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E. O. Fetisov et al.
an environment, depending on the specific system’s solubility, where the clusters
interact with each other requiring a solution theory that may be highly non-ideal.
Below the triple point, there is also the added complication that a nucleating system
will pass through metastable amorphous states as well as various crystalline polymorphs leading to the final stable crystalline state. Characterizing these nucleation
pathways requires careful interpretation of both experiment and theory to accurately
quantify the mechanisms and rates. We will show in this review that the cluster distribution function underlying the nucleation process is very sensitive to the potentials
of mean force between the nucleating species. To this end, we will provide a formal correspondence between dynamical nucleation theory (DNT), CNT, molecular
simulations, interfacial surface energies, and experimental chemical potentials.
Here we will focus our attention on the nucleation of CaCO 3 to showcase the theoretical and computational framework. The nucleation of CaCO 3 from solution has
been studied using a variety of experimental techniques as well as simulations, whose
interpretation has generated spirited debates in the literature concerning the precise
molecular picture of the nucleation process [3, 4]. Specifically, several authors have
pointed to “non-classical” pathways where small clusters are thought to form without
having to overcome the “classical” barrier of formation. Said differently, the small
CaCO 3 clusters formed spontaneously without any nucleation barrier. CNT permits
the existence of small clusters in sub-saturated solutions as the resting or preferred
state, i.e., the concentration of such a small cluster can exceed the monomer concentration. As the solution concentration is increased, not only does the monomer
ion concentration increase but the concentration of clusters increases too. Thus, the
cluster size distribution may shift toward larger sizes making their detection as well
as the likelihood of nucleation more probable. But, as long as the solution is below
its saturation point (with respect to any polymorphic phase, whether amorphous or
crystalline), the cluster distribution function will still decrease as a function of size
as the critical size is approached. The cluster size distribution does not need to be
monotonic even for sub-saturated phases because small clusters may be dominant
(e.g., Coulomb interactions strongly favor ion pairing, but are less important for
the clustering of two ion pairs). The cluster size distribution only becomes monotonic if the solution concentration is sufficiently supersaturated to allow for spinodal
decomposition. Therefore, even though the fact that small clusters (amorphous or
otherwise) may be detected in measurements or simulations does not mean they
are “non-classical”—there can still be a nucleation barrier that the clusters must
overcome before they can increase in size and crystallize. It is essential to fully characterize the nucleation barrier (i.e., reversible work of cluster formation) before the
nucleation can be called “non-classical.” For example, in the ion-induced form of
CNT pre-critical clusters are stable in solution, i.e. they lie at a pre-critical minimum
on the reversible work of formation, all the while the nucleation pathway beyond these
small clusters being entirely “classical.” A similar situation arises for ion-induced
vapor-to-liquid nucleation [5, 6]. The sub-saturated (superheated) vapor phase of
hydrogen fluoride [7, 8] and the sub-saturated and saturated solution phases of alcohols in non-polar solvents [9–11] are also known to contain hydrogen-bonded chains
that bear striking resemblance to the liquid-like ionic polymers observed for CaCO 3
E. O. Fetisov et al.
an environment, depending on the specific system’s solubility, where the clusters
interact with each other requiring a solution theory that may be highly non-ideal.
Below the triple point, there is also the added complication that a nucleating system
will pass through metastable amorphous states as well as various crystalline polymorphs leading to the final stable crystalline state. Characterizing these nucleation
pathways requires careful interpretation of both experiment and theory to accurately
quantify the mechanisms and rates. We will show in this review that the cluster distribution function underlying the nucleation process is very sensitive to the potentials
of mean force between the nucleating species. To this end, we will provide a formal correspondence between dynamical nucleation theory (DNT), CNT, molecular
simulations, interfacial surface energies, and experimental chemical potentials.
Here we will focus our attention on the nucleation of CaCO 3 to showcase the theoretical and computational framework. The nucleation of CaCO 3 from solution has
been studied using a variety of experimental techniques as well as simulations, whose
interpretation has generated spirited debates in the literature concerning the precise
molecular picture of the nucleation process [3, 4]. Specifically, several authors have
pointed to “non-classical” pathways where small clusters are thought to form without
having to overcome the “classical” barrier of formation. Said differently, the small
CaCO 3 clusters formed spontaneously without any nucleation barrier. CNT permits
the existence of small clusters in sub-saturated solutions as the resting or preferred
state, i.e., the concentration of such a small cluster can exceed the monomer concentration. As the solution concentration is increased, not only does the monomer
ion concentration increase but the concentration of clusters increases too. Thus, the
cluster size distribution may shift toward larger sizes making their detection as well
as the likelihood of nucleation more probable. But, as long as the solution is below
its saturation point (with respect to any polymorphic phase, whether amorphous or
crystalline), the cluster distribution function will still decrease as a function of size
as the critical size is approached. The cluster size distribution does not need to be
monotonic even for sub-saturated phases because small clusters may be dominant
(e.g., Coulomb interactions strongly favor ion pairing, but are less important for
the clustering of two ion pairs). The cluster size distribution only becomes monotonic if the solution concentration is sufficiently supersaturated to allow for spinodal
decomposition. Therefore, even though the fact that small clusters (amorphous or
otherwise) may be detected in measurements or simulations does not mean they
are “non-classical”—there can still be a nucleation barrier that the clusters must
overcome before they can increase in size and crystallize. It is essential to fully characterize the nucleation barrier (i.e., reversible work of cluster formation) before the
nucleation can be called “non-classical.” For example, in the ion-induced form of
CNT pre-critical clusters are stable in solution, i.e. they lie at a pre-critical minimum
on the reversible work of formation, all the while the nucleation pathway beyond these
small clusters being entirely “classical.” A similar situation arises for ion-induced
vapor-to-liquid nucleation [5, 6]. The sub-saturated (superheated) vapor phase of
hydrogen fluoride [7, 8] and the sub-saturated and saturated solution phases of alcohols in non-polar solvents [9–11] are also known to contain hydrogen-bonded chains
that bear striking resemblance to the liquid-like ionic polymers observed for CaCO 3