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

Dynamics and evolution of Turbulent Taylor rolls

In many shear- and pressure-driven wall-bounded turbulent flows secondary motions spontaneously develop and their interaction with the main flow alters the overall large-scale features and transfer properties. Taylor-Couette flow, the fluid motion developing in the gap between two concentric cylinders rotating at different angular velocity, is not an exception, and toroidal Taylor rolls have been observed from the early development of the flow up to the fully turbulent regime. In this manuscript we show that under the generic name of ``Taylor rolls'' there is a wide variety of structures that differ for the vorticity distribution within the cores, the way they are driven and their effects on the mean flow. We relate the rolls at high Reynolds numbers not to centrifugal instabilities, but to a combination of shear and anti-cyclonic rotation, showing that they are preserved in the limit of vanishing curvature and can be better understood as a pinned cycle which shows similar characteristics as the self-sustained process of shear flows. By analyzing the effect of the computational domain size, we show that this pinning is not a product of numerics, and that the position of the rolls is governed by a random process with the space and time variations depending on domain size.Comment: Submitted to JF

Control theory for infinite dimensional dynamical systems and applications to falling liquid film flows

In this thesis, we study the problem of controlling the solutions of various nonlinear PDE models that describe the evolution of the free interface in thin liquid films flowing down inclined planes. We propose a control methodology based on linear feedback controls, which are proportional to the deviation between the current state of the system and a prescribed desired state. We first derive the controls for weakly nonlinear models such as the Kuramoto-Sivashinsky equation and some of its generalisations, and then use the insight that the analytical results obtained there provide us to derive suitable generalisations of the controls for reduced-order long-wave models. We use two long-wave models to test our controls: the first order Benney equation and the first order weighted-residual model, and compare some linear stability results with the full 2-D Navier-Stokes equations. We find that using point actuated controls it is possible to stabilise the full range of solutions to the generalised Kuramoto-Sivashinsky equation, and that distributed controls have a similar effect on both long-wave models. Furthermore, point-actuated controls are efficient when stabilising the flat solution of both long-wave models. We extend our results to systems of coupled Kuramoto-Sivashinsky equations and to stochastic partial differential equations that arise by adding noise to the weakly nonlinear models.Open Acces

Thin Film Flow on Functional Surfaces: Stability and Morphology

This thesis explores the stability and morphology of gravity induced film flow down smoothly corrugated rigid substrate, uniformly heated/cooled from below. The problem of interest is complicated by the presence of a free-surface whose location is unknown \textit{a priori}. This complication is overcome by reducing the governing equations of motion and energy to a manageable form within the framework of the well-known long-wave expansion, which exploits the disparity between the horizontal and vertical length scales in order to eliminate the depth-coordinate from the governing formulation. Two methods for implementing a long-wave expansion are considered, with each leading to an asymptotic model of reduced dimensionality. The first is a perturbation series of the fluid velocity and temperature with respect to a small parameter ${\epsilon}$ which represents the disparity between the horizontal and vertical length scales, the second is a power series expansion with respect to the vertical coordinate in which the series truncation is correlated to the number of degrees of freedom with respect to the horizontal coordinate. \par A key feature of the power series method is proof that, for any asymptotic model to be able to accurately resolve the thermodynamics beyond the trivial case of `a flat film flowing down a planar uniformly heated incline', the expansion of the fluid temperature must be quadratic to leading-order in the long-wave expansion. The ensuing analysis reveals why heat transfer models based on the Nusselt linear temperature distribution fail to converge outside of the long-wave limit and details how asymptotic models can be extended to higher-order. Superior predictions are obtained compared with earlier work and reinforced via a series of corresponding solutions to the full governing equations acquired using a purpose written finite element analogue, enabling comparisons of free-surface disturbance and temperature predictions to be made, as well as those of the streamline pattern and temperature contours inside the film. In particular, the free-surface temperature is captured extremely well at moderate Prandtl numbers for film flow down smoothly corrugated substrate. \par Investigation of the stability characteristics of gravity-driven film flow is opened with the classical problem of a thin film flowing down an inclined plate and its associated hydrodynamic stability as described by the Orr-Sommerfeld equation, which reveals the asymptotic methods are not able to fully capture the thermo-capillary effect in the heated/cooled case. The stability problem is extended to film flow over non-planar substrate via Floquet theory, with the interaction between the substrate topography and thermo-capillarity investigated through a set of neutral stability curves. Although no relevant experimental data is currently available for the heated film problem, existing numerical predictions and experimental data concerning the stability behaviour of isothermal film flow are taken as a reference point from which to explore the effect of both heating and cooling