The effect of marangoni convection on fibrous monotectic growth
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I. INTRODUCTION
THE nonvariant monotectic reaction L1 → S ⫹ L2 in which a liquid L1 decomposes simultaneously into a solid S and another liquid L2 is observed in a wide class of alloys exhibiting a so-called miscibility gap in the liquid state.[1,2,3] The reaction is very similar to a eutectic one. Alloys of monotectic composition can be solidified directionally yielding, in a restricted range of velocities and temperature gradients ahead of the solid/liquid interface,[4,5] a fibrous structure. Often, however, the fibers of the minority phase L2 appear as a string of pearls.[4,6–9] Theoretical explanations for fibrous composite growth are usually based on the classical Jackson and Hunt model for eutectics.[10] The most complete description of monotectic growth was recently given by Coriell and co-workers. They extended the model to better treat some specialities of immiscible alloys such as the huge density differences between the phases.[11] Although they thoroughly studied in detail a Jackson and Hunt type analysis of monotectic composite growth for Al-In alloys, they concluded that there is a big discrepancy between theory and experiment and that the reason for this is still unknown. In the approach presented here, we propose an additional mode of mass transport in front of the monotectic interface, because in our mind, the main difference between monotectic and eutectic solidification is the liquid phase state of the (rod) L2 phase growing simultaneously within a nearly perfectly pure solid matrix. We assume that the thermocapillary effect causes convection at the interface of the liquid L2 L. RATKE, Vice Director, is with the Institute of Space Simulation, Germany Aerospace Center, 51170 Cologne, Germany. Contact e-mail: [email protected] This article is based on a presentation given in the symposium “Fundamentals of Solidification” which occurred at the TMS Fall meeting in Indianapolis, Indiana, November 4-8, 2001, under the auspices of the TMS Solidification Committee. METALLURGICAL AND MATERIALS TRANSACTIONS A
phase to the molten L1 matrix. This Marangoni convection induces a flow vortex and has an influence on the solute transport and, thus, the constitutional supercooling of the solidification front. The analytical theory outlined here uses a series of approximations in order to get some insight into the essential effects of Marangoni motion on monotectic composite growth. The numerical model instead leads to a Navier–Stokes equation with appropriate boundary conditions and the diffusion equation, which are solved by the well-known SIMPLE algorithm using a finite-difference scheme on a staggered grid. The influence of the flow field on the constitutional supercooling is examined and the mean supercooling of the solidification front is calculated. Supposing that the composite monotectic structure grows at the minimum supercooling, a new relation between the fiber distance 2R and the solidification velocity v0 is derived. II. OUTLINE OF THE MODEL We set up the model in complete analogy to the Jackson an
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