6 N O R M A L GRAIN GROWTH A N D

$ECON D- PHASE PARTICLES

6.1 N O R M A L G R A I N G R O W T H

Grain growth in polycrystals is best explained in terms of a chemically pure single-phase system. Even in this case, however, the kinetics of movement varies from boundary to boundary, because the grain boundary energy varies with the grain boundary orientation and the grain boundary mobility may not be constant. 1 As a result, grain growth cannot be rigorously analysed even in this simplest system and a number of theories have been proposed. 2 For a fundamental understanding of grain growth, however, the classical theory 3'4 developed on the assumption of constant grain boundary energy is useful.

Figure 6.1 shows a typical, single-phase polycrystalline microstructure. For such a microstructure, the average grain shape is hexagonal 5 but most of the grain boundaries are curved following the number of surrounding grains. The atoms on both sides of a curved grain boundary are under different pressures across the boundary. If the atoms are in local equilibrium, the pressure difference AP is 2yb/Ro, where Ro is the radius of curvature of the grain boundary. Since the growth rate of an average sized grain G must be propor- tional to the average velocity N of grain boundary movement,

n

dG = OtVb -- otJf2

dt

D~- (VP)a - a ~ - ~

D-~ 2yb Vm

R T Ro (6.1)

Here, or, J, S2, Db L, w and k are, respectively, the proportionality constant, atom flux, atom volume, atom diffusion coefficient across the grain boundary, grain

91

92 CHAPTER6 NORMAL GRAIN GROWTH AND SECOND-PHASE PARTICLES

Figure 6.1. Typical microstructure of a single-phase polycrystalline material (sintered alumina).

boundary thickness, and Boltzmann constant (1.3806 J/K). For a given average grain size and grain size distribution, Eq. (6.1) can be rewritten as

d-G D-~2yb Vm dt flRTGw

(6.2)

because the average radius of curvature is proportional to the average grain size. Here, fl is a constant which includes or. Integration of Eq. (6.2) from time to to t gives

--2 -C2 4D-~yb Vm t (6.3) G t - to- fiRTw

Therefore, the grain growth is proportional to the square root of the annealing time. The classical model of Eq. (6.3) was derived for bulk polycrystalline materials but in the case of thin films with two-dimensional grains, the kinetics is reduced by one-half because the pressure difference across the boundary is then AP = yb/Ro.

Classical theory explains the complicated phenomenon of grain growth in a simple manner. It is assumed that the driving force is determined only by the radius of curvature of the grain boundary and that the average grain growth rate is proportional to the average rate of grain boundary movement. The latter assumption is justified only when the grain shape and the grain size distribution are invariable during the grain growth. This condition appears to be satisfied in real microstructures that are free of abnormal grain growth.

6.1 NORMAL GRAIN GROWTH 93

As annealing time increases, the grain size distribution reaches a stationary state and the average grain size increases as a function of the square root of annealing time (tl/2). However, a theoretically rigorous treatment of this phenomenon does not seem to be complete as yet. 2'6'7

The dynamic shape change of grains during growth can be explained in terms of topology. 5 Figure 6.2 shows schematically a two-dimensional micro- structure and an overlapping large hexagonal array. As the grains grow to the size of the large hexagon, the external edges (boundaries) persist while the internal edges have disappeared. In such a process, the overall change in microstructure during grain growth can be simply explained. Two types of corners are present in the microstructure in Figure 6.2; one of these is shared by two polygons (in) while the other has only one polygon (out). Denoting hi and ho as the numbers of each type and using the expression in Section 3.2, the total number of corners in this microstructure, C, is expressed as

C = Y~nPn 1 2 ------~ + -~ hi + -~ ho (6.4)

In addition, since Eb = hi + ho,

E Y~ nPn hi + ho - t ( 6 . 5 )

2 2

Therefore, from Eq. (3.4),

Z (6 - n)P, + ho - hi - 6 (6.6)

This equation implies that after grain growth, the distribution of large grains is determined by the original grouping of small grains. It also indicates that the

Figure 6.2. Hexagons superimposed on a group of tri-connected polygons, s

94 CHAPTER6 NORMAL GRAIN GROWTH AND SECOND-PHASE PARTICLES

average shape of large grains is a hexagon as a result of the disappearance of internal small grains whose average number of edges is also 6.

Whether a specific grain is shrinking or growing during grain growth is analogous to the case of the volume changes occurring in a gas bubble of the same size by gas diffusion through the bubble wall. For simplicity, consider a two-dimensional microstructure. The rate of area change of a grain d A / d t is expressed as 8'9

d A __ yrMyb (n -- 6) (6.7) dt 3

where A is the grain area, M the grain boundary mobility and n the number of edges of the grain. This equation shows that the change of grain size is determined only by the number of edges. The equation also indicates that the area of a six-sided grain is invariable and that the rate of area change of n-sided polygons other than a hexagon is proportional to ( n - 6 ) . This result is acceptable only when the grain boundary movement occurs by the diffusion of atoms across the boundary and is, therefore, continuous. In reality, however, not all grain boundary movement is continuous. A discontinuous movement, such as the instantaneous disappearance of small grains, may occur and the number of edges suddenly changes. 1~ Therefore, Eq. (6.7) can provide only a limited explanation of the kinetics of grain growth. According to a recent investigation on two-dimensional microstructures, 1~ the grains become unstable and disappear when the average number of edges is -~4.5. In the two-dimensional cross-section of a bulk polycrystal, however, many triangular grains are observed and these disappear continuously. Such observations suggest that the cross-sectional microstructure of a bulk polycrystal cannot be considered in terms of a simple two-dimensional array microstructure.

A number of experimental investigations on grain growth of zone-refined polycrystals have been made.2 '11 However, the reported kinetics rarely satisfy the kinetic equation (6.3) and the measured exponent is usually larger than 2. Reports of exponent values >2 in grain growth measurements occur, in general, when the grain size is large, the sample contains an appreciable amount of impurities or the annealing temperature is low. The discrepancy between the theory and experiments can be attributed to impurity drag of grain boundaries (see Chapter 7) even in zone-refined material and a threshold driving force for grain boundary movement. 11 However, the existence of a threshold driving force needs clarification. According to investigations on the effect of boundary structure, the movement of faceted boundaries needs a critical driving force. 12-15 (See Section 9.2.) On the other hand, the presence of a minimum driving force for grain boundary movement is evident in systems with second-phase particles and the boundary drag of such particles is referred to as the Zener (sometimes called Smith-Zener) effect. 16

6.2 ZENER EFFECT 95

6.2 EFFECT OF SECOND-PHASE PARTICLES O N GRAIN GROWTH: ZENER EFFECT

When second-phase particles are present at grain boundaries as shown sche- matically in Figure 6.3, the particles hinder the grain growth. ~6-~8 The drag force of a particle against the boundary movement in Figure 6.3, Fa, is

Fd - - ?'b sin 0 x 2zrr cos 0

= zrryb sin 20 (6.8)

Therefore, the maximum drag force, F~d ax, is 7rr~/b at 0= 45 ~ Thermodynami- cally, the drag force results from a reduction in total grain boundary energy which is achieved by the boundary occupation of the second-phase particles.

If the second-phase particles are randomly distributed, the maximum drag force against grain growth can easily be calculated following Zener's proposal. Assuming that the radii of second-phase particles are constant, r, and their volume fraction is fv, the number of particles attached to a unit area of grain boundary is

4 2rfv / ~ Jrr 3 = (6.9)

2Jrr 2

Then, the maximum drag force against grain boundary movement per unit area of grain boundary, U~a, is

3fvYb F~a = 2r (6.10)

and the work needed for 1 mol of atoms to move across the boundary, i.e. the grain boundary movement is

W= 3L• 2r (6.11)

Figure 6.3. Dragging of grain boundary movement by second-phase particles: the Zener effect.

96 CHAPTER6 NORMAL GRAIN GROWTH AND SECOND-PHASE PARTICLES

Therefore,

dG D~-l[2yb Vm l dt = R T w fiG 2r J (6.12)

Equation (6.12) shows that the driving force for grain growth disappears when the value of the bracketed term becomes zero. In other words, there is a limiting grain size Gt,

4r 2r G t - 3furl or R t - 3fvfl (6.13)

where Rt is the radius of the limiting grain size. When fl is 1/2, this equation is the original equation of Zener. According to Eq. (6.13), the limiting grain size decreases as the volume fraction of second-phase particles is increased and their size reduced. Reduction in limiting grain size, i.e. increase in grain boundary drag with reduction in particle size, is thermodynamically due to a reduction in total grain boundary area and energy with the particle size reduction.

The above calculation, however, exaggerates the grain boundary drag of second-phase particles. In reality, the particles in front of the moving grain boundary do not hinder the boundary movement but facilitate it up to their equator while those behind the boundary drag it up to and even beyond the distance of their radius from the moving boundary. 17'18 Considering this phenomenon, Louat ~7 calculated the limiting grain size and found that the value of Gt is larger than that calculated from Eq. (6.13) for ratios of grain to particle radius <107.

During grain growth, second-phase particles may also grow. 19 According to Eq. (6.23), the limiting grain size increases as the second-phase particles grow. For particle growth by lattice diffusion, r c~ t 1/3 (see Section 15.2.1), and for that by grain boundary diffusion, r c~tl/4. 2~ In these cases, the time dependency of Gt should be similar to that of particle growth. In reality, however, it would be difficult to experimentally confirm the dependency because the assumption of random distribution of second-phase particles may not hold during grain growth.