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

Abstract

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.

First order phase transitions and the thermodynamic limit

“…See, e.g. [34][35][36] for situations where this condition is not fulfilled. We note in passing that in situations where the total concentration is not controlled, the parameter μ becomes a relevant control parameter representing an external field or imposed chemical potential.…”

Self-organized dip-coating patterns of simple, partially wetting, nonvolatile liquids

“…This externally imposed front velocity is a control parameter of the system. Examples for investigations of such triggered pattern formation range from the experimentally and theoretically investigated structure formation in Langmuir-Blodgett films [29][30][31][32][33][34], over the study of Cahn-Hilliard-type model equations in one (1D) and two (2D) dimensions for externally quenched phase separation (e.g., by a moving temperature jump for films of polymer blends or binary mixtures) [35][36][37] to the rigorous mathematical analysis of trigger fronts in a complex Ginzburg-Landau equation as well as in a Cahn-Hilliard and an Allen-Cahn equation [38][39][40]. In the aforementioned systems, a switch from a linearly stable to an unstable state takes place at a certain position within the considered domain.…”

Two-dimensional patterns in dip coating - first steps on the continuation path

“…Time-periodic states that emerge at these Hopf bifurcations are mostly unstable and end at other Hopf bifurcations or at global bifurcations. For details see [37]. In Fig.…”

“…Their exact values are of minor importance. For details about the driven Cahn-Hilliard model and corresponding mesoscopic hydrodynamic models see [51,52,37,53] The finally resulting scaled nondimensional model writes…”

“…The separation into these phases is caused by substrate-induced condensation [34], i.e., by an influence of the molecular interaction with the substrate on the surfactant phase transition. Most aspects of the process are successfully described by a generalized Cahn-Hilliard model for phase separation that entirely neglects hydrodynamic motion [37]. An extensive study in 1d employing numerical path continuation [38] reveals an intricate bifurcation structure consisting of many branches of steady concentration profiles and time-periodic states interconnected by a plethora of local and global bifurcations [37,39] while time simulations in 2d show that 1d patterns are not always transversally stable [37,40].…”