A Mathematical Analysis of Directional Solidification of Aqueous Solutions

Abstract

Horizontal directional solidification techniques have been broadly utilized for the freezing of biological matter under conditions in which the freezing rate during solidification must be controlled and known. Directional solidification is used for diverse applications such as fundamental research on freezing of biological materials, cryopreservation of biological matter, and tissue engineering. This study is motivated by our intent to use directional solidification as a simplified model for the study of three-dimensional (3D) cryoprinting. In evaluating directional solidification in the context of 3D cryoprinting, we realized that current mathematical models of directional solidification are not adequately representative for this purpose, because they are simplified and one-dimensional (1D). Here, we introduce an experimentally verified and more representative two-dimensional (2D) mathematical model of directional solidification that can aid in the fundamental study of freezing of biological matter, in particular during 3D cryoprinting. The mathematical model was used to develop correlations between the freezing rates that a layer of an aqueous solution experiences during directional solidification and the various design parameters such as thickness of the sample and temperature gradients in the substrate. Results show that the freezing rates can be higher than those suggested by the previously used simplified 1D mathematical models. The results can be used for developing simplified models of 3D cryoprinting. In addition, the results suggest that many experimental studies on directional solidification of aqueous solutions and biological matter may require readjustment of analysis, in view of these findings.