2023, Müller, Rüdiger, Landstorfer, Manuel

In this work, the balance equations of non-equilibrium thermodynamics are coupled to Galilean limit systems of the Maxwell equations, i.e., either to (i) the quasi-electrostatic limit or (ii) the quasi-magnetostatic limit. We explicitly consider a volume (Formula presented.), which is divided into (Formula presented.) and (Formula presented.) by a possibly moving singular surface S, where a charged reacting mixture of a viscous medium can be present on each geometrical entity (Formula presented.). By the restriction to the Galilean limits of the Maxwell equations, we achieve that only subsystems of equations for matter and electromagnetic fields are coupled that share identical transformation properties with respect to observer transformations. Moreover, the application of an entropy principle becomes more straightforward and finally helps estimate the limitations of the more general approach based the full set of Maxwell equations. Constitutive relations are provided based on an entropy principle, and particular care is taken in the analysis of the stress tensor and the momentum balance in the general case of non-constant scalar susceptibility. Finally, we summarise the application of the derived model framework to an electrochemical system with surface reactions.

2012, Arkhipov, Rostislav, Pimenov, Alexander, Radziunas, Mindaugas, Vladimirov, Andrei G., Arsenjevi´c, Dejan, Rachinskii, Dmitrii, Schmeckebier, Holger, Bimberg, Dieter

Hybrid mode-locking in a two section edge-emitting semiconductor laser is studied numerically and analytically using a set of three delay differential equations. In this set the external RF signal applied to the saturable absorber section is modeled by modulation of the carrier relaxation rate in this section. Estimation of the locking range where the pulse repetition frequency is synchronized with the frequency of the external modulation is performed numerically and the effect of the modulation shape and amplitude on this range is investigated. Asymptotic analysis of the dependence of the locking range width on the laser parameters is carried out in the limit of small signal modulation. Our numerical simulations indicate that hybrid mode-locking can be also achieved in the cases when the frequency of the external modulation is approximately twice larger and twice smaller than the pulse repetition frequency of the free running passively mode-locked laser fP . Finally, we provide an experimental demonstration of hybrid mode-locking in a 20 GHz quantum-dot laser with the modulation frequency of the reverse bias applied to the absorber section close to fP / 2.

2013, Pimenov, Alexander, Vladimirov, Andrei G., Gurevich, Svetlana V., Panajotov, Krassimir, Huyet, Guillaume, Tlidi, Mustapha

Control of the motion of cavity solitons is one the central problems in nonlinear optical pattern formation. We report on the impact of the phase of the time-delayed optical feedback and carrier lifetime on the self-mobility of localized structures of light in broad area semiconductor cavities. We show both analytically and numerically that the feedback phase strongly affects the drift instability threshold as well as the velocity of cavity soliton motion above this threshold. In addition we demonstrate that non-instantaneous carrier response in the semiconductor medium is responsible for the increase in critical feedback rate corresponding to the drift instability.

2020, Colli, Pierluigi, Gilardi, Gianni, Sprekels, Jürgen

In a recent paper the same authors have proved existence, uniqueness and regularity results for a class of viscous and nonviscous Cahn--Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers in the spectral sense of general linear operators, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space of square-integrable functions on a bounded and smooth three-dimensional domain, and have compact resolvents. Here, for the case of the viscous system, we analyze the asymptotic behavior of the solution as the fractional power coefficient of the second operator tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of the second operator appears.

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.

2015, Dreyer, Wolfgang, Guhlke, Clemens, Landstorfer, Manuel, Neumann, Johannes, Müller, Rüdiger

The Lippmann equation is considered as universal relationship between interfacial tension, double layer charge, and cell potential. Based on the framework of continuum thermo-electrodynamics we provide some crucial new insights to this relation. In a previous work we have derived a general thermodynamic consistent model for electrochemical interfaces, which showed a remarkable agreement to single crystal experimental data. Here we apply the model to a curved liquid metal electrode. If the electrode radius is large compared to the Debye length, we apply asymptotic analysis methods and obtain the Lippmann equation. We give precise definitions of the involved quantities and show that the interfacial tension of the Lippmann equation is composed of the surface tension of our general model, and contributions arising from the adjacent space charge layers. This finding is confirmed by a comparison of our model to experimental data of several mercury-electrolyte interfaces. We obtain qualitative and quantitative agreement in the 2V potential range for various salt concentrations. We also discuss the validity of our asymptotic model when the electrode radius is comparable to the Debye length.

2018, Hennessy, Matthew G., Münch, Andreas, Wagner, Barbara

We present a formulation of the free boundary problem for a hydrogel that accounts for the interfacial free energy and finite strain due to the large deformation of the polymer network during solvent transport across the free boundary. For the geometry of an initially dry layer fixed at a rigid substrate, our model predicts a phase transition when a critical value of the solvent concentration has been reached near the free boundary. A one-dimensional case study shows that depending on the flux rate at the free boundary an initial saturation front is followed by spinodal decomposition of the hydrogel and the formation of an interfacial front that moves through the layer. Moreover, increasing the shear modulus of the elastic network delays or even suppresses phase separation.

2012, Aki, Gonca, Daube, Johannes, Dreyer, Wolfgang, Giesselmann, Jan, Kränkel, Mirko, Kraus, Christiane

In this contribution, we investigate a diffuse interface model for quasi–incompressible flows. We determine corresponding sharp interface limits of two different scalings. The sharp interface limit is deduced by matched asymptotic expansions of the fields in powers of the interface. In particular, we study solutions of the derived system of inner equations and discuss the results within the general setting of jump conditions for sharp interface models. Furthermore, we treat, as a subproblem, the convective Cahn–Hilliard equation numerically by a Local Discontinuous Galerkin scheme.

2010, Dreyer, Wolfgang, Giesselmann, Jan, Kraus, Christiane, Rohde, Christiane

This paper deals with a sharp interface limit of the isothermal Navier-Stokes-Korteweg system. The sharp interface limit is performed by matched asymptotic expansions of the fields in powers of the interface width. These expansions are considered in the interfacial region (inner expansions) and in the bulk (outer expansion) and are matched order by order. Particularly we consider the first orders of the corresponding inner equations obtained by a change of coordinates in an interfacial layer. For a specific scaling we establish solvability criteria for these inner equations and recover the results within the general setting of jump conditions for sharp interface models.

2010, Münch, Andreas, Please, Colin P., Wagner, Barbara

We consider a mathematical model of spin coating of a single polymer blended in a solvent. The model describes the one-dimensional development of the thin layer of the mixture as the layer thins due to flow created by a balance of viscous forces and centrifugal forces and due to evaporation of the solvent. In the model both the diffusivity of the solvent in the polymer and the viscosity of the mixture are very rapidly varying functions of the solvent volume fraction. Guided by numerical solutions an asymptotic analysis reveals a number of different possible behaviours of the thinning layer dependent on the nondimensional parameters describing the system. The main practical interest is in controlling the appearance and development of a ``skin'' on the polymer where the solvent concentration reduces rapidly on the outer surface leaving the bulk of the layer still with high concentrations of solvent. The critical parameters controlling this behaviour are found to be eps the ratio of the diffusion to advection time scales, delta the ratio of the evaporation to advection time scales and exp(-gamma), the ratio of the diffusivity of the initial mixture and the pure polymer. In particular, our analysis shows that for very small evaporation with delta ll exp(-3/(4gamma)) eps^3/4 skin formation can be prevented