114
E. O. Fetisov et al.
we find that the differences in Gibbs free energies between adjacent-sized clusters
converge to the equilibrium solution chemical potential μ 0 .
Thus, we have established a fundamental connection between the discrete cluster statistical mechanics and continuum nucleation thermodynamics. This can be
seen from the convergence of the size dependent cluster free energy differences, the
interfacial surface energy σ (from the slope), and the solution chemical potential
μ 0 (from the intercept). In order to show that our definition of G i ≡ g i − ig 1 and
G i,i−1 = g i,i−1 − g 1 indeed leads to the correct limit of solution chemical potential when i → ∞ we can expand −((g i,i−1 − g 1 ) in terms of K i (see Eq. (4)). All
terms corresponding to individual ions cancel out and we are left with:
− ((g i,i−1 − g 1 ) = k B T ln
n i λ i−1
n i−1 λ i
− k B T ln
n 1
λ 1
.
(12)
When i → ∞,
λ i−1
λ i
→ 1, and after introducing V (total volume) into both fractions
we get:
− ((g i,i−1 − g 1 ) = k B T ln
N i
N i−1
− k B T ln
N 1
λ 1 V
,
(13)
where the first and second terms on the R.H.S. of Eq. (13) are the chemical potentials
of the specific CaCO 3 polymorph and the CaCO 3 monomer, respectively. Therefore
Eq. (13) reduces to −μ 0 , i.e., minus the equilibrium chemical potential differences
in solution as shown in Eq. (11) [37]. Using the standard Gibbs free energies of
formation for the aqueous Ca
2+ and CO
2−
3 ions and CaCO 3 polymorphs (i.e., ACC,
vaterite, aragonite, and calcite) yields
μ 0 = G rxn = −k B T ln K sp = G f [ions] aq − G f [polymorph]
(14a)
= G f [Ca
2+
] aq + G f [CO
2−
3 ] aq − G f [CaCO 3 polymorph],
(14b)
where it has been assumed that the monomer chemical potential is the sum of the
Gibbs free energies of formation for the ions. Using the K sp values for the various
CaCO 3 polymorphs from the work of Lassin et al. [38] we obtain: ACC G rxn =
8.84 kcal/mol to 10.64 kcal/mol (K sp = 10
−6.4 to 10
−7.7 ), vaterite G rxn = 10.91
kcal/mol (K sp = 10
−7.9 ), aragonite G rxn = 11.48 kcal/mol (K sp = 10
−8.31 ), and
calcite G rxn = 11.72 kcal/mol (K sp = 10
−8.48 ). We will see in what follows that
these choices for reference aqueous chemical potentials yield a natural cluster free
energy scale that provides deeper insight into the polymorphic phase transition of
CaCO 3 from aqueous solution.
In Fig. 6 we show the i-cluster Gibbs free energy differences, referenced to the
aqueous chemical potentials, versus i
−1/3 . These clusters represent metastable states
with respect to their bulk polymorphic crystalline phases. Metastable clusters are
inherently difficult to probe experimentally and computationally. This analysis provides means to obtain the essential nucleation ingredients (interfacial energetics and
solubilities) of the metastable clusters. From the slopes and extrapolated intercepts
E. O. Fetisov et al.
we find that the differences in Gibbs free energies between adjacent-sized clusters
converge to the equilibrium solution chemical potential μ 0 .
Thus, we have established a fundamental connection between the discrete cluster statistical mechanics and continuum nucleation thermodynamics. This can be
seen from the convergence of the size dependent cluster free energy differences, the
interfacial surface energy σ (from the slope), and the solution chemical potential
μ 0 (from the intercept). In order to show that our definition of G i ≡ g i − ig 1 and
G i,i−1 = g i,i−1 − g 1 indeed leads to the correct limit of solution chemical potential when i → ∞ we can expand −((g i,i−1 − g 1 ) in terms of K i (see Eq. (4)). All
terms corresponding to individual ions cancel out and we are left with:
− ((g i,i−1 − g 1 ) = k B T ln
n i λ i−1
n i−1 λ i
− k B T ln
n 1
λ 1
.
(12)
When i → ∞,
λ i−1
λ i
→ 1, and after introducing V (total volume) into both fractions
we get:
− ((g i,i−1 − g 1 ) = k B T ln
N i
N i−1
− k B T ln
N 1
λ 1 V
,
(13)
where the first and second terms on the R.H.S. of Eq. (13) are the chemical potentials
of the specific CaCO 3 polymorph and the CaCO 3 monomer, respectively. Therefore
Eq. (13) reduces to −μ 0 , i.e., minus the equilibrium chemical potential differences
in solution as shown in Eq. (11) [37]. Using the standard Gibbs free energies of
formation for the aqueous Ca
2+ and CO
2−
3 ions and CaCO 3 polymorphs (i.e., ACC,
vaterite, aragonite, and calcite) yields
μ 0 = G rxn = −k B T ln K sp = G f [ions] aq − G f [polymorph]
(14a)
= G f [Ca
2+
] aq + G f [CO
2−
3 ] aq − G f [CaCO 3 polymorph],
(14b)
where it has been assumed that the monomer chemical potential is the sum of the
Gibbs free energies of formation for the ions. Using the K sp values for the various
CaCO 3 polymorphs from the work of Lassin et al. [38] we obtain: ACC G rxn =
8.84 kcal/mol to 10.64 kcal/mol (K sp = 10
−6.4 to 10
−7.7 ), vaterite G rxn = 10.91
kcal/mol (K sp = 10
−7.9 ), aragonite G rxn = 11.48 kcal/mol (K sp = 10
−8.31 ), and
calcite G rxn = 11.72 kcal/mol (K sp = 10
−8.48 ). We will see in what follows that
these choices for reference aqueous chemical potentials yield a natural cluster free
energy scale that provides deeper insight into the polymorphic phase transition of
CaCO 3 from aqueous solution.
In Fig. 6 we show the i-cluster Gibbs free energy differences, referenced to the
aqueous chemical potentials, versus i
−1/3 . These clusters represent metastable states
with respect to their bulk polymorphic crystalline phases. Metastable clusters are
inherently difficult to probe experimentally and computationally. This analysis provides means to obtain the essential nucleation ingredients (interfacial energetics and
solubilities) of the metastable clusters. From the slopes and extrapolated intercepts