Classical nucleation theory of ice nucleation: second-order correction of thermodynamic parameters

Accurate estimate of nucleation rate is crucial for the study of ice nucleation and ice-promoting/anti-freeze strategies. Within the framework of Classical Nucleation Theory (CNT), the estimate of ice nucleation rate is very sensitive to thermodynamic parameters, such as chemical potential difference between water and ice $\Delta \mu$ and ice-water interfacial free energy $\gamma$. However, even today, there are still many contradictions and approximations in the estimating of these thermodynamic parameters, introducing large uncertainty to the estimate of the ice nucleation rate. Herein, starting from the basic concepts, for a general solid-liquid crystallization system, we expand the Gibbs-Thomson (GT) equation to second order, and derive the second-order analytical formulas of $\Delta \mu$, $\gamma$ and nucleation barrier $\Delta G$ with combining molecular dynamics (MD) simulations. These formulas describe well the temperature dependence of these thermodynamic parameters. Our results can provide a method of estimating $\Delta \mu$, $\gamma$ and $\Delta G$.