106
E. O. Fetisov et al.
2.2 Determining Cluster Size Distributions
One of the most fundamental microscopic properties of a nucleating system is the
cluster size distribution function. For a binary system of ions, this equilibrium distribution can be readily related to the reversible work of formation of the corresponding
binary clusters [26]:
N
EQ
(i 1 ,i 2 ) = N monomer exp
−
W
rev
(i 1 ,i 2 )
RT
,
(1)
where N
EQ
(i 1 ,i 2 ) is the number of clusters containing i 1 + i 2 ions, N monomer is the average
number of free ions equal to (N (1,0) + N (0,1) )/2, R is the gas constant, and T is the
absolute temperature. In this formulation, the maximum in W
rev with respect to size
corresponds to the critical nucleus and various properties, i.e., nucleation rate, can
be inferred from it. Therefore, it is crucial to be able to obtain the equilibrium size
distribution in a molecular simulation and ensure that it is statistically converged in
order to understand and explain the initial stages of nucleation. The main challenge
comes from the fact that nucleation processes inherently constitute rare events, especially in condensed phases. To overcome it, advanced biased simulation techniques
have to be employed, e.g., forward flux sampling [27]. In this work, we combine our
solution model with Monte Carlo (MC) methods specifically designed to sample the
cluster formation and association/dissociation rare events [28, 29].
MC simulations were performed in the canonical ensemble at T = 298 K with
systems consisting of 300 Ca
2+ cations and 300 CO
2−
3 anions, where each ion is
represented as a single interaction site. It should be noted that the pressure–volume
work associated with cluster formation is negligible for solution-phase nucleation
(due to small differences in the partial molar volumes for ions in the solution and ACC
phases), whereas it is an important contribution to cluster formation in the gas phase.
Use of an implicit-solvent description necessitates use of the canonical ensemble.
To efficiently sample ion addition or removal to and from clusters, aggregationvolume-bias Monte Carlo (AVBMC) moves [28] were used in addition to standard
translational moves; the ratio between attempted translational and AVBMC moves
was set to 4:1. Maximum displacements of translational moves were adjusted during
equilibration to yield an acceptance ratio of 40%; AVBMC moves used a distance
criterion of 2.7 Å ≤ r ≤ 3.9 Å (corresponding to the CIP potential well) to define
the bonding region and explored 32 trial locations for the calculation of Rosenbluth
weights [30], which resulted in an acceptance rate of about 9%.
In the current simulations, a simple Stillinger distance-based criterion is used to
define the ion–ion aggregate size distribution [31]. To this end, an ion belongs to a
given aggregate if its distance to at least one other unlike ion in the aggregate is less
than 4.0 Å. Two different sets of simulations, with and without umbrella sampling,
were carried out. On one hand, simulations without additional umbrella potentials
(unbiased) were used to determine the size (numbers of cations and anions) and shape
distributions of small aggregates. On the other hand, simulations with an additional
E. O. Fetisov et al.
2.2 Determining Cluster Size Distributions
One of the most fundamental microscopic properties of a nucleating system is the
cluster size distribution function. For a binary system of ions, this equilibrium distribution can be readily related to the reversible work of formation of the corresponding
binary clusters [26]:
N
EQ
(i 1 ,i 2 ) = N monomer exp
−
W
rev
(i 1 ,i 2 )
RT
,
(1)
where N
EQ
(i 1 ,i 2 ) is the number of clusters containing i 1 + i 2 ions, N monomer is the average
number of free ions equal to (N (1,0) + N (0,1) )/2, R is the gas constant, and T is the
absolute temperature. In this formulation, the maximum in W
rev with respect to size
corresponds to the critical nucleus and various properties, i.e., nucleation rate, can
be inferred from it. Therefore, it is crucial to be able to obtain the equilibrium size
distribution in a molecular simulation and ensure that it is statistically converged in
order to understand and explain the initial stages of nucleation. The main challenge
comes from the fact that nucleation processes inherently constitute rare events, especially in condensed phases. To overcome it, advanced biased simulation techniques
have to be employed, e.g., forward flux sampling [27]. In this work, we combine our
solution model with Monte Carlo (MC) methods specifically designed to sample the
cluster formation and association/dissociation rare events [28, 29].
MC simulations were performed in the canonical ensemble at T = 298 K with
systems consisting of 300 Ca
2+ cations and 300 CO
2−
3 anions, where each ion is
represented as a single interaction site. It should be noted that the pressure–volume
work associated with cluster formation is negligible for solution-phase nucleation
(due to small differences in the partial molar volumes for ions in the solution and ACC
phases), whereas it is an important contribution to cluster formation in the gas phase.
Use of an implicit-solvent description necessitates use of the canonical ensemble.
To efficiently sample ion addition or removal to and from clusters, aggregationvolume-bias Monte Carlo (AVBMC) moves [28] were used in addition to standard
translational moves; the ratio between attempted translational and AVBMC moves
was set to 4:1. Maximum displacements of translational moves were adjusted during
equilibration to yield an acceptance ratio of 40%; AVBMC moves used a distance
criterion of 2.7 Å ≤ r ≤ 3.9 Å (corresponding to the CIP potential well) to define
the bonding region and explored 32 trial locations for the calculation of Rosenbluth
weights [30], which resulted in an acceptance rate of about 9%.
In the current simulations, a simple Stillinger distance-based criterion is used to
define the ion–ion aggregate size distribution [31]. To this end, an ion belongs to a
given aggregate if its distance to at least one other unlike ion in the aggregate is less
than 4.0 Å. Two different sets of simulations, with and without umbrella sampling,
were carried out. On one hand, simulations without additional umbrella potentials
(unbiased) were used to determine the size (numbers of cations and anions) and shape
distributions of small aggregates. On the other hand, simulations with an additional