Phase Field Crystal Modeling and Classical Density Functional Theory

K. R. Elder1, Nikolas Provatas2, Joel Berry1,3, Peter Stefanovic2, and Martin Grant31Department of Physics, Oakland University, Rochester, MI, 48309-4487

2 Department of Materials Science and Engineering and Brockhouse Institute for Materials Research,McMaster University, Hamilton, ON, Canada L8S-4L7 and

3 Physics Department, Rutherford Building, 3600 rue University,McGill University, Montreal, Quebec, Canada H3A 2T8

(Dated: November 27, 2006)

In this paper the relationship between the classical density functional theory of freezing andphase field modeling is examined. More specifically a connection is made between the correlationfunctions that enter density functional theory and the free energy functionals used in phase fieldcrystal modeling and standard models of binary alloys (i.e., regular solution model). To demonstratethe properties of the phase field crystal formalism a simple model of binary alloy crystallization isderived and shown to simultaneously model solidification, phase segregation, grain growth, elasticand plastic deformations in anisotropic systems with multiple crystal orientations on diffusive timescales.

PACS numbers: 05.70.Ln, 64.60.My, 64.60.Cn, 81.30.Hd

Contents

I. Introduction 1

II. Classical Density Functional Theory ofFreezing 3

III. Analysis of Free Energy Functionals 4A. Pure Materials 4B. Binary Alloys 5

1. Liquid phase properties 52. Crystalline phase properties 6

C. Simple Binary Alloy Model 71. Linear elastic constants 82. Calculation of alloy phase diagrams 8

IV. Dynamics 8

V. Applications 10A. Eutectic and Dendritic Solidification 10B. Epitaxial Growth 10

1. Numerical simulations 11C. Dislocation Motion in Spinodal

Decomposition 121. Numerical Simulations 13

VI. Discussion and Conclusions 14

VII. Acknowledgments 14

VIII. Simple Binary Alloy Model 14

References 15

I. INTRODUCTION

The formalism for calculating equilibrium states wasestablished many years ago by Gibbs, Boltzmann and

others. While this formalism has proved remarkably suc-cessful there are many systems which never reach equilib-rium, mainly due to the existence of metastable or longlived transient states. This is most apparent in solid ma-terials. For example it is very unlikely that the reader issitting a room containing any single crystals except itemsproduced with considerable effort such as the silicon chipsin computers. In fact the vast majority of naturally oc-curring or engineered materials are not in equilibriumand contain complex spatial structures on nanometer,micron or millimeter length scales. More importantlymany material properties (electrical, optical, mechani-cal, etc.) are strongly influenced by the non-equilibriumstructures that form during material processing. For ex-ample the yield strength of a polycrystal varies as theinverse square of the average grain size.

The study of non-equilibrium microstructure forma-tion has seen considerable advances through the useof the phase field approach. This methodology mod-els the dynamics of various continuum fields that col-lectively characterize microstructure in phase transfor-mations. For example, phase field or continuum modelshave been used to simulate spinodal decomposition [1],order-disorder transition kinetics [2], ordering of block-copolymer melts [3], solidification of pure and binary sys-tems [4–8] and many other systems. In these phenomenathe evolution of the appropriate field(s) (e.g., solute con-centration in spinodal decomposition) is assumed to bedissipative and driven by minimizing a phenomenologicalfree energy functional [1].

Advances in the phase field modeling of solidificationphenomena have followed a progression of innovations,beginning with the development of free energies that cap-ture the thermodynamics of pure materials [4–6] and al-loys [7, 8]. Several modification were then proposed [9–11] to simplify numerical simulations and improve com-putational efficiency. Perhaps the most important inno-vation was the development of matched asymptotic anal-

2

ysis techniques that directly connect phase field modelparameters with the classical Stefan (or sharp-interface)models for pure materials or alloys [12–15]. These tech-niques were complimented by new adaptive mesh refine-ment algorithms [16, 17], whose improved efficiency sig-nificantly increased the length scales accessible by numer-ical simulations, thus enabling the first experimentallyrelevant simulations of complex dendritic structures andtheir interactions in organic and metallic alloys [18–22].

A weakness of the traditional phase field methodol-ogy is that it is usually formulated in terms of fieldsthat are spatially uniform in equilibrium. This eliminatesmany physical features that arise due to the periodic na-ture of crystalline phases, including elastic and plasticdeformation, anisotropy and multiple orientations. Tocircumvent this problem traditional phase field modelshave been augmented by the addition of one or moreauxiliary fields used to describe the density of disloca-tions [23–25] continuum stress and strain fields [26, 27]and orientation fields [28–30]. These approaches haveproven quite useful in various applications such as poly-crystalline solidification [24, 28–32]. Nevertheless it hasproven quite challenging to incorporate elasto-plasticity,diffusive phase transformation kinetics and anisotropicsurface energy effects into a single, thermodynamicallyconsistent model.

Very recently a new extension to phase field model-ing has emerged known as the phase field crystal method(PFC) [33–35]. This methodology describes the evolutionof the atomic density of a system according to dissipa-tive dynamics driven by free energy minimization. In thePFC approach the free energy functional of a solid phaseis minimized when the density field is periodic. As dis-cussed in prior publications [33–35] the periodic natureof the density field naturally gives rise to elastic effects,multiple crystal orientations and the nucleation and mo-tion of dislocations. While these physical features are in-cluded in other atomistic approaches (such as moleculardynamics) a significant advantage of the PFC method isthat, by construction, it is restricted to operate on diffu-sive time scales not on the prohibitively small time scalesassociated with atomic lattice vibrations. The approachis similar to the atomic density function theory that wasrecently proposed by Jin and Khachaturyan [36]. In thecase of pure materials the PFC approach has been shown[33, 34] to model many phenomena dominated by atomicscale elastic and plastic deformation effects. These in-clude grain boundary interactions, epitaxial growth andthe yield strength of nano-crystals.

The original PFC model is among the simplest mathe-matical descriptions that can self-consistently combinethe physics of atomic-scale elasto-plasticity with thediffusive dynamics of phase transformations and mi-crostructure formation. Nevertheless, analogously to tra-ditional phase field modeling of solidification, furtherwork is required to fully exploit the methodology. Morespecifically it is important to be able to generalize themethod to more complex situations (binary alloys, faster

dynamics, different crystal structures, etc.), to developmore efficient numerical techniques and to make a di-rect connection of the parameters of the model to ex-perimental systems. Several innovations toward this goalhave already been developed. Goldenfeld et al. [37, 38]have recently derived amplitude equations for the PFCmodel which are amenable to adaptive mesh refinementschemes. This work has the potential to enable simu-lations of mesoscopic phenomena (µm → mm) that areresolved down to the atomic scale and incorporate allthe physics discussed above. Another recent advanceis the inclusion of higher order time derivatives in thedynamics to simulate “instantaneous” elastic relaxation[39]. This extension is important for modeling complexstress propagation and externally imposed strains. Veryrecently, Wu et al. [40, 41] fitted the PFC parametersto experimental data in iron and were able to to showthat the PFC model gives an accurate description of theanisotropy of the surface tension. In addition to this workWu and Karma have also developed a simple and elegantscheme to extend the method to other crystal symmetries(i.e., FCC in three dimensions).

The purpose of this paper is to link the formalism ofclassical density functional theory (DFT) of freezing, asformulated by Ramakrishnan and Yussouff [42] (and alsoreviewed by many other authors, such as Singh [43]) withthe PFC method and to exploit this connection to de-velop a PFC model for binary alloys. The organizationof the paper and a summary of the remaining sections isas follows.

In Section IIA the classical density functional theoryof freezing of pure and binary systems is briefly outlined.In this approach the free energy functional is written interms of the time averaged atomic density field ρ (ρA

and ρB in binary systems) and expanded around a liquidreference state existing along the liquid/solid coexistenceline. Formally the expansion contains the n-point corre-lation functions of the liquid state. In this work the seriesexpansion of the free energy is truncated at the 2-pointcorrelation function, C(~r1, ~r2).

Within this framework it is shown in Section IIIA thatthe PFC model for a pure material can be recovered fromDFT if C(~r1, ~r2) is parameterized by three constants re-lated to the liquid and solid state compressibilities andthe lattice constant. The parameters of the PFC modelcan thus be directly related to the physical constants thatenter the DFT of freezing and the PFC model can beviewed as a simplified form of DFT. In Section IIIB abinary system is considered. Similar to the case of purematerials the free energy expansion of a binary alloy willbe truncated at the 2-point correlation functions whichare then characterized by three parameters. At this levelof simplification it is shown that the “regular” solutionmodel used in materials physics for alloys can be obtaineddirectly from DFT. It is shown that the phenomenologi-cal nearest neighbour bond energies that enter the “reg-ular” solution model are equal to the compressibilitiesthat enter DFT. This section also provides insight into

3

the concentration dependence of various properties of thecrystalline phase of a binary alloy such as the lattice con-stant, effective mobilities and elastic constants.

In Section IIIC a simplified version of the binary alloyfree energy is derived. This is done in order to providea mathematically simpler model that can more transpar-ently illustrate the use of the PFC formalism in simulta-neously modeling diverse processes such as solidification,grain growth, defect nucleation, phase segregation andelastic and plastic deformation. This section also showsthat the free energy of the simplified alloy PFC modelreproduces two common phase diagrams associated withtypical binary alloys in materials science. Some of themore tedious calculations in the derivation of the simpli-fied model are shown in the Appendix (Section VIII).

In Section IV dynamical equations of motion that gov-ern the evolution of the solute concentration and densityfield of the binary alloy are derived. Finally in SectionV the simplified binary alloy model is used to simulateseveral important applications involving the interplay ofphase transformation kinetics and elastic and plastic ef-fects. This includes solidification, epitaxial growth andspinodal decomposition.

II. CLASSICAL DENSITY FUNCTIONALTHEORY OF FREEZING

In this section free energy functionals of pure and bi-nary systems as derived from the classical density func-tional theory of freezing are presented. For a rigoroustreatment o of their derivation the reader is referred tothe work of Ramakrishnan and Yussouff [42] and numer-ous other very closely related review articles by Singh[43], Evans [44] and references therein.

In DFT the emergence of an ordered phase during so-lidification can be viewed as a transition to a phase inwhich the atomic number density, ρ(~r), is highly non-homogenous and possesses the spatial symmetries of thecrystal [43]. This approach implicitly integrates outphonon modes in favour of a statistical view of the or-dered phase that changes on diffusive time scales. Thefree energy functional of a system is expressed in termsof ρ and constitutes the starting point of the PFC model.

In this work the free energy functional, denoted F [ρ],is expanded functionally about a density, ρ = ρl, corre-sponding to a liquid state lying on the liquidus line of thesolid-liquid coexistence phase diagram of a pure materialas shown in Fig. (1a). The expansion is performed inpowers of δρ ≡ ρ− ρl.

As shown by others [42–44] the free energy density canbe written as

Fc/kBT =∫

d~x [ρ(~r) ln (ρ(~r)/ρl)− δρ(~r)]

−∞∑

n=2

1n!

∫ n∏

i=1

d~riδρ(~ri)Cn(~r1, ~r2, ~r3, . . . , ~rn) (1)

FIG. 1: (a) Sample phase diagram. In this figure the shadedarea corresponds to a coexistence region. In the calculationspresented in this paper the correlation functions are takenfrom points along the liquidus line at density ρ`. (b) In thisfigure a ‘typical’ liquid state two-point direct correlation func-tion is sketched. The dashed line represents the approxima-tion used in most of the manuscript.

where Fc is the free energy corresponding to the densityρ(~r) minus that at the constant density ρl, and the Cn

functions are n point direct correlation functions of anisotropic fluid. Formally the correlation functions aredefined by

Cn(~r1, ~r2, ~r3, · · · ) ≡ δnΦ∏i=ni=1 δρ(~ri)

(2)

where Φ[ρ] represents the total potential energy of inter-actions between the particles in the material. A partic-ularly simple proof that Φ is a functional of ρ is given inEvans [44].

For an alloy involving one or more components thefree energy functional of a pure material in Eq. (1) isextended to the form

Fc

kBT=

∑

i

∫d~r

[ρi (~r) ln

(ρi (~r) /ρi

`

)− δρi (~r)]

− 12

∑

i,j

∫d~r1 d~r2 δρi(~r1)Cij δρj(~r2) + · · · (3)

where the sums are over the elements in the alloy, δρi ≡ρi − ρi

`, ρi` is the value of the number density of compo-

nent i on the liquid side of the liquid/solid coexistence

4

line. The function Cij is the two point direct correlationfunction of between components i and j in an isotropicfluid. As in the case of a pure materials it will be assumedthat Cij ≡ Cij(|~r1 − ~r2|). The next term in the expan-sion of Eq. (3) contains the three point correlation, thenext after that, the four point, etc.. In this paper onlytwo point correlations will be considered, but it must bestressed that these higher order correlations maybe cru-cial for some systems, such as Si. Before considering theproperties of the binary alloy free energy in detail it isinstructive to first study the properties of a pure systemand show the connection between this formalism and theoriginal PFC model.

III. ANALYSIS OF FREE ENERGYFUNCTIONALS

A. Pure Materials

In this section the free energy functional of a singlecomponent alloy is considered in the limit that the seriesgiven in Eq. (1) can be truncated at C2, i.e.,

F/kBT =∫

d~r [ρ ln (ρ/ρ`)− δρ]

− (1/2)∫

d~r d~r′ δρC(~r, ~r′) δρ′ (4)

where, for convenience, the subscript ‘2’ has beendropped from the two-point correlation function as hasthe subscript c from Fc. To understand the basic fea-tures of this free energy functional it is useful to expandF in the dimensionless deviation of the density ρ fromits average, ρ, using the re-scaled density

n ≡ (ρ− ρ)/ρ. (5)

Expanding F in powers of n gives,

∆Fρ kBT

=∫

d~r

[n

1− ρ C

2n− n3

6+

n4

12− · · ·

], (6)

where ∆F ≡ F −Fo and Fo is the the free energy func-tional at constant density (i.e., ρ = ρ). For simplicity Cis an operator defined such that n C n ≡ ∫

d~r ′n(~r)C(|~r−~r ′|)n(~r ′). Terms that are linearly proportional to n inthe above integral are identically zero by definition.

To gain insight into the properties of the free energyfunctional in Eq. (6) it is useful to expand the two pointcorrelation function in a Taylor series around k = 0, i.e.,

C = C0 + C2k2 + C4k

4 + · · · . (7)

(in real space this corresponds to C = (C0 − C2∇2 +C4∇4 − · · · )δ(~r − ~r′), where the gradients are with re-spect to ~r′). The function C is sketched for a typicalliquid in Fig. (1b). In what follows only terms up tok4 will be retained. In this manner the properties of the

material are parameterized by the three variable, C0, C2

and C4. To fit the first peak in C, C0, C2 and C4 must benegative, positive and negative, respectively. These vari-ables are related to three basic properties of the material,the liquid phase isothermal compressibility (∼ (1−ρC0)),the bulk modulus of the crystal (∼ ρ C2

2/|C4|) and lat-tice constant (∼ (C2/|C4|)1/2). In other words the k = 0term is related to the liquid phase isothermal compress-ibility, the height of the first peak (Cm in Fig. (1b))is related to the bulk modulus of the crystalline phaseand the position of the first peak determines the latticeconstant.

It is important to note that at this level of simplifica-tion the material is only defined by three quantities whichmay not be enough to fully parameterize any given ma-terial. For example this simple three parameter modelalways predicts triangular symmetry in two dimensionsand BCC symmetry in three dimensions. Other crystalsymmetries can be obtained by using more complicated 2-point correlation functions [40, 41] or by including higherorder correlation functions. In addition it is possible thatfitting to the width, height and position of the first peakin C may lead to a more accurate fitting to experimentaldata.

In two dimensions F is minimized by a triangular lat-tice that can be represented to lowest order by a one-mode approximation as

n = A

(12

cos(

2qy√3

)− cos(qx) cos

(qy√(3)

)). (8)

Substituting Eq. (8) into Eq. (6) and minimizing withrespect to q gives equilibrium wavevector of

qeq =√

3C2/(8|C4|). (9)

or in terms of the equilibrium lattice constant aeq =2π/qeq. When q = qeq, ∆F becomes

∆F =316

∆B A2 − 132

A3 +15512

A4 + · · · , (10)

where ∆F ≡ ∆F/(ρ kB T S), S is the area of a unit cell,∆B ≡ B`−Bs, B` ≡ 1− ρC0 and Bs ≡ ρ (C2)2/(4|C4|).The parameter B` is the dimensionless bulk modulus ofthe liquid state (i.e., B` = κ/(ρ kB T ), where κ is the bulkmodulus of a liquid). The parameter Bs is proportionalto the bulk modulus in the crystalline phase.

Equation (10) indicates that the liquid state is linearlyunstable to the formation of the crystalline phase when∆B < 0. This instability arises from a competition be-tween the elastic energy stored in the liquid and crys-talline phases. It is interesting to note that ∆B can alsobe written,

∆B = (ρs − ρ)/ρs (11)

where ρs = 1/Cm and Cm is the height of the first peakof C as shown in Fig. (1). Written in this form ρs can be

5

thought of as defining the effective spinodal density, i.e.,the average density at which the liquid becomes linearlyunstable to crystallization.

Unfortunately it is difficult to obtain the equilibriumstate (i.e., by solving d∆F/dA = 0) without truncat-ing the infinite series in Eq. (10). If only terms to or-der A4 are retained an analytic approximation can beobtained for the amplitude (Amin) that minimizes F .In this approximation the solution is Amin = 2(1 +√

20ρ/ρs − 19 )/5. Thus solutions for a crystalline stateexist when ρ > 19/20 ρs.

It is also straightforward to calculate the change in en-ergy of the crystalline state upon deformation (i.e., bulk,shear or deviatoric). Details of similar calculations aregiven in previous publications [33, 34]. The result ofthese two-dimensional calculations gives the dimension-less bulk modulus, Bc, of the crystalline phase, i.e.,

Bc =332

ρ(C2)2

|C4|A2

min =38

Bs A2min. (12)

In other words the parameter Bs controls the bulk mod-ulus of the crystalline phase.

These calculations can be easily extended to three di-mensions. As discussed previously this particular ap-proximation for C leads to a BCC crystal in three dimen-sions which can be represented in a one mode approxi-mation as, n = A

(cos (qx) cos (qy) + cos (qx) cos (qz) +

cos (qy) cos (qz)). Substituting this functional form into

the free energy and minimizing with respect to q gives,

q3deq =

√C2/|C4|, and the free energy functional at this q

is,

∆F 3d =38∆BA2 − 1

8A3 +

45256

A4 + · · · . (13)

Thus in this instance the ’spinodal’ occurs at the samedensity as in the two dimensional case. If the series istruncated at A4 the amplitude that minimizes the freeenergy is then A3d

min = 4(1 +√

15ρ/ρs − 14)/15. Thusin this approximations crystalline (BCC) solutions onlyexist if ρ > 14ρs. In addition the elastic constants canalso be calculated in 3D in the usual manner. For ex-ample the dimensionless bulk modulus of the crystallinestate is given by; B3d

c = 3 Bs(A3dmin)2. This calculation

gives the basic functional dependence of the (dimension-less) bulk modulus on Bs and the amplitude. For a moreaccurate calculation higher order fourier components andmore terms in the powers series in n should be retained.

Finally it is useful to consider fixing the density andvarying the temperature. If the liquidus and solidus linesare roughly linear then, ρs can be approximated by alinear function of temperature. In the sample phase di-agram shown in Fig. (1a) the liquidus and solidus linesare roughly parallel and it is likely that the spinodal isalso roughly parallel to these lines. In this case ∆B canbe written,

∆B = α∆T (14)

where ∆T ≡ (T −Ts)/Ts, Ts is the spinodal temperatureand α ≡ (Ts/ρs)(∂ρs/∂Ts), evaluated at ρ = ρ.

B. Binary Alloys

For a binary alloy made up of ‘A’ and ‘B’ atoms thefree energy functional can be written to lowest order interms of the direct correlation functions as,

FkBT

=∫

d~r

[ρA ln

(ρA

ρA`

)− δρA + ρB ln

(ρB

ρB`

)− δρB

]

−12

∫d~r1d~r2

[δρA(~r1)CAA(~r1, ~r2) δρA(~r2)

+δρB(~r1)CBB(~r1, ~r2) δρB(~r2)

+2 δρA(~r1)CAB(~r1, ~r2) δρB(~r2)]

(15)

where δρA ≡ ρA − ρAl and δρB ≡ ρB − ρB

l . It is assumedhere that all two point correlation functions are isotropic,i.e., Cij(~r1, ~r2) = Cij(|~r1 − ~r2|).

In order to make a connection between the alloy freeenergy and standard phase field models it is useful todefine the total number density, ρ ≡ ρA + ρB and a localconcentration field c ≡ ρA/ρ. In terms of these fieldsthe atomic densities can be written, ρA = cρ and ρB =ρ(1 − c). Furthermore it is useful to define ρ = ρl + δρwhere ρl ≡ ρA

l + ρBl and δc = 1/2− c. Substituting these

definitions into Eq. (15) gives,

FkBT

=∫

d~r[ρ ln(ρ/ρ`)− δρ + βδc + Fo

−12δρ

{c CAA + (1− c) CBB

}δρ

+ρ{c ln (c) + (1− c) ln (1− c)

}

+ρc{(CAA + CBB)/2− CAB

}(1− c)ρ

](16)

where β ≡ ρl(CAA −CBB)(ρ + ρl)/2 + ρ ln(ρB` /ρA

` ) andFo ≡ ρ ln(ρ`/(ρA

` ρB` )1/2)−CAA/2((ρA

` )2 +ρl/2(ρl + ρ))−CBB/2((ρB

` )2 + ρl/2(ρl + ρ)).To illustrate the properties of the model in Eq. (16)

it useful to consider two limiting cases, a liquid phase atconstant density and a crystalline phase at constant con-centration. These calculations are presented in followingtwo sub-sections.

1. Liquid phase properties

In the liquid phase ρ is constant on average and in themean field limit can be replaced by ρ = ρ. To simplicitycalculations, the case ρ = ρ ≈ ρ` (or δρ ≈ 0) will nowbe considered. As in the previous section it is useful toexpand the direct correlation functions in Fourier space,i.e.,

Cij = Cij0 + Cij

2 k2 + Cij4 k4 + · · · . (17)

6

where the subscript i and j refer to a particular element.Substituting the real-space counterpart of the Fourier ex-pansion for Cij (to order k2) into Eq. (16) gives,

FC

ρ kBT=

∫d~r

[c ln (c) + (1− c) ln (1− c)

+ρ∆C0

2c(1− c) + γ` δc +

ρ∆C2

2|∇c|2

],(18)

where FC is the total free energy minus a constant thatthat depends only on ρ, ρl

A and ρlB ,

γ` ≡ (BBB` −BAA

` ) + ρ ln(ρB

` /ρA`

), (19)

∆Cn ≡ CAAn + CBB

n − 2CABn . (20)

and Bij` = 1 − ρCij

0 is the dimensionless bulk compress-ibility. Equation (18) is the regular solution model of abinary alloy in the limit γ` = 0. It is also noteworthythat that Eq. (18) implies a temperature dependence ofthe gradient energy coefficient which is consistent withother theoretical [65] and experimental [66] studies.

The coefficient of c(1− c) in Eq. (18) is given by

ρ∆C0 = 2BAB` −BAA

` −BBB` . (21)

This result shows that in the liquid state the ’interaction’energies that enter regular solution free energies are sim-ply the compressibilities (or the elastic energy) associatedwith the atomic species. The γ` term is also quite inter-esting as it is responsible for asymmetries in the phasediagram. Thus Eq. (19) implies that asymmetries canarise from either different compressibilities or differentdensities.

Expanding Eq. (18) around c = 1/2 gives,

∆FC

ρ kBT=

∫d~r

[r`

2δc2 +

u

4δc4 + γ` δc +

K

2|∇c|2

](22)

where, ∆FC ≡ FC − ρkBT∫

d~r(ρ∆C0/8 − ln(2)), u ≡16/3, r` ≡ (4 − ρ∆C0) and K = ρ∆C2. The parameterr` is related only to the k = 0 part of the two-pointcorrelation function and can be written,

r` = 4 + (BAA` + BBB

` − 2BAB` ). (23)

This result implies that the instability to phase segre-gation in the fluid is a competition between entropy (4)and the elastic energy of a mixed fluid (2BAB

` ) with theelastic energy associated with a phase separated fluid(BAA

` + BBB` ). Replacing the dimensionless bulk moduli

with the dimensional version (i.e., B = κ/kBT ), givesthe critical point (i.e. rl = 0) as

T `C = (2κAB − κBB − κAA)/(4kB). (24)

where κ is the dimensional bulk modulus.The properties of the crystalline phase are more com-

plicated but at the simplest level the only real differenceis that the elastic energy associated with the crystallinestate must be incorporated. This is discussed in the nextsection.

2. Crystalline phase properties

To illustrate the properties of the crystalline state, thecase in which the concentration field is a constant is con-sidered. In this limit the free energy functional given inEq. (16) can be written in the form

FkBT

≡∫

d~r

[ρ ln

(ρ

ρ`

)− δρ− 1

2δρ ¯C δρ + G

](25)

where G is a function of the concentration c and ρl andcouples only linearly to δρ. The operator ¯C can be writ-ten as

¯C ≡ c2CAA + (1− c)2CBB + 2c(1− c)CAB . (26)

Thus in the limit that the concentration is constantthis free energy functional is that of a pure materialwith an effective two point correlation function that isan average over the AA, BB and AB interactions. Inthis limit the calculations presented in section III A canbe repeated using the same approximations (i.e., ex-panding ρ around ρ`, expanding ¯C to ∇4 and using aone mode approximation for δρ) to obtain predictionsfor the concentration dependence of various quantities.For example the concentration dependence of the equi-librium wavevector (or lattice constant, Eq. (9)) andbulk modulus Eq. (12) can be obtained by redefiningCn = c2CAA

n + (1− c)2CBBn + 2c(1− c)CAB

n .As a more specific example the equilibrium lattice con-

stant can be expanded around c = 1/2 to obtain in twoor three dimensions,

aeq(δc) = aeq(0) (1 + η δc + · · · ) (27)

where δc = c − 1/2 and η is the solute expansion coeffi-cient given by,

η = (δC4 − δC2)/2 (28)

where

δCn ≡ (CAAn − CBB

n )/ ˆCn (29)

and ˆCn ≡ Cn(δc = 0) = (CAAn + CBB

n + 2CABn )/4.

This line of reasoning can also be used to understandthe influence of alloy concentration on crystallization.Specifically, for the case of an alloy, the terms in Eq.(10) (with A replaced with Amin) become functions ofconcentration, since ∆B and Amin are concentration de-pendent. Here, ∆B can be expanded around c = 1/2,i.e.,

∆B(δc) = ∆B0 + ∆B1 δc + ∆B2 δc2 + · · · (30)

where ∆B0 = Bl0 − Bs

0, ∆B1 = Bl1 − Bs

1 and ∆B2 =Bl

2 −Bs2 are determined in the Appendix (Section VIII).

This would imply that in the crystalline phase the freeenergy has a term of the form, rc(δc)2, where

rc = r` + 3∆B2A2min/8 (31)

7

in two dimensions (in three dimensions the 3/8 factor isreplaced with 3/4). This result indicates that crystalliza-tion (i.e., a non-zero Amin) favours phase segregation, as-suming κAA + κBB < 2κAB . For example, when Bs

2 = 0,the critical temperature increases and can be written,

T cC = T `

C(1 + 3A2min/8), (32)

or T cC = T `

C(1 + 3A2min/4) in three dimensions.

C. Simple Binary Alloy Model

In this section a simple binary alloy model is pro-posed based on a simplification of the free energy inEq. (16). The goal of this section is to develop a math-ematically simple model that can be used to simultane-ously model grain growth, solidification, phase segrega-tion in the presence of elastic and plastic deformation. Tosimplify calculations it is convenient to first introduce thefollowing dimensionless fields,

nA ≡ (ρA − ρA)/ρ

nB ≡ (ρB − ρB)/ρ. (33)

Also, it is convenient to expand in the following two fields,

n = nA + nB

δN = (nB − nA) +ρB − ρA

ρ. (34)

The following calculations will use the field δN insteadof δc. Expanding Eq. (16) around δN = 0 and n = 0gives a free energy of the form

Fρ kBT

=∫

d~r(n

2[B` + Bs

(2R2∇2 + R4∇4

)]n

+t

3n3 − v

4n4 + γδN +

w

2δN2 +

u

4δN4

+L2

2|~∇δN |2 + · · ·

). (35)

Details of this free energy and explicit expressions for B`,Bs, R, γ, w and L are given in the Appendix (SectionVIII). The variables t, v and u are constants. For sim-plicity the calculations presented in this section are for atwo dimensional system.

The transition from liquid to solid is intimately relatedto ∆B = B` − Bs as was the case for the pure materialand can be written in terms of a temperature difference,i.e., Eq. (14). In addition some of the polynomial termsin n and δN have been multiplied by variable coefficients(t, v and u) even though they can be derived exactlyas shown in Appendix VIII. For example the parameterv = 1/3 recovers the exact form of the n4 term. Thisflexibility in the choice of coefficients was done to be ableto match the parameter of the free energy with exper-imental materials parameters. As an example Wu andKarma [40, 41] showed that adjusting the parameter v

can be used to match the amplitude of fluctuations ob-tained in molecular dynamics simulations. With this fitthey are able to accurately predict the anisotropy of thesurface energy of a liquid/crystal interface in iron.

To facilitate the calculation of the lowest order phasediagram corresponding to Eq. (35) it is convenient toassume the concentration field δN varies significantlyover length scales much larger than the atomic num-ber density field n. As a result, the density field canbe integrated out of the free energy functional. Also,in the spirit of keeping calculations as simple as possi-ble without loosing the basic physics contained in themodel, γ = 0 in the free energy. In this instance theone-mode approximation for the total density i.e.,n =A[cos(2qy/

√3)/2−cos(qx) cos(qy

√3)] will be used. Sub-

stituting this expression into Eq. (35) and minimiz-ing with respect to q and A (recalling that δN is as-sumed constant over the scale that n varies) gives qeq =√

3/(2R) and Amin = 4(t+√

t2 − 15v∆B)/15v. The freeenergy that is minimized with respect to amplitude andlattice constant is then,

Fsol =w

2δN2 +

u

4δN4 +

316

∆BA2min −

t

16A3

min

+45v512

A4min. (36)

For mathematical simplicity all further calculationswill be limited to the approximations B` = B`

0+B`2(δN)2

and Bs = Bs0. In this limit analytic expressions can be

obtained for a number of quantities and the free energyfunctional is still general enough to produce for examplea eutectic phase diagram.

It is relatively simple to calculate the solid-solid coex-istance curves by expanding Fsol to order (δN)4, whichthen yields the solid/solid coexistence concentrations atlow temperatures according to

δNcoex = ±√|a|/b (37)

where a ≡ w + 3B`2 (Ao

min)2/8, b ≡ u −6(B`

2)2 Ao

min/(15vAomin− 4t) and Ao

min ≡ Amin(δN = 0)(which is thus a function of ∆B0).The critical temper-ature, ∆BC

0 is determined by setting δNcoex = 0 andsolving for ∆B0, which gives,

∆Bc0 =

(15wv − 2t

√−6B`

2w

)/

(6B`

2

). (38)

To obtain the liquid/solid coexistence lines the freeenergy of the liquid state must be compared to that ofthe solid. The mean field free energy of the liquid stateis obtained by setting n = 0 which gives,

Fliq =w

2δN2 +

u

4δN4. (39)

To obtain the solid-liquid coexistence lines it is usefulto expand the free energy of the liquid and solid statesaround the value of δN at which the liquid and solid

8

states have the same free energy, i.e., when Fsol = Fliq.This occurs when, δNls = ±

√(∆Bls

0 −∆Bo)/B`2, where

∆Bo ≡ Bl0 − Bs

0 and ∆Bls0 ≡ 8t2/(135v) is the lowest

value of ∆Bls0 at which a liquid can coexist with a solid.

To complete the calculations, Fsol and Fliq are expandedaround δNls to order (δN − δNls)2 and Maxwell’s equalarea construction rule can be used to identify the liq-uid/solid coexistence lines. The liquid/solidus lines are;

δNliq/δNls = 1 + G

(1−

√bsol/bliq

)

δNsol/δNls = 1 + G

(1−

√bliq/bsol

). (40)

where, G ≡ −8t2/(135v(4∆B0 − 3∆Bls0 )), bliq = (w +

3uδN2ls)/2 and bsol = bliq +2B`

2(4∆B0−3∆Bls0 )/(5v), for

δNliq > 0, δNsol > 0 and similar results for δNliq < 0,δNsol < 0, since F is a function of δN2 in this example.The calculations in this section and the previous sectionare reasonably accurate when ∆Bls

0 > ∆Bc0, however in

the opposite limit a eutectic phase diagram forms andthe accuracy of the calculations decreases. This case willbe discussed below.

1. Linear elastic constants

As shown in previous publications [33, 34], the elasticconstants can be calculated analytically in a one modeapproximation by considering changes in F as a functionof strain. For the binary model similar calculations canbe made in a constant δN approximation and give,

C11/3 = C12 = C44 = 3Bs(Amin)2/16. (41)

(this calculation can be done for arbitrary dependence ofBs on δN). As expected the elastic constants are directlyproportional to the amplitude of the density fluctuations.This implies that the elastic constants decrease as theliquid solid transition is approached from the solid phase.This result implies both a temperature and concentrationdependence through the dependence of Amin on ∆B. Inaddition to this dependence (which might be consideredas a ’thermodynamic’ dependence) the magnitude of theelastic constants can be altered by the constant Bs.

2. Calculation of alloy phase diagrams

To examine the validity of some of the approximationsfor the phase diagrams made in the previous section,numerical simulations were conducted to determine theproperties of the solid and liquid equilibrium states. Thesimulations were performed over a range of δN values,three values of ∆B0 (0.07, 0.02 and -0.03) and two val-ues of w (0.088 and -0.04). The specific values of theother constants that enter the model are given in thefigure caption of Fig. (2).

FIG. 2: Phase diagram of ∆B0 Vs. δN for the parame-ters Bs

0 = 1.00, B`1 = 0, B`

2 = −1.80, t = 0.60, v = 1.00,w = 0.088, u = 4.00, L = 4.00, γ = 0, Ro = 1.00 andR1 = 1.20 (see Eq. 51 for definitions of Ro and R1). Thesolid line is a numerical solution of the one mode approxi-mation and the dashed lines are from Eq. (37) for the lowersolid/solid coexistence lines and Eq. (40) for the upper liq-uid/solid coexistence lines. The solid points are from numer-ical solutions for the minimum free energy functional given inEq. (35).

In general the numerical results for the free energy, F ,the lattice constant R and bulk modulus agreed quitewell with the analytic one-mode predictions presented inthe previous section for all parameters. Comparisons ofthe analytic and numerical predictions for the phase di-agram are shown in Fig. (2) and (3) for w = 0.088 and−0.04, respectively. As seen in these figures the agree-ment is quite good expect near the eutectic point shownin Fig. (3). In this case, the analytic calculations (Eqns.(40) and (37)) for the coexistence lines breakdown at theeutectic point and higher order terms in δN are neededto accurately predict the phase diagram.

IV. DYNAMICS

To simulate microstructure formation in binary alloys,dynamical equations of motions for the field δN and nneed to be developed. The starting point is the full freeenergy in Eq. (15), written in terms of ρA and ρB , i.e.,F(ρA, ρB). The dynamics of ρA and ρB is assumed to bedissipative and driven by free energy minimization, i.e.,

∂ρA

∂t= ∇ ·

(MA (ρA, ρB)∇ δF

δρA

)+ ξA (42)

∂ρB

∂t= ∇ ·

(MB (ρA, ρB)∇ δF

δρB

)+ ξB (43)

9

FIG. 3: Phase diagram of ∆B0 Vs. δN for the same parame-ters as those used to generate Fig. (2), with the exception thatw = −0.04. The dotted lines below the eutectic temperature,∆BE

0 ≈ 0.028, correspond to metastable states.

where MA and MB are the mobilities of each atomicspecies. In general these mobilities depend on the den-sity of each species. The variables ξA and ξB are con-versed Gaussian correlated noise fields that representthermal fluctuation in the density of species A and B.They satisfy the fluctuation-dissipation theorm < ξ2

i >=−2kBTMi∇2δ(~r − ~r′)δ(t− t′), where Mi is the mobilityof A or B atoms. For simplicity the noise terms will beneglected in what follows.

The free energy F(ρA, ρB) can be equivalently definedin terms of n and δN . This allows the previous equationsto be re-written as

∂n

∂t= ~∇ ·

{M1

~∇δFδn

}+ ~∇ ·

{M2

~∇ δFδ(δN)

}(44)

∂(δN)∂t

= ~∇ ·{

M2~∇δF

δn

}+ ~∇ ·

{M1

~∇ δFδ(δN)

}. (45)

where M1 ≡ (MA + MB)/ρ2 and M2 ≡ (MB −MA)/ρ2.Equations (44) and (45) couple the dynamics of the fieldsδN and n through a symmetric mobility tensor. The de-pendence of the mobilities MA and MB will in generaldepend on local crystal density and the local relative con-centration of species A and B.

For the case of substitutional diffusion between speciesA and B, MA ≈ MB ≡ M . In this limit the dynamicsof n and δN decouple. Moreover if it is further assumedthat that the mobility is a constant, Eqs.( 44) and (45)become

∂n

∂t= Me∇2 δF

δn(46)

∂(δN)∂t

= Me∇2 δFδ(δN)

(47)

FIG. 4: The grey scales in the figure correspond to density(n), concentration (δN) and the local energy density in frames(a,b,c), (d,e,f) and (g,h,i) respectively. The area enclosed bywhite boxes is area shown in figures (a,b,c). The parametersin this simulation are the same as Fig. (3) except L = 1.20and R1/R0 = 1/4 (Ro = 1) at δN = 0 and ∆Bo = 0.02.Figures (a,d,g), (b,e,h) and (c,f,i) correspond to times t =6600, 16200, 49900, respectively.

where the effective mobility Me ≡ 2M/ρ2. In the appli-cations using Eqs. (46) and (47) in the following sections,the dynamics of n and δN are simulated with time re-scaled by t → t ≡ 2Mt/ρ2.

To illustrate the dynamics described by Eqs. (46), (47)and (35), a simulation of heterogenous eutectic crystal-lization from a supercooled homogenous liquid was per-formed. The results of this simulation are shown in Fig.(4). This figure shows the density (n), the density dif-ference (δN) and the local energy density at three timesteps in the solidification process. These figures showliquid/crystal interfaces, grain boundaries, phase segre-gation, dislocations and multiple crystal orientations allin a single numerical simulation of the simple binary alloyPFC model. In this simulation a simple Euler algorithmwas used for the time derivative and the spherical Lapla-cian approximation was used. The grid size was ∆x = 1.1and the time step was ∆t = 0.05. Unless otherwise speci-fied all simulations to follow use the same algorithm, gridsize and time step.

10

V. APPLICATIONS

This section applies the simplified PFC model derivedin Section III C, coupled to the dynamical equations ofmotion derived in Section IV, to the study of elastic andplastic effects in phase transformations. The first appli-cation demonstrates how the PFC alloy model can beused to simulate eutectic and dendritic microstructures.That is followed by a discussion of the effects of com-pressive and tensile stresses in epitaxial growth. Finally,simulations demonstrating the effects of dislocation mo-tion in spinodal decomposition are presented.

A. Eutectic and Dendritic Solidification

One of the most important applications of the al-loy PFC model is the study of solidification microstruc-tures. These play a prominent role in numerous appli-cations such as commercial casting. Traditional phasefield models of solidification are typically unable to self-consistently combine bulk elastic and plastic effects withphase transformation kinetics, multiple crystal orienta-tions and surface tension anisotropy. While some of theseeffects have been included in previous approaches (e.g.surface tension anisotropy) they are usually introducedphenomenologically. In the PFC formalism, these fea-tures arise naturally from DFT.

To illustrate solidification microstructure formation us-ing the PFC formalism, two simulations of Eqs. (46) and(47) were conducted of the growth of a single crystalfrom a supercooled melt in two dimensions. In the firstsimulation a small perturbation in the density field wasintroduced into a supercooled liquid using the parame-ters corresponding to the phase diagram in Fig. (3), ex-cept L = 1.20 and R1/R0 = 1/4. The reduced tempera-ture ∆B0 = 0.0248 and average concentration δN = 0.0.To reduce computational time the size of the lattice wasgradually increased as the seed increased in size. A snap-shot of the seed is shown at t = 480, 000 in Fig. (5a).A similar simulation was conducted for the growth of adendrite from a supercooled melt for reduced temper-ature ∆B0 = 0.04 and δN = 0.0904, with other pa-rameters corresponding to those in Fig. (2), except forL = 1.20 and R1/R0 = 1/4. A sample dendritic struc-ture is shown in Fig. (5b) at t = 175, 000. It should benoted that the dendrite in Fig. (5b) is not six-fold sym-metric about its main trunk because the simulation isinfluenced by the boundaries of the numerical simulationcell, which are four-fold symmetric. Dendrites nucleatedand grown in the centre of the simulation domain retaintheir six-fold symmetry. Finally, it should be stressedthat the side-branching is only qualitatively correct inFig. (5b) as it was initiated by numerical grid noise. Fora more quantitative simulation of nucleation and growthof side-branches, thermal fluctuation need to included inEqs. (46) and (47).

Simulations such as these can play an important role

FIG. 5: The grey-scale in the main portion of both figuresshow the concentration field δN . In the insets, the grey-scaleshows the density field, n, for the small portion of the mainfigure that is indicated by the white boxes. a) Eutectic crystalgrown from a supercooled liquid at ∆B0 = 0.0248 and δN =0.0. The parameters that enter the model are the same as Fig.(3) except L = 1.20 and R1/R0 = 1/4 (Ro = 1). b) Dendritecrystal grown from a supercooled liquid at ∆B0 = 0.04 andδN = 0.0904. The parameters that enter the model are thesame as Fig. (2) except L = 1.20 and R1/R0 = 1/4. In Fig.(b) mirror boundary conditions were used.

in establishing various constitutive relations for use inhigher-scale finite element modeling (FEM) of elasto-plastic effects in alloys during deformation. In particular,traditional FEM approaches often employ empirical orexperimental constitutive models to describe stress-strainresponse in elements that are intended to represent one(or more) grains. These constitutive relations are oftenlimited in their usefulness as they do not self-consistentlyincorporate realistic information about microstructuralproperties that develop during solidification.

B. Epitaxial Growth

Another potential application of the PFC model is inthe technologically important process of thin film growth.The case of heteroepitaxy, the growth of a crystalline film

11

exhibiting atomic coherency with a crystalline substrateof differing lattice constant, has been examined in previ-ous PFC studies of pure systems [33, 34]. These initialworks focused on two of the primary phenomena influenc-ing film quality: (i) morphological instability to bucklingor roughening and (ii) dislocation nucleation at the filmsurface. A third important effect in alloy films, (iii) com-positional instability (phase separation in the growingfilm), requires consideration of multiple atomic speciesand their interaction. The purpose of this section is toillustrate how the binary PFC model addresses such com-positional effects in alloy heteroepitaxy, focusing on thespatial dynamics of phase separation over diffusive timescales.

To date, a number of models of single componentfilm growth incorporating surface roughening, disloca-tion nucleation, or both have been proposed [25, 45–53],and models of binary film growth incorporating surfaceroughening and phase separation have been proposed aswell [54–57]. However, no existing models of binary filmgrowth known to the authors have captured all of theabove important phenomena, and it would be reasonableto expect that new insights into the nature of film growthcould be gained through the simultaneous investigationof all of these growth characteristics. A unified treat-ment of this sort is required for the following reasons.There is clearly a strong link between surface roughen-ing and dislocation nucleation, originating from the factthat dislocations nucleate at surface cusps when the filmbecomes sufficiently rough. It is also known that phaseseparation in the film is significantly influenced by localstresses, which are inherently coupled to surface mor-phology and dislocation nucleation. The dynamics of thegrowth process must then be influenced by the cooper-ative evolution of all three of these phenomena. In thenext subsection numerical simulations will be presentedto show that the binary PFC model produces all of thegrowth characteristics described above, and that each isinfluenced by misfit strain and atomic size and mobilitydifferences between species.

1. Numerical simulations

The physical problem recreated in these simulationsis that of growth of a symmetric (i.e. 50/50 mixture, oraverage density difference δN0 = 0) binary alloy film froma liquid phase or from a saturated vapor phase abovethe bulk coherent spinodal temperature (∆Tc). Growthat temperatures above the miscibility gap is typical ofexperimental conditions and should ensure that phaseseparation is driven by local stresses and is not due tospinodal decomposition. Initial conditions consisted of abinary, unstrained crystalline substrate, eight atoms inthickness, placed below a symmetric supercooled liquidof components A and B. In all the simulations presented,parameters are the same as in Fig. (2) except for L =1.882 and ∆B0 = 0.00886 unless specified in the figure

!" #"

$"%"

FIG. 6: Plots of the smoothed local free energy showing theprogression of the buckling instability, dislocation nucleationand climb towards the film/substrate interface. From a) to d)times shown are t=600, 1050, 1200, and 2550. In this figureε = 0.04, η = −1/4 and MA = MB = 1.

caption. In what follows the misfit strain, ε, is defined as(afilm−asub)/asub, where afilm ≈ aA(1+ηδN0) if in theconstant concentration approximation. For a symmetricmixture of A and B atoms (i.e., δN0 = 0) afilm = (aA +aB)/2.

Periodic boundary conditions were used in the lateraldirections, while a mirror boundary condition was ap-plied at the bottom of the substrate. A constant fluxboundary condition was maintained along the top bound-ary, 120∆x above the film surface, to simulate a finitedeposition rate. Misfit strain was applied to the systemby setting R = 1 in the substrate and R = 1+ ε+ηδN inthe film. This approach yields a film and substrate thatare essentially identical in nature except for this shiftin lattice parameter in the film. Complexities resultingfrom differing material properties between the film andsubstrate are therefore eliminated, isolating the effects ofmisfit strain, solute strain, and mobility differences onthe film growth morphology. The substrate was permit-ted to strain elastically, but was prevented from decom-posing compositionally except near the film/substrate in-terface.

A sample simulation is shown in Fig. (6) demonstrat-ing that the well-documented buckling or Asaro-Tiller-Grinfeld [46, 47] instability is naturally reproduced bythe PFC model. This instability is ultimately suppressedas a cusp-like surface morphology is approached, withincreasingly greater stress developing in surface valleys.The buckling behavior ceases when the local stress in agiven valley imparts on the film a greater energy thanthat possessed by an equivalent film with a dislocation.At this stage a dislocation is nucleated in the surfacevalley and the film surface begins to approach a planarmorphology.

The nature of phase separation within the bulk filmand at the film surface was found to vary with modelparameters, but a number of generalizations applicableto all systems studied have been identified. For the case

12

3

1 2

FIG. 7: Plot of the smoothed local concentration field show-ing lateral phase separation between the surface peaks andvalleys. White: Component A (large, fast), Black: Compo-nent B (small, slow). In this figure ε = −0.02, η = 0.4,MA = 1, MB = 1/4, and t = 3500. See text for discussion ofthe numbered arrows.

a) Compressive

b) Tensile

FIG. 8: Plot of the smoothed local concentration field showingthe nature of the phase separation under opposite signs of ε.In figures a) and b) ε = 0.04 and −0.025 respectively and inboth figures MA = MB = 1 and η = 0.25.

of equal species mobilities (MA = MB) we find that inthe presence of misfit and solute strain, the componentwith greater misfit relative to the substrate preferentiallysegregates below surface peaks (see regions marked 1 and2 in Fig. (7) and Fig. (8)). Larger (smaller) atoms willbe driven toward regions of tensile (compressive) stresswhich corresponds to peaks (valleys) in a compressivelystrained film and to valleys (peaks) in a film under tensilestrain. This coupling creates a lateral phase separationon the length scale of the surface instability and has beenpredicted and verified for binary films [54–62] and analo-gous behavior has been predicted and verified in quantumdot structures [63, 64, 67].

Secondly, again for the case of equal mobilities, thecomponent with greater misfit relative to the substrateis driven toward the film surface (see Fig. (8)). Thisbehavior can also be explained in terms of stress relax-ation and is somewhat analogous to impurity rejection indirectional solidification. The greater misfit componentcan be viewed as an impurity that the growing film wishesto drive out toward the interface. Experimental evidencefrom SiGe on Si [58] and InGaAs on InP [59, 60] verifiesthis behavior as an enrichment of the greater misfit com-ponent was detected at the film surface in both systems.Other models [54–57] have not led to this type of verticalphase separation possibly due to neglecting diffusion inthe bulk films.

The third generalization that can be made is that, inthe case of sufficiently unequal mobilities, the componentwith greater mobility accumulates at the film surface (see

region marked 3 in Fig. (7)). It was found that when thetwo components have a significant mobility difference,typically greater than a 2:1 ratio, the effect of mobility ismore important than the combined effects of misfit andsolute strains in determining which component accumu-lates at the surface. Since Ge is believed to be the moremobile component in the SiGe system, we see that thefindings of Walther et al [58] for SiGe on Si provide ex-perimental support for this claim. They find a significantenrichment of Ge at the film surface, a result that waslikely due to a combination of this mobility driven effectas well as the misfit driven effect described in the sec-ond generalization. Experimental evidence also indicatesthat segregation of substrate constituents into the filmmay occur during film growth [68, 69]. We have similarlyfound that a vertical phase separation is produced nearthe film/substrate interface and is complimented by aphase separation mirrored in direction near defects. Theextent of this phase separation is controlled largely by thebulk mobilities of the two constituents, and to a lesserdegree by η. The complimenting phase separation nearclimbing defects is a transient effect, any traces of whichare dulled once the defect reaches the film/substrate in-terface.

C. Dislocation Motion in Spinodal Decomposition

Spinodal decomposition is a non-equilibrium processin which a linearly unstable homogenous phase sponta-neously decomposes into two daughter phases. An ex-ample of this process in the solid state occurs duringa quench below the spinodal in Fig. (2) when δN = 0.During this process domains of alternating concentrationgrow and coarsen to a scale of tens of nanometers. Spin-odal decomposition is of interest as it is a common mech-anism for strengthening alloys, due to the large numberof interfaces that act to impede dislocation motion.

Solid state strengthening mechanisms, such as spinodaldecomposition, rely critically on the interactions that ex-ist between dislocations and phase boundaries. Cahn wasfirst to calculate that the driving force for nucleation ofan incoherent second phase precipitate is higher on a dis-location than in the bulk solid [70]. A similar resultwas obtained by Dolins for a coherent precipitate withisotropic elastic properties in the solid solution [71]. Huet. al confirmed the results of Cahn and Dolins usinga model that included elastic fields from compositionalinhomogeneities and structural defects [72].

Recent studies of spinodal decomposition have usedphase field models to examine the role of dislocations onalloy hardening [73, 74]. These phase field models couplethe effects of static dislocations to the kinetics of phaseseparation. Leonard and Desai where the first to simu-late the effect of static dislocations on phase boundaries,showing that the presence of dislocations strongly favorsthe phase separation of alloy components [75].

Haataja et al. recently introduced mobile dislocations

13

into a phase field model that couples two burgers vectorsfields to solute diffusion and elastic strain relaxation. Itwas shown that mobile dislocations altered the early andintermediate time coarsening regime in spinodal decom-position [76, 77]. Specifically, it was found that coherentstrains at phase boundaries decrease the initial coarsen-ing rate, they increase the stored elastic energy in thesystem. As dislocations migrate toward moving inter-faces, they relax this excess strain energy, thus increasingthe coarsening rate [76]. The growth regimes predictedby the model in Ref. [76] are in general agreement withseveral experimental studies of deformation on spinodalage hardening [78–81].

1. Numerical Simulations

Spinodal decomposition was simulated usingEqs. (46)and (47), for an alloy corresponding tothe phase diagram in Fig. (2). Simulations began with aliquid phase of average dimensionless density differenceδN0 = 0, which first solidified into a polycrystallinesolid (alpha) phase, which subsequently phase separatedas the reduced temperature (∆Bo) was lowered belowthe spinodal. Fig. (9) shows the concentration anddensity fields for four time sequences during the spinodaldecomposition process. The dots in the figures denotethe locations of dislocation cores. Parameters for thissimulation are given in the caption of Fig. (9). Thespinodal coarsening rate corresponding to the data ofFig. (9) was found to exhibit an early and intermediatetime regime that is slower that its traditional t1/3 be-haviour, while at late times it asymptotically approachest1/3.

The simulations in Fig. (9) contain compositional do-main boundaries and grain boundaries between grainsof different orientations. As a result, the observed dis-location motion is affected by elastic strain energy dueto phase separation and curvature driven grain bound-ary motion. To isolate the effect of phase separation ondislocations, Fig. (10) demonstrates dislocation motionnear a coherent interface in the alloy. As in Ref. [76],coherent strain energy built up due to compositional dif-ferences in the two phases drives the dislocation towardthe compositional boundary.

It is noteworthy that the alloy PFC introduced in thiswork does not incorporate “instantaneous” elastic relax-ation. A proper treatment of rapid relaxation of strainfields requires the model to be extended in a manneranalogous to [39]. However, because of the asymptot-ically slow kinetics of spinodal decomposition and thesmall length scales between domain boundaries, it is ex-pected that this will only influence the time scales overwhich dislocations interact with domain boundaries. Asa result, the general trends depicted in (9) are expectedto be correct.

FIG. 9: Four time sequences in the evolution of the concen-tration field (grey scale), superimposed on the correspondingdensity field. Dislocations are labeled by a square on thedislocation core surrounded by a circle. The time sequence(a)-(d) corresponds to t = 12000, 24000, 60000 and 288000,respectively (in units of ∆t = 0.004). The system size is:1024∆x × 1024∆x, where ∆x = π/4. The average densitydifference δN0 = 0, while L = 2.65, R1/R0 = 1/4 (R0 = 1)and all other parameters are the same as Fig. (2).

FIG. 10: A dislocation migrates toward a coherent phaseboundary thus relaxing mismatch strain. An 800×800 (unitsof ∆x) portion of the actual simulations domain is shown.The data shows four time frames in the motion of the dis-location. Parameters of the simulation are the same as inFig. (9).

14

VI. DISCUSSION AND CONCLUSIONS

In this paper a connection between the density func-tional theory of freezing and phase field modeling wasexamined. More specifically it was shown that the phasefield crystal model introduced in earlier publications [33–35] and the regular solution commonly used in mate-rial science can be obtained from DFT in certain lim-its. These calculations relied on parameterizing the di-rect two-point correlation function that enters DFT bythree quantities related to the elastic energy stored inthe liquid and crystalline phases, as well as the latticeconstant.

In addition, a simplified binary alloy model was de-veloped that self-consistently incorporates many physi-cal features inaccessible in other phase field approaches.The simplified alloy PFC model was shown to be ableto simultaneously model solidification, phase segrega-tion, grain growth, elastic and plastic deformations inanisotropic systems with multiple crystal orientations ondiffusive time scales.

It is expected that the alloy PFC formalism and itsextensions can play an important role in linking materialproperties to microstructure development in a mannerthat fundamentally links the meso-scale to the atomicscale. As such this formalism, particularly when com-bined with novel adaptive mesh techniques in phase-amplitude or reciprocal space, can lead the way to a trulymulti-scale methodology for predictive modeling of ma-terials performance.

VII. ACKNOWLEDGMENTS

K. R. E. acknowledges support from the National Sci-ence Foundation under Grant No. DMR-0413062. N.P. and M. G. would like to thank the National Scienceand Engineering Research Council of Canada for finan-cial support. M. G. also acknowledges support from leFonds Quebecois de la recherche sur la nature et les tech-nologies. J. B. acknowledges support from the RichardH. Tomlinson Foundation.

VIII. SIMPLE BINARY ALLOY MODEL

This appendix goes through the expansion required toarrive at the simplified alloy model presented in sectionIII B. For this calculation the free energy functional inEq. (16) is expanded in the variables n and δN , as de-fined in Eq. (34), up to order four (noting that termsof order n or δN can be dropped since they integrate tozero in the free energy functional as they are all definedaround their average values). In addition it will be as-sumed that δN varies on length scales much larger thann. This is a reasonable on long time (diffusion) timesscales, where solute and host atoms intermix on lengthscales many times larger than the atomic radius. Thisassumption allows terms of order nδN to be eliminated

from the free energy. The result of these expansions andapproximations is that the free energy functional can bewritten as

FρkBT

=∫

d~r[fo + B` n2

2− n3

6+

n4

12+ nF∇2n

+δN

2

(1− CAA + CBB − 2CAB

4

)δN

+δN4

12+

β

2ρ(1− n3)δN + nG∇4n

+dC

4ρ

((ρ− ρ`)2 − ρ2

`n3)

)δN

](48)

where

fo = ln(

ρ

2ρ`

)− (1− ρ`/ρ)− ρCAB

0 /4

−18(ρ + 2ρ2

`/ρ− 4ρ`)(CAA0 + CBB

0 )

B` = 1− ρ ˆC0 + ρ−1(β + ρ2`dC0/2)δN + δN2

F = −ρ ˆC2 + ρ2`dC2 δN/2ρ

G = −ρ ˆC4 + ρ2`dC4 δN/2ρ

dC = CAA − CBB , (49)

while ˆCn ≡ (CAAn + CBB

n + 2CABn )/4 and dCn ≡ CAA

n −CBB

n .The previous equation can finally be cast into a form

similar to that presented in Section III C of the text,

FρkBT

=∫

d~r[fo +

n

2[B` + Bs(2R2∇2 + R4∇4)

]n

− n3

6+

n4

12+

w

2δN2 +

δN4

12+

L2

2|∇δN |2

+ γδN +H4

2δN∇4δN

]

(50)

where Bs = F 2/(2G), R =√

2G/F , w = (1 −∆C0/2),L2 = ∆C2/2, H2 = −∆C4/2 and 2ργ = β(1 − n3) +dC

((ρ− ρ`)2 − ρ2

`n3)/2 (∆Cn as in Eq. 20).

The dependence of the coefficients in Bl, B` and Ron the density difference can be explicitly obtained byexpanding them in δN as well. This gives,

B` = B`0 + B`

1 δN + B`2 δN2

Bs = Bs0 + Bs

1 δN + Bs2 δN2 + · · ·

R = R0 + R1 δN + R2 δN2 + · · · (51)

where B`0 = 1− ρ ˆC0, ρB`

1 = β + ρ2`dC0/2, B`

2 = 1, Bs0 =

−ρ( ˆC2)2/ ˆC4, Bs1 = − ˆC2ρ

2`(

ˆC2dC4 − 2 ˆC4dC2)/4ρ ˆC24 ,

Bs2 = −ρ2

`(ˆC2dC4 − ˆC4dC2)2/8ρ3 ˆC3

4 , R0 =√

2 ˆC4/ˆC2,

R1/R0 = −ρ2`(dC4/

ˆC4 − dC2/ˆC2)/4ρ2, and R2/R0 =

−ρ4`(

ˆC2dC4 − ˆC4dC2)( ˆC2dC4 + 3 ˆC4dC2)/32ρ4 ˆC24

ˆC22 .

15

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