With growing food scarcity and high demands for vaccine storage, advancing spray freeze-drying technology has never been more important for prolonging shelf life of biological and pharmaceutical materials. Particularly, the estimation of nucleation behaviour for both pure substances and binary mixtures has become vital to the optimal thermal design and implementation of spay freeze-drying technology. Notwithstanding that past nucleation frameworks could predict nucleation rate and temperature of droplet solidification, few of them considered extreme surrounding conditions, such as very low ambient temperature below −60 degree Celsius. These environments, however, play a significant role in ascertaining the preservation and storage of chemical and pharmaceutical products, e.g., vaccines and protein drugs. It is therefore of great interest to establish accurate and reliable mathematical framework on simulating nucleation during droplet solidification subjected to ultra-cold conditions. This paper develops a semi-analytical heterogeneous nucleation model and anticipates nucleation phenomena of a suspended droplet under ultra-cold environment. Nucleation temperatures calculated from the presented model are validated against a set of experiments on single suspended droplets for a wide array of ambient temperatures from −20 until −160 degree Celsius. Both pure water and 20% w/w sucrose aqueous solution are examined for these droplets. Cumulative probability distributions of nucleation for both types of droplets over nucleation temperatures are also presented and comparisons are made between the model results and recent experimental data from literatures. Our preliminary findings demonstrate that drastic changes in nucleation temperature for ultra-cold surroundings are the aftermath of alterations in interfacial surface tension. Conventionally, the inter-facial surface tension is defined as a function of supercooling degree only, which fails as surrounding temperature is prescribed below −40 degree Celsius. In this study, the interfacial surface tension is linearly optimized using error minimization with experimental data fit, such that it substantially relates to both the supercooling degree and surrounding temperature under a given environment for pure water. As for sucrose aqueous solution (i.e., an example of binary mixtures), their solute concentration is also a dependent variable of interfacial surface tension. The results indicate that our proposed framework is capable of predicting heterogeneous nucleation in a droplet filled with either pure material or binary mixture. Development of this nucleation model for spray freeze-drying can expedite manufacturing process and reduce expenses in handling, transportation and storage of biological products, thus improving the shelf life of pharmaceuticals and availability of foods at large. Our model can be extended on other pure materials and binary mixtures, which will further be used to facilitate the design and implementation of spray freeze-drying technology for preserving and storing more chemicals and pharmaceutical excipients.

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