Deformation behavior of an omega hardened alpha-beta titanium alloy : Ernest Levine, Steven Hayden & Harold Margolin

Abstract

We develop a model that allows interface kinetics to be incorporated in a simple way in the determination of the dendrite operating state. The model is based on an optimum stability conjecture, according to which the dendrite tip radius is related to the wavelength of the fastest growing Fourier component in a linear stability analysis that includes consideration of interface kinetics. The basic idea is similar to that of the marginal stability hypothesis but leads to a value of the selection parameter σ = 2κd0/(Vϱ2) that decreases as the Peclet number Pe = Vϱ/(2κ) is increased (V = dendrite growth velocity, ϱ = dendrite tip radius, κ = thermal diffusivity and d0 = capillary length). This results in 1/σ being a linear function of 1/ϱ. The Ivantsov equation for the dimensionless supercooling S in terms of Pe is modified to account for capillarity and kinetics by using the dendrite tip temperature instead of the melting temperature in the original Ivantsov equation. This yields a complete parametric solution to the problem, and the resulting “scaling laws” are exhibited for the case of two spatial dimensions. We also reexamine the data of Rubinstein and Glicksman for dendritic growth in pivalic acid for which σ varies by about 100% over the measured range of supercooling. The value of 1/σ is nearly linear in 1/ϱ, in agreement with the optimum stability conjecture. Comparison with the present theory results in a kinetic coefficient of μ = 0.218cm/s·K. The resulting calculated values of V and ϱ are found to be in better agreement with experiment than for constant σ, but still show considerable discrepancy at small supercoolings, probably due to convection. This stability theory is based on a planar geometry rather than on a parabolic geometry, the modification of the Ivantsov relation is crude, and the theory completely neglects treatment of anisotropy of either capillarity or interface kinetics, which are known to be important from microscopic solvability theory. Nevertheless, we identify a dimensionless group m = μ/TMγ(κLV) which must be small for interface kinetics to be important (TM = absolute melting temperature, γ = crystal-melt surface free energy, and LV = latent heat per unit volume).