Grain boundaries, magnetic domain walls, ferroelectric domain walls, anti-phase boundaries, dislocation, etc. all migrate in response to external stimuli. This motion is normally dissipative. In many cases, the intrinsic mobility of the domain wall is not observed because even in the purest materials, the concentration of impurities that interact with the domain walls is sufficient that the mobility is determined by some process associated with the impurities. This phenomena is particularly rich when the impurities are free to diffuse in response to the presence of the domain wall. We present some Monte Carlo simulation results of the motion of domain walls in the presence of diffusing impurities. The Monte Carlo model is based upon several simple extensions of the Ising model. The simulation model shows that there are two regimes of behavior: one in which the domain wall moves by dragging a nearly equilibrium impurity cloud and one in which the domain wall moves fast compared with the diffusion of the impurities. In some cases, there is an abrupt transition between the two. The results are shown to be in rough qualitative agreement with a simple 1-d continuum model, but in rather poor quantitative agreement. We trace the disagreement primarily to the discreteness of the Ising model and nearly all physical systems of interest. We present a model based upon kink nucleation and migration in the presence of impurities that is consistent with the low driving force simulation data.
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