Primary particle melting rates and equiaxed grain nucleation
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ch Fellow, is with the Materials Department, Oxford University, Oxford OX1 3PH, United Kingdom. A. HELLAWELL, Emeritus Research Professor, is with the Department of Metallurgy and Materials Engineering, Michigan Technological University, Houghton, MI 49931. Manuscript submitted July 16, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS B
where NP/V 5 number of particles of radius r, per unit volume. Taking NP/V 5 109 m23 (1 mm23) and r 5 1025 m, any alloy system will suffice; thus, for d-iron, DH ' 2z109Jzm23 and cL 5 5.74z106 Jzm23 K21, yielding DTm 5 1.4z1023 K. The result is similar for materials of similar entropies of fusion. For the purposes of this discussion, we will assume that this is a negligible cooling effect and that such a large relative liquid volume may be regarded as remaining effectively isothermal. Even with an increase in particle density of two orders of magnitude, the effect would still be unimportant. Within the temperature range T0 , TL , TM (Figure 1(a)), i.e., up to the melting point of the solvent component, the melting rate is controlled by solvent/solute diffusion in the liquid, away from/toward the melting particle(s). Above TM, the solid inevitably melts without time for significant material transport, as a pocket of pure solvent, and composition adjustment follows. This is where there is a thermosolutal transition in the melting rate. With significant solid solubility, Figure 1(b), disregarding solid diffusion (but see subsequent discussion), the corresponding transition temperature is the liquidus temperature where the solid melts without composition change, i.e., at k0C0, and is given for any liquid composition, C0, by TT/S5TM 2 mLk0C0
[2]
where mL is the liquidus slope. The locus of the thermosolutal temperature, TT/S, is shown by the broken line in Figure 1(b). Such dispersed particles of solid are generally moving about in the bulk liquid, and their melting causes local convection from thermal and solutal gradients so that thermal and solutal diffusion occur within limited boundary layers, dT and dS (these are not necessarily equal or constant). Simple linear melting rates can be expressed. Above TM in Figure 1(a) and TT/S in Figure 1(b), the solid melts without composition change by supply of latent heat, DH, through the boundary layer of mean thickness dT and the thermally controlled melting rate, vH, rises from zero at TM or TT/S, with the superheat DT1: vHDH 5
kL z DT1 dT
[3]
where kL is the thermal conductivity of the liquid. Below TM or TT/S, the solutally controlled melting rate, vS, rises from zero at the initial equilibrium temperature, T0, to a maximum at TM or TT/S. For the case of negligible solid solubility, (Figure 1(a)) the rate is obtained from the solvent concentration profile of Figure 2(a), assuming steady-state liquid diffusion through a given solutal boundary layer, dS, giving vS 5 2
DL (CL 2 C0) (1 2 C0)dS
[4]
where DL is the liquid diffusion coefficient and the concentrations are as indicated in Figure 2(a). With solid solubility (Figure 1(b)), the si
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