Mean flow in hexagonal convection: stability and nonlinear dynamics

Weakly nonlinear hexagon convection patterns coupled to mean flow are investigated within the framework of coupled Ginzburg-Landau equations. The equations are in particular relevant for non-Boussinesq Rayleigh-B\'enard convection at low Prandtl numbers. The mean flow is found to (1) affect only one of the two long-wave phase modes of the hexagons and (2) suppress the mixing between the two phase modes. As a consequence, for small Prandtl numbers the transverse and the longitudinal phase instability occur in sufficiently distinct parameter regimes that they can be studied separately. Through the formation of penta-hepta defects, they lead to different types of transient disordered states. The results for the dynamics of the penta-hepta defects shed light on the persistence of grain boundaries in such disordered states.Comment: 33 pages, 20 figures. For better figures:http://astro.uchicago.edu/~young/hexmeandi

Theoretical Model for Cellular Shapes Driven by Protrusive and Adhesive Forces

The forces that arise from the actin cytoskeleton play a crucial role in determining the cell shape. These include protrusive forces due to actin polymerization and adhesion to the external matrix. We present here a theoretical model for the cellular shapes resulting from the feedback between the membrane shape and the forces acting on the membrane, mediated by curvature-sensitive membrane complexes of a convex shape. In previous theoretical studies we have investigated the regimes of linear instability where spontaneous formation of cellular protrusions is initiated. Here we calculate the evolution of a two dimensional cell contour beyond the linear regime and determine the final steady-state shapes arising within the model. We find that shapes driven by adhesion or by actin polymerization (lamellipodia) have very different morphologies, as observed in cells. Furthermore, we find that as the strength of the protrusive forces diminish, the system approaches a stabilization of a periodic pattern of protrusions. This result can provide an explanation for a number of puzzling experimental observations regarding cellular shape dependence on the properties of the extra-cellular matrix

Stability of monolayers and bilayers in a copolymer-homopolymer blend model

We study the stability of layered structures in a variational model for diblock copolymer-homopolymer blends. The main step consists of calculating the first and second derivative of a sharp-interface Ohta-Kawasaki energy for straight mono- and bilayers. By developing the interface perturbations in a Fourier series we fully characterise the stability of the structures in terms of the energy parameters. In the course of our computations we also give the Green's function for the Laplacian on a periodic strip and explain the heuristic method by which we found it.Comment: 40 pages, 34 Postscript figures; second version has some minor corrections; to appear in "Interfaces and Free Boundaries

Three basic issues concerning interface dynamics in nonequilibrium pattern formation

These are lecture notes of a course given at the 9th International Summer School on Fundamental Problems in Statistical Mechanics, held in Altenberg, Germany, in August 1997. In these notes, we discuss at an elementary level three themes concerning interface dynamics that play a role in pattern forming systems: (i) We briefly review three examples of systems in which the normal growth velocity is proportional to the gradient of a bulk field which itself obeys a Laplace or diffusion type of equation (solidification, viscous fingers and streamers), and then discuss why the Mullins-Sekerka instability is common to all such gradient systems. (ii) Secondly, we discuss how underlying an effective interface description of systems with smooth fronts or transition zones, is the assumption that the relaxation time of the appropriate order parameter field(s) in the front region is much smaller than the time scale of the evolution of interfacial patterns. Using standard arguments we illustrate that this is generally so for fronts that separate two (meta)stable phases: in such cases, the relaxation is typically exponential, and the relaxation time in the usual models goes to zero in the limit in which the front width vanishes. (iii) We finally summarize recent results that show that so-called ``pulled'' or ``linear marginal stability'' fronts which propagate into unstable states have a very slow universal power law relaxation. This slow relaxation makes the usual ``moving boundary'' or ``effective interface'' approximation for problems with thin fronts, like streamers, impossible.Comment: 48 pages, TeX with elsart style file (included), 9 figure

Particles with selective wetting affect spinodal decomposition microstructures

We have used mesoscale simulations to study the effect of immobile particles on microstructure formation during spinodal decomposition in ternary mixtures such as polymer blends. Specifically, we have explored a regime of interparticle spacings (which are a few times the characteristic spinodal length scale) in which we might expect interesting new effects arising from interactions among wetting, spinodal decomposition and coarsening. In this paper, we report three new effects for systems in which the particle phase has a strong preference for being wetted by one of the components (say, A). In the presence of particles, microstructures are not bicontinuous in a symmetric mixture. An asymmetric mixture, on the other hand, first forms a non-bicontinuous microstructure which then evolves into a bicontinuous one at intermediate times. Moreover, while wetting of the particle phase by the preferred component (A) creates alternating A-rich and B-rich layers around the particles, curvature-driven coarsening leads to shrinking and disappearance of the first A-rich layer, leaving a layer of the non-preferred component in contact with the particle. At late simulation times, domains of the matrix components coarsen following the Lifshitz-Slyozov-Wagner law, $R_1(t) \sim t^{1/3}$.Comment: Accepted for publication in PCCP on 24th May 201