Crystal Dislocations with Different Orientations and Collisions

Abstract

We study a parabolic differential equation whose solution represents the atom dislocation in a crystal for a general type of Peierls-Nabarro model with possibly long range interactions and an external stress. Differently from the previous literature, we treat here the case in which such dislocation is not the superposition of transitions all occurring with the same orientations (i.e. opposite orientations are allowed as well). We show that, at a long time scale, and at a macroscopic space scale, the dislocations have the tendency to concentrate as pure jumps at points which evolve in time, driven by the external stress and by a singular potential. Due to differences in the dislocation orientations, these points may collide in finite time. More precisely, we consider the evolutionary equation

$$(v_\varepsilon)_t=\frac{1}{\varepsilon}\left( \mathcal{I}v_\varepsilon-\frac{1}{\varepsilon^{2s}}W'(v_\varepsilon)+\sigma(t, x)\right),$$

where ${v_\varepsilon=v_\varepsilon(t, x)}$ is the atom dislocation function at time t >  0 at the point ${x \in \mathbb{R}, {\mathcal{I}_{s}}}$ is an integro-differential operator of order ${2s \in (0, 2), W}$ is a periodic potential, ${\sigma}$ is an external stress and ${\varepsilon > 0}$ is a small parameter that takes into account the small periodicity scale of the crystal. We suppose that ${v_\varepsilon(0, x)}$ is the superposition of N−K transition layers in the positive direction and K in the negative one (with ${K \in\{0,\dots,N\}}$); more precisely, we fix points ${x_1^0 < \dots < x_N^0}$ and we take

$$v_\varepsilon(0, x)= \frac{\varepsilon^{2s}}{W''(0)}\sigma(0, x)+\sum_{i=1}^N u\left(\zeta_i\frac{x-x_i^0}{\varepsilon}\right).$$

Here ${\zeta_i}$ is either −1 or 1, depending on the orientation of the transition layer u, which in turn solves the stationary equation ${\mathcal{I}_{s} u=W'(u)}$ . We show that our problem possesses a unique solution and that, as ${\varepsilon \to 0^+}$ , it approaches the sum of Heaviside functions H with different orientations centered at points x _i (t), namely

$$\sum_{i=1}^N H(\zeta_i(x-x_i(t))).$$

The point x _i evolves in time from ${x_i^0}$ , being subject to the external stress and a singular potential, which may be either attractive or repulsive, according to the different orientation of the transitions; more precisely, the speed ${\dot x_i}$ is proportional to

$$\sum_{j\neq i}\zeta_i\zeta_j\frac{x_i-x_j}{2s |x_i-x_j|^{1+2s}}-\zeta_i\sigma(t, x_i).$$

The evolution of such a dynamical system may lead to collisions in finite time. We give a detailed description of such collisions when N = 2, 3 and we show that the solution itself keeps track of such collisions; indeed, at the collision time T _c the two opposite dislocations have the tendency to annihilate each other and make the dislocation vanish, but only outside the collision point x _c , according to the formulas

$$\begin{array}{ll}{} \qquad \lim_{t \rightarrow T_c^-}\lim_{\varepsilon\rightarrow0^+}v_\varepsilon(t,x)=0 \quad {\mbox{when $x\ne x_c$,}}\\ {\rm and}\quad \limsup_{t\rightarrow T_c^-\atop \varepsilon \rightarrow 0^+}v_\varepsilon(t,x_c)\geq1.\end{array}$$

We also study some specific cases of N dislocation layers, namely when two dislocations are initially very close and when the dislocations are alternate. To the best of our knowledge, the results obtained are new even in the model case s = 1/2.