PHYSICAL REVIE& 8 VOLUME 37, NUMBER 13 1 MAY 1988 Modified asymptotic approach to modeling a dilute-binary-alloy solidificatio front James M. Hyman Center for Nonlinear Studies, Theoretical Division, MS M84, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 Any Novick-Cohen Department of Mathematics, Michigan State University, East Lansing, Michigan 48824 Philip Rosenau Deportment of Mechanical Engineering, Technion, Israel Institttte of Technology, Haifa 32MO, Israel (Received 19 June 1987) Directional solidi5cation in the presence of an impurity may be described by a set of impurity- concentration and thermal-difFusion equations coupled at a free boundary. Small deviations of the interface from planarity can be described by a single fourth-order equation. This equation is de- rived by a Iong-wavelength, smaB-amplitude expansion in the limit of a small distribution coef5cient. %e present an alternative asymptotic approach that isolates and preserves the crucially important nonlinearities in their original form, and thus preserves the proper behavior at large am- plitudes during pattern formation. The resulting evolution equation is in better agreement with the physical phenomena of front destabilization and dr&piet creation than are previously presented, models. The formation of different solidiScation patterns is numerically elucidated. I. IN j.icOBUtÃxON AND STA I;RMENT OF THE PRQSI.EM The sohdification front of a dilute binary alloy is sensi- tive to the delicate balance between competing nonlinear processes. In deriving equations that describe the evolu- tion of the front, it is essential to properly preserve these nonlinearities. The amplitude equation derived using standard asymptotic methods, such as in Ref. 1, predicts the solution to blow up within a finite time. Generally speaking, the exact form and nature of the blowup — and in fact, whether or not the solution blows up at all — can be highly dependent on how the nonlinearities are ap- proximated by the asymptotic expansion. To better model the interface, rather than go to a more complex physical model, we developed a refined deriva- tion that preserves the crucial nonhnearities of the full equations and thus overcomes the major difficulty of Ref. 1. The derivation will be outlined in Sec. II, followed by numerical studies in Sec. III and a discussion of the re- sults in Sec. IV. We now give a brief description of the solidi5cation problem. Directional solidification is a process in which an alloy is transported with a fixed velocity V through an exter- nally imposed temperature gradient. This process has many metallurgical applications, including zone refinement. Repeated directional solidification of a dilute binary aHoy (the minor phase here is the impurity) gradu- ally reduces the amount of impurity in the alloy. This occurs primarily because j', the segregation coe@cient (defined as the ratio of the impurity concentration on the solid side to the impurity concentration on the liquid side of a planar solid-liquid interface at thermodynamic equi- librium), is frequently much less than 1. Written in terms of the moving coordinate frame x=z, ie — Vt, and using the diffusion coefficient D of the impurity in the solute to nondimensionalize the problem, the equations for the impurity concentration and the temperature profiles in terms of the spatial var- iables (x,y, z) = (D / V)(x, y, z) and the time variable t =DV t are (Refs. 2-4) hC+Cs — Cg — — 0, z) f bT =0, z «f ET=0, zgQ. (lc) Qn the surface z = f(x, t) we also have the conditions T = 1+MC + ytt = T, R (n VT) — (n. VT}=FV n VC =(E — 1)CV„, (le) where tt is the curvature, n is the unit vector normal to the interface, and V„ is the normal component of the ve- locity of the interface. Here f(x, t) denotes the location of the liquid-solid interface, which will be assumed here to be sharp. The liquid (solid) side of the interface will be located at z & f (z g tIt) The concentr. ation of the impuri- ty C{x, t) has been normalized to be unity at the liquid side of the interface. The temperature T(x, t} [T{x, t)], at the hquid (solid} side of the interface has been normal- ized so that the melting temperature at a planar interface with no impurity will be unity. Here M = rnC'/Tst is the slope of the liquidus line on the phase diagram, Tss is the melting temperature (in kel- vin) of the pure solvent, and C' is the solvent concentra- tion in the liquid at the interface when the interface is 37 7603 19&g The American Physical Society