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Effects of internal stresses and intermediate phases on the coarsening of coherentprecipitates: A phase-field study

M. Asle Zaeema,*, H. El Kadiria,b, M.F. Horstemeyera,b, M. Khafizovc, Z. Utegulovc,d

aCenter for Advanced Vehicular Systems, Mississippi State University, Starkville, MS 39759, United StatesbMechanical Engineering Department, Mississippi State University, Starkville, MS 39762, United Statesc Idaho National Laboratory, Department of Materials Science and Engineering, Idaho Falls, ID 83415, United StatesdNazarbayev University, School, of Science and Technology, Department of Physics, Astana 010000, Kazakhstan

a r t i c l e i n f o

Article history:Received 17 June 2011Received in revised form22 August 2011Accepted 6 September 2011Available online 14 September 2011

Keywords:CahneHilliard phase-field modelCoherent precipitatesCoarseningCompositional strainIntermediate phaseFinite element

a b s t r a c t

Phase stability, topology and size evolution of precipitates are important factors in determining themechanical properties of crystalline materials. In this article, the CahneHilliard type of phase-field modelwas coupled to elasticity equations within a mixed-order Galerkin finite element framework to study thecoarsening morphology of coherent precipitates. The effects of capillarity, particle size and fraction,compositional strain, and inhomogeneous elasticity on the kinetics and kinematics of coherent precip-itates in a binary dual phase crystal admitting a third intermediate stable/meta-stable phase wereinvestigated. The results demonstrated the ability of the model to simulate coarsening under theconcomitant action of Ostwald ripening and mismatch elastic strain mechanisms. Using a phenomeno-logical coarsening power law, coarsening rates were determined to depend on precipitate size andvolume fraction, compositional strain, and strain mismatch between precipitates and the matrix. Resultsalso showed that the necking incubation time between two neighboring precipitates depends inverselyon the precipitate’s initial sizes; however, under fixed volume fraction of precipitates, any increase in theinitial sizes of the precipitates mitigates the coarsening. Meanwhile, the compositional strain and thegrowth of the intermediate stable/meta-stable phase leads to substantial enhancements of precipitatecoarsening.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Precipitation hardening is a well-established practice that hasbeen substantially utilized in materials manufacturing and designfor the development of high-strength alloys [1e7] (e.g., AleCualloys [1], CueCo alloys [2], Niebase alloys [3], AleZn alloys [4, 5],AleSi alloys [6], and NieGe alloys [7]). Thus, literature includesa considerable number of experimentally based studies dedicatedto investigating and controlling the effects of precipitation onmaterials and their mechanical properties [1e13]. However, onlyduring the past two decades some efforts have been undertaken todevelop materials science models able to predict the topologicaland morphological evolution of precipitates [14e35]. Notably,nearly all of these model studies have concerned precipitation ina purely elastic domain [13e17,19e23,27,31,36e40].

In one such example of these precipitation-elasticity models, Lee[21e23] used a discrete atomic method (DAM) to pragmaticallypredict the shape evolution and bifurcation of a single coherentprecipitate embedded in a matrix subjected to an applied stress.When the precipitate-matrix system is isotropic, Lee [21e23] arguedthat, at equilibrium, a soft precipitate adopts a low-symmetricalshape such as a plate or a needle, while a hard precipitate tends tobe spherical. When the precipitate-matrix system tends to bestrongly anisotropic, however, the precipitate shape depends toa great extent on the size, elastic misfit, and interfacial energy of theprecipitate.

In stress-free systems with isotropic or low anisotropic interfacialenergies, thermodynamic equilibrium is achieved when the totalinterfacial area decreases and the curvature gradient is the dominantdriving force, which can lead to Ostwald ripening [24,28,41]. Fanet al. [25] used a diffusive interface approach to study Ostwaldripening in a two-phase system through non-conserved phase-fieldvariables, which represented orientations of precipitates in a secondphase matrix. For the composition of the secondary phase, however,they used a conserved phase-field variable. As Fan et al. [25] did

* Corresponding author. Tel.: þ1 662 325 0126.E-mail address: mohsen@cavs.msstate.edu (M. Asle Zaeem).

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not incorporate the elasticity effects, their approach for Ostwaldripeningwould bemore appropriate for fluids than crystalline solids.

Wang and Khachturyan [26], on the other hand, treated theeffects of coherent elastic strain on the instabilities affectingthe shape of a growing precipitate. These authors used the

time-dependent GinzburgeLandau (TDGL) evolution equation todescribe the growth of an ordered precipitate of an equilibriumcomposition at the expense of a supersaturated disorderedmatrix of a non-equilibrium composition. They showed that,unlike the shape evolution during coarsening which can bedescribed completely by the strain and interfacial energies [27],the shape evolution during growth is additionally controlled bythe supersaturation which provides the driving force for thegrowth.

Using a two-dimensional phase-field model, Warren and Mur-ray [28] examined the effect of particle spacing and interfacethickness on the coalescence and transformation mechanisms.These authors, however, did not suggest any functional dependenceof transformations on the topology and physical properties of theprecipitates.

In this paper, a phase-field-elasticity model is used to examinethe effects of precipitate size and fraction, compositional strain,mismatch elastic strain, and growth of an intermediate phase onthe concomitant transformation, shape evolution, and coarseningrate of a coherent precipitate system. We coupled a conservedphase-field model with elasticity and applied them together toprecipitation within a binary alloy. Numerical simulations wereperformed using a finite element (FE) approach, which has beenused to study transformations in single precipitates [42], phasetransformation and instabilities in multilayer thin solid films[43e45], and instabilities at solid-fluid interfaces [46].

Fig. 1. The non-dimensional triple-well free energy density f , as a function of scaledcomposition c, which provided the energy barriers for transformations illustrated bythe humps and wells.

Fig. 2. Phase-field simulation results showing coarsening of two neighboring precipitates where rl and rr were the initial radii of the left and right precipitates, respectively, d wasthe initial distance between two precipitates (shown in (b) t ¼ 0), and r¼ 30 a non-dimensional length, showing (a) necking mechanisms of closely spaced particles withoutcompositional strain effects, (b) Ostwald ripening mechanisms for relatively widely spaced particles without compositional strain effects, and (c) necking with particle drifts forwidely spaced particles with the effects from compositional strain. The colored bar scales the composition contour. (For interpretation of the references to colour in this figurelegend, the reader is referred to the web version of this article.)

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2. Formulations and simulation method

We coupled the CahneHilliard phase-field model [47] to elas-ticity to study solid state diffusion-controlled phase trans-formation. A binary system was considered, with components Aand B able to form three phases, designated by a, b, and g, andcharacterized by the molar composition of component B, cðx; tÞ inspatial (x), and time domains (t). The total free energy F of a non-homogeneous system with volume V, bounded by surface S, wasconstructed as follows:

Fðc;uÞ ¼ZV

�f ðcÞþUð 3;cÞþ1

2ðVcÞ$K$ðVcÞ

�dV�

ZS

t$udS; (1)

where f(c) is the stress-free Helmholtz free energy density ofa uniform system. The gradient energy coefficient tensor K wasassumed to be symmetric and positive definite, which reduces toa scalar K ¼ kI for isotropic systems, I being the identity tensor, tthe surface traction, and u the displacement vector. In (1), thesurface energy of the outer boundary S was neglected. Uð 3;cÞ is thestrain energy density and 3is the total strain tensor, so classicalkinematics imply the following:

U ¼ 12ð 3� eIÞ : C : ð 3� eIÞ; (2)

3¼ 12ðuVþ VuÞ; (3)

The fourth order partial differential equation governing phase-field/diffusion can be written as follows [42]:

_c ¼ V$ðB$VMÞ in V ; (4)

Q ¼ n$B$VM on S; (5)

where B is the symmetric, positive definite mobility coefficientstensor, which reduces to B ¼ BI for isotropic systems; n is the unitouter normal to S; and Q is the normal flux at the boundaries. Thechemical potential M corresponds to the variational derivative ofthe free energy of the system, as in Eq. (1), with respect to the molenumber or composition. The expression of chemical potential is thekey for the incorporation of multi-physics; thus, we assumeda primary dependence on strain, composition, and compositionsecond derivative Mð 3; c;V2cÞ. The chemical potential should thenbe written as follows [48]:

M ¼ dFðc;uÞdc

¼ f 0 þ U0 � K : ðVVcÞ; (6)

where primes indicate partial derivatives with respect to c.The second order partial differential equation governing

mechanical equilibrium can be obtained from dFðc;uÞ=du ¼ 0,which then takes the following form:

V$C : ðVuÞ ¼ ~C$Ve; ~C ¼ C : I in V ; (7)

where e is the dilatational strain due to composition change [49],and C is the stiffness tensor. The most pragmatic model follows thelinear Vegard’s law [50]:

e ¼ hðc� c0Þ: (8)

The reference composition c0 was taken to be the stress-freecomposition. h is the compositional strain parameter, whichchanges dramatically from alloy to alloy, e.g., h¼ 0.094 for Al-2 at. %Cu and h¼ 0.264 for Fe-5 at. %C [51]. The stress was defined as:

s ¼ vUv 3

¼ C : ð 3� eIÞ: (9)

The elastic properties mismatch between the precipitates andthe parent matrix was captured through the dependence of theelastic constants on the phase-field variable, but a smooth transi-tion along the interface separated the two neighboring phases,a and g. Thus, we have the following:

C ¼ 1�cg � ca

�2h�c� cg

�2Ca þ ðc� caÞ2Cgi; (10)

where cg and ca are the compositions related to the precipitate andthe matrix, respectively, and Cg and Ca are the stiffness tensors of

Fig. 3. Phase-field simulation results showing the necking time versus initial distancebetween two neighboring precipitates for various values of the compositional strainparameter.

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the precipitate and the matrix, respectively. If an intermediatephase b is introduced, the stiffness tensor of the intermediate phaseis assumed to take an intermediate value between Cg andCaðe:g: Cb ¼ ðCg þ CaÞ=2Þ.

In previous analyses [41], we developed a FE method thattreated similar multi-physics phenomena. As the mechanical anddiffusion/phase-field equations in the present case are of differentorders, uniform convergence of the results requires a differentorder of continuity for the interpolation functions. We formulatedand implemented similar mixed-order FE formulations, with bothC0 interpolation functions for the displacement and C1 interpola-tion functions for the concentration. Considering no traction on theboundaries, the weak form of the mechanical equation in (7) can bewritten as follows:ZV

ðVuÞ : C : dðuVÞdV ¼ZV

e~C : dðuVÞdV : (11)

In this equation, the displacement has a first order gradient, soconvergence could only be obtained through C0-type interpolation

functions. Four-node and three-node elements with regular C0-type interpolation functions were used to generate a two-dimensional (2D) mesh. After some mathematical manipulations,the weak form of phase-field/diffusion equations (4) and (5)becomes the following:

ZV

_cdc dV þZV

�f 00Vc$B$dVcþ ðVVcÞ : KB : dVVc

�dV

�ZS

ðVVcÞ : KB$n$dVc dSþZV

�U00 þ se0

�Vc$B$dVc dV

¼ZV

seB : dVVc dV þZS

sen$B$dVc dSþZS

Qdc dS; ð12Þ

where s ¼ ~C : 3. In (12), composition has a second order gradient,so C1-type interpolation functions were needed for convergence.We defined four-node (rectangular) and three-node nonlinearelements based on the one-dimensional Hermite cubic shapefunctions in order to discretize the 2D domain for composition.

Fig. 4. Phase-field simulation results showing coarsening mechanisms of N ¼ 65 randomly distributed precipitates with h ¼ 0:02 and Vfpy0:2. The colored bar scales the

composition contour. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Such C1-type interpolation functions form sixth order polynomialsin 2D. Uniform quadratic convergence, expected for conformingelements, was already achieved from this coupled mixed-order FEmodel [42]. A special time integration scheme was used, whichconsisted of implicit time-stepping and equilibrium/consistencyiterations at each time step. The details of the mixed-order FEmodel and its advantages compared with other numerical models,such as finite difference [52,53] and Fourier-spectral [16,17,19]methods, can be found in [42].

3. Results and discussions

For computational convenience, we used non-dimensionalgroups identified by overbars. Non-dimensional groups enhanceversatility for different binary systems. This non-dimensionaltreatment was reported in [42,52], but some aspects aredescribed in the Appendix. The dimensionless composition c variesbetween �1 and 1. The dimensionless Helmholtz free energydensity of a uniform unstressed system in (1) was assumed to bea triple-well potential that can produce two-phase and three-phasebinary systems; that is,

f ðcÞ ¼ W1ðcþ 1Þ2ðcÞ2ðc� 1Þ2þW2ðcþ 1Þ2ðc� 1Þ2; (13)

where W1 and W2 are dimensionless energy coefficients andc ¼ �1, c ¼ 0, and c ¼ 1 are the mole fractions of component B atwhich f ðcÞ is minimized. Three combinations of W1 and W2 wereconsidered to produce systems with two phasesðW1 ¼ 0; W2 ¼ 0:04Þ, systems with stable intermediate phasesðW1 ¼ 0:4; W2 ¼ �0:04Þ, and systems with a meta-stable inter-mediate phase ðW1 ¼ 0:4; W2 ¼ 0:02Þ. These combinations areillustrated in Fig. 1. These three systems reflect the behavior of theactual phase diagram of a binary system at different temperatures.Each case can be used to study the microstructural evolution ata constant temperature. For cases with intermediate phases, thecoarsening rate during the growth of an intermediate phase wasadditionally controlled by the energy level of the intermediatephase in the phase diagram that provides the driving force for thegrowth. This effect is similar to the supersaturation effectaddressed by the Wang and Khachturyan model [26].

Following [47], the non-dimensional nominal (equilibrium)interface thickness in the absence of elasticity (h ¼ 0) is:

l0 ¼ �cg � ca

�� k

2Dfmax

1=2; (14)

where Dfmax, illustrated in Fig. 1, is known as the chemical energybarrier for phase transformation. The optimal FE mesh size wasdetermined to be about one fifth of the interface width [42], whichguaranteed the uniform quadratic convergence naturally expectedfor conforming elements [54].

Fig. 5. Phase-field simulation results showing the dependence of the average radius ofprecipitates upon time for different values of compositional strain parameter hand fortwo volume fractions of particles (a) Vf

py0:2; (b) Vfpy0:4.

Table 1Coarsening law exponent and coarsening rate constant for two precipitate volumefractions Vf

p and three values of compositional strain parameter h.

h Vfpy0:2 Vf

py0:4

n K n K

0 3.58 0.52 3.49 1.250.02 3.57 0.88 3.53 2.120.05 3.60 1.86 3.52 3.59

Fig. 6. Phase-field simulation results showing the dependence of the coarsening rateconstant K upon precipitate volume fraction Vf

p for different values of compositionalstrain parameter h.

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Isotropic elasticity was adopted for all simulations. The dimen-sionless elastic constants for the a-phase (matrix) from theAppendix are C

a11 ¼ ð1� vÞ=ð1� 2vÞ � C

a44, C

a12 ¼ v=ð1� 2vÞ�C

a44,

and Ca44 ¼ ð1� vÞ=ð1þ vÞ, where v ¼ 0:3. We also defined

a parameter h ¼ Cg44=C

a44, which showed the hardness of the

precipitates compared to that of the matrix for systems withinhomogeneous elastic properties; i.e., precipitates are relativelysoft for h < 1 and hard for h > 1.

3.1. Two-phase systems

In this section, we consider an aeg dual phase system withoutan intermediate phase (solid line in Fig. 1). Both phases had equalisotropic stiffness tensors, (h ¼ 1), leading to precipitates withcircular shape at equilibrium. Fig. 2 shows coarsening between twoadjacent circular precipitates with nearly equal radii. As the spacing

was small, the precipitates in Fig. 2(a) developed necking withoutany contribution from the compositional strain because hwas set atzero. The diffusion distance was so small that surface tensionsquickly triggered local precipitation that bridged the two particles.As the bridge was established, pronounced curvature gradientstook place along the particle/matrix interface with strong locali-zation at the bridge. Curvature gradients drive sintering and resultin establishment of a single particle with uniform curvature at theinterface, ultimately a spherical particle.

In Fig. 2(b) and (c), the spacing between the two precipitateswas set larger to investigate sintering thresholds. In fact, atd ¼ 2r=3, the left-hand side precipitate decreased in size, and theOstwald ripening process took place. This spacing corresponds tothe case where far-field composition gradients drive an abnormalgrowth process, where a particle grows at the expense of anotherone. The dissolved atoms of the smallest particle join directly thelarger particle after upsetting the concentration of the neighboringregions.

The spacing in Fig. 2(b) was further investigated in the presenceof a positive compositional strain and the results are reported inFig. 2(c). Here, the two precipitates turned to coalesce, which forcednecking to take place during the coarsening process. Compositionalstrain and the induced stress drive atoms to high stressed positions.Thus any matter returning to solid solution from the dissolvingparticle will tend to rejoin the same particle preferentially atregions with minimal closer distance to the second particle. Thetwo particles tend thus to coalesce and form a bridge, which willdrive sintering. Sintering, however, is slower at higher composi-tional strains because particles tend to refine and split to escapefrom stress.

Although the final size of the relaxed precipitates in Fig. 2(b) and(c) were approximately the same, sintering in Fig. 2(c) induced the

Fig. 8. Phase-field simulation results showing the dependence of the necking time onthe initial spacing between two precipitates for different values of h, with h ¼ 0:02and rl ¼ rr ¼ 25 at t ¼ 0.

Fig. 9. Phase-field simulation results showing the evolution of precipitates’ radii fordifferent values of h, and with h ¼ 0:02, Vf

p ¼ 0:15, and r ¼ 25 at t ¼ 0.

Table 2Coarsening law exponents and coarsening rate constants for thenonlinear fits shown in

h Vfpy0:15

n K

0.5 3.72 1.141 3.76 0.551.5 3.75 0.23

Fig. 7. Phase-field simulation results showing the dependence of the coarsening rateconstant K upon initial particle radius for different values of compositional strainparameter h when the precipitate volume fraction was kept constant, Vf

py0:2.

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final locations of the final particles in a Lagrangian framework to bedifferent.

In a previous study, we analyzed the effects of compositionalstrain on the phase-field interface thickness and equilibrium size of

precipitates [55]. For the large values of compositional strainparameter (h > 0:1), the equilibrium interface thickness increaseddramatically, while the equilibrium size of single precipitatesdecreased if they did not completely dissolve in the matrix. In this

Fig. 10. Phase-field simulation results showing coarsening of two neighboring particles in a binary system with a stable intermediate phase. Initial conditions corresponded to rl ¼rr ¼ 15 and d ¼ 20. h ¼ 0:05. The colored bar scales the composition contour. (For interpretation of the references to colour in this figure legend, the reader is referred to the webversion of this article.)

Fig. 11. Phase-field simulation results showing coarsening of two neighboring particles with the presence of a meta-stable b-phase. The meta-stable phase served as an inter-mediary that facilitates diffusion and the appearance of a gradient of curvature for necking to take place. Initial conditions corresponded to rl ¼ rr ¼ 25 and d ¼ 10. h ¼ 0:02. Thecolored bar scales the composition contour. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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paper, we set relatively low values for the compositional strainparameter (h � 0:05) to alleviate these limiting cases.

We further investigated the effects of compositional strain onthe necking time between two closely spaced neighboring precip-itates. Fig. 3(a) and (b) exhibit the necking time versus initialdistance between two precipitates for different values of h, withparticle sizes of rl ¼ rr ¼ 25 and rl ¼ rr ¼ 30, respectively. h

reduced the necking time with more pronounced magnitudes forlarger particles. A threshold particle spacing, where necking did nottake place for h ¼ 0:0, was identified between d ¼ 18 and d ¼ 19,instead, Ostwald ripening occurred similarly to that illustrated inFig. 2(b).

To measure the coarsening rate of isotropic systems withcircular precipitates, we analyzed the dependence of the averageradii of precipitates (r) versus time on different values of particlevolume fractions ðVf

pÞ and h. For all simulations, 800� 800dimensionless domains were considered with periodic boundaryconditions. These domains included randomly distributed precipi-tates having initially equal radii (r0 ¼ 25). Fig. 4 is an example

showing coarsening of 65 randomly distributed precipitates withh ¼ 0:02, Vf

py0:2, and uniform initial precipitate radii of r0 ¼ 25.The coarsening power law was taken as follow:

ðrtÞn�ðr0Þn ¼ Kt; (15)

where rt and r0 were the average radii of precipitates at time t and 0,respectively, K the coarsening rate constant function of volumefraction of precipitates, and n the coarsening exponent.

Fig. 5(a) and (b) shows the variation of particle average radiuswith time for two volume fractions of precipitates and three valuesof h. Eq. (15) was used to fit the numerical simulations, and theresults were plotted in solid lines with the estimated values of Kand n in Table 1. The main observed effect is that increasing thevalues of h or the initial particle volume fraction enhanced thecoarsening rate. Fig. 6 illustrates better the effect of particle volumefraction, where Kwas shown to increase almost exponentially withVfp . This results from the decrease in the atomic diffusion distance in

the matrix and mechanisms observed in Fig. 2(c).

Fig. 12. Phase-field simulation results showing coarsening of N ¼ 65 randomly distributed precipitates with compositional strain parameter and volume fraction of particles beingh ¼ 0:02 and Vf

py0:2, respectively. The colored bar scales the composition contour. (For interpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)

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Fig. 7 exhibits the coarsening rate as a function of the initial sizeof precipitates for a constant volume fraction of precipices, Vf

py0:2.An increase in particle size that keeps a constant volume fractionleads to larger spacing between particles, and as such to a decreasein the coarsening rate. This decrease is almost linear for each valueof h. Accordingly, the fitted slopes are functions of h, and wherethese fits cross the horizontal axes (K¼ 0) reveals a system whichinitially had a single large precipitate (zero coarsening rate). Thesize of this single precipitate with Vf

py0:2 and an 800� 800dimensionless large domain is rf ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

800� 800� 0:2=pp

y202.This radius also corresponds to the finial size of the precipitateremaining in these systems if the coarsening would continue for anunlimited time.

3.1.1. Inhomogeneous elasticity (hs1)This section examines the effect of the relative hardness/soft-

ness of precipitates on the coarsening rate using different values ofh. Fig. 8 reveals that necking between two precipitates was muchmore delayed when they were hard than when they were soft. Infact, the mechanical and diffusional constraints between theparticles are much more alleviated for soft particles so processes ofsintering and dissolution are accelerated. For hard precipitates,a threshold for no coarsening could be identified. A limiting case forno or very slow coarsening could also be identified for a system oflow volume fractions of randomly distributed hard precipitates.

The growth kinetics of the precipitates was investigated fordifferent values of h under a relatively low volume fraction ofprecipitates, Vf

p ¼ 0:15, and reported in Fig. 9. Nonlinear-fits led tovalues of n and K presented in Table 2. Increasing h decreased thecoarsening rate, and with a tendency similar to that in Fig. 8, thecoarsening rate was substantially mitigated for systems havinghard precipitates with low volume fractions.

3.2. Growth of a stable/unstable intermediate phase

Growth of an intermediate phase at the interface of differentmetallic phases and under influence of different temperature,magnetic, or electrical fields has been reported in several experi-mental studies [56e62]. Examples include growth of: Ni3Ti, NiTi,NiTi2 phases at the Ni/Ti interface [56], Ag3Sn phase at the Sn/Aginterface [57], Ni3Sn4 phase at the Sn/Ni, Ni/Sn-0.7wt.%Cu, andNi/Sn-3.5wt.%Ag interfaces [58,59]. Also different modeling approacheshave been employed to study the growth of an intermediate phase atthe interfacial regions between two metallic phases [45,52,63e65].However, the effect of intermediate phase growth on the coarseningbehavior of coherent precipitates is not fully comprehended yet.

In this section, we examined the effect of stable/unstable inter-mediate phase growth on the coarsening rate of coherent precipi-tates. Isotropic elasticity with equal stiffness tensors for all threephases was considered; that is, h ¼ 1. Figs. 10 vand 11 showcoarsening between two adjacent circular precipitates. In Fig.10, thetwo precipitates had the same radii and were relatively widelyspaced. The growth of the stable intermediate phase b caused theparticle spacing to decrease, which allowed necking and coarseningto take place. This is in agreement with our previous work [42],where an increasing value of h caused an increase in the growth rateof a stable intermediate phase, which enhanced necking andcoarsening.

Fig. 11 shows simulation results for a two-precipitate systemthat admits formation of an intermediatemeta-stable phase b. Aftera certain extent of initial growth of the b-phase from thematrix andprecipitates, a thin b-phase contained crust was produced on theprecipitates and it could not grow further as the b-phase wasunstable. However, this initial growth of the b-phase caused the

overall precipitating phase to be larger, so the effective spacingdecreased. Further growth of the b-phase enhanced the neckingprocess. However, after completion of the precipitate coarseninginto a single precipitate, only the b-phase at the precipitate crustremained. The thickness of this b-phase crust depends on thechemical energy barrier ðDfmaxÞ and the compositional strain

Fig. 13. Phase-field simulation results showing the evolution of the precipitate averageradius from an initial volume fraction of Vf

py0:2.

Table 3Coarsening law exponent and coarsening rate constants for systems having a stableand a meta-stable intermediate phase with an initial volume fraction of Vf

py0:2.

h Stable b-phase Meta-stable b-phase

n K n K

0 5.52 3850 3.73 1.080.02 5.51 6015 4.52 35.460.05 5.48 9112 5.41 2887

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parameter h. For relatively large values of h (h > 0:05), the growthof the unstable b-phase continues to consume the entire parentphase [43,44]. This overwhelming growth of the b-phase is drivenby the high mismatch elastic strains that tended to reduce the totalfree energy of the system.

Fig. 12 shows an example of a randomly distributed precipitatecoarsening with a stable intermediate phase growth. The initialconditions were chosen to be the same as those used for simula-tions reported in Fig. 4. Initial stages of growth were characterizedby rapid consumption of the g-phase precipitates by the stable b-phase, which then coalesced and increased in size. The rest of thecoarsening process was similar to that of the two-phase systems.Thus, any phase change does not only alter the kinetics of particlegrowth, but also the mechanical and physical properties of thematerial as a whole.

Fig. 13(a) and (b) show the evolution of the precipitate averageradius for different values of h for systems having stable and meta-stable intermediate phases, respectively. In an effort to preventexcessive growth of the third phase (b), we only simulated caseshaving relatively low volume fractions of precipitates ðVf

p � 0:4Þ;otherwise, precipitates with circular shape would not systemicallyform. The fitted curves in Fig. 13(a) and (b) followed the formatgiven in Eq. (15), and the estimated values of K and n for differentvalues of h are shown in Table 3.

These results suggest that increasing h would enhance thecoarsening rate to a degree that generally exceeds the rates ob-tained with the two-phase system (Section 3.1). In fact, theenhanced growth of the b-phase would supplement the increase ofthe effective size of the particle, which causes necking and coars-ening to trigger and process faster. The mechanisms discussedearlier for Fig. 2(c) remain similar in essence. The effect of b-phaseon the coarsening was found to bemore pronounced than the effectof necking or Ostwald ripening. As such, the fitted K and n valueswere markedly different from those fitted in the case of the two-phase system.

In systems admitting growth of a meta-stable b-phase, lowervalues of h led to a behavior identical to the two-phase systems.However, with increasing values of h, a similar behavior to thatrecorded in the stable b-phase growth took place. In fact, for aneffect of the meta-stable phase to be noticeable, larger values of hare needed; otherwise, sufficient penetration within the particlecould not be achieved, and only the particle surface was affected bythe b-phase. As h reached large values, the metal-table b-phasegrowth tended to act in a similar way to that recorded for the stablephase.

4. Conclusions

A phase-field-elasticity model capturing the stability andcoarsening of coherent precipitates was developed to incorporatethe simultaneous actions of precipitate size and volume fraction,precipitate mechanical strength, compositional strain, and growthof an intermediate phase. The model was formulated with non-dimensional groups to increase its applicability. The governingequationswere derived from a coupling between the CahneHilliardphase-field model and elasticity, and were discretized usinga mixed-order finite element model.

The model was able to capture Ostwald ripening, necking andcoarsening while admitting effects from mismatch elastic strains.The results of the simulations can be summarized as follows:

- For the two-phase systems, increasing the compositional strainparameter enhanced necking and coarsening of two neigh-boring precipitates and consequently increased the coarseningrate in systems of randomly distributed precipitates.

- The coarsening rate of soft precipitates was higher than that ofhard precipitates. A volume fraction threshold for coarsening ofhard precipitates to occur could be identified.

- Higher volume fraction of precipitates increased the coars-ening rate.

- When a stable intermediate phase (b-phase) formed in thematrix, the transformation could overwhelmingly spreadwithin the precipitates for relatively high volume fractions ofprecipitates.

- Large values of compositional strain parameter enhanced thegrowth rates of the intermediate phase, and as such boostedthe necking and coarsening of the two neighboring precipitatesor randomly distributed precipitates.

- When a meta-stable intermediate phase grew in the matrix,the compositional strain aided the increasing effect of the b-phase on the coarsening the particle. A lower bound effectcorresponded to the co-existence of the three phases, while anupper bound effect corresponded to the entire consumption ofthe precipitate by the meta-stable phase, aiding the coarseningprocess tremendously.

Acknowledgments

The authors are grateful to Battelle Energy Alliance LLC and ToddAllen for financial support of this work under award number00108032.

Appendix

The scaled composition is:

c ¼ 2c� cg � ca�cg � ca

� ; (A1)

where ca, cb, and cg are the mole fractions of component B at whichf(c) is minimized. c varies between ca ¼ �1 and cg ¼ 1 with anintermediate phase at cb. The characteristic length scale L, thecharacteristic time scale U, and the characteristic energy densityWare respectively:

L ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi

k

r0kBT

r �2

cg � ca

2

; (A2)

U ¼ k

2Bðr0kBTÞ2�

2cg � ca

8; (A3)

W ¼ r0kBT�cg � ca

2

6

; (A4)

r0 is the density of the lattice sites, kB is Boltzmann’s constant, and Tis the temperature. The dimensionless gradient energy coefficient kbecomes 1 for all isotropic systems. Considering isotropic mobility,the dimensionless mobility coefficient B reduces to unity.

The elastic constants are scaled by E=ð1� vÞ, where E is theYoung’smodulus and v is the Poisson ratio of the precipitate phase. Ifan isotropic system is considered, the non-dimensional elasticconstants depend on the Poisson ratio only, givingC11 ¼ ð1� vÞ=ð1� 2vÞ � C44, C12 ¼ v=ð1� 2vÞ � C44, andC44 ¼ ð1� vÞ=ð1þ vÞ. The scaled compositional strainparameter is:

h ¼ h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE

r0kBTð1� vÞ

s �2

cg � ca

2

; (A5)

M. Asle Zaeem et al. / Current Applied Physics 12 (2012) 570e580 579

Author's personal copy

For an isotropic system, and assuming a fixed Poisson ratio, thegeneral system is governed by only the three independent non-dimensional parameters; h, W1 ¼ W1=W , and W2 ¼ W2=W .

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