Two-Dimensional Phase-Field Model Applied to Freezing into Supercooled Melt

Cryopreservation of living cells is a necessary part of many medical procedures such as organ transplants and preservation of sperm and oocytes of endangered species. However, there is at least one apparent contradiction between the concept of cryopreservation and experimental findings that cells and tissues can be damaged by the cryopreservation process itself. Successful cryopreservation was made possible by the addition of glycerol as a cryoprotective agent (CPA). A major portion of the damage is due to and occurs during the supercooling of tissues and cells and their environment. Therefore, a detailed understanding of how supercooling impacts biological environments is important to preventing damage to cells and tissues during cryopreservation. Studies of supercooling are complicated due to the inherent instability associated with supercooling and the influence of surface tension as a stabilizing factor and other parameters associated with the liquid-solid phase-change front. The only method that effectively incorporates surface tension and many other physical phenomena associated with supercooling is the phase-field model in which an extra equation is introduced that solves for a parameter, that is, phase-field parameter that varies from 0 (solid) to 1 (liquid) with its value sharply, but smoothly, changing over the freezing front. A dimensionless set of equations for a two-dimensional (2D) domain is used to model this problem. Parameters such as surface tension and interface kinetic coefficients are included in the model in order to capture the physics of this complex problem. In addition, conditions for nucleation are considered and modeled. The numerical results show a dendritic behavior. To ensure the soundness of the solution, the first law of thermodynamics is applied to our domain. Further, our model answers that the total energy is constant within the boundaries of truncation error. With this model, our results are shown to be in agreement with physical measurements of several different types. We therefore conclude that phase-field model is an effective method in analyzing cryopreservation processes.