Pinning force from multiple second-phase particles in grain growth

A factor that can reduce particle pinning force significantly in grain growth is found when the grain-boundary is pinned by multiple particles. The pinning force, in this case, is a function of particle radius over inter-particle distance. A previously proposed phase-field model for particle pinning is used to validate this predicted pinning force reduction in two and three dimensions. When applied to coherent pinning particles, the same effect is observed in simulations. It is shown that, at application relevant high particle volume fraction, the average grain size is affected by this reduction of pinning force.