Pergamon Scripta Mat&h, Vol. 37, No. 8, pp. Ill l-l 116.1997

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ATOMIC JUMP MODEL FOR MIGRATION OF CURVED GRAIN BOUNDARY

Byung-Nam Kim Research Center for Advanced Science and Technology, The University of Tokyo

4-6- 1 Komaba, Meguro-ku, Tokyo 153, Japan

(Received April 10, 1997) (Accepted June 6,1997)

Introduction

Normal grain growth is defmed as the uniform increase in ‘the average grain size of a polycrystalline aggregate after primary recrystallization. In general, the variation of the radius of grain, r, with time, t, can be represented approximately by

rm -r,” =Kt (1)

where r,, is the initial radius of grain at t = 0, m is a kinetic exponent and K is a constant depending on temperature. Beck et al. (1) and Burke (2) proposed that the average grain size in an ideal system is proportional to a square root of time (m = 2), and many models were suggested supporting m = 2.

Burke and Turnbull (3) assumed that curved grain boundary tends to move toward its center of cur- vature with ai velocity proportional to the curvature. In the model, the pressure difference between adjacent two grains was assumed to be the driving force for the boundary migration. By employing the assumption, Hillert (4) proposed the following equation governing the kinetics of grain boundary migration:

where 01 is a constant, M is the mobility, y is the surface energy of grain boundary and r, is the critical radius. Rq. 2 is identical to the grain boundary velocity in the mean-field model where an isolated grain of radius r is surrounded by a matrix of property rc, resulting in m = 2. Such curvature driven growth is the generally accepted assumption for normal grain growth. In addition, several models and simula- tions employing the assumption show the same parabolic law of grain growth in polycrystals (S-9). Because of relatively few models of m # 2 (lo-12), the curvature itself due to pressure difference is often regarded as a fundamental driving force for the gram boundary migration.

1111

1112 MIGRATION OF GRAIN BOUNDARY Vol. 37, No. 8

In this study, another approach is carried out for the migration of curved grain boundary. The mi- gration velocity is obtained for an isolated circular and spherical grain by constructing the diffision- like atomic jump model. The driving force for the migration is based on the decrease of the grain boundary surface energy due to the reduction of surface area, not on the pressure difference. Although the approximately same kinetic equation is obtained as Eq. 2 for large grain size in the present model, the physical basis of the grain boundary migration is quite different from Eq. 2.

Atomic Jump Model

The migration of curved grain boundary occurs toward its center of curvature as this reduces the boundary area and hence the energy associated with it. An isolated circular grain will reduce the radius until it vanishes, and the curved boundary in which both ends are fixed will be flattened. Burke and Turnbull (3) described such phenomenological migration behavior with the curvature. However, in an atomic level, the migration of grain boundary accompanies inevitably the movement of atoms across the boundary. As in the case of diffusion of vacancies, effective atomic jump occurs into the reverse direction of the boundary migration, which was originally proposed by Tumbull(13) to describe the growth behavior of second phase. If the energy state of the grain boundary becomes lower by the mi- gration, then the reduction of the energy would be the driving force for the boundary migration.

Consider a straight grain boundary migrating by the diffusion-like process as shown in Fig. l(b). Let AG1 be the activation energy for a jump from grain 1 to grain 2, and AG be the energy difference (driving force) of an atom across the boundary. Here, AG for the straight boundary is assumed to be given already, for example, by the gradient of chemical potential. The N atoms of diameter, b, in each side faced on the boundary jump each other across the boundary due to thermal activation. Then, the net number of atoms, N., jumping from grain 1 to grain 2 during the time, At, is

N” dt=fNvexp( -s) [l-exp(-g)] (3)

where v is a mean frequency, k is Boltzman’s constant, T is the temperature, and f is a constant relating to the boundary shape. In the case of the 2-dimensional(2D) straight boundary, f is 0.5 because the gradients of AG and AGr are l-dimensional. If N atoms jump from grain 1 to grain 2, the boundary would move by b in the direction of grain 1. Hence, the migrating velocity of the boundary, dx/dt, by the jump of N, atoms is given by

dx

-=fbvexp(-TF dt AG1)[l-exp(

Providing that the straight boundary of length, 1, has the total driving force, AG’, and AG’ is cat- centrated on the boundary atoms of grain 1, AG per atom would be AGW. Since b, v and AG, is a constant at certain temperature, Eq. 4 can be written as

dx ,=c, [1-exp(+y)] (5)

by introducing a dimensionless constant C,(=exp(-AGJkT)), X(=x/b) and r(=vt). Eq. 5 represents the migrating velocity of the straight boundary in the normal direction by the thermally activated atomic jump process across the boundary.

Vol. 37, No. 8 MIGRATION OF GRAIN BOUNDARY 1113

(4 I !grain 2 ; grain 1 I

04 - grain boundary

I

o+o 0 ; 0 omoatom

I I

OO

Figure 2. Circular grain boundary of radius r and thickness b.

Figure 1. (a) Energy states of atoms adjacent to grain boundary and (b) atomic jump across grain boundary.

Shrinkage of an Isolated Circular Grain

As shown in Fig. 2, consider a 2D isolated circular grain where Ni and N, atoms face on the boundary in the inner and the outer side, respectively. Providing an interatomic distance of 2b, namely, a bound- ary thickness of b independent of grain size, Ni and N, could be obtained to be 2n;(r - b)/b and %(r + b)/b, respectively. The jump of Ni atoms into the outer side across the boundary results in the move- ment of the boundary into the inner side by b. Hence, the change of the grain boundary surface energy (AG’) is 2lcyt, due to the migration of the circular boundary. Assuming again that AG’ is concentrated on the boundary atoms in the inner side, AG = yb2/(r - b) is obtained.

For the isolated circular grain shown in Fig. 2, the net number of atoms, N., jumping from the inner to the outer side is given by

N 0 = Cl [fiNi -f,N, exp( A%

(6)

where fi (=0.5 + bRrc(r - b)) and f, (=0.5 - b/2n(r + b)) are a shape factor of the inner and the outer side atoms, respectively. Since the jump of Ni atoms from the inner side is equivalent to the movement of the grain boundary by b, Eq. 6 can be written as

+c, x(R+l)-1

n(R-1)+1 exp(-’

;-1)-l 1 (7)

by using a dimensionless radius R (=r/b) and a constant C2 (=by/kT), yielding the shrinkage kinetics of the 2D isolated circular grain at constant temperature.

In a similar manner, the shrinkage kinetics of a 3-dimensional(3D) isolated spherical grain can be obtained as

(R+1)(2R+l) (2R-1)

(R-1)(2R-1) exp(-C3 (R_l)2)-1 1 I (8)

1114 MIGRATION OF GRAIN BOUNDARY Vol. 37, No. 8

2 s ’ ’ u” shrinkage

% S

1, growth

? 3 OS . - C2(W

8 - C3(3W

‘0 200 4U0 GO0 800 1000

Patti& Radius. R

Figure 3. Critical value of C, and C, for particle shrinkage.

where Ni = 4x(r - b)*/b’, N, = 4n(r + b)‘/b2, AG’ = 4rrr[lj! - (r - b)‘], AG = yb3(2r - b)/(r - b)‘, fi = [Cl.5 +

b/4(r - b)], f, = [0.5 - b/4(r + b)], and C3 = $‘/kT. y is the grain boundary surface energy per unit length in a 2D, and per unit area in a 3D.

Results and Discussion

Although an isolated grain has a tendency to shrink naturally, the critical value of CZ and Cs for a minus dR/dr can be calculated from Eq. 7 and 8. As shown in Fig. 3, the critical value of CZ and C3 approaches 1.363 and 1.500 respectively with increasing R. The critical CZ and C3 are also affected by the boundary thickness; the thicker boundary of 2b results in the increased critical CZ and CJ to 1.682 and 1.750 at large grain size, respectively, while the difference between the two critical values de- creases. The larger CZ and C3 than the critical ones indicate a minus dR/dr, that is, the shrinkage of the isolated grain. Since CZ and CS are inversely proportional to T, the shrinkage will occur at lower tem- perature than the critical temperature T,. On the other hand, the isolated circular and spherical grains grow at higher temperature than T,. However, the existence of T, is only a prediction from Eq. 7 and 8, and no experimental observations have been reported. If T, is higher than a melting point, a minus dR/dT would be obtained at all solid states. The migration of grain boundary only toward its center of curvature is discussed in this study.

Figure 4 shows the shrinkage behavior of the 3D isolated spherical grain of the initial radius R. = 10000 at lower temperature than T,, calculated from Eq. 8. The physical properties were used of AG, = 160 kJ/mol, y = 2 J/m’ and b = 2.5 x 10-i’ m for y-Fe, which are assumed to be independent of tem- perature. From Eq. 8, T, is 6040 K, and the temperature is represented in Fig. 4 as a normalized one

30 40 50 GO 10 20 30 40 50 GO Time, T /IO” Time, T. llUln

Figure 4. Shrinkage behavior of particle radius (a) a and (b) 1 - (R&J’ with time.

Vol. 37, No. 8 MIGRATION OF GRAIN BOUNDARY 1115

Temperature, T/T,

Figure 5. Variation of K with respect to temperature.

with T,. The shrinking velocity klR/dzl increases with time and temperature. The entire shrinkage behavior is very similar to the results from the Monte Carlo simulation (14).

By applying Eq. 1 to the variation of R with 2, m = 2 is obtained approximately. As shown in Fig. 4(b), 1 - (R&J” is linearly proportional to 2. When R is large enough, Eq. 8 can be approximated by

(9)

which is the same form as Eq. 2 of r, = 00. Hence, the shrinkage behavior of the isolated grain can be represented approximately by Eq. 1 with m = 2.

Figure 5 shows the variation of K, when the results of Fig. 4(b) are approximated by Eq. 1. With increasing T, .the migration velocity increases until it reaches a maximum at T = 0.76T,. The value of K is sufficiently small at T < 0.2T, compared with the value at other temperature. In the range of T < 0.76T,, the dependence of the migration velocity on temperature shows a qualitative consistence with the kinetic prediction based on the thermally activated process.

However, the velocity decreases at T > 0.76T, and approaches zero. K = 0 (dR/dz = 0) means that the net number of jumping atoms across the boundary would be zero. In the curved grain boundary, while the energy of an atom is higher in the inner side by AG than in the outer side, the total number of atoms is larger in the outer side. Therefore, considering that the migration of the boundary in the pres- ent model is governed by both the energy decrease ( AG) and the number of atoms faced on the bound- ary (Ni, N,), the kinetics of the boundary migration can be divided into two regions; AG-governing (T < 0.76TJ andi Ni, N,goveming regions (T > 0.76T,). At T > 0.76T,, the migration velocity decreases with T because of the increased atomic jumps from the outer side.

Summary

The migration behavior of grain boundary was analyzed for 2D and 3D isolated grains by the atomic jump model, where the decrease of the grain boundary surface energy due to the reduction of surface area, not the pressure difference, is regarded as the driving force of the migration. The kinetics both in a 2D and 3D can be represented as an exponential function of grain size. When the grain size is large enough, the variation of the radius is nearly proportional to a square of time.

1116 MIGRATION OF GRAIN BOUNDARY

References

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10. F.N. Rhine-s and K.R. Craig, Metall. Trans. 5A, 413 (1974). 11. L-W. Chen, Acta metall. 35, 1723 (1987). 12. M.P. Anderson, D.J. Srolovitz, G.S. Grest and P.S. Sahni, Actametall. 32,783 (1984). 13. D. Tumbull, Phase Changes, Solids State Physics, Vol. 3,225 (1956). 14. B. Radhakrishnan and T. Zacharia, Metall. Mater. Trans., 26A, 167 (1995).

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