An amplitude equation for the conserved-Hopf bifurcation-Derivation, analysis, and assessment.

We employ weakly nonlinear theory to derive an amplitude equation for the conserved-Hopf instability, i.e., a generic large-scale oscillatory instability for systems with two conservation laws.

Binding potential and wetting behavior of binary liquid mixtures on surfaces.

We present a theory for the interfacial wetting phase behavior of binary liquid mixtures on rigid solid substrates, applicable to both miscible and immiscible mixtures. In particular, we calculate the binding potential as a function of the adsorptions, i.e.

Drops on Polymer Brushes: Advances in Thin-Film Modeling of Adaptive Substrates.

We briefly review recent advances in the hydrodynamic modeling of the dynamics of droplets on adaptive substrates, in particular, solids that are covered by polymer brushes. Thereby, the focus is on long-wave and full-curvature variants of mesoscopic hydrodynamic models in gradient dynamics form. After introducing the approach for films/drops of nonvolatile simple liquids on a rigid smooth solid substrate, it is first expanded to an arbitrary number of coupled degrees of freedom before considering the specific case of drops of volatile liquids on brush-covered solids.

Nonreciprocal Cahn-Hilliard Model Emerges as a Universal Amplitude Equation.

Oscillatory behavior is ubiquitous in out-of-equilibrium systems showing spatiotemporal pattern formation. Starting from a linear large-scale oscillatory instability-a conserved-Hopf instability-that naturally occurs in many active systems with two conservation laws, we derive a corresponding amplitude equation. It belongs to a hierarchy of such universal equations for the eight types of instabilities in homogeneous isotropic systems resulting from the combination of three features: large-scale vs small-scale instability, stationary vs oscillatory instability, and instability without and with conservation law(s).

Stationary broken parity states in active matter models.

We demonstrate that several nonvariational continuum models commonly used to describe active matter as well as other active systems exhibit nongeneric behavior: each model supports asymmetric but stationary localized states even in the absence of pinning at heterogeneities. Moreover, such states only begin to drift following a drift-transcritical bifurcation as the activity increases. Asymmetric stationary states should only exist in variational systems, i.

Stick-slip dynamics in the forced wetting of polymer brushes.

We study the static and dynamic wetting of adaptive substrates using a mesoscopic hydrodynamic model for a liquid droplet on a solid substrate covered by a polymer brush. First, we show that on the macroscale Young's law still holds for the equilibrium contact angle and that on the mesoscale a Neumann-type law governs the shape of the wetting ridge. Following an analytic and numeric assessment of the static profiles of droplet and wetting ridge, we examine the dynamics of the wetting ridge for a liquid meniscus that is advanced at constant mean speed.

Nonequilibrium configurations of swelling polymer brush layers induced by spreading drops of weakly volatile oil.

Polymer brush layers are responsive materials that swell in contact with good solvents and their vapors. We deposit drops of an almost completely wetting volatile oil onto an oleophilic polymer brush layer and follow the response of the system upon simultaneous exposure to both liquid and vapor. Interferometric imaging shows that a halo of partly swollen polymer brush layer forms ahead of the moving contact line.

From a microscopic inertial active matter model to the Schrödinger equation.

Active field theories, such as the paradigmatic model known as 'active model B+', are simple yet very powerful tools for describing phenomena such as motility-induced phase separation. No comparable theory has been derived yet for the underdamped case. In this work, we introduce active model I+, an extension of active model B+ to particles with inertia.

Non-reciprocity induces resonances in a two-field Cahn-Hilliard model.

We consider a non-reciprocally coupled two-field Cahn-Hilliard system that has been shown to allow for oscillatory behaviour and suppression of coarsening. After introducing the model, we first review the linear stability of steady uniform states and show that all instability thresholds are identical to the ones for a corresponding two-species reaction-diffusion system. Next, we consider a specific interaction of linear modes-a 'Hopf-Turing' resonance-and derive the corresponding amplitude equations using a weakly nonlinear approach.

Symmetry-breaking, motion and bistability of active drops through polarization-surface coupling.

Cell crawling crucially depends on the collective dynamics of the acto-myosin cytoskeleton. However, it remains an open question to what extent cell polarization and persistent motion depend on continuous regulatory mechanisms and autonomous physical mechanisms. Experiments on cell fragments and theoretical considerations for active polar liquids have highlighted that physical mechanisms induce motility through splay and bend configurations in a nematic director field.

Gradient-dynamics model for liquid drops on elastic substrates.

The wetting of soft elastic substrates exhibits many features that have no counterpart on rigid surfaces. Modelling the detailed elastocapillary interactions is challenging, and has so far been limited to single contact lines or single drops. Here we propose a reduced long-wave model that captures the main qualitative features of statics and dynamics of soft wetting, but which can be applied to ensembles of droplets.

Suppression of coarsening and emergence of oscillatory behavior in a Cahn-Hilliard model with nonvariational coupling.

We investigate a generic two-field Cahn-Hilliard model with variational and nonvariational coupling. It describes, for instance, passive and active ternary mixtures, respectively. Already a linear stability analysis of the homogeneous mixed state shows that activity not only allows for the usual large-scale stationary (Cahn-Hilliard) instability of the well-known passive case but also for small-scale stationary (Turing) and large-scale oscillatory (Hopf) instabilities.

Two-dimensional localized states in an active phase-field-crystal model.

The active phase-field-crystal (active PFC) model provides a simple microscopic mean field description of crystallization in active systems. It combines the PFC model (or conserved Swift-Hohenberg equation) of colloidal crystallization and aspects of the Toner-Tu theory for self-propelled particles. We employ the active PFC model to study the occurrence of localized and periodic active crystals in two spatial dimensions.

Phase-field-crystal description of active crystallites: Elastic and inelastic collisions.

The active Phase-Field-Crystal (aPFC) model combines elements of the Toner-Tu theory for self-propelled particles and the classical Phase-Field-Crystal (PFC) model that describes the transition between liquid and crystalline phases. In the liquid-crystal coexistence region of the PFC model, crystalline clusters exist in the form of localized states that coexist with a homogeneous background. At sufficiently strong activity (related to self-propulsion strength), they start to travel.

Efficient calculation of phase coexistence and phase diagrams: application to a binary phase-field-crystal model.

We show that one can employ well-established numerical continuation methods to efficiently calculate the phase diagram for thermodynamic systems described by a suitable free energy functional. In particular, this involves the determination of lines of phase coexistence related to first order phase transitions and the continuation of triple points. To illustrate the method we apply it to a binary phase-field-crystal model for the crystallisation of a mixture of two types of particles.

Adaptive stochastic continuation with a modified lifting procedure applied to complex systems.

Many complex systems occurring in the natural or social sciences or economics are frequently described on a microscopic level, e.g., by lattice- or agent-based models.

Thin-film modeling of resting and moving active droplets.

We propose a generic model for thin films and shallow drops of a polar active liquid that have a free surface and are in contact with a solid substrate. The model couples evolution equations for the film height and the local polarization in the form of a gradient dynamics supplemented with active stresses and fluxes. A wetting energy for a partially wetting liquid is incorporated allowing for motion of the liquid-solid-gas contact line.

Bifurcations of front motion in passive and active Allen-Cahn-type equations.

The well-known cubic Allen-Cahn (AC) equation is a simple gradient dynamics (or variational) model for a nonconserved order parameter field. After revising main literature results for the occurrence of different types of moving fronts, we employ path continuation to determine their bifurcation diagram in dependence of the external field strength or chemical potential. We then employ the same methodology to systematically analyze fronts for more involved AC-type models.

Effects of time-periodic forcing in a Cahn-Hilliard model for Langmuir-Blodgett transfer.

The influence of a temporal forcing on the pattern formation in Langmuir-Blodgett transfer is studied employing a generalized Cahn-Hilliard model. The occurring frequency-locking effects allow for controlling the pattern formation process. In the case of one-dimensional (i.

Resting and traveling localized states in an active phase-field-crystal model.

The conserved Swift-Hohenberg equation (or phase-field-crystal [PFC] model) provides a simple microscopic description of the thermodynamic transition between fluid and crystalline states. Combining it with elements of the Toner-Tu theory for self-propelled particles, Menzel and Löwen [Phys. Rev.

Connecting Monotonic and Oscillatory Motions of the Meniscus of a Volatile Polymer Solution to the Transport of Polymer Coils and Deposit Morphology.

We study the deposition mechanisms of polymer from a confined  meniscus of volatile liquid. In particular, we investigate the physical processes that are responsible for qualitative changes in the pattern deposition of polymer and the underlying interplay of the state of pattern deposition, motion of the meniscus, and the transport of polymer within the meniscus. As a model system we evaporate a solution of poly(methyl methacrylate) (PMMA) in toluene.

Modelling of surfactant-driven front instabilities in spreading bacterial colonies.

The spreading of bacterial colonies at solid-air interfaces is determined by the physico-chemical properties of the involved interfaces. The production of surfactant molecules by bacteria is a widespread strategy that allows the colony to efficiently expand over the substrate. On the one hand, surfactant molecules lower the surface tension of the colony, effectively increasing the wettability of the substrate, which facilitates spreading.

Equilibrium Contact Angle and Adsorption Layer Properties with Surfactants.

The three-phase contact line of a droplet on a smooth surface can be characterized by the Young equation. It relates the interfacial energies to the macroscopic contact angle θ. On the mesoscale, wettability is modeled by a film-height-dependent wetting energy f( h).

Collective Cell Migration in Embryogenesis Follows the Laws of Wetting.

Collective cell migration is a fundamental process during embryogenesis and its initial occurrence, called epiboly, is an excellent in vivo model to study the physical processes involved in collective cell movements that are key to understanding organ formation, cancer invasion, and wound healing. In zebrafish, epiboly starts with a cluster of cells at one pole of the spherical embryo. These cells are actively spreading in a continuous movement toward its other pole until they fully cover the yolk.