Stability and shapes of cellular profiles in directional solidification: expansion and matching methods

Abstract

Ideas based on constitutional supercooling suggest that the periodic steady state cellular patterns seen in the directional solidification of systems with small partition coefficient may be unstable if the impurity concentration in the melt just in front of the tips falls into the two phase (miscibility gap) region of the phase diagram. This gives a simple stability criterion relating the position of the tips of the cells to the pulling velocity that is in good qualitative agreement with the limited experimental data available in the cellular regime. Implications of this criterion for a particular class of steady state solutions derived using asymptotic matching methods are explored. These solutions arise from a generalization to finite Péclet number for systems with small partition coefficient of the ideas of Dombre and Hakim relating directional solidification patterns to viscous (Saffman-Taylor) fingers. Families of steady state solutions yielding both small amplitude interface patterns as well as fingerlike solutions with narrow deep grooves are accurately described by the methods discussed herein. A systematic expansion method provides corrections to the classical Scheil shapes for the grooves. However, the stability criterion, as well as other considerations, suggest that the entire class of narrow grooved solutions found by the matching methods may be unstable. Comparison with other numerical work suggests that other branches of narrow grooved solutions exist and are relevant to experiments. Several experimental and numerical tests of these ideas are proposed.

Steady deep-cellular growth in solidification

“…This phenomenon can be well observed with the device, the so-called Hele-Shaw cell (Fig. 1) and has been studied analytically for long time by a number of authors (see [4][5][6]). However, the problems remains.…”

“…Deep-cellular growth in directional solidification is a classic and fundamental subject in condensed matter physics and material science [1][2][3][4][5][6][7][8][9]. This phenomenon can be well observed with the device, the so-called Hele-Shaw cell (Fig.…”

The Growth of a Single Cell/Dendrite in a Directional Solidification Process

“…A criterion to match these two regions remains unclear and some mathematical treatments have been proposed, but all lack the clear physics. [7,12] On the other hand, the complex cell shape can be studied through numerical calculations [4,5,13] and phasefield simulations, [14][15][16] and the calculated shape can be directly compared with experiments. Hunt et al [4,5] have pioneered the computation of cell shapes in a directional solidification process and significant progress has been made ever since the first finite-difference work by McCartney and Hunt.…”

“…For the cellular/ dendritic morphology, the shape of a growing tip and the spacing selection have been studied both theoretically and experimentally. [1][2][3][4][5][6][7][8] For the evolution of a dendrite tip of a pure substance, Ivantsov [1] first solved the equations governing the growth from its undercooled melt and found that the tip should be a paraboloid of revolution and the tip radius should be controlled by the undercooling of the melt bath. The smooth region of a dendrite tip in both undercooled growth and directional growth of alloys is also found to be a paraboloid of revolution.…”

“…The smooth region of a dendrite tip in both undercooled growth and directional growth of alloys is also found to be a paraboloid of revolution. [2,6] However, the shape of a smooth cell in the directional growth of alloys is much more complicated: [5][6][7][8][9][10][11] the tip region may be fit into a Saffman-Taylor finger and the tail part usually follows the Scheil equation. A criterion to match these two regions remains unclear and some mathematical treatments have been proposed, but all lack the clear physics.…”