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Nucleation initiated phase transformation occurs in several natural and man made environments. It is involved in water turning to ice in biological systems,^1,2 solid phase regrowth of amorphous silicon,³ vapor phase growth of semiconductor nanowires,⁴ and templated growth of functional materials.^5,6,7 Therefore, understanding this process in quantitative detail is of broad fundamental and applied interest. Over the past century, classical nucleation theory (CNT) has been widely used to describe the energetics and kinetics of phase transformation.^8,9,10 This phenomenological theory has been successfully applied to a wide array of problems, including pattern formation in epitaxial systems,¹¹ nucleation in polymer material,¹² and formation of optically active oxide nanoparticles.^13,14 One critical feature of this theory is that an energy barrier must be overcome in order to form the stable microscopic nucleus of the new phase in the existing phase. In the case of solid to liquid phase transformation, the physical manifestation of the nucleation barrier is that the solid can be stable to temperatures well above its equilibrium solidification temperature, i.e., superheating.^15,16 Likewise, in liquid to solid transformations, the liquid can be supercooled well below the equilibrium solidification point. This phenomenon was observed three centuries ago in the case of solidification of water to ice by Fahrenheit.¹⁷ Despite the general understanding of phase transformation via CNT, a general shortcoming is its inability to predict certain quantitative features of phase transformation, such as the characteristic undercooling temperature at which the liquid must solidify. Thus, to date, only experimental measurements have been able to establish the magnitude of the characteristic undercooling temperatures, such as those done approximately 60 years ago by Turnbull and Cech^18,19 for a large number of elemental metallic liquids and more recently by Vinet et al.^20,21 These cumulative works as well as more recent measurements have been tabulated in various reviews and books.^10,22

In this work we show that thermodynamics in conjunction with CNT can give useful quantitative information about the energetics of homogeneous nucleation and solidification. The important step that enables this is to account for all energy contributions to the formation of a nucleus, including that arising from the density change during phase transformation. In the case of solidification, the internal Gibbs energy due to the excess volume of the liquid resulting from the density change contributes a substantial component to the total free energy, which, in the past, was typically evaluated only from the interface and volume energies of the two phases involved.^8,9 The resulting modifications in CNT permits prediction of the characteristic undercooling T^* and characteristic undercooling temperature T_UC for elemental metallic liquids, and our results are in very good agreement with the experimental measurements made over the past 60 years. The empirical observation of Turnbull,¹⁸ i.e., that most metals have T^*~0.18  T_MP, is also observed from our theory. This theory also shows that there is a universal character to the minimum nucleation barrier energy and the critical radius. The minimum barrier energy occurs at the temperature T_N~0.27  T_MP for all the elemental liquids investigated, while the temperature dependencies of the barrier energy and critical radius appear similar when expressed as a function of the scaled temperature T_UC/T_MP.