Existence of solutions to an anisotropic degenerate Cahn-Hilliard-type equation

We prove existence of solutions to an anisotropic Cahn-Hilliard-type equation with degenerate diffusional mobility. In particular, the mobility vanishes at the pure phases, which is typically used to model motion by surface diffusion. The main difficulty of the present existence result is the strong non-linearity given by the fourth-order anisotropic operator. Imposing particular assumptions on the domain and assuming that the strength of the anisotropy is sufficiently small enables to establish appropriate auxiliary results which play an essential part in the present existence proof. In addition to the existence we show that the absolute value of the corresponding solutions is bounded by 1

Existence of solutions to an anisotropic degenerate Cahn--Hilliard-type equation

We prove existence of solutions to an anisotropic Cahn-Hilliard-type equation with degenerate diffusional mobility. In particular, the mobility vanishes at the pure phases, which is typically used to model motion by surface diffusion. The main difficulty of the present existence result is the strong non-linearity given by the fourth-order anisotropic operator. Imposing particular assumptions on the domain and assuming that the strength of the anisotropy is sufficiently small enables to establish appropriate auxiliary results which play an essential part in the present existence proof. In addition to the existence we show that the absolute value of the corresponding solutions is bounded by 1

Stability analysis of non-constant base states in thin film equations

We address the linear stability of non-constant base states within the class of mass conserving free boundary problems for degenerate and non-degenerate thin film equations. Well-known examples are the finger-instabilities of growing rims that appear in retracting thin solid and liquid films. Since the base states are time dependent and do not have a simple travelling wave or self-similar form, a classical eigenvalue analysis fails to provide the dominant wavelength of the instability. However, the initial fronts evolve on a slower time-scale than the typical perturbations. We exploit this time-scale separation and develop a multiple-scale approach for this class of stability problems. We show that the value of the dominant wavelength is rapidly attained once the base state has entered an approximately self-similar scaling. We note that this value is different from the one obtained by the linear stability analysis with "frozen modes", frequently found in the literature. Furthermore we show that for the present class of stability problems the dispersion relation behaves linear for large wavelengths, which is in contrast to many other instability problems in thin film flows

Stability analysis of non-constant base states in thin film equations

We address the linear stability of non-constant base states within the class of mass conserving free boundary problems for degenerate and non-degenerate thin film equations. Well-known examples are the finger-instabilities of growing rims that appear in retracting thin solid and liquid films. Since the base states are time dependent and do not have a simple travelling wave or self-similar form, a classical eigenvalue analysis fails to provide the dominant wavelength of the instability. However, the initial fronts evolve on a slower time-scale than the typical perturbations. We exploit this time-scale separation and develop a multiple-scale approach for this class of stability problems. We show that the value of the dominant wavelength is rapidly attained once the base state has entered an approximately self-similar scaling. We note that this value is different from the one obtained by the linear stability analysis with "frozen modes", frequently found in the literature. Furthermore we show that for the present class of stability problems the dispersion relation behaves linear for large wavelengths, which is in contrast to many other instability problems in thin film flows

A phase-field model for solid-state dewetting and its sharp-interface limit

We propose a phase field model for solid state dewetting where the surface energy is weakly anisotropic. The evolution is based on the Cahn-Hilliard equation with degenerate mobility and 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

Der Entnetzungsprozess von dünnen festen Filmen : Modellierung, Analysis und numerische Simulation

This dissertation is devoted to the mathematical study of solid state dewetting and deals with various mathematical topics such as phase field modeling, the derivation of corresponding sharp interface limits, existence of solutions, numerical simulations and linear stability analysis of the dewetting front. We start with the formulation of a two-dimensional anisotropic phase field model for solid state dewetting on a solid substrate. The evolution is described by a Cahn-Hilliard type equation with a bi-quadratic degenerate mobility and a polynomial homogeneous free energy. We propose an anisotropic free boundary condition at the film/substrate contact line which correspond to the natural boundary condition from the variational derivation. We show via matched asymptotic analysis that the resulting sharp interface model is consistent with the pure surface diffusion model. In addition, we show that the corresponding natural boundary conditions at the substrate imply a contact angle condition which is known as Young-Herring condition. We provide an existence result for the present degenerate partial differential equation on a simplified domain with homogeneous Neumann boundary conditions. Under the assumption that the strength of the anisotropy is sufficiently small, we establish certain convexity properties and higher order bounds of the strongly non-linear anisotropic operator. This enables to prove existence of weak solutions. Furthermore, we show that solutions are bounded by one without having a maximum principle. Completing the part which is concerned with the phase field representation, we consider the numerical simulation of the present model, where we apply a diffuse boundary approximation to handle the boundary conditions at the substrate. The reformulated equation can be solved by a standard finite element method. A matched asymptotic analysis shows that solutions of the reformulated equations formally converge to those of the original equations. We provide numerical simulations which confirm this analysis. In addition, we address the previously discussed question of how the mobility influences the evolution and simulate dewetting scenarios for different mobilities and anisotropies. In the last main chapter we consider a generalized class of thin film equations, including the case which corresponds to the small slope approximation of the sharp interface model for isotropic solid state dewetting. We present an improved method for the linear stability analysis of unsteady, non-uniform base states in thin film equations which exploits that the initial fronts evolve on a slower time-scale than the typical perturbations. The result is a unique value for the dominant wavelength which is different from the one obtained by the frequently applied linear stability analysis with "frozen modes". Furthermore we show that for the present class of stability problems the dispersion relation is linear in the long wave limit, which is in contrast to many other instability problems in thin film flows.Die vorliegende Dissertation widmet sich der mathematischen Studie des Entnetzungsprozesses von dünnen festen Filmen und beschäftigt sich mit einer Vielfalt von mathematischen Themen, so wie die Modellierung eines Phasenfeld-Modells, die Herleitung des entsprechenden Sharp Interface Grenzwertes, schwache Lösungstheorie und die lineare Stabilitätsanalyse der Entnetzungsfront. Wir beginnen mit der Formulierung eines Phasenfeldmodells für den Entnetzungsprozess fester Filme auf einem festen Substrat. Die Evolution wird durch eine Gleichung vom Cahn-Hilliard-Typ mit einer biquadratischen degenerierten Mobilität und einer polynomialen homogenen freien Energie beschrieben. Wir fordern anisotrope Randbedingungen am freien Rand an der Film-Substrat Kontaktlinie, die den natürlichen Randbedingungen aus der variationellen Formulierung des Modells entsprechen. Wir zeigen mit Hilfe von "matched asymptotic analysis", dass das resultierende sharp interface Modell konsistent ist mit dem Modell für reine Oberflächendiffusion. Darüber hinaus zeigen wir, dass die dazugehörigen Randbedingungen eine Kontaktwinkel Bedingung implizieren, die auch als Young-Herring Bedingung bekannt ist. Wir zeigen die Existenz von schwachen Lösungen der vorliegenden Differentialgleichung auf einem vereinfachten Gebiet mit homogenen Neumann Randbedingungen. Unter der Bedingung, dass die Stärke der Anisotropie genügend klein ist, zeigen wir bestimmte Konvexität-Eigenschaften und Beschränktheit zu höheren Ordnungen des stark nicht-linearen anisotropen Operators. Das ermöglicht es uns die Existenz von schwachen Lösungen zu beweisen. Des weiteren zeigen wir, dass diese Lösungen durch Eins beschränkt sind ohne dass ein Maximumsprinzip zur Verfügung steht. Den ersten großen Hauptteil, abschließend betrachten wir die numerische Simulation des vorliegenden Modells, wobei wir eine "diffuse boundary" Näherung benutzen um die Randbedingungen am Substrat umzusetzen. Die umformulierte Gleichung kann mit Hilfe von standard- finite Elemente Methoden implementiert werden. Wir zeigen via "matched asymptotic analysis", dass Lösungen des umformulierten Problems formal gegen Lösungen des ursprünglichen Problems konvergieren. Wir zeigen numerische Simulationen, die diese Analysis belegen. Darüber hinaus beschäftigen wir uns mit der Fragestellung der vorangegangen Kapitel, nämlich in welcher Art die Mobilität die Evolution beeinflusst. Dazu zeigen wir Simulationen zu verschiedenen Mobilitäten und Anisotropien. Im letzten Hauptkapitel betrachten wir eine verallgemeinerte Klasse von Dünnfilmgelichungen, die den Fall der "small slope approximation" des entsprechenden sharp interface Modells für Oberflächendiffusion beinhaltet. Wir präsentieren eine verbesserte lineare Stabilitätsanalyse für zeitabhängige Grundzustände in Dünnfilmgelichungen, die ausnutzt dass sich die die Entnetzungsfront auf einer langsameren Zeitskala entwickeln als die typischen Instabilitäten. Das Resultat ist ein eindeutiger Wert für die dominante Wellenlänge, die sich zudem von dem Wert, der durch eine klassische "frozen mode" Analyse erhalten wird, unterscheidet. Des weiteren zeigen wir, dass sich für die vorliegende Klasse von Stabilitätsproblemen die Dispersionsrelation im "long wave limit" linear verhält, was im Gegensatz steht zu vielen anderen Stabilitätsproblemen in dünnen Filmen

Existence of solutions to an anisotropic degenerate Cahn–Hilliard-type equation

We prove existence of solutions to an anisotropic Cahn–Hilliard-type equation with degenerate diffusional mobility. In particular, the mobility vanishes at the pure phases, which is typically used to model motion by surface diffusion. The main difficulty of the present existence result is the strong non-linearity given by the fourth-order anisotropic operator. Imposing particular assumptions on the domain and assuming that the strength of the anisotropy is sufficiently small enables to establish appropriate bounds which allow to pass to the limit in the regularized problem. In addition to the existence we show that the absolute value of the corresponding solutions is bounded by 1

A phase-field model for solid-state dewetting and its sharp-interface limit

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 equatio