# Impediment of grain boundary migration due to particle drag

Metallic materials used in engineering applications frequently contain some volume fraction of particles. These may be impurity particles or particles that are deliberately added to influence the microstructure behavior.

As grain boundaries migrate through the microstructure, driven by boundary curvature and/or stored energy gradients, any particles present will impede - or even prevent - the movement by exerting drag forces on the passing boundaries. Classically this effect is termed Zener drag or Zener pinning. This pinning pressure is usually written as $p=-z_{1}\gamma\frac{f_{\mathrm{V}}^{z_{2}}}{r_{\mathrm{p}}}$

where the negative sign indicates a retarding pressure and where $\gamma$ is the boundary energy while $f_{\mathrm{V}}$ and $r_{\mathrm{p}}$ are the volume fraction of particles and the average particle size, respectively. $z_{1}$ and $z_{2}$ are parameters.

In the original model by Zener (published by C.S. Smith in 1948), the formulation is based on the assumption of rigid grain boundaries between the particles, providing $z_{i}=\left\{3/2,1\right\}$. Later studies have tried to incorporate the tendency of the grain boundary to bow out between particles, arriving at other values of the parameters $z_{1}$ and $z_{2}$.

A mesoscale RVE model of dynamic discontinuous recrystallization, considering particle drag, is established in Hallberg et al. (2014). The model is based on a 3D cellular automaton formulation. See illustration below. Influence on recrystallization kinetics from a small volume fraction of homogeneously distributed particles of size (in microns). The recystallized volume fraction is shown on the vertical axis and time on the horizontal axis. Results are shown at two different temperatures. It is obvious that an increased presence of particles retard the progression of recrystallization. Simulation results from a 3D cellular automata model.