Crossover Scaling of Wavelength Selection in Directional Solidification of Binary Alloys

We simulate dendritic growth in directional solidification in dilute binary alloys using a phase-field model solved with an adaptive-mesh refinement. The spacing of primary branches is examined for a range of thermal gradients and alloy compositions and is found to undergo a maximum as a function of pulling velocity, in agreement with experimental observations. We demonstrate that wavelength selection is unambiguously described by a non-trivial crossover scaling function from the emergence of cellular growth to the onset of dendritic fingers, a result validated using published experimental data.Comment: 4 pages, four figures, submitted to Physical Review Letter

Multiscale Random-Walk Algorithm for Simulating Interfacial Pattern Formation

We present a novel computational method to simulate accurately a wide range of interfacial patterns whose growth is limited by a large scale diffusion field. To illustrate the computational power of this method, we demonstrate that it can be used to simulate three-dimensional dendritic growth in a previously unreachable range of low undercoolings that is of direct experimental relevance.Comment: 4 pages RevTex, 6 eps figures; substantial changes in presentation, but results and conclusions remain the sam

Kinetic cross coupling between non-conserved and conserved fields in phase field models

We present a phase field model for isothermal transformations of two component alloys that includes Onsager kinetic cross coupling between the non-conserved phase field and the conserved concentration field. We also provide the reduction of the phase field model to the corresponding macroscopic description of the free boundary problem. The reduction is given in a general form. Additionally we use an explicit example of a phase field model and check that the reduced macroscopic description, in the range of its applicability, is in excellent agreement with direct phase field simulations. The relevance of the newly introduced terms to solute trapping is also discussed

Growth and Structure of Random Fibre Clusters and Cluster Networks

We study the properties of 2D fibre clusters and networks formed by deposition processes. We first examine the growth and scaling properties of single clusters. We then consider a network of such clusters, whose spatial distribution obeys some effective pair distribution function. In particular, we derive an expression for the two-point density autocorrelation function of the network, which includes the internal structure of a cluster and the effective cluster-cluster pair distribution function. This formula can be applied to obtain information about nontrivial correlations in fibre networks.Peer reviewe

Density correlations in paper

We present an analysis of areal mass density correlations in paper. Using β radiography, the local mass density of laboratory paper sheets has been measured. The real space density autocorrelation function calculated from the data reveals a nontrivial power law type of correlations with the decay exponent being roughly independent of the basis weight of the sheets. However, for low densities we find that correlations may extend at least an order of magnitude beyond the fiber length, whereas for heavier paper they quickly die out.Peer reviewe

Onsager approach to 1D solidification problem and its relation to phase field description

We give a general phenomenological description of the steady state 1D front propagation problem in two cases: the solidification of a pure material and the isothermal solidification of two component dilute alloys. The solidification of a pure material is controlled by the heat transport in the bulk and the interface kinetics. The isothermal solidification of two component alloys is controlled by the diffusion in the bulk and the interface kinetics. We find that the condition of positive-definiteness of the symmetric Onsager matrix of interface kinetic coefficients still allows an arbitrary sign of the slope of the velocity-concentration line near the solidus in the alloy problem or of the velocity-temperature line in the case of solidification of a pure material. This result offers a very simple and elegant way to describe the interesting phenomenon of a possible non-single-value behavior of velocity versus concentration which has previously been discussed by different approaches. We also discuss the relation of this Onsager approach to the thin interface limit of the phase field description.Comment: 5 pages, 1 figure, submitted to Physical Review

Dynamics of driven interfaces near isotropic percolation transition

We consider the dynamics and kinetic roughening of interfaces embedded in uniformly random media near percolation treshold. In particular, we study simple discrete ``forest fire'' lattice models through Monte Carlo simulations in two and three spatial dimensions. An interface generated in the models is found to display complex behavior. Away from the percolation transition, the interface is self-affine with asymptotic dynamics consistent with the Kardar-Parisi-Zhang universality class. However, in the vicinity of the percolation transition, there is a different behavior at earlier times. By scaling arguments we show that the global scaling exponents associated with the kinetic roughening of the interface can be obtained from the properties of the underlying percolation cluster. Our numerical results are in good agreement with theory. However, we demonstrate that at the depinning transition, the interface as defined in the models is no longer self-affine. Finally, we compare these results to those obtained from a more realistic reaction-diffusion model of slow combustion.Comment: 7 pages, 9 figures, to appear in Phys. Rev. E (1998

Universal Dynamics of Phase-Field Models for Dendritic Growth

We compare time-dependent solutions of different phase-field models for dendritic solidification in two dimensions, including a thermodynamically consistent model and several ad hoc models. The results are identical when the phase-field equations are operating in their appropriate sharp interface limit. The long time steady state results are all in agreement with solvability theory. No computational advantage accrues from using a thermodynamically consistent phase-field model.Comment: 4 pages, 3 postscript figures, in latex, (revtex

Fast and Accurate Coarsening Simulation with an Unconditionally Stable Time Step

We present Cahn-Hilliard and Allen-Cahn numerical integration algorithms that are unconditionally stable and so provide significantly faster accuracy-controlled simulation. Our stability analysis is based on Eyre's theorem and unconditional von Neumann stability analysis, both of which we present. Numerical tests confirm the accuracy of the von Neumann approach, which is straightforward and should be widely applicable in phase-field modeling. We show that accuracy can be controlled with an unbounded time step Delta-t that grows with time t as Delta-t ~ t^alpha. We develop a classification scheme for the step exponent alpha and demonstrate that a class of simple linear algorithms gives alpha=1/3. For this class the speed up relative to a fixed time step grows with the linear size of the system as N/log N, and we estimate conservatively that an 8192^2 lattice can be integrated 300 times faster than with the Euler method.Comment: 14 pages, 6 figure

Crossover Scaling in Dendritic Evolution at Low Undercooling

We examine scaling in two-dimensional simulations of dendritic growth at low undercooling, as well as in three-dimensional pivalic acid dendrites grown on NASA's USMP-4 Isothermal Dendritic Growth Experiment. We report new results on self-similar evolution in both the experiments and simulations. We find that the time dependent scaling of our low undercooling simulations displays a cross-over scaling from a regime different than that characterizing Laplacian growth to steady-state growth