topological events in two-dimensional grain growth - Science Direct

Acta metall, mater. Vol. 42, No. 8, pp. 2719-2727, 1994 Pergamon

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TOPOLOGICAL EVENTS IN TWO-DIMENSIONAL GRAIN GROWTH: EXPERIMENTS A N D SIMULATIONS V. E. FRADKOV, M. E. GLICKSMAN, M. PALMER and K. RAJAN Materials Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, U.S.A.

(Received 12 November 1993) Abstract---Grain growth in polycrystals is a process that occurs as a result of the vanishing of small grains. The mean topological class of vanishing two-dimensional (2-D) grains was found experimentally to be about 4.5. This result suggests that most vanishing grains are either 4- or 5-sided. A recent theory of 2-D grain growth is explicitly based on this fact, treating the switchings as random events. The process of shrinking of 4- and 5-sided two-dimensional grains was observed experimentally on polycrystalline films of transparent, pure succinonitrile (SCN), Grain shrinking was studied theoretically and simulated by computer (both dynamic and Monte Cario). It was found that most shrinking grains are topologically stable and remain within their topological class until they are much smaller than their neighbors. We discuss differences which were found with respect to the behavior of 2-D polycrystals, a 2-D ideal soap froth, and a 2-D section of a 3-D grain structure.

1. INTRODUCTION Topological structure of 2-D polycrystals plays a key role in grain growth dynamics. Mullins [1], for example, showed that the rate of change of the area of an individual grain does not depend on the size or geometrical shape of the grain, but depends only on the topological class (number of sides). Specifically, Mullins showed that

da dt

mYn(n

6)

(1)

3

where a and n are the area and topological class of the grain, and M and y are the mobility and surface tension of the grain boundaries (assumed to be uniform). Based on this result, theoretical descriptions and computer simulations of the process were developed [2-18]. The main difficulties in such studies lie in the way the topology of the polycrystalline system changes during evolution. There are two essential topological events in 2-D grain growth: (i) neighbor switchings, which occur when the length of a boundary decreases to zero, and an unstable fourfold vertex appears that instantaneously decomposes into two threefold vertices and a new boundary, and (ii) vanishings of grains with n < 6. The earliest treatment of topological changes during grain growth was suggested by C. S. Smith

tThe special case of a fully symmetric grain may obviously be neglected. :Ht is worth mentioning that there are a few models where vanishings of 4- and 5-sided grains were inevitable because switchings were not present in the models [5, 12, 14, 16-18, 24].

[19]. Smith stated that all grains undergo switchings and that only 3-sided grains undergo vanishings. Moreover, for grains with 3 < n < 6 vanishing is preceded by one or more neighbor switchings, with the shortest boundaries disappearing first.~" Grains with n = 2 were not considered. Smith's view of twodimensional grain growth is widely accepted, and a number of theoretical models use it as a postulate [15, 20-23].:~ A key prediction of Smith's model purportedly appeared to be established experimentally. Specifically, as shown in [3], the grain size, r, dependence of the average topological class n(r), based on data from 2-D sections of a 3-D polycrystal has a limit ~(r)~--li 3. This result implies that all vanishing grains are 3-sided. However, the microstructure of a 2-D section of a 3-D polycrystal is rather different from that of a 2-D polycrystal. A similar investigation of 2-D polycrystals showed that ~i(0)= 4.5 + 0.2 [6, 25]. Thus, there are many small (compared to the average grain size) 4- and 5-sided grains in the evolving grain structure. A model of 2-D grain growth was developed based on equation (1), postulating that switchings occur equiprobably for any boundary in the system, without producing a topological cascade of 5--,4 ~ 3-sided transformations for shrinking grains. Thus, in this model, 4- and 5-sided grains mostly vanish within their topological class [7, 8, 26-28]. The continuity equation for the grain density function F~(a, t) in a phase space of grain area a and topological class n was shown to be

2719

cOt

t- ~aa

(n -- 6)Fn(a , t) = In(a, t)

(2)

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FRADKOV et al.: GRAIN GROWTH IN 2-D MICROSTRUCTURES

where the right-hand side of equation (2), I,(a, t), is a term responsible for topological rearrangements. For equiprobable neighbor switchings I.---- F{[(. + 1 ) F n + l - n F . ] + f l [ ( n + 1)F.+, - 1)F._l]~, n =3,2t . . . .

.I 12~'F{2F2-t-~(3F3--2F2)}

(3)

where the arguments a and t of the density function F.(a, t) are dropped for the sake of simplicity, ~ is the average grain area, F is a rate of vanishing events given by F = -~ ~ (n - 6)2F.(0, t)

(4)

Vn=2

and fl is a parameter giving the ratio of the rate of switching events to that of vanishing events. The results of this model are in good agreement with experiment, but the postulate itself remains unproved.

0 sec

2. IN SITU EXPERIMENT Grain growth in thin transparent films of succinonitrile (SCN) has been observed in situ [29, 30], providing continuous monitoring of the microstructure evolution. A variety of topological rearrangements were observed among the grain arrangements in these experiments, e.g. neighbor switchings, and 3-, 4- and 5-sided grain vanishings [25]. Two-sided grains (lenses) were never observed with this system.t These images do not show any cascades of switchings preceding vanishings. However, any observational method has a limited resolution. Thus, the details of the ultimate topological behavior of a vanishing grain might escape observation. Moreover, in SCN experiments shrinking grains fade out just before vanishing (see Figs I and 2. Similar observations have been obtained for grain growth in thin films of aluminum [31] and zirconium oxide [32]). Most probably it happens due to the finite thickness of the film, which implies that the smallest grains can no longer be assumed two-dimensional. Figure 3 shows the dependence of the mean topological class of grains, ti, vs normalized grain area. The results obtained from the SCN experiment are plotted in Fig. 3(a), as well as those obtained from the experiments on A1 foil [6] (b) and from the Monte Carlo computer simulation discussed below (c). Extrapolation to zero grain area suggests that the average topological class for vanishing grains lies tTh© absence of 2-sided grains may be explained by the fact that there were only about 200 grains visualized initially in this experiment, and the fraction of 2-sided lenses is sufficiently small to make their observation improbable. In fact, in the experiments reported on Al foil [6, 28] the percentage of 2-sided grains was found to be about 0.1%.

30 sec

50 sec Fig. I. Shrinking of a 4-sided grain in 2-D SCN-polycrystals.

FRADKOV et al.:

GRAIN GROWTH IN 2-D MICROSTRUCTURES

between 4 and 5. An alternative explanation of this dependence may be that 4- and 5-sided grains actually do lose sides before vanishing, but for some reason such events occur when the grain areas are already much less than the average area in the system. We propose here, that for any macroscopic statistical description of grain growth, there is no practical difference between these two topologically distinct scenarios. Comparison with the results obtained in a 2-D section of a 3-D polycrystal shows that the situation in 3-D represents fundamentally different behavior. Figure 4 shows that the average number of edges for a vanishing grain ti(0) = 3. For a 2-D section o f a 3-D structure Smith's grain vanishing conjecture is correct, with only 3-sided grains leaving the section (and apparently disappearing), because the most probable limiting 2-D section of a polyhedron is a triangular shape.

2721

0 sec

3. MONTE CARLO SIMULATION The statistics of topological events in 2-D polycrystals with uniform isotropic grain boundaries was investigated using Monte Carlo simulation of a variant of the 2-D Ising model [33]. A doubly periodic 2-D triangular lattice with the values ("spins") assigned to the nodes represents the grain structure. The spins can be either l or 0 (or "boundary" and "bulk", respectively). The initial structure is all "boundary" and includes a number o f " b u l k " nuclei. The Hamiltonian has the form H = ~ Si + restrictions

(5)

where restrictions are topological rules that prevent the onset of trivial behavior wherein the "bulk" regions would grow at the expense of the "boundary" regions until all the space would be filled by "bulk" sites. The restrictions change the behavior completely: they forbid a boundary site (which has higher energy) to transfer to a bulk site if such an event breaks the contiguity of the "boundary" phase (in other words, coalescence of grains is forbidden). Further nucleation of grains and homogeneous nucleation of the "boundary" phase within grains are also forbidden. All restricted local configurations, with the 1 ~ 0 transformation forbidden, are directly enumerated (Fig. 5). All other transformations are permitted.f With this set of restrictions the "bulk" nuclei grow freely until they meet each other, but after that the "boundary" layer between them cannot be destroyed. The system then enters its second stage of evolution, viz. grain growth. The "boundary" layers serve as fin terms of the Hamiltonian the restrictions mean that there exists a set of configurations having extremely high energy, so such configurations can never appear spontaneously during the evolution. Algorithmically, it is much simpler, though, to specify for forbidden transformations.

20 sec

40 sec Fig. 2. Shrinking of a 5-sided grain in a 2-D SCN-polycrystal.

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FRADKOV et al.:

GRAIN GROWTH IN 2-D MICROSTRUCTURES

///

10

(a)

8 7

ff(r/r)

~(al~) 6

z 6 5

5

4 3

2

(b)

01s

I

,!5

9

8

~(a/~)

o'.s

~

l!s

~

2.s

of a 3-D polycrystal [3].

~(a / ~)

3

o

r/r Fig. 4. Mean topological class vs normalized grain radius calculated as radius of area equivalent circle for 2-D section

a/d

;

,.s

*/~/

(c)

! 8~

!

5

linear, corresponding to the so called parabolic growth law for the average grain diameter. Topological class and area distributions fit well to those observed in experiments on A1 foils [6] as well as the dependence of the mean topological class on the area (see again Fig. 3). The main feature of this simulation is that every switching and vanishing event may be recorded during the evolution. Figure 8 shows the probability of a switching event for a grain as a function of its area. The plot shows that small grains do not lose sides faster than large ones. Both in situ experiments and Monte Carlo simulation indicate that in 2-D polycrystals, grains with n = 4 and n = 5 predominantly shrink to vanishing (or almost to vanishing) without losing sides, in contrast to conventional wisdom, and in contrast with observations on 2-D sections of 3-D polycrystals. The underlying dynamic and topological behav-

4

(a)

3

o'.s

~

~'.s~

~

o 1 110 01

2.s

a/#

Fig. 3. Mean topological class of grains vs normalized grain area. Data from experiments on (a) ahiminium foil [6], (b) SeN films [251 and (c) Monte Carlo simulation [331.

grain boundaries. The system is thereafter decreasing its energy by the shrinking of these boundaries. The driving force is, in fact, "capillary" in nature, because each such boundary possesses excess specific energy proportional to its average thickness. If the temperature, T, is low enough (compared to the energy difference between bulk sites and boundary sites), this thickness is about unity, and the higher the temperature the thicker the boundaries (the mean width of the boundaries is approximately proportional to the temperature). Thus, the unwanted effect of basic lattice anisotropy in the system can be controlled. Figure 6 shows a series of configurations of a region of the system with T = 3. The initial number of nuclei was 25,000, with the size of the structure 2000 x 2000 pixels. Figure 7 shows the mean grain area d vs time t. The asymptotic behavior is close to

1 o 011 10

O0 111 O0

01 010 i0

10 010 Ol

ii 010 ii

Ol iii i0

i0 iii Ol

I0 0 i0 i0

Ol Ii0 O0

i0 011 O0

Ol 0 i0 Ol

O0 0 ii i0

O0 ii0 Ol

Ii 010 i0

Ol 111 O0

i0 0 Ii Ol

Ol 0 i0 II

O0 111 I0

i0 110 Ol

ii 010 Ol

Ol 011 i0

O0 111 Ol

i0 010 11

Ol 110 i0

i0 111 O0

ii 0 ii i0

Ol 111 Ol

i0 011 ii

Ol ii0 ii

I0 iii i0

ii 110 Ol

(b) 00 0 0 0 00

11 111 11

Fig. 5. Local configurations used in the Ising-type model providing restrictions on the central site flipping to prevent either (a) grain coalescence or (b) nucleation.

FRADKOV

et al.:

GRAIN GROWTH IN 2-D MICROSTRUCTURES 10000

2723

.....

eooo (sites) 6000

4000

2000

0

40100

0

80100

t

I 12000

16000

(MCS/site)

Fig. 7. Mean grain area as a function of time in the simulation [33].

t = 0 MCS/site

ior in a quasi-2-D polycrystaline film is apparently rather different from the traditional picture that suggests a cascade of switchings occurs before the vanishing of a grain with n > 3. However, neither the simulation nor the experiments allow detailed investigation of the ultimate stage of grain vanishing, because they both do not describe accurately the grains which are physically very small or are comparable to the width of the simulated boundaries. 4. DYNAMIC SIMULATION Direct computer simulation of the evolution of a single 2-D grain shape was performed [34] to elucidate the details of this surprising difference in microstructural behavior. The equation of motion of isotropic boundaries moving via "atomic flow by mean curvature" is v, = myK (6)

t = 3000

MCS/site

where v, is the normal velocity of the element of the boundary, and K is the local curvature of this element. This equation is numerically integrated during the dynamic simulation [34]. Each boundary is represented by an array of points as shown in Fig. 9. The explicit scheme of integration is used, with the displacement of each point proportional to the local curvature evaluated by a 3-point osculating circle for every time step. Between time steps the points are 0.3

0.2

T 0.1

Tt ) 0

+

i

a / d

t = 1 2 0 0 0 MCS/site Fig. 6. Typical evolution of a sample of the microstructure in the Monte Carlo simulation of the Ising-type model [33].

Fig. 8. Frequency f of topological rearrangements n-+(n - l) per grain boundary vs normalized grain area. Error bars are estimated by data scattering during simulation.

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FRADKOV et al.: GRAIN GROWTH IN 2-D MICROSTRUCTURES

Fig. 9. Vicinity of a threefold vertex illustrating the computer simulation model [34]. rearranged in the tangential direction to maintain a roughly uniform boundary point density. If the point density increases due to the shrinkage of the boundary, then the algorithm removes some points. When there is only one point remaining at a boundary and shrinkage continues, then the program executes a switching event; rather than eliminating this last point, the boundary in question changes connections with its adjacent boundaries. In the case of a lens (n -- 2) the program removes the entire grain from the system instead of switching the side. If, on the other hand, the point density decreases due to extension of the boundary, then the algorithm inserts new points using parabolic extrapolation. The average density of points is not constant during the simulation. The density changes to maintain a constant number of points at the boundaries belonging to the shrinking grain. This method maintains good accuracy even for small grains. The co-ordinates of the positions of the triple junctions are chosen to make the limiting angles between the boundaries exactly 120°. Algorithms to perform similar computations were independently developed earlier and implemented in [35-38]. As an initial state for the simulation a single grain is chosen with straight boundaries, shaped as a

Fig. 11. Simulation of 4-sided grain vanishing [34].

Fig. 10. Typical initial configuration for the computer simulation [34],where ¢ is the measure of the environmental asymmetry of the central grain.

perfect square (n -- 4) or pentagon (n = 5). External terminations of the outgoing boundaries are pinned at a circle concentric to the internal grain (Fig. 10). The positions of external points are changed from run to run to investigate the environmental influence on grain shrinking. Independent of this environmental asymmetry, equation (1) holds in this simulation with an accuracy better than 1%. Figure 11 shows that a shrinking 4-sided grain embedded in an asymmetrical surrounding apparently does not tend to lose a side. Rather, the grain

FRADKOV et al.:

2725

GRAIN GROWTH IN 2-D MICROSTRUCTURES

seems to shrink intact. Similar behavior is exhibited by a 5-sided grain. However, the detailed investigation of the behavior of grains just before vanishing shows that they ultimately undergo switchings, consequently becoming 3- and 2-sided (see Fig. 12). Figure 13 shows the ratio of the 4-sided grain area at the moment it loses its first side to the initial area of the grain as a function of the asymmetry of the external configuration. Note, the angle q~ (see Fig. 10) is used as a measure of such asymmetry. One sees that for reasonably asymmetrical configurations the grain shrinks to a very small area before changing the topology. This phenomenon is referred to as the topological stability of shrinking 2-D grains [39]. Note, that although these results formally correspond

100

a

,

a°

1 0 -I

10-2 10.3 10-40

liO

210

310

410

5i0

6i0

710

80

~o (deg.)

Fig. 13. The ratio of the minimal area a 4-sided grain reaches without topological rearrangements a* to the initial area of the grain a0 vs the anisotropy of the surrounding structure measured by the angle