Elastically buckled film-substrate system as a two-dimensional crystal

Compressive mechanical stress exceeding a critical value leads to the formation of periodic surface buckling patterns in film-substrate systems. A comprehensive understanding of this buckling phenomenon is essential in applications where the surface topologies are modulated to achieve multifunctionalities. Here, we reformulate the elastic theory of a film-substrate system by treating the compliant substrate as a nonlinear elastic solid. The resulting elastic free energy functional of the deflection field is shown to be equivalent to a minimal density functional of the phase-field crystal theory plus a Gaussian curvature-related term. The proposed elastic model constructs a phase diagram based on free energy minimization, quantitatively agreeing with the buckling transitions observed in previous experiments. The emerging hexagonal buckling system is shown to be equivalent to a two-dimensional crystal with proper scalings. We further conducted simulations of repeated buckling under cyclic stress to demonstrate a dynamically modulated structural adhesive, which resembles the physical process of repeated crystallization and melting near a critical temperature.