David A. Kessler

Pattern selection in fingered growth phenomena

Abstract A variety of non-equilibrium growth processes are characterized by phase boundaries consisting of moving fingers, often with interesting secondary structures such as sidebranches. Familiar examples are dendrites, as seen in snowflake growth, and fluid fingers often formed in immiscible displacement. Such processes are characterized by a morphological instability which renders planar or circular shapes unstable, and by the competing stabilizing effect of surface tension. We survey recent theoretical work which elucidates how such systems arrive at their observed patterns. Emphasis is placed upon dendritic solidification, simple local models thereof, and the Saffman-Taylor finger in two-dimensional fluid flow, and relate these systems to their more complicated variants. We review the arguments that a general procedure for the analysis of such problems is to first find finger solutions of the governing equations without surface tension, then to incorporate surface tension in a non-perturbative manne...

A Model for Strain-Induced Roughening and Coherent Island Growth

We have investigated the morphological evolution of strained films during growth. Novel Monte Carlo studies, which incorporate linear elasticity, have been performed to simulate film growth with misfit. These studies demonstrate the onset of islanding for sufficiently large misfit. We present an analytic calculation which shows that from the onset of deposition the films are energetically unstable to large-scale islanding. We argue that the kinetics ultimately determines the surface morphology. Dislocations are not necessary for surface lattice relaxation. Support for this picture is inferred from experimental results on a number of strained growth systems.

Pattern selection in three dimensional dendritic growth

Abstract We study the selection of the shape and growth velocity of three dimensional dendritic crystals in cubically anisotropic materials. We demonstrate that aside from minor additional complexities due to the lack of axisymmetry, the recently discovered mechanism of “microscopic solvability” can be extended to these systems and used to find a unique needle crystal solution of the equations of thermal diffusion-controlled solidification. We compare the predictions of this approach with measured growth rates in succinonitrile. Finally, we extend our analysis to determine the ratio of the sidebranch wavelength to the tip radius.

Theory of the spiral core in excitable media

Abstract A rigorous asymptotic spiral solution to an excitable reaction diffusion system is found by separating space into two scaling regions: an “outer region”, having the same spatial scaling as the overall spiral structure, which exhibits a singularity at the spiral tip; and a “core” region around the spiral tip, where space is scaled so as to resolve the tip singularity. The stability of the spiral structure is investigated for both the outer region, which is found to possess no intrinsic instability, and the core region which is found to be unstable. Both the zero and small diffusion systems are found to exhibit qualitatively similar behaviour. The implication of these results for both experiment and simulation, in particular within the context of the observed “meandering” instability, are addressed.

Effect of diffusion on patterns in excitable Belousov-Zhabotinskii systems

Travelling wave patterns occur frequently in chemically reacting systems: these include planar fronts, target patterns and spiral structures. We review the dispersion relation for planar waves, including the effects of diffusion in the “slow” field for a simplified piecewise-linear model, with a focus on the scaling behavior with respect to the relative rate of the “fast” and “slow” reactions. We discuss the implications of these results for spirals, showing the origins of Fife scaling and deriving a boundary-integral formulation of the spiral equations. We discuss the generalization to more realistic models, in particular, the popular Oregonator model for Belousov-Zhabotinskii reactions.

Spiral selection as a free boundary problem

Abstract We present a new formulation of the spiral selection problem for the Belousov-Zhabotinsky reaction. In particular, we focus on deriving an exact integro-differential shape equation and discuss the possible behavior of solutions to this equation. We also present some new results on the asymptotic (far from core) structure of spirals in the Fife scaling regime.

Pattern Formation Far from Equilibrium : The Free Space Dendritic Crystal

The selection of shape and velocity for the dendritic crystal has been an outstanding problem for several decades. Work over the past few years has revealed a new mechanism, that of “microscopic solvability” which resolves these issues. The mechanism relies on a non-perturbative contribution of the microscopic dynamics to create a consistency condition for the macroscopic pattern which in turn has a unique solution. This talk will focus on explaining the problem and its solution and furthermore on the generality of our approach, which to-date has been shown to apply to a variety of interfacial pattern forming systems.

Discrete set selection of Saffman–Taylor fingers

A study of the linear stability of the discrete set of steady‐state Saffman–Taylor finger solutions at finite surface tension is presented. It is shown by explicit computation that members of the set aside from the lowest width finger are linearly unstable. This completes the demonstration that finite surface tension effects determine uniquely the allowed interfacial pattern in the steady‐state regime.

The geometrical model of dendritic growth: The small velocity limit

Abstract We present a systematic analysis of the geometrical model of dendritic growth in the small velocity limit. Velocity selection is demonstrated analytically and the allowed velocities are explicitly calculated as a function of anisotropy.

Computational approach to steady-state eutectic growth

Abstract We formulate a boundary integral approach to the determination of periodic steady-state eutectic growth patterns. The numerical implementation of this method allows us to compute the band of allowed velocities for a given undercooling. We illustrate how this works in a simplified symmetric eutectic solidifying exactly at the eutectic composition.