The Statistical Mechanics of Solution-Phase Nucleation …
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that within the framework of a reduced model we can, in principle, converge the sampling of the cluster distribution function that is a central molecular scale descriptor in
our theory of solution-phase nucleation. However, exploration of rates of nucleation
to final crystal polymorphs requires extensions to this approach where the water
molecules are treated explicitly. Indeed, in a recent study Jiang et al. have computed
rates of nucleation of NaCl using advanced sampling techniques, e.g., forward flux
sampling in conjunction with an all-atom simulation based on a MM representation of
interactions [27]. Advantages to the NaCl system afford the use of simulation supercells large enough to access the experimental solubility in addition to being small
enough to satisfy the computational demands of the forward-flux sampling methods
[27]. Nevertheless, the concepts and protocol described here become important when
relying on representations of interaction based in quantum mechanics to describe the
emergent interactions for nucleation where all-atom representations in conjunction
with enhanced sampling methods are still out of reach for realistic systems. One of
the important results from the model presented here is prediction (and experimental
verification) that nucleation of CaCO 3 proceeds through a dense liquid-like phase
characterized by a smaller interfacial surface energy before ultimately transforming
to a stable solid polymorph of CaCO 3 [18]. To the extent that this is a general pathway
for other nucleating salts requires further investigation focusing on the consistencies
of the approach presented here with all-atom approaches of Jiang et al. [27]. Important future research will be focused on the circumstances where the reduced model
presented here fails and an all-atom representation is necessary, e.g., when chemical
bonds are broken or made or excited electronic states are involved. This will, in
part, be predicated on the relative stability of the dense liquid phase compared to the
desired solid structure.
Having outlined the statistical mechanical formalism of homogeneous nucleation
and its connection to molecular simulation, in detail, we can address some recent
concerns raised by Gebauer et al. [19]. First and foremost, there is a clear and welldefined difference between reversible work of formation and absolute free energy
of clusters and these terms are not interchangeable. Gebauer et al., however, indeed
use them interchangeably, which in turn leads to an overall confusion of how species
with equilibrium constants of formation greater than unity (or consequently, negative
absolute free energies) can be unstable in molecular simulations and how reversible
work of formation can depend on concentration. This confusion probably stemmed
from the work of Wallace et al. [35] as mentioned previously. We hope the interested
reader is now able to better understand the statistical mechanical rigor required to
discriminate between cluster free energetics and the reversible work of cluster formation. Secondly, a lot of discussion in the aforementioned perspective is dedicated
to standard states. Fundamentally, thermodynamics (including the CNT formulation) does not require a standard state nor do species in an experimental system or a
simulation box know anything about standard states. The power of molecular simulations (and statistical mechanics underlying them) is the ability to yield macroscopic
thermodynamic properties from microscopic sampling of both enthalpic (intra- and
intermolecular interactions) and entropic (volume) properties of a molecular system.
Both of these terms are correctly accounted for in our simulations. CNT, even with its
117
that within the framework of a reduced model we can, in principle, converge the sampling of the cluster distribution function that is a central molecular scale descriptor in
our theory of solution-phase nucleation. However, exploration of rates of nucleation
to final crystal polymorphs requires extensions to this approach where the water
molecules are treated explicitly. Indeed, in a recent study Jiang et al. have computed
rates of nucleation of NaCl using advanced sampling techniques, e.g., forward flux
sampling in conjunction with an all-atom simulation based on a MM representation of
interactions [27]. Advantages to the NaCl system afford the use of simulation supercells large enough to access the experimental solubility in addition to being small
enough to satisfy the computational demands of the forward-flux sampling methods
[27]. Nevertheless, the concepts and protocol described here become important when
relying on representations of interaction based in quantum mechanics to describe the
emergent interactions for nucleation where all-atom representations in conjunction
with enhanced sampling methods are still out of reach for realistic systems. One of
the important results from the model presented here is prediction (and experimental
verification) that nucleation of CaCO 3 proceeds through a dense liquid-like phase
characterized by a smaller interfacial surface energy before ultimately transforming
to a stable solid polymorph of CaCO 3 [18]. To the extent that this is a general pathway
for other nucleating salts requires further investigation focusing on the consistencies
of the approach presented here with all-atom approaches of Jiang et al. [27]. Important future research will be focused on the circumstances where the reduced model
presented here fails and an all-atom representation is necessary, e.g., when chemical
bonds are broken or made or excited electronic states are involved. This will, in
part, be predicated on the relative stability of the dense liquid phase compared to the
desired solid structure.
Having outlined the statistical mechanical formalism of homogeneous nucleation
and its connection to molecular simulation, in detail, we can address some recent
concerns raised by Gebauer et al. [19]. First and foremost, there is a clear and welldefined difference between reversible work of formation and absolute free energy
of clusters and these terms are not interchangeable. Gebauer et al., however, indeed
use them interchangeably, which in turn leads to an overall confusion of how species
with equilibrium constants of formation greater than unity (or consequently, negative
absolute free energies) can be unstable in molecular simulations and how reversible
work of formation can depend on concentration. This confusion probably stemmed
from the work of Wallace et al. [35] as mentioned previously. We hope the interested
reader is now able to better understand the statistical mechanical rigor required to
discriminate between cluster free energetics and the reversible work of cluster formation. Secondly, a lot of discussion in the aforementioned perspective is dedicated
to standard states. Fundamentally, thermodynamics (including the CNT formulation) does not require a standard state nor do species in an experimental system or a
simulation box know anything about standard states. The power of molecular simulations (and statistical mechanics underlying them) is the ability to yield macroscopic
thermodynamic properties from microscopic sampling of both enthalpic (intra- and
intermolecular interactions) and entropic (volume) properties of a molecular system.
Both of these terms are correctly accounted for in our simulations. CNT, even with its