The Statistical Mechanics of Solution-Phase Nucleation …
107
umbrella potential acting on only one specific aggregate (biased) were used to probe
the size and shape distributions of larger aggregates along the nucleation pathway.
For the unbiased set, 16 independent runs were carried out for each concentration.
At least 200,000 MC cycles (where a cycle consists of N = 600 randomly selected
MC moves) were used for equilibration, and at least 1,000,000 MC cycles were
used for the production runs. In the biased case, one specific ion was tagged to
define the nucleating aggregate, and self-adaptive umbrella sampling [29] was used
to flatten the size distribution for this particular cluster, thereby providing access to
reliable statistics for much larger aggregates. Eight independent runs were carried out
consisting of at least 200,000 MC cycles for the refinement of the umbrella potential,
followed by at least 700,000 MC cycles for production runs.
3 Results and Discussion
3.1 The Initial Stages of Nucleation of CaCO 3
Using our reduced model in conjunction with the AVBMC method we are able to
determine fully converged cluster concentrations, n (i 1 ,i 2 ) , at different total CaCO 3
concentrations, including in the “supersaturated” region. The term “supersaturated”
is used loosely here, because the crystalline state is ill-defined for the PMF-based
models ; hence, the saturation limit cannot be determined for this model. The resulting
cluster populations are shown in Fig. 2a where it is clear that the populations are
dominated by free ions (n (1,0) and n (0,1) ) and ion pairs (n (1,1) ), and the concentrations
of all larger species are orders of magnitude smaller. It should be noted that, in
principle, the cluster distribution does not have to be symmetric with respect to the
number of cations and anions, i.e., n (2,1) and n (1,2) can be different. However, in our
simulations we found that statistically all populations are symmetric; therefore, we
are only showing one half of the obtained values. Additionally, the total concentration,
n total , corresponds to the total concentration of Ca
2+ or CO
2−
3 species in a given
simulation and not just the sum of free ions.
We can also compute equilibrium constants of formation of CaCO 3 clusters as
defined by:
K
AVBMC
(i 1 ,i 2 )
=
n (i 1 ,i 2 )
n
i 1
(1,0) n
i 2
(0,1)
.
(2)
As is obvious from Eq. 2, the choice of concentration unit affects the numerical value
of K (i 1 ,i 2 ) because i 1 + i 2 monomeric ions in the denominator form a single aggregate in the numerator. Changing from units of [M] to units of [mM] would decrease
K (i 1 ,i 2 ) by a factor of 10
3(i 1 +i 2 −1) . Hence, one cannot equate K (i 1 ,i 2 ) > 1 with a negative
reversible work of formation, W (i 1 ,i 2 ) = −RT ln (N (i 1 ,i 2 ) /N monomer ), that is computed
from a ratio of the number of species. The computed values of K (i 1 ,i 2 ) (using molar
concentrations) are shown in Fig. 2b. It can be seen that over a broad range of con-
107
umbrella potential acting on only one specific aggregate (biased) were used to probe
the size and shape distributions of larger aggregates along the nucleation pathway.
For the unbiased set, 16 independent runs were carried out for each concentration.
At least 200,000 MC cycles (where a cycle consists of N = 600 randomly selected
MC moves) were used for equilibration, and at least 1,000,000 MC cycles were
used for the production runs. In the biased case, one specific ion was tagged to
define the nucleating aggregate, and self-adaptive umbrella sampling [29] was used
to flatten the size distribution for this particular cluster, thereby providing access to
reliable statistics for much larger aggregates. Eight independent runs were carried out
consisting of at least 200,000 MC cycles for the refinement of the umbrella potential,
followed by at least 700,000 MC cycles for production runs.
3 Results and Discussion
3.1 The Initial Stages of Nucleation of CaCO 3
Using our reduced model in conjunction with the AVBMC method we are able to
determine fully converged cluster concentrations, n (i 1 ,i 2 ) , at different total CaCO 3
concentrations, including in the “supersaturated” region. The term “supersaturated”
is used loosely here, because the crystalline state is ill-defined for the PMF-based
models ; hence, the saturation limit cannot be determined for this model. The resulting
cluster populations are shown in Fig. 2a where it is clear that the populations are
dominated by free ions (n (1,0) and n (0,1) ) and ion pairs (n (1,1) ), and the concentrations
of all larger species are orders of magnitude smaller. It should be noted that, in
principle, the cluster distribution does not have to be symmetric with respect to the
number of cations and anions, i.e., n (2,1) and n (1,2) can be different. However, in our
simulations we found that statistically all populations are symmetric; therefore, we
are only showing one half of the obtained values. Additionally, the total concentration,
n total , corresponds to the total concentration of Ca
2+ or CO
2−
3 species in a given
simulation and not just the sum of free ions.
We can also compute equilibrium constants of formation of CaCO 3 clusters as
defined by:
K
AVBMC
(i 1 ,i 2 )
=
n (i 1 ,i 2 )
n
i 1
(1,0) n
i 2
(0,1)
.
(2)
As is obvious from Eq. 2, the choice of concentration unit affects the numerical value
of K (i 1 ,i 2 ) because i 1 + i 2 monomeric ions in the denominator form a single aggregate in the numerator. Changing from units of [M] to units of [mM] would decrease
K (i 1 ,i 2 ) by a factor of 10
3(i 1 +i 2 −1) . Hence, one cannot equate K (i 1 ,i 2 ) > 1 with a negative
reversible work of formation, W (i 1 ,i 2 ) = −RT ln (N (i 1 ,i 2 ) /N monomer ), that is computed
from a ratio of the number of species. The computed values of K (i 1 ,i 2 ) (using molar
concentrations) are shown in Fig. 2b. It can be seen that over a broad range of con-