numerical simulation of grain growth in liquid phase sintered materials—ii. study of isotropic...

Pergamon Acra mafer. Vol. 46, No. 4, pp. 1343-1356, 1998

0 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved

Printed in Great Britain PII: S1359-6454(97)00268-l 1359-6454/98 $19.00 + 0.00

NUMERICAL SIMULATION OF GRAIN GROWTH IN LIQUID PHASE SINTERED MATERIALS-II. STUDY OF

ISOTROPIC GRAIN GROWTH

V. TIKARE’ and J. D. CAWLEY’ ‘Sandia National Laboratories, Albuquerque, NM 87185-1405, U.S.A. and *Case Western Reserve

University, Cleveland, OH 44106, U.S.A.

(Received 9 January 1996; accepted 11 July 1997)

Abstract-The Potts model was used to study grain growth in liquid phase sintered materials. Its appli- cation to the study of isotropic grain growth by Ostwald ripening in a fully wetting system will be presented. The interpretation of the simulation results will be described and discussed. It was found that the set of simulation parameters used gave diffusion-controlled grain growth for solid fraction ranging from 0.30 to 0.90. The grain size distribution varied with solid fraction, becoming broader and more peaked with increasing solid fraction. The skewness was near zero at solid fraction of 0.41 and shifted to larger grain sizes with increasing solid fraction. This shift in the skewness of grain size distribution is not predicted by previous analytical or numerical models; however, it is consistent with experimental data collected by Fang and Patterson [Acta metall., 1993, 41, 20171 in the W-Ni-Fe system. 0 1998 Acta Metallurgica Inc.

1. INTRODUCTION

Liquid phase sintering is generally described as occurring in three stages, rearrangement, solution- reprecipitation and microstructural coarsening. Rearrangement leads to very rapid densification in the early stages and contributes little in the later stages of sintering; thus, it is typically thought of as occurring in the first stage. The second mechanism, solution-reprecipitation, occurs throughout the sintering cycle, but is generally considered as occurring in the second stage of sintering as it dominates densification after rearrangement has ceased. What is often termed the third stage of liquid phase sintering, microstructural coarsening, occurs predomi- nantly after densification is complete. In fully wetting systems, grain growth occurs by the solution-reprecipitation mechanism and is therefore appropriately considered in the context of Ostwald ripening.

The third stage of sintering, grain growth by Ostwald ripening, is the focus of this work. Many have modeled grain growth by Ostwald ripening using analytical and numerical techniques. Greenwood [l] was the first to consider coarsening of grains dispersed in a matrix and was able to predict power law behavior for grain growth:

R” - R;f = Kt (1)

where t is time, R is average grain radius at time t, R,-, is average grain radius at time t = 0, K is a constant and the grain growth exponent, n = 3. Lifshitz and Slyozov [2] and Wagner [3] indepen-

dently modeled Ostwald ripening and were able to predict the power law given in equation (1) and a steady state grain size distribution. Their combined work is known as the LSW theory. Wagner, in the same work, distinguished between two rate controlling processes, diffusion controlled and interfacial- reaction controlled grain growth. He showed that the two rate controlling processes gave different grain growth exponents (n = 2 for interfacial-reaction control and n = 3 for diffusion control) and different grain size distributions. The LSW theory made large strides in addressing the essence of Ostwald ripening, but failed in predicting the details of grain growth because of approximations made to render the problem mathematically tractable. The most significant assumptions were that the distance between grains was much larger than the grain size (or equivalently, the volume fraction of grains was virtually zero) and the concentration in the matrix is a mean concentration (mean field theory).

A few investigators modified LSW theory to include finite volume fractions of grains by introdu- cing a sphere of influence around grains [46] and chemical rate of growth for grains of different sizes [7]. Both these approaches make approximations about grain geometries, such as spherical grains, with all grains or grains of the same size experiencing the same environment. Furthermore, they are mean field theories. Voorhees and Glicksman [8] and later Yao et al. [9] used numerical techniques to solve the multi-particle diffusion problem. This technique did not assume a mean field concentration, but the diffusion equation they

1343

1344 TIKARE and CAWLEY: GRAIN GROWTH II. ISOTROPIC

solved assumes that the distance between particles was large compared to the particle dimension. Marqusee and Ross [lo] also incorporated finite grain fractions into the LSW theory using numerical techniques. They limited their investigation to the case of grain fraction < 0.10, since their model is most accurate in this regime. Most technologi- cally important systems however, have higher solid fractions. DeHoff [ll] has modeled the Ostwald ripening using an original method. He considers the evolution of the aggregate curvature of the two- phase system with each grain communicating with each of its neighbors. However, conversion of this aggregate curvature to stereological quantifiable parameters such as average grain size and grain size distributions has not been done. Fan and Chen [12] have recently proposed a model to study Ostwald ripening by finite difference solution of the Ginzburg-Landau equation. This model is still in its development stage and few kinetic and topologi- cal results have been obtained.

All these investigators have found that grain size, R, scales with time as R3mzt for diffusion-controlled grain growth. All have reported a grain size distribution that is skewed to smaller grain sizes (negative skewness) at very low solid fraction. As the solid fraction increases the grain size broadens and the skewness increases (becomes less negative), but never changes to a positive skewness at any solid fraction.

In this investigation, the Potts model described in the previous paper [ 131 is used to simulate grain growth by Ostwald ripening. This simulation technique is able to eliminate many of the approximations made by previous models. It is used to study two-dimensional grain growth in a fully wetting, isotropic system.

2. SIMULATION PROCEDURE

Grain growth was simulated using the Potts model similar to the one used by Srolovitz et al. [14] and described in the previous paper [13]. The microstructure was bitmapped on a two-dimensional, square lattice with periodic boundary conditions in the X- and Y-directions. The two-phase system was simulated by populating the lattice with two-component, canonical ensemble. The A-component could assume 1 of Q (= 100) degenerate states and B could only assume 1 state. The Hamiltonian for the simulations was defined as the sum of interaction energies for each site, i, with each of its eight first and second nearest neighbors, j, as:

H = $ F E (qi$ qj) j=1 i=l

where E(qi, qj) is the neighbor interaction energies for site i and j. A-sites and B-sites were allowed to exchange places using equation (2) as the equation

of state via the classical Metropolis algorithm [1.5]. The Metropolis algorithm uses Boltzmann statistics to determine the exchange probability, w as follows

for AH > 0

and (3) w =l for AH<0

where kB is the Boltzmann constant, and T is absol- ute temperature and has units of bond energy divided by the Boltzmann constant. In this paper, all temperatures will be given as kB T with units of energy rather than just temperature. The number of iterations is given in units of Monte Carlo step, MCS. At 1 MCS, the number of attempted exchanges was equal to the total number of sites in the simulation space. It has been shown that MCS scales linearly with time by some constant, r, for such simulations [16, 171; thus, MCS may be considered a measure of time in arbitrary units for the purposes of this simulation. Two grain growth mechanisms, coalescence and direct-exchange, were explicitly prevented in these grain growth simulations because they are artifacts of the simulation technique. A complete description and discussion of these mechanisms along with a step-by-step description of the simulation technique is given in the previous paper.

All grain growth simulations were run under fully wetting (EAB= 1.0, EAiAjc2.5, EaiAj=O, EBB=O, or equivalently, yss = (2.5~~1) and the total number of orientations, Q = 100. The composition ranged from mole fraction of component A XA= 0.30 to 0.90. The temperature ranged from kB T = 0 to kB TC, the liquidus temperature. The simulation lattice size used was 300 x 300 sites.

3. RESULTS OF GRAIN GROWTH SIMULATIONS

3.1. Kinetics

The grain growth simulation run at composition XA= 0.70 will be used to present detailed results. The microstructures generated for the composition, XA= 0.70 at temperature kB T = 1.3, as the simulation progressed, are shown in Fig. 1. The microstructures at 100,000 MCS and increasing temperature, kB T = 1.2, 1.3, 1.4 and 1.5 are shown in Fig. 2. These figures illustrate that grain growth does occur and growth rates increase with increasing temperature.

Data obtained from such simulations are plotted as the arithmetic mean of grain radii vs MCS on logarithmic scales for temperatures kB T = 0.9 and 1.3 in Fig. 3(a). Two features are immediately obvious from this figure. First, two distinct regions are apparent; the early stage of the simulation was characterized by a rapid, non-linear increase in grain size followed by a second stage characterized

TIKARE and CAWLEY: GRAIN GROWTH II. ISOTROPIC 1345

Fig. I(a and b)


Fig. ka T

1. =

Simulated microstructures illustrating grain growth by Ostwald ripening at temper 1.3, composition XA =0.70 and (a) 3978; (b)ll,O88; (c) 35,532 and (d) 100,000 MCS. The continuous feature is the liquid matrix surrounding grains which are the grey features.

dark


Fig. 2(a and b).

TIKARE and CAWLEY: GRAIN GROWTH II. ISOTROPIC

Fig. 2. Simulated microstructures after 100,000 MCS for composition X*=0.70 at temper kB T = (a) 1.2, (b) 1.3, (c) 1.4 and (d) 1.5.

‘atwe


I = T=IJJ

10’

loo 3 ld 103 10' lo5

ia)

0.6

0.5

0.4

0.3

0.2

0.1

0.0 lo2 10" lo4 10"

(b) Tie, MCS

Fig. 3. (a) Grain growth curves for the simulation run at composition XA = 0.70 at temperature, kn T = 0.9 and 1.3. (b) The corresponding solution concentrations of the

liquid matrix during the simulations.

by a linear region which suggests a power law behavior. Second, the slope during the second stage increased with increasing temperature, that is, the grain growth exponent increased with increasing temperature.

The origin of the two regions is considered first. Examination of the microstructures shown in Figs 1 and 2 revealed that component A was dissolved in B to form the liquid solution of A-sites and B-sites throughout the simulations. A plot of liquid solution concentration as a function of logarithmic MCS for the same simulations shown in Fig. 3(a) is shown in Fig. 3(b) for temperatures kB T = 0.9 and 1.3. The solution concentration was highly supersa- turated at the start of the simulation, since the simulation lattice was populated by random assign- ment of q at each lattice site and virtually all A- sites were single A-sites, which were considered dissolved in the liquid. As the simulation progressed, the solution concentration decreased dramatically and grain size increased until the solution concentration reached a plateau in the second region. A comparison of Fig. 3(a) and (b) reveals that the onset of the plateau in Fig. 3(b) occurred towards the end of the transition from the first region to the second region seen in Fig. 3(a). These data indicate that nucleation of grains occurred at the beginning

of the transition region followed by rapid precipitation of the A component on these nuclei during the transition region. Grain growth by Ostwald ripening (growth of some grains at the expense of others) occurred in the second stage of the simulation where the solution concentration remained relatively constant. The inverse of the grain growth slope in the second stage of the simulation is the grain growth exponent.

The results of grain growth simulation are sum- marized in Fig. 4(a), a plot of grain growth exponents as a function of temperature for the various compositions. The grain growth exponent, FZ, decreased with increasing temperature up to temperature, ka T = 1.3 and remained constant at n = 3 at temperatures kB T 2 1.3. This temperature dependence of 12 at lower temperatures is attributed to the lack of sufficient thermal energy at the lower temperatures. At higher temperatures, grain growth occurred with the expected kinetic behavior for diffusion-controlled grain growth.

By varying the temperature and composition, grain growth can be studied over a range of solid fractions. The grain growth exponents given in Fig. 4(a) are plotted as a function of solid fraction in Fig. 4(b). Diffusion-controlled kinetics were obtained from 0.30 to 0.90 solid fraction.

Composition @ 90%A

0 &JXA

0 70ZA

t bO%A

x 50RA

0.0 0.5 1.0 1.5 2.0 (1)

Temperature, %T

c 5*o *

0.0 0.2 0.3 0.5 0.7 0.8 1.0 (bl

Solid Fraction

Fig. 4. (a) Grain growth exponents as a function of simulation temperature for various compositions. (b) Grain

growth exponent as a function of solid fraction.


lE+5 0 400 . - 9O%A A A - 0 8O%A 350 0

8 8E+4- ??7O%A E _ 0 6O%A 0 “. 300 d 6E+4-;2; . 9 250 0 O .

200 g 3

4E+4 - A . 0

-3

2 0 0 A

g 150 ?? A

c 0

0 o

g

$ 0

2E+4 - A : ; 8

: 100 . A

??50

0EO0°11,8,,1’111’tt1’,1,‘1,1’11’.

i ; Q

0 ~~~‘~~1’~~~1~~~‘~~~‘~~~‘~~1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

w Temperature, &T (b) Temperature, &T

Fig. 5. (a) The number of MCS at the end of precipitation as a function of temperature. (b) Mean grain size at the end of precipitation for various compositions.

Further examination of Fig. 3(a) and (b) revealed that the number of MCS at the end of precipitation, rp, and the mean grain size after precipitation, R,, increased with increasing temperature. The number of MCS and the average grain size after precipitation, rp and R,, are plotted as a function of temperature for various compositions in Fig. 5(a) and (b), respectively, where t,, was defined as the MCS at the onset of the plateau in Fig. 3(b). Both, rp and R, increased with increasing temperature.

The number of MCS to completion of precipitation, rr,, was also a function of composition. As component A decreased and B increased, from 20% B and up, rp increased. The supersaturation of the liquid solution decreased with increasing temperature and increasing component B. As the supersaturation decreased, the driving force for nucleation also decreased yielding longer nucleation times.

At very low mole fraction of B, around 10% B, in increased with decreasing B. This behavior is attributed to an artifact of the simulation technique used. Only neighboring sites occupied by different components are allowed to exchange places. When the mole fraction of B was very low, the probability of randomly choosing one B-site to exchange with a neighboring A-site is very low. Thus, evolution of the microstructure which occurred by A-sites exchanging places with B-sites was very slow.

At a given composition, as the liquidus temperature was approached, studying grain growth became increasingly computationally demanding because the MCS at the end of precipitation were long, rp > 100,000 MCS, and the mean grain sizes at the onset of grain growth were large. These factors bounded the upper temperature region for grain growth simulations. The lower temperature bound

of ks T> 1.2 was constant for all compositions, the upper temperature bound varied with composition.

At compositions of X, 0.40, meaningful grain growth simulation is not feasible because the upper temperature bound converges with the lower one. For example, rp at composition X*=0.30 and temperature kB T = 1.3 is greater than 100,000 MCS. At lower temperatures, kB T < 1.3, grain growth by Ostwald ripening did not occur. The temperature and composition range for practical simulation is given in Fig. 6. To obtain grain growth results at low solid fractions (#J < 0.30), the simulation must be run at high temperatures and/or at compositions of low XA. Under these conditions, the MCS at the end of precipitation and the grain size after precipitation is very large. Thus, the number of MCS and

0

1

I a, * I I I I I I I I I I I I.

20 40 60 80 100

Composition, B%

Fig. 6. The range of temperatures and compositions over which the Potts model simulations can be run.


the lattice size necessary to grain growth at low soli- difications must be much larger than that used in this investigation. Numerical computations on such a large scale are not yet practical on workstation- class computers.

3.2. Grain size distributions

The grain size distributions obtained from the simulations are presented in this section. A single simulation of size 300 x 300 did not generate a sufficient number of grains to plot smooth grain size distribution histograms; therefore, multiple simulations under the same conditions were run to generate large numbers of grains. Five simulation conditions were chosen for further study. They were compositions, XA = 0.90, 0.80, 0.70, 0.60 and 0.50 at temperature kn T = 1.3. At the composition, X*=0.90, four simulations were run; at X*=0.80,

six simulations; and at X*=0.70, 0.60 and 0.50, seven simulations for each composition. At 100,000 MCS, multiple simulations yielded a minimum of 788 grains for compositions, XA = 0.50 and a maxi- mum of 2780 grains at XA = 0.90.

The grain size distributions of all the composition studied were examined after 20,000 MCS. Nucleation in these simulations occurred between 9000 MCS for X*=0.90 and 0.50 to 4000 MCS for XA =O.SO [see Fig. 5(a)]; thus, all simulations were well into the grain growth region at 20,000 MCS. Dynamic scaling of the grain size distribution was achieved for each of the five compositions after 20,000 MCS.

The grain size distributions at 100,000 MCS for varying solid fraction are shown in Fig. 7(a)-(e). The grain size distribution varied with the solid fraction. It became broader and more peaked with

r

Grain Size, R/CR>

Fig. 7. Grain size distributions after 100,000 MCS for solid fraction 4 = (a) 0.89, (b) 0.78, (c) 0.66, (d) 0.53 and (e) 0.41.


increasing solid fraction. The peak of the grain size distributions remained at R just under R/(R) for all the different compositions. The grain size distribution was skewed to the larger sizes for solid fraction = 0.89. As the solid fraction decreased, the skewness shifted to the smaller sizes.

In order to quantitize these characteristics, the variance, skewness and kurtosis of the grain size distributions were calculated. Skewness and kurtosis were calculated as, ,~s/&‘~ and ,u4/&3, respectively, where px is the xth moment of the grain size distribution. Variance was normalized by the mean squared, pz/$, to compare variances of different distributions directly to each other. These quantities are plotted as a function of MCS for the different solid fractions in Fig. 8(a-c).

The variance increased slightly with increasing MSC and with decreasing solid content. The small change in variance can be explained by the truncation of small grains due to the digitized nature of the simulations. The small grains in a steady-state grain size distribution are produced by shrinking grains. In the simulations, there were two sources of the small grains. One was from larger grains which had shrunk. The second and predominant source of these small grains was the formation of subcritical nuclei which were forming and redissolving constantly. Consider the grain size distribution of the grain

0.12

0.11 8 B .LI 0.10

3 r

0.09

008 . LJ-d-l- 2E+4 4E+4 6E+4 8E+4 lE+S

(P) Time, MCS

0.1 I 1

-0.5 A

-0.6 I I I I I 1

2E+4 4E+4 6E+4 8E+4 lE+5

(c) Time, MCS

growth simulation at composition X*=0.70 and temperature kB T = 1.3. Some relevant grain size data are tabulated in Table 1 at 20,000 and 100,000 MCS.

The number of grains of area a = 2 after 100,000 MCS was 878 out a total of 2522, which is disproportionately high. Clearly, 878 grains had not shrunk to a size of a = 2; rather they had precipi- tated out of solution and would redissolve as the simulations continued. Likewise, 61 grains of size a = 3, was disproportionately high as were 1089 grains of size a = 2 and 90 grains of size u = 3 after 20,000 MCS. For this reason all grains of size a < 4 were not included in the grain size distribution analysis. This had the effect of truncating the grain size at R/(R) < 0.131 for the grain size distribution after 100,000 MCS and at R/ (R) < 0.232 after 20,000 MCS. Since, the truncation occurred at a higher relative size, R/(R) at 20,000 MCS than at 100,000 MCS, the variance was slightly decreased. To confirm this hypothesis, the grain size distribution at 100,000 MCS was truncated at R(R) < 0.232 which corresponds to size a = 13. The variance of this distribution decreased from 0.111 to 0.100 and the variance at 20,000 MCS was 0.094. Thus, the small change in variance as the simulation progresses is attributed to the truncation of the small grains.

_oo;f ,‘,A; ; ,a, p 1 . @I2E+4 4E+4 6E+4 8E+4 lE+5

Time, MCS

Solid fraction Q 0.89

. 0.70

0 0.60

I 0.03

n 0.41

Fig. 8. The (a) variance, (b) skewness and (c) kurtosis of the grain size distribution as a function of MCS at different solid fractions.


Table 1. Average grain size and the number of grains between sire a = 2 and 9 at 20,000 and 100,000 MCS for composition XA = 0.70 and temperature ks T = 1.3

20.000 MCS, (R) = 8.64 100,000 MCS. (R) = 15.28

Grain size. area Number of grains 2 1089 3 90 4 15 5 3 6 9 I 9 8 13 9 9 Total number of 4961

R W) 0.164 0.201 0.232 0.259 0.284 0.306 0.327 0.347

Number of grains 878

61 5 3 0 1 0 I

1583

RI(R) 0.093 0.113 0.131 0.146 0.160 0.173 0.185 0.196

The small change in variance with solid fraction was also due to truncation of small grains. The average grain size increased with decreasing solid fraction. At high solid fractions, when the average grain size was small, variance was depressed due to truncation. The variance of the grain size distribution did not change much with simulation time or with solid fraction.

The skewness remained constant as the simulation progressed, but increased with increasing solid fraction. The increase is skewness with increasing solid fraction was consistent with the grain size distribution shown in Fig. 7(a-e) and was due to the shifting of grain size distribution toward the right. The truncation of small grains increased skewness of grain size distributions with lower average grain size, (R).Consider the example in Table 1. The skewness of the grain size distribution at 100,000 MCS was 0.09 and the skewness of the same data truncated at size a < 13 was 0.14. This variation in skewness was small compared to the variation in skewness at different solid fractions. Thus, truncation had little effect on skewness.

The kurtosis also remained constant as the simulation progressed, but increased with increasing solid fraction indicating that the grain size distribution was more peaked at higher solid fractions. This confirms the trend seen in Fig. 7(aae). Truncation of small grains decreased the kurtosis slightly; however, like skewness the difference between kurtosis of different solid fraction was larger than the change in kurtosis due to truncation.

4. DISCUSSION

The grain growth exponent, n, was constant at n = 3 at temperatures kn T> 1.2. As temperature decreased below kB T = 1.2, the grain growth exponent increased until n = cc at kB T = 0. The increase in the grain growth exponent indicates that grain growth was increasingly inhibited with decreasing temperature. The process required for grain growth which was highly temperature depen- dent was dissolution. For solution to occur, an A- site had to detach itself from a grain. This process

was always accompanied by an increase in the inte- gral energy. Thus, sufficient thermal energy had to be present to overcome this energy barrier. At kB T = 0, no thermal energy was present in the system and solution could not occur; therefore, grain growth could not occur and n = co. At temperatures kB T = 1.3, there was enough thermal energy in the system to simulate grain growth by the solution precipitation method. This behavior is in con- trast with that of a real liquid phase sintered material. In a real system, if the grain growth mechanism were temperature independent, n = 3 at all temperatures, but grain growth constant in equation (1) K -+ 0 as T decreased to the solidus temperature.

The range of grain sizes obtained from the simulations increased with increasing solid fraction. This trend was in agreement with the predictions of existing grain growth theories. The broadening of the grain size distribution occurred because a few grains grew larger. The largest grain size increased from 1.9(R) at solid fraction 4 = 0.41 to 2.2(R) at 4 = 0.89. Although, this spread was larger than grain size distributions predicted by all grain growth theories except for Voorhees and Glicksman’s [8]; the overall spread in the grain size distribution was approximately in the range predicted by grain growth theories.

In the simulations, skewness of the grain size distribution at solid fraction 4 = 0.41 was almost zero and increased gradually to 0.3 at $ = 0.89. This is very different from the predictions of grain growth theories. At 4 -+ 0, the LSW theory predicts a grain size distribution with negative skewness, The theories which consider finite solid fractions, predict an increase in skewness with increasing solid fraction. but none predict a change from negative to positive skewness as observed in the simulations.

Another significant difference between the simulations and grain growth theories was that the grain size distributions obtained from simulations were more peaked (higher kurtosis) at higher solid fractions. All grain growth theories predict a flattening of the peak with increasing solid fraction.


The grain size distribution obtained from the simulation at solid fraction $J = 0.53 was compared to that predicted by Voorhees and Glicksman [8] in Fig. 9(a). While the spread in grain sizes from the simulations was in agreement with that predicted by Voorhees and Glicksman, the skewness was quite different. The simulation grain size distribution at solid fraction $J = 0.66 was compared to experimental data collected by Fang and Patterson [19] for the W-14Ni-6Fe system at solid fraction 4 = 0.60 in Fig. 9(b). The two grain size distributions are in excellent agreement. The range of grain sizes, the skewness of the distribution and the peak of the distribution all agree well. The grain size distribution at solid fraction 4 = 0.89 was compared to that obtained by Holm [18] in simulated single phase systems in Fig. 9(c). Holm used the Potts model developed by Srolovitz et al. [14]. Her simulation of a single phase system may be

considered for the purposes of the simulations used in this investigation to occur at the limit as solid fraction 4 + 1. The broadening in the grain size distribution was greater than that observed for a solid fraction of 0.89. The skewness was larger (skewed to larger grain sizes) than that observed at solid fraction of 0.89. Thus the trends observed in spread and skewness with increasing solid fraction continued as solid fraction 4 + 1.

The advantage of the simulation technique presented in this investigation over that of analytical models was the incorporation of spatial distribution of grains; grain shape accommodation and solute gradients between grains were inherent to the technique. Previous models and simulations of grain growth by Ostwald ripening did not consider the spatial arrangement of grains during the grain growth process. They assumed spherical grains, made no allowance for changes in size and shape of

1.2

SoUd

1.0 - 0.8

P 8 & E

0.4

(a) Grain Size, Iv<R, Grain Size, RI-CR>

0.0 0.5 LO 15 2.0 2.5 3.0 3.5

Grain Size, R/-c%

Fig. 9. Comparison of simulated grain size distributions to that (a) predicted by Voorhees and Glicksman, (b) experimentally determined by Fang and Patterson, and (c) simulated by Holm in the

single phase polycrystalline systems.


Fig. 10. The microstructure after 100,000 MCS for solid fraction 4 = (a) 0.89 and (b) 0.41


grains due to spatial arrangement and assumed a mean field solute concentration around grains. The exceptions who did consider spatial arrangement of grains were Voorhees and Glicksman [8] and Yao et al. [9]. They considered diffusion gradients around spherical grains, but only at relatively low grain fractions where interactions were minimal. DeHoff s communication model [l l] was able to consider both deviation in curvature of grains and interaction of gradients between grains, but corre- lating curvature parameters and interaction parameters to non-spherical grain shapes has not been done.

This ability of the simulation to incorporate spatial features may be the reason it was able to predict the change in skewness from left to right with increasing solid fraction. The microstructures at 100,000 MCS for solid fraction 4 = 0.89 and 0.41, shown in Fig. 10 (a and b), revealed that as solid fraction increased, considerable grain shape accommodation occurred. At solid fraction 4 = 0.89, many grains had flat edges and sharp corners in other regions. These features will modify both the local driving force for grain growth and the local solute gradients in the liquid matrix which in turn may influence the grain size distribution to give positive skewness at high solid fractions.

5. CONCLUSIONS

The Potts model simulation of grain growth by Ostwald ripening presented in this paper is able to eliminate many of the approximations and simplifi- cations made by previous models. The Potts model made no assumptions about the solution gradients between grains; rather it relied on the ability of the Monte Carlo technique to simulate random walk. Most of the previous models assumed grains were spherical at all solid fractions. The model presented here did not assume that grains were of any specific shape and allowed the simulation to actually evolve a microstructure. Thus, it was able to consider shape accommodation of grains as a function of solid fraction. This simulation is, however, limited to study grain growth at higher solid fractions, at solid fractions greater than 0.30. Lower solid fraction simulation can be run, but the computational time required is large.

Diffusion-controlled grain growth kinetics with power law behavior was obtained at solid fraction 4 > 0.30. Grain size distributions varied with solid fraction; it became broader and more peaked with increasing solid fraction. The grain size distribution had skewness of zero at solid fraction 4 = 0.41 and shifted increasingly to larger grain sizes with increasing solid fraction. The grain size distributions predicted by the Potts model are in better agreement with experimentally measured grain size distributions than previous Ostwald ripening models.

Acknowledgements-This work was performed at NASA’s Lewis Research Center under a cooperative grant with CWRU (NCC3-139) and completed by one of us (V.T.) at Sandia National Laboratories supported by the U.S. Department of Energy under contract number DE-AC04- 94LA85000.

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numerical simulation of grain growth in liquid phase sintered materials—ii. study of isotropic...

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