Pattern Selection in Solidification
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METALLURGICAL microstructures are relatively simple examples of a broad and extremely interesting class of natural pattern-forming phenomena. Other examples involve flow-patterns in hydrodynamic systems, patterns formed by diffusion of chemical reactants, a n d - - at the highest level of complexity--biological systems. These problems are very difficult, and the basic principles required for their solution seem not yet to be understood. What is needed at the moment are models which are sufficiently tractable to allow exploration of difficult mathematical questions, but which are not so simple that they lose all experimental relevance. It seems possible that metallurgical microstructures may be ideal sources of such models. In this lecture I shall talk principally about patterns formed during directional solidification of alloys: cellular structures, lamellar eutectics, e t c . ; but I shall also be concerned in an important way with the growth of free dendrites. The advantage of the directional geometry is that one can perform an exact linear-stability analysis for an initially flat solidification front, 1 and one can carry the nonlinear calculation far enough to see explicitly the existence of the continuous family of stable cellular patterns which may emerge when the plane front is driven beyond its limit of stability. 2'3'4 The general scheme for this kind of calculation is the following. For any given system there exists some dimensionless group of parameters, denoted here by u, which serves as a control parameter. In the case of directional solidification of a dilute alloy, u is proportional to the growth rate and the solute concentration and inversely proportional to the thermal gradient at the front. The amplification rate to for infinitesimal sinusoidal deformations of wave number k is shown schematically in Figure 1 for various values of u. For u greater than some critical value u,., to becomes positive within some finite band of wave numbers and the system is unstable against the corresponding deformations. To carry the calculation further, one must use some kind of nonlinear technique. 2 This is usually a perturbation method valid only for small values of u - u,.; but more J. S. LANGER is with the Institute for Theoretical Physics, University of California at Santa Barbara, Santa Barbara, CA 93106. This paper is based on a presentation made at the symposium "Establishment of Microstructural Spacing during Dendritic and Cooperative Growth" held at the annual meeting of the AIME in Atlanta, Georgia on March 7, 1983 under the joint sponsorship of the ASM-MSD Phase Transformations Committee and the TMS-AIME Solidification Committee. METALLURGICALTRANSACTIONS A
ambitious procedures have been described in the literature. 3'5 It is important at this stage that one not only look for stationary solutions of the nonlinear equations of motion but also test these solutions for stability against deformations which change both their amplitudes and periodicities. The most dangerous instabilities, that is, the deformat
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