Substrate-mediated pattern formation in monolayer transfer: a reduced model

The formation of regular stripe patterns during the transfer of surfactant monolayers from water surfaces onto moving solid substrates can be understood as a phase decomposition process under the influence of the effective molecular interaction between the substrate and the monolayer, also called substrate-mediated condensation (SMC). To describe this phenomenon, we propose a reduced model based on an amended Cahn-Hilliard equation. A combination of numerical simulations and continuation methods is employed to investigate stationary and time-periodic solutions of the model and to determine the resulting bifurcation diagram. The onset of spatiotemporal pattern formation is found to result from a homoclinic and a Hopf bifurcation at small and large substrate speeds, respectively. The critical velocity corresponding to the Hopf bifurcation can be calculated by means of the marginal stability criterion for pattern formation behind propagating fronts. In the regime of low transfer velocities, the stationary solutions exhibit snaking behavior.