Effects of Finite Rate Phase Transformation Kinetics on The Steady-State Solidification Front Propagation Speed in Undercooled Pure Liquids

Abstract

This novel approach to modeling the steady-state solidification of undercooled pure liquids is based upon first principles. Continuum equations are used to describe a volumetrically averaged, coexisting mixture of solid and liquid in the thin phase transformation zone between regions of pure liquid and pure solid. These equations are coupled with a dynamic equilibrium based rate law that describes temperature dependent phase transformation kinetics. The time scale associated with finite rate phase transformation is found to depend on a solidification activation energy, thermal energy, and the enthalpy of fusion. The model leads naturally to an eigenvalue problem for the solidification front propagation speed. In addition, the variation of the volumetrically averaged liquid fraction across the solidification zone is predicted.

The effect of finite–rate heat release on the solidification of an undercooled pure liquid

“…whereH =h/kT M represents the ratio of phase-change enthalpy for an atom to the average thermal energy of an atom at temperature T M , and = kT M /E F represents the ratio of average thermal energy to the activation energy for freezing. Estimates of E F from the literature (Jackson & Chalmers 1956), as well as calculations using the mesoscale model to match Willnecker's front speed data (Norris 1993), indicate that the magnitude of is approximately 10 −3 , hence it will be considered a small parameter.…”

“…For example, the specific total energy at any given point is the average of the specific total energy distributed in both the liquid and solid within δV. Similar definitions are developed for the specific internal energy, specific enthalpy and all other variables and parameters (Norris 1993). The characteristic length scale of δV, denoted O(δV 1/3 ), must be much smaller than the conduction length scale, yet only slightly larger than the length scale for which the addition or removal of a small portion of solid material within δV will not significantly alter the values of the averaged variables.…”

“…As a first step in the development of a finite-rate theory, the mesoscale model is used to describe a planar solidification zone moving into an undercooled liquid initially at a temperature T ∞ < T M where the latter quantity denotes the melting temperature. The details of the derivation are given in Norris (1993). The reference frame is fixed to the material and the temperature distribution T is a function of the spatial coordinate x and time t. Pure undercooled liquid at temperature T ∞ exists initially in a half-space x 0.…”

“…Here, the individual liquid-to-solid and solid-to-liquid transformations are described by Arrhenius-type kinetic relations with dimensional molecular 'freezing' and 'melting' activation energies E F and E M = E F +h, respectively, whereh is the enthalpy of fusion per molecule. The dimensionless net rate of solidification for the mixture of liquid and solid can then be written as (Jackson & Chalmers 1956;Norris 1993)…”

“…The coefficient A in (2.12) arises from the volumetric averaging process. It is in effect a mixture material function which accounts for the detailed solid-liquid interface morphological geometry within the small volume δV (Norris 1993), as well as for the atomic site availability, atomic jump frequencies and atomic surface densities on those same solid-liquid interfaces (Ashby & Jones 1986;Jackson & Chalmers 1956). In the present formulation effort, A is treated as a material constant; however, it is recognized that its dependence on morphology should be exploited in future efforts.…”