Novel Paradigms in Nonclassical Nucleation Theory

Abstract

The last 25 years have seen an explosion of interest in the subject of nucleation driven by new experimental techniques and computer simulation methods. The theoretical community has struggled to keep pace with this onslaught. One of the main reasons is an adherence to the paradigm of classical nucleation theory—a theory more than 80 years old and yet still the dominant conceptual framework. In this chapter, a fundamentally new approach to the theoretical description of nucleation is outlined. It is based on the understanding that nucleation is a nonequilibrium process for which equilibrium approximations are of limited applicability. Nevertheless, the theory reproduces classical nucleation theory (CNT) in appropriate limits while offering the flexibility to fully describe with the new paradigms arising from the experiment and simulation.

Appendix: Proof of the First Nucleation Theorem

To prove the first nucleation theorem, we first recall some elementary—but exact—results from finite-temperature density functional theory (Lutsko 2010). From the theory behind DFT, we learn that the statistical mechanics of a single-component classical system can be entirely formulated in terms of the local number density $\rho \left (\mathbf{r}\right )$. There exists a functional of this density of the form $\varPsi \left [\rho \right ] = F\left [\rho \right ] -\mu N$ such that the equilibrium density distribution, $\rho _{eq}\left (\mathbf{r}\right )$, minimizes $\varPsi \left [\rho \right ]$, e.g.,

$$\displaystyle{ 0 = \left. \frac{\delta \varPsi \left [\rho \right ]} {\delta \rho \left (\mathbf{r}\right )}\right \vert _{\rho _{eq}\left (\mathbf{r}\right )} }$$

(2.25)

In the expression for Ψ, the first term F has no dependence on the chemical potential, while in the second term, it is important to remember that N is simply the integral of the density over all space, so that the only independent quantities are the local density, the chemical potential, and the temperature (which does not change and so is not indicated). Furthermore, the free energy is simply the functional evaluated at the equilibrium density $\varOmega _{eq} =\varPsi \left [\rho _{eq}\right ]$. Although it is not an equilibrium state, this is assumed to hold true as well for the critical cluster which has density $\rho _{c}\left (\mathbf{r}\right )$ and free energy $\varOmega _{c} =\varPsi \left [\rho _{c}\right ]$. Hence, we have that

$$\displaystyle{ \frac{d\left (\varOmega _{c} -\varOmega _{eq}\right )} {d\mu } =\int \left (\!\left. \frac{\delta \varPsi \left [\rho \right ]} {\delta \rho \left (\mathbf{r}\right )}\right \vert _{\rho _{c}\left (\mathbf{r}\right )}\frac{\partial \rho _{c}\left (\mathbf{r}\right )} {\partial \mu } -\left. \frac{\delta \varPsi \left [\rho \right ]} {\delta \rho \left (\mathbf{r}\right )}\right \vert _{\rho _{eq}\left (\mathbf{r}\right )}\frac{\partial \rho _{eq}\left (\mathbf{r}\right )} {\partial \mu } \!\right )d\mathbf{r} -\left (N_{c}-N_{eq}\right ) }$$

(2.26)

Since the left-hand side demands the total derivative with respect to the chemical potential, there are two contributions: the first due to the change in the equilibrium density distribution when the chemical potential changes and the second due to the explicit factor of chemical potential in the functional Ψ. Now, since both $\rho _{c}\left (\mathbf{r}\right )$ and $\rho _{eq}\left (\mathbf{r}\right )$ satisfy Eq. (2.25), the first term is identically zero leaving

$$\displaystyle{ \frac{d\left (\varOmega _{c} -\varOmega _{eq}\right )} {d\mu } = -\left (N_{c} - N_{eq}\right ) }$$

(2.27)

which is the desired result.