Pattern Formation In Solidification Research Articles

Mathematics of pattern growth in condensed matter

Mathematics of pattern growth in condensed matter

Phase-field calculations of pattern formation in solidification of binary alloys

The cellular growth of Ni x Cu1−x and Ge1−x Si x has been studied by means of a phase-field method. Concerning the wavelength as a function of the cooling velocity our results for Ni x Cu1−x are in good agreement with those obtained by other authors. The computations for Ge1−x Si x are more challenging, because the structures of interest are much larger. Studies of the cellular growth for different temperature gradients and diffusion coefficients have been performed.

Pattern Formation in Solidification of Chondrules using A Three Dimensional Phase Field Model

Pattern Formation in Solidification of Chondrules using A Three Dimensional Phase Field Model

Phase-field modeling of binary alloy solidification with coupled heat and solute diffusion.

A phase-field model is developed for simulating quantitatively microstructural pattern formation in solidification of dilute binary alloys with coupled heat and solute diffusion. The model reduces to the sharp-interface equations in a computationally tractable thin-interface limit where (i). the width of the diffuse interface is about one order of magnitude smaller than the radius of curvature of the interface but much larger than the real microscopic width of a solid-liquid interface, and (ii). kinetic effects are negligible. A recently derived antitrapping current [Phys. Rev. Lett. 87, 115701 (2001)]] is used in the solute conservation equation to recover precisely local equilibrium at the interface and to eliminate interface stretching and surface diffusion effects that arise when the solutal diffusivities are unequal in the solid and liquid. Model results are first compared to analytical solutions for one-dimensional steady-state solidification. Two-dimensional thermosolutal dendritic growth simulations with vanishing solutal diffusivity in the solid show that both the microstructural evolution and the solute profile in the solid are accurately modeled by the present approach. Results are then presented that illustrate the utility of the model for simulating dendritic solidification for the large ratios of the liquid thermal to solutal diffusivities (Lewis numbers) typical of alloys.

Pattern formation in solidification

Regular patterns form in many solidification processes. Examples occur during lamella and rod-like eutectic growth and when single-phase cells or dendrites are formed. The scale and regularity of the microstructure can determine the properties of the cast materials and is thus important practically. The purpose of the present paper is to show that common features occur in all processes.Steady-state analysis indicates that a wide range of possible spacings could occur during eutectic, cellular or dendritic growth. The degree of freedom is removed by considering the mechanism determining the minimum and maximum spacing on a specimen. It is found that the minimum spacing occurs when the array first becomes stable for a lamella or rod-like eutectic, for cell growth and for some dendrites. For low temperature gradient, high-velocity dendrites, the minimum spacing is determined by thespacing when the dendrites irst become near enough to interact. The maximum spacing for eutectics andfor cells is determined by tip splitting. The maximum spacing for dendrites occurs when a tertiary arm becomes a new primary. Good agreement is obtained between theory and experiment using this approachto predict spacing limits. The average spacing on a specimen can approach either limit depending o past history. The two extreme spacings are found to span the spacing of the minimum undercooling foreutectic and cellular growth and this allows an average spacing to be estimated using a single condition.It is concluded that three conditions are necessary to form regular structures. A mechanism must exist to eliminate members of the array when the spacing is too small. A mechanism must exist to form new members of the array when the spacing is too wide. The structure must be stable to fluctuations in the range of spacing between the two limits.

Pattern formation in solidification

Regular patterns form in many solidification processes. Examples occur during lamella and rodlike eutectic growth and when single phase cells or dendrites are formed. The scale and regularity of the microstructure can determine the properties of the cast materials and is thus important practically. The purpose of the present paper is to show that common features occur in all processes. Steady state analysis indicates that a wide range of possible spacings could occur during eutectic, cellular, or dendritic growth. The degree of freedom is removed by considering the mechanism determining the minimum and maximum spacing on a specimen. It is found that the minimum spacing occurs when the array first becomes stable for a lamella or rodlike eutectic, for cell growth, and for some dendrites. For low temperature gradient, high velocity dendrites the minimum spacing is determined by the spacing when the dendrites first become near enough to interact. The maximum spacing for eutectics and for cells is determined by tip splitting. The maximum spacing for dendrites occurs when a tertiary arm becomes a new primary. Very good agreement is obtained between theory and experiment using this approach to predict spacing limits. The average spacing on a specimen can approach either limit depending on past history. The two extreme spacings are found to span the spacing of the minimum undercooling for eutectic and cellular growth and this allows an average spacing to be estimated using a single condition. It is concluded that three conditions are necessary to form regular structures. A mechanism must exist to eliminate members of the array when the spacing is too small. A mechanism must exist to form new members of the array when the spacing is too wide. The structure must be stable to fluctuations in the range of spacing between the two limits.

Boundary-layer model of pattern formation in solidification

We propose and investigate the properties of a model of pattern formation in crystal growth. The principal dynamical variables in this model are the curvature of the solidification front and the thickness (or heat content) of a thermal boundary layer, both taken to be functions of position along the interface. This model is mathematically much more tractable than the realistic, fully nonlocal version of the free-boundary problem, and still recaptures many of the features that seem essential for studying dendritic behavior, for example. In this paper we describe analytic properties of the model. Preliminary numerical solutions produce snowflakelike patterns similar to those seen in nature.

Shape instabilities and pattern formation in solidification: A new method for numerical solution of the moving boundary problem

When a solidification front is advancing into a region of supercooled liquid, its shape is subject to instabilities which can lead to complex modes of growth. The evaluation of the growth pattern is determined by the interaction of the driving force of the instability due to heat diffusion with the restabilizing force due to surface tension, and its full development is a highly nonlinear process. A new method for solving the two-phase heat diffusion problem under these conditions is described. The method is based on a single-domain treatment (weak formulation) of the moving boundary problem, with the temperature field near the interface approximated by a piecewise linear function in the spirit of the finite-element technique. Calculated results for a model problem are in excellent agreement with the predictions of an analytical approximation.