116
E. O. Fetisov et al.
83.0 ergs/cm
2 inferred from a calcite crystallization experiment [43]. Comparing the
DFT and MM PMFs from Fig. 1, one can see that the deeper MM CIP relative to the
DFT CIP is reflected in the ∼3 kcal/mol free energy difference between the MM and
DFT results in Fig. 6. The agreement between the DFT ACC model intercept and the
range of experimental ACC intercepts validates and underscores the fidelity of the
DFT ACC model compared to the MM ACC model. The MM ACC model appears
to overbind the CaCO 3 monomer to the cluster by ∼1.5 to 3.3 kcal/mol relative to
the range of ACC phases; however, it does show convergence close (∼0.4 kcal/mol)
to the calcite phase.
As the small clusters approach their ultimate final crystalline polymorph, they
pass through various distorted or amorphous configurations, including the presence
of trapped water molecules. Since different direct pathways (e.g., ions directly to
calcite) may not exist for each stable crystalline CaCO 3 polymorph, characterizing
their free energetic progress along various traversing nucleation pathways leading to
those polymorphs provides a means of understanding the mechanism of crystallization. Furthermore, when the cluster free energy differences are plotted this way, one
can see: (1) the natural cluster free energy difference scale, (2) how they yield interfacial surface energies and converge to the bulk chemical potentials to yield K sp values,
(3) that the cluster free energetics of the polymorphs lie close to one another (or may
actually overlap), (4) that they are extremely sensitive to the underlying PMF shown
in Fig. 1 or, more generally, the cluster free energy differences shown in Fig. 6, and
(5) that they may be used as progress variables for nucleation pathways to crystallization. Importantly, it should be stressed that this analysis embraces and is consistent
with a “classical” picture of nucleation albeit for a complicated unary polymorphic
system where the water is playing an essential role.
Taken together, these results are interpreted to mean that calcium and carbonate ions are able to explore lower nucleation barriers by evolving first through the
hydrated ACC polymorph, through the anhydrous ACC polymorph, then through the
intermediate vaterite polymorph prior to forming the calcite polymorph. In this indirect way, the ions do not have to surmount such a large nucleation barrier associated
with the direct pathway to calcite, but can do so gradually by exploiting other lower
barrier pathways. It should be remembered that the DFT ACC model is a reduced
mean field model that includes water implicitly and thus cannot describe the anhydrous polymorph and the crystalline CaCO 3 polymorphs where the water molecules
are absent. We are currently working toward constructing interaction potentials where
the hydration/dehydration of the Ca
2+ and CO
2−
3 ions occurs more naturally allowing
the ions to traverse the various CaCO 3 polymorphs as needed.
4 Summary and Future Overview
In summary, we have provided a theoretical and computational framework to connect
the molecular details to macroscopic outcomes germane to the initial stages of nucleation in saturated salt solutions. The advantage to the approach presented herein is
E. O. Fetisov et al.
83.0 ergs/cm
2 inferred from a calcite crystallization experiment [43]. Comparing the
DFT and MM PMFs from Fig. 1, one can see that the deeper MM CIP relative to the
DFT CIP is reflected in the ∼3 kcal/mol free energy difference between the MM and
DFT results in Fig. 6. The agreement between the DFT ACC model intercept and the
range of experimental ACC intercepts validates and underscores the fidelity of the
DFT ACC model compared to the MM ACC model. The MM ACC model appears
to overbind the CaCO 3 monomer to the cluster by ∼1.5 to 3.3 kcal/mol relative to
the range of ACC phases; however, it does show convergence close (∼0.4 kcal/mol)
to the calcite phase.
As the small clusters approach their ultimate final crystalline polymorph, they
pass through various distorted or amorphous configurations, including the presence
of trapped water molecules. Since different direct pathways (e.g., ions directly to
calcite) may not exist for each stable crystalline CaCO 3 polymorph, characterizing
their free energetic progress along various traversing nucleation pathways leading to
those polymorphs provides a means of understanding the mechanism of crystallization. Furthermore, when the cluster free energy differences are plotted this way, one
can see: (1) the natural cluster free energy difference scale, (2) how they yield interfacial surface energies and converge to the bulk chemical potentials to yield K sp values,
(3) that the cluster free energetics of the polymorphs lie close to one another (or may
actually overlap), (4) that they are extremely sensitive to the underlying PMF shown
in Fig. 1 or, more generally, the cluster free energy differences shown in Fig. 6, and
(5) that they may be used as progress variables for nucleation pathways to crystallization. Importantly, it should be stressed that this analysis embraces and is consistent
with a “classical” picture of nucleation albeit for a complicated unary polymorphic
system where the water is playing an essential role.
Taken together, these results are interpreted to mean that calcium and carbonate ions are able to explore lower nucleation barriers by evolving first through the
hydrated ACC polymorph, through the anhydrous ACC polymorph, then through the
intermediate vaterite polymorph prior to forming the calcite polymorph. In this indirect way, the ions do not have to surmount such a large nucleation barrier associated
with the direct pathway to calcite, but can do so gradually by exploiting other lower
barrier pathways. It should be remembered that the DFT ACC model is a reduced
mean field model that includes water implicitly and thus cannot describe the anhydrous polymorph and the crystalline CaCO 3 polymorphs where the water molecules
are absent. We are currently working toward constructing interaction potentials where
the hydration/dehydration of the Ca
2+ and CO
2−
3 ions occurs more naturally allowing
the ions to traverse the various CaCO 3 polymorphs as needed.
4 Summary and Future Overview
In summary, we have provided a theoretical and computational framework to connect
the molecular details to macroscopic outcomes germane to the initial stages of nucleation in saturated salt solutions. The advantage to the approach presented herein is