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 project aims to 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 interfacial 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

In this project we developed a robust and computational friendly framework to tackle the problem of droplet freezing. The model included a comprehensive treatment of heterogeneous nucleation and dendritic growth and couples the non-equilibrium phase-change process with the quasi-steady perturbation analysis for equilibrium freezing stage and 1-D transient heat conduction models for liquid supercooling and solid subcooling stage.

For heterogeneous nucleation model, the interfacial surface energy, chemical potential and contact angle were found to be very important parameters in accurately estimating the nucleation rates. The nucleation temperatures and times were found to be in good agreement with the experimental results from the literature. The results from the dendritic growth model were also found to be in good agreement with the experimental results. The growth velocity and interface undercooling during the recalescence process took into account the non-equilibrium thermodynamics during the crystal growth process and explained the rapid formation of an opaque mixture of ice and water droplet at the end of recalescence stage as reported in the experiments. 

The use of effective latent heat in the perturbation series solution for the equilibrium freezing stage resulted in an accurate prediction for mass fraction evolution, temperature and freezing times of the droplet. Even though singularity was observed in the perturbation solution near the center of the droplet, it did not affect the prediction of the final freezing times. The freezing time was predicted to be a strong function of the Biot and Stefan number during the equilibrium freezing stage. 

The model presented in the study can be applied to several engineering problems which involve subcoolings of upto 30 Kelvin with confidence.  Applications pertaining to spray freezing, ice accretion, pharmaceuticals, meteorology and freeze drying are of particular relevance to this study. The model can also be successfully used as a subgrid model for more complicated high-fidelity simulations of crystal growth and spray freezing process. A possible extension of this work would be to implement this framework to other geometrical configurations, materials and porous media.

With the ever-increasing engineering and biological applications of rapid droplet solidification on super hydrophobic surfaces, developing accurate dendritic growth models has become vital at lower bulk temperatures. In addition to the thermal diffusion, the interface kinetics and curvature effects become non-negligible at such low temperatures and thus should be taken into account. In this study, we presented a novel crystal growth model coupled with a heterogeneous nucleation model to investigate the dynamics of recalescence stage during droplet freezing. 

A statistical framework was developed for the sensitivity analysis which uses Monte-Carlo method to quantify the influence of self-diffusivity and interface kinetic factor on crystal growth rate, rigorously. Further, the model parameters were optimized using the gradient-based method. The proposed crystal growth model along with the optimized parameters, can reliably simulate linear, and non-linear interface kinetics for metastable water of supercooling up to 30 [K]. Our key findings demonstrated that the dendritic growth rate is a strong function of the type of diffusivity expression, diffusivity parameters, and the interface kinetics factor. These findings accurately captured the recalescence dynamics for the droplet freezing of pure Lennard-Jones liquids and, thus, are of importance to several physicochemical, biological, and engineering applications.