The nonlinear dynamics of solidification of a binary melt with a nonequilibrium mushy region

Abstract

A new theoretical analysis of the solidification process of a binarymelt with a mushy region, in which the temperature is below the equilibrium temperature, is presented. The mushy region consists of the liquid and the growing spherical particles. The equations of heat and mass of the solute and the kinetic equation for the particle size distribution function are used. The nucleation rate and the crystal growth rate are phenomenologically defined. An approximate expression for a portion of solid phase in the mushy region is given by the investigation of the kinetic equation. As a result, the dynamics of the nonequilibrium mushy region is determined by two nonlinear integro-differential equations with appropriate boundary conditions. An approximate analytical solution of this system of equations is obtained for the quasistationary regime of solidification. The temperature, concentration distribution, and the distribution function are defined for the mushy region. The method can be used for analysing the solute distribution pattern in solid materials obtained by solidification from a melt.

On the theory of the unsteady-state growth of spherical crystals in metastable liquids

“…Non-stationary contributions entering in the growth rates of spherical particles, their radii, and the distribution of temperature and concentration fields must be taken into account in the theoretical description of the intermediate stage of phase transformation in melts and solutions (the analytical description of crystal growth in dilute solutions is presented in appendix A). For this reason, many theories of growth of crystals or droplike aggregates in supersaturated solutions, supercooled melts, mushy layers, colloids, magnetic fluids and other physical systems (see, among others, [18][19][20][21][22][23][24][25][26][27]) should be reconsidered with allowance for the non-stationary contributions to the particle growth rates.…”

The influence of non-stationarity and interphase curvature on the growth dynamics of spherical crystals in a metastable liquid

“…First, in the case of stationary growth (normal∂T/normal∂t=0) in the absence of curvature (χfalse→0) and attachment kinetics (μkfalse→normal∞), we obtain from (2.1) [43] dRdt=β∗normalΔT1+β∗qTR, where qT=LV/λl. This formula contains two limiting cases frequently used in theoretical and experimental investigations [2,11,12,44–48] dRdt=β∗normalΔT, β∗qTR≪12emand2emdRdt=normalΔTqTR, β∗qTR≫1.} Note that the fir...…”

Nucleation and evolution of spherical crystals with allowance for their unsteady-state growth rates

“…Estimating β 2 * ∆t/(aΛ) ∼ 10 −2 t s −1 as a typical case for metallic melts, we conclude that the second term in expressions (18) becomes substantial at times t 10 s after nucleation of a certain particle. In other words, the commonly used approximation of the particle growth rate V(t) ≈ β * ∆ [13,14,22,[26][27][28][29] is only the rough estimate that describes the main contribution only. In order to obtain a more precise description of the real nucleation process one can use next terms in the asymptotic expansion of R(t) and V(t) found in expressions (18).…”