Dislocation dynamics on deformable surfaces

Abstract

We develop a fully coupled theoretical description of dislocation dynamics on deformable crystalline surfaces, using continuum modeling and the amplitude-phase-field crystal (APFC) framework extended to curved geometries. We derive a general kinematic expression for dislocation velocity directly from the complex-amplitude evolution equations, which is also applicable to deformed surfaces through curvature-modified differential operators. From numerical simulations, we show that even small out-of-plane deformations reshape the phenomenology of defect motion through curvature-induced self-propulsion, modified glide directions, and non-classical defect-defect interactions. Our results show how surface geometry profoundly influences defect dynamics and establish the surface-APFC model as a powerful framework for predicting and interpreting curvature-defect coupling across a wide range of systems, from stiff but deformable layers to soft matter surfaces and membranes that retain crystalline order.

Figures (3)

• Figure 1: Dislocation dipole. (a): Real and imaginary parts of an amplitude carrying topological charge. Isolines $\Phi = 0.3\,\Phi_\mathrm{MAX}$ enclose the dislocation cores. $\phi_0$ is the (real) value of the amplitude in the relaxed bulk. (b): Field $\Phi$ in the square region highlighted in panel (a) by the dashed line. The black solid line corresponds to the isoline $\Phi = 0.3\,\Phi_\mathrm{MAX}$. The orange arrow represents the velocity of the dislocation core calculated via Eq. \ref{['eq:vv']}. (c): Same setup as panel (b), but considering a deformable surface. The color map shows the height profile around the rightmost defect. The black solid line corresponds to the isoline $\Phi = 0.3\,\Phi_\mathrm{MAX}$. The orange arrow represents the velocity of the dislocation core calculated via Eq. \ref{['eq:vv']}. (d): Trajectories of dislocation dipoles on deformable surfaces with variable bending stiffness $\kappa$.
• Figure 2: Effect of the curvature on dipole motion. (a) Illustration of the initial conditions. The density plot shows the double-Gaussian height profile. Pink dots mark the initial positions of the dislocations. We define four configurations: dislocations starting from the flat, equidistant to the Gaussian profiles (E), starting from the flat, with one defect closer to the Gaussian than the other (S$+$, S$-$), and starting at the peak of the Gaussian profiles (P). The initial distance between the dislocations is the same in all configurations. (b) Distance between the dislocations over time in the four configurations illustrated in panel (a) as well as for a flat surface (Flat).
• Figure 3: Velocity of a single dislocation due to surface deformation in the linear core approximation. The color maps show height profiles, with quivers indicating the direction of dislocation motion. One-dimensional plots show the height profile and the $v_y$ component of the velocity across a vertical cut at $x=0$. (a,b): Sinusoidal profile. We identify asymmetric stable equilibrium points on the slopes and unstable equilibrium points at the peaks. (c,d): Steep 1D Gaussian. Asymmetric stable equilibrium points on the slopes, one unstable equilibrium point at the peak. (e,f): Shallow 1D Gaussian. The height profile is rescaled by a factor of 200 for plotting purposes. One unstable equilibrium point at the peak. (g,h): Steep 2D Gaussian. Opposite signs of the height profile produce the same velocity field. We identify one stable equilibrium point (red dot), one unstable equilibrium point (green), and one saddle point (pink). The 1D cut is identical to panel (d). (i): Shallow 2D Gaussian. Only one unstable equilibrium point (green dot). The 1D cut is identical to panel (f).