Phase-field modelling of the effect of density change on solidification revisited: Model development and analytical solutions for single component materials

TL;DR

This paper develops a physically consistent phase-field model for solidification shrinkage, incorporating density changes and fluid flow effects, with analytical solutions for interface behavior, enhancing understanding of solid-liquid interface dynamics.

Contribution

The paper introduces a new phase-field model with quasi-incompressible hydrodynamics for solidification, including analytical solutions for interface properties and effects of density contrast.

Findings

Fluid flow affects solidification front speed during shrinkage and expansion.

The model's equilibrium interface is independent of density-phase coupling.

Results agree with previous theoretical predictions.

Abstract

In this paper the development of a physically consistent phase-field theory of solidification shrinkage is presented. The coarse-grained hydrodynamic equations are derived directly from the N-body Hamiltonian equations in the framework of statistical physics, while the constitutive relations are developed in the framework of the standard Phase-field Theory, by following the variational formalism and the principles of non-equilibrium thermodynamics. To enhance the numerical practicality of the model, quasi-incompressible hydrodynamic equations are derived, where sound waves are absent (but density change is still possible), and therefore the time scale of solidification is accessible in numerical simulations. The model development is followed by a comprehensive mathematical analysis of the equilibrium and propagating 1-dimensional solid-liquid interfaces for different density-phase couplings. It is shown, that the fluid flow decelerates/accelerates the solidification front in case of shrinkage/expansion of the solid compared to the case when no density contrast is present between the phases. Furthermore, such a free energy construction is proposed, in which the equilibrium planar phase-field interface is independent from the density-phase coupling, and the equilibrium interface represents an exact propagating planar interface solution of the quasi-incompressible hydrodynamic equations. Our results are in excellent agreement with previous theoretical predictions.