Klaus Kassner

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.

Comparison of phase-field models for surface diffusion

The description of surface-diffusion controlled dynamics via the phase-field method is less trivial than it appears at first sight. A seemingly straightforward approach from the literature is shown to fail to produce the correct asymptotics, albeit in a subtle manner. Two models are constructed that approximate known sharp-interface equations without adding undesired constraints. Numerical simulations of the standard and a more sophisticated model from the literature as well as of our two models are performed to assess the relative merits of each approach. The results suggest superior performance of the models in at least some situations.

Influence of external flows on crystal growth: numerical investigation.

We use a combined phase-field-lattice-Boltzmann scheme [Medvedev and Kassner, Phys. Rev. E 72, 056703 (2005)] to simulate nonfaceted crystal growth from an undercooled melt in external flows. Selected growth parameters are determined numerically. For growth patterns at moderate to high undercooling and relatively large anisotropy, the values of the tip radius and selection parameter plotted as a function of the Péclet number fall approximately on single curves. Hence, it may be argued that a parallel flow changes the selected tip radius and growth velocity solely by modifying (increasing) the Péclet number. This has interesting implications for the availability of current selection theories as predictors of growth characteristics under flow. At smaller anisotropy, a modification of the morphology diagram in the plane of undercooling versus anisotropy is observed. The transition line from dendrites to doublons is shifted in favor of dendritic patterns, which become faster than doublons as the flow speed is increased, thus rendering the basin of attraction of dendritic structures larger. For small anisotropy and Prandtl number, we find oscillations of the tip velocity in the presence of flow. On increasing the fluid viscosity or decreasing the flow velocity, we observe a reduction in the amplitude of these oscillations.

Stability of travelling fronts in a piecewise-linear reaction-diffusion system

A stability analysis of fronts is performed analytically for a Rinzel-Keller-type model with equal diffusion constants. Exact solutions are obtained for the propagating front, the bifurcation diagram of the front velocity and the growth rate of disturbances. The effect of the most unstable eigenmode on the front shape is displayed. Moreover, we give the eigenvalues of the stability operator and show explicitly how a non-moving front becomes unstable at the bifurcation point while a moving front becomes stable.

Phase-field approach to crystal growth in the presence of strain

Abstract A phase-field model for the dynamics of a strained solid in contact with its melt is developed. The sharp-interface limit of this model reduces to the continuum equations describing the Asaro–Tiller–Grinfeld instability. In addition, it allows the derivation of generalizations of these equations. The dynamics of extended systems are investigated via the phase-field approach which contains an inherent lower length cutoff, thus avoiding cusp singularities. General initial conditions lead to coarsening. For periodic states, this arises via a series of approximate period doublings, the beginning of which can be demonstrated analytically.

Spatial geometry of the rotating disk and its non-rotating counterpart

A general relativistic description of a disk rotating at constant angular velocity is given. It is argued that conceptually this direct approach poses fewer problems than the special relativistic one. For observers on the disk, the geometry of their proper space is hyperbolic. This has interesting consequences concerning their interpretation of the geometry of a non-rotating disk having the same radius. The influence of clock synchronization on spatial measurements is discussed.

Analytic solutions for monotonic and oscillating fronts in a reaction–diffusion system under external fields

Abstract A piecewise linear reaction–diffusion system including an external field is considered. Analytic solutions are obtained for the propagating front, the front velocity, the perturbed front and the growth rate of perturbations. It is found that, when the ratio of the time scales, null-cline parameter and the external field are small enough, the moving front has an oscillating behaviour, similar to that of a front in an oscillatory medium. A comparison with an oscillatory front invading an unstable state at small velocity is made and it is argued that there are crucial differences between the two cases. In particular, we find that for fronts oscillating about a stable state both concentrations must oscillate in two-variable models, whereas fronts oscillating about an unstable state can approach it with one of the two variables being monotonic.

Vicinal surfaces: growth structures close to the instability threshold and far beyond

We introduce a new numerical approach to step flow growth, making use of its analogies to dendritic growth. Concentrating on the situation close to the instability threshold of step growth, nonlinear evolutionary equations for the steps on a vicinal surface can be derived in a multiple-scale analysis. This approach retains the relevant nonlinearities sufficiently close to the threshold. Our simulations recover and visualize these findings. However, on the basis of our simulations we further report results on the behaviour far from the threshold. Step propagation is treated as a moving-boundary problem based on the Burton-Cabrera-Frank (Burton W K, Cabrera N and Frank F C 1951 Phil. Trans. R. Soc. A 243 299) model. Our method handles the problem in a fully dynamical manner without any quasistatic approximations. Furthermore, it allows for overhangs.

Confinement effects in dendritic growth

We study the interplay between crystal orientation and confinement in diffusion-limited growth. Growth of the overall morphology in the direction of minimal surface stiffness has been investigated in some detail by various authors, the most recent advances being summarized by Brener et al (Brener E, Muller-Krumbhaar H and Temkin D 1996 Phys. Rev. E 54 2714). Here, we consider competing influences, each trying to impose a different growth direction. The simplest possible situation giving rise to such a competition is growth in a channel with a mismatch between the orientation of the channel walls and surface tension anisotropy. Analysing this situation, we find a new structure and gain further insight into the problem of morphological stability. Another case is that of periodic boundary conditions, where the same angle of misorientation can be used to describe growth of a tilted array of finger-shaped crystals. It is found that the transition between dendritic and doublonic structures is affected by the tilt.

Selection theory of free dendritic growth in a potential flow.

The Kruskal-Segur approach to selection theory in diffusion-limited or Laplacian growth is extended via combination with the Zauderer decomposition scheme. This way nonlinear bulk equations become tractable. To demonstrate the method, we apply it to two-dimensional crystal growth in a potential flow. We omit the simplifying approximations used in a preliminary calculation for the same system [Fischaleck, Kassner, Europhys. Lett. 81, 54004 (2008)], thus exhibiting the capability of the method to extend mathematical rigor to more complex problems than hitherto accessible.