Pinning of grain boundaries by deformable particles
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I.
INTRODUCTION
INTERACTION between rigid spherical particles and a grain boundary has been the subject of several investigations in the past. 11-SjThe force, f~, that a rigid planar boundary exerts on a spherical particle has been derived by Zener. tu fs = 7rrs 3,gb sin 2~b
[1]
where rs and 3,gb are the particle radius and surface energy of the grain boundary, respectively. The angle ~b is the angle which the radial line drawn to the point of boundary intersection makes with the plane of horizon. In a subsequent treatment by Ashby et a l . , tn] the grain boundary is assumed to be flexible and the angle which the boundary makes with the particle, the contact angle 0, remains constant as the boundary bypasses the particle. This allows the shape of the grain boundary and the force acting on the particle to be derived as the boundary expands through the gap between the particles. f s = 27rrs3,g b c o s t~ c o s ( 8 - ~b)
y~ sin
81 =
")/2
sin 02
r 0 = ( A - - ~ B ) 1/3
[3] [4]
[51
where V = 4 / 3 7rr 3 and A and B are 71"
A = m (1 - c o s 81)2(2 + COS 81) 3 sin 3 81 7r B = -
S. NOURBAKHSH, Associate Professor, is with the Department of Metallurgy and Materials Science, Polytechnic University, Brooklyn, NY 11201. Manuscript submitted May 3, 1991.
-
3 sin 3 82
(1 - cos 02)2(2 + cos 8z)
[6]
[7]
It should be noted that when the particle is spherical and rigid, the extension of the grain boundary does not necessarily pass through the center of the particle. The stand-off angle, as defined in Figure 1(a), is given by ~b# = 0 -
PARTICLE INTERACTION
Figure 1(a) depicts a spherical particle lying on a planar grain boundary. The surface tensions, the particle surface energies, 3,1 and 3'2, in grains 1 and 2, and the grainboundary energy, acting at the point of contact between the boundary and the particle, are not balanced along a
METALLURGICAL TRANSACTIONS A
Yl COS 81 + 3,2 COS 02 = 3,gb
[2]
In the derivation of the force, the particle was assumed to be a rigid sphere. The aim of the work presented is to calculate the force for the case when the boundary is pinned by deformable particles. This situation is expected to occur in two-phase systems in which the rate of self-diffusion in the particle is equal to or higher than the rate of self-diffusion in the matrix. The effect of the interphase energy on the required driving force to unpin a grain boundary, from a square array of particles, as well as on the rate of boundary migration will be presented. II. DEFORMABLE GRAIN-BOUNDARY
direction parallel to the boundary. If self-diffusion within the particle is allowed to take place, the shape of the particle will change in response to these forces. The particle would elongate and assume a lenticular shape, as shown in Figure l(b). The particle's surface energy in each grain is assumed to be independent of the interphase boundary orientation. Since two different interphase boundary energies are used, the results obtained are applicable to three-phase systems. In two-phase systems, Yl is equal to Y2. The an
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