Precipitate particles on grain growth in metals 299 The driving force for grain growth was assumed to be 2y/i20. This force, however, relates to a contracting spherical surface with no attached interfaces. Thus Zener’s ultimate criterion, p = M , where / is the volume fraction of precipitate particles, may be expected to over estimate the driving force for grain growth, because the diminution of one inter face usually involves the expansion of some other attached interfaces. In fact, substitution of typical precipitate particle sizes, and volume fractions, into the Zener equation indicates a much larger equilibrium grain size than is observed. This observation confirms that the driving force for grain growth is overestimated in the Zener model.
300 T. Gladman depends on the shape of the boundary in the region of the particle and on the circle of contact between the boundary and the particle. It has been assumed that the distortion from a planar boundary varies inversely as the distance from the particle, y(s-x) = k, (1) and that the interfacial angles between the particle and grain A and the particle and grain B are equal, 0 — (f> (2) These assumptions are supported by experimental evidence, typical of which are the micrographs shown in figure 2, plate 12. The equation for the grain boundary grain A grain B pinning particle pinning distortion diametral position rigid planar boundary distorted boundary Figure 1. Model for unpinning.
Precipitate particles on grain growth in metals 301 The solution to this integral is 2U-i \na sinh-ij?<£^L2j_ (1 + (5) 2(s-x) ) j L1 where a — 5[r2_ (15)2]£ This equation describes the area of the distorted boundary between the limits Lx and L2. It now remains to consider the physical interpretation of these limits. The lower limit will occur at the junction between the grain boundary and the change in boundary energy (r2y ergs) rigid flexible boundary boundary boundary displacement, r (cm) Figure 3. Energy change due to unpinning.
302 T. Gladman away from the diametral position have been calculated and are shown in the form of a normalized graph in figure 3. The maximum rate of change of energy has been estimated at 3-96ry erg/cm or approximately fry erg/cm. This may be compared with the pinning force derived of wry by Zener (1949); these results are very similar.
Precipitate particles on grain growth in metals 303 (b) Grain growth When a grain grows and absorbs neighbouring grains, there are two principal sources of energy change. First, the energy of the system is increased by the expan sion of the interface of the growing grain, and, secondly, the energy of the system is decreased by the elimination of the grain interfaces of the grains which are absorbed by the growing grain. There are certain simplifying assumptions which have to be made before these energy changes can be evaluated. The energy change due to the growth of a cubo-octahedral grain in an environment of similar grains would be discontinuous; the mathematical expression of a discontinuous energy Figure 5. Model for grain growth. Grain = grain B0.
304 T. Gladman The grain, however, also increases its own area of boundary. The increase in bound ary area, Ac, is given by Ac « (13) Thus the net change in area, An, of grain boundaries per unit area of growing interface when a grain increases the radius from to is An = s(2lR -3l2R0).(14) When R/R0 < 1-33, there is an increase in boundary area, and when R/R0 > 1*33, there is a decrease in grain boundary area. There is no doubt that this is the theo retical basis of grain growth and explains why large grains grow at the expense of small grains.
Precipitate particles on grain growth in metals 305 where Ep is the pinning energy derived in §2. Values of ET were obtained using a digital computer program. A typical sequence of energy change occurring when a boundary is unpinned, is shown in figure 6. It is obvious that unless the rate of de crease in energy due to grain growth exceeds the maximum rate of increase in unpinned ‘ZOO 400 600 boundary displacement, s (A) Figure 6. Energy changes during unpinning. / = 0-0005; = 0-0014 cm; r = 350 A; 2 = 1-5; y = 800 erg/cm2 (D. McLean, private communication).
306 T. Gladman Strictly, the critical particle radius, from the point of view of grain growth, would be defined by the activation energy barrier being reduced to a level at which thermally activated release could occur. If a thermally activated release could occur at a maximum of 5eV, this would be equivalent to 0-1 x 10-10 erg. The difference in particle radius between H — 0-1 x 10~10 erg and TI — Oerg is too small to be measured, and therefore negligible. Thus, the critical particle size r* may be defined by the condition H = 0 and this occurs when [dEp/ds]m^ = - [dEJds],(19) where [d^/p/ds]max is the maximum rate of increase in energy involved in the un pinning process, and [dEJds] is the rate of change of energy due to grain growth.