Friction pressure method for simulating solute drag and particle pinning in a multiphase-field model

In many polycrystalline materials, second phase particles and solute atoms impose a drag pressure on the motion of grain boundaries. The drag effect occurs on a scale comparable to the particle diameter and interface thickness. However, to simulate grain growth with numerical efficiency one requires a model that captures the drag pressure on the interfaces but does not resolve the fine particles or solute segregation spike. In this paper, a multiphase-field model is proposed to simulate the evolution of microstructure under constant and velocity dependent drag pressures. The accuracy of the model is confirmed in comparison with analytical expressions for a shrinking circular grain. Application of the model is presented for grain growth in two dimensions under particle pinning. Measuring curvature of grain-boundary segments reveals that in the completely pinned structure, the average driving pressure is not equal to but lower than the pinning pressure. Considering this effect, the predicted limiting grain size is about three times larger than that assumed in conventional mean-field theories. Based on this observation, a correction factor is introduced for these mean-field models. The proposed phase-field formulation is also applied to simulate grain growth in the presence of solute drag. The grain growth kinetics follows a phenomenological relationship that can be described with a power law with a time exponent in the range 0.35–0.50. The deviation of the time exponent from 0.5, associated with ideal grain growth, and its correlation with the solute drag parameters is discussed.