eq. (66), dlv¼0 can be given a similar interpre-tation as the Kirkendall diffusion coefficient d for a perfect crystal. According to Eq. (66) and Eq. (67) ,dlv¼0 relates the crystal frame velocity, ml, to a concentration gradient of B atoms and therfore mea-sures the susceptibility of the alloy to the Kirkendall effect under the thermodynamic constraint thatlv ¼ 0 throughout the solid. In a thermodynamically and kinetically ideal alloy (CA ¼ CB), it can beshown that dlv¼0 is zero and no vacancy fluxes will emerge as a result of concentration gradients ofA and B.

Local Kirkendall shifts of the lattice frame in solids containing a continuous distribution of vacancysources and sinks only occur in regions having concentration gradients. In regions far away from theinterdiffusion zone of a diffusion couple, concentration gradients are absent and hence no fluxes of Aand B exist. The crystal frame in these regions is then locally fixed to the laboratory frame. The localmigration of the lattice frame within the interdiffusion zone can be measured by attaching inert mark-ers to the lattice frame, as was first done by Kirkendall [78,77].

For diffusion couples, it is often desirable to work with flux expressions formulated within the lab-oratory frame of reference, as it is in this frame that measurements of concentration profiles are made.The flux of B atoms within the laboratory frame, eJB, is equal to the flux of B in the crystal frame, JB,(given by Eq. (59)) plus a flux of B due to a local shift of the crystal frame with velocity v l [75,69,2,76]

~JB ¼ JB þ xBv l

X

� �¼ �xBJA þ xAJB; ð71Þ

where the last identity is only valid as xV ! 0. A similar expression can be formulated for ~JA. As be-fore, we can apply Eqs. (58) and (59) to eliminate JA and JB and Eq. (58) to relate rCA to rCB for anequilibrium vacancy concentration, yielding
eJB ¼ �eDrCB; ð72Þ
where

eD ¼ xB

eLAA

xA�eLAB

xB

!þ xA

eLBB

xB�eLAB

xA

!" #xB

kBTdlB

dxB

� �lV¼0

: ð73Þ

This expression can be viewed as a Fickian flux equation for interdiffusion within the laboratory frameof reference. eD is commonly referred to as the interdiffusion coefficient. If the equilibrium vacancy con-


centration is independent of alloy composition then rCA ¼ �rCB and the interdiffusion coefficienttakes the familiar form

eD ¼ xBDA þ xADB; ð74Þ

as a result of Gibbs-Duhem and the fact that dxA ¼ �dxB when xV is independent of alloy composition.If the vacancy concentration is not independent of alloy concentration, this identity no longer holds[113,114] and Eq. (73) must be used to calculate the interdiffusion coefficient eD. The interdiffusioncoefficient eD describes intermixing between A and B in the laboratory frame of reference under thethermodynamic constraint that lV ¼ 0 at each point of the solid. It therefore is different from k�,which describes intermixing in a perfect crystal within the crystal frame of reference (in alloys havinga dilute vacancy concentration).

3.2. Comparison between the perfect crystal and the uniform source-sink approximation

3.2.1. Thermodynamically ideal alloyIt is useful to compare the intermixing coefficient for a perfect-crystal with a dilute concentration

of vacancies ðk�Þ and the interdiffusion coefficient within the uniform source-sink approximation ðeDÞ.Both metrics describe the same physical phenomenon of intermixing between A and B of a binary al-loy. Nevertheless, k� describes intermixing within a perfect crystal in the crystal frame of referencewhile eD describes intermixing in the laboratory frame of reference of a crystal containing abundantvacancy sources and sinks to maintain a uniform equilibrium vacancy concentration.

An appropriate starting point for a comparison between k� and eD is the ideal reference alloy, whichis both thermodynamically ideal as well as kinetically ideal (i.e., CA ¼ CB). For this alloy, it can be ver-ified using Eqs. (25), (62) and (33) that k� and eD are not only independent of concentration, but areactually equal to each other as xV ! 0. This is due to the absence of the Kirkendall effect in a kinet-ically ideal alloy. In the absence of initial gradients in the vacancy concentration, it is impossible togenerate a vacancy flux in a kinetically ideal alloy as a result of gradients of A and B concentrations.Therefore the crystal need not locally expand or contract to accommodate local supersaturations invacancy concentration. The lattice frame of reference does not move relative to the laboratory frameof reference and eD therefore describes the same mode of intermixing as k�.

The two metrics of interdiffusion become concentration dependent and diverge from each otheronce a thermodynamically ideal alloy deviates from kinetic ideality (i.e. CA – CB). This is illustratedin Fig. 10. While eD and k� approximate each other in the dilute limits, they deviate at intermediateconcentrations. In an alloy with CB ¼ 100� CA where A has similar properties to Al at 600 K(xV ¼ 1:4� 10�6;DEb ¼ 600 meV and m� ¼ 4� 1013 Hz, see Section 2.5), the difference is more thanan order of magnitude when using the analytical kinetic-transport coefficients eLij derived by Molekoet al. [79] (see Appendix B). For this alloy, eD exhibits a maximum at intermediate concentration.

The difference between eD and k� for alloys having large CB=CA ratios can be understood as follows.An alloy having very different vacancy exchange frequencies for the two components will exhibit astrong Kirkendall effect. Once a percolating network of the fast diffuser is established within a perfectcrystal, the vacancies can predominantly wander along this network, exchanging rarely with the slowdiffuser and thereby leading to a reduction in the efficiency of intermixing. Hence, k� can decrease asthe concentration of the fast diffuser is increased (Fig. 10a and b). If, however, abundant vacancysources and sinks are available, intermixing can be mediated by the Kirkendall effect. The spatialredistribution of the fast diffuser can then occur, not by intermixing with the slow diffuser withinthe crystal, but by exchanging with net vacancy fluxes. The fast diffuser then accumulates at grainboundaries and climbing dislocations where their concentration is low. The departure of fast diffusersfrom regions in which they have a high concentration causes local contraction of the crystal, while thearrival of the fast diffuser in regions in which their concentration is low causes local expansion of thecrystal. Hence eD, which describes this mechanism of interdiffusion, can be large in alloys in which thecomponents have very different hop frequencies.

For comparison, Fig. 10, also shows the interdiffusion coefficient eDDarken calculated within the Dar-ken approximation in which Eq. (65) is used as an approximation for DA and DB. In a random alloy,



10-13

10-12

10-11

diffu

sion

coe

ffici

ents

(cm

2 /s)

10-10

ΓB=10xΓA

(a)

D~

λ−

10-13

10-12

10-11

diffu

sion

coe

ffici

ents

(cm

2 /s)

10-10

(b)

ΓB=100xΓA

λ−

D~

Fig. 10. Various metrics of interdiffusion for a random alloy with (a) CB ¼ 10� CA and (b) CB ¼ 100� CA . The dashed line is eD,the circles are eDDarken calculated within the Darken approximation (i.e., the self-diffusion coefficients evaluated according to Eq.(65)) and the solid line corresponds to k� .


eDDarken, which neglects cross correlations between the trajectories of different diffusing atoms, is lessthan eD.

3.2.2. Thermodynamically non-ideal alloyWe again consider fcc Alð1�xÞLix for the purpose of comparing eD and k� in a thermodynamically non-

ideal alloy. Fig. 11 shows k�; eD and eDDarken as a function of Li concentration x, calculated at 600 K. Thevarious interdiffusion metrics do not differ substantially except within the L12 phase in which eDDarken

is now larger than both eD and k�. However, the concentration interval in which the solid solution ofAlð1�xÞLix is stable, is limited to x < 0:15. A comparison with the thermodynamically ideal alloys ofFig. 10a and b, shows that for small concentrations of the fast diffuser, the difference between eDand k� is also small in thermodynamically ideal alloys.

3.3. Discrete distribution of sources and sinks

So far, we have considered two extremes of substitutional diffusion: binary diffusion in a perfectcrystal and binary diffusion assuming a continuous and dense distribution of vacancy sources. Withinthe continuous source approximation, the presence of vacancy sources is assumed to only impose athermodynamic constraint on the driving forces for diffusion resulting from the condition that

10-14

10-13

10-12

10-11

0 0.1 0.2 0.3

Diff

usio

n co

effic

ient

(cm

2 /s)

two phase

region

L12

solid solution


Fig. 11. Various metrics of intermixing for the metastable fcc Alð1�xÞLix alloy. Grey circles are eD, filled circles are eD calculatedwithin the Darken approximation (in the solid solution, the filled circles are below the grey circles) and the empty squares arek�.


lV ¼ 0 throughout the solid. Any role that a high density of vacancy sources may play in promotingshort-circuit diffusion is neglected. The uniform source/sink approximation is conventionally madein textbook treatments of substitutional diffusion [2,69,76] and is also commonly used in quantitativeanalyses of diffusion couples [112,115].

A treatment of substitutional diffusion in a solid that contains a low density of vacancy sources andsinks is possible at a continuum level if we consider the alloy as consisting of regions that are perfectcrystals (e.g., grains) explicitly decorated and or bounded by discrete vacancy sources and sinks (dis-locations, grain boundaries and surfaces). Diffusion within the grains is then to be described with theperfect crystal Fickian flux expressions (35) while the vacancy sources and sinks serve as boundaryconditions to Eq. (35). This approach requires an accurate characterization of the spatial distributionof dislocations and grain boundaries within the solid.

We can distinguish between ideal vacancy sources, which maintain an equilibrium vacancy con-centration in its immediate vicinity, and non-ideal sources that require some degree of vacancy super-saturation or depletion before vacancies are annihilated or created. Dislocations accommodatevacancies through climb processes, which usually requires the recombination of partial dislocationsthrough thermal fluctuations. The effectiveness of this climb process is therefore sensitive to thestacking fault energy of the solid. Dimensionality of the source can be another factor that plays a rolein the effectiveness of accommodating vacancy supersaturations. Dislocations are one-dimensionalsources while grain boundaries are two-dimensional sources. Although dislocations are usually moreabundant and uniformly distributed throughout a solid as compared to grain boundaries, only simu-lations containing explicit sources and sinks will accurately indicate the relative effectiveness be-tween dislocations and grain boundaries in accommodating supersaturations in vacancy. A roughestimate for the grain size at which grain boundaries would dominate dislocations as vacancy sourcescan be obtained when simplifying assumptions, such as a simple geometry of the grain, are made[121].

As an illustration of a continuum simulation with explicit sources, we consider below an idealizeddiffusion couple initially containing equally spaced parallel grain boundaries and investigate the de-gree with which the evolution of the concentration profile predicted for a perfect crystal deviates fromthat of a solid with a continuous and dense distribution of sources.

Grain boundaries that act as ideal vacancy sources will locally pin the vacancy concentration at itsequilibrium value. Furthermore, the concentration profiles of A and B will be continuous across eachgrain boundary if they are ideally permeable to both A and B. These constraints on the various concen-trations impose a set of boundary conditions on the continuity equation for diffusion within each grain

Fig. 12.with CB

vacancyfor a crwere in


@CA

@t¼ rðDAArCA þ DABrCBÞ

@CB

@t¼ rðDBArCA þ DBBrCBÞ

ð75Þ

as described in Ref. [116]. The continuity equations within each grain can be solved numerically with afinite-difference scheme, accounting for the composition dependence of the diffusion coefficients.

The annihilation or creation of vacancies at each source can lead to local expansions or contractionsof the crystal. Accounting for this in a numerical simulation is possible by adjusting the distance be-tween grid points adjacent to the source by an amount that ensures that the vacancy concentration atthe source maintains its equilibrium value. The technical details of this numerical approach to treatdiscrete sources and sinks within a diffusion couple are described in Ref. [116].

Fig. 12 shows a concentration profile at intermediate time in the laboratory frame of referencepredicted with a continuum simulation of a diffusion couple initially containing five equal-sizedgrains (dashed line). The horizontal axis y in Fig. 12 is the position in dimensionless length (scaledwith the total length of the crystal) and the vertical axis bCB is the dimensionless concentration(scaled with lattice site density) as a function of position y. The hats indicate dimensionless quanti-ties. The fast diffuser B is initially located within the left half of the diffusion couple, while the slowdiffuser A occupies the right half of the diffusion couple. The diffusion coefficients used to describetransport within the grains are for a thermodynamically ideal alloy with CB ¼ 100� CA. Notice thatthe concentration profile is not smooth in the vicinity of a vacancy source. The spikes in concentra-tion around the vacancy source arise from a difference in vacancy exchange frequencies between Aand B atoms. The fast diffuser B accumulates at the source where vacancies are created and is de-pleted around sources (sinks) that annihilate vacancies. Uphill diffusion, whereby atoms diffuse uptheir concentration gradient, is only possible in a strictly binary system if the diffusion coefficientis negative. The apparent uphill diffusion in Fig. 12, emerges because each grain is effectively a ter-nary system and the large flux of vacancies in the vicinity of the vacancy sources preferentially ex-changes with the fast diffuser.

The presence of explicit vacancy sources also leads to a Kirkendall shift of the diffusion couple. Fur-thermore, differences in vacancy fluxes at the ends of each grain can lead to a Kirkendall-effect-in-

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

dimensionless position y

CBˆ

CBˆ pc

CBˆ cs

ˆ

Concentration profiles in the laboratory frame of reference of a diffusion couple for a thermodynamically ideal alloy¼ 100� CA at intermediate time. The solid black curve labeled bCpc

B is for a perfect crystal without internal or externalsources, the grey curve labeled bC cs

B is for a crystal with a continuous vacancy source distribution and the dashed curve isystal containing 5 grains. The positions of the grain boundaries and surfaces are indicated by filled circles. The grainsitially equal in size.


duced coarsening phenomenon [116], whereby some grains shrink in size and others grow, even in theabsence of capillarity effects (which are not present in this case since each grain boundary is assumedplanar). In the example of Fig. 12, the left two grains shrink due to vacancy ejection at sources in theinitially B-rich (left) side, while the other three grains grow due to vacancy injection at sources there.In a three-dimensional crystal with grains having complex three-dimensional shapes, any Kirkendallshift as well as grain size and shape change will cause stresses within the solid and increase the driv-ing force for void formation and decohesion along grain boundaries.

The profile of Fig. 12 illustrates the role of the Kirkendall effect in mediating intermixing in thepresence of vacancy sources. Due to a difference by a factor of 100 in vacancy exchange frequenciesbetween A and B, the fast diffuser B migrates to the A-rich regions by predominantly exchanging witha net vacancy flux (see Fig. 6a for the CB ¼ 100� CA alloy, showing that /, which measures the fractionof the vacancy flux that exchanges with B, is close to 1 above xB ¼ 0:2). The influx of B atoms in the A-rich regions is accommodated by a lengthening of the A-rich grains. The depletion of B on the B-richside is accommodated by a shrinkage of the B-rich grains. Due to the discrete nature of the vacancysources, enrichment of B on the A rich side occurs predominantly in the vicinity of the sources (spikesin concentration), while depletion of B in the B-rich region also occurs close to the sources and resultsin a contraction of the adjacent grains. Interdiffusion in the presence of vacancy sources clearly relieson the Kirkendall effect for alloys with a large CB=CA ratio, occurring to an important extent by an ex-change of the fast diffuser with net vacancy fluxes and a rigid shift of the slow diffuser through Kir-kendall-effect-induced grain shrinkage or growth.

Also illustrated in Fig. 12, are concentration profiles of the diffusion couple at intermediate timecalculated for a perfect crystal without any vacancy sources, bCpc

B , and within the continuous sourceapproximation, bCcs

B . The concentration profile for bCpcB ðyÞ was obtained by solving the continuity Eq.

(75) with zero fluxes at the surfaces. The concentration profile bCcsB ðyÞ is determined by solving a con-

tinuity equation applied to the interdiffusion flux expression, Eq. (72), for the uniform source sinkapproximation

@CcsB

@t¼ reDrCcs

B ð76Þ

within the laboratory frame of reference. In both continuity equations, the diffusion coefficients Dij

and eD are derived from the same set of transport coefficients eLij and therefore describe the same sys-tem, but within different approximations with respect to the treatment of vacancy sources and sinks.Fig. 12 shows that the profile with a finite number of vacancy sources is intermediate between thetwo extremes, bCpc

B and bCcsB . Intermixing for an alloy with a large difference in hop frequencies be-

tween A and B is the slowest in a perfect crystal, where hops of both A and B atoms are required.Intermixing increases with the density of vacancy sources. and becomes most effective when the va-cancy source density is high enough to regulate an equilibrium vacancy concentration throughoutthe solid.

As noted in Section 3.2.1, the perfect-crystal interdiffusion coefficient k� is equal to the continuoussource approximation interdiffusion coefficient eD for the kinetically ideal reference alloy ðCA ¼ CBÞ inthe dilute vacancy limit, as it does not exhibit the Kirkendall effect. Hence concentration profilesdetermined for diffusion couples containing discrete vacancy sources will be identical to those pre-dicted within the continuous source approximation in kinetically ideal alloys.

The time dependence of the concentration profile of a diffusion couple with discrete vacancysources should converge to that predicted by the uniform source approximation as the density of dis-crete sources is increased. A measure of the deviation between simulation results that account for ex-plicit sources and the solution using the continuous source approximation is a norm of the differencein the concentration profiles defined as

NDbC ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRbL0bCB yð Þ � bCcs

B yð Þ� �2

dybLvuuut ð77Þ


where bL is the dimensionless length of the one-dimensional diffusion couple (i.e., bL ¼ 1Þ; bCBðyÞ is theconcentration from the simulation with discrete sources and sinks and bC cs

B ðyÞ is the concentration pre-dicted within the uniform source-sink approximation.

Fig. 13 illustrates the norm, Eq. (77), as a function of a dimensionless time, t, for diffusion coupleshaving different numbers of grains with grain boundaries acting as ideal vacancy sources. The timeevolution of the norm N

DbC for thermodynamically ideal alloys depends on the CA=CB ratio when plot-ted versus the dimensionless time, defined according to

Fig. 13.with cowith (a

t ¼ ta2CA

L2 ; ð78Þ

where L is the total length of the diffusion couple and a is the atomic hop distance. The actual timedependence of the norm for a thermodynamically ideal alloy can be extracted from Fig. 13 by multi-plying the dimensionless time t by L2=a2CA. Since the initial concentration profiles are identical for

0 4x105 8x105 12x1050.00

0.04

0.08

0.12

0.16

0.0

0.1

0.2

0.3

0.4

NΔC^

NΔC^

t = t a2ΓA/L2^

0 4x105 8x105 12x105

t = t a2ΓA/L2^

perfect crystal

2 grains

4 grains

10 grains

perfect crystal

2 grains

4 grains

10 grains

(a)

(b)

The norm of the difference between the concentration profiles in systems with discrete sources and that in a systemntinuous sources as a function of dimensionless time for a variety of source densities for thermodynamically ideal alloys) CB ¼ 10� CA and (b) CB ¼ 100� CA .


each diffusion couple (left half pure B and right half pure A), the norm in all cases is zero at t=0. Asdiffusion proceeds, N

DbC initially increases, reaches a maximum and then decreases to zero with

increasing time. In the long time limit, all diffusion couples converge to a uniform concentration pro-file. The norms in Fig. 13 are overall smaller and converge more rapidly to zero as the source densityincreases, showing, as expected, that the time evolution of the concentration profile approaches thatpredicted by the continuous source approximation with increasing density of vacancy sources. Fig. 13also shows that the deviation between the discrete vacancy source concentration profile and that pre-dicted within the continuous source approximation increases with the CB=CA ratio.

It is also instructive to consider the time required to reach equilibrium in a planar diffusion coupleas a function of the number of discrete vacancy sources. The following norm

Fig. 14.numbediffuser

g ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRbL0 ðbCBðy; tÞ � bCeq

B Þ2dy

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRbL

0 ðbCBðy;0Þ � bCeqB Þ

2dy

r ð79Þ

measures the deviation of a diffusion couple from its final equilibrium state characterized by a uni-form concentration of all its components: in the initial state g ¼ 1 while in the final equilibrium stateg ¼ 0. Fig. 14 illustrates the elapsed time (normalized according to Eq. (78)) for the norm, Eq. (79), toreach g ¼ 0:001 as a function of the number of grains in a one-dimensional diffusion couple. The lim-iting case corresponds to the continuous source (cs) approximation. The simulations were performedfor a diffusion couple of length L ¼ 1 lm using values for CA, a and the vacancy concentration xV rep-resentative of Al. The fast diffuser B was assumed to have a hop frequency CB ¼ 10� CA. As is clearfrom Fig. 14, the time to reach equilibrium converges quite rapidly to that of the continuous sourceapproximation, leveling off around 10 grains (e.g., 11 discrete vacancy sources). For this particularsimulation, this implies that continuous source approximation behavior is nearly reached when planarvacancy sources are separated from each other by 100 nm. Larger separations between vacancysources for this model system lead to equilibration times that are sensitive to the inter-vacancy sourcespacing. These results clearly illustrate a transition of diffusion behavior from a perfect-crystal modelto a continuous-vacancy-source model as the vacancy source density increases.

0 5 10 15 cs

6

7

8

x105

t

number of grains

Elapsed time to reach the final equilibrium state of a planar diffusion couple (defined by g ¼ 0:001) as a function of ther of grains that act as discrete vacancy sources. The fast diffuser hop frequency is 10 times larger than that of the slowðCB ¼ 10� CAÞ. The continuous source approximation is denoted by (cs).


4. Conclusion

The aim of this paper has been to clarify the link between macroscopic transport properties in bin-ary substitutional alloys and atomistic hop frequencies that can be calculated from first principles. Wehave restricted the focus to crystalline binary substitutional solids in which diffusion is mediated byvacancies. The presence of vacancies in a binary crystalline alloy renders the solid a ternary system. Acomplicating factor in real substitutional alloys is the presence of vacancy sources and sinks, whichcan annihilate and create vacancies. Diffusion in a binary substitutional alloy is sensitive to the densityof vacancy sources and sinks and different metrics for diffusion can be formulated depending onwhether the alloy more closely approximates a perfect crystal or whether it can be assumed to containa continuous distribution of vacancy sources and sinks to maintain an equilibrium vacancy concentra-tion throughout the solid.

Our starting point has been a description of a formalism to calculate kinetic-transport coefficientsin a perfect crystal without vacancy sources and sinks. The diffusion coefficients in non-dilute sub-stitutional alloys are derived from a matrix product of a thermodynamic factor and a kinetic factor.Here, we have derived expressions relating the thermodynamic factor to the Hessian of the free en-ergy of substitutional solids and shown how this Hessian can be calculated within the grand canon-ical ensemble using Monte Carlo simulations applied to a cluster expansion of the configurationalenergy of the alloy. The kinetic factor is a matrix of Onsager kinetic-transport coefficients, whichcan be calculated with Kubo-Green expressions, first derived by Allnatt [67–69] for diffusion in a per-fect crystal. These expressions are based on ensemble averages of the square of the displacement ofthe center of mass of diffusing A and B atoms at equilibrium. With this statistical mechanical formal-ism laid out, it is possible to predict perfect-crystal diffusion coefficients of binary substitutional al-loys from first principles using well-established techniques from alloy theory based on the clusterexpansion formalism.

While the matrix of diffusion coefficients describing transport in a crystalline binary alloy is essen-tial input for realistic continuum simulations of diffusional processes, it does not provide direct infor-mation about the dominant modes of diffusion in a substitutional solid, namely intermixing andvacancy fluxes that can lead to the Kirkendall effect. In this respect Kehr et al. [51] demonstrated that,for perfect crystalline binary solids having regular solution model thermodynamics, the eigenvalues ofthe diffusion coefficient matrix are related to the vacancy mobility and the rate of intermixing in thedilute-vacancy limit. Here, we have extended the analysis of Kehr et al. [51] and introduced two addi-tional metrics derived from the diffusion coefficient matrix, / and d, that provide additional insightinto the dominant modes of binary substitutional diffusion. These metrics, which serve to characterizethe Kirkendall effect, have been defined such that they also measure the deviation of the alloy fromboth thermodynamic and kinetic ideality.

Knowledge of perfect crystal kinetic-transport coefficients and diffusion coefficients also allows usto analyze diffusion in solids containing vacancy sources and sinks such as grain boundaries andclimbing dislocations. This is possible if any contribution to transport from short-circuit diffusiondue to crystalline defects can be neglected. At the continuum level, two approaches were describedto account for vacancy sources and sinks. The first, which is the standard textbook treatment of sub-stitutional diffusion, relies on the assumption that the solid contains a sufficiently dense vacancysource and sink distribution to regulate an equilibrium vacancy concentration throughout the solid.Within this approximation, two metrics are defined: One measures the drift of the crystal frame of ref-erence relative to the laboratory frame of reference and a second describes interdiffusion within thelaboratory frame of reference. An alternative approach to account for vacancy sources and sinks withina continuum description is to assume a discrete distribution of sources which serve as boundary con-ditions to the Fickian flux expressions for a perfect crystal.

We have shown here that the interdiffusion coefficient in the continuous source/sink approxima-tion differs substantially from the intermixing coefficient of a perfect crystal in alloys with a high sus-ceptibility to the Kirkendall effect. In a perfect crystal, intermixing requires both the fast and slowdiffuser to migrate in order to dissipate a concentration gradient among the components of the solid.In a solid containing a continuous distribution of vacancy sources and sinks, interdiffusion can be


mediated by the Kirkendall effect whereby only the fast diffuser migrates and the slow diffuser isredistributed in space by a rigid shift of the lattice.

With the formalism of substitutional diffusion established, we are now in a position to explore a widevariety of alloys from first principles to shed light on the relationship between complex atomic hopmechanisms and macroscopic metrics for diffusion. This will provide guidance in the development ofphysically motivated mobility matrices to conveniently and self-consistently parameterize experimen-tal diffusion data [117]. At the macroscopic level, three-dimensional continuum simulations of perfectcrystals surrounded and decorated by one and two dimensional vacancy sources and sinks will resolveimportant questions about the precise role of these defects on diffusion. Such simulations will indicatecritical densities of sources required before the commonly used continuous source/sink approximationbecomes valid. They will also indicate how sensitive substitutional diffusion is to the nature of vacancysources and sinks (e.g., one dimensional versus two dimensional) [120].

Acknowledgments

The authors would like to acknowledge helpful discussions with John Morral and William Boettin-ger. AVDV acknowledges support from NSF under Grant Nos. DMR 0605700 and DMR 0748516. H.C. Y.and K. T. acknowledge support from NSF under Grant Nos. DMR-0502737 and DMS-05112324 .

Appendix A. Here we derive an expression for the thermodynamic factor for perfect crystal dif-fusion in terms of particle number fluctuations that can be calculated with grand canonical MonteCarlo simulations. The thermodynamic factor is defined by

eH ¼ 1kBT

@~lA

@xA

@~lA

@xB

@~lB

@xA

@~lB

@xB

0BB@1CCA: ð80Þ

It is convenient to consider the inverse of this matrix, which can be expressed as

eH�1 ¼ kBT

@xA

@~lA

@xA

@~lB

@xB

@~lA

@xB

@~lB

0BB@1CCA: ð81Þ

Within the grand canonical ensemble, this matrix can be expressed in terms of the variances of thenumber of A and B atoms within the perfect crystal at fixed ~lA and ~lB according to (see for exampleHill [99])

eH�1 ¼ 1M

hN2Ai � hNAi2 hNANBi � hNAihNBi

hNANBi � hNAihNBi hN2Bi � hNBi2

!: ð82Þ

Taking the inverse of this matrix yields Eq. (24).

Appendix B. Moleko et al. [79] derived a set of analytical expressions for eLij for a random alloy inwhich CA – CB. In the notation and formalism developed here, these expressions take the form

eLAA ¼ xV xAqa2CA 1� 2xBCA

K

� �;

eLBB ¼ xV xBqa2CB 1� 2xACB

K

� �;

eLAB ¼2qa2CACBxV xAxB

K

ð83Þ


where

K ¼ 12ðF þ 2ÞðxACA þ xBCBÞ � CA � CB þ 2ðxACB þ xBCAÞ

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ðF þ 2ÞðxACA þ xBCBÞ � CA � CB

� �2

þ 2FCACB

sð84Þ

and

F ¼ 2f1� f

: ð85Þ

As with the kinetically ideal expressions for eLij, f is the correlation factor for a single component solidhaving the crystal structure of the A–B alloy, and is a function of xV .

As has been shown by Hartmann et al. [55] and illustrated in Fig. 2, these coefficients agree verywell with kinetic Monte Carlo approximations of the eLij. The circles in Fig. 2 were calculated using aMonte Carlo cell having dimensions of 8 � 8 � 8 primitive fcc unit cells, containing one vacancy. Thismeans that a vacancy never meets other vacancies in these simulations, which is a reasonable approx-imation in the dilute vacancy limit.

The analytical expressions, Eq. (83), show that for a thermodynamically ideal alloy, the eLij scale lin-early with the vacancy concentration for dilute vacancy concentrations. This dependence on vacancyconcentration should persist for thermodynamically non-ideal alloys as long interactions among dif-ferent vacancies are negligible (i.e. atomic transport is dominated by single vacancy hops). The eLij cantherefore be calculated at a convenient vacancy concentration in kinetic Monte Carlo simulations andthen adjusted to a value consistent with the actual vacancy concentration by multiplying the calcu-lated eLij by ðxactual

v =xsimulationv Þ where xsimulation

v is the vacancy concentration in the simulation cell andxactual

v is the vacancy concentration in the actual system.Both ensemble averages and time averages were performed to numerically evaluate the Kubo-

Green expressions, Eq. (32), plotted in Fig. 2. At each concentration, 4000 different random initial con-figurations were used. For each one of these initial configurations, 1000 Monte Carlo passes were per-formed, of which the first 500 were equilibriation steps for each random starting configuration. EachMonte Carlo pass corresponds to a sequence of hops of the vacancy equal to the number of lattice sitesin the simulation cell. Averaging was performed by sampling values for eLij at different times for all thedifferent initial configurations.

Appendix C. In this appendix, we show that for alloys having a dilute vacancy concentration, k�

and d scale with xV , while kþ and / do not. We demonstrate this for alloys in which the kinetic-transport coefficients eLij are proportional to xV in the dilute vacancy limit (see Appendix B).

It is useful to express the chemical potentials more explicitly in terms of the concentration vari-ables. elA ¼ lA � lV , where both lV and lA can be written as lo

V ðTÞ þ kBT lnðcV xV Þ andlo

AðTÞ þ kBT lnðcAxAÞ, respectively. The activity coefficients cV and cA are each a function of xA and xB

and serve as a measure for the deviation from ideality (i.e., cV and cA equal 1 in an ideal solution).The chemical potential elA can be expressed as
elA ¼ lo
A � loV þ kBT ln cA þ kBT ln xA � kBT ln cV � kBT ln xV : ð86Þ

elB can be written in a similar way. The natural logarithms of the activity coefficients ci are referred toas excess partial molar free energies and can be Taylor expanded in terms of the independent concen-tration variables around a reference concentration [118,119]. Our scaling analysis of kþ; k�; d and / re-lies on the assumption that the Taylor expansions of ln ci do not diverge in the limit of a dilute vacancyconcentration xV ! 0 (i.e., they do not scale as ln xV or 1=xb

V where b is a positive number). We willrefer to such alloys as ‘‘thermodynamically well-behaved.”

Using expressions for ~li in the form of Eq. (86), we can write the elements of the thermodynamicfactor matrix such as HAA as


eHAA ¼@

~lA

kBT

� �@xA

¼ 1xVþ 1

xAþ fAA; ð87Þ

where fAA is defined according to

fij ¼@ ln ci

@xj� @ ln cV

@xj: ð88Þ

Note that all partial derivatives with respect to xA are taken at constant xB and constant number of lat-tice sites M. In a similar manner

~HBB ¼@

~lB

kBT

� �@xB

¼ 1xVþ 1

xBþ fBB; ð89Þ

while

eHAB ¼@

~lA

kBT

� �@xB

¼ 1xVþ fAB;

~HBA ¼@

~lB

kBT

� �@xA

¼ 1xVþ fBA:

ð90Þ

With the elements of the thermodynamic factor matrix written as in Eqs. (87), (89) and (90), we canexpress the elements of the diffusion coefficient matrix as

DAA ¼1xVðeLAA þ eLABÞ þ eLAA

1xAþ fAA

� �þ eLABfAB;

DAB ¼1xVðeLAA þ eLABÞ þ eLAAfAB þ eLAB

1xBþ fBB

� �;

DBA ¼1xVðeLBB þ eLABÞ þ eLBBfAB þ eLAB

1xAþ fAA

� �;

DBB ¼1xVðeLBB þ eLABÞ þ eLBB

1xBþ fBB

� �þ ~LABfAB:

ð91Þ

We can then use these expressions for Dij to determine the scaling with vacancy concentration ofd;/; k� and kþ.

C.1. Scaling of d

Substituting the above Dij into d ¼ DBB þ DAB � DBA � DAA, we obtain

d ¼ �ðeLAA þ eLABÞ1xAþ fAA

� �þ ðeLAB þ eLBBÞ

1xBþ fBB

� �� ðeLBB � eLAAÞfAB: ð92Þ

In the dilute vacancy limit, the eLij scale linearly with the vacancy concentration xV while the coeffi-cients of eLij in this expression to first order should be independent of xV . This is true for the terms1=xA and 1=xB and should be true for the fij if the alloy is thermodynamically well-behaved as definedabove. Hence d to first order scales linearly with xV .

C.2. Scaling of /

The parameter / is defined as the ratio of DBA, an off-diagonal diffusion coefficient for perfect crys-tal binary diffusion, and the sum of the off-diagonal diffusion coefficients DAB þ DBA. Using Eq. (91), wecan write


DAB þ DBA ¼1xVðeLAA þ eLBB þ 2eLABÞ þ ðeLAA þ eLBBÞfAB þ eLAB

1xAþ fAA þ

1xBþ fBB

� �: ð93Þ

The first term in Eq. (93), ðeLAA þ eLBB þ 2eLABÞ=xV , is independent of the vacancy concentration, as the eLij

in the dilute vacancy limit are proportional to xV . The other terms do scale linearly with xV in the dilutevacancy limit. Likewise for DBA, the first term in Eq. (91), ðeLBB þ eLABÞ=xV , is independent of xV . The ratioof DBA to DAB þ DBA therefore does not scale with the vacancy concentration.

C.3. Scaling of kþ and k�

To determine the scaling with xV of kþ and k�, we follow the same approach used by Kehr et al. [51],which they applied to a regular-solution model. The eigenvalues of the diffusion coefficient matrix Dij

are related to its elements according to

k� ¼ DAA þ DBB

21�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4

DABDBA � DAADBB

ðDAA þ DBBÞ2

s !ð94Þ

Using the Eq. (91) for the Dij, the following relations can be derived

DAA þ DBB ¼1xVðeLAA þ eLBB þ 2eLABÞ þ OðxV Þ; ð95Þ

which to first order in xV is independent of the vacancy concentration, while

DABDBA � DAADBB ¼ �1xVðeLAA

eLBB � eL2ABÞ �

1xAþ 1

xBþ fAA þ fBB � 2f AB

� �þ Oðx2

V Þ: ð96Þ

This expression scales linearly with the vacancy concentration xV . The second factor in Eq. (96) can bewritten as partial derivatives of chemical potentials according to

Hmix ¼1

kBT@ð~lA � ~lBÞ

@xA� @ð

~lA � ~lBÞ@xB

� �¼ 1

kBT@2g@x2

A

þ @2g@x2

B

� 2@2g

@xA@xB

!; ð97Þ

where the second equality follows from the fact that ~li ¼ @g=@xi with gðxA; xBÞ being the free energynormalized per crystal site of the solid containing A, B and vacancies. The second equality in Eq. (97)shows that ~Hmix is equal to the curvature of gðxA; xBÞ along the direction of a constant vacancy concen-tration, i.e., dxA ¼ �dxB.

The ratio of diffusion coefficients appearing under the square root in Eq. (94)

DABDBA � DAADBB

ðDAA þ DBBÞ2¼�xV

eLAAeLBB � eL2

AB

� �1xAþ 1

xBþ fAA þ fBB � 2f AB

� �ðeLAA þ eLBB þ 2eLABÞ

þ Oðx2V Þ ð98Þ

scales linearly with xV and is much smaller than one. This remains true even when xA ! 0 or xB ! 0where 1=xA or 1=xB diverge respectively. In these dilute limits, the alloy can be treated as thermody-namically ideal, allowing us to equate the kinetic coefficients ~Lij to the analytical expressions of Mole-ko et al. [79] (see Eq. (83) in Appendix B). The analytical expressions of eLAA and eLAB contain a factor xA,which cancel the 1=xA term, while the analytical expressions of eLBB and eLAB contain a factor xB, whichcancel the 1=xB term. Hence even in the dilute limits (xA ! 0 or xB ! 0), the ratio, Eq. (98), scaleswith xV allowing us to Taylor expand the square root in Eq. (94) in the dilute vacancy limit. Keepingall terms up to order xV , yields expressions, similar to the results of Kehr et al. [51],

kþ ¼ 1xVðeLAA þ eLBB þ 2eLABÞ þ OðxV Þ ð99Þ

and

k� ¼eLAAeLBB � eL2

ABeLAA þ 2eLAB þ eLBB

eHmix þ Oðx2V Þ: ð100Þ


This shows that for dilute vacancy concentrations, kþ is independent of xV , while k� is linearly propor-tional to xV .

As was pointed out by Kehr et al. [51], the expression for k�, Eq. (100), can be viewed as a factor-ization of the perfect-crystal interdiffusion coefficient into a product of a kinetic factor and a thermo-dynamic factor, Hmix. The derivation here extends the treatment of Kehr et al. [51] beyond a simpleregular solution model.

Appendix D. Here we derive relationships between the matrix of diffusion coefficients,DAA;DAB;DBA and DBB, and the metrics for perfect crystal substitutional diffusion, kþ; k�; d and /. Theparameters kþ and k� are the eigenvalues of the diffusion coefficient matrix and are given by Eq. (94)while d and / are defined in terms of the Dij according to Eqs. (38) and (39). By inverting these fourrelations, the following expressions can be derived for DAA;DAB;DBA and DBB in terms of kþ; k�; d and /

DAA ¼12ðkþ þ k�Þ � d� 2/� 1

1� /

� �DAB

� �DBB ¼

12ðkþ þ k�Þ þ dþ 2/� 1

1� /

� �DAB

� �DAB ¼ ð1� /Þ �ð2/� 1Þdþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ � k�Þ2 � 4/ð1� /Þd2

q� �DBA ¼

/1� /

DAB:

ð101Þ

For the kinetically ideal reference alloy, d ¼ 0, and the above expressions reduce to

DAA ¼ ð1� /Þkþ þ /k�;

DBB ¼ /kþ þ ð1� /Þk�;DAB ¼ ð1� /Þðkþ � k�Þ;DBA ¼ /ðkþ � k�Þ:

ð102Þ

For an alloy that is not thermodynamically and kinetically ideal the following relations emerge for al-loys with a dilute vacancy concentration

DAA ¼ ð1� /Þkþ þ /k� � 2/ð1� /Þd;DBB ¼ /kþ þ ð1� /Þk� þ 2/ð1� /Þd;DAB ¼ ð1� /Þðkþ � k� � ð2/� 1ÞdÞ;DBA ¼ /ðkþ � k� � ð2/� 1ÞdÞ:

ð103Þ

These expressions are derived from Eq. (101) by Taylor expanding the square root and neglecting allterms of order x2

V and higher. The above expressions are therefore accurate to first order in the vacancyconcentration with an error of order Oðx2

V Þ.

References

[1] Christian JW. The theory of transformations in metals and alloys: an advanced textbook in physical metallurgy. Oxford,New York: Pergamon Press; 2002.

[2] Balluffi RW, Allen SM, Carter WC. Kinetics of materials. Hoboken (NJ): Wiley-Interscience; 2005.[3] Hillert M. Jernkontorets Ann 1957;141:757.[4] Cahn JW. Acta Metall 1961;9:795.[5] Lifshitz IM, Slyozov VV. J Phys Chem Solids 1961;19:35.[6] Cook HE, de Fontaine D, Hilliard JE. Acta Metall 1969;17:765.[7] Voorhees PW. Annu Rev Mater Sci 1992;22:197.[8] Hillert M. Acta Mater 1999;47:4481.[9] Soisson F, Martin G. Phys Rev B 2000;62:203.

[10] Moore KT, Johnson WC, Howe JM, Aaronson HI, Veblen DR. Acta Mater 2002;50:943.[11] Thornton K, Ågren J, Voorhees PW. Acta Mater 2003;51:5675.[12] Hafner J, Wolverton C, Ceder G. MRS Bull 2006;31:659.[13] Sanchez JM, Ducastelle F, Gratias D. Physica A 1984;128:334.


[14] Mohri T, Sanchez JM, de Fontaine D. Acta Metall 1985;33:1171.[15] Rabe KM, Joannopoulos JD. Phys Rev Lett 1987;59:570.[16] Wei SH, Ferreira LG, Zunger A. Phys Rev B 1990;41:8240.[17] Sanchez JM, Stark JP, Moruzzi VL. Phys Rev B 1991;44:5411.[18] Laks DB, Ferreira LG, Froyen S, Zunger A. Phys Rev B 1992;46:12587.[19] Zhong W, Vanderbilt D, Rabe KM. Phys Rev Lett 1994;73:1861.[20] de Fontaine D. In: Ehrenreich H, Turnbull D, editors. Solid state physics. New York: Academic Press; 1994. p. 33.[21] Sluiter MHF, Watanabe Y, de Fontaine D, Kazazoe Y. Phys Rev B 1996;53:6136.[22] Waghmare UV, Rabe KM. Phys Rev B 1997;55:6161.[23] Ozolins V, Wolverton C, Zunger A. Phys Rev B 1998;57:6427.[24] Van der Ven A, Aydinol MK, Ceder G, Kresse G, Hafner J. Phys Rev B 1998;58:2975.[25] Asta M, Foiles SM, Quong AA. Phys Rev B 1998;57:11265.[26] Ozolins V, Asta M. Phys Rev Lett 2001;86:448.[27] Wolverton C, Ozolins V. Phys Rev Lett 2001;86:5518.[28] van de Walle A, Ceder G. J Phase Equilib 2002;23:348.[29] van de Walle A, Asta M. Model Simul Mater Sci Eng 2002;10:521.[30] Mohri T, Chen Y. Mater Trans 2002;43:2104.[31] Van der Ven A, Ceder G. Phys Rev B 2005;71:054102.[32] Wang JW, Wolverton C, Muller S, Liu ZK, Chen LQ. Acta Mater 2005;53:2759.[33] Hart GLW, Blum V, Walorski MJ, Zunger A. Nat Mater 2005;4:391.[34] Sluiter MHF, Colinet C, Pasturel A. Phys Rev B 2006;73:174204.[35] Zhou F, Maxisch T, Ceder G. Phys Rev Lett 2006;97:155704.[36] Xu Q, Van der Ven A. Intermetallics 2009;17:319.[37] Van der Ven A, Ceder G, Asta M, Tepesch PD. Phys Rev B 2001;64:184307.[38] Van der Ven A, Ceder G. Phys Rev Lett 2005;94:045901.[39] Van der Ven A, Thomas JC, Xu Q, Swoboda B, Morgan D. Phys Rev B 2008;78:104306.[40] Kikuchi R, Sato H. J Chem Phys 1969;51:161.[41] Kikuchi R, Sato H. J Chem Phys 1970;53:2702.[42] Sato H, Kikuchi R. Phys Rev B 1983;28:648.[43] Sato H, Ishikawa T, Kikuchi R. J Phys Chem Solids 1985;46:1361.[44] Mohri T. Acta Metall Mater 1990;38:2455.[45] Nastar M, Dobretsov V Yu, Martin G. Phil Mag A 2000;80:155.[46] Nastar M, Clouet E. Phys Chem Chem Phys 2004;6:3611.[47] Nastar M, Barbe V. Adv Eng Mater 2006;8:1239.[48] Murch GE. In: Murch GE, Nowick AS, editors. Diffusion in crystalline solids. Orlando, Florida: Academic Press; 1984. p.

379.[49] Murch GE, Thorn RJ. Phil Mag 1977;35:493.[50] Kehr KW, Kutner R, Binder K. Phys Rev B 1981;23:4931.[51] Kehr KW, Binder K, Reulein SM. Phys Rev B 1989;39:4891.[52] Tringides M, Gomer R. Surface Sci 1984;145:121.[53] Uebing C, Gomer R. J Chem Phys 1991;95:7626.[54] Belova IV, Murch GE. Phil Mag A 2000;80:599.[55] Hartmann MA, Weinkamer R, Fratzl P, Svoboda J, Fischer FD. Phil Mag 2005;85:1243.[56] Chandler D. Introduction of modern statistical mechanics. Oxford University Press; 1987.[57] Onsager L. Phys Rev 1931;37:405.[58] Onsager L. Phys Rev 1931;38:2265.[59] Callen HB, Welton TA. Phys Rev 1951;83:34.[60] Callen HB, Greene RF. Phys Rev 1952;86:702.[61] Green MS. J Chem Phys 1952;20:1281.[62] Green MS. J Chem Phys 1954;22:398.[63] Kubo R. J Phys Soc Jpn 1957;12:570.[64] Kubo R, Yokota M, Nakajima S. J Phys Soc Jpn 1957;12:1203.[65] Zwanzig R. J Chem Phys 1964;40:2527.[66] Zwanzig R. Annu Rev Phys Chem 1965;16:67.[67] Allnatt AR. J Chem Phys 1965;43:1855.[68] Allnatt AR. J Phys C: Solid State Phys 1982;15:5605.[69] Allnatt AR, Lidiard AB. Atomic transport in solids. Cambridge: Cambridge University Press; 1993.[70] Bortz AB, Kalos MH, Lebowitz JL. J Comput Phys 1975;17:10.[71] Bulnes FM, Pereyra VD, Riccardo JL. Phys Rev E 1998;58:86.[72] Andersson JO, Ågren J. J Appl Phys 1992;72:1350.[73] Engström A, Höglund L, Ågren J. Metall Mater Trans A 1994;25:1127.[74] Borgenstam A, Engström A, Höglund L, Ågren J. J Phase Equilib 2000;21:269.[75] Howard RE, Lidiard AB. Rep Prog Phys 1964;27:161.[76] Glicksman ME. Diffusion in solids. Wiley-Interscience Publication; 2000.[77] Kirkendall EO. Trans Am Inst Min Metall Eng 1942;147:104.[78] Smigelskas AD, Kirkendall EO. Trans Am Inst Min Metall Eng 1947;171:130.[79] Moleko LK, Allnatt AR, Allnatt EL. Phil Mag A 1989;59:141.[80] de Groot SR, Mazur P. Non-equilibrium thermodynamics. New York: Dover Publications; 1984.[81] Cahn JW, Larche FC. Scripta Met 1983;17:927.[82] Larche FC, Cahn JW. Acta Metall 1985;33:331.


[83] Voorhees PW, Johnson WC. Solid State Phys 2004;59:1.[84] Lupis CHP. Chemical thermodynamics of materials. New Jersey: Prentice Hall P T R; 1983.[85] Ceder G. Comput Mater Sci 1993;1:144.[86] Craievich PJ, Weinert M, Sanchez JM, Watson RE. Phys Rev Lett 1994;72:3076.[87] Sliwko VL, Mohn P, Schwarz K, Blaha P. J Phys Cond Matter 1996;8:799.[88] Craievich PJ, Sanchez JM, Watson RE, Weinert M. Phys Rev B 1997;55:787.[89] Souvatzis P, Eriksson O, Katsnelson MI, Rudin SP. Phys Rev Lett 2008;100:095901.[90] Bhattacharya J, Van der Ven A. Acta Mater 2008;56:4226.[91] Ozolins V. Phys Rev Lett 2009;102:065702.[92] Newman MEJ, Barkema GT. Monte Carlo methods in statistical physics. USA: Oxford University Press; 1999.[93] Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E. J Chem Phys 1953;21:1087.[94] Hohenberg P, Kohn W. Phys Rev B 1964;136:B864.[95] Kohn W, Sham LJ. Phys Rev 1965;140:1133.[96] Jones RO, Gunnarsson O. Rev Mod Phys 1989;61:689.[97] Parr RG, Yang W. Density-functional theory of atoms and molecules. Oxford University Press; 1989.[98] van de Walle A, Ceder G. Rev Mod Phys 2002;74:11.[99] Hill TL. Statistical mechanics. New York: Dover Publications; 1987.

[100] Eyring H. J Chem Phys 1935;3:107.[101] Hanggi P, Talkner P, Borkovec M. Rev Mod Phys 1990;62:251.[102] Flynn CP, Stoneham AM. Phys Rev B 1970;1:3966.[103] Vineyard GH. J Phys Chem Solids 1957;3:121.[104] LeClaire AD, Lidiard AB. Phil Mag 1956;1:518.[105] Manning JR. Phys Rev B 1971;4:1111.[106] Belova IV, Murch GE. Phil Mag A 2000;80:1469.[107] Murch GE, Belova IV. Phil Mag A 2000;80:2365.[108] Peterson NL. In: Nowick AS, Burton JJ, editors. Diffusion in solids: recent developments. New York: Academic; 1975. p.

115.[109] Kao CR, Chang YA. Intermetallics 1993;1:237.[110] Herzig C, Divinski S. Intermetallics 2004;12:993.[111] Darken LS. Trans Am Inst Min (Metall) Eng 1948;175:184.[112] Höglund L, Ågren J. Acta Mater 2001;49:1311.[113] Qin Z, Murch GE. Phil Mag A 1995;71:323.[114] Bencze L, Raj DD, Kath D, Oates WA, Singheiser L, Hilpert K. Metall Mater Trans B 2004;35:867.[115] Boettinger WJ, McFadden GB, Coriell SR, Sekerka RF, Warren JA. Acta Mater 2005;53:1995.[116] Yu HC, Yeon DH, Van der Ven A, Thornton K. Acta Mater 2007;55:6690.[117] Campbell CE, Boettinger WJ, Kattner UR. Acta Mater 2002;50:775.[118] Lupis CPH, Elliot JF. Acta Met 1966;14:529.[119] Lupis CPH. Acta Met 1968;16:1365.[120] Yu HC, Van der Ven A, Thornton K. Appl Phys Lett 2008;93:091908.[121] Yu HC, Yeon DH, Li X, Thornton K. Acta Mater 2009, in press, doi:10.1016/j.actamat.2009.07.033.

http://dx.doi.org/10.1016/j.actamat.2009.07.033

progress in materials scienceweb.mit.edu/ceder/publications/van_der_ven_ceder_2010.pdf · a. van...

Documents