Solving the classical nucleation theory with respect to the surface energy

An essential parameter of the Classical Nucleation Theory (CNT) is the surface energy between a critical-size nucleus and the ambient phase, sigma. In condensed matter, this parameter cannot be experimentally determined independently of CNT. A common practice to obtain sigma is to assume a model for its temperature-dependence and perform a regression of the CNT equation against experimental nucleation data. The drawback of this practice is that assuming the temperature-dependence of sigma adds a bias to the analysis. Nonetheless, this practice is common because an analytical solution of the Classical Nucleation Theory with respect to sigma is not possible considering common expressions of this theory. In this article, a general numerical solution to this problem using the Lambert W function is proposed, tested, and compared with typical regression methods. The major advantage of the proposed method is that there is no need to assume a model for the temperature-dependence of sigma.