Pattern Formation in Monolayer Transfer Systems with Substrate-Mediated Condensation

Abstract

The formation of regular stripe patterns during transfer of surfactant monolayers onto solid substrates is investigated. Two coupled differential equations describing the surfactant density and the height profile of the water subphase are derived within the lubrication approximation. If the transfer is carried out in the vicinity of a first order phase transition of the surfactant, the interaction with the substrate plays a key role. This effect is included in the surfactant free-energy functional via a height-dependent external field. Using transfer velocity as a control parameter, a bifurcation from a homogeneous transfer to regular stripe patterns arranged parallel to the contact line is investigated in one and two dimensions. Moreover, in the two-dimensional case, a secondary bifurcation to perpendicular stripes is observed in a certain control parameter range.

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

“…A recently developed model describes the monolayer transfer process using a set of two coupled nonlinear partial differential equations for the time evolution of the height of the liquid sub-phase and the density of the covering surfactant layer [19,20]. The hydrodynamics of the thin liquid layer is treated within the framework of lubrication theory [21] and is coupled to the monolayer thermodynamics via a surfactant equation of state.…”

“…The results obtained using this model indicate that in this case the hydrodynamics in the meniscus is not essential for the formation of stripe patterns. Instead, the patterns form by phase decomposition under the influence of an external field that describes the interaction between the substrate and the monolayer [19,20]. Therefore, this effect may be captured by a simplified model that allows for a more profound mathematical analysis and, as a consequence, a deeper understanding of the physical mechanisms.…”

“…Two coefficients, M 1 and M 2 , characterize the free energy of the surfactant layer that is modeled as a symmetric double-well centered around the critical density (see [20] for details). By choosing S(H ) = B (H ), where (H ) = dH (H ) with integration constant zero is the potential of the substrate-liquid interaction and B is a positive coupling constant, SMC acts on length scales comparable to the substrate-liquid interaction [19,20]. Note that the model given by equations (1) and (2) assumes that the monolayer covered surface is perfectly mobile; that is, the rigidity of the surfactant layer is not taken into account.…”

“…In a time simulation, parallel stripes always emerge first, i.e. they occur at least as transient states that then transform into perpendicular stripes through an orientation transition [19,20]. Note that the perpendicular stripe pattern is stationary in the laboratory frame, while the parallel stripe pattern is time periodic and consists of domains that are perpetually generated and grow at the contact line, detach from it and finally are advected with the moving substrate.…”

Self-assembly of metastable langmuir monolayers on planar solid surfaces

“…A similar theoretical model that described sub strate induced condensation in terms of an external field acting on a monolayer with a force, which depends on the distance between the monolayer and the solid surface (i.e., on the thickness of a dividing liquid film), was used in [96][97][98]. The authors employed numerical simulation to show that still more complex periodic structures can be formed due to syn chronization of hydrodynamic conditions and the process of condensation during the transfer of a monolayer onto a surface coated with a periodic pat tern preliminarily fabricated on it [98].…”