  Water freezing in particle suspensions widely exists in nature. As a typical
physical system of free boundary problem, the spatiotemporal evolution of the
solid/liquid interface not only origins from phase transformation but also from
permeation flow in front of ice. Physical models have been proposed in previous
efforts to describe the interface dynamic behaviors in unidirectional freezing
of particle suspensions. However, there are several physical parameters
difficult to be determined in previous investigations dedicated to describing
the spatiotemporal evolution in unidirectional freezing of particle
suspensions. Here, based on the fundamental momentum theorem, we propose a
consistent theoretical framework to address the unidirectional freezing process
in the particle suspensions coupled with the effect of water permeation. An
interface undercooling-dependent pushing force exerted on the compacted layer
with a specific formula is derived based on the surface tension. Then a dynamic
compacted layer is considered and analyzed. Numerical solutions of the
nonlinear models reveal the dependence of system dynamics on some typical
physical parameters, particle radius, initial particle concentration in the
suspensions, freezing velocity and so on. The system dynamics are characterized
by interface velocity, interface undercooling and interface recoil as functions
of time. The models allow us to reconsider the formation mechanism of ice
spears in freezing of particle suspensions in a simpler but novel way, with
potential implications for both understanding and controlling not only ice
formation in porous media but also crystallization processes in other complex
systems.
