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Simulations of grain impingement and recrystallization kinetics
OOOl-6160/87 $3.00+O.OO Copyright 0 1987Pergamon Journals Ltd
Acrn metal/. Vol. 35, No. 6, pp. 1377-1390,1987 Printed in Great Britain. All rights reserved
SIMULATIONS OF GRAIN IMPINGEMENT RECRYSTALLIZATION KINETICS
AND
c. w. PRICE Chemistry and Materials Science Department, Lawrence Livermore National Laboratory, University of California, Livermore, CA 94.550, U.S.A. (Receiued 31 March 1986) Abstract-A simplified computer model based on regular distributions of geometric grains was used to simulate grain impingement that occurs during recrystallization reactions. The model assumes instantaneous nucleation with all grains growing at the same rate. Impingement geometry was studied as a function of grain shape, grain symmetry, and distribution geometry. Kinetic recrystallization plots were generated from the geometric-impingement data by relating the radius or size parameter of the grain to an appropriate time scale. These kinetic simulations permitted analysis of existing kinetic models and isolated some critical limitations of the models. R&sum&-Nous avons utilise un modtle simplifie, base sur des distributions reguliers de grains geombtriques, pour simuler la rencontre des grains qui se produit au tours des reactions de recristallisation. Le. modele suppose un germination instantanee ou tous les grains grossissent a la meme vitesse. Nous avons itudie la geomttrie de la rencontre en fonction de la forme et de la symttrie du grain ainsi que de la geometric de la distribution. Nous avons produit des diagrammes de recristallisation cinttique a partir des donntes geomitriques de la rencontre, en reliant le rayon ou le parametre de taille du grain a une &chelle des temps approprite. Ces simulations de cinetique ont permis l’analyse des mod&es cinttiques existants et ils ont mis en evidence quelques limitations critiques de ces modeles. Zusammenfassung-Mit einem vereinfachten Computermodell, welches auf einer regelmigigen Verteilung der Kiimer wahrend der Regeometrischer Kijmer aufbaut, wurde das Aufeinandertreffen kristallisationsreaktion simuliert. Das Model1 nimmt instantane Keimbildung an, wobei alle Khmer mit derselben Rate wachsen. Die Geometrie des Aufeinandertreffens wurde in Abhlngigkeit von der Komform, der Komsymmetrie und der Verteilungssymmetrie untersucht. Kinetische Rekristallisationsdiagramme wurden aus diesen Daten hergestellt, indem der Radius oder der Griigenparameter des Kornes auf eine geeignete Zeiskala bezogen wurde. Diese Simulierung der Kinetik erlaubte, die vorhandenen kinetischen Modelle zu analysieren und einige kritische Einschriinkungen der Modelle aufzufinden.
INTRODUCTION
Mathematical treatments of recystallization kinetics must compensate for grain impingement that occurs during recrystallization. As recrystallized grains grow and impinge on neighboring grains, the impingement restricts continued growth of the grains and retards the reaction rate as measured by the volume fraction of recystallized material that forms. The impingement geometry is dictated by both the shapes of recrystallized grains and the distribution of nucleation sites. Since deformation is a necessary precursor for both nucleation of recrystallized grains and sustained growth during recrystallization, the distribution of nucleation sites is strongly dependent upon both the deformed grain geometry and the defect structure induced by deformation. Consequently, impingement geometry is quite complex. One of the first formal treatments of impingement geometry was presented independently by Johnson and Mehl [1] and Avrami [2-4]. They compensated for grain impingement by using an extended-volume concept based on the unrestricted growth of a ran-
dom distribution of spheres. The extended volume was defined as the total volume of the spheres with no correction for overlapping. The expression for the actual volume, V,, at a reaction time of t = r was developed by Avrami [3] in terms of the extended volume, V,, : V, = 1 - exp( - V,,).
(1)
This concept along with an assumption of lineargrowth kinetics yielded the well-known kinetic relation f = 1 - exp(Bt”)
(2)
where f is the volume fraction of recrystallized material, and B and n are constants for isothermal reactions. A more recent treatment of recrystallization kinetics by Speich and Fisher [5] was based on an empirical relation betweenfand the interfacial area, A, between recrystallized and unrecrystallized material A = U-(1
-f)
(3)
where K, is a constant. Extensive experimental mea-
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surements offand A performed by Speich and Fisher conformed reasonably well to equation (3). Use of equation (3), an empirical relation suggested by English and Backofen [6] for inverse time-dependent recrystallization growth rates, and the Cahn-Hagel concept of surface-controlled growth [7] yielded the kinetic relation f/(1 -f)
= Kt”
(4)
where K is a constant for isothermal reactions. Limitations of both models were discussed recently [8]. Formulation of the Johnson-Mehl-Avrami (J-M-A) model on linear growth rates restricts equation (2) to only the early stages of recrystallization, but numeric analysis indicates that even this limited agreement is questionable because of the insensitivity of logarithmic plots at small values off. In contrast, the Speich-Fisher (S-F) model in equation (4) shows reasonable agreement with experimental kinetic data over a wide range off (from f = 0.02 to f = 0.95 in their work). However, Cahn [9] used stereological relations to show that equation (3) was not rigorously correct at either extreme off. To further explore the significance and the limitations of equation (3), a simplified computer model was developed [lo] to study grain impingement. The model was based on the growth of a regular, cubic distribution of spheres with equal radii. Although the model is an obvious oversimplification of actual grain structures, it provides rigorous and tractable calculations of f and A along with precisely defined impingement geometry. Computations of f and A produced by the model were in reasonable agreement with both equation (3) and the experimental data generated by Speich and Fisher. The computer model also supported Cahn’s criticism of equation (3) by showing that the slopes of the simulated growth curves are infinite at f = 0 and f = 1 in the model but are & and -&, respectively, for equation (3). This paper formally relates the computer model to the extended-volume concept, extends the model to the impingement of other geometric shapes, and uses the model to simulate recrystallization kinetics. Important objectives of this paper are to relate impingement geometry to recrystallization kinetics and to isolate limitations of the kinetic models represented by equations (2) and (4). DESCRIPTION OF THE COMPUTER MODEL The computer model [lo] consists of a regular distribution of spheres of radius r on an infinite, simple cubic lattice and corresponds to a distribution of hypothetical nucleation sites on a regular, infinite lattice with instantaneous nucleation. For this study, it was extended to other geometric shapes and lattice geometries. In the model, all nuclei are assumed to grow at the same rate, with growth ceasing at the point of impingement. The symmetry of the distribution lattice permitted the model to be based on
IMPINGEMENT
computations offand A for a single, geometric grain growing in a cell that conformed to the geometry of the distribution lattice. The cell was constructed as a Wigner-Seitz primitive cell [ll] with (200) planes. The lengths of the cell axes were 2a, 2b, and 2c in the x, y, and z directions, respectively. The axial lengths were adjusted to yield a cell of unit volume. The faces of this cell coincide with the planes of impingement of the sphere with the juxtaposed spheres; hence, the unit-volume cell is termed the impingement cell. The symmetry of the orthogonal impingement cell permitted computations to be performed in the positive octant of the impingement cell; this is termed the computation ceN. The origin of the computation cell was at (0,0,0),and the axes of the cell extended to (a, O,O), (0, b, 0), and (O,O, c). The simulated geometric grain was permitted to grow beyond the faces of the cell, but the volume and surface area of the grain were computed only for the portion of the grain contained within the computation cell. Grain shapes simulated were spheres, ellipsoids, and bipyramids. Bipyramids were constructed with either { 111) or (110) planes. The {ill} bipyramid was simulated with a single (111) plane in the computation cell. The (110) bipyramid required intersecting (101) and (011) planes in the computation cell. The following equations were used to form the geometric grains Spheresx*+y*+z*=r*
(5)
Ellipsoids(x/Z)* + (y/m)* + (z/n)* = r*
(6)
{111) Bipyramidsx/l+y/m+z/n=r
(7)
{1IO} BipyramidsPlane l-x/l
+ z/n = r
(84
Plane 2-y/m
+ z/n = r
(8b)
In equations (5X8), r is the radius or size parameter of the geometric grain. In equations (6X8), I, m, and n determine the relative axial intercepts of the ellipsoid or bipyramid and establish the axial symmetry of the simulated grain. To simplify computations off and A for spheres and ellipsoids, the cell was divided into finite elements extending from the x-y plane of the computation cell, as described previously [lo]. For spheres and ellipsoids, reasonable plotting accuracy was achieved with a 1000 x 1000 network of elements. The planar surfaces of the bipyramids permitted using simple geometric computations that significantly improved the computation efficiency. For this paper, geometric-growth curves are defined as plots of A against $ They were simulated by incrementing r from zero to a value that yieldedf = 1. The model was extended to kinetic simulations by assuming the growth rate of the simulated grain to be defined by the growth rate of r. This assumption
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permitted construction of simulated kinetic from the geometric impingement data. IMPINGEMENT
OF GRAIN
plots
The impingement geometry is dictated by the geometry of overlapping of the extended or full volumes of the impinging grains. The impingement geometry changes as the number of grains impinging at a given point changes. Major impingement points (IPs) were previously defined as those points of intersection of the simulated grain with the impingement cell that correspond to changes in impingement geometry [lo]. In the computer model, these IPs occur at the comer positions of the computation cell. They are numbered sequentially in the order of occurrence with increasing r. The change in the geometry of overlapping at an IP causes a corresponding change in the slope of the growth curve. This change is conveniently described in the usual terms of hard and soft impingement for phase transformations [12]. Hard impingement occurs when a spherical or ellipsoidal surface becomes tangent to one of the planes of the impingement cell or an edge of a planar grain coincides with one of the planes. The resultant overlap correction is much greater for A than forfand causes a sharp break in the growth curve. Soft impingement results in relatively small overlap corrections for both f and A and causes only a slight break in the growth curve. The impingement geometry for a sphere growing in a cubic impingement cell was described previously [IO]. As r increases, IP 1 occurs simultaneously at the equivalent positions of (f, 0, 0), (0, i, 0), and (0,0,i) for r = l/2. The sphere is tangent to the planes of the impingement cell, and the impingement is hard. IP 2 occurs simultaneously at the equivalent positions of (f, f, 0), (i, 0, f) and (0, f, f) for r = ,/?/2. The sphere is tangent to the midpoints of the edges of the impingement cell, and the impingement is soft. The third and final IP occurs at (f, f, i) for r = ,/3/2 and f = 1 and corresponds to completion of the hypothetical recrystallization reaction. As the symmetry of a simulated grain varies from the symmetry of the impingement cell, additional IPs are encountered. Thus, a sphere growing in a tetragonal impingement cell has four IPs, and a sphere growing in an orthorhombic cell has six IPs, if one ignores the final IP in each case. Impingement geometries are correlated with shapes of growth curves in the section on “Impingement Simulations.” CORRELATION
WITH EXTENDED CONCEPT
of a random distribution {equation (9) in Ref. 121) F, = F, ex- v,,,+
GEOMETRY
1379
IMPINGEMENT
v,,,-
of overlapping ... +(-l)““V,,,
spheres (9)
where V,, as defined in equation (1), is the actual volume after correcting for overlapping, and V,,, is the sum of the extended or full volume of all segments formed by m overlapping spheres. Specifically, Vi exis the sum of the full volume of all spheres in Avrami’s model, whereas in the geometric computer model, V,,, is simply the full volume of the model grain. The overlap geometry for a sphere growing in a cubic impingement cell is shown in Fig. 1; the extended-volume concept is shown schematically at the top, and the geometric model at the bottom. The model sphere is located at (O,O, 0) and labled A. Three juxtaposed spheres at (0, 1, 0), (1, 1, 0), and (1, 0,O)are labeled B, C, and D, respectively. Representative segments formed by the overlapping of spheres A, B, C, and D are individually cross-hatched in the figures. Each segment corresponds to an extended volume and can be identified with the lowercase letters at each intersection. Superscripts are used to identify the set of overlapping spheres that form each segment; Vfc, is the total volume of sphere A, ViqeXis segment ae, Vtz is segment bg, V;‘e”x” is segment bef, and If;::” is segment c&J Because of the symmetry of the model, only the extended volumes associated with sphere A need to be considered. Vmex in equation (9) is formed by summing over all segments formed by m - 1 spheres that overlap sphere A. Dividing the sum by m to eliminate redundancy yields
(10) where 2 is the coordination number of all sets of overlapping spheres that form identical segments; superscript a identifies a representative set of overlapping spheres that forms a specific segment; and the summation is over all unique sets of a. Since impingement does not occur in the computer model until IP 1, V, = V, exup to the first IP. Between IP 1 and IP 2, only segments ae form on each of the
VOLUME
Before the impingement simulations are discussed, it is desirable to formally relate the impingement geometry used in the computer model to the extended-volume concept outlined by Avrami [2]. Avrami developed a relation based on the geometry
Fig. 1. Overlap geometry for extended volume corrections (top) and geometrical corrections (bottom).
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six faces of the impingement cube, and equation (9) yields V* = V*cx-5V*B 2 2ex. (11) This relation is formally equivalent to the geometric computation which subtracts the volume of the six spherical segments corresponding to segment hj in Fig. 1 from the volume of sphere A. To retain identity with 2 and m in equation (lo), the fractions have not been reduced. Between IP 2 and IP 3, equation (9) assumes the form V, = Vf,, - f Viqex- y V$$ + y VfeF 12
+$V$:D+g,cx
To simplify shows
equation
ACD _
(12), simple
12 ABCD 4 v,,, .
superposition
Spheres in cubic and tetragonal impingement cells
tv~~=;v~~c+:v;qep+;v~~o-~v,qcD. Substituting
(12)
curves of the spheres and ellipsoids were divided into 25 equal steps up to and including the final IP. For plotting, A was normalized to the maximum area for each growth curve. Computations also were performed for f and A at the IPs, and each IP is labeled “X” on the geometric-growth curves. Also, IPs are listed in Table 1 for spheres growing in tetragonal impingement cells and in Table 2 for ellipsoids growing in a cubic cell in the order of occurrence for increasing r. The tables include the values of r, f, A, and normalized A. Since one of the objectives of this study was to compare the simulated growth curves with equation (3), equation (3) was normalized to a maximum area of A = 1 by setting K, = 4 and superimposed on some of growth curves.
equation
(13)
(13) into equation (12) yields
K = V?ex--;v;:++4v4qBo.
(14)
Equation (14) is formally equivalent to the geometric computation that subtracts the volumes of the 6 spherical segments corresponding to segment hi from the volume of sphere A and adds the 12 volumes corresponding to segment ijk, since these volumes are subtracted twice when the volumes corresponding to segment hi are subtracted. Equations (11) and (14) verify that the extended-volume concept in equation (9) and the geometric computation used in the computer model are equivalent for regular geometric distributions. IMPINGEMENT
SIMULATIONS
To provide reasonable computation times for the finite-element computations, the geometric-growth Table 1. Major c/a
IP
0.1
1 2 3 4
3 4
(0,O.c) (a, 0, c) (0, b, c)
4
0.9 2 3 4 1.0 2
1
5
3 4 1.5
10.0
(0,0,cl (0.0, 0)
2 3 4 1
2
1.1
impingementpoints for
Impingement positions (0, b, 0) (a. 0, c) (0, b, c) (a, b, 0) (0.0, c) @, 0, 0) (0, b, 0) (a, 0, c) (0, b, c) (a. b, 0) (0, 0, c) (0.0,0) (0, b, 0) (a. 0, c) (0, b, c) (a. b, 0) (a, 0, 0) (0, 6, 0) (0, 0, c) (a, b. 0) (a, 0, c) (0, b, c) (a, 0, 0) (0, b, 0) (0.0, c) (a. b, 0) (a, 0, c) (0, b, c) (a, 0, 0) (0, b, 0) (a. b, 0) (0, 0, c) (a. 0, c) (O,b, c) @,O,O) (0, b.0)
OS
2 (a,b,O)
The effect of slight variations in impingement geometry is demonstrated for spheres growing in slightly tetragonal impingement cells in Figs 2 and 3 for c/a = 0.9 and 1.1, respectively. These figures correspond to the IPs identified in Table 1. The corresponding geometric-growth curves are plotted in Fig. 4, along with the growth curve for a sphere in a cubic impingement cell. Figure 4 also includes a normalized plot of equation (3) to emphasize the asymmetric shift of the simulated growth curves from the S-F parabolic relation. This shift is a consequence of the decrease in surface curvature as r increases. The maximum in the growth curve for a sphere growing in a cubic cell occurs at IP 1, which is hard. The sphere is tangent to the six (200) planes of the impingement cell at IP 1. Impingement at IP 2, which is soft, causes only a slight break in the growth curve. Decreasing c/a to 0.9 shifts the normalized growth curve slightly above that for the cubic cell up to IP 1 because the maximum area for this geometry is less spheres in tetragonal cells r
f
A
A IA,,,
0.108 1.077 1.083 1.523 0.315 0.630 0.704 0.891 0.466 0.518 0.697 0.732 0.500 0.707 0.484 0.533 0.685 0.720 0.437 0.618 0.655 0.787 0.232 0.328 2.321 2.332
0.005 0.783 0.790 1.000 0.131 0.720 0.869 0.998 0.424 0.573 0.953 0.979 0.524 0.965 0.476 0.618 0.941 0.972 0.349 0.758 0.830 0.986 0.052 0.114 0.997 0.999
0.146 1.464 1.317 0.003 1.247 2.501 1.475 0.101 2.730 3.038 0.947 0.513 3.142 0.762 2.948 2.917 1.132 0.638 2.397 1.986 1.885 0.423 0.678 0.561 0.432 0.093
0.100 1.000 0.936 0.002 0.499 1.000 0.590 0.040 0.899
1.000 0.312 0.169 1.oOO 0.243 1.000 0.989 0.384 0.216 l.Ow 0.829 0.786 0.176
I.000
0.827 0.638 0.137
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Table 2. Maior imoinnementoointsforelliusoids with I = m = 1.0in a cubiccell
e/a
IP
10
1 2 3 4
1.5
1 (O,O,c) 2 3 4
1.1
k&O) (a,O,c)(O,b,c)
1 (a,0, 0) (0,b,0)
2 3 4 0.1
(a.O,0) CO,b,0) (a,O,c)(O,b,c) (a,b,O)
1 (a>O,O)(0,b,0) 2 (O,O,c)
3 4 0.5
(a>O,O)(0,b,0) (a,O,c)(O,b,c) k&O)
1 (O,O,c) 2 3 4
0.9
Impingementpositions (O,O,c) kO.0) CO,b,O) (a>O,c)CO,b,c) (G&O)
k&O) (O,O,c) @,O,c) (O,b,c)
1 (a,O,O)(0,b,‘3 2 3 4
(a,b,O) (O,O,c) (a,O,c) CO,& c)
than that for the cubic case (Table 1). At this point, Fig. 2(a), the sphere becomes tangent to the (002) plane of the computational cell; the impingement is hard; and a significant break occurs in the growth curve. The growth curve continues to a maximum at IP 2, Fig. 2(b), where the sphere becomes tangent to the (200) and (020) planes, again in hard impingement. This maximum shifts to largerfthan the maximum for the cubic cell. Impingements at IP 3 and IP 4 [Fig. 2(c) and (d), respectively] are soft and shift upward and downward on the growth curve,
(a) 1P 1, f = 0.424
T
/
0.050 0.500 0.502 0.707 0.333 0.500 0.601 0.707 0.455 0.500 0.676 0.707 0.500 0.556 0.707 0.747 0.500 0.707 1.000 1.118 0.500 0.707 5.OOO 5.025
0.005 0.783 0.790 1.000 0.233 0.669 0.909 0.994 0.433 0.569 0.955 0.978 0.471 0.628 0.938 0.938 0.262 0.569 0.911 0.992 0.052 0.114 0.997 l.OOO
A 0.248 3.185 2.783 0.005 1.883 3.014 1.424 0.235 2.765 3.070 0.942 0.535 2.939 2.925 1.185 0.635 2.170 2.180 2.054 0.447 1.162 2.008 2.000 0.429
0.078 l.OOO 0.874 0.002 0.625 1.000 0.472 0.078 0.901 I.000 0.307 0.174
I .ooo
0.995 0.403 0.216 0.951 0.955 0.900 0.196 0.579 l.OOO 0.996 0.214
respectively, from the corresponding soft IP 2 for the cubic cell. Increasing c/n to 1.1 shifts the normalized growth curve even higher. The maximum in the growth curve occurs at IP 1 and shifts to smaller f than that for the cubic impingement cell. At this point, Fig. 3(a), the sphere is tangent to the (200) and (020) planes in hard impingement. The curve remains relatively flat to IP 2, Fig. 3(b), where the sphere becomes tangent to the (002) planes of the impingement cell, then the growth curve breaks sharply downward. Again, the
(b) IP 2.f = 0.573
(al IP 1, f=
(cl 1P 3, f= 0.953
Al&,x
(d)
IP
0.476
(b) IP 2,f = 0.616
4, f=o.wg
Fig. 2. Impingement geometry of a sphere in a tetragonal cell with c/u = 0.9; IP is the impingement point number, and f the volume fraction.
(c)IP 3.f = 0.941
(d) IP 4,f= 0.972
Fig. 3. Impingement geometry of a sphere in a tetragonal cell with c/a = 1.1.
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.: : :
E
:’
1
:
P
.’
0.
0
g
SIMULATIONS OF GRAIN IMPINGEMENT
‘..8
x
9
“...,g
0
B
“....,,. 8 -
0
.E
5 c
;j
;
0.5 -
G 2
-
E
jj .il
-
B $
2
..*..,J -
q
0
Tang~nal
Major impingsment point
+
.I,” -
cubic cell, c/a = 1.0
x
’
:, m
cell, cla = 1.1
A .... 1
Tetqonal
:,a_
cell, da = 0.9
+
t!
Eauation (3) ’
0
0.0
’
0.5
0
1
1
’
1.0
c/a
Volume fraction
Fig. 4. Normalized geometric-growth curves for a sphere in a cubic cell and in tetragonal cells with c/a = 0.9 and 1.1.
Fig. 6. Variation of the maximum area of impinging spheres in tetragonal cells as a function of c/a.
soft IPs in Fig. 3(c) and (d) are shifted upward and downward on the growth curve, respectively, from the corresponding soft IP2 for the cubic cell. Both normalized growth curves for the tetragonal cells in Fig. 4 expand outward from that for the cubic cell. Changes in the growth curves for larger variations in c/a are shown in Fig. 5 for c/a ratios of 0.1, 0.5, 1.5, and 10. Correlation of these curves with the impingement simulations and the data in Table 1 reveals that the growth-curve maximum is dictated by the IP at (a, 0,O) and (0, b, 0), regardless of its order of occurrence. As c/a increases from c/a = 1, the maximum in the growth curve continues to occur at the first IP, but shifts to lower J The impingement also becomes increasingly hard, causing a more severe downward break in the growth curve. The soft IP at (a, b,O) eventually shifts from the third to the second IP, but hard impingement continues to occur at (0,0,c), and this IP continues to shift to higherf. As c/u decreases from c/u = 1, the maximum in the
growth curve continues to occur at (a, 0,O) and (0, b, 0) and shifts to slightly larger J: The absolute magnitude of the maximum area in a growth curve, A,,, changes significantly as the impingement geometry changes. The change in A,,, is not reflected in the normalized plots in Figs 4 and 5, but it is apparent in the data recorded in Table 1 and Fig. 6, in which A,,, is plotted as a function of c/u for spheres growing in tetragonal cells. Ellipsoids in a cubic impingement cell
Growth curves for ellipsoids growing in cubic cells are roughly similar to growth curves for spheres growing in tetragonal cells when n -I for the ellipsoid equals c/u for the tetragonal cell. This is demonstrated in Fig. 7 for ellipsoids with n = 10, 1.5, 0.5, and 0.1 and with 1 and m fixed at 1.0 in equation (6). The impingement data recorded in Table 2 include data for n = 0.9 and 1.1 for comparison with similar c/u ratios in Table 1. Impingement geometries for 1.0
0
L v
m
0
A”~oocm 0
0
0
0=
n
n a
0
0
A A
0.0 0.0
0
’
’
8
’
0.5
8
’
’
0
0.0 1.0
Volume fraction
Fig. 5. Normalized geometric-growth curves for a sphere in tetragonal cells with c/a = 10, 1.5, 0.5, and 0.1.
I
q
Ellipsoid, n =
0
Ellipsoid, n =
0
Ellipsoid, n =
A
Elliiid,
X
Major impingement point
I
9
I
n = 10
’
0.5
0
’
I
I
1.0
Volume fraction
Fig. 7. Normalized geometric-growth curves for ellipsoids withI=1.0,m=1.0,andn=0.1,0.5,1.5,and10inacubic cell.
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SIMULATIONS
C
b
IP l,f=0.262
f = 0.411
OF GRAIN IMPINGEMENT
IP2,
1383
e
d
f=O.569
IP 3, f=0.911
IP4,
f=0.992
Fig. 8. Impingement of an ellipsoid with I = 1.0, M = 1.0, and n = 0.5 in a cubic cell.
a
C
d
f = 0.496
IP 3, f = 0.997
b
IP 1, f =0.052
IP2,f=0.114
e IP4,
fal.O
Fig. 9. Impingement of an ellipsoid with I = 1.0, m = 1.0, and n = 0.1 in a cubic cell.
n = 0.5, 0.1, and 10 are shown in Figs 8, 9, and 10, respectively. As n decreases from n = 1, the ellipsoid flattens in the z direction, as shown in Figs 8 and 9. IP 1 occurs at the midpoints of the (200) and (020) planes, Figs 8(a) and 9(a). The impingement is hard, but as n decreases and the ellipsoid becomes more oblate, the curvature increases at the intersection of the z = 0 plane, as is evident in Fig. 9(a), and the hardness decreases. Initially, the maximum in the growth curve occurs at IP 1, and the curve is relatively flat between the IP 1 and IP 2, both of which are hard. As n continues to decrease, (a, b, 0) becomes the second IP, and the maximum shifts to a position between the first and second IPs. This is evident in Fig. 7 for n = 0.5; Fig. 8(b) shows the corresponding impingement geometry at the maximum. At this point, the surface of the ellipsoid within the cell has maximum curvature. As r increases beyond this point, the surface flattens and A decreases. Because of the flatness of the ellipsoid within the impingement cell, very little change occurs in A between IP 2 and IP 3 in Fig. 8(c) and (d), respectively. IP 3 is hard as the ellipsoid becomes tangent to the (002) planes of the impingement cell. As r increases through IP 3, A decreases rapidly through IP 4 [Fig. 8(e)]. With further decrease in n, the maximum in the growth curve occurs at or slightly after IP 2. This is evident in Fig. 7 for the n = 0. I curve. Since the ellipsoid is relatively flat, and the curvature in the z direction remains fairly constant with further increase in r, the growth curve remains relatively flat between IP 2 and IP 3. The
growth curve then decreases rather rapidly through IP4 (Fig. 7). As n increases from n = 1 and the ellipsoid becomes prolate, IP 1 occurs at (0,0,c), and IP 2 occurs at (a, 0,O) and (0, b, 0). Both impingements are initially hard, but as n continues to increase, the curvature of the prolate ellipsoid increases at its intersection with the z axis, and IP 1 transforms to soft
C
IP 3,
f = 0.790
Fig. 10. Impingement of an ellipsoid with I = 1.0, m = 1.0, and n = 10 in a cubic cell.
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OF GRAIN IMPINGEMENT (a, b, 0), (a, 0, c), and
0.0:
0.0
I
I
-
{ 1111 Bipyramid
‘...
Equation
’
I
’
(3)
’
0.5 Volume
I
I
’ 1.0
fraction
Fig. 15. Normalized geometric-growth curve of a {Ill} bipyramid in an orthorhombic cell with m/l = b/a = 0.75 and n/l = c/a = 0.25. Bipyramids Geometric growth curves for bipyramids show many similarities to those for spheres and ellipsoids. They are equally sensitive to variations in symmetry, and a single normalized curve results when the axial-
symmetry constraint is imposed. The normalized geometric-growth curve for a {ill} bipyramid in an orthorhombic cell with m/l = b/a = 0.75 and n/l = c/a = 0.25 is shown in Fig. 15. These are the same axial constraints used for the ellipsoid in Fig. 12 and the corresponding geometric-growth curve in Fig. 13. The impingement geometry, Fig. 16, also is similar to that for the ellipsoid in Fig. 12. IP 1, Fig. 16(a), occurs simultaneously at (a, 0, 0), (0, b, 0), and (O,O, c); IP 2, Fig. 16(c), occurs simultaneously at
(0, b, c). In contrast to the ellipsoid, however, both IPs are soft. This is a consequence of effectively infinite curvature at the apexes. A,,, occurs when the {111} face of the bipyramid passes through the center of the computation cell. At this position, Fig. 16(b), the bipyramid intersects the midpoints of the edges of the computation cell. This occurs at f = 0.5 for all axial symmetries. The growth curve is exactly symmetric about this point because the center of the computation cell is a point of inversion symmetry for planes of a given orientation that are equidistant from the center. When the symmetry varies from the axial-symmetry constraint, the geometric-growth curve remains symmetric aboutf = 0.5, but the curve distorts either inward or outward, depending on the relative axial symmetries. The geometric-growth curve for the (1 IO} bipyramid in an orthorhombic cell with m/l = b/a = 0.75 and n/l = c/a = 0.25 (Fig. 17) has only one IP other than the final IP. It occurs when the edges of the bipyramid in the x-y plane become tangent to the (200) and (020) planes, and the apexes become tangent to the (002) planes. Again, the impingement of the apexes is soft, but when the edges become tangent to the impingement planes, the impingement is hard. This produces an asymmetric growth curve with IP 1 at the maximum in the growth curve. KINETIC SIMULATIONS Johnson-Mehl-Avrami
simulations
Simulated J-M-A plots were generated from the geometric impingement data by relating r in equations (5)-(g) to a linear growth rate and assuming a
Ib!
f = 0.167
b)
t = 0.5
IP 2 f = 0.833
Fig. 16. Impingement of a {111)bipyramid in an orthormbic cell conforming to the symmetry constraint, m/l = b/a = 0.75 and n/l = c/a = 0.25. A.M.35,6-M
1386
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OF GRAIN
IMPINGEMENT
O-
z
*; -l-
C > r
-2 -
+ + +
0” J -3 -
- - -
(111)
Bipyramid
Extended
volume
Corrected
volume
/
.....
0.0
0.0
’
1
’
Equation
’
(3)
’
’
’
’
’
0.5 Volume
Fig. 19. Simulated J-M-A plots for the {11l} bipyramid growing in a cubic impingement cell, the corresponding extended volume (V,,) and the corrected volume (V,).
1.0
fraction
Fig. 17. Normalized geometric-growth curve of a (100) bipyramid in an orthorhombic cell with m/l = b/a = 0.75 and n/l = c/a = 0.25.
total reaction time of 2500 s. The J-M-A relation in equation (2) should be linear with a slope of n for plots of log (ln[l/(l -f)]} against log t. Figure 18 is the resultant J-M-A plot for the sphere growing in a cubic impingement cell. The dashed line drawn through the first and fifth points indicates linear correspondence to equation (2). It has a slope of 3.0, which is the slope predicted by equations (1) and (2) for instantaneous nucleation. The simulated data in Fig. 18 coincide with the dashed line up to about f = 0.1. This is the extent of the apparent limited correspondence of J-M-A recrystallization plots to linear behavior [8]. At higher values of f, the simulated kinetic curve deviates slightly upward. However, this deviation is both significantly less and in the direction opposite to the severe deviation consistently displayed in experimental J-M-A plots for f > 0.1. Kinetic curves generated for both the {111) and the {1lo} bipyramids growing in cubic impingement cells were nearly identical to that of the sphere but with slightly less upward deviation. >,I
!
I I ,1111,
I
1-
+ ,’ /
o/
=
’
T
-lc
: r
III1
/
/
/
0.9999 0.99
-
0.90 0.59
_
10-l
-
10-Z
J’
k 2
-
/’
0 1 1 ‘1f11’
-4-“# 102
! 3
f’
-3 -
5
'g E
/*
_*-
-
,
103
10-a
Since equations (11) and (14) verify equivalency of the computer model and the extended-volume concept, kinetic simulations based on the extendedvolume concept were compared with the simulated geometric data. Both V,, = V, cI:and V, obtained from V,, using equation (1) are plotted in Fig. 19 using the same time scale converted from the size parameter, r, for the {111) bipyramid growing in a cubic cell. As expected, the curve for I’, coincides identically with the simulated curve up to IP 1 and then rises above the simulated curve. In all cases conforming to the axial-symmetry constraint, the corresponding V, plot coincides with the latter extrapolation of the first and fifth simulated points (within plotting accuracy) and n = 3, as predicted by equations (1) and (2). The positive deviation of the geometric data from V, demonstrates that equation (1) overcompensates for impingement of the modeled geometric grains. For the sphere growing in a cubic impingement cell, this overcompensation reaches a maximum of about 26% atf= 0.7, but it is only 6.6% atf= 1.0. These values are somewhat less for both bipyramids. When the axial symmetry was constrained to m/l = b/a and n/l = c/a, the simulated kinetic plots were identical to the appropriate simulated shape growing in a cubic cell. As axial symmetries vary from the axial-symmetry constraint, significant deviation occurs from linearity, as shown in Fig. 20 for a sphere growing in a tetragonal impingement cell with c/a = 10. The corrected volume is plotted as the dashed line in Fig. 20. This demonstrates that the kinetic curves are sensitive to the shape of the geometric-growth curves, and it also indicates that variations from the axial-symmetry constraint are not realistic impingement geometries for recrystallization. Speich-Fisher
I III, 104
Time (SW)
Fig. 18. Simulated Johnson-Mehl-Avrami (J-M-A) plot for the sphere in a cubic impingement cell using a linear growth rate.
simulations
The English-Backofen inverse time-dependent growth rate [6] used in the S-F relation was G = K,/t
W)
where G is the growth rate, K, is essentially a constant
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SIMULATIONS
OF GRAIN IMPINGEMENT
1387
negative values of r,, yield incubation times greater than 1.0 s. However, this interpretation must be /’ tempered by the empirical nature of equation (15) /’ ,’ oand the fact that it does not hold for the early stages = , - 8% r, 8 /;+++ of recrystallization. 5 _ 10-l .i The deviation from linearity in Fig. 21 occurs i= -1 ,+* r: * because A and fin equation (3) are not geometrically c ,,+' m -2 , related through r. To analyze the strong divergence + + + Sphere in tetragonalIO-* f s cell with c/a = 10 4’ at both extremes of the plot in Fig. 21, A from the z - -Corrected volume - 10-3 -3 simulated data was converted to a new volume fraction using equation (3). The divergence was reI I III I b I11111 -4J”I duced but was not corrected to linearity, since t was 104 103 102 related to the geometric r. This analysis demonstrates Time (set) that the breakdown of equation (3) at f= 0 and f= 1.0, which actually is a manifestation of the Fig. 20. Simulated J-M-A plot for a sphere in a tetragonal impingement cell with c/a = 10. nongeometric relation off and A ‘in equation (3), is the source of the divergence. In fact, at f = 1.0, r,, in equation (16) specifies that the plot in Fig. 21 is asymptotic to t = 87 s at r = 0, while at f = 1.0, that English and Backofen found to be independent f/(1 -f) becomes infinite, and the plot is asymptotic of temperature and strain except during the initial However, this strong divstages of growth, and t is the time. Relating r to t by to t = 2876 s at r = $/2. assuming dr = G dt and integrating from limits of to ergence is accentuated in the kinetic plots by the disproportionate expansion of f at both extremes, to t and r,, to r and it occurs outside of the useful range stipulated for log(t) = (r - r,)/K,. (16) equation (4). In this form, r,, is effectively the value of r at to = 1.0. Modified Johnson-Mehl-Avrami simulation The S-F relation in equation (4) should be linear Since these simulations indicate that a major limiwith a slope of n in plots of log[f/(l -f)] against log t. For S-F simulations using the data for the tation of the J-M-A relation is the assumption of sphere in a cubic impingement cell, equation (16) was linear growth, the relation was modified using the fit to a minimum time of 100 s and a maximum time inverse time-dependent growth-rate in equation (15) of 2500 s with values of r,, = - 1.11 and KB = 0.247 in and the relation of r to t in equation (16). Assuming the dimensionless geometry of the model. The re- instantaneous nucleation in conformance with the computer model, the number of active nuclei (n,) is sulting S-F plot is shown in Fig. 21. To indicate linear behaviour corresponding to equation (4), the constant, and V,, assumes the form dashed line in Fig. 21 was constructed through Points I’,, = XsK:[ln(t) + r,,/K,13 (17) 5 and 20. These points correspond to the extremes where x is a geometric factor. Using B = xniK: and suggested by Speich and Fisher of f = 0.02 and f = 0.95. The simulated data are reasonably close to substituting equation (18) into equation (1) modifies relation for inverse time-dependent linear behavior within this range and yield values of the J-M-A growth rate to n = 2.11 and K = 1.3 x 10m6 in equation (4). These values are in reasonable agreement with experimental 1 -f= exp{ -B[ln(t) - r,,/KE]‘}. (18) data for vanadium at 1173 K [8]. Equation (16) implies that a finite incubation period does exist, and Under these assumptions, plots of log ln[ (1 -f )-‘I vs log [In(t) + r,,/K,] were nearly identical to the plot in Fig. 18 for linear growth. The initial linear portion 5 of each plot had a slope of 3.0, as predicted by 0.9999 4 equation (19). When experimental data for vanadium 0.999 3 E 2 ’ - 0.99 .g [8] were fit to equation (18), reasonably linear plots were obtained, but the slope varied with temperature. - 0.90 E 1 : This indicates that r, is a function of temperature, - 0.50 E z 0 10-l 2 0” -1 and equation (18) is therefore not considered to be J 10-2 8 -2 practical for recrystallization kinetics. ;/‘+ 10-S -3 The J-M-A relation was modified using other _+ 10-d -4 time-dependent relations such as the t -2 dependence I I I I I11111 I I I Illll_ -5 suggested by Li [13] for recovery reactions. Each 103 102 104 modification yielded a reasonably linear plot with a Time bed slope of 3 and a fit essentially identical to that in Fig. Fig. 21. Simulated Speich-Fisher (S-F) plot for the sphere growing in a cubic impingement c& the growth rate is 18, but the experimental vanadium data did not inversely proportional to time. conform to these plots. This demonstrates that the 7,
l-
I
I,
,
I
f,“”
1 ,b
11
1””
-
-
+
&
#
-
0.9999 0.99
1388
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SIMULATIONS
computer model is useful for evaluating the geometric basis for kinetic models, but the temporal basis requires careful correlation with experimental data. DISCUSSION
A distinct advantage of the computer model used in this study is that it produced well-defined impingement geometries for various shapes of simulated geometric grains. Consequently, the model permitted impingement geometries to be studied in sufficient detail to determine how grain shape and symmetry affect impingement geometry and the shape of the geometric-growth curves. Since the computer model is an obvious oversimpli~cation of actual grain geometries, no attempt was made to exhaustively explore all relevant geometries pertaining to recrystallization. However, grain geometries that appear to be most relevant to recrystallization reactions in real materials conform to the axial-symmetry constraint of m/l = b/a and n/l = c/a.? These geometries yield a common normalized geometric-growth curve that is in reasonable agreement with the experimental S-F data forfand A. Furthermore, these geometries yield kinetic simulations that are in reasonable agreement with equations (2) and (4), whereas nonconforming simulated grain geometries do not. The importance of the axial-symmet~ constraint is that the S-F relation in equation (3) appears to be a generalized form for the growth of clusters of equiaxed recrystallized grains. The grains that nucleate first during recrystallization have a nearly isotropic shape 16, 141. In highly deformed materials, these grains usually nucleate in clusters along grain boundaries or dense deformation bands [6, W-161, and impingement and coalescence of these grains occur at an early stage. This initial impingement effectively constitutes hard impingement and should cause a break or discontinuity in the growth curve. Indeed, such a di~ontin~ty appears to exist in the SF experimental curve at approximately f = 0.1, although a more extensive break appears to occur in the S-F data betweenf = 0.8 and f = 0.9 that is not explained by this model. After this early impingement within clusters, each cluster surface should conform to the geometric shape dictated by the axialsymmetry constraint. The impingement geometry would then be governed by this cluster shape, and equation (3) would apply to the growth and impingement of the clusters. In fact, this cluster model suggests that nucleation is implicitly treated by equation (3). In lightly deformed materials, grains usually nucleate at grain corners [6, 14,151 without forming clusters, and equation (3) would apply to the growth and impingement of individual grains. In real materials, the axial-symmetry constraint is ?Possible exceptions are materials such as oxide-dispersionstrengthened metals in which the dispersions constrain grain-boundary movement.
OF GRAIN IMPINGEMENT
related to the deformation geometry. An important aspect of the axial-symmet~ constraint and the ~~s~lli~-cluster concept is that the cluster-surface area per unit volume increases significantly as deformation increases. This is shown for the tetragonal case in Fig. 14. In fact, the c/a ratio in Fig. 14 correlates with cross rolling for c/a < 1 and extrusion or wire drawing for c/a > 1. Since the increase in surface area indicated in Fig. 14 is substantial, the Cahn-Hagel concept implies that this should be a significant contribution to the fact that recrystallization rates increase with the amount of deformation. A similar contribution would be expected from the initial grain size, since the cluster density and surface area at a given rec~stalliz~ volume fraction increase substantially as the grain size decreases. The extent of these contributions will require careful analysis of experimental data and is beyond the scope of this paper. Geometric-growth curves of regular geometric shapes could be a closer match to the ex~~mental S-F curve if the simulated grains have a compound geometric shape similar to the curved polyhedral grain faces treated previously by Cahn [9]. The S-F experimental growth curve is skewed to the right in their plot of A vs $ Simulated geometric-growth curves for spherical and ellipsoidal grains conforming to the axial-symmetry constraint are skewed even further to the right because of the decrease in surface curvature with increasing r. In contrast, growth curves for {111) bipyramids conforming to the constraint are symmetric about f = 0.5. This suggests that a compound geometric shape composed of a regular polyhedron with faces that have either spherical or ellipsoidal curvature should provide a more realistic geometric-growth curve. This compound geometric shape is a more realistic approximation of real grain structures than are spherical and ellipsoidal shapes. Impingement geometry in real materials is affected by the distribution of grain sizes. The distribution of sizes should smear out breaks in the growth curve associated with IPs, and soft IPs should disappear. Since most materials have a relatively uniform distribution of recrystallized grain sizes, however, evidence of hard impingement should remain. This is particularly true for geometries conforming to the axialsymmetry constraint with the single hard IP, and it also agrees with the general shape of the S-F experimental growth curve. Recent work [ 171indicates that recrystallization in dispersion-strengthened metals may comply with the linear-growth kinetics of the J-M-A model. However, the kinetic simulations using the J-M-A relation support the previous contention [8, lo] that the major limitation of the J-M-A model for recrystallization in most other materials is the use of linear-growth kinetics. Correiating the computer model with the extends-vol~e concept used in the J-M-A relation demonstrates that the extended-
PRICE:
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OF GRAIN IMPINGEMENT
volume concept does provide reasonable compensation for grain impingement. However, modification of the J-M-A relation using l/t or other timedependent growth kinetics does not appear to yield practical relations. Kinetic simulations using the S-F relation were in reasonable agreement with equation (4) in the range between f = 0.02 and f = 0.95. In practice, the useful range can be extended from f= 0.01 to f= 0.99 [S] before the breakdown of equation (3) becomes significant. The simulated data do show noticeable deviation from the linear behavior predicted by equation (4) within this range. This deviation is a consequence of the fact thatfand A in equation (3) are not geometrically related. However, the absolute magnitude of this deviation is roughly of the same order of magnitude as the scatter of experimental data, and this deviation should not detract from the use of the S-F relation for practical applications within the stipulated range. Rigorous mathematical treatment of grain impingement will require different mathematical expressions between each set of IPs, but this will result in a complex set of expressions that is unnecessarily cumbersome for modeling continuous kinetic curves. Such complexity is not justified in view of the as-yet limited understanding of atomistic mechanisms in recrystallization reactions. Therefore, both the extended-volume concept in equation (1) and the empirical S-F relation in equation (3) appear to be the best available geometric relations for impingement compensation in deriving kinetic relations. However, using equation (1) in the J-M-A model invokes the apparent anomaly of using only volumetric terms to model a surface reaction, as previously discussed by Gokhale et al. [ 181.On the other hand, the S-F relation in equation (3) suffers from the fact that f and A are only empirically and not geometrically related. However, major advantages of equation (3) are that (a) it is compatible with the Cahn-Hagel [7] concept of surface controlled reactions, and (b) the cluster model discussed above indicates that nucleation is implicit in the empirical model. In contrast, nucleation is not considered in equation (1) and must be treated explicitly in equation (2), which is another limitation of the J-M-A model in view of the limited understanding of nucleation mechanisms. The simplified computer model has clarified some important aspects of grain impingement and recrystallization kinetics, and it should remain useful for analysis of new kinetic models as they are developed. Unfortunately, the computer model is necessarily restricted to instantaneous nucleation, but variable nucleation rates undoubtedly are important in recystallization reactions, as reviewed by Hu [19]. Other computer models that generate more-realistic grain structures [20,21] should accommodate variable nucleation rates, and these models also should be considered for future recrystallization simulations.
SUMMARY
1389
AND CONCLUSIONS
The simplified computer model, which is based on the growth of equisized, regular, geometric grains in a cubic distribution, provides surprisingly reasonable simulations of impingement geometry and recrystallization kinetics. Impingement geometry in the computer model is dictated by the hard and soft nature of the major impingement points, the relative geometries of the grains and the distribution lattice, and the surface curvature of the grain. The impingement geometry defines the shape of the geometric-growth curve. Grain geometries that conform to the axial-symmetry constraint yield impingement geometries that are expected to be most relevant to recrystallization. They also yield geometric-growth curves that reasonably approximate the experimental growth curve presented by Speich and Fisher. In real materials, major impingement points should smear out to some extent, but the general shape of the geometric-growth curve should be retained. When growth rates are related to the radius or size parameter of the grains, geometric-growth curves define simulated, kinetic, recrystallization plots based on time dependence and plotting parameters associated with specific kinetic models. The computer model demonstrates that the Johnson-Mehl-Avrami extended-volume concept does provide reasonable compensation for grain impingement. With the possible exception of dispersionstrengthened metals, the major limitations of the J-M-A relation for recrystallization kinetics is the assumption of linear growth; the fact that the relation is based on volumetric terms also appears to be a limitation, since recrystallization is effectively a surface reaction. The major limitation of the Speich-Fisher relation is that it has an empirical basis in which f and A are not geometrically related. The absolute magnitude of errors generated from this limitation is not large, however, and the relation is useful for practical applications within the range off = 0.01 to f = 0.99. The breakdown of equation (3) becomes significant only outside
of this range.
Acknowledgements-This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract No. W-7405ENG-48. The author gratefully acknowledges the editorial assistance of J. M. Bruner; the assistance and cooperation of W. L. Barmore, who provided the HewlettPackard 9845B computer and plotting system on which this work was performed; and the advice offered by R. W. Cahn.
REFERENCES 1. W. A. Johnson and R. F. Mehl, Trans. Am. Inst. Min. Engrs 135, 416 (1939). 2. M. Avrami, J. them. Phys. 7, 103 (1939). 3. 4.
M. Avrami, J. them. Phys. 8, 212 (1940). M. Avrami, J. them. Phys. 9, 212 (1941).
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5. G. R. Speich and R. M. Fisher, in Recrystailization, Grain Growth and Textures (edited by H. Margolin), p. 563. Am. Sot. Metals, Metals Park, Ohio (1966). 6. A. T. English and W. A. Backofen, Trans. metall. Sot. A.I.M.E. 230, 396 (1964). 7. J. W. Cahn and W. Hagel, in Decomposition of Austenite by D@iisional Processes (edited by V. F. Zackay and H. I. Aaronson). D. 131. Interscience. New York (1960). 8. C. W. Price, Scripta metaii. 19, 669 (1985). . ’ 9. J. W. Cahn, Trans. metall. Sot. A.Z.M.E. 239, 610 (1967). 10. C. W. Price, Scripta metall. 19, 785 (1985). 11. C. Kittel, Introduction to Solid State Physics, 3rd edn, p. 12. Wiley, New York (1968). 12. J. W. Christian, The Theory of Transformation in Metals and Alloys, p. 442. Pergamon, Oxford (1965). 13. J. C. M. Li, in Recrystallization, Grain Growth and Textures (edited by H. Margolin), p. 45. Am. Sot. Metals, Metals Park, Ohio (1966). 14. P. K. Ambalal and C. W. Price, in Microstructurd I,.
IS. 16.
17. 18. 19.
20.
21.
Science, Vol. 13, (edited by S. A. Sheils, C. Bagnall, R. E. Witkowski and G. F. Vander Voort), p. 313. Am. Sot. Metals, Metals Park, Ohio (1986). _ J. W. Cahn. Acta metall. 4. 449 (1956). R. A. Vandermeer and P.‘Gordon, in Recovery and Recrystallization of Metals (edited by L. Himmel), p. 211. Interscience, New York (1963). I. Beden, Scripta metall. 20, 1 (1986). A. M. Gokhale, C. V. Iswaran and R. T. DeHoff, Metali. Trans. lOA, 1239 (1980). H. Hu, in Metallurgical Treatises (edited by J. K. Tien and J. F. Elliott), p. 385. T.M.S.-A.I.M.E., Warrendale, Pa (1981). K. W. Mahin, in Proc. of the 1976 Internat. Conf on Computer Simulation for Materials Applications (edited by R. J. Arsenault, J. R. Beeler Jr and J. A. Simmons), p. 39. National Bureau of Standards, Gaithersburg, Md (1976). M. P. Anderson, D. J. Srolovitx, G. S. Grest and P. S. Sahni, Acta metall. 32, 783 (1984).
Acrn metal/. Vol. 35, No. 6, pp. 1377-1390,1987 Printed in Great Britain. All rights reserved
SIMULATIONS OF GRAIN IMPINGEMENT RECRYSTALLIZATION KINETICS
AND
c. w. PRICE Chemistry and Materials Science Department, Lawrence Livermore National Laboratory, University of California, Livermore, CA 94.550, U.S.A. (Receiued 31 March 1986) Abstract-A simplified computer model based on regular distributions of geometric grains was used to simulate grain impingement that occurs during recrystallization reactions. The model assumes instantaneous nucleation with all grains growing at the same rate. Impingement geometry was studied as a function of grain shape, grain symmetry, and distribution geometry. Kinetic recrystallization plots were generated from the geometric-impingement data by relating the radius or size parameter of the grain to an appropriate time scale. These kinetic simulations permitted analysis of existing kinetic models and isolated some critical limitations of the models. R&sum&-Nous avons utilise un modtle simplifie, base sur des distributions reguliers de grains geombtriques, pour simuler la rencontre des grains qui se produit au tours des reactions de recristallisation. Le. modele suppose un germination instantanee ou tous les grains grossissent a la meme vitesse. Nous avons itudie la geomttrie de la rencontre en fonction de la forme et de la symttrie du grain ainsi que de la geometric de la distribution. Nous avons produit des diagrammes de recristallisation cinttique a partir des donntes geomitriques de la rencontre, en reliant le rayon ou le parametre de taille du grain a une &chelle des temps approprite. Ces simulations de cinetique ont permis l’analyse des mod&es cinttiques existants et ils ont mis en evidence quelques limitations critiques de ces modeles. Zusammenfassung-Mit einem vereinfachten Computermodell, welches auf einer regelmigigen Verteilung der Kiimer wahrend der Regeometrischer Kijmer aufbaut, wurde das Aufeinandertreffen kristallisationsreaktion simuliert. Das Model1 nimmt instantane Keimbildung an, wobei alle Khmer mit derselben Rate wachsen. Die Geometrie des Aufeinandertreffens wurde in Abhlngigkeit von der Komform, der Komsymmetrie und der Verteilungssymmetrie untersucht. Kinetische Rekristallisationsdiagramme wurden aus diesen Daten hergestellt, indem der Radius oder der Griigenparameter des Kornes auf eine geeignete Zeiskala bezogen wurde. Diese Simulierung der Kinetik erlaubte, die vorhandenen kinetischen Modelle zu analysieren und einige kritische Einschriinkungen der Modelle aufzufinden.
INTRODUCTION
Mathematical treatments of recystallization kinetics must compensate for grain impingement that occurs during recrystallization. As recrystallized grains grow and impinge on neighboring grains, the impingement restricts continued growth of the grains and retards the reaction rate as measured by the volume fraction of recystallized material that forms. The impingement geometry is dictated by both the shapes of recrystallized grains and the distribution of nucleation sites. Since deformation is a necessary precursor for both nucleation of recrystallized grains and sustained growth during recrystallization, the distribution of nucleation sites is strongly dependent upon both the deformed grain geometry and the defect structure induced by deformation. Consequently, impingement geometry is quite complex. One of the first formal treatments of impingement geometry was presented independently by Johnson and Mehl [1] and Avrami [2-4]. They compensated for grain impingement by using an extended-volume concept based on the unrestricted growth of a ran-
dom distribution of spheres. The extended volume was defined as the total volume of the spheres with no correction for overlapping. The expression for the actual volume, V,, at a reaction time of t = r was developed by Avrami [3] in terms of the extended volume, V,, : V, = 1 - exp( - V,,).
(1)
This concept along with an assumption of lineargrowth kinetics yielded the well-known kinetic relation f = 1 - exp(Bt”)
(2)
where f is the volume fraction of recrystallized material, and B and n are constants for isothermal reactions. A more recent treatment of recrystallization kinetics by Speich and Fisher [5] was based on an empirical relation betweenfand the interfacial area, A, between recrystallized and unrecrystallized material A = U-(1
-f)
(3)
where K, is a constant. Extensive experimental mea-
1377
1378
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SIMULATIONS
OF GRAIN
surements offand A performed by Speich and Fisher conformed reasonably well to equation (3). Use of equation (3), an empirical relation suggested by English and Backofen [6] for inverse time-dependent recrystallization growth rates, and the Cahn-Hagel concept of surface-controlled growth [7] yielded the kinetic relation f/(1 -f)
= Kt”
(4)
where K is a constant for isothermal reactions. Limitations of both models were discussed recently [8]. Formulation of the Johnson-Mehl-Avrami (J-M-A) model on linear growth rates restricts equation (2) to only the early stages of recrystallization, but numeric analysis indicates that even this limited agreement is questionable because of the insensitivity of logarithmic plots at small values off. In contrast, the Speich-Fisher (S-F) model in equation (4) shows reasonable agreement with experimental kinetic data over a wide range off (from f = 0.02 to f = 0.95 in their work). However, Cahn [9] used stereological relations to show that equation (3) was not rigorously correct at either extreme off. To further explore the significance and the limitations of equation (3), a simplified computer model was developed [lo] to study grain impingement. The model was based on the growth of a regular, cubic distribution of spheres with equal radii. Although the model is an obvious oversimplification of actual grain structures, it provides rigorous and tractable calculations of f and A along with precisely defined impingement geometry. Computations of f and A produced by the model were in reasonable agreement with both equation (3) and the experimental data generated by Speich and Fisher. The computer model also supported Cahn’s criticism of equation (3) by showing that the slopes of the simulated growth curves are infinite at f = 0 and f = 1 in the model but are & and -&, respectively, for equation (3). This paper formally relates the computer model to the extended-volume concept, extends the model to the impingement of other geometric shapes, and uses the model to simulate recrystallization kinetics. Important objectives of this paper are to relate impingement geometry to recrystallization kinetics and to isolate limitations of the kinetic models represented by equations (2) and (4). DESCRIPTION OF THE COMPUTER MODEL The computer model [lo] consists of a regular distribution of spheres of radius r on an infinite, simple cubic lattice and corresponds to a distribution of hypothetical nucleation sites on a regular, infinite lattice with instantaneous nucleation. For this study, it was extended to other geometric shapes and lattice geometries. In the model, all nuclei are assumed to grow at the same rate, with growth ceasing at the point of impingement. The symmetry of the distribution lattice permitted the model to be based on
IMPINGEMENT
computations offand A for a single, geometric grain growing in a cell that conformed to the geometry of the distribution lattice. The cell was constructed as a Wigner-Seitz primitive cell [ll] with (200) planes. The lengths of the cell axes were 2a, 2b, and 2c in the x, y, and z directions, respectively. The axial lengths were adjusted to yield a cell of unit volume. The faces of this cell coincide with the planes of impingement of the sphere with the juxtaposed spheres; hence, the unit-volume cell is termed the impingement cell. The symmetry of the orthogonal impingement cell permitted computations to be performed in the positive octant of the impingement cell; this is termed the computation ceN. The origin of the computation cell was at (0,0,0),and the axes of the cell extended to (a, O,O), (0, b, 0), and (O,O, c). The simulated geometric grain was permitted to grow beyond the faces of the cell, but the volume and surface area of the grain were computed only for the portion of the grain contained within the computation cell. Grain shapes simulated were spheres, ellipsoids, and bipyramids. Bipyramids were constructed with either { 111) or (110) planes. The {ill} bipyramid was simulated with a single (111) plane in the computation cell. The (110) bipyramid required intersecting (101) and (011) planes in the computation cell. The following equations were used to form the geometric grains Spheresx*+y*+z*=r*
(5)
Ellipsoids(x/Z)* + (y/m)* + (z/n)* = r*
(6)
{111) Bipyramidsx/l+y/m+z/n=r
(7)
{1IO} BipyramidsPlane l-x/l
+ z/n = r
(84
Plane 2-y/m
+ z/n = r
(8b)
In equations (5X8), r is the radius or size parameter of the geometric grain. In equations (6X8), I, m, and n determine the relative axial intercepts of the ellipsoid or bipyramid and establish the axial symmetry of the simulated grain. To simplify computations off and A for spheres and ellipsoids, the cell was divided into finite elements extending from the x-y plane of the computation cell, as described previously [lo]. For spheres and ellipsoids, reasonable plotting accuracy was achieved with a 1000 x 1000 network of elements. The planar surfaces of the bipyramids permitted using simple geometric computations that significantly improved the computation efficiency. For this paper, geometric-growth curves are defined as plots of A against $ They were simulated by incrementing r from zero to a value that yieldedf = 1. The model was extended to kinetic simulations by assuming the growth rate of the simulated grain to be defined by the growth rate of r. This assumption
PRICE:
SIMULATIONS
permitted construction of simulated kinetic from the geometric impingement data. IMPINGEMENT
OF GRAIN
plots
The impingement geometry is dictated by the geometry of overlapping of the extended or full volumes of the impinging grains. The impingement geometry changes as the number of grains impinging at a given point changes. Major impingement points (IPs) were previously defined as those points of intersection of the simulated grain with the impingement cell that correspond to changes in impingement geometry [lo]. In the computer model, these IPs occur at the comer positions of the computation cell. They are numbered sequentially in the order of occurrence with increasing r. The change in the geometry of overlapping at an IP causes a corresponding change in the slope of the growth curve. This change is conveniently described in the usual terms of hard and soft impingement for phase transformations [12]. Hard impingement occurs when a spherical or ellipsoidal surface becomes tangent to one of the planes of the impingement cell or an edge of a planar grain coincides with one of the planes. The resultant overlap correction is much greater for A than forfand causes a sharp break in the growth curve. Soft impingement results in relatively small overlap corrections for both f and A and causes only a slight break in the growth curve. The impingement geometry for a sphere growing in a cubic impingement cell was described previously [IO]. As r increases, IP 1 occurs simultaneously at the equivalent positions of (f, 0, 0), (0, i, 0), and (0,0,i) for r = l/2. The sphere is tangent to the planes of the impingement cell, and the impingement is hard. IP 2 occurs simultaneously at the equivalent positions of (f, f, 0), (i, 0, f) and (0, f, f) for r = ,/?/2. The sphere is tangent to the midpoints of the edges of the impingement cell, and the impingement is soft. The third and final IP occurs at (f, f, i) for r = ,/3/2 and f = 1 and corresponds to completion of the hypothetical recrystallization reaction. As the symmetry of a simulated grain varies from the symmetry of the impingement cell, additional IPs are encountered. Thus, a sphere growing in a tetragonal impingement cell has four IPs, and a sphere growing in an orthorhombic cell has six IPs, if one ignores the final IP in each case. Impingement geometries are correlated with shapes of growth curves in the section on “Impingement Simulations.” CORRELATION
WITH EXTENDED CONCEPT
of a random distribution {equation (9) in Ref. 121) F, = F, ex- v,,,+
GEOMETRY
1379
IMPINGEMENT
v,,,-
of overlapping ... +(-l)““V,,,
spheres (9)
where V,, as defined in equation (1), is the actual volume after correcting for overlapping, and V,,, is the sum of the extended or full volume of all segments formed by m overlapping spheres. Specifically, Vi exis the sum of the full volume of all spheres in Avrami’s model, whereas in the geometric computer model, V,,, is simply the full volume of the model grain. The overlap geometry for a sphere growing in a cubic impingement cell is shown in Fig. 1; the extended-volume concept is shown schematically at the top, and the geometric model at the bottom. The model sphere is located at (O,O, 0) and labled A. Three juxtaposed spheres at (0, 1, 0), (1, 1, 0), and (1, 0,O)are labeled B, C, and D, respectively. Representative segments formed by the overlapping of spheres A, B, C, and D are individually cross-hatched in the figures. Each segment corresponds to an extended volume and can be identified with the lowercase letters at each intersection. Superscripts are used to identify the set of overlapping spheres that form each segment; Vfc, is the total volume of sphere A, ViqeXis segment ae, Vtz is segment bg, V;‘e”x” is segment bef, and If;::” is segment c&J Because of the symmetry of the model, only the extended volumes associated with sphere A need to be considered. Vmex in equation (9) is formed by summing over all segments formed by m - 1 spheres that overlap sphere A. Dividing the sum by m to eliminate redundancy yields
(10) where 2 is the coordination number of all sets of overlapping spheres that form identical segments; superscript a identifies a representative set of overlapping spheres that forms a specific segment; and the summation is over all unique sets of a. Since impingement does not occur in the computer model until IP 1, V, = V, exup to the first IP. Between IP 1 and IP 2, only segments ae form on each of the
VOLUME
Before the impingement simulations are discussed, it is desirable to formally relate the impingement geometry used in the computer model to the extended-volume concept outlined by Avrami [2]. Avrami developed a relation based on the geometry
Fig. 1. Overlap geometry for extended volume corrections (top) and geometrical corrections (bottom).
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SIMULATIONS OF GRAIN IMPINGEMENT
six faces of the impingement cube, and equation (9) yields V* = V*cx-5V*B 2 2ex. (11) This relation is formally equivalent to the geometric computation which subtracts the volume of the six spherical segments corresponding to segment hj in Fig. 1 from the volume of sphere A. To retain identity with 2 and m in equation (lo), the fractions have not been reduced. Between IP 2 and IP 3, equation (9) assumes the form V, = Vf,, - f Viqex- y V$$ + y VfeF 12
+$V$:D+g,cx
To simplify shows
equation
ACD _
(12), simple
12 ABCD 4 v,,, .
superposition
Spheres in cubic and tetragonal impingement cells
tv~~=;v~~c+:v;qep+;v~~o-~v,qcD. Substituting
(12)
curves of the spheres and ellipsoids were divided into 25 equal steps up to and including the final IP. For plotting, A was normalized to the maximum area for each growth curve. Computations also were performed for f and A at the IPs, and each IP is labeled “X” on the geometric-growth curves. Also, IPs are listed in Table 1 for spheres growing in tetragonal impingement cells and in Table 2 for ellipsoids growing in a cubic cell in the order of occurrence for increasing r. The tables include the values of r, f, A, and normalized A. Since one of the objectives of this study was to compare the simulated growth curves with equation (3), equation (3) was normalized to a maximum area of A = 1 by setting K, = 4 and superimposed on some of growth curves.
equation
(13)
(13) into equation (12) yields
K = V?ex--;v;:++4v4qBo.
(14)
Equation (14) is formally equivalent to the geometric computation that subtracts the volumes of the 6 spherical segments corresponding to segment hi from the volume of sphere A and adds the 12 volumes corresponding to segment ijk, since these volumes are subtracted twice when the volumes corresponding to segment hi are subtracted. Equations (11) and (14) verify that the extended-volume concept in equation (9) and the geometric computation used in the computer model are equivalent for regular geometric distributions. IMPINGEMENT
SIMULATIONS
To provide reasonable computation times for the finite-element computations, the geometric-growth Table 1. Major c/a
IP
0.1
1 2 3 4
3 4
(0,O.c) (a, 0, c) (0, b, c)
4
0.9 2 3 4 1.0 2
1
5
3 4 1.5
10.0
(0,0,cl (0.0, 0)
2 3 4 1
2
1.1
impingementpoints for
Impingement positions (0, b, 0) (a. 0, c) (0, b, c) (a, b, 0) (0.0, c) @, 0, 0) (0, b, 0) (a, 0, c) (0, b, c) (a. b, 0) (0, 0, c) (0.0,0) (0, b, 0) (a. 0, c) (0, b, c) (a. b, 0) (a, 0, 0) (0, 6, 0) (0, 0, c) (a, b. 0) (a, 0, c) (0, b, c) (a, 0, 0) (0, b, 0) (0.0, c) (a. b, 0) (a, 0, c) (0, b, c) (a, 0, 0) (0, b, 0) (a. b, 0) (0, 0, c) (a. 0, c) (O,b, c) @,O,O) (0, b.0)
OS
2 (a,b,O)
The effect of slight variations in impingement geometry is demonstrated for spheres growing in slightly tetragonal impingement cells in Figs 2 and 3 for c/a = 0.9 and 1.1, respectively. These figures correspond to the IPs identified in Table 1. The corresponding geometric-growth curves are plotted in Fig. 4, along with the growth curve for a sphere in a cubic impingement cell. Figure 4 also includes a normalized plot of equation (3) to emphasize the asymmetric shift of the simulated growth curves from the S-F parabolic relation. This shift is a consequence of the decrease in surface curvature as r increases. The maximum in the growth curve for a sphere growing in a cubic cell occurs at IP 1, which is hard. The sphere is tangent to the six (200) planes of the impingement cell at IP 1. Impingement at IP 2, which is soft, causes only a slight break in the growth curve. Decreasing c/a to 0.9 shifts the normalized growth curve slightly above that for the cubic cell up to IP 1 because the maximum area for this geometry is less spheres in tetragonal cells r
f
A
A IA,,,
0.108 1.077 1.083 1.523 0.315 0.630 0.704 0.891 0.466 0.518 0.697 0.732 0.500 0.707 0.484 0.533 0.685 0.720 0.437 0.618 0.655 0.787 0.232 0.328 2.321 2.332
0.005 0.783 0.790 1.000 0.131 0.720 0.869 0.998 0.424 0.573 0.953 0.979 0.524 0.965 0.476 0.618 0.941 0.972 0.349 0.758 0.830 0.986 0.052 0.114 0.997 0.999
0.146 1.464 1.317 0.003 1.247 2.501 1.475 0.101 2.730 3.038 0.947 0.513 3.142 0.762 2.948 2.917 1.132 0.638 2.397 1.986 1.885 0.423 0.678 0.561 0.432 0.093
0.100 1.000 0.936 0.002 0.499 1.000 0.590 0.040 0.899
1.000 0.312 0.169 1.oOO 0.243 1.000 0.989 0.384 0.216 l.Ow 0.829 0.786 0.176
I.000
0.827 0.638 0.137
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1381
OF GRAIN IMPINGEMENT
Table 2. Maior imoinnementoointsforelliusoids with I = m = 1.0in a cubiccell
e/a
IP
10
1 2 3 4
1.5
1 (O,O,c) 2 3 4
1.1
k&O) (a,O,c)(O,b,c)
1 (a,0, 0) (0,b,0)
2 3 4 0.1
(a.O,0) CO,b,0) (a,O,c)(O,b,c) (a,b,O)
1 (a>O,O)(0,b,0) 2 (O,O,c)
3 4 0.5
(a>O,O)(0,b,0) (a,O,c)(O,b,c) k&O)
1 (O,O,c) 2 3 4
0.9
Impingementpositions (O,O,c) kO.0) CO,b,O) (a>O,c)CO,b,c) (G&O)
k&O) (O,O,c) @,O,c) (O,b,c)
1 (a,O,O)(0,b,‘3 2 3 4
(a,b,O) (O,O,c) (a,O,c) CO,& c)
than that for the cubic case (Table 1). At this point, Fig. 2(a), the sphere becomes tangent to the (002) plane of the computational cell; the impingement is hard; and a significant break occurs in the growth curve. The growth curve continues to a maximum at IP 2, Fig. 2(b), where the sphere becomes tangent to the (200) and (020) planes, again in hard impingement. This maximum shifts to largerfthan the maximum for the cubic cell. Impingements at IP 3 and IP 4 [Fig. 2(c) and (d), respectively] are soft and shift upward and downward on the growth curve,
(a) 1P 1, f = 0.424
T
/
0.050 0.500 0.502 0.707 0.333 0.500 0.601 0.707 0.455 0.500 0.676 0.707 0.500 0.556 0.707 0.747 0.500 0.707 1.000 1.118 0.500 0.707 5.OOO 5.025
0.005 0.783 0.790 1.000 0.233 0.669 0.909 0.994 0.433 0.569 0.955 0.978 0.471 0.628 0.938 0.938 0.262 0.569 0.911 0.992 0.052 0.114 0.997 l.OOO
A 0.248 3.185 2.783 0.005 1.883 3.014 1.424 0.235 2.765 3.070 0.942 0.535 2.939 2.925 1.185 0.635 2.170 2.180 2.054 0.447 1.162 2.008 2.000 0.429
0.078 l.OOO 0.874 0.002 0.625 1.000 0.472 0.078 0.901 I.000 0.307 0.174
I .ooo
0.995 0.403 0.216 0.951 0.955 0.900 0.196 0.579 l.OOO 0.996 0.214
respectively, from the corresponding soft IP 2 for the cubic cell. Increasing c/n to 1.1 shifts the normalized growth curve even higher. The maximum in the growth curve occurs at IP 1 and shifts to smaller f than that for the cubic impingement cell. At this point, Fig. 3(a), the sphere is tangent to the (200) and (020) planes in hard impingement. The curve remains relatively flat to IP 2, Fig. 3(b), where the sphere becomes tangent to the (002) planes of the impingement cell, then the growth curve breaks sharply downward. Again, the
(b) IP 2.f = 0.573
(al IP 1, f=
(cl 1P 3, f= 0.953
Al&,x
(d)
IP
0.476
(b) IP 2,f = 0.616
4, f=o.wg
Fig. 2. Impingement geometry of a sphere in a tetragonal cell with c/u = 0.9; IP is the impingement point number, and f the volume fraction.
(c)IP 3.f = 0.941
(d) IP 4,f= 0.972
Fig. 3. Impingement geometry of a sphere in a tetragonal cell with c/a = 1.1.
1382
PRICE:
.: : :
E
:’
1
:
P
.’
0.
0
g
SIMULATIONS OF GRAIN IMPINGEMENT
‘..8
x
9
“...,g
0
B
“....,,. 8 -
0
.E
5 c
;j
;
0.5 -
G 2
-
E
jj .il
-
B $
2
..*..,J -
q
0
Tang~nal
Major impingsment point
+
.I,” -
cubic cell, c/a = 1.0
x
’
:, m
cell, cla = 1.1
A .... 1
Tetqonal
:,a_
cell, da = 0.9
+
t!
Eauation (3) ’
0
0.0
’
0.5
0
1
1
’
1.0
c/a
Volume fraction
Fig. 4. Normalized geometric-growth curves for a sphere in a cubic cell and in tetragonal cells with c/a = 0.9 and 1.1.
Fig. 6. Variation of the maximum area of impinging spheres in tetragonal cells as a function of c/a.
soft IPs in Fig. 3(c) and (d) are shifted upward and downward on the growth curve, respectively, from the corresponding soft IP2 for the cubic cell. Both normalized growth curves for the tetragonal cells in Fig. 4 expand outward from that for the cubic cell. Changes in the growth curves for larger variations in c/a are shown in Fig. 5 for c/a ratios of 0.1, 0.5, 1.5, and 10. Correlation of these curves with the impingement simulations and the data in Table 1 reveals that the growth-curve maximum is dictated by the IP at (a, 0,O) and (0, b, 0), regardless of its order of occurrence. As c/a increases from c/a = 1, the maximum in the growth curve continues to occur at the first IP, but shifts to lower J The impingement also becomes increasingly hard, causing a more severe downward break in the growth curve. The soft IP at (a, b,O) eventually shifts from the third to the second IP, but hard impingement continues to occur at (0,0,c), and this IP continues to shift to higherf. As c/u decreases from c/u = 1, the maximum in the
growth curve continues to occur at (a, 0,O) and (0, b, 0) and shifts to slightly larger J: The absolute magnitude of the maximum area in a growth curve, A,,, changes significantly as the impingement geometry changes. The change in A,,, is not reflected in the normalized plots in Figs 4 and 5, but it is apparent in the data recorded in Table 1 and Fig. 6, in which A,,, is plotted as a function of c/u for spheres growing in tetragonal cells. Ellipsoids in a cubic impingement cell
Growth curves for ellipsoids growing in cubic cells are roughly similar to growth curves for spheres growing in tetragonal cells when n -I for the ellipsoid equals c/u for the tetragonal cell. This is demonstrated in Fig. 7 for ellipsoids with n = 10, 1.5, 0.5, and 0.1 and with 1 and m fixed at 1.0 in equation (6). The impingement data recorded in Table 2 include data for n = 0.9 and 1.1 for comparison with similar c/u ratios in Table 1. Impingement geometries for 1.0
0
L v
m
0
A”~oocm 0
0
0
0=
n
n a
0
0
A A
0.0 0.0
0
’
’
8
’
0.5
8
’
’
0
0.0 1.0
Volume fraction
Fig. 5. Normalized geometric-growth curves for a sphere in tetragonal cells with c/a = 10, 1.5, 0.5, and 0.1.
I
q
Ellipsoid, n =
0
Ellipsoid, n =
0
Ellipsoid, n =
A
Elliiid,
X
Major impingement point
I
9
I
n = 10
’
0.5
0
’
I
I
1.0
Volume fraction
Fig. 7. Normalized geometric-growth curves for ellipsoids withI=1.0,m=1.0,andn=0.1,0.5,1.5,and10inacubic cell.
PRICE:
a
SIMULATIONS
C
b
IP l,f=0.262
f = 0.411
OF GRAIN IMPINGEMENT
IP2,
1383
e
d
f=O.569
IP 3, f=0.911
IP4,
f=0.992
Fig. 8. Impingement of an ellipsoid with I = 1.0, M = 1.0, and n = 0.5 in a cubic cell.
a
C
d
f = 0.496
IP 3, f = 0.997
b
IP 1, f =0.052
IP2,f=0.114
e IP4,
fal.O
Fig. 9. Impingement of an ellipsoid with I = 1.0, m = 1.0, and n = 0.1 in a cubic cell.
n = 0.5, 0.1, and 10 are shown in Figs 8, 9, and 10, respectively. As n decreases from n = 1, the ellipsoid flattens in the z direction, as shown in Figs 8 and 9. IP 1 occurs at the midpoints of the (200) and (020) planes, Figs 8(a) and 9(a). The impingement is hard, but as n decreases and the ellipsoid becomes more oblate, the curvature increases at the intersection of the z = 0 plane, as is evident in Fig. 9(a), and the hardness decreases. Initially, the maximum in the growth curve occurs at IP 1, and the curve is relatively flat between the IP 1 and IP 2, both of which are hard. As n continues to decrease, (a, b, 0) becomes the second IP, and the maximum shifts to a position between the first and second IPs. This is evident in Fig. 7 for n = 0.5; Fig. 8(b) shows the corresponding impingement geometry at the maximum. At this point, the surface of the ellipsoid within the cell has maximum curvature. As r increases beyond this point, the surface flattens and A decreases. Because of the flatness of the ellipsoid within the impingement cell, very little change occurs in A between IP 2 and IP 3 in Fig. 8(c) and (d), respectively. IP 3 is hard as the ellipsoid becomes tangent to the (002) planes of the impingement cell. As r increases through IP 3, A decreases rapidly through IP 4 [Fig. 8(e)]. With further decrease in n, the maximum in the growth curve occurs at or slightly after IP 2. This is evident in Fig. 7 for the n = 0. I curve. Since the ellipsoid is relatively flat, and the curvature in the z direction remains fairly constant with further increase in r, the growth curve remains relatively flat between IP 2 and IP 3. The
growth curve then decreases rather rapidly through IP4 (Fig. 7). As n increases from n = 1 and the ellipsoid becomes prolate, IP 1 occurs at (0,0,c), and IP 2 occurs at (a, 0,O) and (0, b, 0). Both impingements are initially hard, but as n continues to increase, the curvature of the prolate ellipsoid increases at its intersection with the z axis, and IP 1 transforms to soft
C
IP 3,
f = 0.790
Fig. 10. Impingement of an ellipsoid with I = 1.0, m = 1.0, and n = 10 in a cubic cell.
PRICE:
SIMULATIONS
1385
OF GRAIN IMPINGEMENT (a, b, 0), (a, 0, c), and
0.0:
0.0
I
I
-
{ 1111 Bipyramid
‘...
Equation
’
I
’
(3)
’
0.5 Volume
I
I
’ 1.0
fraction
Fig. 15. Normalized geometric-growth curve of a {Ill} bipyramid in an orthorhombic cell with m/l = b/a = 0.75 and n/l = c/a = 0.25. Bipyramids Geometric growth curves for bipyramids show many similarities to those for spheres and ellipsoids. They are equally sensitive to variations in symmetry, and a single normalized curve results when the axial-
symmetry constraint is imposed. The normalized geometric-growth curve for a {ill} bipyramid in an orthorhombic cell with m/l = b/a = 0.75 and n/l = c/a = 0.25 is shown in Fig. 15. These are the same axial constraints used for the ellipsoid in Fig. 12 and the corresponding geometric-growth curve in Fig. 13. The impingement geometry, Fig. 16, also is similar to that for the ellipsoid in Fig. 12. IP 1, Fig. 16(a), occurs simultaneously at (a, 0, 0), (0, b, 0), and (O,O, c); IP 2, Fig. 16(c), occurs simultaneously at
(0, b, c). In contrast to the ellipsoid, however, both IPs are soft. This is a consequence of effectively infinite curvature at the apexes. A,,, occurs when the {111} face of the bipyramid passes through the center of the computation cell. At this position, Fig. 16(b), the bipyramid intersects the midpoints of the edges of the computation cell. This occurs at f = 0.5 for all axial symmetries. The growth curve is exactly symmetric about this point because the center of the computation cell is a point of inversion symmetry for planes of a given orientation that are equidistant from the center. When the symmetry varies from the axial-symmetry constraint, the geometric-growth curve remains symmetric aboutf = 0.5, but the curve distorts either inward or outward, depending on the relative axial symmetries. The geometric-growth curve for the (1 IO} bipyramid in an orthorhombic cell with m/l = b/a = 0.75 and n/l = c/a = 0.25 (Fig. 17) has only one IP other than the final IP. It occurs when the edges of the bipyramid in the x-y plane become tangent to the (200) and (020) planes, and the apexes become tangent to the (002) planes. Again, the impingement of the apexes is soft, but when the edges become tangent to the impingement planes, the impingement is hard. This produces an asymmetric growth curve with IP 1 at the maximum in the growth curve. KINETIC SIMULATIONS Johnson-Mehl-Avrami
simulations
Simulated J-M-A plots were generated from the geometric impingement data by relating r in equations (5)-(g) to a linear growth rate and assuming a
Ib!
f = 0.167
b)
t = 0.5
IP 2 f = 0.833
Fig. 16. Impingement of a {111)bipyramid in an orthormbic cell conforming to the symmetry constraint, m/l = b/a = 0.75 and n/l = c/a = 0.25. A.M.35,6-M
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SIMULATIONS
OF GRAIN
IMPINGEMENT
O-
z
*; -l-
C > r
-2 -
+ + +
0” J -3 -
- - -
(111)
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Fig. 19. Simulated J-M-A plots for the {11l} bipyramid growing in a cubic impingement cell, the corresponding extended volume (V,,) and the corrected volume (V,).
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Fig. 17. Normalized geometric-growth curve of a (100) bipyramid in an orthorhombic cell with m/l = b/a = 0.75 and n/l = c/a = 0.25.
total reaction time of 2500 s. The J-M-A relation in equation (2) should be linear with a slope of n for plots of log (ln[l/(l -f)]} against log t. Figure 18 is the resultant J-M-A plot for the sphere growing in a cubic impingement cell. The dashed line drawn through the first and fifth points indicates linear correspondence to equation (2). It has a slope of 3.0, which is the slope predicted by equations (1) and (2) for instantaneous nucleation. The simulated data in Fig. 18 coincide with the dashed line up to about f = 0.1. This is the extent of the apparent limited correspondence of J-M-A recrystallization plots to linear behavior [8]. At higher values of f, the simulated kinetic curve deviates slightly upward. However, this deviation is both significantly less and in the direction opposite to the severe deviation consistently displayed in experimental J-M-A plots for f > 0.1. Kinetic curves generated for both the {111) and the {1lo} bipyramids growing in cubic impingement cells were nearly identical to that of the sphere but with slightly less upward deviation. >,I
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Since equations (11) and (14) verify equivalency of the computer model and the extended-volume concept, kinetic simulations based on the extendedvolume concept were compared with the simulated geometric data. Both V,, = V, cI:and V, obtained from V,, using equation (1) are plotted in Fig. 19 using the same time scale converted from the size parameter, r, for the {111) bipyramid growing in a cubic cell. As expected, the curve for I’, coincides identically with the simulated curve up to IP 1 and then rises above the simulated curve. In all cases conforming to the axial-symmetry constraint, the corresponding V, plot coincides with the latter extrapolation of the first and fifth simulated points (within plotting accuracy) and n = 3, as predicted by equations (1) and (2). The positive deviation of the geometric data from V, demonstrates that equation (1) overcompensates for impingement of the modeled geometric grains. For the sphere growing in a cubic impingement cell, this overcompensation reaches a maximum of about 26% atf= 0.7, but it is only 6.6% atf= 1.0. These values are somewhat less for both bipyramids. When the axial symmetry was constrained to m/l = b/a and n/l = c/a, the simulated kinetic plots were identical to the appropriate simulated shape growing in a cubic cell. As axial symmetries vary from the axial-symmetry constraint, significant deviation occurs from linearity, as shown in Fig. 20 for a sphere growing in a tetragonal impingement cell with c/a = 10. The corrected volume is plotted as the dashed line in Fig. 20. This demonstrates that the kinetic curves are sensitive to the shape of the geometric-growth curves, and it also indicates that variations from the axial-symmetry constraint are not realistic impingement geometries for recrystallization. Speich-Fisher
I III, 104
Time (SW)
Fig. 18. Simulated Johnson-Mehl-Avrami (J-M-A) plot for the sphere in a cubic impingement cell using a linear growth rate.
simulations
The English-Backofen inverse time-dependent growth rate [6] used in the S-F relation was G = K,/t
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where G is the growth rate, K, is essentially a constant
PRICE:
SIMULATIONS
OF GRAIN IMPINGEMENT
1387
negative values of r,, yield incubation times greater than 1.0 s. However, this interpretation must be /’ tempered by the empirical nature of equation (15) /’ ,’ oand the fact that it does not hold for the early stages = , - 8% r, 8 /;+++ of recrystallization. 5 _ 10-l .i The deviation from linearity in Fig. 21 occurs i= -1 ,+* r: * because A and fin equation (3) are not geometrically c ,,+' m -2 , related through r. To analyze the strong divergence + + + Sphere in tetragonalIO-* f s cell with c/a = 10 4’ at both extremes of the plot in Fig. 21, A from the z - -Corrected volume - 10-3 -3 simulated data was converted to a new volume fraction using equation (3). The divergence was reI I III I b I11111 -4J”I duced but was not corrected to linearity, since t was 104 103 102 related to the geometric r. This analysis demonstrates Time (set) that the breakdown of equation (3) at f= 0 and f= 1.0, which actually is a manifestation of the Fig. 20. Simulated J-M-A plot for a sphere in a tetragonal impingement cell with c/a = 10. nongeometric relation off and A ‘in equation (3), is the source of the divergence. In fact, at f = 1.0, r,, in equation (16) specifies that the plot in Fig. 21 is asymptotic to t = 87 s at r = 0, while at f = 1.0, that English and Backofen found to be independent f/(1 -f) becomes infinite, and the plot is asymptotic of temperature and strain except during the initial However, this strong divstages of growth, and t is the time. Relating r to t by to t = 2876 s at r = $/2. assuming dr = G dt and integrating from limits of to ergence is accentuated in the kinetic plots by the disproportionate expansion of f at both extremes, to t and r,, to r and it occurs outside of the useful range stipulated for log(t) = (r - r,)/K,. (16) equation (4). In this form, r,, is effectively the value of r at to = 1.0. Modified Johnson-Mehl-Avrami simulation The S-F relation in equation (4) should be linear Since these simulations indicate that a major limiwith a slope of n in plots of log[f/(l -f)] against log t. For S-F simulations using the data for the tation of the J-M-A relation is the assumption of sphere in a cubic impingement cell, equation (16) was linear growth, the relation was modified using the fit to a minimum time of 100 s and a maximum time inverse time-dependent growth-rate in equation (15) of 2500 s with values of r,, = - 1.11 and KB = 0.247 in and the relation of r to t in equation (16). Assuming the dimensionless geometry of the model. The re- instantaneous nucleation in conformance with the computer model, the number of active nuclei (n,) is sulting S-F plot is shown in Fig. 21. To indicate linear behaviour corresponding to equation (4), the constant, and V,, assumes the form dashed line in Fig. 21 was constructed through Points I’,, = XsK:[ln(t) + r,,/K,13 (17) 5 and 20. These points correspond to the extremes where x is a geometric factor. Using B = xniK: and suggested by Speich and Fisher of f = 0.02 and f = 0.95. The simulated data are reasonably close to substituting equation (18) into equation (1) modifies relation for inverse time-dependent linear behavior within this range and yield values of the J-M-A growth rate to n = 2.11 and K = 1.3 x 10m6 in equation (4). These values are in reasonable agreement with experimental 1 -f= exp{ -B[ln(t) - r,,/KE]‘}. (18) data for vanadium at 1173 K [8]. Equation (16) implies that a finite incubation period does exist, and Under these assumptions, plots of log ln[ (1 -f )-‘I vs log [In(t) + r,,/K,] were nearly identical to the plot in Fig. 18 for linear growth. The initial linear portion 5 of each plot had a slope of 3.0, as predicted by 0.9999 4 equation (19). When experimental data for vanadium 0.999 3 E 2 ’ - 0.99 .g [8] were fit to equation (18), reasonably linear plots were obtained, but the slope varied with temperature. - 0.90 E 1 : This indicates that r, is a function of temperature, - 0.50 E z 0 10-l 2 0” -1 and equation (18) is therefore not considered to be J 10-2 8 -2 practical for recrystallization kinetics. ;/‘+ 10-S -3 The J-M-A relation was modified using other _+ 10-d -4 time-dependent relations such as the t -2 dependence I I I I I11111 I I I Illll_ -5 suggested by Li [13] for recovery reactions. Each 103 102 104 modification yielded a reasonably linear plot with a Time bed slope of 3 and a fit essentially identical to that in Fig. Fig. 21. Simulated Speich-Fisher (S-F) plot for the sphere growing in a cubic impingement c& the growth rate is 18, but the experimental vanadium data did not inversely proportional to time. conform to these plots. This demonstrates that the 7,
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PRICE:
SIMULATIONS
computer model is useful for evaluating the geometric basis for kinetic models, but the temporal basis requires careful correlation with experimental data. DISCUSSION
A distinct advantage of the computer model used in this study is that it produced well-defined impingement geometries for various shapes of simulated geometric grains. Consequently, the model permitted impingement geometries to be studied in sufficient detail to determine how grain shape and symmetry affect impingement geometry and the shape of the geometric-growth curves. Since the computer model is an obvious oversimpli~cation of actual grain geometries, no attempt was made to exhaustively explore all relevant geometries pertaining to recrystallization. However, grain geometries that appear to be most relevant to recrystallization reactions in real materials conform to the axial-symmetry constraint of m/l = b/a and n/l = c/a.? These geometries yield a common normalized geometric-growth curve that is in reasonable agreement with the experimental S-F data forfand A. Furthermore, these geometries yield kinetic simulations that are in reasonable agreement with equations (2) and (4), whereas nonconforming simulated grain geometries do not. The importance of the axial-symmet~ constraint is that the S-F relation in equation (3) appears to be a generalized form for the growth of clusters of equiaxed recrystallized grains. The grains that nucleate first during recrystallization have a nearly isotropic shape 16, 141. In highly deformed materials, these grains usually nucleate in clusters along grain boundaries or dense deformation bands [6, W-161, and impingement and coalescence of these grains occur at an early stage. This initial impingement effectively constitutes hard impingement and should cause a break or discontinuity in the growth curve. Indeed, such a di~ontin~ty appears to exist in the SF experimental curve at approximately f = 0.1, although a more extensive break appears to occur in the S-F data betweenf = 0.8 and f = 0.9 that is not explained by this model. After this early impingement within clusters, each cluster surface should conform to the geometric shape dictated by the axialsymmetry constraint. The impingement geometry would then be governed by this cluster shape, and equation (3) would apply to the growth and impingement of the clusters. In fact, this cluster model suggests that nucleation is implicitly treated by equation (3). In lightly deformed materials, grains usually nucleate at grain corners [6, 14,151 without forming clusters, and equation (3) would apply to the growth and impingement of individual grains. In real materials, the axial-symmetry constraint is ?Possible exceptions are materials such as oxide-dispersionstrengthened metals in which the dispersions constrain grain-boundary movement.
OF GRAIN IMPINGEMENT
related to the deformation geometry. An important aspect of the axial-symmet~ constraint and the ~~s~lli~-cluster concept is that the cluster-surface area per unit volume increases significantly as deformation increases. This is shown for the tetragonal case in Fig. 14. In fact, the c/a ratio in Fig. 14 correlates with cross rolling for c/a < 1 and extrusion or wire drawing for c/a > 1. Since the increase in surface area indicated in Fig. 14 is substantial, the Cahn-Hagel concept implies that this should be a significant contribution to the fact that recrystallization rates increase with the amount of deformation. A similar contribution would be expected from the initial grain size, since the cluster density and surface area at a given rec~stalliz~ volume fraction increase substantially as the grain size decreases. The extent of these contributions will require careful analysis of experimental data and is beyond the scope of this paper. Geometric-growth curves of regular geometric shapes could be a closer match to the ex~~mental S-F curve if the simulated grains have a compound geometric shape similar to the curved polyhedral grain faces treated previously by Cahn [9]. The S-F experimental growth curve is skewed to the right in their plot of A vs $ Simulated geometric-growth curves for spherical and ellipsoidal grains conforming to the axial-symmetry constraint are skewed even further to the right because of the decrease in surface curvature with increasing r. In contrast, growth curves for {111) bipyramids conforming to the constraint are symmetric about f = 0.5. This suggests that a compound geometric shape composed of a regular polyhedron with faces that have either spherical or ellipsoidal curvature should provide a more realistic geometric-growth curve. This compound geometric shape is a more realistic approximation of real grain structures than are spherical and ellipsoidal shapes. Impingement geometry in real materials is affected by the distribution of grain sizes. The distribution of sizes should smear out breaks in the growth curve associated with IPs, and soft IPs should disappear. Since most materials have a relatively uniform distribution of recrystallized grain sizes, however, evidence of hard impingement should remain. This is particularly true for geometries conforming to the axialsymmetry constraint with the single hard IP, and it also agrees with the general shape of the S-F experimental growth curve. Recent work [ 171indicates that recrystallization in dispersion-strengthened metals may comply with the linear-growth kinetics of the J-M-A model. However, the kinetic simulations using the J-M-A relation support the previous contention [8, lo] that the major limitation of the J-M-A model for recrystallization in most other materials is the use of linear-growth kinetics. Correiating the computer model with the extends-vol~e concept used in the J-M-A relation demonstrates that the extended-
PRICE:
SIMULATIONS
OF GRAIN IMPINGEMENT
volume concept does provide reasonable compensation for grain impingement. However, modification of the J-M-A relation using l/t or other timedependent growth kinetics does not appear to yield practical relations. Kinetic simulations using the S-F relation were in reasonable agreement with equation (4) in the range between f = 0.02 and f = 0.95. In practice, the useful range can be extended from f= 0.01 to f= 0.99 [S] before the breakdown of equation (3) becomes significant. The simulated data do show noticeable deviation from the linear behavior predicted by equation (4) within this range. This deviation is a consequence of the fact thatfand A in equation (3) are not geometrically related. However, the absolute magnitude of this deviation is roughly of the same order of magnitude as the scatter of experimental data, and this deviation should not detract from the use of the S-F relation for practical applications within the stipulated range. Rigorous mathematical treatment of grain impingement will require different mathematical expressions between each set of IPs, but this will result in a complex set of expressions that is unnecessarily cumbersome for modeling continuous kinetic curves. Such complexity is not justified in view of the as-yet limited understanding of atomistic mechanisms in recrystallization reactions. Therefore, both the extended-volume concept in equation (1) and the empirical S-F relation in equation (3) appear to be the best available geometric relations for impingement compensation in deriving kinetic relations. However, using equation (1) in the J-M-A model invokes the apparent anomaly of using only volumetric terms to model a surface reaction, as previously discussed by Gokhale et al. [ 181.On the other hand, the S-F relation in equation (3) suffers from the fact that f and A are only empirically and not geometrically related. However, major advantages of equation (3) are that (a) it is compatible with the Cahn-Hagel [7] concept of surface controlled reactions, and (b) the cluster model discussed above indicates that nucleation is implicit in the empirical model. In contrast, nucleation is not considered in equation (1) and must be treated explicitly in equation (2), which is another limitation of the J-M-A model in view of the limited understanding of nucleation mechanisms. The simplified computer model has clarified some important aspects of grain impingement and recrystallization kinetics, and it should remain useful for analysis of new kinetic models as they are developed. Unfortunately, the computer model is necessarily restricted to instantaneous nucleation, but variable nucleation rates undoubtedly are important in recystallization reactions, as reviewed by Hu [19]. Other computer models that generate more-realistic grain structures [20,21] should accommodate variable nucleation rates, and these models also should be considered for future recrystallization simulations.
SUMMARY
1389
AND CONCLUSIONS
The simplified computer model, which is based on the growth of equisized, regular, geometric grains in a cubic distribution, provides surprisingly reasonable simulations of impingement geometry and recrystallization kinetics. Impingement geometry in the computer model is dictated by the hard and soft nature of the major impingement points, the relative geometries of the grains and the distribution lattice, and the surface curvature of the grain. The impingement geometry defines the shape of the geometric-growth curve. Grain geometries that conform to the axial-symmetry constraint yield impingement geometries that are expected to be most relevant to recrystallization. They also yield geometric-growth curves that reasonably approximate the experimental growth curve presented by Speich and Fisher. In real materials, major impingement points should smear out to some extent, but the general shape of the geometric-growth curve should be retained. When growth rates are related to the radius or size parameter of the grains, geometric-growth curves define simulated, kinetic, recrystallization plots based on time dependence and plotting parameters associated with specific kinetic models. The computer model demonstrates that the Johnson-Mehl-Avrami extended-volume concept does provide reasonable compensation for grain impingement. With the possible exception of dispersionstrengthened metals, the major limitations of the J-M-A relation for recrystallization kinetics is the assumption of linear growth; the fact that the relation is based on volumetric terms also appears to be a limitation, since recrystallization is effectively a surface reaction. The major limitation of the Speich-Fisher relation is that it has an empirical basis in which f and A are not geometrically related. The absolute magnitude of errors generated from this limitation is not large, however, and the relation is useful for practical applications within the range off = 0.01 to f = 0.99. The breakdown of equation (3) becomes significant only outside
of this range.
Acknowledgements-This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract No. W-7405ENG-48. The author gratefully acknowledges the editorial assistance of J. M. Bruner; the assistance and cooperation of W. L. Barmore, who provided the HewlettPackard 9845B computer and plotting system on which this work was performed; and the advice offered by R. W. Cahn.
REFERENCES 1. W. A. Johnson and R. F. Mehl, Trans. Am. Inst. Min. Engrs 135, 416 (1939). 2. M. Avrami, J. them. Phys. 7, 103 (1939). 3. 4.
M. Avrami, J. them. Phys. 8, 212 (1940). M. Avrami, J. them. Phys. 9, 212 (1941).
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5. G. R. Speich and R. M. Fisher, in Recrystailization, Grain Growth and Textures (edited by H. Margolin), p. 563. Am. Sot. Metals, Metals Park, Ohio (1966). 6. A. T. English and W. A. Backofen, Trans. metall. Sot. A.I.M.E. 230, 396 (1964). 7. J. W. Cahn and W. Hagel, in Decomposition of Austenite by D@iisional Processes (edited by V. F. Zackay and H. I. Aaronson). D. 131. Interscience. New York (1960). 8. C. W. Price, Scripta metaii. 19, 669 (1985). . ’ 9. J. W. Cahn, Trans. metall. Sot. A.Z.M.E. 239, 610 (1967). 10. C. W. Price, Scripta metall. 19, 785 (1985). 11. C. Kittel, Introduction to Solid State Physics, 3rd edn, p. 12. Wiley, New York (1968). 12. J. W. Christian, The Theory of Transformation in Metals and Alloys, p. 442. Pergamon, Oxford (1965). 13. J. C. M. Li, in Recrystallization, Grain Growth and Textures (edited by H. Margolin), p. 45. Am. Sot. Metals, Metals Park, Ohio (1966). 14. P. K. Ambalal and C. W. Price, in Microstructurd I,.
IS. 16.
17. 18. 19.
20.
21.
Science, Vol. 13, (edited by S. A. Sheils, C. Bagnall, R. E. Witkowski and G. F. Vander Voort), p. 313. Am. Sot. Metals, Metals Park, Ohio (1986). _ J. W. Cahn. Acta metall. 4. 449 (1956). R. A. Vandermeer and P.‘Gordon, in Recovery and Recrystallization of Metals (edited by L. Himmel), p. 211. Interscience, New York (1963). I. Beden, Scripta metall. 20, 1 (1986). A. M. Gokhale, C. V. Iswaran and R. T. DeHoff, Metali. Trans. lOA, 1239 (1980). H. Hu, in Metallurgical Treatises (edited by J. K. Tien and J. F. Elliott), p. 385. T.M.S.-A.I.M.E., Warrendale, Pa (1981). K. W. Mahin, in Proc. of the 1976 Internat. Conf on Computer Simulation for Materials Applications (edited by R. J. Arsenault, J. R. Beeler Jr and J. A. Simmons), p. 39. National Bureau of Standards, Gaithersburg, Md (1976). M. P. Anderson, D. J. Srolovitx, G. S. Grest and P. S. Sahni, Acta metall. 32, 783 (1984).