~cra merail. Vol. 33, NO. 12, pp. 2233-2247, 1985 ooo1-6160/85 t3.00 +o.oo printed in Great Britain. All rights rcsxved Copyright 0 1985 Pcrgamon Press Ltd

COMPUTER SIMULATION OF GRAIN GROWTH-V. ABNORMAL GRAIN GROWTH

D. J. SROLOVITZ,‘J G. S. CREST’ and M. P. ANDERSON’ ‘Corporate Research Science Laboratory, Exxon Research and Engineering Company, Annandale,

NJ 08801 and %coretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A.

(Received 24 September 1984)

Abstract-Monte Carlo computer simulation techniques have been utilized to investigate abnormal grain growth in a two dimensional matrix. The growth of abnormally large grains is mod&d under two conditions: (a) where the driving force is provided solely by curvature and (b) where the driving force is provided by the difference in the gas-metal surface energy bctwccn grains of dil%xent cryrtrllographic orientation. For curvature driven growth three cases arc consider& (a) the growth of abnormally large grains in microstructures without grain growth restraints, (b) the growth of abnormally large grains in microstructures with particle dispersions, and (c) grain growth in a particle pinned microstruaure in which a sudden decrease in the number of particles occurs. In all these cases, the initiation of abnormal @n growth/secondary elization is not found to occur. In systems free from grain growth ratmints the norrml grain size distribution is very robust and strongly resistant to perturbations. For m which contain particle dispersions strong pinning of the grain boundaries is always observed. However, when a preferred surface energy orientation is introduced, abnormal grain growth/secondary mtion does take plaa. The microstru&ml evolution observed during secondary naystrllipton is in good correspondence with experiment. The area fraction of secondary grains exhibits sigmoidal behavior as a function of time, and is characterized by an Avrami exponent of 1.8 f 0.3 when fit to a mod&d Avmmi equation.

R&umCNous avons utili.6 la technique de simulation sur ordinateur de Monte Carlo pour &udicr la croissance anormale dcs grains dans une matice bidiicnsionnelle. Nous avons mod&I& la c&sancc anode da grains dans deux conditions: (a) la force motricc cst foumie uniquemcnt par Ia anubure et(b)laforcemotrice~fournie~~~~d’Cnergie&lasurfaceg;lEmsCtrlpourdagninr d’orientations di&cntes. Dans k cas d’une croissance par courburc, nous m trois cu: (a) la croiwance de grains anormalunat gros dans dcs microstructures sans entrave i la m da grains, (b) la croissancc de grains anonnalanent gros dans du microstructures avccdesdispcnIonsdepartia@ et(c)lacroisMnades~duuunemicrostructureancrCepardes~~ctduuIrqod)eil# produit un abaisscment soudain du nombre de particulcs. Dans aucun cas, nous n’wom trouvh de &but cmbancc anommk da grains (ra&allisation sccondairc). Dans its syst&na sans cntravcd h rcputition normale de la taille dcs graim est tr& stable et r6siste t&s fortement aux pcrhubations. Dans la syst&ma contenant une dispemion de par&&s, on observe toujours un fort ancrage da joint, de graIna_ C!4pcndant, lorsqu’on intmduit une orientation d%ncrgie nupcriiciclle p+entieII~+ Ia m anode dcs grains (rec&dIisation sccondairc) se produit. L’Cvolution de la rmaortrudun observ& au co~de~~~ti~~~~enbonaccordavec:l’exptriena. Lafractiond~surfacedagrains sccondairea pr&nte un comportcment sigmoidal en fonction du tcmps et eUe eat cam&is& par un exposant d’Awami de I,8 f 0.3 quand on l’ajustc $ unc Equation d’Avrami mod%&.

m_Das anomalc Kornwachstum in ciner zwcidimcnsionalcn Matrix wurdc mit ciner MonteCarlo-Simtition untcrmtcht, Das Wachstum anomal groDer KBrner wird untcr zweI Dedineungen modcllicrt: (a) die trcibende Kmft fuhrt ausscbliel3lich von der Kriimmung her und (b) die trcibende Km!I cntstcht aus dcr differcnz in dcr Encrgie dcr Gas-Metall-GrcnzBiiche zwischcn Kiimcm untcrachiallicher kristallograftscher Oric&mng. Fiir den Fall der Ktimmung w&en drci F%lle untcrschicdex (a) das Wachstum anomal groRcr K6mcr in Miistrukturen ohne Einschriinkungen im Komwachstum, @) das Wachstum anomal gro&r Kbmcr in Mikrostrukturen mit ciner Tcilcbendispcrsion, und (c) Komwach- stum in cincr durch T&hen vcrankcrtcn Mikrostmktur, in der tin pl&licher Abfti in der T&hen&l dntritt. In all diucn FUlen tritt dcr Bcginn anomalcn Komwachstums/sekundgrcr Rek&aI&tion nicht auf. In den Systemen ohne EnschrUung da Komwachstums ist die notmale Vcrteihmg dcr Komgri&n s&r robust und s&r widerstandsfiihig gcgcniibcr StCrungcn. Bei den Systemen Systemen mit Teilcben- dispersionen wird immer eine starke Vcrankcrung dcr Komgrcnzcn bcobachtet. Wird jcdoch eine encrgctisch bevorzugte Oberll5achenoricnticrung cingcflihrt, dann tritt anomales Komwachsh& sckundgrc RekrisUisation auf. Die Entwicklung dcr Mikrostruktur W&rend dcr Sekundgrtn Rekristaliisation stimmt gut mit dan Experiment &b&n. Dcr Flkhenateil da aehmd&a K&IKX zeigt tin s-f&m&s Vcrhaltcn in Abhitngigkeit von dcr ikit; dicscs ist &amkt&icrt durch &en Avrami- Exponenten von 1.8 f 0.3. wenn tine modiicrte Avrami-Glcichung angepagt wird.

1. INTRODUCTION crystalline aggregate at& primary recrystallization

Grain growth is the term used to describe the increase is complete. Two diRerent types of grain growth

in grain size which occurs upon annealing a poly- phenomena have been distinguish. Normal grain growth is said to occur when the microstructure

2233

2234 SROLOVITZ er al.: COMPUTER SIMULATION OF GRAIN GROWTH-V

exhibits a uniform increase in grain size [ 11. This type

of growth is a steady state kinetic process character- ized by time invariance of the normalized grain size distribution function F(R/R) and the topological distribution function P(N,), where R is the grain radius and N, is the number of grain edges. In this case the mean grain size R increases with time as ~1.

R = kt” (1)

with isothermal annealing, where k is a constant which is a function of temperature and n ij 0.5. The steady state evolution of the microstructure during normal grain growth may be described in terms of (1) a driving force which is associated with the reduction in total grain boundary energy and (2) the rate theory model of boundary migration. Through the use of computer simulation employing these basic assump tions we have shown that all of the characteristic features of normalgrain growth (topology, kinetics, grain size distribution, grain size-topology relation- ship) can be properly reproduced. Our basic model procedure and properties of the microstructure are presented in papers I and II (Refs [3,41). Extension of the normal grain growth model to include growth in either the presence of a particle dispersion or with anisotropic grain boundary energies is described in papers III and IV (Refs (5, a]), respectively.

The second type of grain growth phenomenon which can occur after primary recrystallization is abnormal grain growth. Some confusion exists in the literature over the proper definition of this term and the associated driving force. In the context of the present paper abnormal grain growth is equivalent to secondary recrystallization as defined by Detert [7]. Detert defines secondary recrystallization as the rapid increase in sim of a few grains in the recrystalhzed microstructure such that topology is not time invari- ant and the maximum grain size increases at a rate much faster than the arithmetic mean. This type of growth is a transient process which can be described in some cases by the kinetic model of Johnson and Mehl [8] and Avrami (91.

,X = 1 - expj--g(t)] (2)

where X is the volume fraction of secondary re- crystallized grains, and g(t) is a general function of time. It is commonly observed that secondary recrystallixation requires normal grain growth to be strongly impeded, with the exception of a few grains which act as nuclei for secondary recrystallization [7, IO]. Inhibition of normal grain growth is attributed to a number of mechanisms:

(1) Grain boundary grooving in the case of thin lllms and sheet materials.

(2) Particle pinning of boundaries. (3) Texture inhibition in a material with

strong preferred orientation. (4) Impurity inhibition.

The growth of individual secondary grains into the matrix of stable recrystallized grains has been charac- terized as exhibiting a linear increase in radius with time, with the growth velocity obeying an Arrhenius temperature dependence [ 11,121. The driving force is generally assumed to be provided by the associated reduction in total grain boundary energy, as for normal grain growth. However, an additional driving force has been experimentally demonstrated in thin Illms and sheet materials, which arises from orjen- tation dependence of the gas-metal surface energies [13-l!q.

The most complete theoretical treatment of ab- normal grain growth/secondary recrystallization has been given by Hillert [la]. Hillert discusses secondary recrystallization in the context of a mean field for- malism, in which the driving force is derived solely from the decrease in total grain boundary energy. He distinguishes two ways in which abnormal grain growth can occur. In the first process grain growth takes place in the absence of an inhibition mechan- ism, but primary recrystallization has resulted in a broad initial grain size distribution. Those re- crystallized grains having a size greater than 21 in two dimensions or 9/4 R in three dimensions are predicted to exhibit abnormal growth. The second condition under which Hillert treats secondary recrystallization is in the presence of a particle dispersion. In this case he derives necessaq (but not sufllcient) requirements for initiation. He predicts that abnormal grain growth can develop in a material if three conditions are simultaneously satisfied:

(1) Normal grain growth cannot take place due to pinning by second phase particles.

(2) The microstructure has pinned at a mean grain size such that the absolute value of the Zener back stress due to the particles is less than the absolute value of the average capillar- ity pressure.

(3) There is at least one grain with a size much larger than the mean such that the capillarity pressure associated with its bound- ary is less than the difference between the absolute value of the average capillary pressure and the absolute value of the Zener back stress.

Hillert suggests that these conditions might be established in the microstructure if normal grain growth has proceeded to pinning at a tixed number and size of particles, and then a time dependent decrease in the Zener back stress occurs due to a re- duction in the number particles (either by dissolution or Cktwald ripening).

Hunderi and Ryum have also examined abnormal grain growth by two separate procedures [17,181. In the first method they performed a computer experiment in which an array of spherical bubbles underwent grain growth in the presence of a Zener back stress [17]. They found complete pinning of the system for large values of the Zener drag, but

indications of a direct transition to abnormal grain growth for small values. They also considered the effect of Zener drag on grain growth by obtaining time dependent solutions to the Hillert mean field equations [18]. Starting with Hillert’s normal grain size distribution, they numerically integrated the mean field equations to obtain the kinetics and the grain sire distribution. In this case pinning was predicted for all values of the Zener drag and no transition to abnormal grain growth occurred.

In summary, these treatments suggest that second- ary recrystallization can occur in one of three ways:

(1) If an abnormally large grain (greater than 2R in two dimensions or 914 R in three dimensions) is introduced into the grain ensemble in a system free from grain growth restraint.

(2) If particle retardation of grain growth takes place but the Zener drag is small and the system has not completely pinned.

abnormal grain growth are nearly identical to those developed and employed by the present authors in previous studies of normal grain growth [3-6]. In short, a continuum microstNCtUrc iS mapped onto a two dimensional triangular lattice containing 40,000 lattice sites (see Ref. [3]). Each lattice site is assigned a number, S,, which correspoads to the orientation of the grain in which it is embedded. The number of discrete grain orientations is Q. Lattice sites which are adjacent to neighboring sites having different grain orientations are regarded as being part Of the grain boundary, whilst a site surrounded by sites with the same grain orientation is in the grain interior. The grain boundary energy is specified by associating a positive energy with grain boundary sites and Nero

energy for sites in the grain interior, according to

Ei = -J 1 (&,s, - 1) (3) M

(3) If the microsttucture is completely pinned by particles but a time dependent decrease in Zener drag (through a reduction in the number of par- ticles by dissolution or ripening) takes place.

The models used to make these predictions assume that grains are spherical and growing in an average environment. Furthermore, the only driving force considered is that due to reduction in total grain boundary energy.

In the present paper we report a new approach to abnormal grain growth utilizing a Monte Carlo com- puter simulation technique. With this procedure we are able to include in our model grain boundary topology, detailed local environment, complex boundary-particle interactions and gas-metal surface energy effects. The first part of the paper deals with curvature driven growth. We examine for a two dimensional matrix (a) the growth of abnormally large grains in microstructures without grain growth restraints, (b) the growth of abnormally large grams in microstructures with particle dispersions, (c) grain growth in a pinned microstructure in which a sudden decrease in the aumber of particles occurs. The second part of the paper deals with grain growth in two dimensional systems in which the surface energy of a particular crystallographic plane is lower than the remainder.

Where 6, is the Kronecker delta, the sum is taken over nearest neighbor (NI) sites and J is a positive constant that sets the energy scale of the simulation. In the present simulations Q was generally taken to be 48, with exceptions as noted below. The grain boundary energy in this case is nearly isotropic. Previous simulations have indicated that a reasonable degree of anisotropy in the grain boundary energy does not have a strong effect on microstructure or kinetics (see paper rv) [6].

The kinetics of boundary motion are simulated employing a Monte Carlo technique in which a lattice site is sehted at random and its orientation is randomly changed to one of the other grain oriea- tations. The change in energy associated with the change in orientation is evaluated. If the change in energy is less than or equal to zero, the re-orieutatioa is accepted. However, if the change in energy is greater that zero, the re-orientation is accepted with a probability

P = exp - (AE/kT). (4)

We find that for curvature driven growth initiation of secondary recrystallization does not occur. In systems which are free from grain growth restraints the normal grain size distributioa is very robust and strongly resistant to perturbations. For systems which coataia particle dispersions strong pinning of the grain boundaries is always observed However, when a preferred surface energy orientation is intro- duced, abnormal grain growth does take place.

2. SIMULATION PROCEDURE

Time, in these simulations, is related to the aumher of re-orientation attempts. N m-orientation attempts is arbitrarily used as the unit of time and is refd to as 1 Monte Carlo Step (IVES), where N is the number of lattice sites (40,000). The conversioa from MCS to real time has an implicit activation energy factor, c - wILr, which corresponds to the atomic jump frequency. Since the quoted times are normal&d by the jump frequency, the only effect of choosing T r 0 (as done in the present simulations) is to restrict the accepted m-orientation attempts to those which lower the energy of the system. Re-orientation of a site at a grain boundary corresponds to boundary migration. The boundary velocity determined in this manner yields kinetics that are formally equivalent to the rate theory model [3]. While this procedure is the

The procedures employed in the simulations of basis for our simulation, several improvements to the algorithm have been added for the sake of efIicient

SROLOVITZ ef al.: COMPUTER SIMULATION OF GRAIN GROWTH-V 2235

2236 SROLOVITZ et al.: COMPUTER SIMULATION OF GRAIN GROWTH--V

utilization of computer resources (see Ref. (191 for details).

In simulating abnormal grain growth/secondary recrystalkation for curvature driven growth (se&on III), several different procedures were employed. First, a normal grain growth microstructure is allowed to develop (as dkussed in paper I) and evolve for 1OOOMCS. This gives a normal grain growth grain sixe distribution, which has an apparent cut-off (maximum) grain sixe between 2.5 and 3 times the mean grain radius. Sina previous theories predict that a grain which is much larger than the mean grain size (and larger than the cut-off grain sixe) may undergo abnormal grain growth [la], a large grain was introduced into the normal grain growth micro- structure. This was done by replacing the grains at the center of the microstructure with one large circular grain. The radius of the new grain was chosen to be 5, 10 15 or 20 times the mean grain radius. Following the introduction of the large grain, the Monte Carlo procedure was started and the evolution of the microstructum was monitored to determine if abnormal growth occur&.

The remahing procedures employed to study curvature driven growth relate to abnormal grain growth in a material containing a dispersion of second phase particles. As a starting point for these studies, a microstructum was produced by allowing grain growth to occur in a system containing a 2.5% area fraction of particles. The details of how this microstrucmm was produced may be found in paper III. After approx. 10,000 MCS, grain growth had stopped and the microstructure was pinned. Experimentally, it has been observed that in pinned microstrWures, the removal of a fraction of the particles by dissolution or Ostwald ripening can result in abnormal grain growth [7,20]. Three methods were employed to simulate this type of behavior. In the first, approx. 40% of the par- ticles were removed at random and the micro- structure was aIlowed to evolve. In the second, all of the particles in one small region of the micro- structure were removed and the Monte Carlo pro- cedure was started. In the last of the abnormal grain growth studies, a large grain is artificially introduced into the pinned structure and the evolution was monitored to determine whether abnormal growth OCCUTS.

In abnormal grain growth in thin films or sheets it is observed that the surfaa energy of a grain depends on both the crystallography of the grain’s surface planes and on the atmosphere [ 13151. If, in a given atmosphere, certain grains have a much lower energy than surrounding grains (due to the surface energy effect), it is speculated that these grains will grow faster than the others-resulting in abnormal grain growth. In the next set of simulations investigating abnormal grain growth (section IV), the energy of a favorably oriented grain is lowered by an amount proportional to the surface area (in two dimensions).

This leads to the following site energy

~=-HC&At-~z(6scr-l) (5) k *I

where the second term on the right is identical to that in equation (3), and corresponds to grain boundary energy. H is the magnitude of the energy by which individual. sites in favorably oriented grains are favored and the delta function in the first term is one if S, is one of the favored orientations and zero if it is not.

In this study, the normal grain growth simulation procedure [quation (311 was run with Q -48 for 1000 MCS to obtain a normal grain sixe distribution function. Then each grain was relabeled ‘such that each grain had a unique grain orientation. This yielded a structure with approx. 1000 grains and hena, Q was approx. 1000. Qk @r equation (5)] was then a list of favored grain orientations, chosen at random from the Q grains. &cause Q was so large, ordinary Monte Carlo simulation techniques would be too slow for cilicient computing and therefore the more efhcient continuous time method (see Ref. [19D was employed. For convenience, the time has been resealed for this very large Q case to that of the standard Q = 48 simulations. It was found that the grain boundary velocity was nearly independent of H (equation (5)], except for very small grains (l-3 sites) and very large values of H. H - 0.1 J was employed for the study of surface energy effects described in section IV.

3. CURVATURE DRIVEN GROWTH

The first simulation performed investigates the stability of the grain size distribution function devel- oped during normal grain growth. In this case H = 0 in equation (5) and the only driving fora is that provided by curvature. A microstructure produced by the grain growth simulation technique described in papers I and II with Q = 48 was allowed to undergo normal grain growth for 1000 MCS. It was then modified by artificially creating a large circular grain in its center. Initial circular grain sixes of 5,10,15 and 20 times the mean grain size (radius) were used. The resulting microstructures are displayed in Fig. 1 for three different times for the case where &/K is initially 5. & is the initial radius of the large circular grain and K is the mean grain radius, excluding the large circular grain. This figure shows that although the initial grain is much larger than all its neighbors, its growth rate is slower. This is evident from the figure, where it can be seen that by approximately 15,000 MCS other grains in the structure are of comparable size.

The time dependence of the area of the initially circular grain, A,, is plotted as a function of time in Fig. 2. The data show that, for a given initial &/II, the large grain grows very slowly at first then gradu-

SROLOVITZ et crf.: COMPUTER SIMULATION OF GRAIN GROWTH-V 2231

0.15 -

0.0 1.0 2.0 3.0 4.0

Wd

Fig. 2. The time dependence of the logarithm of the ratio of thearcasofthelargegain,~,toibinitial~A,(r=O). The data is presented for initial huge grain radii (at 1000 MCS) of (a) 5, (b) 10 and (c) 20 R. Data for two sinu~lations

arc plotted for each initial grain radius.

ally increases as its’ growth rate. The slow starting growth is associated with the readjustment of the grain shape from a nonphysical circle to one which is compatible with the remainder of the microstructure. Over the time range of the simulations power law growth for the abnormal grain is never observed, as evidenced by the curvature in the log-log plots of Fig. 2. The time averaged growth rate for the abnormal grain is significantly lower than the mean growth rate for the remaining microstructure. Further, the data show that the time averaged growth rate dareases as the initial RJR is made larger. These observations suggest that the abnormal gram may be a temporary feature of the microstructure. That is, the abnormal grain grows at a rate slower than the remainder of the microstructure until it is absorbed into the normal grain sixe distribution. To check this possibility, the ratio of the area of the large grain, 4, to the mean grain area, 2, was monitored for simulations employ- ing different initial RJR values. For any given b/K, A,,/Z could only be followed for a limited interval before the number of grains became too small to provide statistically significant results. Therefore, the initial RJR values for successive simulations were chosen so that the &/A values measured at the end of one simulation coincided with the starting &?’ values for the next. These data are displayed in Fig. 3. This figure clearly shows that the difference in size between the initially circular grain and the mean grain sixe of the remainder of the microstructum is decreasing with thne. To further check this aspect, one simulation was run employing a much smaller initial average gram sixe by allowing normal grain growth to evolve for only 100 MCS per site. An abnormally large grahr of initial sixe &/R = 10 was then introduced and the simulation restarted. The microstructural development was followed until the abnormal grain was completely absorbed into the normal grain size distribution.

2238 SROLOWTZ et ~1.: COMPUTER SIMULATION OF GRAIN GROWTH-V

Figures 1 and 3 demonstrate that the normal grain growth grain size distribution function for curvature driven is remarkably stable against perturbations. Furthermore, these results contradict suggestions that abnormal grain growth is a direct consequence of the instability of the grain size distribution function with respect to large grain sizes. The present simulation indicates that the experimental observa- tions of large grain initiation of abnormal grain growth are due to some feature not normally associ- ated with curvature driven growth; for example, strain energy effects, impurity effects, diffusional relaxation of mechanical constraint, etc. The theor- etical basis for the speculation that abnormal grain growth is initiated by unusually large grains may be traced to instabilities in the mathematical solutions for the grain size distribution functions in the mean field theories of grain growth (e.g. Ref. [la]). These points are further addressed in section V below.

Since abnormally large grains do not of themselves lead to abnormal grain growth, it was decided to investigate abnormal grain growth in the presence of particle dispersions. In such systems, it has been suggested that particle disappearance (dissolution, coarsening, etc.) can lead to abnormal growth (7,161. Such a situation has been investigated by using a microstnmture where grain growth has become com- pletely stopped, due to the presence of the particles. Following complete pinning of the microstructure a&r 10,000 MCS, the concentration (area fraction) of particles was reduced from 2.5 to 1.5%. The resultant microstructure is displayed in Fig. 4 for three di&ent times. This microstructural time sequence clearly shows that abnormal grain growth does not develop. The evolution of the mean grain size following the change in particle concentration, at 10,000 MCS, is plotted in Fig. 5. This figure indicates

Fig. 3. The time dependence of the ratio of the areas of the large grain, A,, to the mean grain size, A. The data is presented for initial large grain radii (at 1000 MCS) of (a) 5, (b) 10 and (c) 20 R. The data in this plot were averaged

over two simulations for each grain radius.

SROLOVITZ ef al.: COMPUTER SIMULATION OF GRAIN GROWTH-V

200 t f

i :

j # I

i 2.5% W.SH

. . . I I

100 -I__- 1 I. ~ __.J 1.0 3.0 5.0 7.0 9.0

1( 10')

Fig. 5. The mean grain size (ama) for the microstructure in Fig. 4 plotted against time.

that the change in particle concentration results in transient growth, which quickly saturates. For the simulation results plotted in Fig. 5, the ratio of the tinal grain area to the initial grain area was approx. 2.0 as compared with the ratio of the initial to final particle concentration of 1.8. These results indicate that the transient growth saturates at a final grain area which is consistent with the results of previous simulations. Those simulations showed that the final grain area scales inversely with area fraction of particles in two dimensions.

Unlike in the simulation just described, particle disappearance in real systems may occur preferen- tially in one part of the microstructure due to non- uniformities in temperature, solute concentration, stress, etc. This case has also been examined to determine if the observed abnormal growth in par- ticle dispersions is due to the unpinning of a small group of grains, while the remainder of the micro- structure stays pinned. The results of such a simu- lation are presented in the microstructural time sequence of Fig 6. In this case, it is seen that grain growth proceeds within the particle free region, while the remainder of the structure remains pinned. Even in the region just a few grains away from the particle free region, grain growth is completely restrained. Since abnormal grain growth does not seem to be initiated by unusually large grains nor by the removal of particles and the subsequent grain growth in a small region of a particle pinned system, it should not he surprising that abnormal growth does not occur when a vary large grain is artificially introduced into a pinned microstructure. A single micrograph resulting’ from this type of study is shown in Fig. 7 where, indeed, the large grain rapidly becomes pinned.

While abnormal grain growth does not result from perturbations to the grain size distribution, as shown

2240 SROLOVITZ et al.: COMPUTER SIMULATION OF GRAIN GROWTH-V

1=30#00 Mcs

Fig. 7. The final pinned microstmcture resulting from the introduction of 8 large grain at the center of an otherwise pinned microstructure. The starting structure was produced bygraingrowthinthepreaence of a 2.5% particle dispersion

which pinned a1 10,000 MCS.

above, another possible source of abnormal growth is the break away of a large grain’s boundary from the particle dispersion. In our previous simulation of grain growth in the pre3ence of particles we showed that the condition for the particles to stop grain growth, is that there must be, on average, one particle per grain edge. Assuming the large grain is very much larger than the surrounding grains, the critical grain boundary con&ration is tie that schematically illustrated in Fig. 8. In tb;s highly idealized depiction of the microstructure around the large grain, it is easily seen that the particle behaves like a vertex where three grain boundaries meet. Provided that the small grains are of approximately the same size, the angle that the grain boundaries make at the particles is, on average, 120”. For the growth of the large grain to continue a passing angle of greater than or equal to 120” is required. Since passing angles are typically of order only a few degrees, no bypassing can occur and the growth of the large grain is stopped. This simple argument supports the simulation results and suggests that elimination of grain boundaries can not be the predominant driving force for abnormal growth.

4. SURFACE ENERGY DRIVEN GROWTH

Since the above simulation results suggest that the driving force for abnormal grain growth is not the elimination of grain boundaries, other types of driving foras must be examined. In particular, volu- metric driving forces may be important. These types of driving forces include elastic strain, residual dis- location density and in two dimensions, differences in surface energy. For the latter case, experimental studies on sheet materials have indicated that modi- fication of the surface energy by changing gaseous atmospheres can result in abnormal growth [13-151. It is Speculated that the mechanism by which abnor- mal growth occurs is the reduction in energy of grains

with “favored” crystallographic orientations with

respect to the surface. In b.c.c. materials the favored crystallographic orientations are either (110) or (100). In this section, grain growth is studied for this type process, i.e. where the energy of certain grain orientations is lowered with respect to the other or “nonfavored” grains. As is customary in the litera- ture nonfavored grains are referred to as matrix grains and favored grains are referred to as second- aries [lo].

In these studies, a grain growth microstructure, produced by a 1000 MCS run of the simulation, procedures outlined in I and II, was employed and a fraction, J of the grains are randomly selected as initial secondaries. f was chosen as either 2, 5, 10 or 20%. The simulation procedure described in section

II above with E, given by equation (5) (H # 0) was employed and the simulation procedure was stopped when the total area fraction of secondary grains, X, was 1.

Figure 9 shows the microstructural evolution of a stru&ure where initially 2% of the grains were stc- ondaries. The shaded regions in this figure indicate matrix grains. Figure 10 shows similar micrographs for which~titially the fraction of secondary grains,f, is 10%. It is seen that the secondaries grow at a much more rapid rate then do the matrix grains ,and that after a relatively short time, the entire microstructure is composed of secondary grains. Comparison of Figs 9 and 10 indicated that the time required to transform the entire structure is a function of the initial percentage of secondary grains. This is made more quantitative in Fig. 11 where the total area fraction qf secondary grains, X, is plotted against time for two runs each at f = 2,5,10 and 20%. Figure 11 clearly shows that the time required for complete transformation decreases with increasing initial j

While the abovq data shows that X initially in- creases linearly with time, experimentally it is often observed that an incubation period occurs prior to the onset of abnormal growth [lo]. This discrepancy may be accounted for in two ways. The lirst possi- bility is that the incubation period re&cts kinetic processes related to developing an equilibrium sur- face energy with respect to the furnace environment

Large Grain

Fig. 8. idealized illustration of a large grain adjacent’to a fine,grain structure that is pinned by particles. This type of

configuration may only be valid in two dimensions.

SROLOVITZ et al.: COMPUTER SIMULATION OF GRAIN GROWTH-V 2241

Fig. 9. Microstructure undergoing abnormal grain growth. The shaded grains are the matrix grains and the unshaded grains are the secondaries. The initial microstructure was produced by a 1000 MCS run of the normal grain growth simulation procedure. For this simulation 2% of the grains were initially selected

at random to be secondary grains.

(i.e. adsorption). The second possibility is that in real systems there exists a threshold for the detection of very small grains, whereas in the simulation grains of arbitrary size are measurable.

The plot of x vs t can be analyxed in terms of the transformation kinetics model of Johnson and Mehl, and Avrami [equation (211. Although g(t) is a function of the particular transformation model, transformation kinetics are often analyzed by fitting the experimental data to equation (2) where g(f) = UP. While during most types of trans- formations for which the JMA analysis is employed, X = 0 at 1= 0, in this type of abnormal grain growth a certain fraction of the grains are favorably oriented at t = 0. Therefore, equation (2) must be modified to reproduce the proper value of X at t = 0. The following modification of equation (2) is employed to extract an Avrami exponent, p

X=l-be-“’ (6)

p may he most easily extracted from the simula- tion data by measuring the slope on a plot of log(log[b/(l - x)]) vs log(~). Such a plot is shown in Fig. 12. An important feature of this plot is its nonlinearity. This nonlinearity indicates that the JMA analysis. as modified in equation 6, is not an exact fit to the simulation data. At early times, p in all cases has a value of approx. 1.3. If p is extracted by fitting the data in Fig. 11 to equation (6) over the entire range of the data, it is found that the p values show no systematic variation with the initial

fraction of secondary grains (f). By’averaging the p values obtained from fitting the individual curves, p = 1.8 f 0.3 is obtained.

The increasing slope of Fig. 12 with increasing time is predominantly due to a deficiency in the JMA equation [equation (2)]. According to that analysis X-r1 only as r tends toward co. In reality, however, X always goes to 1 in finite times. When X = 1, the abscissa in Fig. 12, log(log[b/(l - X)j), is mdefined. Apart from this inherent flaw in the JMA analysis, one can still extract a p value from equation (6) in order to compare with experiment or simple theory. Application of JMA to a system with site saturated nucleation (all nuclei formed at r -0) and constant boundary velocity yields an exponent p of 2. While grain boundary velocity is constant in these simu- lations (see below), the assumption of site saturated nucleation is only partially fttl6lled. Although in these abnormal grain growth simulations all of the favored grains (nuclei) initiate growth at the same time, their starting size is not initially zero as it is for true nuclei. Nonetheless, the theoretical value of p is within less than one standard deviation of the mean p value extracted from the simulation.

In order to determine the nature,of growth of the secondary grains into an array of matrix grains a simulation was performed in which one large second- ary grain is embedded into a normal grain growth microstructure of matrix grains. The evolution of this microstructure is shown in Fig. 13. The radius of the secondary grain is plotted against time in Fig. 14 for

2242 SROLOVITZ et cl.: COMPUTER SIMULATION OF GRAIN GROWTH-V

0.6

I

0.4

0.2

0.0

1.0 6.0 11.0

t(rl0')

Fig. 11. Fractional area of secondary grains, X, vs time. The results of two simulations (solid and open circles) are displayed, in which (a) 2%. (b) PA, (c) 1Ok and (d) 20% of

the grains were initially secondaries.

three different initial secondary grain sizes. This figure indicates that the grain radius (&a#) increases linearly with time, as observed experi- mentally [l 1,121. Since the secondary grain remains roughly circular, it is possible to relate the grain boundary velocity to the evolution of grain size: i.e. Y = d&/dz. Hence the grain boundary velocity for a secondary grain growing into the matrix is a con- stant. The value of this constant is independent of the initial size of the secondary grain (see Fig. 14). It is interesting to coinpare the microstructure for this simulation (Fig. 13) with that for a large unfavored

4.0 I I I

2.0 3.0 4.t

Fig. 12. Graphical representation of equation (11) for the data shown in Fig. 10. The value n in the modified JMA equation is extracted from the slope in this plot. The quantity b in the abscissa label equals the area fraction of- matrix grains at the start of abnormal grain growth. The results of two simulations (solid and open circles) in which (a) 2%. (b) So/,, (c) loo/, and (d) 20% of the grains were

initially secondaries.

SROLOVITZ ef a/.: COMPUTER SIMULATION OF GRAIN GROWTH-V 2243

1=2,mOMcs 1=6,000 t=10,000

Fig. 13. The time evolution of a large secondary grain which was embedded in an array of matrix grains at I = 1000 MCS. The initial radius of the secondary grain is 5 times the mean grain radius.

grain growing into an array of unfavored grains (Fig. returns to normal grain growth. During times inter- 1). While the unfavored grain grows in such a way mediate between these two extremes the secondary that it is soon becomes lost in the microstructure, the grains are competing both with the matrix grains and secondary or favored grain grows rapidly. It would other secondary grains. An individual grain may be appear that Fig. 13 is more a case of abnormal subject to both types of competition, or undergo only growth than is Fig. 1. This point will be addressed in one type of competition. Figures 9 and 10 indicate section V, below. that both situations occur.

Since all secondary grains are favored from the beginning of the simulation, when the secondary grains are still small and hence not nearest neighbors, the mean size of the secondaries, &, should scale with the second power of time. A plot of AX vs time is display+ in Fig. 15. This figure shows that initially that d17,..dt increases with time, than becomes a constant and &ally starts to decrease. At early times (near t = 1000 MCS in Fig. 15) all of the curves corresponding to different values off are congruent and increase, consistent with the observations on single secondary grains (Fig. 14). At late times, very few or no matrix grains remain and the system

The evolution of the mean grain size for the matrix grains shows a very different history than that for the secondary grains. This is shown in Fig. 16 where the mean grain size of the mati grains, 4 is plotted vs time. Initially, A is dominated by normal grain growth and the kinetics are as in normal grain growth (see I). At intermediate times the is a competition between continuing grain growth and grain elimin- ation due to the growth of the secondary grains. At very late times, the matrix grains are surrounded by the secondary, growing grains and hence; the areas of the matrix grains quickly decay. The sharp peak in Fig. 16 at late times occurs beamse there are only a

60

40

30

20

10

01 I 1 I I

0.0 5.0 10.0

t(xw

Fig. 14. The radius of a large secondary grain, 3, embedded inanarrayofmatrixgrainsvstime.Curvesa,bandc correspond to initial favored grain radii of 5. 10 and 20 times the mean grain radius after 1000 MCS of normal grain

growth, respectively.

Fig. IS. The mean grain size of the secondary grains ,& vs time. The results of two simulations (solid and open circks) are displayed for initial (I = 1000 MCS) m&xtruU with an area fraction of secondary grains of (a) p/, (b) ;y

(c) loO/* and (d) 2W/W

2600

500

0 1;o 6.0 11:o

i(XlW)

2244 SROLOVITZ CI (II.: COMPUTER SIMULATION OF GRAIN GROWTH-V

200. I I I .

a 160 -

. l _ l O.

. .

. . 0 I t I 0.

1.0 6.0 11.0

YXlo')

Fig. 16. The mean grain sixe of the matrix grains A vs time. The results of two simulations (solid and open circles) are displayed for an initial (t = 1000 MC!I) microstructure with

an initial area fraction of secondary grains of 2%.

few matrix grains remaining. Immediately before the peak there are two matrix grains in the micro- structure, a large one and a small one, with mean area intermediate between the two grain areas. When the small shrinks and disappears, the A is just the area of the large grain and hence A undergoes a large positive

0.3

1 A L

discontinuous jump. The large matrix grain then shrinks rapidly yielding the decay of the peak.

While during the simulation both d and 1, are undergoing substantial changes, the total grain size distribution is a weighted average of these two types of behaviors. At I = 0, the grain size distribution function is just the normal grain growth distribution function, Fig. 17(a). At intermediate times, the grain size distribution function consists of the huge second- ary grains and the much smaller matrix grains. This results in a broad grain size distribution function, as seen in Fig. 17 (b) and (d) where 5Q of the area is occupied by secondary grains and f is 5 and 200/,, respectively. When the transformation is’ complete, the secondary grains compete against each other by normal grain growth and the grain size distribution functions, Figs. 17 (c) and (e), have the same width as those before the onset of abnormal grain growth.

5. DISCUSSION

Abnormal grain growth in the absence of growth restraints, where the driving force is provided solely by curvature, is predicted to occur when a grain is formed with a size larger than the cut-off in the normal grain size distribution function. However, the simulation results unequivocably demonstrate that abnormal grain growth does not occur when grains with R, > kt are introduced. Even with the intro-

6

sow

0.3

[ C

8”’ 1 n 100%

B u. 0.1

&T-J==- -0.5 0.0 0.5 1.0 logdRl?i)

Fig. 17. The grain sire distribution corresponding to (a) a normal grain growth microstructure at 1000 MCS Figures b and c correspond to the microstructure having initially (t = 1000MCS) 2% secondary grains for X = 50 and 1 ooO/& Figures d and e are corresponding distributions for the microstructure having initially loO/, secondary grains. The distributions are subject to increasingly poor statistics with increasing

percent secondary grains due to fewer grains in the microstructure.

SROLOVITZ el al.: COMPUTER SIMULATION OF GRAIN GROWTH-V

duction of grains having Ri = 20R = IO&,, the normal grain size distribution is restored through the continual’decrease of R/k. These results strongly suggest (contrary to common belief and the results of mean field calculations) that the normal grain size distribution is very robust and is stable against even large perturbations,

It is also generally believed that abnormal grain growth requires normal grain growth to be strongly impeded with the exception of a few grains which act as nuclei for the transformation. The inhibition of normal grain is attributed to, among other factors, particle pinning of grain boundaries. Hillert’s analysis suggests that there are two critical grain sizes in particle containing systems: R, is the mean grain size at which normal grain growth is stopped and R, is the smallest individual grain size above which abnormal growth will occur. As the volume fraction of particles decreases, R2 is progressively reduced from infinity to the size of the hugest grain in the population. This then is able to grow provided that it is in contact with grains smaller than R,. This approach was tested in the simulation in two ways: (1) an abnormally large grain was introduced into a pinned structure at 6xed particle concentration and (2) the area fraction of particles in a pinned micro- structure was decreased so as to reduce R2 and increase R,. In neither of these cases; however, did abnormal grain growth occur. These results imply that the inhibition of normal growth and the intro- duction of abnormally large grains is not a sufficient condition for the instigation ofabnormal growth. This further demonstrates the remarkable stability of the normal grain size distribution when the driving force is provided by curvature.

Lwe Gntin

Experimental observations in thin sheet materials show that when abnormal grain growth occurs, it is often associated with grain boundary motion in a direction opposed to that of the grain boundary curvature [21]. This is a clear indication that growth of secondary grains is not solely due to capillarity. Furtbermore, examination of the time dependence of the size of an abnormally growing grain indicates that the grain boundary velocity is a constant [ 11,121. Both of these features are consistent with the specu- lation that abnormal growth is a result of a driving force associated with a volume distributed energy, much as is observed in primary recrystallization. For abnormal growth in thin sheets, the weight of experimental evidence identifies the volume energy with the difference in surface energy between grains of different crystallographic orientation.

Fig. 18. One possible route for secondary recrystallization in a three dimensional matrix in thich normal grain growth inhibition has occurred. (a) Idealized residual strains result- ing from primary recrystaliization which a~ held in place by compatibility constraints. Plus (+) corresponds to a grain in compression, minus (-) corresponds to a grain in tension and 0 represents the stress free state. In normal grain growth these stresses can bc uniformly relax4 however, when grain growth is prevented the strain energy is retained. In this cast strain energy is reduced by the growth of the aecond- ary grain, (b). which averages the individual strain, to a

common value characteristic of the large grain.

initial dislocation density is spatially nonuniform, local variations in the density result in a distribution of strains which are held in place by compatibility constraints. This is schematically illustrated in Fig. 18(a). In particle free systems at elevated tempera- tures, differences in strain between neighboring grains may easily be eliminated by gram boundary sliding (e.g. see Ref. [22]). However, when normal grain growth has been stopped by a particle dispersion, grain boundary sliding can be substantially slowed by the presence of the particles on the boundary [22]. In effect the role of the particles is to retain the strain energy in the system.

In bulk materials, the volumetric driving forces When a large grain is formed, however, it finds may be present due to elastic strains remaining itself in a configuration similar to that shown in Fig. after primary recrystallization is complete. These 18(b). The advance of the grain boundary separating elastic strains may arise from the overall decrease in the small, strained grains from the huge grain reduces dislocation density as a result of the primary re- the elastic energy of the system by averaging the crystallization process. Removal of the dislocations strains in the small grains to a common value char- results in a change in the density of the material due acteristic of the large grain. The condition for the to anharmonic effects (see the Appendix). Because the advancing grain boundary to reduce the energy of the

A.M. 33,,2--1

2246 SROLOVITZ et al.: COMPUTER SIMULATION OF GRAIN GROWTH-V

system is given by

v&+v,c:+v~:+... > (V, + V, + V, + . . .)(E)’ (7)

where Vs and 6 are the volume and strain in the large gram and V, and e, are the volume and strain in the grains lost to the advancing boundary. I is the volume averaged strain after passage of the bound- ary. These strain energy considerations give rise to a driving force corresponding to a volumetric energy (strain energy). Furthermore, the model is consistent with the observation that abnormal growth in bulk materials occurs preferentially in systems with a particle dispersion and also accounts for the necessity of having large grains.

While the above picture of abnormal growth is based upon strain energy effects resulting from stored elastic energy due to recrystallization, other sources of volumetric energy are possible. Additional volu- metric energies include: residual dislocation density, solute redistribution, magnetic anisotropy, etc.

Aeknowlcdguncnu-The authors would like to thank I.-W. Chcn for his suggestion to include surface energy driven growth in our study of the abnormsl grain growth problem.

REFERENCES

1. F. Hacasner, Recrystallization of Metallic Materti (alital by F. Haessncr), p. 3. Ricdmr, Stuttgart (1978).

2. F. Hausner and S. Hofmann, Recrystuffizarion o/ Met&c hfareri& (edited by F. Hacssncr). p. 76. Ricdcrcr. Stuttgart (i978). -

3. M. P. Anderson. D. J. Srolovitz G. S. Grest and P. S. Sahni, Acta me&. 32, 783 (1984).

4. D. J. Srolovitz., M. P. Anderson, P. S. Sahni and G. S. Grest. Acla metall. rZ, 793 (1984).

5. D. J. Srolovitz, M. P. Anderson, G. S. Grest and P. S. !&hni. Acta metall. 32. 1429 (19841.

6. G. S. Grest, D. J. Sroiovitz &d M: P. Anderson, Acra meraM. 33, 509 (1985).

7. K. Dttcrt, Recrystallization of Merallic Materials (c&cd by F. Haessner), p. 97. Ricderer, Stuttgart (1978).

8. W. A. Johnson and R. F. M&l, Trans. Am. Inst. Min. Engrs 135,416 (1939).

9. M. Avrami, /. them. pl?yJ. 7, 1103 (1939); ibid. S, 212 (1950); Ibid 9, 177 (1941).

10. C. G. Dunn and J. L. Walter, Recrysrallization. Grain Growth ad Textures (edited by H. Margolin), p. 461. Am. Sot. Mefas., Metals Park. Ohio (1966).

11. F. D. Rosi, B. H. Alexander and C. A. Dube, Trans. Am. INI. Min. I%grs 194, 189 (1952).

12. J. L Walter, J. appt. Phys. 36, 1213 (1965). 13. J. L. Walter and C. G. Dunn, Acta mefall. 4 497 (1960). 14. F. H. Buttncr, E. R. Funk and H. Udin, Lphys. Chem.

56, 657 (1952). 15. D. Kohlcr. 1. appl. Phys. Suppl. 31, 408s (1960). 16. M. Hilkrt, Acta mefoll. 13, 227 (1965). 17. 0. Hunderi and N. Ryum, Acta meroll. 29.1737 (1981). 18. 0. Hundcri and N. Ryum. Acta metall. 30,739 (1982). 19. P. S. Sahni, D. 1. Srolovitz, 0. S. Grcst. M. P. Anderson

and S. A. Safran. Phys. Rev. B 28, 2705 (1983). 20. P. A. Beck, M. L. Holzworth and P. R. Sperry, Trans.

Am. Inst. Min. Engrs 108. 163 (1949). 21. J. L. Walter and C. G. Dunn, Truns. Am. Ins~. Min.

Engrs 218, 1033 (1960). 22. R. Raj and M. F. Ashby. Me/a//. Trans. 2, 113 (1971).

23. G. Albenga, Atti Accad. Sci., Torino, Cl. Sci. Fi. Mar. Natur. 54, 864 (1918119).

24. C. Zcner, Trans. Am. Insr. Min. Engrs 147,361 (1942). 25. A. Sceger and P. Haascn. Phil. Mag. 3, 470 (1950). 26. J. P. Hirth and J. Lothe, Theory of Disbcafions. Wiley,

New York (1982).

APPENDIX

In this appendix we estimate the elastic strain energy remaining in a crystal following the annihilation of a large number of dislocations, presumably due to primary n- crystallization. Within the con&s of linear elastic theory, a change in dislocation density doa not dmngc the density of the material. This result may be derived from a theorem due to Albenga [23], which states that the average stress in an unloaded. equilibrium body is zero. However, in real crystals it is ncessary to also consider nonlinear elastic terms to account for the anbarmonicity of the interatomic interactions. The importance of the nonlinear terms to an elastic analysis may be guagcd by the magnitude of the Gruneisen constant, y

1 dB _ -- y-2 ap ( >

I

where B is the bulk modulus and P is the hydrostatic stress. Employing Zen&s second order elastic formula for the

relationship between dilation and strain energy [24], Sccger and Haasen [2Sj estimated the volume expansion per unit Ien* of dislocation, aV, as

(AZ)

(A3)

where b is the Burgers vector, v is Poisson’s ratio, G is the shear modulus and R and r, arc the outer and inner cutoff radii, respectively. Replacing R by (1/2)p’“, where p is the dislocation density, shows that the dilatational strain, 8. scales as p ln p. Sina the second term in equation (A3) is nearly identical to equation (AZ), equation (A2) will be employed in the estimate of the strain. The strains asscei- ated with dislocation annihilation arc listed in Table 1 for Fe, Ni. Al and Cu and dislocation densities between lo’ and 10” cm-*. The physical constants used in preparing Table I were found in Rcfs [2s] and [Za].

In dctumining whether strain energy will dominate grain boundary energy as a driving force for grain growth it is useful to compare the magnitude of that energies. The

Table 1. Sti 8 and critical radius R, for metal dirlocatba delld&a for four Iypiral meti

Disloulion density Metal

kni’~ FC Ni Al CU

10’0 8 R, 1.6~10' 8x10-' 3x1@

IO" 8 6x IO-' 2x IO-' 3x IO-' I x IO-'

4 4x10-' 4x10-' 1x10-~ 1x10-'

IO" 8 4x10-' 2x10-' 2x lo-' 1x10-' R. 8x IO-' 6x IO-' 3x IO-' 1 x IO-'

SROLOVITZ et al.: COMPUTER SIMULATION OF GRAIN GROWTH-V 2241

grain boundary scales as R,, is found below which grain boundary energy is domi-

E ,2 nant and above which the strain energy due to dislocation

On 2R (A4) anninilation is dominant

and the strain energy scales as R=3v ’ K8’ 646)

E‘$ (AS) R, is tabulated in Table 1. R, is seen to decrease with increasing dislocation density prior td primary recrystalliz-

Combining equations (A4) and (AS), a circular grain size, ation.