Coarsening rates for the dynamics of slipping droplets

We derive reduced finite dimensional ODE models starting from one dimensional lubrication equations describing coarsening dynamics of droplets in nanometric polymer film interacting on a hydrophobically coated solid substrate in the presence of large slippage at the liquid/solid interface. In the limiting case of infinite slip length corresponding in applications to free films a collision/absorption model then arises and is solved explicitly. The exact coarsening law is derived for it analytically and confirmed numerically. Existence of a threshold for the decay of initial distributions of droplet distances at infinity at which the coarsening rates switch from algebraic to exponential ones is shown

Weak solutions to lubrication equations in the presence of strong slippage

The existence of global weak solutions is proved for one-dimensional lubrication models that describe the dewetting process of nanoscopic thin polymer films on hydrophobyzed substrates and take account of large slippage at the polymer-substrate interface. The convergence of these solutions as either the Reynolds number or the capillarity goes to zero, as well as their limiting behaviour as the slip length goes to zero or infinity are investigated

Surface Energies Arising in Microscopic Modeling of Martensitic Transformations

In this paper we construct and analyze a two-well Hamiltonian on a 2D atomic lattice. The two wells of the Hamiltonian are prescribed by two rank-one connected martensitic twins, respectively. By constraining the deformed configurations to special 1D atomic chains with position-dependent elongation vectors for the vertical direction, we show that the structure of ground states under appropriate boundary conditions is close to the macroscopically expected twinned configurations with additional boundary layers localized near the twinning interfaces. In addition, we proceed to a continuum limit, show asymptotic piecewise rigidity of minimizing sequences and rigorously derive the corresponding limiting form of the surface energy

Thin film models for active gels

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 dynamcs in the bulk as well as wall defects and thus presents a significant extension of previous models based on the Lesli-Erickson-Parodi theory. Analytic expression are derived that reveal the interplay of anchoring conditions, film thickness and active terms and their control of transitions of flow structure.Comment: 33 pages, 3 figure

Weak solutions to lubrication systems describing the evolution of bilayer thin films

The existence of global nonnegative weak solutions is proved for coupled one-dimensional lubrication systems that describe the evolution of nanoscopic bilayer thin polymer films that take account of Navier-slip or no-slip conditions at both liquid-liquid and liquid-solid interfaces. In addition, in the presence of attractive van der Waals and repulsive Born intermolecular interactions existence of positive smooth solutions is shown

Thermal rupture of a free liquid sheet

We consider a free liquid sheet, taking into account the dependence of surface tension on temperature, or concentration of some pollutant. The sheet dynamics are described within a long-wavelength description. In the presence of viscosity, local thinning of the sheet is driven by a strong temperature gradient across the pinch region, resembling a shock. As a result, for long times the sheet thins exponentially, leading to breakup. We describe the quasi one-dimensional thickness, velocity, and temperature profiles in the pinch region in terms of similarity solutions, which posses a universal structure. Our analytical description agrees quantitatively with numerical simulations

Coarsening dynamics of slipping droplets

This paper studies the late phase dewetting process of nanoscopic thin polymer films on hydrophobized substrates using some recently derived lubrication models that take account of large slippage at the polymer-substrate interface. The late phase of this process is characterized by the slow-time coarsening dynamics of arrays of droplets that remain after rupture and the initial dewetting phases. For this situation a reduced system of ordinary differential equations is derived from the lubrication model for large slippage using asymptotic analysis. This extends known results for the no-slip case. On the basis of the reduced model, the role of the slippage as a control parameter for droplet migration is analysed and several new qualitative effects for the coarsening process are identified

Asymptotics for the spectrum of a thin film equation in a singular limit

In this paper the linear stability properties of the steady states of a no-slip lubrication equation are studied. The steady states are configurations of droplets and arise during the late-phase dewetting process under the influence of both destabilizing van der Waals and stabilizing Born intermolecular forces, which in turn give rise to the minimum thickness \eps of the remaining film connecting the droplets. The goal of this paper is to give an asymptotic description of the eigenvalues and eigenfunctions of the problem, linearized about the one-droplet solutions, as \eps\to 0. For this purpose, corresponding asymptotic eigenvalue problems with piecewise constant coefficients are constructed, such that their eigenvalue asymptotics can be determined analytically. A comparison with numerically computed eigenvalues and eigenfunctions shows good agreement with the asymptotic results and the existence of a spectrum gap to a single exponentially small eigenvalue for sufficiently small \eps

Liquid crystal defects in the Landau-de Gennes theory in two dimensions-beyond the one-constant approximation

We consider the two-dimensional Landau-de Gennes energy with several elastic constants, subject to general $k$-radially symmetric boundary conditions. We show that for generic elastic constants the critical points consistent with the symmetry of the boundary conditions exist only in the case $k=2$. In this case we identify three types of radial profiles: with two, three of full five components and numerically investigate their minimality and stability depending on suitable parameters. We also numerically study the stability properties of the critical points of the Landau-de Gennes energy and capture the intricate dependence of various qualitative features of these solutions on the elastic constants and the physical regimes of the liquid crystal system