Finite element method for epitaxial growth with thermodynamic boundary conditions

We develop an adaptive finite element method for island dynamics in epitaxial growth. We study a step-flow model, which consists of an adatom (adsorbed atom) diffusion equation on terraces of different height, thermodynamic boundary conditions on terrace boundaries including anisotropic line tension, and the normal velocity law for the motion of such boundaries determined by a two-sided flux, together with the one-dimensional (possibly anisotropic) "surface" diffusion of edge-adatoms along the step-edges. The problem is solved using two independent meshes: a two-dimensional mesh for the adatom diffusion and a one-dimensional mesh for the boundary evolution. A penalty method is used in order to incorporate the boundary conditions. The evolution of the terrace boundaries includes both the weighted/anisotropic mean curvature flow and the weighted/anisotropic surface diffusion. Its governing equation is solved by a semi-implicit front-tracking method using parametric finite elements

Return Radius and volume of recrystallized material in Ostwald Ripening

Within the framework of the LSW theory of Ostwald ripening the amount of volume of the second (solid) phase that is newly formed by recrystallization is investigated. It is shown, that in the late stage, the portion of the newly generated volume formed within an interval from time $t_0$ to $t$ is a certain function of $t/t_0$ and an explicit expression of this volume is given. To achieve this, we introduce the notion of the {\it return radius} $r(t,t_0)$, which is the unique radius of a particle at time $t_0$ such that this particle has -- after growing and shrinking -- the same radius at time $t$. We derive a formula for the return radius which later on is used to obtain the newly formed volume. Moreover, formulas for the growth rate of the return radius and the recrystallized material at time $t_0$ are derived

Morphological stability of electromigration-driven vacancy islands

The electromigration-induced shape evolution of two-dimensional vacancy islands on a crystal surface is studied using a continuum approach. We consider the regime where mass transport is restricted to terrace diffusion in the interior of the island. In the limit of fast attachment/detachment kinetics a circle translating at constant velocity is a stationary solution of the problem. In contrast to earlier work [O. Pierre-Louis and T.L. Einstein, Phys. Rev. B 62, 13697 (2000)] we show that the circular solution remains linearly stable for arbitrarily large driving forces. The numerical solution of the full nonlinear problem nevertheless reveals a fingering instability at the trailing end of the island, which develops from finite amplitude perturbations and eventually leads to pinch-off. Relaxing the condition of instantaneous attachment/detachment kinetics, we obtain non-circular elongated stationary shapes in an analytic approximation which compares favorably to the full numerical solution.Comment: 12 page

A variational formulation of anisotropic geometric evolution equations in higher dimensions

Accepted versio

Gitter-Quantenfeldtheorien mit Quantensymmetrie

Cover and Contents Introduction 1. DHR-superselection theory 2. Quantum groups as symmetry algebras 3. Lattice models and amplified DHR-theory 4. Overview and summary of results Chapter 1. Diagonal crossed products by duals of quantum groups 1. Coactions and crossed products 2. Two-sided coactions and diagonal crossed products 3. Generating matrices 4. Quantum group spin chains and lattice current algebras Chapter 2. Diagonal crossed products by duals of quasi-quantum groups 1. Quasi-quantum groups 2. Coactions of quasi-quantum groups 3. Two-sided coactions 4. Left and right diagonal crossed products 5. Generating matrices 6. Proofs Chapter 3. Generalization to weak quasi-quantum groups 1. Weak quasi-quantum groups 2. Diagonal crossed products Chapter 4. The quantum double D(G) 1. D(G) as a quasi-bialgebra and D(G)-coactions 2. The quasitriangular quasi-Hopf structure 3. The twisted double of a finite group 4. The monodromy algebra Chapter 5. Quantum group spin chains and lattice current algebras 1. Two-sided crossed products 2. Quantum group spin chains 3. Lattice current algebras 4. Representation theory 5. Proofs Appendix A. Representation theoretic interpretation Appendix B. Graphical calculus 1. Basic definitions 2. The antipode image of the R-matrix 3. The antipode in the quantum double D(G) 4. Graphical description of the diagonal crossed product Conclusions and outlook Bibliography Curriculum VitaeIn low dimensional quantum field theories the global (gauge) symmetry can in general not be described by an ordinary group but by some more general algebraic object such as quantum groups or generalizations thereof. In this thesis we construct 1+1 - dimensional lattice quantum field theories - socalled quantum group spin chains and lattice current algebras - whose global symmetry is given by some quantum group at roots of unity. The main problem in constructing these models stems from the fact that the semisimple quotients of quantum groups at roots of unity are no longer coassiciative and have to be described by weak quasi-quantum groups. To solve this problem we introduce a new mathematical construction, the so-called diagonal crossed product of an algebra M with the dual of a quantum group G. We give a natural generalization of this construction to the case where G is a quasi-Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e., where the coproduct is non-unital). In these cases our diagonal crossed product will still be an associative algebra, even though the analogue of an ordinary crossed product in general is not well defined as an associative algebra. In the case M = G we obtain an explicit definition of the quantum double D(G) for (weak) quasi-Hopf algebras G. We prove that D(G) is itself a (weak) quasi- triangular quasi-Hopf algebra and we give explicit formulas for the coproduct, the antipode and the R-matrix. Moreover we show that any diagonal crossed product naturally admits a two-sided D(G)-coaction. We then apply our formalism to construct quantum spin chains and lattice current algebras based on a weak quasi-Hopf algebra as iterated diagonal crossed products. This contains the important cases of truncated quantum groups at roots of unity. Both lattice models admit the quantum double D(G) as a localized cosymmetry. We investigate the representation theory of these models. In particular we show that irreducible representations of lattice current algebras (based on a semisimple weak quasi Hopf algebra G) are in one- to-one correspondence with the irreducible representations of the quantum double D(G).Quantenfeldtheorien in niederen Raum-Zeit-Dimensionen (kleiner als 4) zeigen das Phänomen der Quantensymmetrie, d.h. die globale (Eich-)Symmetrie läßt sich nicht mehr durch eine Gruppe beschreiben, sondern man benötigt allgemeinere allgebraische Objekte wie zum Beispiel Quantengruppen. In dieser Arbeit werden 1+1 dimensionale Gitterquantenfeldtheorien - sogenannte Quantengruppenspinketten und Gitterstromalgebren - konstuiert, deren globale Symmetrie durch Quantengruppen an den Einheitswurzeln gegeben ist. Die Hauptschwierigkeit bei dieser Konstruktion rührt von der Tatsache her, daß die halbeinfachen Quotienten von Quantengruppen an den Einheitswurzeln nicht mehr koassoziativ sind. Sie besitzen die Struktur einer schwachen Quasi- Quantengruppe. Wir führen deswegen eine neue mathematische Konstruktion, das sogenannte diagonale verschränkte Produkt einer Algebra M mit der dualen einer Quantengruppe G ein. Diese Konstruktion läßt sich auf natürliche Weise auf Quasi-Hopfalgebren im Sinne von Drinfeld oder noch allgemeiner im Sinne von Mack and Schomerus (i.e. mit nicht unitalem Koprodukt) verallgemeinern. In diesen Fällen ergibt unser diagonales verschränktes Produkt immer noch eine assoziative Algebra obwohl eine entsprechende Verallgemeinerung des gewöhnlichen verschränkten Produkts im allgemeinen zu keiner wohldefinierten assoziativen Algebra führt. Der Fall M = G führt zu einer Definition des Quantendoppels D(G) einer (schwachen) Quasi-Hopfalgebra G. Wir zeigen, daß D(G) selbst eine schwache quasitrianguläre Quasi-Hopfalgebra ist. Wir geben explizite Formeln für das Koprodukt, für die Antipode und für die R-Matrix an. Außerdem zeigen wir, daß jedes diagonale verschränkte Produkt auf natürliche Weise eine zweiseitige D(G)-Kowirkung besitzt. Dann benutzen wir unseren Formalismus zur Konstruktion von Quantenspinketten und Gitterstromalgebren als iterierte diagonale verschränkte Produkte und erreichen so unser Ziel der Konstruktion dieser Modelle an den Einheitswurzeln. Auf beiden Gittermodellen wirkt das Quantendoppel als lokalisierte Kosymmetrie. Zum Schluß untersuchen wir die Darstellungstheorie der konstruierten Quantenketten. Insbesondere zeigen wir, daß die irreduziblen Darstellungen einer Gitterstromalgebra eineindeutig den irreduziblen Darstellungen des Quantendoppels D(G) der zugrundegelegten schwachen Quasi- Hopfalgebra G zugeordnet werden können

Diagonal Crossed Products by Duals of Quasi-Quantum Groups

A two-sided coaction ffi : M! G\Omega M\Omega G of a Hopf algebra (G; \Delta; ffl; S) on an associative algebra M is an algebra map of the form ffi = (\Omega id M ) ffi ae = (id M\Omega ae) ffi , where (; ae) is a commuting pair of left and right G-coactions on M, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra G on M by / and . , respectively, we define the diagonal crossed product M ./ G to be the algebra generated by M and G with relations given by 'm = (' (1) . m / S \Gamma1 (' (3) )) ' (2) ; m 2 M; ' 2 G : We give a natural generalization of this construction to the case where G is a quasi--Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e., where the coproduct \Delta is non-unital). In these cases our diagonal crossed product will still be an associative algebra structure on M\Omega G extending M j M\Omega 1, even though the analogue of an ordinary crossed ..

A Level Set Approach to Anisotropic Flows with Curvature Regularization

Modeling and simulation of faceting effects on surfaces are topics of growing importance in modern nanotechnology. Such effects pose various theoretical and computational challenges, since they are caused by non-convex surface energies, which lead to ill-posed evolution equations for the surfaces. In order to overcome the illposedness, regularization of the energy by a curvature-dependent term has become a standard approach, which seems to be related to the actual physics, too. The use of curvature-dependent energies yields higher order partial differential equations for surface variables, whose numerical solution is a very challenging task. In this paper we investigate the numerical simulation of anisotropic growth with curvature-dependent energy by level set methods, which yield flexible and robust surface representations. We consider the two dominating growth modes, namely attachment-detachment kinetics and surface diffusion. The level set formulations are given in terms of metric gradient flows, which are discretized by finite element methods in space and in a semi-implicit way as local variational problems in time. Finally, the constructed level set methods are applied to the simulation of faceting of embedded surfaces and thin films