Velocity selection problem for combined motion of melting and solidification fronts

We discuss a free boundary problem for two moving solid-liquid interfaces that strongly interact via the diffusion field in the liquid layer between them. This problem arises in the context of liquid film migration (LFM) during the partial melting of solid alloys. In the LFM mechanism the system chooses a more efficient kinetic path which is controlled by diffusion in the liquid film, whereas the process with only one melting front would be controlled by the very slow diffusion in the mother solid phase. The relatively weak coherency strain energy is the effective driving force for LFM. As in the classical dendritic growth problems, also in this case an exact family of steady-state solutions with two parabolic fronts and an arbitrary velocity exists if capillary effects are neglected. We develop a velocity selection theory for this problem, including anisotropic surface tension effects. The strong diffusion interaction and coherency strain effects in the solid near the melting front lead to substantial changes compared to classical dendritic growth.Comment: submitted to PR

Linear stability analysis of an insoluble surfactant monolayer spreading on a thin liquid film

Recent experiments by several groups have uncovered a novel fingering instability in the spreading of surface active material on a thin liquid film. The mechanism responsible for this instability is yet to be determined. In an effort to understand this phenomenon and isolate a possible mechanism, we have investigated the linear stability of a coupled set of equations describing the Marangoni spreading of a surfactant monolayer on a thin liquid support. The unperturbed flows, which exhibit simple linear behavior in the film thickness and surfactant concentration, are self-similar solutions of the first kind for spreading in a rectilinear geometry. The solution of the disturbance equations determines that the rectilinear base flows are linearly stable. An energy analysis reveals why these base flows can successfully heal perturbations of all wavenumbers. The details of this analysis suggest, however, a mechanism by which the spreading can be destabilized. We propose how the inclusion of additional forces acting on the surfactant coated spreading film might give rise to regions of adverse mobility gradients known to produce fingering instabilities in other fluid flows

Dynamics and stability of vortex-antivortex fronts in type II superconductors

The dynamics of vortices in type II superconductors exhibit a variety of patterns whose origin is poorly understood. This is partly due to the nonlinearity of the vortex mobility which gives rise to singular behavior in the vortex densities. Such singular behavior complicates the application of standard linear stability analysis. In this paper, as a first step towards dealing with these dynamical phenomena, we analyze the dynamical stability of a front between vortices and antivortices. In particular we focus on the question of whether an instability of the vortex front can occur in the absence of a coupling to the temperature. Borrowing ideas developed for singular bacterial growth fronts, we perform an explicit linear stability analysis which shows that, for sufficiently large front velocities and in the absence of coupling to the temperature, such vortex fronts are stable even in the presence of in-plane anisotropy. This result differs from previous conclusions drawn on the basis of approximate calculations for stationary fronts. As our method extends to more complicated models, which could include coupling to the temperature or to other fields, it provides the basis for a more systematic stability analysis of nonlinear vortex front dynamics.Comment: 13 pages, 8 figure

Instabilities in crystal growth by atomic or molecular beams

The planar front of a growing a crystal is often destroyed by instabilities. In the case of growth from a condensed phase, the most frequent ones are diffusion instabilities, which will be but briefly discussed in simple terms in chapter II. The present review is mainly devoted to instabilities which arise in ballistic growth, especially Molecular Beam Epitaxy (MBE). The reasons of the instabilities can be geometric (shadowing effect), but they are mostly kinetic or thermodynamic. The kinetic instabilities which will be studied in detail in chapters IV and V result from the fact that adatoms diffusing on a surface do not easily cross steps (Ehrlich-Schwoebel or ES effect). When the growth front is a high symmetry surface, the ES effect produces mounds which often coarsen in time according to power laws. When the growth front is a stepped surface, the ES effect initially produces a meandering of the steps, which eventually may also give rise to mounds. Kinetic instabilities can usually be avoided by raising the temperature, but this favours thermodynamic instabilities. Concerning these ones, the attention will be focussed on the instabilities resulting from slightly different lattice constants of the substrate and the adsorbate. They can take the following forms. i) Formation of misfit dislocations (chapter VIII). ii) Formation of isolated epitaxial clusters which, at least in their earliest form, are `coherent' with the substrate, i.e. dislocation-free (chapter X). iii) Wavy deformation of the surface, which is presumably the incipient stage of (ii) (chapter IX). The theories and the experiments are critically reviewed and their comparison is qualitatively satisfactory although some important questions have not yet received a complete answer.Comment: 90 pages in revtex, 45 figures mainly in gif format. Review paper to be published in Physics Reports. Postscript versions for all the figures can be found at http://www.theo-phys.uni-essen.de/tp/u/politi

Gradient flow approach to an exponential thin film equation: global existence and latent singularity

In this work, we study a fourth order exponential equation, $u_t=\Delta e^{-\Delta u},$ derived from thin film growth on crystal surface in multiple space dimensions. We use the gradient flow method in metric space to characterize the latent singularity in global strong solution, which is intrinsic due to high degeneration. We define a suitable functional, which reveals where the singularity happens, and then prove the variational inequality solution under very weak assumptions for initial data. Moreover, the existence of global strong solution is established with regular initial data.Comment: latent singularity, curve of maximal slope. arXiv admin note: text overlap with arXiv:1711.07405 by other author