- Email: [email protected]
3D Monte-Carlo simulation of texture-controlled grain growth
Acta Materialia 51 (2003) 1019–1034 www.actamat-journals.com
3D Monte-Carlo simulation of texture-controlled grain growth O.M. Ivasishin a,∗, S.V. Shevchenko a, N.L. Vasiliev a, S.L. Semiatin b b
a Institute for Metal Physics, 36 Vernadsky Str., 03142 Kiev, Ukraine Air Force Research Laboratory, AFRL/ML, Wright-Patterson Air Force Base, OH 45433-7817 USA
Received 17 July 2002; received in revised form 14 October 2002; accepted 24 October 2002
Abstract A three-dimensional (3D) Monte-Carlo (MC) routine was developed to quantify the interaction of grain growth and texture development during annealing. The program included special software to enable the input of the initial grain structure and texture and incorporated a description of the misorientation-dependence of the grain-boundary mobility. Outputs from the model quantified the evolving texture in terms of pole figures or crystallite orientation distribution functions and statistics on the grain structure such as the grain-size distribution and boundary-misorientation distribution function. The MC routine was applied to establish grain growth and texture development in materials with random or strongly textured starting conditions and isotropic or anisotropic grain-boundary mobility. Depending on the starting condition and material properties, normal grain growth or a behavior characterized by alternating cycles of fast and slow grain growth was predicted. 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Grain growth; Texture; Potts model
1. Introduction The control of microstructure in metallic alloys and ceramics is key to improving their mechanical properties. The prediction of microstructural evolution due to recrystallization, grain growth, etc. starting from an arbitrary initial structure is important for materials design and processing. For example, grain growth in polycrystalline aggre-
Corresponding author. Tel.: +380-44-444-2210; fax: +38044-444-0120. E-mail address: [email protected] (O.M. Ivasishin). ∗
gates has been studied extensively, and many important physical phenomena have been modeled. In early models [1–4], the main parameters which characterize grain statistics and features of the network of grain boundaries were included only in an average way. Much more detailed models based on both analytical [5] and computer-simulation [6–8] approaches were reported later. Atkinson [9] reviewed the classical theories, models, and computer simulations. In most cases, however, such simulations have been restricted to two dimensions (2D). Recently, a number of studies [10–14] have been conducted to develop three-dimensional (3D) computer models of grain growth. However, none
1359-6454/03/$30.00 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6454(02)00505-0
1020
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
of these 3D grain-growth simulations took into account the evolution of the textural state at each step of the process. As was recently shown in a series of very careful experiments [15], grain growth and crystallographic texture evolution during the annealing of metals are closely related. Although a number of previous studies of grain growth [6,8,13,14,16–19] have utilized the Monte-Carlo technique, none have addressed the coupling of grain growth and texture evolution in a quantitative fashion. In addition, all of the previous models have made various assumptions to simplify the calculations. These include the use of a two-dimensional, rather than three-dimensional, approach [6,8,13,16–19], use of a limited number of possible grain orientations or the use of integers to represent different orientations rather than Euler angles [6,8,16,18,19], and the disregard or simplification of the dependence of grain-boundary energy and mobility on misorientation [14,16,20]. Even with these approximations, a number of interesting observations have been made. For example, Grest et al. [19] showed that the grain-growth exponent may vary from approximately 2 to 4, depending on the magnitude of the anisotropy of the grain boundary energy with misorientation. In subsequent work, Rollet et al. [16] suggested that grain growth can lead to a marked change in texture. In Ref. [17], Ono et al. concluded that grain-growth behavior as a whole depends on the competition between the overall reduction of system energy and the motion of individual boundaries as controlled by local configurations. The present work was undertaken to expand upon and clarify some of the issues raised in previous research on the Monte-Carlo modeling of grain growth. The objective of the present work, therefore, was to develop a Monte-Carlo (MC) modeling approach with which the interaction of grain growth and texture evolution could be quantitatively described for three-dimensional polycrystalline aggregates. For this purpose, a technique was developed to enable the determination of pole figures, orientation distribution functions (ODFs), sections of ODFs, the volume fractions of specific texture components, the boundary-misorientation distribution function (BMDF), and the grain-size
distribution function for the model volume (or any part of the model volume) at any stage of grain growth. To make the simulations physically realistic, large three-dimensional model volumes with a large number of possible orientations distributed uniformly throughout Euler space were used. The capability of incorporating the misorientation-dependence of grain-boundary energy and mobility was also built into the simulation routine.
2. Modeling approach The grain boundaries of polycrystalline solids form complex three-dimensional network structures. At elevated temperatures, grain-boundary motion occurs to reduce the total grain-boundary energy. This process results in a continuous change in the nature of the grain-boundary network [11]. The velocity of grain-boundary migration is the main factor that controls grain-growth kinetics. The local velocity of boundary migration depends on the grain-boundary energy and mobility and on the local grain-boundary curvature [21]. The velocity v of a grain-boundary segment characterized by energy g, mobility M, and radius R is given by [5]: v ⫽ γMR⫺1,
(1)
and the average migration rate of a boundary between two grains u and m is: ⫺1 vum ⫽ gumMum(R⫺1 u ⫺ Rm ),
(2)
in which (R ⫺R ) is the average curvature for this grain boundary. Small grains disappear as a consequence of boundary motion, and the average size of the remaining grains increases. Despite the fact that the grain-growth process is a geometrically complex phenomenon, a vast amount of experimental data has shown that a simple mathematical relation expresses the statistical behaviour of a large number of grains, i.e., ⫺1 u
⫺1 m
Dn⫺Dn0 ⫽ Kt exp(⫺Q / RgT)
(3)
in which Q is the activation energy, Do and D denote the initial and final average grain sizes, t is the annealing time, Rg is the gas constant, T is the absolute temperature, n is the grain growth
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
exponent, and K represents the rate constant. For so-called normal grain growth during which boundary motion is not impeded and the grain boundaries maintain a self-similar network structure, the square of the mean grain size increases linearly with time (i.e., grain-growth exponent n ⫽ 2) [21,22]. The main features of the present model for texture-controlled grain growth can be divided into those aspects related to the geometry of the problem and the actual kinetics of grain growth. 2.1. Geometric features of the model The geometry-related features of the model include descriptions of the model domain, including its grain structure and texture, and procedures for setting the initial microstructure and texture. 2.1.1. Model domain and grain structure In the present work, the model domain was formed by a three-dimensional cubic array of model units (MUs), each of which represented a point in a cubic lattice. In all calculations, the length of the side of the cubic lattice was set equal to 250 MU, thus yielding a domain volume of 2503 ⫽ 15,625,000 MU3. This volume is substantially greater than that used in Refs. [14] (2,000,000) and [19] (~5,000,000) and thus allowed the introduction of a larger number of grains in the initial microstructure. By this means, textures can be described more precisely using an orientation distribution function (ODF), and the interaction of texture evolution and grain growth can be studied to longer times. As in previous work utilizing the MC approach [6,8,16,18], the material domain did not contain grain boundaries per se. Rather, the grain boundary position was associated with the space between two sites having unlike orientations. Each grain was characterized by a volume equal to the number of MUs within the grain. However, the actual grain size was taken to be equal to the diameter of a sphere containing the same volume. The smallest initial grain was assumed to comprise ~35–40 MUs. With this assumption, small clusters (with volumes below 10 MU3) that arise during simulations were not taken into account in calculating the
1021
microstructure statistics because the local grainboundary curvature has no physical meaning in such cases. Coalescence of grains with the same orientation was not allowed in the simulations. 2.1.2. Crystallographic orientations and misorientations Crystallographic orientations were specified in three-dimensional Euler space by the rotation angles j (⫺90 ⬍ ϕ ⬍ 90), q (0 ⬍ θ ⬍ 180) and c (0 ⬍ χ ⬍ 180). These three values characterized the orientation of the grain-coordinate system (KG unit vectors) relative to the fixed reference directions of the model domain (KMD unit vectors parallel to the RD, TD, and ND) (Fig. 1a). Orientation space was divided into segments of 2 × 2 × 2 degrees, giving rise to 729,000 possible orientations. Thus, each MU was assigned a number L (0 ⬍ L ⬍ 729,000) corresponding to its specific interval in Euler space, gi ⫹ dgi. The orientation distribution function f(g) (ODF) characterizing the textural state of material was then expressed as: f(g) ⫽
冘
→
→
f(R,g); f(R,g ⫽ gi ⫹ dgi) ⫽ 1; MD
(4)
→
f(R,g ⫽ gi ⫹ dgi) ⫽ 0, in which MD denotes the model domain, and R is the spatial location in Cartesian coordinates. A special procedure was developed to create the initial texture state in the model domain and to replicate the specified ODF as close as possible. In this regard, special attention was paid to ensure that grains with orientations belonging to different texture components were distributed randomly, i.e., there were no local textures in the model domain that differed greatly from the specified ODF. The misorientation angle between two neighboring grains with orientations g1(j1, q1, c1) and g2(j2, q2, c2) was determined as “orientation distance” in Euler space (Fig. 1b). The explicit expressions for the determination of the orientation distance ε are: ε ⫽ arccos((A⫺1) / 2);
(5)
A ⫽ cos(ϕ1⫺ϕ2)cos(χ1⫺χ2)[1 ⫹ cosθ1cosθ2]⫺sin(ϕ1⫺ϕ2)sin(χ1
(6)
1022
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
then be taken into account through the corresponding symmetry of the grain-boundary mobility dependence on misorientation. With this definition of ε, the BMDF is defined as P(ε,dε), or the fraction of boundaries with misorientation angles between ε and ε ⫹ dε [24]. The BMDF describes the probability of finding a boundary with a given misorientation angle. In model calculations, the value of dε was chosen to be 2°. Thus, for ε ⫽ 0ⴰ, for example, P(ε,dε) was the probability of finding a misorientation greater than 0° and less than 2°. 2.2. Kinetic features of the model The kinetic features of the model comprised descriptions of the grain-boundary energy/mobility and the MC procedure for describing grain growth, incorporating appropriate time and temperature variables.
Fig. 1. Description of crystallographic orientations and misorientations between grains: (a) Euler angles and (b) orientation distance ε.
⫺χ2)[cosθ1 ⫹ cosθ2] ⫹ [cos(ϕ1⫺ϕ2) ⫹ cos(χ1⫺χ2)]sinθ1sinθ2 ⫹ cosθ1cosθ2 Eqs. (5) and (6) are equivalent to the commonly used algorithm [14,23] without the selection of minimum angle allowed by crystal lattice symmetry, however. By using the full spectrum of ε values (0ⱕεⱕp), the boundary misorientation spectra of complex textures can be conveniently described. The crystal symmetry of material can
2.2.1. Grain boundary energy/mobility The local velocity of a grain boundary is often assumed to be proportional to the grain boundary energy and mobility, i.e., Eq. (2). In most cases, however, it is not possible to separate the influence of these two factors on grain growth [25,26]. For this reason, the dependence of grain growth on mobility and boundary energy is often treated by introducing the potential V(gi, gj), in which gi denotes the discrete, local crystal orientations available in the modeling space. Hence, the effective mobility of the boundary between grains having orientations i and j and misorientation e is given by: Mij ⫽ M0V(gi,gj) ⫽ M0V(ε).
(7)
It is assumed in the present model that M 0 ⫽ M max ⫽ 1, and V(gi, gj) 苸[0,1]. Hence, the present model specifies V(gi, gj) as a function of ε. The existing experimental data for the dependence of grain-boundary energy and mobility on local lattice misorientation [27–30] are contradictory in many cases. From a qualitative perspective, however, all of the measurements are similar to those shown in Fig. 2a. The number of maxima in the misorientation interval [0, p] depends on the
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
1023
2.2.2. MC routine To simulate the kinetics of boundary motion with the MC technique, a MU was selected at random, and a trial orientation was also chosen at random from one of the orientations of its neighborhood. This is an essential feature used to optimize MC routines such as that proposed by Anderson et al. [6]. Trials with all possible orientations in the modeling domain were impossible in the present formulation because of the very large L value and thus the excessive computer time that would have been required. Hence, there were two different possible states of the modeling domain after the MC trial—the same as the previous one and the state in which the MU has changed its orientation to the chosen one. The next step of the MC procedure consisted of the comparison of the free energies of these two states. The model used is the simple Hamiltonian: H ⫽ ⫺J(dLiLj⫺1),
Fig. 2. Misorientation dependence of the grain-boundary mobility factor: (a) literature values for fcc metals and (b) assumed values for MC simulations (Eq. (8)) corresponding to 1: S→⬁: S ⫽ 10250, 3: S ⫽ 4000, and 4: S ⫽ 2250.
type of crystal structure. In the general case without restrictions due to a specific crystal symmetry, a phenomenological dependence which describes the function V(gi, gj) with m maxima at misorientations ⑀ ⫽ Gi, i ⫽ 1, 2, …, m is given by the following relations: V(gi,gj) ⫽ V(ε) ⫽ 1⫺(Gi⫺ε)2 / S;e苸[0;Gi ⫹ 0.5(Gi+1⫺Gi)]
i ⫽ 1;
V(gi,gj) ⫽ V(ε) ⫽ 1⫺(Gi⫺ε)2 / S;苸[Gi⫺0.5(Gi⫺Gi⫺1);Gi ⫹ 0.5(Gi+1⫺Gi)]
i ⫽ 2,3,…,m⫺1;
V(gi,gj) ⫽ V(ε) ⫽ 1⫺(Gi⫺ε)2 / S;e苸[Gi⫺0.5(Gi⫺Gi⫺1);p]
i ⫽ m.
(8) For cubic crystals (Fig. 2b), G 1 ⫽ 45° and G 2 ⫽ 135°; for hexagonal close-packed metals, G 1 ⫽ 30°, G 2 ⫽ 90°, and G 3 ⫽ 150°.
(9)
in which Li is the orientation of site i, Lj is the orientation of the neighboring site j, and δhl is the Kronecker delta. The sum is taken over all neighboring sites within the specified ‘control volume’ around the MU that is inspected; this volume is characterized by a search radius SR. The shape of the control volume was assumed to be cubic, centered around the selected MU, and with a side length of 2SR ⫹ 1 in model units (Fig. 3). Thus, nearest neighbor pairs contribute J to the system energy when they are of unlike orientation and zero otherwise. The transition probability, W, is then given by: W⫽
再
Mijexp(⫺⌬G / kbT); ⌬G ⬎ 0,
Mij
⌬Gⱕ0,
(10)
in which ⌬G is the change in energy caused by the change of orientation, Mij is the grain boundary mobility, and kb is Boltzmann’s constant. Successful transitions at a grain boundary to the orientation of a nearest-neighbor grain correspond to boundary migration. The MC routine can be modified readily to include the discrete dependence of grain boundary energy and grain-boundary mobility on misorientation by incorporating a boundary-energy factor into the calculation of H and ⌬G in Eqs. (9) and (10) [31].
1024
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
faster). Furthermore, a choice of SR ⬎ 2 (Fig. 3c) would have led to very extensive calculations and would have undesirably increased the probability of incorporating the influence of distant grains and grains boundaries. As in past MC investigations, the unit of time in the simulations was the Monte-Carlo “step” (MCS). During one MCS, the number of simulation procedures done in accordance with Eqs. (9) and (10) was equal to the number of sites in the modeling domain, i.e., 2503. 2.2.3. Temperature- and time-scale factors As can be deduced from Eqs. (1) and (2), the real time scale of the model can be expressed as: t ⫽ B(T)·l·tmcs,
Fig. 3. Definition of the search radius, SR, for MC simulations of grain growth: (a) SR ⫽ 1, (b) SR ⫽ 2, and (c) SR ⫽ 3.
In all calculations, the sampling neighborhood, or control volume, was characterized by SR ⫽ 2. Thus, 124 nearest MUs were taken into account in the cubic three-dimensional array for the calculation of Eq. (9) (Fig. 3b). This is the best value for SR. Choosing the SR ⫽ 1 (Fig. 3a) would have led to a very rough capture of the local grainboundary curvature (but made the calculations
(11)
in which t is time, tmcs is modeling time (i.e., number of MCS), l is the length of the MU, and B is the time-scale factor. The phenomenological expression (11) can be used to obtain the value of the time-scale factor by a comparison of experimental measurements of the initial grain size and the rate of normal grain growth during isothermal annealing for a non-textured material with the calculated grain-growth rate in MC simulations. Another way to estimate the magnitude of the time-scale factor would be to measure the velocity of an isolated grain boundary (with constant curvature) at temperature T0 in a real material and then compare this velocity with the kinetics of a comparable grain boundary in the modeling domain of a simulation. For instance, careful experiments with such boundaries have been performed for pure aluminum by Winning et al. [27]. The velocity of grain-boundary migration is closely related to the rate of diffusion at the boundary. This results in an Arrhenius dependence of grain-boundary velocity on temperature. Hence, temperature can be incorporated into the MC model by considering a time-scale factor which is a function of absolute temperature, i.e., B(T) ⫽ B0(T0)(V0(T0) / V(T)),
(12a)
thus leading to the following relation: t ⫽ B0(T0)exp[⫺(Qa / R)(1 / T0⫺1 / T)]· l·tmcs,
(12b)
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
1025
where Qa denotes the activation energy for grainboundary migration. The calculations described below were done for T→0, t(1 MCS)→⬁. Thus, it was not necessary to account for the real time-temperature scale. Nevertheless, because all calculations were done with a fixed modeling domain and the same MC routine, the comparison of kinetics for different cases was possible in terms of MCS as the relative measure of time and the MU as the relative measure of length.
3. Results and discussion A series of simulations was run to assess the validity of the three-dimensional MC routine in Fig. 5. MC predictions of the shape (in cross section) of an isolated spherical grain (case 1): (1) initial grain boundary, and the grain boundary after: (2) 200 MCS, (3) 500 MCS, (4) 1000 MCS, (5) 1500 MCS, and (6) 1900 MCS.
quantifying the interaction of grain growth and texture evolution. These simulations described the shrinkage of an isolated grain in a homogeneous matrix and the effects of starting texture and anisotropy of the grain-boundary mobility on grain growth. The latter results are reported in terms of grain growth kinetics, grain size distributions, BMDFs, and (100) pole figures. 3.1. Case 1: shrinkage of an isolated grain
Fig. 4. MC predictions for the shrinkage of an isolated spherical grain (case 1): (a) kinetics of shrinkage and (b) grain-boundary velocity.
As an initial validation of the MC routine, the shrinkage of an isolated spherical grain in a homogeneous matrix was modeled. The objectives of this trial case were to ensure that the use of the Hamiltonian in Eq. (9) could replicate the boundary migration rate described by Eq. (2) and to determine the extent of the statistical “noise” associated with the migration-rate calculations. In addition, the case was used to check the quality of the random-number generator used in the MC routine and to verify that the expected isotropic shrinkage could be modeled. The grain had a starting diameter of 200 MU
1026
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
Fig. 6. MC predictions of texture evolution for case 2: (a) initial texture, and textures after (b) 100 MCS and (c) 1000 MCS; and for case 3: (d) initial texture, and textures after (e) 100 MCS and (f) 1000 MCS. Iso-intensity lines correspond to 1, 2, 5, and ⬎10 times random.
and was placed in a model domain that was 250 × 250 × 250 MU in extent. The matrix had a crystallographic orientation different from that of the grain. The mobility factor of the grain boundary was assumed to be 1 (i.e., the maximum value).
Fig. 7. Predicted grain-growth kinetics for case 2 (solid line) and case 3 (broken line).
Simulation results for case 1 are shown in Figs. 4 and 5. A plot of the kinetics of grain shrinkage (Fig. 4a) indicated that the initial grain disappeared completely after 1955 MCS. In addition, within a high degree of accuracy, the boundary migration rate was proportional to the boundary curvature (=inverse diameter) (Fig. 4b), in agreement with Eq. (2). The statistical fluctuation in the migration rate at a given curvature was approximately 10%. The occurrence of such statistical noise is a result of the random, mutually independent selection of MU elements during each discrete MC trial. The statistical nature of the calculations is seen also in cross-sections through the grain (Fig. 5). By and large, the predicted grain shape remained spherical during the entire simulation. However, a small amount of distortion developed due to the discrete character of the model. For example, at the end of the simulation (1900 MCS), the centroid of the grain was located slightly off-center, i.e., at X ⫽ 0, Y ⫽ 1, Z ⫽ 1 MU. Thus, the deviation from
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
1027
Table 1 Volume fractions of grains with orientations within region A (VA) and region B (VB) in Fig. 12a Time, MCS
VA
VB
VA+VB
0 100 350 650
0.052 0.080 0.095 0.123
0.052 0.070 0.074 0.087
0.104 0.150 0.169 0.210
Fig. 8. MCS.
Predicted grain-size distribution for case 2 after 1000
Fig. 9. Predicted grain-boundary misorientation distribution functions for: (1) case 2 and (2) case 3. For each case, the three lines correspond to 0, 100, and 1000 MCS.
Fig. 10. MC predictions for case 4: (a) kinetics showing the retardation of grain growth due to texture formation (broken lines are the best fit for the time intervals 0–600 MCS and 900– 1500 MCS) and (b) BMDFs: (1) initial condition, (2) 10 MCS, (3) 100 MCS, and (4) 650 MCS
1028
Fig. 11.
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
Predicted grain-size distributions for case 4: (a) Initial condition, and after (b) 10 MCS, (c) 100 MCS, and (d) 650 MCS.
uniform shrinkage was equal to or less than 0.8% (=1/125) throughout the modeling domain. 3.2. Grain growth in untextured (case 2) and textured (case 3) material with misorientationindependent grain-boundary mobility Two MC simulations were conducted for graingrowth conditions in which the grain-boundary mobility was assumed to be independent of misorientation. The initial microstructure was assumed to have an average grain size of 5.43 MU and a log-normal grain-size distribution. For case 2, the starting material was assumed to be untextured. For case 3, the texture comprised two strong components, A and B, that were 60° apart and each of which were Gaussian in Euler space having a half-width of 4° in all three angular dimensions. The volume fractions of the two components were 0.10 and 0.90, respectively. The (100) pole figures for the initial textures in these two cases are shown in Fig. 6a and d, respectively. For case 3, only the
(100) poles (and not the (010) and (001) poles) are shown on the pole figures for clarity. This is sufficient in view of the method of representing orientation distance and of calculating misorientations and the associated boundary energies and mobilities. As expected, the grain-growth kinetics for the two different cases were found to be the same. From plots of the average grain size versus time (Fig. 7), the grain-growth exponents (i.e., value of n in Eq. (3)) were found to be 2.02 for case 2 and 2.00 for case 3. Because of the statistical nature of MC simulations, one can conclude that the exponents in both cases were both essentially equal to 2.0, the value that characterizes classical normal grain-growth kinetics. MC predictions of the texture (in terms of (100) pole figures) for cases 2 and 3 are summarized in Fig. 6. For case 2, the texture remained essentially isotropic (Fig. 6 b and c). Furthermore, the grainsize distribution for this case (Fig. 8) was close to log-normal. For case 3, the texture remained essen-
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
1029
Fig. 12. Predicted texture evolution for case 4 after: (a) 100 MCS, (b) 350 MCS, (c) 650 MCS, and (d) 1500 MCS. Iso-intensity lines correspond to 1, 2, 5, 10, and ⬎30 times random.
tially unchanged as well (Fig. 6e and f), and the grain-size distribution was also nearly log-normal at the end of the simulation. The boundary-misorientation distribution functions (BMDF) for cases 2 and 3 are shown in Fig. 9. There were significant differences in the BMDFs for the random and highly textured starting conditions for each time during the simulations. The BMDF of the randomly textured polycrystal followed the so-called Mackenzie distribution [13,24]. The BMDF of the highly-textured condition had two maxima in positions close to 0 and 60°. However, there was no evidence of any significant changes in the BMDFs during grain growth for either starting texture. Deviations from the initial BMDFs were statistical in nature and increased when the total number of grains within the modeling domain decreased. 3.3. Case 4: grain growth in initially randomtextured material with misorientation-dependent grain-boundary mobility Grain-growth kinetics and texture evolution in a material that had an initial random texture and a
misorientation-dependent grain-boundary mobility (case 4) were quantified using the MC approach. The mobility-factor dependence followed Eq. (8) with S ⫽ 2250 (Fig. 2b). At short annealing times, the predicted graingrowth exponent was equal to 1.99 (Fig. 10a). After ~900 MCS, however, the grain growth exponent was predicted to increase to 2.58. In addition, the corresponding grain-size distribution gradually deviated from log-normal behavior as grain-growth occurred (Fig. 11). After 100 MCS (Fig. 11c), the fraction of small grains had increased. This trend can be explained by the fact that small grains disappear slowly when they are surrounded by large grains of the same texture component. At yet longer times (Fig. 11d), several local maxima, instead of a single maximum, had developed in the grain-size distribution. The evolution of the grain-size distribution can also be rationalized in terms of the gradual transformation of the BMDF in which some evidence of texture formation was also seen (Fig. 10b). The hypothesis that a noticeable texture had formed in this case was confirmed by the corresponding pole figures
1030
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
Fig. 14. Comparison of the predicted grain-growth kinetics for case 5 (solid line) and normal grain-growth kinetics (broken line).
Fig. 13. BMDFs for the circled areas in Fig. 12a for case 4: (a) only grains with orientation A and (b) grains with orientations A and B, multiplied by corresponding volume fractions (Fig. 12, Table 1). (1) Initial condition, (2) after 100 MCS, (3) after 350 MCS, (4) after 650 MCS.
generated by the MC routine. There was an indication of the formation of a cube-on-corner texture in the (100) pole figures (Fig. 12). The results in Fig. 10b also suggested that microstructure evolution during the first 10 MCS led to a corresponding increase in the number of low-mobility boundaries with misorientations close to 0, 90, and 180°. The subsequent evolution of the BMDF was more complicated, however. This interpretation required the analysis of the evolution of separate texture components (i.e., volume fractions, grain-size distributions, misorientation function). For this purpose, a special procedure was incorporated into the model. Using this procedure, it was found that texture components labeled A and B in Fig. 12a gradually developed from
the initial random texture (Fig. 13a). This led to a transformation of the “local” BMDFs for specific texture components (e.g., Fig. 13a for component A) and an increase in the volume fraction of grains of the two specific orientations (Table 1). Taken together, these effects caused a significant increase in the density of boundaries at certain angles (Fig. 13b). The maximum at small angles, which is also evident in Fig. 10b, comes from misorientations within each of the A and B components (i.e., AA and BB boundaries), while the maximum associated with the high angular misorientation has resulted from boundaries of the AB type. 3.4. Case 5: grain growth and texture evolution in material with an initial two-component texture and misorientation-dependent grain-boundary mobility The final MC simulation (case 5) focussed on grain growth in a material whose initial texture consisted of two components, i.e., the same as the initial texture for case 3. As in case 4, the mobilityfactor dependence for this case followed Eq. (8) with S ⫽ 2250 (Fig. 2b). The predicted kinetics of grain growth for case 5 are shown in Fig. 14. Periods of fast and slow grain growth were apparent. There were also pronounced variations in texture as grain growth occurred as seen in the (100) pole figures in Fig.
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
1031
Fig. 15. Predicted texture evolution for case 5: (a) initial condition and after (b) 15 MCS, (c) 100 MCS, (d) 600 MCS, (e) 1200 MCS, and (f) 2000 MCS. Iso-intensity lines correspond to 1, 2, 5, 10, and ⬎30 times random.
15. (As for case 3, only the (100) poles (and not the (010) and (001) poles) are shown on the pole figures for case 5 for clarity.) The texture changes were characterized by periodic interchanges of the volume fractions of the two components. Furthermore, the periods of rapid grain growth in Fig. 14 were found to be associated with the times at which there was a large fraction of high-angle boundaries (e.g., at 15 MCS and 1200 MCS, as shown in Fig. 16). These are the times at which the volume fractions of the two texture components were approximately equal. The MC-simulated texture evolution can be explained as follows. Because of their lower volume fraction, the A grains initially had a much higher probability of being surrounded by B grains rather than by other A grains and thus of having high mobility A–B boundaries. Hence, small A grains were rapidly consumed by large B grains, but large A grains were able to grow rapidly. Because the B grains were in contact preferably with other B grains, they had mainly low mobility B–B boundaries; therefore, most of the B grains participated only slightly in the initial stages of
grain growth. This led to a very different grainsize distribution for the modeling domain as a whole (Fig. 17a) compared to that for those grains with the A texture component (Fig. 17b). After the A grains had consumed a majority of the volume, the growth rate of the A grains decreased, because there was now a high probability that each was surrounded by other A grains. As B had become the minority component, the growth rate of B grains exceeded that of the A grains. Over the long times, such phenomena produced cyclic changes in the relative volume fractions of the A and B texture components. Such periodic evolution of a two-component texture has been previously predicted analytically (Fig. 18a) [5] and confirmed experimentally in very recent work on the beta annealing of a titanium alloy [15]. The results of the present MC model (Fig. 18b) are in good agreement with the previous theoretical calculations as well as with the experimental findings. A similar behavior was also deduced in Ref. [32] in which the evolution of a two-component texture was modelled for the case comprising a strong (axial) component and a weak
1032
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
Fig. 17. Predicted grain-size distributions for case 5 after 15 MCS: (a) for the entire material and (b) for the grains belonging to texture component A. Fig. 16. Predicted variation of the BMDF for case 5 over time intervals of (a) 0–100 MCS and (b) 600–2000 MCS.
4. Summary and conclusions
consist of any combination of initial microstructure, texture, and grain-boundary mobility. Furthermore, the approach provides the ability to describe and control all model parameters at each stage of grain growth. Several simulations were performed to confirm the validity of the approach and to analyze some specific cases of grain growth. The following conclusions were drawn from this work:
A new modeling approach, which includes an optimised MC simulation routine and procedures for the control of the modeling domain microstructure and texture, and its associated computer program were developed to establish the interaction of grain growth and texture evolution. The input can
1. When the grain-boundary mobility is independent of misorientation, normal grain growth with a grain-growth exponent of 2.0 occurs irrespective of whether the starting material has an initial random or sharp texture. In these cases, the grain-size distribution remains log-
component consisting of a background distribution of random orientations.
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
1033
growth can be correlated to the cyclic changes in the texture. Growth is fast or slow when the volume fractions of the two texture components are approximately equal or noticeably unequal, respectively. 4. The 3D Monte-Carlo method described in this paper can be modified readily to include the discrete dependences of grain-boundary energy (and mobility) on misorientation by incorporating a boundary-energy factor into the calculations.
Acknowledgements The present work was supported by the Air Force Office of Scientific Research (AFOSR) and the AFOSR European Office of Aerospace Research and Development (AFOSR/EOARD) within the framework of STCU Partner Projects P041 and P-057A. The encouragement of the AFOSR program managers (Drs C.H. Ward and C.S. Hartley) is greatly appreciated.
References
Fig. 18. Predictions of the volume fraction of the texture components and the grain size as a function of annealing time for a two-component initial texture: (a) results from Ref. [5] and (b) results from the present work.
[1] [2] [3] [4] [5] [6]
normal in character, and the texture does not change. 2. For a material with an initial random texture and misorientation-dependent grain-boundary mobility, initial grain growth begins in normal fashion (i.e., n ⫽ 2.0), but the grain-growth exponent increases slightly (to n~2.6) at longer times. The slowing of grain growth is a result of the formation of a non-random texture. 3. For a material with a marked initial texture containing two components and misorientationdependent grain-boundary mobility, the behavior is characterized by alternating periods of fast and slow grain growth. These periods of
[7] [8] [9] [10] [11] [12] [13] [14] [15]
Hillert M. Acta metall. 1965;13:277. Louat NP. Acta metall. 1974;22:721. Rhines FR, Craig KR. Metall. Trans. 1974;5:413. Kurtz SF, Carpay FM. J. Appl. Phys. 1980;51:5725. Eichelkraut H, Abbruzzese G, Lucke K. Acta metall. 1988;36:55. Anderson MP, Grest GS, Srolovitz DJ. Phil. Mag. 1989;B59:293. Kawasaki K, Nagai T, Nakashima K. Phil. Mag. 1989;B60:399. Anderson MP, Srolovitz DG, Grest GS, Sahni PS. Acta metall. 1984;32:784. Atkinson HV. Acta metall. 1988;36:469. Yang Z, Sista S, Elmer JW, Debroy T. Acta mater. 2000;48:4813. Wakai F, Enomoto N, Ogawa H. Acta mater. 2000;48:1297. Xiaoyan S, Guoquan L, Nanju G. Scripta mater. 2000;43:355. Holm EA, Hassold GN, Miodownik MA. Acta mater. 2001;49:2981. Miodownik MA, Godfrey AW, Holm EA, Hughes DA. Acta mater. 1999;47:2661. Ivasishin OM, Semiatin SL, Markovsky PE, Shevchenko SV, Ulshin SV. Mater. Sci. Eng. A 2002;A337:88.
1034
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
[16] Rollet AD, Srolovitz DJ, Anderson MP. Acta metall. 1989;37:1227. [17] Ono N, Kimura K, Watanabe T. Acta mater. 1999;47:1007. [18] Holm EA, Zacharopoulos N, Srolovitz DJ. Acta mater. 1998;46:953. [19] Grest GS, Srolovitz DJ, Anderson MP. Acta metall. 1985;33:509. [20] Hwang NM, Sung BL, Doh YK. Scripta mater. 2001;44:1153. [21] Gordon P. Energetics in metallurgical phenomena, vol. 1, p. 207. New York: Gordon and Breach, 1962. [22] Dunn CG, Walter JL. In: Recrystallization, grain growth, and textures. Metals Park, OH: ASM; 1966. p. 483. [23] Zhu G, Mao W, Yu Y. Scripta mater. 2000;42:37. [24] Mackenzie JK. Biometrika 1958;45:229.
[25] Weaire D, McMurry S. In: Ehrenreich M, Spaepen F, editors. Solid state physics: advances in research and applications, vol. 50. San Diego, CA: Academic Press; 1996. p. 1. [26] Gordon P, Vandermeer RA. In: Recrystallization, grain growth, and textures. Metals Park, OH: ASM; 1966. p. 20-5. [27] Winning M, Gottstein G, Shvindlerman LS. Acta mater. 2002;50:353. [28] Wolf D. Acta metall. mater. 1990;38:781. [29] Wolf D. Acta metall. mater. 1990;38:791. [30] Shewmon PGI. In: Recrystalization, grain growth, and textures. Metals Park, OH: ASM; 1966. p. 165. [31] Holm EA, Battaile CC. JOM 2001;53:20. [32] Ivasishin OM, Shevchenko SV, Vasiliev NL, Semiatin SL. Metallofiz. Noveishie Teckhnol. 2001;23:1569.
3D Monte-Carlo simulation of texture-controlled grain growth O.M. Ivasishin a,∗, S.V. Shevchenko a, N.L. Vasiliev a, S.L. Semiatin b b
a Institute for Metal Physics, 36 Vernadsky Str., 03142 Kiev, Ukraine Air Force Research Laboratory, AFRL/ML, Wright-Patterson Air Force Base, OH 45433-7817 USA
Received 17 July 2002; received in revised form 14 October 2002; accepted 24 October 2002
Abstract A three-dimensional (3D) Monte-Carlo (MC) routine was developed to quantify the interaction of grain growth and texture development during annealing. The program included special software to enable the input of the initial grain structure and texture and incorporated a description of the misorientation-dependence of the grain-boundary mobility. Outputs from the model quantified the evolving texture in terms of pole figures or crystallite orientation distribution functions and statistics on the grain structure such as the grain-size distribution and boundary-misorientation distribution function. The MC routine was applied to establish grain growth and texture development in materials with random or strongly textured starting conditions and isotropic or anisotropic grain-boundary mobility. Depending on the starting condition and material properties, normal grain growth or a behavior characterized by alternating cycles of fast and slow grain growth was predicted. 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Grain growth; Texture; Potts model
1. Introduction The control of microstructure in metallic alloys and ceramics is key to improving their mechanical properties. The prediction of microstructural evolution due to recrystallization, grain growth, etc. starting from an arbitrary initial structure is important for materials design and processing. For example, grain growth in polycrystalline aggre-
Corresponding author. Tel.: +380-44-444-2210; fax: +38044-444-0120. E-mail address: [email protected] (O.M. Ivasishin). ∗
gates has been studied extensively, and many important physical phenomena have been modeled. In early models [1–4], the main parameters which characterize grain statistics and features of the network of grain boundaries were included only in an average way. Much more detailed models based on both analytical [5] and computer-simulation [6–8] approaches were reported later. Atkinson [9] reviewed the classical theories, models, and computer simulations. In most cases, however, such simulations have been restricted to two dimensions (2D). Recently, a number of studies [10–14] have been conducted to develop three-dimensional (3D) computer models of grain growth. However, none
1359-6454/03/$30.00 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6454(02)00505-0
1020
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
of these 3D grain-growth simulations took into account the evolution of the textural state at each step of the process. As was recently shown in a series of very careful experiments [15], grain growth and crystallographic texture evolution during the annealing of metals are closely related. Although a number of previous studies of grain growth [6,8,13,14,16–19] have utilized the Monte-Carlo technique, none have addressed the coupling of grain growth and texture evolution in a quantitative fashion. In addition, all of the previous models have made various assumptions to simplify the calculations. These include the use of a two-dimensional, rather than three-dimensional, approach [6,8,13,16–19], use of a limited number of possible grain orientations or the use of integers to represent different orientations rather than Euler angles [6,8,16,18,19], and the disregard or simplification of the dependence of grain-boundary energy and mobility on misorientation [14,16,20]. Even with these approximations, a number of interesting observations have been made. For example, Grest et al. [19] showed that the grain-growth exponent may vary from approximately 2 to 4, depending on the magnitude of the anisotropy of the grain boundary energy with misorientation. In subsequent work, Rollet et al. [16] suggested that grain growth can lead to a marked change in texture. In Ref. [17], Ono et al. concluded that grain-growth behavior as a whole depends on the competition between the overall reduction of system energy and the motion of individual boundaries as controlled by local configurations. The present work was undertaken to expand upon and clarify some of the issues raised in previous research on the Monte-Carlo modeling of grain growth. The objective of the present work, therefore, was to develop a Monte-Carlo (MC) modeling approach with which the interaction of grain growth and texture evolution could be quantitatively described for three-dimensional polycrystalline aggregates. For this purpose, a technique was developed to enable the determination of pole figures, orientation distribution functions (ODFs), sections of ODFs, the volume fractions of specific texture components, the boundary-misorientation distribution function (BMDF), and the grain-size
distribution function for the model volume (or any part of the model volume) at any stage of grain growth. To make the simulations physically realistic, large three-dimensional model volumes with a large number of possible orientations distributed uniformly throughout Euler space were used. The capability of incorporating the misorientation-dependence of grain-boundary energy and mobility was also built into the simulation routine.
2. Modeling approach The grain boundaries of polycrystalline solids form complex three-dimensional network structures. At elevated temperatures, grain-boundary motion occurs to reduce the total grain-boundary energy. This process results in a continuous change in the nature of the grain-boundary network [11]. The velocity of grain-boundary migration is the main factor that controls grain-growth kinetics. The local velocity of boundary migration depends on the grain-boundary energy and mobility and on the local grain-boundary curvature [21]. The velocity v of a grain-boundary segment characterized by energy g, mobility M, and radius R is given by [5]: v ⫽ γMR⫺1,
(1)
and the average migration rate of a boundary between two grains u and m is: ⫺1 vum ⫽ gumMum(R⫺1 u ⫺ Rm ),
(2)
in which (R ⫺R ) is the average curvature for this grain boundary. Small grains disappear as a consequence of boundary motion, and the average size of the remaining grains increases. Despite the fact that the grain-growth process is a geometrically complex phenomenon, a vast amount of experimental data has shown that a simple mathematical relation expresses the statistical behaviour of a large number of grains, i.e., ⫺1 u
⫺1 m
Dn⫺Dn0 ⫽ Kt exp(⫺Q / RgT)
(3)
in which Q is the activation energy, Do and D denote the initial and final average grain sizes, t is the annealing time, Rg is the gas constant, T is the absolute temperature, n is the grain growth
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
exponent, and K represents the rate constant. For so-called normal grain growth during which boundary motion is not impeded and the grain boundaries maintain a self-similar network structure, the square of the mean grain size increases linearly with time (i.e., grain-growth exponent n ⫽ 2) [21,22]. The main features of the present model for texture-controlled grain growth can be divided into those aspects related to the geometry of the problem and the actual kinetics of grain growth. 2.1. Geometric features of the model The geometry-related features of the model include descriptions of the model domain, including its grain structure and texture, and procedures for setting the initial microstructure and texture. 2.1.1. Model domain and grain structure In the present work, the model domain was formed by a three-dimensional cubic array of model units (MUs), each of which represented a point in a cubic lattice. In all calculations, the length of the side of the cubic lattice was set equal to 250 MU, thus yielding a domain volume of 2503 ⫽ 15,625,000 MU3. This volume is substantially greater than that used in Refs. [14] (2,000,000) and [19] (~5,000,000) and thus allowed the introduction of a larger number of grains in the initial microstructure. By this means, textures can be described more precisely using an orientation distribution function (ODF), and the interaction of texture evolution and grain growth can be studied to longer times. As in previous work utilizing the MC approach [6,8,16,18], the material domain did not contain grain boundaries per se. Rather, the grain boundary position was associated with the space between two sites having unlike orientations. Each grain was characterized by a volume equal to the number of MUs within the grain. However, the actual grain size was taken to be equal to the diameter of a sphere containing the same volume. The smallest initial grain was assumed to comprise ~35–40 MUs. With this assumption, small clusters (with volumes below 10 MU3) that arise during simulations were not taken into account in calculating the
1021
microstructure statistics because the local grainboundary curvature has no physical meaning in such cases. Coalescence of grains with the same orientation was not allowed in the simulations. 2.1.2. Crystallographic orientations and misorientations Crystallographic orientations were specified in three-dimensional Euler space by the rotation angles j (⫺90 ⬍ ϕ ⬍ 90), q (0 ⬍ θ ⬍ 180) and c (0 ⬍ χ ⬍ 180). These three values characterized the orientation of the grain-coordinate system (KG unit vectors) relative to the fixed reference directions of the model domain (KMD unit vectors parallel to the RD, TD, and ND) (Fig. 1a). Orientation space was divided into segments of 2 × 2 × 2 degrees, giving rise to 729,000 possible orientations. Thus, each MU was assigned a number L (0 ⬍ L ⬍ 729,000) corresponding to its specific interval in Euler space, gi ⫹ dgi. The orientation distribution function f(g) (ODF) characterizing the textural state of material was then expressed as: f(g) ⫽
冘
→
→
f(R,g); f(R,g ⫽ gi ⫹ dgi) ⫽ 1; MD
(4)
→
f(R,g ⫽ gi ⫹ dgi) ⫽ 0, in which MD denotes the model domain, and R is the spatial location in Cartesian coordinates. A special procedure was developed to create the initial texture state in the model domain and to replicate the specified ODF as close as possible. In this regard, special attention was paid to ensure that grains with orientations belonging to different texture components were distributed randomly, i.e., there were no local textures in the model domain that differed greatly from the specified ODF. The misorientation angle between two neighboring grains with orientations g1(j1, q1, c1) and g2(j2, q2, c2) was determined as “orientation distance” in Euler space (Fig. 1b). The explicit expressions for the determination of the orientation distance ε are: ε ⫽ arccos((A⫺1) / 2);
(5)
A ⫽ cos(ϕ1⫺ϕ2)cos(χ1⫺χ2)[1 ⫹ cosθ1cosθ2]⫺sin(ϕ1⫺ϕ2)sin(χ1
(6)
1022
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
then be taken into account through the corresponding symmetry of the grain-boundary mobility dependence on misorientation. With this definition of ε, the BMDF is defined as P(ε,dε), or the fraction of boundaries with misorientation angles between ε and ε ⫹ dε [24]. The BMDF describes the probability of finding a boundary with a given misorientation angle. In model calculations, the value of dε was chosen to be 2°. Thus, for ε ⫽ 0ⴰ, for example, P(ε,dε) was the probability of finding a misorientation greater than 0° and less than 2°. 2.2. Kinetic features of the model The kinetic features of the model comprised descriptions of the grain-boundary energy/mobility and the MC procedure for describing grain growth, incorporating appropriate time and temperature variables.
Fig. 1. Description of crystallographic orientations and misorientations between grains: (a) Euler angles and (b) orientation distance ε.
⫺χ2)[cosθ1 ⫹ cosθ2] ⫹ [cos(ϕ1⫺ϕ2) ⫹ cos(χ1⫺χ2)]sinθ1sinθ2 ⫹ cosθ1cosθ2 Eqs. (5) and (6) are equivalent to the commonly used algorithm [14,23] without the selection of minimum angle allowed by crystal lattice symmetry, however. By using the full spectrum of ε values (0ⱕεⱕp), the boundary misorientation spectra of complex textures can be conveniently described. The crystal symmetry of material can
2.2.1. Grain boundary energy/mobility The local velocity of a grain boundary is often assumed to be proportional to the grain boundary energy and mobility, i.e., Eq. (2). In most cases, however, it is not possible to separate the influence of these two factors on grain growth [25,26]. For this reason, the dependence of grain growth on mobility and boundary energy is often treated by introducing the potential V(gi, gj), in which gi denotes the discrete, local crystal orientations available in the modeling space. Hence, the effective mobility of the boundary between grains having orientations i and j and misorientation e is given by: Mij ⫽ M0V(gi,gj) ⫽ M0V(ε).
(7)
It is assumed in the present model that M 0 ⫽ M max ⫽ 1, and V(gi, gj) 苸[0,1]. Hence, the present model specifies V(gi, gj) as a function of ε. The existing experimental data for the dependence of grain-boundary energy and mobility on local lattice misorientation [27–30] are contradictory in many cases. From a qualitative perspective, however, all of the measurements are similar to those shown in Fig. 2a. The number of maxima in the misorientation interval [0, p] depends on the
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
1023
2.2.2. MC routine To simulate the kinetics of boundary motion with the MC technique, a MU was selected at random, and a trial orientation was also chosen at random from one of the orientations of its neighborhood. This is an essential feature used to optimize MC routines such as that proposed by Anderson et al. [6]. Trials with all possible orientations in the modeling domain were impossible in the present formulation because of the very large L value and thus the excessive computer time that would have been required. Hence, there were two different possible states of the modeling domain after the MC trial—the same as the previous one and the state in which the MU has changed its orientation to the chosen one. The next step of the MC procedure consisted of the comparison of the free energies of these two states. The model used is the simple Hamiltonian: H ⫽ ⫺J(dLiLj⫺1),
Fig. 2. Misorientation dependence of the grain-boundary mobility factor: (a) literature values for fcc metals and (b) assumed values for MC simulations (Eq. (8)) corresponding to 1: S→⬁: S ⫽ 10250, 3: S ⫽ 4000, and 4: S ⫽ 2250.
type of crystal structure. In the general case without restrictions due to a specific crystal symmetry, a phenomenological dependence which describes the function V(gi, gj) with m maxima at misorientations ⑀ ⫽ Gi, i ⫽ 1, 2, …, m is given by the following relations: V(gi,gj) ⫽ V(ε) ⫽ 1⫺(Gi⫺ε)2 / S;e苸[0;Gi ⫹ 0.5(Gi+1⫺Gi)]
i ⫽ 1;
V(gi,gj) ⫽ V(ε) ⫽ 1⫺(Gi⫺ε)2 / S;苸[Gi⫺0.5(Gi⫺Gi⫺1);Gi ⫹ 0.5(Gi+1⫺Gi)]
i ⫽ 2,3,…,m⫺1;
V(gi,gj) ⫽ V(ε) ⫽ 1⫺(Gi⫺ε)2 / S;e苸[Gi⫺0.5(Gi⫺Gi⫺1);p]
i ⫽ m.
(8) For cubic crystals (Fig. 2b), G 1 ⫽ 45° and G 2 ⫽ 135°; for hexagonal close-packed metals, G 1 ⫽ 30°, G 2 ⫽ 90°, and G 3 ⫽ 150°.
(9)
in which Li is the orientation of site i, Lj is the orientation of the neighboring site j, and δhl is the Kronecker delta. The sum is taken over all neighboring sites within the specified ‘control volume’ around the MU that is inspected; this volume is characterized by a search radius SR. The shape of the control volume was assumed to be cubic, centered around the selected MU, and with a side length of 2SR ⫹ 1 in model units (Fig. 3). Thus, nearest neighbor pairs contribute J to the system energy when they are of unlike orientation and zero otherwise. The transition probability, W, is then given by: W⫽
再
Mijexp(⫺⌬G / kbT); ⌬G ⬎ 0,
Mij
⌬Gⱕ0,
(10)
in which ⌬G is the change in energy caused by the change of orientation, Mij is the grain boundary mobility, and kb is Boltzmann’s constant. Successful transitions at a grain boundary to the orientation of a nearest-neighbor grain correspond to boundary migration. The MC routine can be modified readily to include the discrete dependence of grain boundary energy and grain-boundary mobility on misorientation by incorporating a boundary-energy factor into the calculation of H and ⌬G in Eqs. (9) and (10) [31].
1024
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
faster). Furthermore, a choice of SR ⬎ 2 (Fig. 3c) would have led to very extensive calculations and would have undesirably increased the probability of incorporating the influence of distant grains and grains boundaries. As in past MC investigations, the unit of time in the simulations was the Monte-Carlo “step” (MCS). During one MCS, the number of simulation procedures done in accordance with Eqs. (9) and (10) was equal to the number of sites in the modeling domain, i.e., 2503. 2.2.3. Temperature- and time-scale factors As can be deduced from Eqs. (1) and (2), the real time scale of the model can be expressed as: t ⫽ B(T)·l·tmcs,
Fig. 3. Definition of the search radius, SR, for MC simulations of grain growth: (a) SR ⫽ 1, (b) SR ⫽ 2, and (c) SR ⫽ 3.
In all calculations, the sampling neighborhood, or control volume, was characterized by SR ⫽ 2. Thus, 124 nearest MUs were taken into account in the cubic three-dimensional array for the calculation of Eq. (9) (Fig. 3b). This is the best value for SR. Choosing the SR ⫽ 1 (Fig. 3a) would have led to a very rough capture of the local grainboundary curvature (but made the calculations
(11)
in which t is time, tmcs is modeling time (i.e., number of MCS), l is the length of the MU, and B is the time-scale factor. The phenomenological expression (11) can be used to obtain the value of the time-scale factor by a comparison of experimental measurements of the initial grain size and the rate of normal grain growth during isothermal annealing for a non-textured material with the calculated grain-growth rate in MC simulations. Another way to estimate the magnitude of the time-scale factor would be to measure the velocity of an isolated grain boundary (with constant curvature) at temperature T0 in a real material and then compare this velocity with the kinetics of a comparable grain boundary in the modeling domain of a simulation. For instance, careful experiments with such boundaries have been performed for pure aluminum by Winning et al. [27]. The velocity of grain-boundary migration is closely related to the rate of diffusion at the boundary. This results in an Arrhenius dependence of grain-boundary velocity on temperature. Hence, temperature can be incorporated into the MC model by considering a time-scale factor which is a function of absolute temperature, i.e., B(T) ⫽ B0(T0)(V0(T0) / V(T)),
(12a)
thus leading to the following relation: t ⫽ B0(T0)exp[⫺(Qa / R)(1 / T0⫺1 / T)]· l·tmcs,
(12b)
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
1025
where Qa denotes the activation energy for grainboundary migration. The calculations described below were done for T→0, t(1 MCS)→⬁. Thus, it was not necessary to account for the real time-temperature scale. Nevertheless, because all calculations were done with a fixed modeling domain and the same MC routine, the comparison of kinetics for different cases was possible in terms of MCS as the relative measure of time and the MU as the relative measure of length.
3. Results and discussion A series of simulations was run to assess the validity of the three-dimensional MC routine in Fig. 5. MC predictions of the shape (in cross section) of an isolated spherical grain (case 1): (1) initial grain boundary, and the grain boundary after: (2) 200 MCS, (3) 500 MCS, (4) 1000 MCS, (5) 1500 MCS, and (6) 1900 MCS.
quantifying the interaction of grain growth and texture evolution. These simulations described the shrinkage of an isolated grain in a homogeneous matrix and the effects of starting texture and anisotropy of the grain-boundary mobility on grain growth. The latter results are reported in terms of grain growth kinetics, grain size distributions, BMDFs, and (100) pole figures. 3.1. Case 1: shrinkage of an isolated grain
Fig. 4. MC predictions for the shrinkage of an isolated spherical grain (case 1): (a) kinetics of shrinkage and (b) grain-boundary velocity.
As an initial validation of the MC routine, the shrinkage of an isolated spherical grain in a homogeneous matrix was modeled. The objectives of this trial case were to ensure that the use of the Hamiltonian in Eq. (9) could replicate the boundary migration rate described by Eq. (2) and to determine the extent of the statistical “noise” associated with the migration-rate calculations. In addition, the case was used to check the quality of the random-number generator used in the MC routine and to verify that the expected isotropic shrinkage could be modeled. The grain had a starting diameter of 200 MU
1026
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
Fig. 6. MC predictions of texture evolution for case 2: (a) initial texture, and textures after (b) 100 MCS and (c) 1000 MCS; and for case 3: (d) initial texture, and textures after (e) 100 MCS and (f) 1000 MCS. Iso-intensity lines correspond to 1, 2, 5, and ⬎10 times random.
and was placed in a model domain that was 250 × 250 × 250 MU in extent. The matrix had a crystallographic orientation different from that of the grain. The mobility factor of the grain boundary was assumed to be 1 (i.e., the maximum value).
Fig. 7. Predicted grain-growth kinetics for case 2 (solid line) and case 3 (broken line).
Simulation results for case 1 are shown in Figs. 4 and 5. A plot of the kinetics of grain shrinkage (Fig. 4a) indicated that the initial grain disappeared completely after 1955 MCS. In addition, within a high degree of accuracy, the boundary migration rate was proportional to the boundary curvature (=inverse diameter) (Fig. 4b), in agreement with Eq. (2). The statistical fluctuation in the migration rate at a given curvature was approximately 10%. The occurrence of such statistical noise is a result of the random, mutually independent selection of MU elements during each discrete MC trial. The statistical nature of the calculations is seen also in cross-sections through the grain (Fig. 5). By and large, the predicted grain shape remained spherical during the entire simulation. However, a small amount of distortion developed due to the discrete character of the model. For example, at the end of the simulation (1900 MCS), the centroid of the grain was located slightly off-center, i.e., at X ⫽ 0, Y ⫽ 1, Z ⫽ 1 MU. Thus, the deviation from
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
1027
Table 1 Volume fractions of grains with orientations within region A (VA) and region B (VB) in Fig. 12a Time, MCS
VA
VB
VA+VB
0 100 350 650
0.052 0.080 0.095 0.123
0.052 0.070 0.074 0.087
0.104 0.150 0.169 0.210
Fig. 8. MCS.
Predicted grain-size distribution for case 2 after 1000
Fig. 9. Predicted grain-boundary misorientation distribution functions for: (1) case 2 and (2) case 3. For each case, the three lines correspond to 0, 100, and 1000 MCS.
Fig. 10. MC predictions for case 4: (a) kinetics showing the retardation of grain growth due to texture formation (broken lines are the best fit for the time intervals 0–600 MCS and 900– 1500 MCS) and (b) BMDFs: (1) initial condition, (2) 10 MCS, (3) 100 MCS, and (4) 650 MCS
1028
Fig. 11.
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
Predicted grain-size distributions for case 4: (a) Initial condition, and after (b) 10 MCS, (c) 100 MCS, and (d) 650 MCS.
uniform shrinkage was equal to or less than 0.8% (=1/125) throughout the modeling domain. 3.2. Grain growth in untextured (case 2) and textured (case 3) material with misorientationindependent grain-boundary mobility Two MC simulations were conducted for graingrowth conditions in which the grain-boundary mobility was assumed to be independent of misorientation. The initial microstructure was assumed to have an average grain size of 5.43 MU and a log-normal grain-size distribution. For case 2, the starting material was assumed to be untextured. For case 3, the texture comprised two strong components, A and B, that were 60° apart and each of which were Gaussian in Euler space having a half-width of 4° in all three angular dimensions. The volume fractions of the two components were 0.10 and 0.90, respectively. The (100) pole figures for the initial textures in these two cases are shown in Fig. 6a and d, respectively. For case 3, only the
(100) poles (and not the (010) and (001) poles) are shown on the pole figures for clarity. This is sufficient in view of the method of representing orientation distance and of calculating misorientations and the associated boundary energies and mobilities. As expected, the grain-growth kinetics for the two different cases were found to be the same. From plots of the average grain size versus time (Fig. 7), the grain-growth exponents (i.e., value of n in Eq. (3)) were found to be 2.02 for case 2 and 2.00 for case 3. Because of the statistical nature of MC simulations, one can conclude that the exponents in both cases were both essentially equal to 2.0, the value that characterizes classical normal grain-growth kinetics. MC predictions of the texture (in terms of (100) pole figures) for cases 2 and 3 are summarized in Fig. 6. For case 2, the texture remained essentially isotropic (Fig. 6 b and c). Furthermore, the grainsize distribution for this case (Fig. 8) was close to log-normal. For case 3, the texture remained essen-
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
1029
Fig. 12. Predicted texture evolution for case 4 after: (a) 100 MCS, (b) 350 MCS, (c) 650 MCS, and (d) 1500 MCS. Iso-intensity lines correspond to 1, 2, 5, 10, and ⬎30 times random.
tially unchanged as well (Fig. 6e and f), and the grain-size distribution was also nearly log-normal at the end of the simulation. The boundary-misorientation distribution functions (BMDF) for cases 2 and 3 are shown in Fig. 9. There were significant differences in the BMDFs for the random and highly textured starting conditions for each time during the simulations. The BMDF of the randomly textured polycrystal followed the so-called Mackenzie distribution [13,24]. The BMDF of the highly-textured condition had two maxima in positions close to 0 and 60°. However, there was no evidence of any significant changes in the BMDFs during grain growth for either starting texture. Deviations from the initial BMDFs were statistical in nature and increased when the total number of grains within the modeling domain decreased. 3.3. Case 4: grain growth in initially randomtextured material with misorientation-dependent grain-boundary mobility Grain-growth kinetics and texture evolution in a material that had an initial random texture and a
misorientation-dependent grain-boundary mobility (case 4) were quantified using the MC approach. The mobility-factor dependence followed Eq. (8) with S ⫽ 2250 (Fig. 2b). At short annealing times, the predicted graingrowth exponent was equal to 1.99 (Fig. 10a). After ~900 MCS, however, the grain growth exponent was predicted to increase to 2.58. In addition, the corresponding grain-size distribution gradually deviated from log-normal behavior as grain-growth occurred (Fig. 11). After 100 MCS (Fig. 11c), the fraction of small grains had increased. This trend can be explained by the fact that small grains disappear slowly when they are surrounded by large grains of the same texture component. At yet longer times (Fig. 11d), several local maxima, instead of a single maximum, had developed in the grain-size distribution. The evolution of the grain-size distribution can also be rationalized in terms of the gradual transformation of the BMDF in which some evidence of texture formation was also seen (Fig. 10b). The hypothesis that a noticeable texture had formed in this case was confirmed by the corresponding pole figures
1030
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
Fig. 14. Comparison of the predicted grain-growth kinetics for case 5 (solid line) and normal grain-growth kinetics (broken line).
Fig. 13. BMDFs for the circled areas in Fig. 12a for case 4: (a) only grains with orientation A and (b) grains with orientations A and B, multiplied by corresponding volume fractions (Fig. 12, Table 1). (1) Initial condition, (2) after 100 MCS, (3) after 350 MCS, (4) after 650 MCS.
generated by the MC routine. There was an indication of the formation of a cube-on-corner texture in the (100) pole figures (Fig. 12). The results in Fig. 10b also suggested that microstructure evolution during the first 10 MCS led to a corresponding increase in the number of low-mobility boundaries with misorientations close to 0, 90, and 180°. The subsequent evolution of the BMDF was more complicated, however. This interpretation required the analysis of the evolution of separate texture components (i.e., volume fractions, grain-size distributions, misorientation function). For this purpose, a special procedure was incorporated into the model. Using this procedure, it was found that texture components labeled A and B in Fig. 12a gradually developed from
the initial random texture (Fig. 13a). This led to a transformation of the “local” BMDFs for specific texture components (e.g., Fig. 13a for component A) and an increase in the volume fraction of grains of the two specific orientations (Table 1). Taken together, these effects caused a significant increase in the density of boundaries at certain angles (Fig. 13b). The maximum at small angles, which is also evident in Fig. 10b, comes from misorientations within each of the A and B components (i.e., AA and BB boundaries), while the maximum associated with the high angular misorientation has resulted from boundaries of the AB type. 3.4. Case 5: grain growth and texture evolution in material with an initial two-component texture and misorientation-dependent grain-boundary mobility The final MC simulation (case 5) focussed on grain growth in a material whose initial texture consisted of two components, i.e., the same as the initial texture for case 3. As in case 4, the mobilityfactor dependence for this case followed Eq. (8) with S ⫽ 2250 (Fig. 2b). The predicted kinetics of grain growth for case 5 are shown in Fig. 14. Periods of fast and slow grain growth were apparent. There were also pronounced variations in texture as grain growth occurred as seen in the (100) pole figures in Fig.
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
1031
Fig. 15. Predicted texture evolution for case 5: (a) initial condition and after (b) 15 MCS, (c) 100 MCS, (d) 600 MCS, (e) 1200 MCS, and (f) 2000 MCS. Iso-intensity lines correspond to 1, 2, 5, 10, and ⬎30 times random.
15. (As for case 3, only the (100) poles (and not the (010) and (001) poles) are shown on the pole figures for case 5 for clarity.) The texture changes were characterized by periodic interchanges of the volume fractions of the two components. Furthermore, the periods of rapid grain growth in Fig. 14 were found to be associated with the times at which there was a large fraction of high-angle boundaries (e.g., at 15 MCS and 1200 MCS, as shown in Fig. 16). These are the times at which the volume fractions of the two texture components were approximately equal. The MC-simulated texture evolution can be explained as follows. Because of their lower volume fraction, the A grains initially had a much higher probability of being surrounded by B grains rather than by other A grains and thus of having high mobility A–B boundaries. Hence, small A grains were rapidly consumed by large B grains, but large A grains were able to grow rapidly. Because the B grains were in contact preferably with other B grains, they had mainly low mobility B–B boundaries; therefore, most of the B grains participated only slightly in the initial stages of
grain growth. This led to a very different grainsize distribution for the modeling domain as a whole (Fig. 17a) compared to that for those grains with the A texture component (Fig. 17b). After the A grains had consumed a majority of the volume, the growth rate of the A grains decreased, because there was now a high probability that each was surrounded by other A grains. As B had become the minority component, the growth rate of B grains exceeded that of the A grains. Over the long times, such phenomena produced cyclic changes in the relative volume fractions of the A and B texture components. Such periodic evolution of a two-component texture has been previously predicted analytically (Fig. 18a) [5] and confirmed experimentally in very recent work on the beta annealing of a titanium alloy [15]. The results of the present MC model (Fig. 18b) are in good agreement with the previous theoretical calculations as well as with the experimental findings. A similar behavior was also deduced in Ref. [32] in which the evolution of a two-component texture was modelled for the case comprising a strong (axial) component and a weak
1032
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
Fig. 17. Predicted grain-size distributions for case 5 after 15 MCS: (a) for the entire material and (b) for the grains belonging to texture component A. Fig. 16. Predicted variation of the BMDF for case 5 over time intervals of (a) 0–100 MCS and (b) 600–2000 MCS.
4. Summary and conclusions
consist of any combination of initial microstructure, texture, and grain-boundary mobility. Furthermore, the approach provides the ability to describe and control all model parameters at each stage of grain growth. Several simulations were performed to confirm the validity of the approach and to analyze some specific cases of grain growth. The following conclusions were drawn from this work:
A new modeling approach, which includes an optimised MC simulation routine and procedures for the control of the modeling domain microstructure and texture, and its associated computer program were developed to establish the interaction of grain growth and texture evolution. The input can
1. When the grain-boundary mobility is independent of misorientation, normal grain growth with a grain-growth exponent of 2.0 occurs irrespective of whether the starting material has an initial random or sharp texture. In these cases, the grain-size distribution remains log-
component consisting of a background distribution of random orientations.
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
1033
growth can be correlated to the cyclic changes in the texture. Growth is fast or slow when the volume fractions of the two texture components are approximately equal or noticeably unequal, respectively. 4. The 3D Monte-Carlo method described in this paper can be modified readily to include the discrete dependences of grain-boundary energy (and mobility) on misorientation by incorporating a boundary-energy factor into the calculations.
Acknowledgements The present work was supported by the Air Force Office of Scientific Research (AFOSR) and the AFOSR European Office of Aerospace Research and Development (AFOSR/EOARD) within the framework of STCU Partner Projects P041 and P-057A. The encouragement of the AFOSR program managers (Drs C.H. Ward and C.S. Hartley) is greatly appreciated.
References
Fig. 18. Predictions of the volume fraction of the texture components and the grain size as a function of annealing time for a two-component initial texture: (a) results from Ref. [5] and (b) results from the present work.
[1] [2] [3] [4] [5] [6]
normal in character, and the texture does not change. 2. For a material with an initial random texture and misorientation-dependent grain-boundary mobility, initial grain growth begins in normal fashion (i.e., n ⫽ 2.0), but the grain-growth exponent increases slightly (to n~2.6) at longer times. The slowing of grain growth is a result of the formation of a non-random texture. 3. For a material with a marked initial texture containing two components and misorientationdependent grain-boundary mobility, the behavior is characterized by alternating periods of fast and slow grain growth. These periods of
[7] [8] [9] [10] [11] [12] [13] [14] [15]
Hillert M. Acta metall. 1965;13:277. Louat NP. Acta metall. 1974;22:721. Rhines FR, Craig KR. Metall. Trans. 1974;5:413. Kurtz SF, Carpay FM. J. Appl. Phys. 1980;51:5725. Eichelkraut H, Abbruzzese G, Lucke K. Acta metall. 1988;36:55. Anderson MP, Grest GS, Srolovitz DJ. Phil. Mag. 1989;B59:293. Kawasaki K, Nagai T, Nakashima K. Phil. Mag. 1989;B60:399. Anderson MP, Srolovitz DG, Grest GS, Sahni PS. Acta metall. 1984;32:784. Atkinson HV. Acta metall. 1988;36:469. Yang Z, Sista S, Elmer JW, Debroy T. Acta mater. 2000;48:4813. Wakai F, Enomoto N, Ogawa H. Acta mater. 2000;48:1297. Xiaoyan S, Guoquan L, Nanju G. Scripta mater. 2000;43:355. Holm EA, Hassold GN, Miodownik MA. Acta mater. 2001;49:2981. Miodownik MA, Godfrey AW, Holm EA, Hughes DA. Acta mater. 1999;47:2661. Ivasishin OM, Semiatin SL, Markovsky PE, Shevchenko SV, Ulshin SV. Mater. Sci. Eng. A 2002;A337:88.
1034
O.M. Ivasishin et al. / Acta Materialia 51 (2003) 1019–1034
[16] Rollet AD, Srolovitz DJ, Anderson MP. Acta metall. 1989;37:1227. [17] Ono N, Kimura K, Watanabe T. Acta mater. 1999;47:1007. [18] Holm EA, Zacharopoulos N, Srolovitz DJ. Acta mater. 1998;46:953. [19] Grest GS, Srolovitz DJ, Anderson MP. Acta metall. 1985;33:509. [20] Hwang NM, Sung BL, Doh YK. Scripta mater. 2001;44:1153. [21] Gordon P. Energetics in metallurgical phenomena, vol. 1, p. 207. New York: Gordon and Breach, 1962. [22] Dunn CG, Walter JL. In: Recrystallization, grain growth, and textures. Metals Park, OH: ASM; 1966. p. 483. [23] Zhu G, Mao W, Yu Y. Scripta mater. 2000;42:37. [24] Mackenzie JK. Biometrika 1958;45:229.
[25] Weaire D, McMurry S. In: Ehrenreich M, Spaepen F, editors. Solid state physics: advances in research and applications, vol. 50. San Diego, CA: Academic Press; 1996. p. 1. [26] Gordon P, Vandermeer RA. In: Recrystallization, grain growth, and textures. Metals Park, OH: ASM; 1966. p. 20-5. [27] Winning M, Gottstein G, Shvindlerman LS. Acta mater. 2002;50:353. [28] Wolf D. Acta metall. mater. 1990;38:781. [29] Wolf D. Acta metall. mater. 1990;38:791. [30] Shewmon PGI. In: Recrystalization, grain growth, and textures. Metals Park, OH: ASM; 1966. p. 165. [31] Holm EA, Battaile CC. JOM 2001;53:20. [32] Ivasishin OM, Shevchenko SV, Vasiliev NL, Semiatin SL. Metallofiz. Noveishie Teckhnol. 2001;23:1569.