Effect of Fine Second-Phase Particles

It is assumed that a polycrystalline solid contains randomly dispersed fine second-phase particles, also known as fine inclusions or precipitates, which are insoluble and immobile in the system. When a grain boundary encounters a fine particle during its movement driven due to its curvature, it will be pinned by the particle, while the rest portion of the grain boundary continues to move. The grain boundary will break away only as the rest portion has moved for a sufiftciently far distance. Therefore, if the grain boundary encounters a sufiicient number of particles, it could be entirely pinned, i.e., the motion of the grain boundary is completely inhibited. This scenario can be described by using two models. 

If the pinning particles are assumed to be monosized, spherical, insoluble, immobile and randomly distributed in the polycrystalline solid, for a grain boundary with principal radii of curvature, a and 02, the driving force per unit area for the motion of the grain boundary is given by [73]  

25 Interaction of a grain boundary with an immobile particle, a Approach of the boundary toward the particle, b Interaction between the grain boundary and the particle leading to a retarding force on the boundary, c Detailed geometry of the particle-grain boundary interaction. Reproduced with permission from [4]. Copyright 2003, CRC Press 

Therefore, the retarding force is the product of the perimeter of contact and the grain-boundary tension, which is in an opposite direction to that of the grainboundary migration. At 8 = 45°, sin cos = 1/2, so that the retarding force is maximized, which is given by 

If there are Nj inclusion particles per unit area in the grain boundary, the maximum retarding force per unit area of the boundary is given by 

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