Finite Element Simulations of 3D Zener Pinning

An original model, based on a variational formulation for boundary motion by viscous drag, is developed to simulate single grain boundary motion and its interaction with particles. The equations are solved by a 3D finite element method to obtain the instantaneous velocity at each triangular element on the boundary surface, before, during and after contact with one or more particles. After validation by comparison with some simple, analytical and numerical cases, it is adapted to model curvature driven grain growth. For single phase material, the single grain boundary model closely matches the grain coarsening kinetics of a 3D multi boundary vertex model. In the presence of spherical incoherent particles the growth rate slows down to give a growth exponent of 2.5. When the boundary is anchored there is a significantly higher density, by a factor of 4, of particles on the boundary than the density predicted by the classic Zener analysis, and many particles exert less than this Zener drag force. As a result the Zener drag is increased by a factor of about 2.2. The limiting grain radius is compared with some experimental results.