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An analysis of the kinetics of stochastic grain growth
Scripta
METAL~URGICA
Vol. 22, pp. 7 8 5 - 7 8 7 , 1988 P r i n t e d in the U . S . A .
Pergamon Press plc All rights reserved
AN ANALYSIS OF THE KINETICS OF STOCHASTIC GRAIN GROWTH
K. J. Kurzydlowski and K. Tangri Metallurgical Sciences Laboratory, The University of Manitoba WINNIPEG, Manitoba, Canada R3T 2N2 ( R e c e i v e d D e c e m b e r 14, 1987) ( R e v i s e d M a r c h 29, 1988)
Introduction In the classical models of grain growth (for example [I]) it has been postulated that the process has a strictly deterministic character, i.e., the rate of growth of an individual grain is a known function of both its size and the mean grain size of the polycrystal. This approach is based on the assumption that surface tension is the driving force for the migration of grain boundaries and that it is proportional to the curvature of the boundaries which in turn is inversely proportional to the size of a given grain. A characteristic on time as
of the classical
models is that they predict the dependence of grain size
R ~ A tn
(I)
where n = I/2. However, experimental studies (for example [2], see also [4]) indicate that the values of n are significantly lower, in the range from I/4 to I/3. This discrepancy between the experimental results and the theoretical prediction may be explained on the basis that the classical approach to grain growth ignores certain aspects of the properties of grain boundaries. For example, Louat [3] pointed out that grain boundary migration cannot be simply related to the size of the grains. The migration of an individual boundary is not controlled by the size of the grain but by other factors, such as the tendency for a boundary to achieve a low energy orientation. In such an extreme case grains will grow or shrink independently of their size. of process is termed stochastic grain growth.
This type
Analytical treatment of stochastic grain growth has been presented by Louat [3]. It is based on an analogy between the diffusion of particles in real space and the movement of grains in the space of grain sizes. This analysis also leads to the conclusion that the mean grain size is proportional to the square root of time which is the same dependence as predicted by the classical models. The main limitation of Louat's model [3] is the applicability of the diffusion equations to describe the changes in the distribution function of grain size. These equations are derived to describe the changes in the position of particles which move independently of each other while the "movement" of grains involves the constrained movement of the interface between neighbouring grains. The growth of a given grain (i.e. its movement towards higher sizes in the space of grain size) is accompanied by the shrinkage of a neighbouring grain (movement towards lower sizes). Grest et al. [4] analyzed the process of grain growth considering a combination of stochastic and systematic grain growth. The stochastic contribution resulted from the assumption that the grain boundary migration depends on the anisotropy of grain boundary energy. Using a computer simulation they found that the exponent n in equation (I) decreases from 0.42 to 0.25 with
785 0036-9748/88 $ 3 . 0 0 + .00 Copyright (c) 1988 P e r g a m o n P r e s s
plc
786
STOCHASTIC
GRAIN
GROWTH
Vol.
22,
No.
6
increasing ansotropy of energy. In another study Chen [4] argued that n can in some circumstances be as low as I/6. These results seem to be inconsistent with both the classical models and the solution for stochastic growth derived by Louat [3] which predict n being equal to I/2. The aim of the present communication is to provide a simple analytical treatment of stochastic grain growth which can be used as an alternative to the estimation given by Louat [3].
Analysis The analysis given below deals with two cases. The first considers purely stochastic growth and the other assumes superpositlon of stochastic and deterministic growth. In purely stochastic grain growth the boundaries migrate randomly (outwards or inwards). This leads to random variations (jumps) in the volume of an individual grain. In grain volume space grains exhibit one-dimensional Brownian motion. C o n s i d e r a p o l y c r y s t a l w h i c h c o n t a i n s n o grains at time t o = O. Growth of some grains required the continuous decrease in the size of some other grains and their eventual annihilation. At time t~ = t the polycrystal contains n~ grains and the ratio n~/n o is inversely proportional to the r a t i o of the mean volume of grains at t o = 0 and t~ = t (v~/Vo). This means that the p r o c e s s can be viewed as growth of n~ grains in a matrix containing (n o - n~) grains. Under c o n d i t i o n s of s t o c h a s t i c g r o w t h the mean i n c r e a s e in the volume of n~ growing grains can be estimated from the theory of Brownian motion such that:
(2) S t o c h a s t i c grain g r o w t h a l s o i m p l i e s that at time t o = 0 the grains which at time t~ fill the p o l y c r y s t a l , r e p r e s e n t a r a n d o m s a m p l e of the p o p u l a t i o n of n o grains. These n~ grains are neither statistically smaller nor larger than the rest of the grains. Thus, their total volume at t o = 0 is given as n~v o, and at t~ = t is equal to n~vl. Using (2) one may obtain n~vl = nlv O + niAv = n1(v o + ~/t)
(3)
Dividing by n, results in the following relation v, - T O = ~/t The mean v o l u m e v~Is by d e f i n i t i o n diameter of grains, by
(4)
related
to the mean
g r a i n size, defined as an equivalent
R - (~)~I, Substituting
(5)
(5) into (4) gives the following relationship for stochastic grain growth:
(R) ~I~
-
(Ro)~I3
=
aft
(6)
For large values of t (Ro << R), R is asymptotically proportional to t ~/6, which agrees with one of the estimations derived by Chen [4]. The classical models which assume that the process of growth involves systematic changes in the size of grains predict a value of n equal to I/2. On the other hand, purely stochastic growth results in n = I/6. Thus, it can be expected that if these two types of grain growth occur simultaneously, the exponent n may have any value in the range of I/6 to I/2 depending on the relative contributions to the overall change in the mean size of grains. For example, if it is assumed that both types of growth operate independently and make equal contributions, the mean grain size will change with time t in the following manner: R(t) = a't I/6 + b't I/2 Depending
on the
value
of time t, e q u a t i o n
(7) (7) can be approximated by the relationship R ~ t n
Vol.
22,
No. ,6
STOCHASTIC
GRAIN
GROWTH
with the exponent n ranging from I/6 to I/2. An exponent mean in Equation 7 is replaced by a geometric mean:
787
n = I/3 comes about if an arithmetic
R(t) = ~at I/6 • bt I/2 - t I/'
(8)
The use of a geometric mean is rationalized by the fact that the two types of grain growth are expected to be intimately coupled as they both involve migration of grain boundaries. A formal Justification for intermittent processes was given quenched unstable states.
the use of by Furukawa
a geometric mean to describe the kinetics [6] who studied the growth rate of droplets
of in
A value of n can even be lower than I/3 if the contribution of systematic growth decreases with increasing mean grain size. However, a quantitative description of the effect requires some additional assumption, which would diminish general character of the analysis. In summary, the analysis of stochastic growth based on the fundamental results from the theory of Brownian motion shows that the kinetics of the process is approximated by a power law with the exponent n = I/6. The analysis also considers super position of stochastic and non-stochastic changes in the size of grains leading to the exponent n = I/3 which is in agreement with the experimental results.
Acknowledgements This work as supported
by NSERC Canada.
References I. 2. 3. 4. 5. 6.
M. Hillert, Acta. Metall., 13, 227. (1965). R . M . German, Metallography, 11, 235, (1978). N . P . Louat, Acta. Metall., 22, 721, (1974). I-Wei Chen, Aeta. Metall., 35, 1733, (1987). G . S . Grest, D. J. Srolovitz and M. P. Anderson, H. Furukawa, Physics Letters, 98A, 361, (1983).
Acta. Metall. 33, 509,
(1985).
METAL~URGICA
Vol. 22, pp. 7 8 5 - 7 8 7 , 1988 P r i n t e d in the U . S . A .
Pergamon Press plc All rights reserved
AN ANALYSIS OF THE KINETICS OF STOCHASTIC GRAIN GROWTH
K. J. Kurzydlowski and K. Tangri Metallurgical Sciences Laboratory, The University of Manitoba WINNIPEG, Manitoba, Canada R3T 2N2 ( R e c e i v e d D e c e m b e r 14, 1987) ( R e v i s e d M a r c h 29, 1988)
Introduction In the classical models of grain growth (for example [I]) it has been postulated that the process has a strictly deterministic character, i.e., the rate of growth of an individual grain is a known function of both its size and the mean grain size of the polycrystal. This approach is based on the assumption that surface tension is the driving force for the migration of grain boundaries and that it is proportional to the curvature of the boundaries which in turn is inversely proportional to the size of a given grain. A characteristic on time as
of the classical
models is that they predict the dependence of grain size
R ~ A tn
(I)
where n = I/2. However, experimental studies (for example [2], see also [4]) indicate that the values of n are significantly lower, in the range from I/4 to I/3. This discrepancy between the experimental results and the theoretical prediction may be explained on the basis that the classical approach to grain growth ignores certain aspects of the properties of grain boundaries. For example, Louat [3] pointed out that grain boundary migration cannot be simply related to the size of the grains. The migration of an individual boundary is not controlled by the size of the grain but by other factors, such as the tendency for a boundary to achieve a low energy orientation. In such an extreme case grains will grow or shrink independently of their size. of process is termed stochastic grain growth.
This type
Analytical treatment of stochastic grain growth has been presented by Louat [3]. It is based on an analogy between the diffusion of particles in real space and the movement of grains in the space of grain sizes. This analysis also leads to the conclusion that the mean grain size is proportional to the square root of time which is the same dependence as predicted by the classical models. The main limitation of Louat's model [3] is the applicability of the diffusion equations to describe the changes in the distribution function of grain size. These equations are derived to describe the changes in the position of particles which move independently of each other while the "movement" of grains involves the constrained movement of the interface between neighbouring grains. The growth of a given grain (i.e. its movement towards higher sizes in the space of grain size) is accompanied by the shrinkage of a neighbouring grain (movement towards lower sizes). Grest et al. [4] analyzed the process of grain growth considering a combination of stochastic and systematic grain growth. The stochastic contribution resulted from the assumption that the grain boundary migration depends on the anisotropy of grain boundary energy. Using a computer simulation they found that the exponent n in equation (I) decreases from 0.42 to 0.25 with
785 0036-9748/88 $ 3 . 0 0 + .00 Copyright (c) 1988 P e r g a m o n P r e s s
plc
786
STOCHASTIC
GRAIN
GROWTH
Vol.
22,
No.
6
increasing ansotropy of energy. In another study Chen [4] argued that n can in some circumstances be as low as I/6. These results seem to be inconsistent with both the classical models and the solution for stochastic growth derived by Louat [3] which predict n being equal to I/2. The aim of the present communication is to provide a simple analytical treatment of stochastic grain growth which can be used as an alternative to the estimation given by Louat [3].
Analysis The analysis given below deals with two cases. The first considers purely stochastic growth and the other assumes superpositlon of stochastic and deterministic growth. In purely stochastic grain growth the boundaries migrate randomly (outwards or inwards). This leads to random variations (jumps) in the volume of an individual grain. In grain volume space grains exhibit one-dimensional Brownian motion. C o n s i d e r a p o l y c r y s t a l w h i c h c o n t a i n s n o grains at time t o = O. Growth of some grains required the continuous decrease in the size of some other grains and their eventual annihilation. At time t~ = t the polycrystal contains n~ grains and the ratio n~/n o is inversely proportional to the r a t i o of the mean volume of grains at t o = 0 and t~ = t (v~/Vo). This means that the p r o c e s s can be viewed as growth of n~ grains in a matrix containing (n o - n~) grains. Under c o n d i t i o n s of s t o c h a s t i c g r o w t h the mean i n c r e a s e in the volume of n~ growing grains can be estimated from the theory of Brownian motion such that:
(2) S t o c h a s t i c grain g r o w t h a l s o i m p l i e s that at time t o = 0 the grains which at time t~ fill the p o l y c r y s t a l , r e p r e s e n t a r a n d o m s a m p l e of the p o p u l a t i o n of n o grains. These n~ grains are neither statistically smaller nor larger than the rest of the grains. Thus, their total volume at t o = 0 is given as n~v o, and at t~ = t is equal to n~vl. Using (2) one may obtain n~vl = nlv O + niAv = n1(v o + ~/t)
(3)
Dividing by n, results in the following relation v, - T O = ~/t The mean v o l u m e v~Is by d e f i n i t i o n diameter of grains, by
(4)
related
to the mean
g r a i n size, defined as an equivalent
R - (~)~I, Substituting
(5)
(5) into (4) gives the following relationship for stochastic grain growth:
(R) ~I~
-
(Ro)~I3
=
aft
(6)
For large values of t (Ro << R), R is asymptotically proportional to t ~/6, which agrees with one of the estimations derived by Chen [4]. The classical models which assume that the process of growth involves systematic changes in the size of grains predict a value of n equal to I/2. On the other hand, purely stochastic growth results in n = I/6. Thus, it can be expected that if these two types of grain growth occur simultaneously, the exponent n may have any value in the range of I/6 to I/2 depending on the relative contributions to the overall change in the mean size of grains. For example, if it is assumed that both types of growth operate independently and make equal contributions, the mean grain size will change with time t in the following manner: R(t) = a't I/6 + b't I/2 Depending
on the
value
of time t, e q u a t i o n
(7) (7) can be approximated by the relationship R ~ t n
Vol.
22,
No. ,6
STOCHASTIC
GRAIN
GROWTH
with the exponent n ranging from I/6 to I/2. An exponent mean in Equation 7 is replaced by a geometric mean:
787
n = I/3 comes about if an arithmetic
R(t) = ~at I/6 • bt I/2 - t I/'
(8)
The use of a geometric mean is rationalized by the fact that the two types of grain growth are expected to be intimately coupled as they both involve migration of grain boundaries. A formal Justification for intermittent processes was given quenched unstable states.
the use of by Furukawa
a geometric mean to describe the kinetics [6] who studied the growth rate of droplets
of in
A value of n can even be lower than I/3 if the contribution of systematic growth decreases with increasing mean grain size. However, a quantitative description of the effect requires some additional assumption, which would diminish general character of the analysis. In summary, the analysis of stochastic growth based on the fundamental results from the theory of Brownian motion shows that the kinetics of the process is approximated by a power law with the exponent n = I/6. The analysis also considers super position of stochastic and non-stochastic changes in the size of grains leading to the exponent n = I/3 which is in agreement with the experimental results.
Acknowledgements This work as supported
by NSERC Canada.
References I. 2. 3. 4. 5. 6.
M. Hillert, Acta. Metall., 13, 227. (1965). R . M . German, Metallography, 11, 235, (1978). N . P . Louat, Acta. Metall., 22, 721, (1974). I-Wei Chen, Aeta. Metall., 35, 1733, (1987). G . S . Grest, D. J. Srolovitz and M. P. Anderson, H. Furukawa, Physics Letters, 98A, 361, (1983).
Acta. Metall. 33, 509,
(1985).