The Stefan problem in a thermomechanical context with fracture and fluid flow

The classical Stefan problem, concerning mere heat-transfer during solid-liquid phase transition, is here enhanced towards mechanical effects. The Eulerian description at large displacements is used with convective and Zaremba-Jaumann corotational time derivatives, linearized by exploiting the additive Green-Naghdi’s decomposition in (objective) rates. In particular, the liquid phase is a viscoelastic fluid while creep and rupture of the solid phase is considered in the Jeffreys viscoelastic rheology exploiting the phase-field model, exploiting a concept of slightly (so-called “semi”) compressible materials. The $L^1$-theory for the heat equation is adopted for the Stefan problem relaxed by allowing for kinetic superheating/supercooling effects during the solid-liquid phase transition. A rigorous proof of existence of week solutions is provided for an incomplete melting, exploiting a time-discretisation approximation.

The temperature dependence of bismuth structures under high pressure

It is unclear whether there is a liquid-liquid phase transition or not in the bismuth melt at high temperature and high pressure, if so, it is necessary to confirm the boundary of the liquid-liquid phase transition and clarify whether it is a first-order phase transition. Here based on X-ray absorption spectra and simulations, the temperature dependence of bismuth structures has been investigated under different pressures. According to the similarity of characteristic peaks of X-ray absorption near edge structure (XANES) spectra, we have estimated that the possible temperature ranges of liquid-liquid phase transition are 779 K ~ 799 K at 2.74 GPa and 859 K ~ 879 K at 2.78 GPa, 809 K ~ 819 K at 3.38 GPa and 829 K ~ 839 K at 3.39 GPa and 729 K ~ 739 K at 4.78 GPa, respectively. Using ab initio molecular dynamics (AIMD) simulations, we have obtained the stable structures of the bismuth melt at different temperatures and pressures and calculated their electronic structures. Meanwhile, two stable phases (phase III-like and phase IV-like) of bismuth melts are obtained from different initial phases of bismuth solids (phase III and phase IV) under the same condition (3.20 GPa and 800 K). Assuming that the bismuth melt undergoes a phase transition from IV-like to III-like between 809 K and 819 K at 3.38 GPa, the calculated electronic structures are consistent with XANES spectra, which provides a possible explanation for the first-order liquid-liquid phase transition.

Verification of a mathematical model for the solution of the Stefan problem using the mushy layer method

The use of solar energy has limitations due to its periodic availability: solar plants do not operate at night and are ineffective in dull weather. The solution of this problem involves the introduction of energy storage and duplication systems into the conversion loop. Among the energy storage systems, solid–liquid phase transition modules have significant energy, ecologic, and cost advantages. Physical processes in modules of this type are described by a system of non-stationary nonlinear partial differential equations with specific boundary conditions at the phase interface. The verification of a method for solving the Stefan problem for a heat-storage material is presented in this paper. The use of the mushy layer method made it possible to simplify the classical mathematical model of the Stefan problem by reducing it to a nonstationary heat conduction problem with an implicit heat source that takes into account the latent heat of transition. The phase transition is considered to occur in an intermediate zone determined by the solidus and liquidus temperatures rather than in in infinite region. To develop a Python code, use was made of an implicit computational scheme in which the solidus and liquidus temperatures remain constant and are determined in the course of numerical experiments. The physical model chosen for computer simulation and algorithm verification is the process of ice layer formation on a water surface at a constant ambient temperature. The numerical results obtained allow one to determine the temperature fields in the solid and the liquid phase and the position of the phase interface and calculate its advance speed. The algorithm developed was verified by analyzing the classical analytical solution of the Stefan problem for the one-dimensional case at a constant advance speed of the phase interface. The value of the verification coefficient was determined from a numerical solution of a nonlinear equation with the use of special built-in Python functions. Substituting the data for the physical model under consideration into the analytical solution and comparing them with the numerical simulation data obtained with the use of the mushy layer method shows that the results are in close agreement, thus demonstrating the correctness of the computer algorithm developed. These studies will allow one to adapt the Python code developed on the basis of the mushy layer method to the calculation of heat storage systems with a solid-liquid phase transition with account for the features of their geometry, the temperature level, and actual boundary conditions.

Simulating diffusion in the conditions of vapor-liquid phase transition by the molecular dynamics method

The molecular dynamics calculations of diffusion coefficients in binary Lennard-Jones systems have been carried out. The parameters of Lennard-Jones potentials correspond to argon and krypton atoms. The universal dependence of the reduced diffusion coefficient of krypton atoms on density for the homogeneous systems of low and middle densities is found. The deviations of the diffusion coefficients from the universal function are observed for the systems in the vapor – liquid phase transition region. The simulations have shown that almost all krypton atoms have situated inside the liquid phase of argon. Special numerical experiments have shown that the nanodroplets of argon are formed as a result of homogeneous nucleation and then the krypton atoms are captured by these droplets. This phenomenon decreases the diffusion coefficient of krypton atoms greatly.