H. Müller-Krumbhaar

Phase Field Modeling of Fast Crack Propagation

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.

Diffusion and relaxation kinetics in stochastic models for crystal growth

The combined influence of relaxation and diffusion processes on the dynamic behavior of stochastic models for crystal growth is systematically investigated. The models are the kinetic Ising model, the discrete Gaussian model, and the SOS–Kossel model. The relaxation kinetics are introduced by single‐site transition probabilities to account for adsorption and evaporation. The diffusion kinetics are introduced by two‐site transition probabilities to account for nearest‐neighbor exchange. The kinetic equations are studied by Monte Carlo simulation, quasichemical(pair) approximation (QCA), and high temperature expansion. The diffusion is found to generally enhance the growth rate of the crystal. In the range of validity of the QCA, i.e., outside the nucleation regime, the results are in very good quantitative agreement with previous Monte Carlo simulations. In the limit of infinite diffusion speed the growth rate approaches the Wilson–Frenkel rate. The local roughness or surface energy is reduced if surface d...

MORPHOLOGY DIAGRAM OF POSSIBLE STRUCTURES IN DIFFUSIONAL GROWTH

A theory for the morphology diagram of possible structures in two-dimensional diffusional growth is presented. The main control parameters are undercooling Δ, anisotropy of surface tension e and the strength of noise. Basic patterns are dendrites and seaweed. 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. The possible extension of these results to three-dimensional growth is shortly discussed.

Morphology transitions during non-equilibrium growth. II: Morphology diagram and characterization of the transition

Abstract In a preceding paper we have presented a new diffusion-transition approach to study pattern formation in systems described by a conserved order parameter on a square lattice. Here we describe and analyze two of the different morphologies observed during growth far from equilibrium: the dense branching morphology (DBM) and the dendritic morphology. Both have been found to represent clearly distinct morphological “phases”. They can be characterized by their envelope: convex for DBM and concave for dendritic morphology. They both propagate at constant velocity. The velocity scales with different powers of the chemical potential for the two different morphologies. For the DBM, the branch width is proportional to the diffusion length. The transitions between the morphologies and their growth behavior are studied as a function of the chemical potential and the macroscopic driving force (supersaturation).

Morphology transitions during non-equilibrium growth

Abstract We present a diffusion-transition scheme to study the penetration of a stable phase into a meta-stable one in systems described by a conserved order parameter. This approach is inspired by the specific example of solidification from supersaturated solution, for which we can take advantage of new experimental observations on surface kinetics. In this paper we present the approach and a study of solid-liquid equilibrium. The average shapes are compared with those evaluated by the Wulff construction. We calculate the fluctuations of the interface about the average shape as well as the temporal fluctuations in the diffusion field. Based on this, we propose a new strategy for experimental study of the kinetics of the phase transition. In part two we will present the morphologies observed in the simulations during non-equilibrium growth, focusing on the dense branching and dendritic morphologies, on their shape preserving envelope and on the transitions between them.

Structure formation in diffusional growth and dewetting

Abstract The morphology diagram of possible structures in two-dimensional diffusional growth is given in the parameter space of undercooling Δ versus anisotropy of surface tension ϵ . The building block of the dendritic structure is a dendrite with a 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 show the analogy of diffusional growth with dewetting patterns of a fluid film on a substrate. We also describe the structures and velocities of fractal dendrites and doublons destroyed by noise. The extension of these results to three-dimensional growth is briefly discussed.

Crack propagation as a free boundary problem.

A sharp interface model of crack propagation as a phase transition process is discussed. We develop a multipole expansion technique to solve this free boundary problem numerically. We obtain steady state solutions with a self-consistently selected propagation velocity and shape of the crack, provided that elastodynamic effects are taken into account. Also, we find a saturation of the steady state crack velocity below the Rayleigh speed, tip blunting with increasing driving force, and a tip splitting instability above a critical driving force.

Elastic Interaction between Surface Defects in Thin Layers

› defect separation), we discover that defects in thin layers may either attract or repeleach other depending on the direction (though elastic deformation is isotropic) with respect to the localgeometric force distribution caused by the defect. Moreover, the force distribution fixes the exponent inthe power law 1y

Morphology and selection processes in diffusion-controlled growth patterns

The evolution and selection of fractal and compact growth patterns is described. As a particular example the growth of a supercritical nucleus occurring during typical first-order phase transitions and its behaviour at long times and large lengths is discussed. While the generic growth mode is the dendritic growth, we discuss also the recently discovered doublon-growth in two dimensions and the occurrence of fractal growth at low driving forces. Finally numerical and analytical results on the competition of two or more growth patterns along a growth front are presented.

Fluctuation effects on dendritic growth morphology

Dendrites are the typical patterns for many anisotropic growth processes. A detailed understanding of their dynamics appears to be crucial for a proper classification of various growth morphologies. In particular the morphology transitions occurring for varying anisotropy were predicted to depend upon fluctuations. In the present investigation we compare analytical and numerical results on the stability of dendrites under influence of external fluctuations. In particular we confirm the previous ideas that the dendrites are linearly stable under influence of noise even in the limit of extremely small but nonzero anisotropy. This supports the concept of a smooth change-over from compact to fractal dendrites and finally to fractal seaweed whose internal length scale was predicted to depend on noise.