  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.
