THE JOURNAL OF CHEMICAL PHYSICS 143, 044505 (2015)
Temperature-dependent solubilities and mean ionic activity coefficients
of alkali halides in water from molecular dynamics simulations
Zoltan Mester and Athanassios Z. Panagiotopoulosa)
Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, USA
(Received 18 May 2015; accepted 6 July 2015; published online 27 July 2015)
The mean ionic activity coefficients of aqueous KCl, NaF, NaI, and NaCl solutions of varying concentrations have been obtained from molecular dynamics simulations following a recently developed
methodology based on gradual insertions of salt molecules [Z. Mester and A. Z. Panagiotopoulos,
J. Chem. Phys. 142, 044507 (2015)]. The non-polarizable ion models of Weerasinghe and Smith
[J. Chem. Phys. 119, 11342 (2003)], Gee et al. [J. Chem. Theory Comput. 7, 1369 (2011)], Reiser
et al. [J. Chem. Phys. 140, 044504 (2014)], and Joung and Cheatham [J. Phys. Chem. B 112, 9020
(2008)] were used along with the extended simple point charge (SPC/E) water model [Berendsen
et al., J. Phys. Chem. 91, 6269 (1987)] in the simulations. In addition to the chemical potentials
in solution used to obtain the activity coefficients, we also calculated the chemical potentials of
salt crystals and used them to obtain the solubility of these alkali halide models in SPC/E water.
The models of Weerasinghe and Smith [J. Chem. Phys. 119, 11342 (2003)] and Gee et al. [J.
Chem. Theory Comput. 7, 1369 (2011)] provide excellent predictions of the mean ionic activity
coefficients at 298.15 K and 1 bar, but significantly underpredict or overpredict the solubilities.
The other two models generally predicted the mean ionic activity coefficients only qualitatively.
With the exception of NaF for which the solubility is significantly overpredicted, the model of
Joung and Cheatham predicts salt solubilities that are approximately 40%-60% of the experimental
values. The models of Reiser et al. [J. Chem. Phys. 140, 044504 (2014)] make good predictions
for the NaCl and NaI solubilities, but significantly underpredict the solubilities for KCl and NaF.
We also tested the transferability of the models to temperatures much higher than were used to
parametrize them by performing simulations for NaCl at 373.15 K and 1 bar, and at 473.15 K
and 15.5 bar. All models overpredict the drop in the values of mean ionic activity coefficients with
increased temperature seen in experiments. The present results, together with earlier calculations
for a number of models for NaCl aqueous solutions at 298.15 K, point to the strong need for
development of improved intermolecular potential models for classical simulations of electrolyte
solutions. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4926840]
I. INTRODUCTION
Aqueous electrolytes are important for industrial,1–3
biological,4–9 and geothermal10,11 applications, but predictions
of their properties using phenomenological or theoretical
methods are frequently of limited general applicability. One
such method is the Debye-Hückel limiting law, which is
exact in the limit of zero concentration but starts becoming
inaccurate at moderate salt concentrations. The extended
Debye-Hückel12 and Davies13 equations are empirical modifications of the Debye-Hückel limiting law, which allow mean
ionic activity coefficients to be calculated for concentrations up to around 0.1 and 1 mol/kg, respectively, for 1:1
electrolytes. To obtain accurate predictions at even higher
concentrations, one must use models (e.g., Refs. 14–17)
which rely on a number of parameters fitted to experimental
data.
The most common measure of thermodynamic nonideality in electrolyte solutions is the mean ionic activity coefficient
a)[email protected]
0021-9606/2015/143(4)/044505/10/$30.00
γ XY (for salt species XY and cation X + and anion Y − when
the salt is dissolved), which quantifies the deviation of the salt
chemical potential from ideal solution behavior. The chemical
potentials for a 1:1 electrolyte in water can be converted to
mean ionic activity coefficients using
β µ XY = β µ†XY + 2 ln mγ XY ,
(1)
µ†XY
where
is the Henry’s law (infinite dilution) standard chemical potential of the salt, β = 1/k BT is the inverse thermal
energy, m is the concentration of salt in terms of moles of solute
per kg of solvent, T is temperature, and kB is Boltzmann’s
constant. The mean ionic activity coefficient is a difficult quantity to obtain from explicit-solvent simulations because of the
challenges in free energy calculations for strongly interacting
aqueous solutions, and also because the infinite dilution reference state is not directly accessible in simulations with finite
systems.
Some previous molecular-based calculations of mean
ionic activity coefficients were for implicit-solvent (so-called
“primitive”) models to avoid sampling problems with explicit
water.18,19 However, explicit-solvent models are expected to
143, 044505-1
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Z. Mester and A. Z. Panagiotopoulos
be more transferrable to different concentrations and temperatures from the ones used in optimizing the model parameters.
Most previous simulation-based calculations of mean ionic
activity coefficients for explicit-solvent models have relied on
obtaining the Henry’s law standard chemical potential µ†XY by
proposing a function for γ XY with several fitting parameters,
and fitting Equation (1) to the simulated salt chemical potential
data with µ†XY acting as an additional fitting parameter. By
contrast, in our prior work,20 we obtained µ†XY by adjusting
its value to make the mean ionic activity coefficient at the
lowest salt concentration we could simulate (0.01 mol/kg)
match the value predicted by the Debye-Hückel limiting law.
Excellent agreement was seen in Ref. 20 with the results of
Moučka et al.21 for the mean ionic activity coefficients of
NaCl using the Joung and Cheatham (JC) model22 for the
ions and the extended simple point charge (SPC/E) model23
for water. An earlier study of Sanz and Vega24 had obtained
activity coefficients for NaCl using the Smith and Dang (SD)
model25 for ion–water interactions, the Born-Huggins-MayerTosi-Fumi (TF) model26–29 for ion–ion interactions, and the
SPC/E model23 for water, but is of lower accuracy than the
more recent studies. There have been no prior explicit-solvent
simulation studies of mean ionic activity coefficients for salts
other than NaCl, to the best of our knowledge, which is
remarkable given the importance of specific-ion effects in
aqueous solutions.30 There have also been no prior explicitsolvent simulations of mean ionic activity coefficients for any
salt at elevated temperatures; the temperature dependence of
the activity coefficients is connected to the excess partial molar
enthalpies through standard thermodynamic expressions.31
The solubility of a salt in water is easily obtained experimentally and is available for many components at different
temperatures; however, its accurate determination from simulation requires a match of solution to the crystal chemical
potential, which is challenging because of the low slope of
the solution chemical potential with respect to concentration.
There have been several explicit-solvent simulation studies of
the solubilities for a variety of ion and water model combinations. These include Ferrario et al.32 on KF, Sanz and Vega
on NaCl and KF,24 Aragones et al.,33 Paluch et al.,34,35 Lísal
et al.,36 Moučka et al.,37 Moučka et al.,38 and Mester and
Panagiotopoulos on NaCl,20 and Moučka et al.39 on a variety
of alkali halides. The temperature dependence of the solubility
is related to the difference of partial molar enthalpy for the salt
in solution and its crystal state; once again, there have been no
prior simulation studies, with the exception of Sanz and Vega24
who performed this study for KF, of how well ion and water
models reproduce the temperature effect on salt solubilities.
Moučka et al.40 and Jiang et al.41 calculated the solubility and mean ionic activity coefficients of NaCl using the
polarizable BK342 and SWM4-DP43 water models along with
the polarizable AH electrolyte models (Ref. 44 for electrolyte
parameters obtained for the BK3 water model and Ref. 43
for electrolyte parameters obtained for the SWM4-DP water
model). While both studies make good predictions for mean
ionic activity coefficients using BK3 water, the computational
efficiency of using non-polarizable models still makes them
useful tools to study the solution behavior of salts. Jiang et al.41
used the same method for calculating the mean ionic activity
J. Chem. Phys. 143, 044505 (2015)
coefficients as Mester and Panagiotopoulos20 and the current
study, and saw computational times that were about a factor of
10 longer.
In order to extend the range of components and temperatures for which data on activity coefficients and solubilities are
available from atomistic models, we study here the solubilities
and mean ionic activity coefficients of KCl, NaF, NaI, and NaCl
using molecular dynamics simulations. We refine the methodology of our prior work,20 which involves gradually turning
on the interactions of an anion-cation pair inserted into the
solution. The solubilities are determined by finding the concentrations where the chemical potentials of the salts in solution
equal that of the solid crystals. The solid chemical potentials
are obtained using the Einstein molecule approach of Vega
and Noya.45 We compare the results for solubilities and mean
ionic activity coefficients predicted by the electrolyte models
of Weerasinghe and Smith,46 Gee et al.,47 Reiser et al.,48 and
Joung and Cheatham22 in SPC/E water23 at a temperature
of 298.15 K and pressure of 1 bar to experiments—of these
models, only the activity coefficients of the Joung-Cheatham
model for NaCl have been previously studied.20,39 To test the
transferability of the models to other temperatures, we also
perform mean ionic activity coefficient and solubility calculations using all three models for NaCl at 373.15 K and 1 bar
and 473.15 K and 15.5 bar.
The structure of this paper is as follows. Section II defines
the molecular models used to simulate the water and electrolytes. In Section III, we summarize the methods, originally
described in Ref. 20, for calculating the chemical potentials of
salt in solution, and for converting the salt chemical potentials
into mean ionic activity coefficients. In this section, we also
describe the procedure based on Refs. 45 and 49 for calculating
the chemical potentials of the salt crystals. Section IV contains
the mean ionic activity coefficient and solubility results for
NaCl, NaI, NaF, and KCl obtained using the models of Weerasinghe and Smith,46 Gee et al.,47 Reiser et al.,48 and Joung and
Cheatham22 in SPC/E water.23 Conclusions are presented in
Section V.
II. MOLECULAR MODELS
We used the SPC/E23 water model in combination with the
ion models of Weerasinghe and Smith46 for Na+ and Cl− and
Gee et al.47 for K+, F−, and I− (both abbreviated to “KBFF”
from this point onwards, as these models were derived from
Kirkwood-Buff integrals), Reiser et al.48 (“RDVH”), and JC22
to calculate the chemical potentials of the salts in solution.
Even though there is a large number of more recent models for
water that perform better than SPC/E for many properties of the
pure substance,50 we are constrained by the fact that ion models
need to be specifically developed for use with a given water
model, and most of the prior work on ion model development
has been for the SPC/E or similar models. The intermolecular
interactions comprise of Lennard-Jones (LJ) and Coulombic
terms. The interaction energy of LJ sites is given by
 ( σ ) 12 ( σ ) 6
ij 
ij
−
,
ui j,LJ = 4ϵ i j 
r i j 
 r i j

(2)
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Z. Mester and A. Z. Panagiotopoulos
J. Chem. Phys. 143, 044505 (2015)
where r i j is the distance between sites i and j, ϵ i j is the welldepth, and σi j is the size. The Coulombic interaction is given
by
ui j,Coul =
qi q j
,
4πκ 0r i j
(3)
where qi is the charge of site i and κ 0 is the dielectric permittivity of free space. Interaction parameters for unlike atoms
in the RDVH and JC models are computed from the LorentzBerthelot combining rules,
ϵij =
√
ϵ iϵ j,
σi j =
σi + σ j
,
2
(4)
where ϵ i and σi are the LJ well-depth and size of particle i.
Interaction parameters for unlike atoms in the KBFF models
are computed from geometric combining rules,
ϵij =
√
ϵ iϵ j,
√
σi j = s i j σi σ j ,
III. SIMULATION METHODS
A. Chemical potential in solution and mean ionic
activity coefficients
The chemical potential of salt of species XY in solution,
indicated by µ XY , is calculated as a function of temperature T,
pressure P, number of water molecules Nw, number of cations
N X +, and number of anions NY − according to our previous
work,20 which relied on obtaining the free energy change by
gradually increasing the interactions of an added cation-anion
pair with the Nw, N X + − 1, and NY − − 1 system. The chemical
ig
potential is expressed in terms of an ideal µ XY and excess part
ex
µ XY ,
ig
β µ XY = β µ XY + β µex
XY .
The ideal part is given by
ig
β µ XY = β µ0X + + β µY0 − + 2 ln
(5)
where s i j = 1 except for the cation-oxygen interactions where
the values are s i j = 0.8 for cation being K and s i j = 0.75 for
cation being Na.
We used values for the Lennard-Jones interaction parameters proposed in the original articles for KBFF,46,47 RDVH,48
and JC22 ion models and SPC/E23 water. The geometry,
location of the charge sites, and values of the charges for
SPC/E water are from Ref. 23. The charges qi are −e for
the anions and e for the cations where e is the value of the
elementary charge. The interaction parameters for the electrolytes and the interaction parameters, charges, and geometry
of the water molecule used in the present study are summarized in Tables I and II, respectively, of the supplementary
material.51
µex
XY =

N XY
,
βP0 ⟨V ⟩
(7)
where P0 = 1 bar is the standard state pressure, V is volume,
N XY = N X + = NY − is the number of salt molecules in the system. The standard molar chemical potentials µ0X + and µY0 −
are taken from the NIST-JANAF thermochemical tables52 and
are summarized in Table III of the supplementary material.51
The excess part is calculated by incrementally increasing the
interactions of the added cation-anion pair with the system
and calculating the free energy change for each increment.
The LJ interactions are increased to their full value before any
attempt is made to increase the Coulombic interactions. This
early introduction of core-repulsion prevents divergence in the
potential due to overlap of particles caused by electrostatic
attractions. Accordingly, the excess part of the chemical potential can be expressed as
µvdW
XY,k +
k
(6)

µCoul
XY,l ,
(8)
l


dV e−β PV dr N X +dr NY −dr Nwe−βU−βΩ(λ k +1)


β µLJ
=
−
ln
,
XY,k
dV e−β PV dr N X +dr NY −dr Nwe−βU −βΩ(λ k )


dV e−β PV dr N X +dr NY −dr Nwe−βU −βΩ(λ=1)−βΥ(φ l+1)


β µCoul
=
−
ln
.
XY,l
dV e−β PV dr N X +dr NY −dr Nwe−βU −βΩ(λ=1)−βΥ(φ l )
U r Nw, r N X +−1, r NY −−1 is the potential of the Nw, N X + − 1,
and NY − − 1 molecule system. The Lennard-Jones potential
of the added cation-anion pair (indicated by indices i and j,
respectively) is given by

Ω r Nw, r N X +, r NY −, λ = λ
uk i,LJ (r k i )
k,i
+λ

uk j,LJ r k j
(11)
k, j
and the Coulombic potential of the added cation-anion pair is
given by
(9)
(10)

uk i,Coul (r k i )
Υ r Nw, r N X +, r NY −, φ = φ
k,i,k, j

+φ
uk j,Coul r k j
k,i,k, j
+ φ ui j,Coul r i j ,
2
(12)
where λ = [0, 1] and φ = [0, 1] are scaling parameters. The
scaling of the Coulombic interactions presented in Eq. (12) is
equivalent to scaling the charges of the added cation-anion pair
by φ. For a full derivation of Eqs. (6)–(12), see our previous
work.20
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Z. Mester and A. Z. Panagiotopoulos
J. Chem. Phys. 143, 044505 (2015)
The incremental excess chemical potentials µLJ
and
XY,k
given by Eqs. (8) and (9), respectively, are calculated
using the Bennett acceptance ratio (BAR) method.20,53–58 µvdW
XY,l
is obtained from
µCoul
XY,l
⟨ f ( βΩ (λ k ) − βΩ (λ k+1) + C)⟩k+1
nk
,
=
⟨ f ( βΩ (λ k+1) − βΩ (λ k ) − C)⟩k
nk+1
nk
C = − β µvdW
.
XY,k − ln
nk+1
(13)
(14)
µCoul
is obtained from
XY,l
⟨ f ( βΥ (φl ) − βΥ (φl+1) + C)⟩l+1
nl
=
,
⟨ f ( βΥ (φl+1) − βΥ (φl ) − C)⟩l
nl+1
nl
,
C = − β µCoul
XY,l − ln
nl+1
(15)
(16)
where nk is the number of statistically independent samples
at state k and f (x) = 1/ (1 + e x ). The number of statistically
independent samples is obtained from the correlation functions
according to the procedure of Chodera et al.59
For a lower variance estimate of excess chemical potential
µLJ
at values of coupling parameters λ near 0, we introduce
XY,k
soft-core potentials to the LJ potentials of the added cationanion pair to remove the divergence of the LJ interactions at
r i j → 0. The soft-core potentials are obtained by replacing r i j
in the LJ potential given by Eq. (2) with r isoft
j , which is given
by
(
)
6
6 1/6
r isoft
,
(17)
j = θσ i j (1 − λ) + r i j
where θ = 0.5 is a parameter of the equation. Since the softcore interactions are identical to the original LJ potentials at
λ = 0 and λ = 1, the soft-core potentials produce the correct
free energy estimation over this range.
Bennett53 obtained expressions for the statistical uncertainties through the derivation of the free energy estimate. The
statistical uncertainties in µLJ
and µCoul
can be obtained
XY,k
XY,l
through these equations. However, since the uncertainties for
µvdW
and µvdW
do not add independently due to both
XY,k
XY,k+1
sharing the k + 1 data series, we use a bootstrapping algorithm20,56,60 for obtaining the statistical errors in the entire
excess chemical potential µex
XY . For each value of λ and φ,
we randomly selected n values with replacements from each
set of n uncorrelated samples constructed from our simulation
data according to Ref. 59, and calculated the chemical potential
µ XY . We repeated this procedure 200 times and obtained the
standard deviation of the µ XY estimates to compute the statistical error.
The simulated chemical potentials µ XY are converted to
mean ionic activity coefficient values, γ XY , using Eq. (1).
In our previous work,20 we had determined the Henry’s law
standard state chemical potentials µ†XY by adjusting their value
so that γ XY equals the value predicted by the Debye-Hückel
limiting law61 at the lowest concentration we can reasonably
simulate, (i.e., 0.01 mol/kg). At this low concentration, there
is about a 2% difference between the experimental value of the
mean ionic activity coefficient and the Debye-Hückel limiting
law prediction. While the bias that this produced in our previous results20 is lower than their statistical uncertainty, we can
get a more accurate prediction by adjusting µ†XY so that γ XY
equals the value predicted by the Davies equation,13
)
( √
m
ln γ XY = −A
√ − 0.2m ln 10,
1+ m
A=
1.824 × 106
(κT)3/2
,
(18)
(19)
where κ is the relative permittivity predicted by the water
model. The advantages of the Davies model are that it does
not contain any fitting parameters and that it predicts essentially the exact experimentally measured value for γNaCl at
0.01 mol/kg and 298.15 K, using the experimental value of the
relative permittivity (78.4962) for water. The values of the relative permittivity κ in Eq. (18) are obtained from simulations.
For SPC/E water at 298.15 K and 1 bar, we use the value of
relative permittivity κ calculated in Ref. 20. For SPC/E water
at 373.15 K and 1 bar and 473.15 K and 15.5 bar, we performed
new calculations of the relative permittivity κ.
The statistical errors of ln γ XY were obtained according
to the process in Ref. 20. The uncertainty of the ln γ XY value
at 0.01 mol/kg reflects the uncertainty in the relative permittivity κ since we assign the value of the Henry’s law (infinite
dilution) standard chemical potential of the salt µ†XY so that
ln γ XY matches the value predicted by the Davies equation.
This causes the statistical error of the salt chemical potential in
the solution µ XY at 0.01 mol/kg and κ to be propagated to µ†XY .
The statistical error values of ln γ XY at higher concentrations
contain both the propagated errors of µ†XY and µ XY at the
concentration of interest.
For the salt chemical potential calculations, the scaling
parameters were λ = [0, 0.1, . . . , 1] and φ = [0, 0.05, . . . , 1].
The systems were run with potentials consistent with their
scaling parameters. The simulations were performed with
the GROMACS63 open-source molecular dynamics package,
version 4.6.5 (available for download from gromacs.org). At
298.15 K and 1 bar, we equilibrated for 2 ns and used a
20 ns production period in the NPT ensemble using the
velocity-Verlet integration method with time steps of 2 fs,
the Nose-Hoover thermostat64,65 with time constant 1 ps,
and the Martyna-Tuckerman-Tobias-Klein (MTTK) barostat66
with time constant 2 ps. At 373.15 K and 1 bar and 473.15 K
and 15.5 bar, we equilibrated for 2 ns and used a 10 ns
production period. The cutoff radii for the LJ and Coulombic
potentials were 0.9 nm, the cutoff radius for the neighbor list
1.15 nm, and the neighbor list was updated every 10 steps.
The potentials were shifted at the inner cutoff radius. Our
simulation boxes contained 500 water molecules except for
concentrations m = 0.01 and 0.06 mol/kg, which contained
5000 and 1000 water molecules, respectively. We used the
particle-mesh Ewald summation67,68 to account for the longranged nature of the Coulombic interactions. We set the
Fourier spacing to 0.1 nm, the relative strength of the Ewaldshifted direct potential at the cutoff to 1 × 10−6, and the order
of the interpolating function to 6. The simulations for the
relative permittivities κ were run with 5000 water molecules
for 5 ns after 2 ns equilibration periods. Configurations were
saved every 0.04 ps. Standard long-range energy and pressure
corrections were applied for the Lennard-Jones potentials in
these simulations. In our previous paper,20 we verified that the
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044505-5
Z. Mester and A. Z. Panagiotopoulos
J. Chem. Phys. 143, 044505 (2015)
chemical potentials calculated with 5000 water molecules produced the same results as systems using 500 water molecules.
Each simulation run for each λ and φ combination of 22 ns for
systems of 500 and 1000 water molecules took approximately
6 h and 9 h, respectively, on 4 cores. For 5000 water
molecules, each 22 ns simulation took approximately 20 h on
8 cores.
B. Chemical potential of the salt crystal and solubility
The chemical potentials of salt crystals can be calculated
with either the Einstein crystal method of Frenkel and Ladd69
or the Einstein molecule method of Vega and Noya.45 Both
methods obtain the Helmholtz free energy of a solid from
the difference in free energy with respect to an ideal Einstein
crystal reference system, for which the free energy is analytically known. The ideal Einstein crystal consists of ideal-gas
particles attached via harmonic springs to lattice positions.
The free energy measurement is performed by calculating
the free energy change of transforming the non-interacting
particles in the Einstein crystal to fully interacting anions and
cations, and, subsequently, measuring the free energy change
of removing the springs from the system. In the Einstein
crystal method,69 pseudo-divergences in the thermodynamic
integration for measuring the free energy of removing the
springs are dealt with by constraining the center-of-mass of
the system to its initial position. Vega and Noya45 solve the
problem of pseudo-divergence by fixing the position of one
particle.
The calculation of free energy begins by measuring the
density of the crystal, starting from the anions and cations
arranged in a fcc lattice, via simulations in the isothermalisobaric NPT ensemble. The simulation provides the volume
of the crystal for the canonical NVT ensemble simulations
for obtaining the Helmholtz free energy. Due to its ease of
implementation, we use the Einstein molecule method of Vega
and Noya.45 The particle and equilibrium
lattice positions
are
given by {r0, r1, . . . , r N −1} and r0,0, r0,1, . . . , r0, N −1 , respectively, where N is the total number of particles. Since we
fix the position of the 0th particle,
we express the positions
as r0;
r1,
r2, . . . ,
r N −1 and r0,0;
r0,1,
r0,2, . . . ,
r0, N −1 , where

ri = ri − r0 and
r0,i = r0,i − r0. In this framework, the potential
of the ideal Einstein crystal is given by
N
−1

N −1 N −1
2

UEin. 
r ,
r0
= λS
ri − 
r0,i ,
(20)
i=1
where λ S is the spring constant. The free energy of the ideal
Einstein crystal is given by Refs. 45 and 49,
β A0 =
3
β 1/3 λ S +
1
βN µ0X + + µY0 − + N ln *
0 2/3
2
2
, π(P ) (
)
3
βλ SV 2/3
− ln
,
2
πN 2/3
(21)
The free energy change that comes from transforming
the ideal gas particles in the Einstein crystal into anions and
cations with LJ and Coulombic interactions is obtained using
free energy perturbation according to
 N −1 −βU [r N −1]−βU [r N −1,r N −1]
Ein.
0
d
r e Int.
β∆A XY,1 = −N ln

N
−1
N
−1
d
r N −1e−βUEin.[r ,r0 ]
−βU = −N ln e Int. Ein.
(22)
where UInt. is the potential from the LJ and Coulombic interactions. As Eq. (22) suggests, ∆A XY,1 is obtained by performing
canonical NVT simulations for the ideal Einstein crystal, and
calculating the average of e−βUInt. in that ensemble.
We use thermodynamic integration to calculate the free
energy change of removing the springs from the fully interacting crystal. The potential of a system along the integration path
is given by
N −1
N −1 N −1
N −1 N −1 ,
(23)
r
+ ζUEin. 
r ,
r0
U 
r ,
r0 ; ζ = UInt. 
where ζ = [0, 1] determines the strength of the contribution of
the springs to the potential. The free energy change calculated
from thermodynamic integration is
N −1 N −1 
 1
∂U 
r ,
r0 ; ζ
dζ
∆A XY,2 = −
∂ζ
0
ζ
 1
N −1 N −1
=−
UEin. 
r ,
r0
dζ,
(24)
ζ
0
where ⟨⟩ζ indicates ensemble average in a system with the
potential given by Eq. (23). The integral is performed using
a 25 point Legendre-Gauss quadrature.
By adding all the free energy contributions, we obtain the
Helmholtz free energy of the salt crystal according to
crystal
A XY
= A0 + ∆A XY,1 + ∆A XY,2.
(25)
The Helmholtz free energy is converted to chemical potential
with
crystal
crystal
µ XY
=
A XY
,
Npairs
(26)
where Npairs = N/2 is the number of ion pairs in the system.
We use a value of 400 000 kBT/nm2 for the spring constant
λ S. Simulations to obtain the chemical potentials of salt crystals were performed with an in-house developed Monte Carlo
(MC) code. All simulations used 4 × 106 MC steps for equilibration and 5 × 107 MC steps for production. In the isothermalisobaric NPT ensemble, we attempted on average one volume
move for every 999 translation moves. The maximum size
of the moves for both volume and translation was chosen
to produce approximately 30% acceptance. The long-ranged
nature of the Coulombic interactions was accounted for using
traditional Ewald summation. The cutoff for both the short
range part of the Coulombic interactions and LJ interactions
was half the length of the box. No long-range corrections were
used for the LJ interactions in the crystal free energy calculations. We dealt with the error resulting from the neglected longrange interactions by extrapolating the density and chemical
potential of the crystal to the thermodynamic limit. Isothermalisobaric NPT simulations were performed with 256, 500, and
864 ion pairs. The simulated densities were plotted versus 1/N
and a linear equation fitted through the points. For systems with
an ensemble average length of a one side of the box less than
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Z. Mester and A. Z. Panagiotopoulos
J. Chem. Phys. 143, 044505 (2015)
3 nm for 500 ion pairs, simulations with 500, 864, and 1372 ion
pairs were performed. The density extrapolated to 1/N → 0
is used in the canonical NVT simulations. While ∆A XY,2 does
not depend on system size, ∆A XY,1 is calculated at the constant
extrapolated density with 256, 500, and 864 or 500, 864, and
1372 ion pairs. We plot the ∆A XY,1/N free energy versus 1/N
and extrapolate it to 1/N → 0 with a linear fit. The chemical
potential of the crystal is constructed with this extrapolated
value of ∆A XY,1. In the isothermal-isobaric NPT ensemble,
simulations of 54 × 106 MC steps with 500 ion pairs took
approximately 8 h on 4 cores. In the canonical NVT ensemble,
the same simulations took about 6 h.
The statistical errors in the densities of individual isothermal-isobaric NPT simulations were estimated by block averaging over 10 blocks. The errors in the extrapolated densities
were estimated by drawing densities from Gaussian distributions with the previously measured standard deviations and
fitting lines through the new data sets. This procedure was
performed 1000 times for each density measurement. The standard deviations of these new extrapolated densities were taken
as the statistical errors of the densities in the thermodynamic
limit.
The measurement of statistical error in the chemical potential of the crystal was performed by running 4 simulations
of the NaCl crystal with the JC electrolyte model with the
simulation volumes used in the canonical NVT ensemble
obtained from drawing densities from a Gaussian distribution
specified by the previously measured crystal density and the
corresponding standard deviation. Due to the computational
expense of this procedure, we treated the statistical error of the
chemical potential of the NaCl crystal with the JC electrolyte
model as the typical error for all the crystals we studied.
The solubilities predicted by the ion models were obtained
by, first, adjusting B, b, C, and D parameters so that the
equation
ln γ XY
β µ XY = β µ†XY + 2 ln m + 2 ln γ XY ,
)
(
√
A m
= ln (10) −
√ + bm + Cm2 + Dm3
1+B m
(27)
(28)
matches the simulation data, where A is given by Eq. (19).
Subsequently, we solve for the concentration m where the µ XY
predicted by the fitted equation equals the crystal chemical
crystal
potential µ XY . The statistical errors in the solubility predictions are estimated by drawing µ XY for all concentrations
from a Gaussian distribution with the simulation mean value
and standard deviation, recalculating µ†XY , and fitting B, b, C,
and D to these new values. By repeating this procedure 1000
times and calculating the standard deviation of the resulting
solubilities, we get an estimation of the error in solubilities.
FIG. 1. Natural logarithm of the mean activity coefficient of NaCl lnγ NaCl
versus the square root of the molality m 1/2 at 298.15 K and 1 bar using the
KBFF, RDVH, SD, and JC ion models with SPC/E water. Experimental data
are from Hammer and Wu.12 The dotted curves are fits based on Eq. (28).
Data for the calculation of ln γ NaCl for the JC and SD models are from
Ref. 20. Statistical uncertainties for ln γ NaCl are between 0.03 and 0.06
(68.3% confidence interval).
higher concentrations (m ≥ 10 mol/kg) where the statistical
uncertainties are 0.06–0.1. The values of ln γ XY and chemical
potentials µ XY , along with their statistical uncertainties, are
tabulated in the supplementary material;51 simulated values
of relative permittivity along with corresponding experimental
values62 for all the temperatures we considered are also listed.
The calculations of the chemical potentials from which
mean ionic activity coefficients were obtained (see Eq. (7))
and the calculations of the salt crystal chemical potentials (see
Eq. (21)) use the standard molar chemical potentials µ0X + and
µY0 −, which were taken from the NIST-JANAF thermochemical
tables52 and are summarized in Table III of supplementary
material.51 The precise values of µ0X + and µY0 −, while neither
affecting the mean ionic activity coefficients nor the predicted
solubility values of the salt, allow direct comparison of chemical potential values to previous work20,21,38,39 and the simulated
salt crystal chemical potentials to experiment (see Tables I-IV).
The values of Henry’s law standard state µ†XY are dependent on
the precise values of µ0X + and µY0 −. With our choice of standard
molar chemical potentials, we can compare the values of µ†XY
obtained from simulation to experiment70 (see Table V of the
supplementary material51).
IV. RESULTS AND DISCUSSION
Figures 1-4 show the natural logarithm of the mean ionic
activity coefficient, ln γ XY , as a function of the square root
of molality m1/2 for NaCl, NaF, NaI, and KCl, respectively,
at 298.15 K and 1 bar, and compare them to experimental
values from Ref. 12. Statistical uncertainties for ln γ XY are
0.03–0.06 (68.3% confidence interval), except for NaI at the
FIG. 2. Natural logarithm of the mean activity coefficient of NaI ln γ NaI
versus the square root of the molality m 1/2 at 298.15 K and 1 bar using
the KBFF,46,47 RDVH,48 and JC ion models with SPC/E water. Experimental
data are from Hammer and Wu.12 The dotted curves are fits based on Eq. (28)
using an additional fourth order term. Statistical uncertainties for ln γ NaI are
between 0.03 and 0.06 (68.3% confidence interval), except at the higher concentrations (m ≥ 10 mol/kg) where the statistical uncertainties are 0.06–0.1.
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Z. Mester and A. Z. Panagiotopoulos
J. Chem. Phys. 143, 044505 (2015)
FIG. 3. Natural logarithm of the mean activity coefficient of NaF ln γ NaF
versus the square root of the molality m 1/2 at 298.15 K and 1 bar using the
KBFF,46,47 RDVH,48 and JC ion models with SPC/E water. Experimental data
are from Hammer and Wu.12 The dotted curves are fits based on Eq. (28).
Since all the data are at low concentration, the fit does not use the C and
D parameters. Statistical uncertainties for ln γ NaF are between 0.03 and 0.06
(68.3% confidence interval).
FIG. 4. Natural logarithm of the mean activity coefficient of KCl ln γ KCl
versus the square root of the molality m 1/2 at 298.15 K and 1 bar using the
KBFF,46,47 RDVH,48 and JC ion models with SPC/E water. Experimental data
are from Hammer and Wu.12 The dotted curves are fits based on Eq. (28). Statistical uncertainties for ln γ KCl are between 0.03 and 0.06 (68.3% confidence
interval).
Values of mean ionic activity coefficients from simulations with the KBFF model show remarkable agreement with
the experimental values. Specifically, the NaCl, NaF, and KCl
results match experimental data for nearly the whole range of
concentrations within their statistical uncertainties. For NaI,
however, agreement is less satisfactory: while the differences
between experimental and simulated values at low and high
concentrations are, e.g., within 1-3 standard deviations of the
simulated values, the disagreement is greater in the middle
of the concentration range (e.g., approximately 10 standard
deviations of the simulated values at 6 mol/kg).
For comparison, we include here results for NaCl obtained
with the SD25 model, which provided the best predictions
of mean ionic activity coefficients in our previous work.20
Figure 1 shows that the KBFF model provides considerably
better predictions than the SD model. Results for the SD and
JC models were constructed from chemical potential values
obtained in our previous study.20 Using the Davies equation
for obtaining the Henry’s law standard state chemical potential
µ†NaCl instead of the Debye-Hückel equation raises the values
of ln γNaCl by 0.01, which is considerably smaller than the
statistical uncertainties.
The values of mean ionic activity coefficient are slightly
underestimated for all models at low concentrations due to
the relative permittivity κ for SPC/E water being lower than
the experimental value (73 vs. 78.4962). The RDVH and JC
models provide good predictions for the mean ionic activity coefficients up to about 1 mol/kg salt concentration. For
NaCl and NaI, these models significantly overestimate the
mean ionic activity coefficients at high concentrations. For
KCl, the JC model significantly overestimates the mean ionic
activity coefficients at high concentrations, and the RDVH
model significantly underestimates the mean ionic activity
coefficients at high concentrations. The RDVH and KBFF
models underestimate the mean ionic activity coefficients for
NaF compared to the experimental values, but the disagreement is not great. It should be noted here, however, that
the range of concentrations for NaF is narrower than for the
crystal
TABLE I. NaCl crystal densities ρ, chemical potentials µ NaCl , and solubilities for the KBFF, RDVH, SD,
and JC models and experiment at 298.15 K and 1 bar, 373.15 K and 1 bar, and 473.15 K and 15.5 bar.
Experimental chemical potentials are from the NIST-JANAF thermochemical tables,52 and solubilities are from
the CRC Handbook of Chemistry and Physics.72 The experimental solubility at 473.15 K is from Cohen.73 The
experimental chemical potential value at 473.15 K is at a pressure of 1 bar. Numbers in parentheses indicate
statistical uncertainties at the 68.3% confidence level for the corresponding quantity, in units of the last decimal
point listed. For example, 1707.1(1) means 1707.1 ± 0.1.
Quantity
KBFF
RDVH
JC
SD
Experiment
ρ kg/m3
crystal
µ NaCl (kJ/mol)
Solubility (mol/kg)
2109.5(1)
−407.521(2)
0.88(2)
T = 298.15 K
1707.1(1)
2010.7(1)
−384.910(2)
−384.060(2)
5.69(7)
3.71(4)
1932.4(1)
−384.019(2)
0.63(1)
2165
−384.024
6.15
ρ kg/m3
crystal
µ NaCl (kJ/mol)
Solubility (mol/kg)
2091.4(1)
−399.730(2)
0.89(2)
T = 373.15 K
1694.0(1)
1993.8(1)
−377.220(2)
−376.251(2)
5.21(9)
3.01(5)
1916.4(1)
−376.241(2)
0.73(1)
−377.112
6.69
ρ kg/m3
crystal
µ NaCl (kJ/mol)
Solubility (mol/kg)
2066.8(1)
−388.831(2)
0.81(2)
T = 473.15 K
1676.3(1)
1970.8(1)
−366.480(2)
−365.351(2)
5.1(1)
2.46(5)
1894.7(1)
−365.331(2)
0.78(2)
−367.595
7.92
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Z. Mester and A. Z. Panagiotopoulos
J. Chem. Phys. 143, 044505 (2015)
crystal
crystal
TABLE II. NaI crystal densities ρ, chemical potentials µ NaI , and solubilities for the KBFF, RDVH, and JC models and experiment at 298.15 K
and 1 bar. Sources for experimental data and representation of statistical
uncertainties are the same as in Table I.
TABLE IV. KCl crystal densities ρ, chemical potentials µ KCl , and solubilities for the KBFF, RDVH, and JC models and experiment at 298.15 K
and 1 bar. Sources for experimental data and representation of statistical
uncertainties are the same as in Table I.
Quantity
Quantity
KBFF
RDVH
JC
Experiment
3877.9(2)
3504.0(2)
3613.8(1)
3670
ρ
µ NaI (kJ/mol)
−304.773(2) −315.078(2) −288.182(2) −284.572
Solubility (mol/kg) >14
13.8(2)
7.71(6)
12.28
kg/m3
other salts because of the lower experimental solubility of
NaF.
The agreement with the experimental values for the KBFF
model makes sense since part of the fitting was done with simulation and experimental values of the Kirkwood-Buff integrals
at a concentration of 4 mol of salt per liter of solution at 300 K
and 1 atm. Weerasinghe and Smith46 and Gee et al.47 calculate
the derivative of the natural logarithm of the mean ionic activity
coefficient with respect to the salt concentration of the solution
from the Kirkwood-Buff integrals. The NaCl, KCl, and NaF
values for this quantity are consistent with experiment within
the uncertainties of the simulation values. While showing fairly
good agreement with the experimental data, the NaI simulation values for this quantity show some inconsistencies with
the experimental values outside of its statistical uncertainties
consistent with the results we obtained for the mean ionic
activity coefficients.
The RDVH48 model was obtained from reparametrization
of the Deublein et al.71 model, which was fitted to densities
at 293.15 K and 1 bar. Deublein et al.71 found that the LJ
energy parameters have little effect on this property. As a
result, Reiser et al.48 adjusted the LJ energy parameters to also
accurately predict the self-diffusion coefficients of the cations
and anions and the first maximum of the radial distribution
functions of water around ions in the temperature range of
293.15 K–298.15 K and at a pressure of 1 bar. Joung and
Cheatham22 obtained their parameters for the JC model by
balancing lattice energies and lattice constants of the salt crystals and hydration free energies of solvated ions. The lack of
relevant high concentration properties used in these fits may
at least partially explain the poor performance of the model
for the mean ionic activity coefficients above concentrations
of 1 mol/kg.
Tables I-IV show the densities, chemical potentials, and
solubilities of salt crystals. The densities that correspond to the
lattice constants predicted by Joung and Cheatham22 for their
JC model are in good agreement with our results. Likewise,
crystal
TABLE III. NaF crystal densities ρ, chemical potentials µ NaF , and solubilities for the KBFF, RDVH, and JC models and experiment at 298.15 K
and 1 bar. Sources for experimental data and representation of statistical
uncertainties are the same as in Table I.
Quantity
KBFF
ρ kg/m
µ NaF (kJ/mol)
Solubility
(mol/kg)
3
1966.5 (1)
−505.412 (2)
0.0142 (4)
RDVH
JC
Experiment
1981.0 (1)
2372.3 (1)
2558
−530.221 (2) −539.779 (2) −545.081
<0.011
>1.33
0.99
ρ
µ KCl (kJ/mol)
Solubility
(mol/kg)
kg/m3
KBFF
RDVH
1984.8 (1)
−419.023 (2)
0.55 (1)
1608.1 (1)
−401.306 (2)
0.126 (3)
JC
Experiment
1903.8 (1)
1984
−406.943 (2) −408.761
2.65 (5)
4.77
the density results for the KBFF crystals are also consistent
with the results of Gee et al.47 within their predicted statistical
uncertainties. While both of these studies were performed
at 300 K and 1 atm with simulations of finite-sized crystals
(i.e., not the thermodynamic limit), the level of accuracy to
which the data were reported makes these differences with
our study irrelevant. Moučka et al.39 obtained the densities of
NaCl and KCl with the JC model as 2010.79 ± 0.06 kg/m3 and
1903.17 ± 0.06 kg/m3, respectively. While the NaCl density
values are consistent with ours, the KCl density values are
outside the statistical uncertainties. In a separate work, Moučka
et al.38 predicted values for the density of NaCl using the SD
model which are consistent with our predictions. The KBFF
model is shown to yield, in the same work, 2112 kg/m3 for
the density of NaCl. This significantly differs from both our
results and the density predicted by Gee et al.47 The density of 2108 kg/m3 predicted by the latter work is consistent
with our results within the reported statistical uncertainties.
As a check on our results, we repeated several of our density
measurements using the GROMACS simulation package,63
and obtained the same results as with our Monte Carlo code
(e.g., 1903.4 kg/m3 for KCl with the JC model for both simulation methods performed with 500 ion pairs). The values of
chemical potentials and densities of the crystals as a function
of the number of ion pairs in the system can be found in the
supplementary material.51
We tested the correctness of our method of calculating
crystal chemical potentials by running a chemical potential calculation at the same temperature (298 K) volume
(24.13 nm3), and number of ion pairs (500) as Aragones et al.49
We obtained −383.962 ± 0.002 kJ/mol for the chemical
potential, which compares favorably to the result of Aragones
et al.49 of −383.9 ± 0.2 kJ/mol. Our results for the chemical
potentials are consistently about 0.3 kJ/mol higher than the
values given by Moučka et al.38,39
The crystal chemical potentials are used to determine
the solubilities of the salts predicted by the ion and water
models. The solubility values are also shown in Tables I-IV.
The solubility predictions are generally poor. The KBFF model
significantly underpredicts the solubilities except for NaI, for
which the solubility is overestimated. The RDVH model makes
fairly good predictions for NaCl and NaI, but significantly underpredicts the NaF and KCl solubilities. The JC predicts solubilities in the 40%-60% range of the experimental solubility
except for NaF where the model overpredicts the solubility by
a significant amount.
In order to confirm the validity of our approach, we
have compared the solubilities for NaCl predicted by several
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TABLE V. Solubility results of NaCl from several studies using the SPC/E
water model and the SD, SD + TF,26–29 and JC electrolyte models. While
some of the studies are performed at 298 K and 1 bar and others at 298.15 K
and 1 bar, given the number of significant figures in the simulation results,
there is no difference between the predicted solubilities due to the temperature. Uncertainties indicated as in Table I.
Solubility (mol/kg)
Study
Sanz and Vega24
Aragones et al.33
Paluch et al.34,35
Moučka et al.37
Moučka et al.38
Mester and Panagiotopoulos20
Current study
SD
0.9(4)
0.61
0.61(1)
0.63(1)
SD + TF
5.4(8)
4.3(3)
0.8
3.6
3.57(5)
JC
4.8(3)
3.64
3.59(4)
3.71(4)
studies for a variety of ion models with the SPC/E23 water
model in Table V. Table V shows that the disagreements in
the crystal chemical potentials between this work and the
works of Moučka et al.38,39 produce only small differences
in the predicted solubilities. Our previous study20 used our
fractional ion insertion method for obtaining solution chemical potentials. The solution chemical potentials were then
compared against the crystal chemical potential results of
Moučka et al.38,39 to calculate the solubility. The current
study uses the values from Tables I-IV for crystal chemical
potentials. The solubility results of both of our studies show
strong agreement with the studies of Moučka et al.38,39 We
verified in our previous work20 that this is due to the good
agreement in solution chemical potential values predicted by
our fractional insertion method and the osmotic ensemble
Monte Carlo simulation method.36,37,39 The studies of Sanz
and Vega24 and Aragones et al.33 give different predictions
for solubility than these studies due to a significant difference
in the solution chemical potential predictions. The solubility
result of Paluch et al.,34,35 based on a self-adaptive WangLandau transition-matrix Monte Carlo method in the expanded
isothermal-isobaric ensemble, is considerably different from
the results of our previous study. This is due to the incorrect
value obtained for the simulated salt crystal free energy. The
FIG. 6. Natural logarithm of the mean activity coefficient of NaCl ln γ NaCl
versus the square root of the molality m 1/2 at 473.15 K and 15.5 bar using the
KBFF,46,47 RDVH,48 SD, and JC ion models with SPC/E water. Experimental
data are closely approximated from the Pitzer equation.74 The dotted curves
are fits based on Eq. (28). Statistical uncertainties for ln γ NaCl are between
0.03 and 0.06 (68.3% confidence interval).
solution salt chemical potential results are consistent within
the statistical uncertainties compared to our previous study.20
However, the comparison is tenuous due to the large statistical
uncertainties in the values of solution chemical potentials of
salt obtained by Paluch et al.34,35 (i.e., ∼2-4 kJ/mol).
We tested the transferability of the predictions of the ion
models to higher temperatures by running simulations for NaCl
at 373.15 K and 1 bar (Figure 5) and 473.15 K and 15.5 bar
(Figure 6). Both simulations are at the experimental saturation
conditions of water. Figures 5 and 6 show that both the experimental and simulated ln γNaCl curves are lowered by temperature. The simulated data show a considerably stronger decrease
with temperature than the experimental data. This indicates
poor transferability of the models to higher temperatures.
Since the RDVH and JC models both overestimate the
mean ionic activity coefficients at high concentration, the
predictions of these models become better with temperature.
For the 473.15 K case, there is very good agreement between
the experimental and simulation results. Since the SD results
at 298.15 K overestimate the mean ionic activity coefficients
at high concentration to a lesser extent, the 373.15 K results
are fairly consistent with experimental results. At 473.15 K,
the SD model greatly underestimates the mean ionic activity
coefficient. The KBFF model gets the mean ionic activity coefficient correct at 298.15 K, and the predictions get considerably
worse at higher temperatures.
Table I shows that the salt models have poor transferability
in terms of solubility as well. The JC and RDVH models predict
a decrease in solubility with temperature, which is opposite of
the trend seen in experiment. The KBFF and SD models show
little temperature dependence on solubility.
V. CONCLUSIONS
FIG. 5. Natural logarithm of the mean activity coefficient of NaCl ln γ NaCl
versus the square root of the molality m 1/2 at 373.15 K and 1 bar using the
KBFF,46,47 RDVH,48 SD, and JC ion models with SPC/E water. Experimental
data are closely approximated from the Pitzer equation.74 The dotted curves
are fits based on Eq. (28). Statistical uncertainties for ln γ NaCl are between
0.03 and 0.06 (68.3% confidence interval).
In this work, we have used molecular dynamics simulations with gradual insertion of ion pairs to obtain the mean
ionic activity coefficients of NaCl, NaI, NaF, and KCl aqueous
solutions at standard conditions of 298.15 K and 1 bar, with the
ions models of Gee et al.46,47 (KBFF), Reiser et al.48 (RVDH),
and JC22 in SPC/E water23 at a temperature of 298.15 K and
pressure of 1 bar to experiments. We showed that the KBFF
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Z. Mester and A. Z. Panagiotopoulos
model predicts mean ionic activity coefficients that are consistent with experimental results. However, generally, this model
underpredicts the solubilities by a large amount. Furthermore,
with a few exceptions, the other two models also provide
poor predictions for solubilities, in addition to not correctly
predicting the mean ionic activity coefficients at high concentrations. All the models for NaCl also show poor transferability
to higher temperatures with the predictions for mean ionic
activity coefficient changing considerably more as temperature
increases than seen in experiment.
The accurate predictions of mean ionic activity coefficient
by the KBFF model at room temperature obtained in the present study are in sharp contrast to the results of our earlier
study20 for a number of other common ion models, which
failed to represent the activity coefficients of NaCl correctly.
These new results point to the possibility of using simple
non-polarizable models to accurately predict the properties of
aqueous electrolytes. This is the subject of ongoing work.
The SPC/E model of water used in the present study is
not considered to be among the most accurate of fixed-pointcharge models currently available,50 even if it continues to be
widely used in simulations. Unfortunately, ion models need to
be specifically developed to be compatible with a given water
model; they are generally not expected to be transferrable to
other water models. Thus, significant future work would be
required to obtain optimized ion models for use in conjunction
with more modern water potentials.
ACKNOWLEDGMENTS
Financial support for this work has been provided by the
Department of Energy, Office of Basic Energy Sciences, under
Award No. DE-SC0002128.
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