Michael H. Köpf

Pattern Formation in Monolayer Transfer Systems with Substrate-Mediated Condensation

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.

Structure Formation by Dynamic Self‐Assembly

This review summarizes the work conducted in the last decade on the fabrication of mesostructured patterns, which have lateral dimensions within the nano- and microscales, over a wafer-scaled size by means of dynamic self-assembly using Langmuir-Blodgett (LB) transfer or dip-coating. First, strategies to form mesostructures from a homogeneous Langmuir monolayer with controlled shape, size, and patterns alignment will be presented, followed by a detailed theoretical explanation of the pattern formation. In addition, the patterning of nanocrystals and other chemicals with LB transfer or other dynamic processes, such as dip-coating, will be summarized.

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.

Modelling Pattern Formation in Dip-Coating Experiments

We briefly review selected mathematical models that describe the dynamics of pattern formation phenomena in dip-coating and Langmuir-Blodgett transfer experiments, where solutions or suspensions are transferred onto a substrate producing patterned deposit layers with structure length from hundreds of nanometres to tens of micrometres. The models are presented with a focus on their gradient dynamics formulations that clearly shows how the dynamics is governed by particular free energy functionals and facilitates the comparison of the models. In particular, we include a discussion of models based on long-wave hydrodynamics as well as of more phenomenological models that focus on the pattern formation processes in such systems. The models and their relations are elucidated and examples of resulting patterns are discussed before we conclude with a discussion of implications of the gradient dynamics formulation and of some related open issues.

Thin film dynamics with surfactant phase transition

A thin liquid film covered with an insoluble surfactant in the vicinity of a first-order phase transition is discussed. Within the lubrication approximation we derive two coupled equations to describe the height profile of the film and the surfactant density. Thermodynamics of the surfactant is incorporated via a Cahn-Hilliard–type free-energy functional which can be chosen to describe a transition between two stable phases of different surfactant density. Within this model, a linear stability analysis of stationary homogeneous solutions is performed, and drop formation in a film covered with surfactant in the lower density phase is investigated numerically in one and two spatial dimensions.

Controlled nanochannel lattice formation utilizing prepatterned substrates

Solid substrates can be endued with self-organized regular stripe patterns of nanoscopic length scale by Langmuir-Blodgett transfer of organic monolayers. Here we consider the effect of periodically prepatterned substrates on this process of pattern formation. It leads to a time periodic forcing of the oscillatory behavior at the meniscus. Utilizing higher-order synchronization with this forcing, complex periodic patterns of predefined wavelength can be created. The dependence of the synchronization on the amplitude and the wavelength of the wetting contrast is investigated in one and two spatial dimensions, and the resulting patterns are discussed. Furthermore, the effect of prepatterned substrates on the pattern selection process is investigated.

Emergence of the bifurcation structure of a Langmuir–Blodgett transfer model

We explore the bifurcation structure of a modified Cahn–Hilliard equation that describes a system that may undergo a first-order phase transition and is kept permanently out of equilibrium by a lateral driving. This forms a simple model, e.g., for the deposition of stripe patterns of different phases of surfactant molecules through Langmuir–Blodgett transfer. Employing continuation techniques the bifurcation structure is numerically investigated using the non-dimensional transfer velocity as the main control parameter. It is found that the snaking structure of steady front states is intertwined with a large number of branches of time-periodic solutions that emerge from Hopf or period-doubling bifurcations and end in global bifurcations (sniper and homoclinic). Overall the bifurcation diagram has a harp-like appearance. This is complemented by a two-parameter study in non-dimensional transfer velocity and domain size (as a measure of the distance to the phase transition threshold) that elucidates through which local and global codimension 2 bifurcations the entire harp-like structure emerges.

Impact of temperature on the LB patterning of DPPC on mica.

The influence of the subphase temperature on the stripe pattern formation during Langmuir-Blodgett transfer (LB patterning) is investigated in a combined experimental and theoretical study. According to our experiments on the LB transfer of dipalmitoylphosphatidylcholine (DPPC) on planar mica substrates, even small temperature changes between 21.5 and 24.5 °C lead to significant changes in the monolayer patterns. For a constant surface pressure and dipper speed, the width of the stripes and the overall spatial period of the patterns increase with increasing subphase temperature. Because the stripe patterns are ascribed to alternating monolayer domains in the liquid-expanded and the liquid-condensed phases, the working regime for the formation of stripes is found to depend strongly on the respective surface pressure-area isotherm. These experimental findings are in accordance with the results of a theoretical investigation based on a model that takes hydrodynamics and the monolayer thermodynamics into account.

Dynamics of a thin liquid film with surface rigidity and spontaneous curvature.

The effect of rigid surfaces on the dynamics of thin liquid films that are amenable to the lubrication approximation is considered. It is shown that the Helfrich energy of the layer gives rise to additional terms in the time-evolution equations of the liquid film. The dynamics is found to depend on the absolute value of the spontaneous curvature, irrespective of its sign. Due to the additional terms, the effective surface-tension can be negative and an instability at intermediate wavelengths is observed. Furthermore, the dependence of the shape of a droplet on the bending rigidity as well as on the spontaneous curvature is discussed.

Stressed states and persistent defects in confined nematic elastica

We analyse the variety of solutions, both defect-free and containing defects in a confined nematic elastomer, arising as a result of competition between nematic ordering and elastic stresses. Phase field analysis predicts bifurcation of solutions with symmetric and antisymmetric distribution of nematic alignment from a perfectly aligned stressed state. The antisymmetric branch always has a lower energy, as confirmed by numerical computations away from the bifurcation point. At still higher deviations, states containing persistent defect pairs become preferable but defect-free states and states with defects coexist in a wide parametric range.