| Abstract: |
| Patterned thin gold lines when undergoing solid-state dewetting break up into linear grain sections interspersed with occasionally larger ``abacus`` particles. We model this system using coupled geometric flows, where the exterior surfaces of the grains evolve via Surface Diffusion $V = -\Delta_{\Gamma(t)}H$ and the internal grain boundaries evolve via Mean Curvature Flow $V = A H$. In this talk we first motivate the problem using these experimental observations. Next, we describe the steady states for this mixed-order system. By assuming axi-symmetry, we can systematically construct composite equilibria by piecing together Delaunay surfaces --- specifically, combining unduloidal, cylindrical and spherical exterior surfaces partitioned by planar or catenoidal internal grain boundaries. We then introduce a stability analysis by first linearising the coupled (nonlinear) problem about these composite steady states, deriving a Jacobi operator whose strict positivity implies asymptotic stability of the linearised problem. Using this approach and building upon the classical Rayleigh stability criteria, we obtain stability predictions for steady-state configurations consisting of near-cylindrical and near-spherical grains meeting at a single planar grain boundary. |
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