Phase field study of precipitate growth: Effect of misfit strain and interface curvature

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Acta Materialia 57 (2009) 3947–3954 www.elsevier.com/locate/actamat

Phase field study of precipitate growth: Effect of misfit strain and interface curvature R. Mukherjee a, T.A. Abinandanan a,*, M.P. Gururajan b b

a Department of Materials Engineering, Indian Institute of Science, Bangalore 560 012, India Department of Applied Mechanics, Indian Institute of Technology-Delhi, Hauz Khas, New Delhi 110 016, India

Received 1 March 2009; received in revised form 25 April 2009; accepted 29 April 2009 Available online 27 May 2009

Abstract We have used phase field simulations to study the effect of misfit and interfacial curvature on diffusion-controlled growth of an isolated precipitate in a supersaturated matrix. Treating our simulations as computer experiments, we compare our simulation results with those based on the Zener–Frank and Laraia–Johnson–Voorhees theories for the growth of non-misfitting and misfitting precipitates, respectively. The agreement between simulations and the Zener–Frank theory is very good in one-dimensional systems. In two-dimensional systems with interfacial curvature (with and without misfit), we find good agreement between theory and simulations, but only at large supersaturations, where we find that the Gibbs–Thomson effect is less completely realized. At small supersaturations, the convergence of instantaneous growth coefficient in simulations towards its theoretical value could not be tracked to completion, because the diffusional field reached the system boundary. Also at small supersaturations, the elevation in precipitate composition matches well with the theoretically predicted Gibbs–Thomson effect in both misfitting and non-misfitting systems. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Phase field modeling; Phase transformation kinetics; Precipitation

1. Introduction In their classic study of diffusional growth of an isolated precipitate into a supersaturated matrix, Zener [1] and Frank [2] established the parabolic growth law: the square of the precipitate size (radius) increases linearly with time. They used what is now referred to as a ‘‘sharp interface” model. Fig. 1 shows a schematic of a composition profile in the precipitate p and matrix m phases during growth. Local equilibrium at the (sharp) interface yields the matrix interfacial composition, cmI , which is used as a boundary condition for solving the diffusion equation; the far-field composition, c1 is the other boundary condition. Neglecting interface curvature (capillarity) effects, cmI is the same as cme , the equilibrium matrix composition obtained from the

*

Corresponding author. Tel.: +91 8022932676; fax: +91 8023600472. E-mail address: [email protected] (T.A. Abinandanan).

phase diagram, and the precipitate grows under a supersaturation of ðc1 –cme Þ. The Zener–Frank (ZF) theory has been extended to study diffusional growth problems in a variety of settings; see Ref. [3] for a recent review. Laraia, Johnson and Voorhees (LJV) extended the theory to diffusional growth of elastically misfitting precipitates [4]; the work of LJV built on the work of Eshelby [5,6], Larche and Cahn [7–10], Johnson and Alexander [11], and Leo and Sekerka [12]. The main result of the LJV study is that the parabolic growth law continues to be valid for precipitates with misfit, but the growth coefficient depends on (i) misfit, (ii) the elastic moduli of the two phases, (iii) interfacial stress and (iv) compositional stress. In the sharp interface model used by LJV, the effect of dilatational misfit is to raise the matrix interfacial composition from cme to cmE , and therefore to decrease the supersaturation (by a constant factor) to c1 –cmE . Thus, neglecting capillarity effects, the LJV theory predicts that the growth

1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.04.056

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R. Mukherjee et al. / Acta Materialia 57 (2009) 3947–3954

(a)

(b)

Fig. 1. Schematic of the composition profile in the matrix and precipitate phases during growth in (a) non-misfitting and (b) misfitting systems. The solid curve in the matrix shows the profile considered in the theories of (a) ZF and (b) LJV. Capillarity modifies the profile, as indicated by dashed curve.

coefficient a in misfitting systems is the same as that in nonmisfitting systems with ðc1 –cmE Þ as the supersaturation. This conclusion is valid for systems in which diffusivity is independent of the state of stress; our study is restricted to these systems (as already mentioned, LJV have studied other stress-induced effects). Direct experimental verification of the LJV results is extremely difficult, not only due to the ‘‘isolated particle” assumption, but also due to lack of reliable data on stress effects on diffusion. Moreover, the assumption of isotropic interfacial energy can also be difficult to realize experimentally in crystalline alloys. Thus, simulations can act as ‘‘computer experiments” for validating the LJV results, since their parameters can be so tuned to make the computational

model resemble that used in the LJV theory as closely as possible. Thus, the first goal of our study is a critical comparison of the sharp interface results of LJV with those from our simulations based on a (diffuse interface) phase field model. In ZF and LJV theories, which neglect the effect of interface curvature, the diffusion fields at different times are selfsimilar. This self-similarity is broken when capillarity effect is included, since cmI (in Fig. 1a and b) is size dependent. Capillarity, then, not only decreases the growth rate a, but also renders it size-dependent. The second goal of this study is to evaluate the extent of this reduction in a. Our results also allow us to address two other issues: 1. Our simulation results on systems with low supersaturation can be used to verify the predictions of Johnson and Alexander [11] and of Leo and Sekerka [12] on the composition of a misfitting precipitate. 2. Many phase field models (including the one used in this study) make use of a phase field variable g whose primary role is to help distinguish one phase from the other. In some settings, such as precipitation of an ordered phase, g may be associated with a long-range order parameter. In other settings (e.g. in the present work, or in solidification), g lacks a direct physical significance, and its role in the model is justified through mathematical convenience. In this study, we show first that our computational model with a phase field variable g reproduces well-known analytical results from ZF theory in one-dimensional (1D) systems in which capillary effects are absent. Our study is organized as follows: Section 2 presents an outline of our phase field model. In Section 3, we first establish the essential correctness of the use of a phase field variable g through a critical comparison of its results against those of 1D ZF theory, in which capillarity and elastic effects are absent. We then use the same model to study the growth of non-misfitting and misfitting circular precipitates, and critically compare our results with those from ZF and LJV theories. These results are analyzed in order to elucidate the elastic and capillarity effects. These results are critically discussed in Section 4, followed by a set of conclusions in Section 5. 2. Phase field model 2.1. Formulation We consider a binary alloy at constant temperature; an isolated precipitate of phase p grows into a matrix phase m. We normalize compositions in such a way that the scaled equilibrium compositions of m and p phases are cme ¼ 0 and cpe ¼ 1, respectively. A two-phase microstructure in this alloy is described in terms of composition cðr; tÞ (conserved variable) and order parameter gðr; tÞ (non-conserved variable) in a periodic domain. The order parameter field g is

R. Mukherjee et al. / Acta Materialia 57 (2009) 3947–3954

defined in such a way that g ¼ 0 in the m-phase and g ¼ 1 in the p-phase; see Eq. (6) below. The microstructural evolution in our system is governed by the Cahn–Hilliard equation [13] and the Allen–Cahn equation [14]: @c ¼ r  Mrl; @t @g dðF =N V Þ ¼ L ; @t dg

l¼

Parameters m

Non-dimensional values

Elastic parameters

G m d eT /ðgÞ bðgÞ C ijkl eff

800 0.3 0.5, 1.0 and 2.0 0.01 g3 ð10  15g þ 6g2 Þ  12 g3 ð10  15g þ 6g2 Þ ijkl ijkl 1 2 ðC m þ C p Þ

Simulation parameters

Dx Dy Dt System size

0.4 0.4 0.2 20000 in 1D 1024  1024 in 2D 6 108

Allowed error in displacements

ð3Þ

The total free energy of the system F is assumed to be given by the sum of a chemical contribution F ch and an elastic contribution F el : F ¼ F ch þ F el :

ð4Þ

The chemical contribution F ch is given by the following functional: Z 2 2 ð5Þ F ch ¼ N V ½f ðc; gÞ þ jc ðrcÞ þ jg ðrgÞ dX; X

where f ðc; gÞ is the bulk free energy per atom, jc and jg are the gradient energy coefficients for gradients in composition and order parameter, respectively, and N V is the number of atoms per unit volume. The bulk free energy density f ðc; gÞ is given by, 2

f ðc; gÞ ¼ f m ðcÞð1  W ðgÞÞ þ f p ðcÞW ðgÞ þ P g2 ð1  gÞ ;

ð6Þ

where P, a constant, sets the height of the free energy barrier between the m-phase and p-phase. In Eq. (6), f m ðcÞ and f p ðcÞ stand for free energies of m (matrix) and p (precipitate) phases, respectively. In our work, they are assumed to take the following simple forms: f m ðcÞ ¼ Ac2 ;

ð7Þ 2

f ðcÞ ¼ Bð1  cÞ ;

ð8Þ

with positive constants A and B. In Eq. (6), W ðgÞ is an interpolation function [15], given by: 8 for g < 0; > < 0; ð9Þ W ðgÞ ¼ g3 ð10  15g þ 6g2 Þ; for 0 6 g 6 1; > : 1; for g > 1: The elastic contribution to the free energy is: Z 1 F el ¼ rel ; el dX 2 X ij ij

ð10Þ

where 0 eel ij ¼ eij  eij ;

Parameter type

ð2Þ

dðF =N V Þ : dc

p

Table 1 Parameters used in the simulations.

ð1Þ

where M is the atomic mobility, l is the chemical potential and L is the relaxation coefficient for the order parameter. Chemical potential l is defined as the variational derivative of the total free energy per atom, F, with respect to the local composition c,

3949

ð11Þ

rel and el are elastic stress and strain tensors, respectively, and e0 is the the position-dependent eigenstrain (misfit strain). The total strain eij is:   1 @ui @uj ; ð12Þ þ eij ¼ 2 @rj @ri where ui is the the displacement field. Assuming both the m and p phases to be linear elastic, we have: el rel kl ¼ C ijkl eij ;

ð13Þ

where C ijkl is the elastic modulus tensor. The stress field rel ij obeys the equation of mechanical equilibrium: rel ij;j ¼ 0:

ð14Þ

The eigenstrain e0ij and the elastic moduli C ijkl are assumed to depend on the order parameter as follows: e0ij ðgÞ ¼ bðgÞeT dij ; C ijkl ðgÞ ¼

C eff ijkl

þ /ðgÞDC ijkl ;

ð15Þ ð16Þ

where eT is a constant that determines the strength of the eigenstrain, dij is the Kronecker delta, bðgÞ and /ðgÞ are scalar (interpolation) functions, where C eff ijkl is an ‘‘effective” modulus, C pijkl and C mijkl are the elastic moduli tensor of the p and m phases, respectively, and DC ijkl ¼ C pijkl  C mijkl . All the model parameters used in our simulations are listed in their non-dimensional form in Table 1. Our non-dimensionalization procedure is the same as that used in Ref. [16]. For a configuration at time t, the equation of mechanical equilibrium, Eq. (14), is solved using an iterative Fourier spectral technique with periodic boundary conditions (described in detail in Gururajan and Abinandanan [16]; see also Refs. [17–21]) to yield the elastic stress and strain fields; these, in turn, are used to integrate the Cahn–Hilliard (Eq. (1)) and Allen–Cahn (Eq. (2)) equations over a time step Dt to yield the configuration at t þ Dt. For this time integration step, we use a semi-implicit Fourier

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R. Mukherjee et al. / Acta Materialia 57 (2009) 3947–3954

spectral technique, due to Chen and Shen [22]; the Fourier transforms are performed using the freeware called FFTW, developed by Frigo and Johnson [23]. Microstructural evolution is simulated through a repeated application of this procedure on the new configuration at the end of each time step.

(a)

3. Results In our simulations, we place a small particle of initial radius Ro and composition cpe at the centre of our simulation cell with periodic boundary conditions; the initial composition outside the particle is set to c1 everywhere; with the normalizing scheme we have used, c1 is numerically the same as the supersaturation (or equilibrium volume fraction): n ¼ ðc1  cme Þ=ðcpe  cme Þ. As the simulation proceeds, we track the particle radius, operationally defined as the distance from the centre where g ¼ 0:5, and hence the growth coefficient a. The simulations end when either the diffusion field or the elastic stress field from neighbouring simulation cells begin to overlap; the values of a obtained just before the simulations end are used in the plots. They are also listed in Table 1 along with Rf , the final precipitate radius, and nE ¼ ðc1 –cmE Þ=ðcpe –cme Þ, the effective (i.e. elasticity-corrected) supersaturation.

(b)

3.1. Zener–frank theory We first compare our 1D simulation results with those from the ZF theory. In Fig. 2, we have plotted the growth coefficient a against matrix supersaturation n. The data points from our 1D simulations are in excellent agreement with the analytical results of ZF theory for 1D systems (the lowest curve) for all supersaturations. This agreement is not just in the growth coefficient, but also in the composition profile in the matrix phase in Fig. 3a.

1D 2D 3D 1D 2D

1.6

α

1.2

0.8

0.4

0 0

0.2

0.4

0.6

0.8

ξ Fig. 2. Dependence of growth coefficient a on matrix supersaturation n. The three curves are from the ZF (sharp interface) theory for 1D, 2D and 3D growth. The data points are from our 1D and 2D phase field simulations.

Fig. 3. A comparison of scaled composition profiles in (a) a 1D and (b) a 2D system with n ¼ 0:4. The dashed curve in the matrix is from the ZF sharp interface theory, while the solid and dotted curves are from phase field simulations at two different particle sizes. The distance on the x-axis is scaled by the instantaneous particle size. In the legend, S and P stand for sharp interface theory and phase field simulations, respectively.

Thus, this excellent agreement in the baseline case— the one without interface curvature and without misfit strains—is our main evidence for the essential correctness of the use of a phase field variable g in our model. Fig. 2 also displays a comparison of the 2D growth coefficients from the ZF theory and our simulations. Values of a from our simulation are in excellent agreement with those from ZF theory for a supersaturation of n ¼ 0:4 (within 0.4% at a final particle size of Rf ¼ 60), but become less so at lower supersaturations. At n ¼ 0:1, growth coefficients from ZF theory and our simulations differ by about 13% at Rf ¼ 21. The main conclusion from these results is that particles of smaller size, which experience a stronger effect due to interface curvature, grow slower than predicted by ZF theory. As shown in Fig. 1a, the curvature effect may be explained using the elevation of matrix interfacial composition from cme from zero to cmI , which results in a reduced supersaturation. Since the reduction in supersaturation

R. Mukherjee et al. / Acta Materialia 57 (2009) 3947–3954 1.8

c∞ = 0.1 , S c∞ = 0.4 , S c∞ = 0.1 , P c∞ = 0.4 , P

1.6 1.4

3951 No Misfit, S Misfit, S Misfit, P

1.6

1.2

1.2

α

α

1

0.8

0.8 0.6

0.4

0.4 0.2

0

0 0

10

20

30

40

50

60

0

70

0.2

0.4

0.6

0.8

ξ

R Fig. 4. Effect of capillarity on instantaneous growth coefficient a, plotted as a function of instantaneous particle radius R in 2D, non-misfitting systems. The horizontal lines are from the ZF sharp interface theory (without considering capillarity effects), and the data points are from our 2D phase field simulations.

Fig. 5. Dependence of growth coefficient a on matrix supersaturation n in a misfitting system with d ¼ 0:5. The dashed curve is for LJV sharp interface theory in 2D, and the data points are from 2D simulations. The solid curve, from the 2D ZF theory for non-misfitting precipitates, is also shown for comparison.

for a given particle size is a larger fraction of the original supersaturation in a less concentrated matrix, the curvature effect on growth coefficient is much higher at n ¼ 0:1 than at n ¼ 0:4. Thus, in Fig. 4, in which the instantaneous values of growth coefficient are plotted against instantaneous particle size, the data points from our simulations are much closer to the ZF result in the latter, even at a particle size of R ¼ 20. We return to a discussion of the curvature effect in Section 4. Similar to the 1D case in Fig. 3a, the composition profile in the matrix at two different sizes obtained from our simulations show good agreement, in Fig. 3b, with that from the ZF theory for a matrix composition of n ¼ 0:4.

and simulations is about 18% at Rf ¼ 21. Thus, the combined effect of misfit and capillarity in Fig. 5 mimics that of capillarity alone in Fig. 2. In Fig. 6, we find a good agreement between the simulated (scaled) composition profile for two different times with that for the LJV model. This agreement is similar to that for the non-misfitting case in Fig. 3b.

In these simulations, we have used a dilatational misfit of 1%. The precipitate and matrix phases are elastically isotropic, with a Poisson’s ratio of 0.3. The shear moduli of the two phases, however, can be different. We have considered three different cases with ðGp =Gm Þ ¼ d ¼ 0:5, 1.0 and 2.0, though, for the sake of clarity, we present our plots only for d ¼ 0:5. As we mentioned in Section 1, if curvature effects are neglected, the growth coefficient vs. supersaturation curve (dashed curve in Fig. 5) from the LJV theory for the misfitting case is obtained by shifting the ZF curve (solid curve, for the non-misfitting case) to the right by cmE . In Fig. 5, the growth coefficients a from our simulations are plotted against supersaturation, along with the LJV result for a system with elastically soft precipitates ðd ¼ 0:5Þ. The agreement is quite good at high supersaturations (with a difference of just 0.1% at a particle size of Rf ¼ 57 in an alloy with n ¼ 0:4Þ. With decreasing solute concentration, however, the agreement becomes worse; at n ¼ 0:1, for example, the difference between the LJV theory

A more detailed assessment of the role of capillarity is possible by considering cpI , the precipitate interfacial composition.  In the  sharp interface model, the difference  DcpI ¼ cpI  cpe , is proportional to DcmI ¼ cmI  cme :

S time R 35, P R 55, P

1

0.8

0.6

c

3.2. Effect of a misfit strain: LJV theory

3.3. Effect of interface curvature

0.4

0.2

0 1

2

3

Scaled Distance Fig. 6. A comparison of scaled composition profiles in a 2D misfitting system with n ¼ 0:4. The dashed curve in the matrix is from the ZF sharp interface theory, while the solid and dotted curves are from phase field simulations at two different particle sizes. The distance on the x-axis is scaled by the instantaneous particle size.

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R. Mukherjee et al. / Acta Materialia 57 (2009) 3947–3954

(a)

S c∞ = 0.1, P c∞ = 0.4, P

1.08

cI

p

1.06

1.04

1.02

1 0

0.02

0.04

0.06

0.08

0.1

1/R (b) 1.08

cIp

1.06

1.04

1.02 S P, c∞ = 0.1 P, c∞ = 0.4

1 0

0.02

0.04

0.06

0.08

0.1

1/R Fig. 7. Dependence of precipitate composition on particle radius in (a) non-misfitting and (b) misfitting systems. The straight line is for cpI , precipitate composition at a curved interface, given by the Gibbs– Thomson effect (see Eqs. (17) and (18)). The data points represent the composition at the centre of the precipitate in simulations with n ¼ 0:1 (open circles) and n ¼ 0:4 (filled circles).

DcpI

Wm ¼ p DcmI W

ð17Þ

where W ¼ ½@ 2 f =@c2 ce in the designated phase, and DcmI is given by the generalized Gibbs–Thomson effect [24]: DcmI ¼

rmij ðemij  epij Þ þ ½rpij ðepij  eT dij Þ  rmij emij =2 vc þ p : ðcpe  cme ÞWm ðce  cme ÞWm

ð18Þ

In the above equation, rij and eij are the stresses and strains, respectively, in the designated phase, eT is the dilatational eigenstrain and v is the mean interface curvature. In our 2D simulations, v ¼ 1=R; from Eqs. (7) and (8) (with the parameter values of A ¼ B ¼ 1), we get Wm ¼ Wp ¼ 2. For the values of misfit, shear modulus and Poisson’s ratio used in this study (see Table 1), the (constant) elasticity contribution to DC mI —the first term on the right-hand side of Eq. (18)) — is 0.044, 0.056 or 0.066 for systems with d ¼ ðGp =Gm Þ ¼ 0:5, 1.0 or 2.0, respectively.

In simulations using diffuse interface models, interfacial compositions cpI and cmI cannot be measured, as the composition changes continuously across an interface region of finite width. However, if we assume that the chemical potential gradients are negligible inside the precipitate, the composition anywhere in the precipitate should be the same as cpI . In Fig. 7, we have plotted cpI , obtained from simulations against the inverse of particle size R in (a) non-misfitting and (b) misfitting systems. In Fig. 7a, the straight line represents the Gibbs–Thomson result (Eq. (18) without the stress and strain terms), and the data points from our simulations are compositions at the centre of the precipitate; for clarity, we have presented data only for two supersaturations: n ¼ 0:1 and n ¼ 0:4. This figure shows that, in a growth setting, particles growing in more concentrated alloys display a smaller Gibbs–Thomson effect. In Fig. 7b, we compare the theoretical value of cpI (the dashed line) with those from simulations for n ¼ 0:1 and n ¼ 0:4. This figure also shows that the Gibbs–Thomson effect is less completely realized for more supersaturated alloys, leading to a better match between a from simulations and that from theory (see Figs. 2 and 5). Thus, Fig. 7 reveals yet another reason for the closer agreement between ZF theory and our 2D simulations for more concentrated alloys: Gibbs–Thomson effect is less completely realized in systems with a higher supersaturation. At n ¼ 0:1, even though our results were obtained in a growth setting, the small supersaturation ensures that the precipitate experiences a large part of the Gibbs–Thomson effect. Thus, in the misfitting case, DcpI ¼ 0:066 from our simulations (at Rf ¼ 21) matches well with the theoretical result of DcpI ¼ 0:067 calculated from Eq. (18). This agreement in Fig. 7b may be taken as an indirect verification of the predictions of Johnson and Alexander [11] and Leo and Sekerka [12] on the phase compositions across a sharp interface. 4. Discussion As we mentioned in Section 1, the primary aim of this article is to use our ‘‘computer experiments” to validate the LJV theory of elastic stress effects during precipitate growth. The following features of our phase field model make it resemble closely the alloy model used in the LJV theory: isotropic interfacial energy, isotropic atomic mobility, constant atomic diffusivity in the matrix (and in the precipitate too, though this is not an essential part of the LJV theory), isotropic elastic moduli, and constant misfit in the precipitate phase. While the LJV theory includes cases where composition gradients may engender stress, our model parameters are such that they do not. However, interfacial curvature (and hence the Gibbs– Thomson effect) is one feature which is always present in phase field simulations (except in 1D), but absent in the LJV and ZF theories. Its primary consequence is to dampen the growth rate. However, this dampening effect keeps decreasing with increasing particle size, and the

R. Mukherjee et al. / Acta Materialia 57 (2009) 3947–3954 Table 2 Comparison of growth coefficients from theory ðat Þ and simulations ðas Þ. System

n

nE

Rf

at

as

100  ðas  at Þ=at

No misfit

0.1 0.2 0.3 0.4

0.1 0.2 0.3 0.4

21 35 48 60

0.369 0.628 0.897 1.199

0.321 0.604 0.885 1.194

13.0 3.8 1.3 0.4

Misfit, d ¼ 0:5

0.1 0.2 0.3 0.4

0.056 0.156 0.256 0.356

21 32 44 57

0.246 0.514 0.775 1.061

0.202 0.502 0.776 1.060

17.9 02.3 +0.1 0.1

Misfit, d ¼ 1:0

0.1 0.2 0.3 0.4

0.044 0.144 0.244 0.344

19 31 43 56

0.208 0.482 0.742 1.023

0.145 0.469 0.744 1.031

30.3 02.7 +0.3 +0.8

Misfit,d ¼ 2:0

0.1 0.2 0.3 0.4

0.034 0.134 0.234 0.334

16 30 42 55

0.178 0.458 0.717 0.996

0.068 0.444 0.722 1.003

61.8 3.1 +0.7 +0.7

instantaneous growth coefficient keeps rising towards the theoretical values (e.g. Fig. 4). It is clear from Figs. 2 and 5 that capillarity has a larger dampening effect on a at lower supersaturations. In misfitting systems, the available (effective) supersaturation nE ¼ ðc1  cmE Þ=ðcpe  cme Þ is smaller when the precipitate is stiffer (see Table 2). Thus we find that a values from simulations are smaller than theoretical values by 18%, 30% and 62%, when d ¼ 0:5; 1:0 and 2.0, respectively, in systems with n ¼ 0:1. Our 2D simulations (with and without misfit) show that the Gibbs–Thomson effect is only partially realized during growth, especially at large supersaturations. Since the Gibbs–Thomson effect dampens the growth rate, its partial realization helps bring the growth rate closer to the theoretically predicted rate in these alloys—see Figs. 2 and 5 for 2D growth. Clearly, with higher (lower) atomic mobility in the p phase, the Gibbs–Thomson effect can be realized more (less) completely. Thus, if the atomic mobility in the p phase is higher (lower), the approach of the growth coefficient (in Fig. 4) towards the ZF or LJV result will be delayed (hastened) to larger precipitate sizes. Our simulations, however, cannot address this issue, since they use a constant mobility in both m and p phases. We now turn to other similar studies that compared results from sharp and diffuse interface models in a kinetic setting. Specifically, we mention Chen and co-workers, who have compared their phase field results on diffusion fields around a precipitate with those obtained by solving a diffusion equation. However, the focus of their work was on diffusion fields in ternary systems [25] or on lengthening and thickening of plate-like precipitates [26]. Thermally induced fluctuations in composition (and order parameter) may induce instabilities in the shape of a growing precipitate; in addition to the classical Mullins– Sekerka instability [27,28], stress-induced shape bifurcation [29,30] may also be caused by fluctuations. These shape

3953

instabilities are promoted at larger sizes. In order to test the role of fluctuations in our system, we introduced a random noise of magnitude 0:05% in both c and g at each time step in our 2D simulations with a circular particle. At the end of these simulations the particle remains circular, indicating that the fluctuations have not had any effect on particle shape in the size regime explored in this study. 5. Conclusions

(1) Using a our phase field simulations as computer experiments, we have validated the LJV theory of growth of misfitting particles in systems in which composition differences do not engender stress. (2) Capillarity effects decrease instantaneous growth rate; this dampening effect is more pronounced at smaller sizes and at smaller supersaturations. (3) At constant particle size, capillarity has a smaller effect in alloys with larger supersaturations, because the Gibbs–Thomson effect is less completely realized. (4) At low supersaturations, the Gibbs-Thomson effect is more completely realized, and our results on precipitate compositions are in agreement with those predicted from the theory of thermodynamics of stressed solids. Acknowledgements The authors thank the Volkswagen Foundation for financial support for this work. They also thank Prof. V. Jayaram and Dr. H. Ramanarayan for discussions. References [1] Zener C. Theory of growth of spherical precipitates from solid solution. J Appl Phys 1949;20(10):950–3. [2] Frank FC. Radially symmetric phase growth controlled by diffusion. Proc Roy Soc London. Ser A, Math Phys Sci 1950;201(1067):586–99. [3] Sekerka RF, Wang S-L. Moving phase boundary problems. In: Aaronson HI, editor. Lectures on the theory of phase transformations. Warrendale (PA): TMS of AIME; 1999. p. 231–84. [4] Laraia VJ, Johnson WC, Voorhees PW. Growth of a coherent precipitate from a supersaturated solution. J Mater Res 1988;3(2): 257–66. [5] Eshelby JD. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc Roy Soc London. Ser A, Math Phys Sci (1934–1990) 1957;241(1226):376–96. [6] Eshelby JD. Elastic inclusions and inhomogeneities. In: Sneddon N, Hill R, editors. Progress in solid mechanics, vol. II; 1961. p. 87–140. [7] Larche F, Cahn JW. A linear theory of thermochemical equilibrium of solids under stress. Acta Metal 1973;21(8):1051–63. [8] Larche F, Cahn J. The effect of self-stress on diffusion in solids. Acta Metall 1982;30(10):1835–46. [9] Cahn JW, Larche FC. A simple model for coherent equilibrium. Acta Metall 1984;32(11):1915–23. [10] Larche FC, Cahn JW. The interactions of composition and stress in crystalline solids. Acta Metall 1985;33(3):331–57. [11] Johnson WC, Alexander JID. Interfacial conditions for thermomechanical equilibrium in two-phase crystals. J Appl Phys 1986;59(8): 2735–46.

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