2007, King, John R., Münch, Andreas, Wagner, Barbara

The topic of this study concerns the stability of the three-phase contact-line of a dewetting thin liquid film on a hydrophobised substrate driven by van der Waals forces. The role of slippage in the emerging instability at the three-phase contact-line is studied by deriving a sharp-interface model for the dewetting thin film via matched asymptotic expansions. This allows for a derivation of travelling waves and their linear stability via eigenmode analysis. In contrast to the dispersion relations typically encountered for the finger-instabilty, where the dependence of the growth rate on the wave number is quadratic, here it is linear. Using the separation of time scales of the slowly growing rim of the dewetting film and time scale on which the contact line destabilises, the sharp-interface results are compared to earlier results for the full lubrication model and good agreement for the most unstable modes is obtained.

2014, Dziwnik, Marion, Münch, Andreas, Wagner, Barbara

We propose a phase field model for solid state dewetting in form of a Cahn-Hilliard equation with weakly anisotropic surface energy and a degenerate mobility together with a free boundary condition at the film-substrate contact line. We derive the corresponding sharp interface limit via matched asymptotic analysis involving multiple inner layers. The resulting sharp interface model is consistent with the pure surface diffusion model. In addition, we show that the natural boundary conditions, as indicated from the first variation of the total free energy, imply a contact angle condition for the dewetting front, which, in the isotropic case, is consistent with the well-known Young's equation

2015, Ahnert, Tobias, Münch, Andreas, Niethammer, Barbara, Wagner, Barbara

The stability of two-dimensional Poiseuille flow and plane Couette flow for concentrated suspensions is investigated. Linear stability analysis of the twophase flow model for both flow geometries shows the existence of a convectively driven instability with increasing growth rates of the unstable modes as the particle volume fraction of the suspension increases. In addition it is shown that there exists a bound for the particle phase viscosity below which the two-phase flow model may become ill-posed as the particle phase approaches its maximum packing fraction. The case of two-dimensional Poiseuille flow gives rise to base state solutions that exhibit a jammed and unyielded region, due to shear-induced migration, as the maximum packing fraction is approached. The stability characteristics of the resulting Bingham-type flow is investigated and connections to the stability problem for the related classical Bingham-flow problem are discussed.

2014, Ahnert, Tobias, Münch, Andreas, Wagner, Barbara

A new two-phase model for concentrated suspensions is derived that incorporates a constitutive law combining the rheology for non-Brownian suspension and granular flow. The resulting model naturally exhibits a Bingham-type flow property. This property is investigated in detail for the simple geometry of plane Poiseuille flow, where an unyielded or jammed zone of finite width arises in the center of the channel. For the steady state of this problem, the governing equation are reduced to a boundary value problem for a system of ordinary differential equations and the dependence of its solutions are analyzed by using phase-space methods. For the general time-dependent case a new drift-flux model is derived for the first time using matched asymptotic expansions that take account of the boundary layers at the walls and the interface between the yielded and unyielded region. Using the drift-flux model, the behavior of the suspension flow, in particular the appearance and evolution of unyielded or jammed regions is then studied numerically for different choices of the parameters.

2010, Grote, Marcus, Palumberi, Viviana, Wagner, Barbara, Barbero, Andrea, Martin, Ivan

Growth factors have a significant impact not only on the growth dynamics but also on the phenotype of chondrocytes (Barbero et al. , J. Cell. Phys. 204, pp. 830-838, 2005). In particular, as chondrocyte populations approach confluence, the cells tend to align and form coherent patches. Starting from a mathematical model for fibroblast populations at equilibrium (Mogilner et al., Physica D 89, pp. 346-367, 1996), a dynamic continuum model with logistic growth is developed. Both linear stability analysis and numerical solutions of the time-dependent nonlinear integro-partial differential equation are used to identify the key parameters that lead to pattern formation in the model. The numerical results are compared quantitatively to experimental data by extracting statistical information on orientation, density and patch size through Gabor filters.

2021, Shiri, Roghayeh, Schmeller, Leonie, Seemann, Ralf, Peschka, Dirk, Wagner, Barbara

We investigate the spinodal dewetting of a thin liquid polystyrene (PS) film on a liquid polymethylmethacrylate (PMMA) subtrate. Following the evolution of the corrugations of the PS film via in situ measurements by atomic force microscopy (AFM) and those of the PS-PMMA interface via ex situ imaging, we provide a direct and detailed comparison of the experimentally determined spinodal wavelengths with the predictions from linear stability analysis of a thin-film continuum model for the bilayer system. The impact of rough interfaces and fluctuations is studied theoretically by investigating the impact of different choices of initial data on the unstable wavelength and on the rupture time. The key factor is the mode selection by initial data perturbed with correlated colored noise in the linearly unstable regime, which becomes relevant only for liquid bilayers to such an extent. By numerically solving the mathematical model, we further address the impact of nonlinear effects on rupture times and on the morphological evolution of the interfaces in comparison with experimental results.

2020, Hennessy, Matthew G., Celora, Giulia L., Münch, Andreas, Waters, Sarah L., Wagner, Barbara

An asymptotic framework is developed to study electric double layers that form at the inter-face between a solvent bath and a polyelectrolyte gel that can undergo phase separation. The kinetic model for the gel accounts for the finite strain of polyelectrolyte chains, free energy ofinternal interfaces, and Stefan?Maxwell diffusion. By assuming that the thickness of the doublelayer is small compared to the typical size of the gel, matched asymptotic expansions are used toderive electroneutral models with consistent jump conditions across the gel-bath interface in two-dimensional plane-strain as well as fully three-dimensional settings. The asymptotic frameworkis then applied to cylindrical gels that undergo volume phase transitions. The analysis indicatesthat Maxwell stresses are responsible for generating large compressive hoop stresses in the double layer of the gel when it is in the collapsed state, potentially leading to localised mechanicalinstabilities that cannot occur when the gel is in the swollen state. When the energy cost of in-ternal interfaces is sufficiently weak, a sharp transition between electrically neutral and chargedregions of the gel can occur. This transition truncates the double layer and causes it to have finitethickness. Moreover, phase separation within the double layer can occur. Both of these featuresare suppressed if the energy cost of internal interfaces is sufficiently high. Thus, interfacial freeenergy plays a critical role in controlling the structure of the double layer in the gel.

2018, Peschka, Dirk, Haefner, Sabrina, Jacobs, Karin, Münch, Andreas, Wagner, Barbara

Thin liquid polymer films on hydrophobic substrates are susceptable to rupture and formation of holes, which in turn initiate a complex dewetting process that eventually evolves into characteristic stationary droplet patterns. Experimental and theoretical studies suggest that the specific type of droplet pattern largely depends on the nature of the polymer-substrate boundary condition. To follow the morphological evolution numerically over long time scales and for the multiple length scales involved has so far been a major challenge. In this study a highly adaptive finite-element based numerical scheme is presented that allows for large-scale simulations to follow the evolution of the dewetting process deep into the nonlinear regime of the model equations, capturing the complex dynamics including shedding of droplets. In addition, the numerical results predict the previouly unknown shedding of satellite droplets during the destabilisation of liquid ridges, that form during the late stages of the dewetting process. While the formation of satellite droplets is well-known in the context of elongating fluid filaments and jets, we show here that for dewetting liquid ridges this property can be dramatically altered by the interfacial condition between polymer and substrate, namely slip.

2011, Jachalski, Sebastian, Huth, Robert, Kitavtsev, Georgy, Peschka, Dirk, Wagner, Barbara

We investigate stationary solutions of flows of thin liquid bilayers in an energetic formulation which is motivated by the gradient flow structure of its lubrication approximation. The corresponding energy favors the liquid substrate to be only partially covered by the upper liquid. This is expressed by a negative spreading coefficient which arises from an intermolecular potential combining attractive and repulsive forces and leads to an ultra-thin layer of thickness e. For the corresponding lubrication models existence of stationary solutions is proven. In the limit e to 0 matched asymptotic analysis is applied to derive sharp-interface models and the corresponding contact angles, i.e. the Neumann triangle. In addition we use G-convergence and derive the equivalent sharp-interface models rigorously in this limit. For the resulting model existence and uniqueness of energetic minimizers are proven. The minimizers agree with solutions obtained by matched asymptotics.

2017, Kitavtsev, Georgy, Münch, Andreas, Wagner, Barbara

In this study we present a free-boundary problem for an active liquid crystal based on the Beris-Edwards theory that uses a tensorial order parameter and includes active contributions to the stress tensor to analyse the rich defect structure observed in applications such as the Adenosinetriphosphate (ATP) driven motion of a thin film of an actin filament network. The small aspect ratio of the film geometry allows for an asymptotic approximation of the free-boundary problem in the limit of weak elasticity of the network and strong active terms. The new thin film model captures the defect dynamics in the bulk as well as wall defects and thus presents a significant extension of previous models based on the Leslie-Erickson-Parodi theory. Analytic expressions are derived that reveal the interplay of anchoring conditions, film thickness and active terms and their control of transitions of flow structure.