Simulation study of effects of initial particle size distribution on dissolution

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Acta Materialia 57 (2009) 316–325 www.elsevier.com/locate/actamat

Simulation study of effects of initial particle size distribution on dissolution G. Wang a,b, D.S. Xu a,*, N. Ma b, N. Zhou b, E.J. Payton b, R. Yang a, M.J. Mills b, Y. Wang b b

a Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China Department of Materials Science and Engineering, The Ohio State University, 2041 College Road, Columbus, OH 43210, USA

Received 21 March 2008; received in revised form 1 September 2008; accepted 7 September 2008 Available online 6 October 2008

Abstract Dissolution kinetics of c0 particles in binary Ni–Al alloys with different initial particle size distributions (PSD) is studied using a threedimensional (3D) quantitative phase field model. By linking model inputs directly to thermodynamic and atomic mobility databases, microstructural evolution during dissolution is simulated in real time and length scales. The model is first validated against analytical solution for dissolution of a single c0 particle in 1D and numerical solution in 3D before it is applied to investigate the effects of initial PSD on dissolution kinetics. Four different types of PSD, uniform, normal, log-normal and bimodal, are considered. The simulation results show that the volume fraction of c0 particles decreases exponentially with time, while the temporal evolution of average particle size depends strongly on the initial PSD. Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Kinetics; Dissolution; Modeling; Phase field models; Nickel alloys

1. Introduction The first step in precipitation hardening treatment is homogenization, where an alloy is heated up to a singlephase region on the phase diagram to dissolve precipitates and remove chemical non-uniformity formed during alloy preparation. The homogenized alloy is then quenched to room temperature and subsequently aged at an intermediate temperature in the two-phase region of the phase diagram to precipitate out second-phase particles in a controllable fashion. Microstructure evolution during isothermal aging or continuous cooling in Ni-base superalloys has been studied extensively using analytical, experimental and simulation means [1,2]. However, much less attention has been paid to microstructure evolution during heating and the corresponding dissolution and homogenization

*

Corresponding author. Tel.: +86 24 23971946; fax: +86 24 23891320. E-mail address: [email protected] (D.S. Xu).

kinetics. Any undissolved precipitates and unremoved chemical non-uniformities in the parent phase during homogenization will greatly affect the precipitation process upon cooling. Furthermore, precipitate dissolution is coupled with grain growth during homogenization treatments of polycrystalline materials. In order to predict grain size, which controls the subsequent phase transformations upon cooling, an accurate prediction of dissolution kinetics and particle size distribution (PSD) change of second-phase particles during dissolution process is indispensable. A quantitative description of dissolution kinetics was first derived by Carslaw and Jaeger [3], who presented an exact solution in one-dimensional (1D) space for a single precipitate in an infinite or semi-infinite matrix with a planar interface. Shortly after, the dissolution kinetics of a spherical particle in 3D was considered by Thomas and Whelan [4], who treated the dissolution process as a reversal of the growth process. However, later studies [5–7] revealed that dissolution in 3D cannot be regarded as a reversal of growth, and the dissolution kinetics does not follow the parabolic law. Aaron et al. [6] showed that there

1359-6454/$34.00 Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2008.09.010

G. Wang et al. / Acta Materialia 57 (2009) 316–325

is no exact solution for precipitate dissolution in 3D, and the analytical solution derived by Whelan [5] under assumptions of stationary interface and local equilibrium at the interface offers the most accurate analytical solution. Aaron and Kotler [7] also considered the effect of curvature on dissolution kinetics and concluded that, in most alloy systems, the effect is negligible unless the difference in concentration at the precipitate–matrix interface and that in the matrix far away from the interface is extremely small. All these early analytical models are limited to precipitate dissolution in an infinite matrix under the assumption of constant diffusivity. Brown [8] investigated the dissolution of a single planar, cylindrical or spherical precipitate with a concentration-dependent diffusivity. While the analytical approaches [3–8] were limited to single-particle systems, numerical methods have been applied to precipitate dissolution in multi-particle systems [9–18], and the stationary interface approximation was removed. Uniform and log-normal PSD have been considered in these studies, and the effect of inter-particle spacing on dissolution kinetics was investigated explicitly. This study systematically investigates the effect of PSD on dissolution kinetics in multi-particle systems using a 3D quantitative phase field model where the Kim–Kim–Suzuki (KKS) treatment of interfaces [19–21] is implemented. In particular, four different types of PSD of c0 particles in a binary Ni–Al alloy are considered, and model inputs are obtained directly from thermodynamic and atomic mobility databases of the alloy. The phase field method has been applied extensively in recent years in simulating microstructural evolution during various materials processes (for recent reviews, see Refs. [22–26]). The method has been shown to converge to analytical solutions when the interfacial width asymptotically approaches the sharp or thin interface limit [27–31]. Recent advances [19–21,32,33] have made it possible to model quantitatively microstructure evolution using the phase field approach at experimentally relevant length and time scales. With input from CALPHAD thermodynamic and DICTRA kinetic databases, the dissolution of a globular a precipitate in Ti–Al–V has been simulated quantitatively in 2D using the phase field method, and the results agree well with DICTRA simulations [34]. With the thermodynamic data linked to JMatPro, the diffusion-controlled dissolution of primary particles in aluminum alloys was simulated in 1D by the phase field method, where an adaptive mesh-free algorithm was implemented [35], and the results obtained are also in excellent agreement with that obtained from the fronttracking method. The paper is organized as follows. The quantitative phase field model is described in Section 2. Validation of the model against existing analytical and numerical solutions of single-particle dissolution in 1D and 3D, respectively, and simulation results of the effect of initial PSD on particle dissolution kinetics in multi-particle systems are presented in Section 3. The simulation results are discussed in Section 4, and conclusions are given in Section 5.

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2. Phase field model The phase field model for precipitation of L12 ordered phase from an fcc solid solution has been well established [36,37]. In the model, a two-phase mixture of the ordered and disordered phases is characterized by concentration and long-range order (LRO) parameter fields. The local free energy of the system as a function of the concentration and LRO parameter fields can be obtained directly from the CALPHAD technique [38–40]. For the convenience of implementing the KKS model [19–21], however, the polynomial approximation employed in [34,41] is adopted in the current study. Since particle dissolution in binary Ni–Al alloy is considered, one concentration field (molar fraction) X and one LRO parameter field g are used to describe the c and c0 two-phase microstructure, and the presence of multiple antiphase domains of c0 phase is ignored. Even though particle coalescence and formation of antiphase domain boundaries during precipitate growth may significantly alter precipitate morphology and growth kinetics [42–44], they are not encountered during precipitate dissolution. The normalized LRO parameter (by its maximum value) changes within the range [0, 1], with g = 0 describing the c phase and g = 1 the c0 phase. With these simplifications, the total free energy of the system can be written as: Z h i 1 j Gm þ jrgj2 dV ð1Þ G¼ Vm V 2 where Vm is the molar volume (assumed constant in the current study), j is the gradient energy coefficient, and Gm is the molar free energy given by Gm ðT ;X ;gÞ ¼ ð1  pðgÞÞfc ðT ;X Þ þ pðgÞfc0 ðT ;X Þ þ xg2 ð1  gÞ

2

ð2Þ where p(g) = g3(10  15g + 6g2), x is the height of the hump on the Gm  g plot, and fc and fc0 are the molar free energy of the c phase and c0 phase, respectively, which can be obtained directly from the CALPHAD thermodynamic databases [38–40]. The temporal evolutions of the LRO parameter field and the concentration field are governed by the time-dependent Ginzburg–Landau equation [45] and the generalized diffusion equation (Cahn–Hilliard equation) [46]: og dG ¼ L ot dg   1 oX dG ¼ r  MðT ; X ; gÞr dX V 2m ot

ð3Þ ð4Þ

where L is the kinetic coefficient characterizing the evolution of the LRO parameter, and M is the chemical mobility. Since the chemical mobility of atoms in the ordered c0 intermetallic phase is much smaller than that in the c phase (practically zero in the DICTRA database), M is assumed to have the following values in the simulations:

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 M¼

Mc

ðg < 0:5Þ

0

ðg P 0:5Þ

ð5Þ

where Mc is the chemical mobility of Al atoms in the c phase in Ni–Al. According to [47], Mc can be described by the following expression: Mc ¼

1 ½ð1  X Þ2 XM c ðAlÞ þ X 2 ð1  X ÞM c ðNiÞ Vm

oX ~ r~f~ 0 Þ ð11Þ ¼ r~  ðM os og ~ 0 ½ðf~ c0  f~ c Þ  f~ 0 ðX c0  X c Þ þ 2xgð1 ~ ~ r~2 gg  gÞð1  2gÞ  j ¼ Lfp os ð12Þ

fc 0 G0

~¼ ; j

f0 G0

0.0341 0.0132 400 1444 K 0.12 lm2 s1

simulations of single-particle dissolution in order to compare the simulation results with the analytical or numerical solutions obtained from sharp interface models for validation. Eqs. (11) and (12) are solved numerically using an explicit algorithm, and periodic boundary conditions are applied along all dimensions. 3. Simulation results 3.1. 1D simulation of single-particle dissolution

with p0 = dp/dg = 30g2(1  g)2. Finally Eqs. (9) and (10) are converted into their dimensionless forms for the convenience of numerical integration:

where s ¼

~ j ~ x ~ L T D

ð8Þ

oX ð9Þ ¼ V m r  ðMrf 0 Þ ot og L 0 ¼ fp ½ðfc0  fc Þ  f 0 ðX c0  X c Þ þ 2xgð1  gÞð1  2gÞ  jr2 gg ot Vm ð10Þ

j G0 l20

0.1 lm 0.5 J m2 1018 m2 mol J s1 104 J mol1 105 m3 mol1

ð7Þ

Based on the above equations, the following new governing equations are obtained:

tM 0 G0 l20

l0 r M0 G0 Vm

ð6Þ

where Mc(Al) and Mc(Ni) are the atomic mobility of Al and Ni in the c phase, respectively, and they can be obtained directly from the DICTRA mobility database. In order to overcome the length scale limit of the conventional phase field model [32,33], the KKS model [20] is implemented, where the following relationships hold in the interfacial regions: X ¼ ð1  pðgÞÞX c þ pðgÞX c0 ofc ofc0 ¼ ¼ f0 oX c oX c0

Table 1 Parameters used in phase field simulations

2

~ ¼ V mM ; L ~ ¼ Ll0 ; f~ c ¼ ; M V mM0 M0

fc G0

;

o ~ ¼ Gx0 and r~ ¼ ½oðx=l f~ c0 ¼ ; f~ 0 ¼ ; x ; o ; o . M0 0 Þ oðy=l0 Þ oðz=l0 Þ and G0 are the normalization factors for mobility and energy, respectively, and l0 is the mesh size. As the KKS model removes the dependence of the interfacial energy on the chemical free energy, a relatively large mesh size l0 = 0.1 lm is adopted. The interface thickness k is set as five times the mesh size. Note that the interfacial width is much larger than the experimentally observed values, and its effect on the dissolution kinetics will be discussed in Section 4. All the input parameters used in the simulations are listed in Table 1. The kinetic coefficient for the LRO parameter was chosen in such a way as to ensure a diffusion-controlled dissolution process. For simplicity, an isothermal aging at T = 1444 K (above the c0 solvus) is considered. The thermodynamic and mobility data at this temperature are extracted from the ThermoCalc and DICTRA databases [48]. It should be noted that a constant diffusivity of Al in the c matrix is used in the

The system size is chosen to be 800 mesh points (80 lm) and the initial particle size is 40 mesh points (4 lm). The initial concentrations of Al in the matrix and precipitate are set at 0.163 and 0.235, respectively, which corresponds to the equilibrium compositions of the two phases at 1366 K. The system is brought to 1444 K instantaneously, and the dissolution kinetics and concentration profiles are obtained by solving the phase field equations. The results are shown in Fig. 1a and b, respectively, by discrete symbols. For a 1D diffusion-controlled dissolution process with a constant inter-diffusivity D, there is an analytical solution for the interface position x as a function of time t for a dissolving particle with a planar interface in a semi-infinite matrix [3]: pffiffiffiffiffi ð13Þ x ¼ x0 þ 2h Dt where x0 is the initial interface position at t = 0, and h is a constant that is related to the supersaturation k by the following relationship: k¼

X M  X ec pffiffiffi ¼ ph expðh2 ÞerfcðhÞ X ec0  X ec

ð14Þ

where X ec and X ec0 are the equilibrium concentration of Al in the c and c0 phase, respectively, and XM is the initial matrix concentration of Al far away from the c/c0 interface. As the system size is much larger than the precipitate size, XM remains constant during the entire dissolution process. Then k is calculated to be 0.18. The value of k increases monotonously with h, so one can get the value of h for the specific case by solving the above transcendental equation. Also the theoretical solution gives the concentration profile in the matrix phase [3]:   r  xðtÞ pffiffiffiffiffi ð15Þ X ðr; tÞ ¼ X M þ ðX c  X M Þerfc 2 Dt where r is the distance from the origin, and x(t) is the interface position at time t. Combining Eqs. (13) and (15), one

G. Wang et al. / Acta Materialia 57 (2009) 316–325

Fig. 1. Comparisons of dissolution kinetics and concentration profiles of Al at different time between phase field simulation (discrete symbols) and analytical solution (solid lines) in 1D: (a) particle width vs time; (b) concentration profiles of Al at t = 200 and 600 s.

obtains an analytical expression of the concentration profile:   r  x0 ð16Þ X ðr; tÞ ¼ X M þ ðX c  X M Þerfc pffiffiffiffiffi  h 2 Dt Comparisons of the particle size as a function of time and the concentration profile of Al at different times between the phase field simulation and the analytical solution are shown in Fig. 1. The agreement is good. 3.2. 3D simulation of single-particle dissolution In 3D the simulation is carried out in a cube with side length 25.6 lm (256 mesh points) and a spherical c0 particle with diameter 4 lm (40 mesh points) located in the center of the cube. The system is large enough compared with the particle size so that no soft impingement occurs during the dissolution, i.e., the concentration field in the matrix far away from the particle is constant. The dissolution kinetics and the concentration profiles along a radial direction

319

Fig. 2. Dissolution kinetics of a single c0 particle in 3D using phase field method: (a) particle radius vs time; (b) concentration profiles of Al along radial direction at t = 0, 20, 40, 60 and 80 s.

obtained from the simulation are shown in Fig. 2. Results obtained with different starting particle sizes are shown in Fig. 3. It can be seen clearly from Fig. 3b that the plots of R20  R2 (where R0 is the initial particle radius at time = 0) vs time are non-linear, which confirms that precipitate dissolution in 3D does not follow the parabolic law. Under the assumption of stationary interface and local equilibrium at the interface, Whelan [5] proposed the following equation for dissolution of a spherical precipitate in an infinite matrix: rffiffiffiffiffi dR kD D ¼ þk ð17Þ dt R pt where k is the supersaturation defined by Eq. (14). However, the kinetics of such a moving interface problem can be obtained by solving the diffusion equation numerically using the method developed by Murray and Landis [49]. The comparison of the dissolution kinetics obtained from these two methods with that obtained from the phase field simulation is shown in Fig. 4. It is readily seen that the

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phase field prediction agrees well with the numerical solution using the Murray–Landis method. Obviously, the stationary interface approximation that becomes accurate only in the asymptotic limit of k approaching zero is invalid in the current case (i.e., k = 0.18). 3.3. 3D simulation of multi-particle dissolution and effects of initial PSD on dissolution kinetics

Fig. 3. Dissolution kinetics of single c0 particle with different initial radius, i.e., 1.0, 1.5, 2.0, 2.5 and 3.0 lm: (a) particle radius vs time; (b) R20  R2 vs time. R is the particle radius, and R0 is the initial particle radius at t = 0.

To address problems in the heat treatment of engineering alloys, it is important to understand the dissolution process in multi-particle systems with different PSD. The simulation study is carried out in 3D with 360 mesh points (36 lm) along each dimension. Spherical c0 particles are generated with a random spatial distribution in the c matrix. It is assumed that the alloys are equilibrated at T = 1173 K. Therefore, the c and c0 phases have initial concentration of Xc = 0.139 and Xc0 = 0.231, respectively, and the initial volume fraction of the c0 particles is 23.23% if the average concentration of Al in the system is set at 0.16. The alloys are subsequently up-quenched to T = 1444 K, that is, above the c0 solvus. The chemical mobility as a function of concentration is calculated by Eqs. (5) and (6). In this study, four types of initial PSD are investigated: (a) uniform distribution (all particles are of the same size); (b) normal distribution; (c) log-normal distribution; and (d) bimodal distribution (two peaks exist in the PSD), which are illustrated in Fig. 5. The initial average particle size (APS) in all cases is 1 lm. Temporal evolutions of the microstructure and PSD during dissolution for the case with an initial log-normal PSD are shown in Figs. 6 and 7, respectively. The volume fraction change as a function of time is shown in Fig. 8, where the discrete symbols represent phase field simulation results, while the solid and dashed lines represent fittings to the simulation data by the following exponential function: f ¼ f0 expðKtn Þ

Fig. 4. Comparison of dissolution kinetics of single c0 particle in 3D: open squares are from phase field simulation, dashed line is from Whelan’s model, and solid line is from numerical solution based on the Murray and Landis method of moving boundary problem.

ð18Þ

where f0 and f are the volume fractions of c0 particles at the start and time t, and K and n are constants. It can be seen from Fig. 8 that the above equation characterizes well the temporal evolution of precipitate volume fraction in all the four cases considered. The values of K and n obtained for the four different cases are listed in Table 2. Fig. 9 shows the temporal evolution of the APS. It can be seen that different initial PSD lead to different time-evolution of the APS. In the case of uniform and normal PSD, the APS decreases monotonously, but the former exhibits a much faster dissolution rate than the latter. The total time for complete dissolution of all particles in the case of normal distribution is about five times that in the case of uniform distribution. More interestingly, the APS in the case of log-normal or bimodal initial PSD is not a monotonic function of time. It decreases at the beginning and then increases, before it finally decreases monotonically. In the cases of normal and log-normal initial PSD, APS decreases much more slowly as compared to the cases of uniform and bimodal PSD.

G. Wang et al. / Acta Materialia 57 (2009) 316–325

Fig. 5. The different initial PSD of c0 particles used in the phase field simulations: (a) uniform; (b) normal; (c) log-normal; (d) bimodal.

Fig. 6. Microstructural evolution during dissolution in an alloy with initial log-normal PSD: (a) t = 0 s; (b) t = 5 s; (c) t = 10 s; (d) t = 20 s.

321

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Fig. 7. Temporal evolution of PSD for the case with initial log-normal PSD.

Fig. 8. Temporal evolution of particle volume fraction with different PSD: the discrete symbols are from phase field simulations and the lines are from exponential fitting.

Fig. 9. Temporal evolution of average particle radius.

time, following Eq. (18) in all cases studied, with different K and n values in each case (Table 2). This includes the single-particle systems shown in Section 3.2 and the one that will be discussed below (Fig. 10) because, with the application of the periodical boundary condition, a single-particle system can be regarded as a multi-particle system with uniform PSD and uniform spatial distribution. Among all the PSD considered, the uniform one yields the fastest dissolution (Figs. 8 and 9). In order to gain further insights into this simulation result, two additional simulations are carried out, where two periodic cubic arrays of eight particles with the same particle volume fraction are considered. The side length of the 3D computational system is 80 mesh points (8 lm). In one of the simulation, all eight particles have the same radius of 15 mesh points (1.5 lm), while in the other, four small particles have the same radius of 10 mesh points (1 lm) and four large particles have the same radius of 18 mesh points (1.8 lm). The

Table 2 Values of K and n for different distributions PSD

K

N

Uniform Normal Log-normal Bimodal

0.7648 0.565 0.6132 0.5833

0.688 0.5843 0.5171 0.502

4. Discussion A 3D quantitative phase field model was applied to study the dissolution of c0 particles in Ni–Al alloys with different initial PSD. The simulation results clearly show the strong dependence of the dissolution kinetics in terms of both precipitate volume fraction change (Fig. 8 and Table 2) and APS change (Fig. 9) on the initial PSD. The precipitate volume fraction is found to decay exponentially with

Fig. 10. Comparison of dissolution kinetics between uniform and random spatial distribution of precipitates in 3D, with the same initial uniform PSD.

G. Wang et al. / Acta Materialia 57 (2009) 316–325

particles in the latter are arranged in such a way that the large particles are surrounded by the small ones, and the small ones are surrounded by the large ones. The dissolution kinetics of the two systems are compared in Fig. 11. It is clear that the uniform PSD leads to faster dissolution kinetics. This can be explained by the effect of particle coarsening. In the case of non-uniform PSD, the small particles feed the growth of large ones because of the Gibbs– Thompson effect, which slows down the dissolution of the large particles as well as the overall dissolution rate. The broader the PSD, the stronger the effect. In the case of uniform PSD, all particles have the same size initially and dissolve at the same rate, limiting the Gibbs–Thompson effect to the minimum. It is worth mentioning that Eq. (18) describes the dissolution kinetics only when the temperature is above the solvus. If the heat-treatment temperature is below the solvus, this equation cannot be satisfied at a late stage. In this case, a more general form could be used f ¼ feq þ ðf0  feq Þ expðKtn Þ

ð19Þ

where feq is the final equilibrium volume fraction of precipitates at the aging temperature. Systematic simulation studies are currently being carried out to validate Eq. (19) for dissolution kinetics at temperatures below c0 solvus. The APS as a function of time is even more sensitive to the initial PSD (Fig. 9). It is well known that, in the case of volume-diffusion controlled precipitate growth, the average precipitate size as a function of time will follow a parabolic growth law until soft impingement occurs [50]. However, the situations are completely different for precipitate dissolution. As shown in Fig. 9, it is impossible to obtain a general relationship between the APS and time. During a dissolution process, the APS is determined by a complicated interplay between the disappearance of particles that increases APS and the shirking of particles that decreases APS. Such interplay depends strongly on the initial PSD. In the case of uniform PSD, for example, all particles dis-

Fig. 11. Comparison of dissolution kinetics between uniform PSD and non-uniform PSD, with the same uniform particle spatial distribution.

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solve at similar rates, because they are of the same size initially, and disappearance of particles occurs at a similar time, i.e., towards the end, leading to fast decay of APS with time. In addition, the lack of particle coarsening also accelerates the dissolution process, as discussed earlier. In the other three cases, large particles dissolve more slowly compared with small ones (see Figs. 3 and 6) because of particle coarsening and also the fact that large particles are more prone to soft impingement (more solutes need to diffuse away). Therefore, the ratio of particle disappearance rate over particle shrinkage rate will be larger in these three cases than that in the case of uniform PSD, leading to relatively slow decaying of APS with time. Among the three cases of normal, log-normal and bi-modal PSD, the time-evolution of the APS will depend on the detailed process of the particle disappearance and shrinkage. In the case of normal PSD, for example, the number frequency of large particles is equal to that of small ones. Thus, small particles will continue to disappear, while large ones continue to shrink, leading to monotonic decay of APS with time. In the cases of log-normal and bi-modal PSD, the number frequency of large particles is much smaller than that of small particles. The oscillation of APS as a function of time shown in Fig. 9 can be readily understood from the change in the balance between particle disappearance and shrinkage. In the case of log-normal PSD, for example, it can be seen from Fig. 7 that the PSD peak shifts from right to left at the beginning (e.g., at t = 5 s). This indicates that each particle reduces its size, but few particles disappear during this stage, so the APS change is dominated by particle shrinkage at this time, and it decreases with time. At later times, small particles start to disappear, while large ones decrease their size slowly (see Fig. 6), thus the distribution peak shifts back and the APS increases (Fig. 9). In this case, the change in APS is dominated by particle disappearance. When the dissolution proceeds (at t = 20 s, for example), large particles become smaller ones and dissolve faster, while small ones continue to disappear at more or less the same rate, the change of APS will be dominated by particle shrinkage again, and it starts to decrease. In the case of bimodal PSD, when the small particle population disappears at the early stages, the dissolution process is completely dominated by the shrinkage of the large particle population. This is why the overall decaying rate of APS in the case of bimodal PSD is much higher than that for the log-normal PSD. Note that the APS does not change much from t = 10 s to t = 40 s in the cases of normal and log-normal PSD, which indicates that the dissolution of large particles and disappearance of small ones almost balance each other. Relatively constant APS has been observed during dissolution of c0 particles in a Ni-base superalloy in the experimental study [51]. In addition to PSD, particle spatial distribution may also affect the dissolution kinetics. To investigate this effect, a 3D simulation of a single-particle system with a spherical c0 particle located in the center of a cube is carried out in order to make comparison with the case of uniform PSD

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presented in Section 3.3, where the spatial distribution is random. The particle radius (10 mesh points or 1 lm) and the side length of the cube (26 mesh points or 2.6 lm) are so chosen to match the initial precipitate volume fraction and particle size of case (a) in Section 3.3. Therefore, the only difference between the two systems is the spatial distribution, i.e., uniform vs random. The dissolution kinetics is shown in Fig. 10. It can be seen that the effect of particle spatial distribution on dissolution kinetics is relatively small compared with the effect of initial PSD. At the beginning, the two systems have similar dissolution rates, but the particle dissolution in the random spatial distribution system becomes slower than that with uniform distribution at later stages. The reason is that soft impingement occurs early on locally for particles close by in the former, and clusters of precipitates may behave more or less like large particles, which will slow down the overall dissolution rate. In the latter, soft impingement among all particles occurs at the same time, and all particles disappear at the same time in the end. In a previous study of precipitate growth in 2D [52], however, a more significant effect from soft impingement has been observed. In order to compare quantitatively such an effect in 2D vs that in 3D, a dissolution study was carried out in 2D with random and uniform spatial distributions. The aging temperature was chosen as T = 1273 K, and the corresponding area fraction of the precipitates was 11.6%, very close to that used in Ref. [52] (11.5%). The system size was 1024  1024, which was also the same as in Ref. [52]. All particles were of the same size initially. The simulation results are shown in Fig. 12. It can be seen that the uniform spatial distribution leads to faster dissolution kinetics, and the effect of spatial distribution is much stronger than that shown in Fig. 10. This is not surprising, because the effect of soft impingement should depend on particle dimensionality. The lower the dimensionality, the stronger the effect.

The advantages of using the phase field approach are that it can easily incorporate arbitrary spatial correlation and variation in a multi-phase microstructure, arbitrary particle shapes and volume fractions, anisotropy in interfacial energy [53–59], and long-range elastic interactions arising from lattice misfit or external stress [60–63]. As a matter of fact, the phase field model can take experimentally reconstructed 3D microstructure [64] as a direct input and predict microstructural evolution during particle dissolution, including temporal evolution of shape, spatial distribution, PSD, particle volume fraction and APS. It is easy to extend the current model to multi-component systems and the thermodynamic and mobility databases, and other physical properties of the system such as interfacial energy, lattice misfit and elastic constants can be incorporated self-consistently into the model, which provides a solid basis for quantitative material-specific simulations and predictions. The current study has resulted in functional dependence of volume fraction of precipitates on time that could be used as a fast-acting model in industrial practice. The particles have been assumed to have a spherical shape in the current study. However, previous studies [5,11,65] have shown that particle shape has little effect on the dissolution kinetics. The shape of the diffusion field some distance away from a dissolving particle was found to be approximately spherical, even for a disc-like precipitate [65]. The present simulations used a relatively large interfacial energy that may result in a relatively strong contribution from the Gibbs–Thompson effect. Quantitative study of such an effect will be carried out in a separate work. As the initial PSD plays an important role in determining precipitate dissolution kinetics, it must be measured when studying particle dissolution processes. From the point of view of practical application, this particle dissolution characteristics will have a significant effect on grain size control by particle pinning [66] and other constitutive relationships of mechanical properties [67] which are determined by both particle volume fraction and particle size. 5. Conclusions

Fig. 12. Comparison of dissolution kinetics between uniform and random spatial distribution of precipitates in 2D, with the same initial uniform PSD.

The dissolution kinetics of c0 particles in binary Ni–Al alloys was studied by 3D phase field simulations. The simulation results confirm that precipitate dissolution does not follow the parabolic law, and the temporal evolution of APS is very sensitive to the initial PSD. The volume fraction of particles decays exponentially with time during dissolution, irrespective of the initial PSD, but the dissolution rate depends strongly on the initial PSD, with the uniform PSD yielding the fastest dissolution kinetics. In comparison with PSD, precipitate spatial distribution has a relatively small effect on precipitate dissolution. As there is no exact analytical solution to the problem of precipitate dissolution and the best approximate analytical solution under the assumption of stationary interface deviates significantly from the numerical solutions of moving interface problems for large supersaturation, computer sim-

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ulation seems to be the only means of providing a quantitative description of the process. The comparisons with analytical and numerical solutions obtained from the sharp interface approach have shown that the phase field model is able to make quantitative predictions of precipitate dissolution in an arbitrary microstructure at the experimentally relevant length and time scales. It also allows one to extract functional forms of precipitate volume fraction as a function of time. However, no functional form can be developed for the change in APS as a function of time. Because in many situations, such as Zener pinning of grain growth and precipitation hardening, both volume fraction and APS are required, one has to rely on computer simulations to obtain all the parameters needed to make a prediction. Acknowledgements The authors express appreciation to Drs. D. Whitis, D. Mourer and D. Wei for invaluable discussions, and to Prof. J. Morral for helpful comments on the manuscript. The supports of GE Aviation (G.W., E.J.P., M.J.M. and Y.W.), the Ministry of Science and Technology of China under Grant No. 2006CB605104 and the Natural Science Foundation of China under Grant Nos. 50471079 and 50631030 (G.W., D.S.X. and R.Y.) are gratefully acknowledged. References [1] Haasen P. Phase transformation in materials. In: Cahn RW, Haasen P, Kramer EJ, editors. Materials science and technology: a comprehensive treatment, vol. 5. Weinheim: VCH; 1991. [2] Monajati H, Jahazi M, Bahrami R, Yue S. Mater Sci Eng A 2004;373:286. [3] Carslaw HS, Jaeger JC. Conduction of heat in solids. 2nd ed. Oxford: Oxford University Press; 1959. [4] Thomas G, Whelan MJ. Philos Mag 1961;6:1103. [5] Whelan MJ. Met Sci J 1969;3:95. [6] Aaron HB, Fainstein D, Kotler GR. J Appl Phys 1970;41:4404. [7] Aaron HB, Kotler GR. Metall Trans 1971;2:393. [8] Brown LC. J Appl Phys 1976;47:449. [9] Tanzilli RA, Heckel RW. Trans AIME 1968;242:2313. [10] Baty UL, Tanzilli RA, Heckel RW. Metall Trans A 1986;17A:1651. [11] Tundal UH, Ryum N. Metall Trans A 1992;23A:433. [12] Vermolen FJ, Vuik K, van der Zwaag S. Mater Sci Eng A 1998;246:93. [13] Vermolen FJ, Vuik K, van der Zwaag S. Mater Sci Eng A 1998;254:13. [14] Vermolen FJ. Ph.D. thesis, Delft Univ Tech; 1998. [15] Chen SP, Vossenberg MS, Vermolen FJ, van de Langkruis J, van der Zwaag S. Mater Sci Eng A 1999;272:250. [16] Durbin TL. Ph. D Thesis, Georgia Inst Tech; 2005. [17] Javierre E, Vuik C, Vermolen FJ, Segal A. J Comput Phys 2007;224:222. [18] Vermolen FJ, Javierre E, Vuik C, Zhao L, van der Zwaag S. Comput Mater Sci 2007;39:767. [19] Tiaden J, Nestler B, Diepers HJ, Steinbach I. Phys D 1998;115:73. [20] Kim SG, Kim WT, Suzuki T. Phys Rev E 1999;60:7186. [21] Beckermann C, Diepers HJ, Steinbach I, Karma A, Tong X. J Comput Phys 1999;154:468. [22] Chen LQ, Wang Y. JOM 1996;48:13. [23] Wang Y, Chen LQ. Simulation of microstructural evolution using the phase field method. In: Methods in materials research, a current protocols. New York: John Wiley; 2000.

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