The effect of temperature gradients on stress development during cryopreservation via vitrification

This study addresses the problem of thermal stress development in bulky specimens during cryopreservation via vitrification (vitreous means glassy in Latin). While this study is a part of an ongoing effort to associate the developing mechanical stress with the relevant physical properties of the cryopreserved media and to its the thermal history, the current paper focuses exclusively on the role of temperature gradients. Temperature gradients arise due to the high cooling rates necessary to facilitate vitrification; the resulting nonuniform temperature distribution leads to differential thermal strain, possibly resulting in cracking. The cooling rate is assumed constant on the outer surface in this study, and the material properties are assumed constant. It is demonstrated that under these assumptions, mechanical stress develops only when the temperature distribution in the specimen approaches thermal equilibrium at a cryogenic storage temperature. It is shown that the maximum possible stresses for a given cooling rate can be computed with a simple thermoelastic analysis; these stresses are associated with cooling to sufficiently low temperatures, and are independent of the variation of viscosity with temperature. Analytic estimates for these stresses are obtained for several idealized shapes, while finite element analysis is used to determine stresses for geometries used in cryopreservation practice. Stresses that develop under a wider range of storage temperatures are also studied with finite element analysis, and the results are summarized with suitable normalizations. It is found that no stresses arise if cooling ceases above the set temperature, which defines the transition from viscous-dominated to elastic-dominated behavior; the set temperature is determined principally by the dependency of viscosity upon temperature. Strategies for rapidly reaching low temperatures and avoiding high stresses are inferred from the results.