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The morphology diagram of possible structures in two-dimensional diffusional growth is given in the parameter space of undercooling ⌬ versus anisotropy of surface tension ⑀. 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.

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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.PACS numbers: 62.20. Mk, 46.50.+a, 47.54.+r Understanding the day-to-day phenomenon of fracture is a major challenge for solid state physics and materials science. Starting with the early idea of Griffith [1], who realized that crack growth is a competition between a release of elastic energy and an increase of surface energy, various approaches have been developed to describe the striking features of cracks [2]. Usually, the motion of cracks is understood on the level of breaking bonds at sharp tips, and obviously theoretical predictions depend sensitively on the underlying empirical models of the atomic properties (see for example [3]). Plastic effects, however, lead to extended crack tips (finite tip radius r 0 ), and it is conceivable that for example fracture in gels can be described macroscopically. Then a full modeling of fracture should not only determine the crack speed but also the crack shape self-consistently.Recent phase field models go beyond the microscopic limit of discrete models with broken rotational symmetry, and encompass much of the expected behavior of cracks [4,5]; these models are close in spirit though different in details with respect to earlier approaches [6]. However, the scale of the appearing patterns is always dictated by the phase field interface width, and thus these models have problems in the sharp interface limit. Other descriptions are based on macroscopic equations of motion but suffer from inherent finite time singularities which do not allow steady state crack growth unless the tip radius is limited by the phase field interface width [7]. Numerical approaches which are not based on a phase field provide a selection mechanism by the introduction of complicated nonlinear terms in the elastic energy for high strains in the tip region [8], requiring additional parameters.It is therefore highly desirable to look for minimal models of fracture which are free from microscopic details and which are based on well established thermodynamical concepts. This is also motivated by experimental results showing that many features of crack growth are rather generic [9]; among them is the saturation of the steady state velocity appreciably below the Rayleigh speed and a tip splitting for high applied tension.Already in our previous publication [10] we emphasized a connection between fracture mechanics and elastically induced surface diffusion processes: the AsaroTiller-Grinfeld (ATG) instability [11] appears to be a good starting point for the quest f...

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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.