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Stability of a Planar Interface During Solidification of a Dilute Binary Alloy
1.J. W. Rutter and B. Chalmers, Can. J. Phys. 31, 15 (1953).
2.W. A. Tiller, J. W. Rutter, K. A. Jackson, and B. Chalmers, Acta Met. 1, 428 (1953).
3.W. A. Tiller and J. W. Rutter, Can. J. Phys. 34, 729 (1956).
4.D. E. Temkin, Dokl. Akad. Nauk. SSSR, 133, 174 (1960).
4.[English transl.: D. E. Temkin, Soviet Phys.‐Doklady 5, 609 (1960)].
5.The fractional change of amplitude should be small in a time equal to the relaxation time of the diffusion field (assumed larger than that of the thermal field) i.e., where λ is the wavelength of the sinusoidal perturbation and D is the solute diffusion coefficient in the liquid.
6.Changes of density upon solidification are ignored in this treatment.
7.For the rippled interface, the plane may be rigorously identified as the plane coinciding with the average position (in the z direction) of the interface and, therefore, traveling with the average velocity V (in the z direction) of the interface. The experimental conditions required to assure that V is indeed constant are discussed immediately after Eqs. (13) and (14).
8.W. W. Mullins and R. F. Sekerka, J. Appl. Phys. 34, 323 (1963).
9.Allowing for the influence of curvature on the partition coefficient k of solute between the two phases and on the slope m leads to results which are the same as those obtained here within the approximation of dilute solution theory.
10.Equation (4) may be viewed as the simultaneous expression of heat balance and mass balance on a differential scale at the interface; diffusion in the solid is ignored.
11.In verifying the boundary conditions expressed by Eqs. (5), the exponentials in Eqs. (6)–(8) may be expanded and all resulting terms in powers of δ higher than the first may be discarded; e.g.,
12.If the liquid is supercooled, and may he negative leading to a thermal gradient instability. This case has been previously discussed by Mullins and Sekerka.8
13.It is cubic in the variable
14.The condition that the old CS criterion predict instability in conflict with our prediction of absolute stability is that and simultaneously hold; from Eq. (27) this is equivalent to the condition assuming the same values as above and
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