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Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

In an effort to unify the various phase-field models that have been used to study solidification, we have developed a class of phase-field models for crystallization of a pure substance from its melt. These models are based on an entropy functional, as in the treatment of Penrose and Fife, and are therefore thermodynamically consistent inasmuch as they guarantee spatially local positive entropy production. General conditions are developed to ensure that the phase field takes on constant values in the bulk phases. Specific forms of a phase-field function are chosen to produce two models that bear strong resemblances to the models proposed by Langer and Kobayashi. Our models contain additional nonlinear functions of the phase field that are necessary to guarantee thermodynamic consistency.

Thermodynamically-consistent phase-field models for solidification

Real-time observations were made of the shape change from pyramids to domes during the growth of germanium-silicon islands on silicon (001). Small islands are pyramidal in shape, whereas larger islands are dome-shaped. During growth, the transition from pyramids to domes occurs through a series of asymmetric transition states with increasing numbers of highly inclined facets. Postgrowth annealing of pyramids results in a similar shape change process. The transition shapes are temperature dependent and transform reversibly to the final dome shape during cooling. These results are consistent with an anomalous coarsening model for island growth.

Transition states between pyramids and domes during Ge/Si island growth

Growth and self-organization of SiGe nanostructures

Our aim in this article is to review and discuss the Cahn–Hilliard equation, as well as some
of its variants. Such variants have applications in, e.g., biology and image inpainting.

The Cahn–Hilliard equation and some of its variants

New types of stationary solutions of a one-dimensional driven sixth-order Cahn-Hilliard type equation that arises as a model for epitaxially growing nano-structures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially non-monotone solutions in the limit of small driving force strength which is the deposition rate in case of epitaxial growth. These solutions extend the family of the monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert $W$ function. Using phase space analysis for the corresponding fifth-order dynamical system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input. Additionally, our approach is first demonstrated for the related but simpler driven fourth-order Cahn-Hilliard equation, also known as the convective Cahn-Hilliard equation.

Stationary solutions of driven fourth- and sixth-order Cahn-Hilliard type equations

We study a sixth order Cahn--Hilliard type equation that arises as a model for the faceting of a growing surface. We show existence of global-in-time weak solutions in two space dimensions, assuming periodic boundary conditions. We also establish exponential-in-time a priori estimates on the $H^3$ norm of solutions. These bounds enable us to prove the uniqueness of weak solutions. We also show the regularizing effect of the equation on the data.

/pdf/global-weak-solutions-to-a-sixth-order-cahn-hilliard-type-3bffriqywv.pdf

Global Weak Solutions to a Sixth Order Cahn--Hilliard Type Equation

New types of stationary solutions of a one-dimensional driven sixth-order Cahn–Hilliard-type equation that arises as a model for epitaxially growing nanostructures, such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially nonmonotone solutions in the limit of small driving force strength, which is the deposition rate in case of epitaxial growth. These solutions extend the family of the monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert W function. Using phase-space analysis for the corresponding fifth-order dynamical system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input. Additionally, our approach is first demonstrated for the related but simpler driven f...

Stationary Solutions of Driven Fourth- and Sixth-Order Cahn–Hilliard-Type Equations

A higher order convective Cahn--Hilliard-type equation that describes the faceting of a growing surface is considered with periodic boundary conditions. By using a Galerkin approach, the existence ...

/pdf/on-a-higher-order-convective-cahn-hilliard-type-equation-2qzrxgp3sc.pdf

On a higher order convective Cahn-Hilliard-type equation

/pdf/from-bell-shapes-to-pyramids-a-reduced-continuum-model-for-3vs1c2mi64.pdf

Maciek D. Korzec

Papers

Stationary solutions of driven fourth- and sixth-order Cahn-Hilliard type equations

Global Weak Solutions to a Sixth Order Cahn--Hilliard Type Equation

Stationary Solutions of Driven Fourth- and Sixth-Order Cahn–Hilliard-Type Equations

On a higher order convective Cahn-Hilliard-type equation

From bell shapes to pyramids: A reduced continuum model for self-assembled quantum dot growth