Modeling and numerical simulations of dendritic crystal growth

Competition between random genetic drift and natural selection play a central role in evolution: Whereas nonbeneficial mutations often prevail in small populations by chance, mutations that sweep through large populations typically confer a selective advantage. Here, however, we observe chance effects during range expansions that dramatically alter the gene pool even in large microbial populations. Initially well mixed populations of two fluorescently labeled strains of Escherichia coli develop well defined, sector-like regions with fractal boundaries in expanding colonies. The formation of these regions is driven by random fluctuations that originate in a thin band of pioneers at the expanding frontier. A comparison of bacterial and yeast colonies (Saccharomyces cerevisiae) suggests that this large-scale genetic sectoring is a generic phenomenon that may provide a detectable footprint of past range expansions.

https://www.pnas.org/content/pnas/104/50/19926.full.pdf

Genetic drift at expanding frontiers promotes gene segregation

Crystal growth under non-equilibrium conditions can give rise to complex patterns which are generically similar to those found in processes such as viscous fingering, aggregation and electrochemical deposition. Recent theoretical understanding focuses on the interplay between the macroscopic driving force associated with the phase transition and the microscopic interfacial dynamics.

/pdf/the-formation-of-patterns-in-non-equilibrium-growth-1wldk2omnj.pdf

The formation of patterns in non-equilibrium growth

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...

Pattern selection in fingered growth phenomena

Solidification microstructures and solid-state parallels: Recent developments, future directions

The morphology diagram of possible structures in two-dimensional diffusional growth is given in the parameter space of undercooling \ensuremath{\Delta} versus anisotropy of surface tension \ensuremath{\epsilon}. The building block of the dendritic structure is a dendrite with parabolic tip, and the basic element of the seaweed structure is a doublon. The transition between these structures shows a jump in the growth velocity. We also describe the structures and velocities of fractal dendrites and doublons destroyed by noise. We introduce a renormalized capillary length and density of the solid phase and use scaling arguments to describe the fractal dendrites and doublons. The resulting scaling exponents for the growth velocity and the different length scales are expressed in terms of the fractal dimensions for surface and bulk of these fractal structures. All the considered structures are compact on length scales larger than the diffusion length and they show fractal behavior on intermediate length scales between the diffusion length and a small size cutoff which depends on the strength of noise. \textcopyright{} 1996 The American Physical Society.

Structure formation and the morphology diagram of possible structures in two-dimensional diffusional growth.

/pdf/fractal-and-compact-growth-morphologies-in-phase-transitions-kdlb2ge07b.pdf

Fractal and compact growth morphologies in phase transitions with diffusion transport.

/pdf/numerical-simulation-of-dendritic-growth-z7ywkl82e3.pdf

Numerical simulation of dendritic growth.

We propose a phase diagram for the selection of growth patterns in systems with a conserved quantity which evolve at asymptotically constant growth rate The occurrence of different growth forms like fractal, compact or dendritic, and the various transitions between them are characterized by scaling relations

http://iopscience.iop.org/article/10.1209/0295-5075/17/6/010/pdf

Kinetic Phase Diagram and Scaling Relations for Stationary Diffusional Growth

We present a continuum theory which predicts the steady state propagation of cracks. The theory overcomes the usual problem of a finite time cusp singularity of the Grinfeld instability by the inclusion of elastodynamic effects which restore selection of the steady state tip radius and velocity. We developed a phase-field model for elastically induced phase transitions; in the limit of small or vanishing elastic coefficients in the new phase, fracture can be studied. The simulations confirm analytical predictions for fast crack propagation.

/pdf/phase-field-modeling-of-fast-crack-propagation-44u4rncf3d.pdf

Heiner Müller-Krumbhaar

Papers

Structure formation and the morphology diagram of possible structures in two-dimensional diffusional growth.

Fractal and compact growth morphologies in phase transitions with diffusion transport.

Numerical simulation of dendritic growth.

Kinetic Phase Diagram and Scaling Relations for Stationary Diffusional Growth

Phase field modeling of fast crack propagation.