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Saric, D., Kohns, M., & Vrabec, J. (2020). Dielectric constant and density of aqueous alkali halide

solutions by molecular dynamics: A force field assessment. The Journal of Chemical Physics, 152(16),

164502. https://doi.org/10.1063/1.5144991

Denis Saric, Maximilian Kohns, Jadran Vrabec

Dielectric constant and density of aqueous

alkali halide solutions by molecular

dynamics: A force field assessment

Accepted manuscript (Postprint)Journal article |

Dielectric Constant and Density of Aqueous Alkali Halide Solutions by

Molecular Dynamics: A Force Field Assessment

Denis Saric,1Maximilian Kohns,2and Jadran Vrabec3, a)

1)Thermodynamics and Energy Technology, University of Paderborn, 33098 Paderborn,

Germany

2)Laboratory of Engineering Thermodynamics, Technische Universit¨at Kaiserslautern, 67633 Kaiserslautern,

Germany

3)Thermodynamics and Process Engineering, Technical University Berlin, 10587 Berlin,

Germany

(Dated: 26 March 2020)

The concentration dependence of the dielectric constant and the density of 11 aqueous alkali halide solutions

(LiCl, NaCl, KCl, RbCl, CsCl, LiI, NaI, KI, CsI, KF, CsF) is investigated by molecular simulation. Predic-

tions using eight non-polarizable ion force fields combined with the TIP4P/εwater model are compared to

experimental data. The influence of the water model and the temperature on the results for the NaCl brine is

also addressed. The TIP4P/εwater model specifically improves the accuracy of dielectric constant predictions

compared to the SPC/E water model. The solution density is predicted well by most ion models. Almost all

ion force fields qualitatively capture the decline of the dielectric constant with increasing concentration for all

solutions and with increasing temperature for NaCl brine. However, the sampled dielectric constant is mostly

in poor quantitative agreement with experimental data. These results are related to the microscopic solution

structure, ion pairing and ultimately the force field parameters. Ion force fields with excessive contact ion

pairing and precipitation below the experimental solubility limit generally yield higher dielectric constant

values. An adequate reproduction of the experimental solubility limit should therefore be a prerequisite for

further investigations of the dielectric constant of aqueous electrolyte solutions by molecular simulation.

Keywords: molecular simulation; dielectric constant; density; electrolyte solutions; force fields; ion pairing

I. INTRODUCTION

The unusually high dielectric constant of water under

standard conditions of temperature and pressure (ε=

78.5)1is one of its anomalies,2which partly explains the

dissolution of 1:1 salts in liquid water at low and mod-

erate concentrations into single ions by preventing their

aggregation into crystals.3A substantial drop of the di-

electric constant, usually caused by increasing temper-

ature or salt concentration, may lead to the formation

of ion pairs as a stable species in aqueous alkali halide

solutions.4

Despite its significance, the dielectric constant of even

the simplest aqueous monovalent salt solutions is poorly

studied by experiment and sufficiently available only for

11 of the 20 alkali halide brines.5–19 In fact, the dielectric

constant is not directly measurable in the laboratory.20

It has to be extrapolated from the frequency-dependent

dielectric constant in a non-trivial procedure, which re-

quires elaborate measurement instruments, dielectric re-

laxation models and a complex estimation of model

parameters.21–23

An alternative to access this property, which was

widely employed for water in the last three decades, is

molecular modeling and simulation. The quality of the

underlying force field models for solvent and solutes plays

the central role for the accuracy of predictions with this

a)Electronic mail: vrabec@tu-berlin.de

approach. However, an extensive computational effort is

required for a statistically sound sampling of the dielec-

tric constant even for pure water.24–26

Water and ions are primarily modeled by non-

polarizable Lennard-Jones (LJ) sites with superimposed

point charges. The temperature dependence of the di-

electric constant of water27 or the recently found anoma-

lously low dielectric constant of confined water28 was

qualitatively reproduced with these simple models. How-

ever, the quantitative accuracy of such predictions is

rather disappointing.

Commonly employed water potentials in studies of

aqueous electrolyte solutions, such as SPC/E (ε= 70.7

at 298 K)29 or TIP4P (ε= 53 at 293 K)30, underesti-

mate the dielectric constant. The TIP4P/200531 water

model, despite its superiority over other non-polarizable

potentials,32 misses the dielectric constant by about 25 %

(ε= 58 at 298 K). The general failure of non-polarizable

water models in dielectric constant simulations was es-

sentially attributed to the application of the same par-

tial charges to empirically fit two distinct water molecule

properties (potential energy and dipole moment).33

In the search for optimal force field parameters for

alkali halide ions, low concentration experimental data

have mostly been used in fitting procedures.34–41 For ex-

ample, the hydration free energy of the ions at infinite

dilution was a popular choice among many groups.34–39

In an attempt to achieve reasonable predictions for the

concentration range where cation-anion interactions be-

come important, crystal properties37,42–44 or activity

derivatives39,42,43 were considered. For a more compre-

hensive discussion of common parametrization strategies

employed for molecular modeling of aqueous electrolyte

solutions, the interested reader is referred to the review

of Nezbeda et al.45

In general, however, these strategies yield unsatisfac-

tory predictions for solution properties, such as density46,

water self-diffusion coefficient47 or salt activity48,49. In

fact, the underprediction of the solubility limit of alkali

halide salts,45,50–54 resulting in excessive ion pairing55–57

and/or premature crystallization,58–60 is a well-known

problem of non-polarizable ion force fields. The cause

may lie in unbalanced force field parameters.58,61 As crys-

tallization is a slow process that takes place on time scales

much longer than those typically accessed with molecular

simulation, it is possible that many reported simulation

results actually do not correspond to equilibrium states.

An accurate prediction of basic solution properties,

such as density, does not only serve as an indica-

tion for an adequate representation of the microscopic

structure,62 but is also necessary to correctly predict

other properties.63 The lack of experimental dielectric

constant data, the inaccuracy of water models for dielec-

tric constant predictions and generally inadequate ion

force fields may explain why the dielectric constant of

monovalent salt solutions was rarely studied with molec-

ular simulation. After all, the prospect of dielectric con-

stant investigations has recently improved dramatically

by the development of water force fields parametrized to

the dielectric constant (such as TIP4P/ε64 or TIP4Q65),

methods for implicitly accounting for polarizability of

non-polarizable models (such as charge scaling66) and

sufficient computational power for long simulations that

are required for the dielectric constant sampling. A more

detailed review of current trends in molecular modeling

and simulation of aqueous electrolyte solutions was pro-

vided by Smith et al.67

There are only a few studies concerning dielectric con-

stant calculations of aqueous alkali halide solutions by

simulation.43,68–72 Ion models used in these studies gen-

erally predict the decrease of the dielectric constant with

increasing salt concentration qualitatively well, but un-

derestimate the experimental values quantitatively. For

example, Pethes70 simulated the dielectric constant, den-

sity and self-diffusion coefficient of water and ions using

29 ion force fields for LiCl up to the experimental sol-

ubility limit of the according brine. In her study, most

of the ion force fields overestimated the experimental di-

electric constant at moderate concentrations. In a very

recent study of aqueous NaCl solutions, a reduced dielec-

tric constant decline at higher concentrations for some

ion models was linked to excessive contact ion pairing.72

The few comprehensive studies in the literature on

molecular simulation of the dielectric constant of aque-

ous electrolyte solutions cover only one or a few salts. No

study of the dielectric constant covering a wide range of

alkali halide salts in aqueous solutions, together with a

comparison of several different ion force fields, has been

presented to date. The main objective of this paper is

thus to provide such a comprehensive comparison. The

performance of eight non-polarizable ion force fields com-

bined with the TIP4P/εwater model was assessed with

respect to the prediction of the dielectric constant and

density of 11 aqueous alkali halide solutions (LiCl, NaCl,

KCl, RbCl, CsCl, LiI, NaI, KI, CsI, KF, CsF). These

salts were chosen as they are the only ones out of the

20 alkali halides for which reliable experimental data are

available for comparison.

The present approach is as follows: first, the role of the

underlying water model was investigated by comparing

dielectric constant predictions for aqueous NaCl solutions

at 298 K, using different ion force fields together with

the TIP4P/εor the SPC/E water model. As TIP4P/ε

yielded better predictions for all investigated ion force

fields, all subsequent studies were carried out with that

water model. Second, the temperature dependence of the

dielectric constant was addressed, again using aqueous

NaCl solutions as an example. Third, a comprehensive

assessment of eight ion force fields for the alkali halides

was carried out. All 11 salts for which experimental data

are available for comparison were considered. Finally, the

predictions for the dielectric constant were linked to ion

pairing and salt solubility in light of the ion force field

parameters.

II. METHODOLOGY

A. Molecular force fields

Throughout this work, ions and water molecules were

represented by pairwise additive, non-polarizable force

fields of the LJ 12-6 type with superimposed point

charges. The interaction energy between particles iand

jis then given by

uij = 4εij "σij

rij 12

−σij

rij 6#+

l=1

m=1

4πε0

qlqm

rlm

(1)

where σij and εij are the LJ parameters for size and

energy, respectively, rij the site-site distance, qland

qmmagnitudes of point charges of the ions or water

molecules separated by the distance rlm,Ne

ithe total

number of charges of particle iand ε0the vacuum per-

mittivity. All interactions between unlike LJ sites were

specified by the Lorentz-Berthelot (LB) combining rules

σij =σii +σjj

2,(2)

εij =√εiiεjj .(3)

In most of the present simulations, water-water inter-

actions were described with the TIP4P/εwater model.64

This force field is a rigid body with a single LJ interac-

tion site for the oxygen atom, two positive charges for

the hydrogen atoms and one negative charge placed on a

virtual site M on the H-O-H bisector. The force field pa-

rameters of the TIP4P/εand the SPC/E water models

are given in Table S.I (Supplementary material). The

LJ size and energy parameters of the TIP4P/εdiffer

only slightly from those of the TIP4P/2005 force field.

TIP4P/εwas parametrized to reproduce the experimen-

tal density maximum of liquid water and the dielectric

constant at ambient temperature. The significantly lower

charge magnitude qH= 0.527 e and virtual site offset

dOM = 0.105 ˚

A of this model compared to TIP4P/2005

(0.5564 e, 0.1546 ˚

A) result in an excellent coverage of the

dielectric constant over a wide range of temperature and

pressure. Both models give good predictions for most

liquid state and vapor-liquid equilibrium properties of

water.

In the present work, ions were modeled by one LJ

site and a single point charge of fixed magnitude +1 e

(cations) or −1 e (anions) in their center. Various ion

force fields were studied, which differ by a factor of up to

two for the size parameter σand by up to four orders of

magnitude for the energy parameter ε, as shown in Tables

S.II and S.III. A summary of parametrization properties

and water models employed in the fitting procedures for

the ion models is given in Table I. It should be noted that

in the original works of Jensen and Jorgensen35, Reif and

H¨unenberger36 and Gee et al.42,43 geometric combination

rules were employed for the unlike LJ parameters.

As none of the ion force fields studied in the present

work was adjusted explicitly for the use with the

TIP4P/εwater model, some remarks are given in the

following about the transferability of these ion models to

other water models than the ones employed in the fit.

Transferability of the RDVH model parameters, fitted

to reproduce the reduced liquid solution density, to four

other water models was shown in the original study.40

Moreover, these model parameters were used success-

fully for density predictions of methanolic73, ethanolic74

and water+methanol75 alkali halide solutions. Consis-

tent ion force fields from Gee et al. showed similar results

for various properties with different water models. Mao

and Pappu developed solvent-independent force field pa-

rameters that were fitted to crystal lattice properties.44

Ion models from Joung and Cheatham parametrized with

the TIP4P-Ew76 water model showed a good compatibil-

ity with the TIP4P/2005 force field for aqueous NaCl77

and LiCl78 solutions due to the similar LJ geometry and

parameters of TIP4P-family models. Hence, combining

different ion model sets with the TIP4P/εwater model

appears to be reasonable.

B. Simulation details

An extended version of the molecular simulation

tool ms279–81 was employed for the present calcula-

tions. Simulations of dielectric constant and density of

electrolyte solutions were performed in the isothermal-

isobaric (NpT) ensemble by molecular dynamics. Eleven

alkali halide brines (LiCl, NaCl, KCl, RbCl, CsCl, LiI,

NaI, KI, CsI, KF, CsF) were sampled at 298 K, 1 bar

and three concentrations for which experimental data are

available, using eight different ion model sets together

with TIP4P/εwater. The remaining nine alkali halide

salts (LiF, LiBr, NaF, NaBr, KBr, RbF, RbBr, RbI,

CsBr) were not considered due to the lack of experimen-

tal data for comparison.

A cubic simulation volume with periodic boundary

conditions contained a total of 1000 ion and solvent parti-

cles throughout. The composition of the solution is given

in terms of the true ion mole fraction

xion =nC

2nC+nW

,(4)

where nCand nWare the mole numbers of cations and

water, respectively. For the alkali halide brines of this

study, the true ion mole fraction ranges from 0.007 to

0.095 mol mol−1.

The pressure was kept constant by the Andersen baro-

stat with a piston mass of 2.2·109kg m−4. Velocity

scaling was used to control the temperature. Following

an equilibration of 5·105time steps, production runs of

107time steps were carried out. Simulation runs over

more than 5-6 ns are needed for convergence of the di-

electric constant.24 Newton’s equations of motion were

solved with a Gear predictor-corrector scheme of fifth or-

der with a timestep of ∆t= 1.2 fs. The average runtime

of a single simulation was eight hours on 4 compute nodes

with 40 parallel cores per node on two Intel Xeon Gold

6148 CPU.

Electrostatic long-range contributions were calculated

by Ewald summation with a real space convergence pa-

rameter κ= 5.6, up to 10 k-vectors in each Cartesian

direction and conducting boundary conditions. The LJ

cut off distance and the real-space cutoff were equal with

13.8 ˚

Statistical uncertainties were estimated by means of

the block averaging method by Flyvbjerg and Petersen82

with a block length of 5000 time steps. Visual snap-

shots of molecular configurations were recorded every

5000 time steps.

Moreover, the radial distribution function (RDF) of

several salts in water (NaCl, KCl, KI and LiI) was sam-

pled to investigate the microscopic structure of the solu-

tions. The number of contact ion pairs (CIP) for a 1:1

electrolyte solution can be computed from the cation-

anion RDF g+,−(r) with

nCIP = 4πρxion Zrmin

g+,−(r)r2dr, (5)

where ρ=N/V is the particle number density of the

studied solution and rmin the position of the first mini-

mum of the cation-anion RDF.

In order to characterize the deviations between simula-

tion results and experimental data for a given brine, the

mean relative error (MRE) was calculated by

TABLE I. Summary of ion force fields considered in this work with water models used for their parametrization.

Ion models Abbr. Parametrization properties Water

model

Combining

rule

Horinek et al.34 HMN-SaHydration free energy, hydration entropy SPC/E -

Fyta et al.38,39 FytabHydration free energy, activity derivatives SPC/E LB

Jensen-Jorgensen35 JJ Hydration free energy, ion-water RDF TIP4P geometric

Reif-H¨unenberger36 RH-LcHydration free energy SPC, SPC/E geometric

Joung-Cheatham37 JC-TdHydration free energy, lattice energy, lattice

constant, binding energy, ion-water RDF

SPC/E, TIP3P,

TIP4P-Ew

Gee et al.42,43 KBFF Activity derivatives, ionic radii, lattice

constant, ion-water RDF

SPC/E geometric

Reiser et al.40,41 RDVH Relative density, ion self-diffusion

coefficient, ion-water RDF

SPC/E LB

Mao-Pappu44 MP Lattice energy, ion-ion distance - LB

aThree parameter sets with different LJ energy parameters were given in the original work. The one with the smallest energy parameter

was studied here. This ion force field does not provide parameters for the Rb+cation.

bThis ion force field does not provide parameters for the Li+and Rb+cations. A modified LB combination rule was employed for

cation-anion interactions in the original work.

cThree parameter sets based on different experimental hydration free energies of the proton and the SPC/E water model were given in

the original work (LE, ME, HE). The one for the lowest experimental hydration free energy (LE) was used here.

dTIP4P-Ew water model specific ion parameters were used in the present work.

MRE = 1

i=1

|zsim −zexp|

zexp

,(6)

where nis the number of data points per ion force field.

C. Sampling of the dielectric constant

The dielectric constant of non-polarizable fluids was

computed from fluctuations83 of the total dipole moment

M=PN

i=1 µiin the simulation volume by

ε=4π

3kBThViM2−hMi2+ 1,(7)

where kBis the Boltzmann constant and µithe individ-

ual dipole moment vector of the molecule. The angled

brackets denote the ensemble average.

Long simulations bring the average dipole moment

hMito zero,25 as expected in the absence of an exter-

nal field,84 so that the term hMi2in Eq. (7) vanishes.

III. RESULTS AND DISCUSSION

In order to evaluate the performance of the studied ion

and water models with respect to dielectric constant and

molar density predictions, simulations were performed as

described in Section II for a total of 327 state points. The

results of 261 simulations were compared to experimen-

tal dielectric constant and density data in Figures 1 to 5.

For the remaining simulations, indications of supersatu-

ration or precipitation were found, as discussed further

below. Tabulated numerical values of simulated proper-

ties together with their statistical uncertainties are listed

in Tables S.IV to S.IX (Supplementary material).

The experimental data show that with rising salt con-

centration, the dielectric constant of the solution first

decreases linearly (for dilute solutions) and then non-

linearly (for higher concentrations).85 This phenomenon

is called dielectric decrement and is primarily explained

by the reduction of orientational correlation between wa-

ter molecules in the proximity of ions.69,86 It should be

pointed out that some experimental data were either de-

livered by a single research group or may be in disagree-

ment with other available data. Therefore, some caution

regarding the reliability of these data should be taken.

The specific density of all aqueous alkali halide solu-

tions is an increasing linear function of ion mole fraction

at a given temperature.87 For many brines, the molar

density shows a similar trend. Exceptions to this are the

alkali iodides and CsCl, for which the high molar mass

of at least one of the ions causes the molar density to

decrease with concentration.

In order to investigate the sensitivity of the ion force

fields to a change of the water model, aqueous NaCl solu-

tions were studied, combining each of the ion force fields

with the SPC/E or with the TIP4P/εwater model. Re-

sults for the dielectric constant and the molar density

under ambient conditions of temperature and pressure

are shown in Figure 1 for three salt concentrations.

The experimental dielectric constant of pure water (ε

= 78.5)1was predicted excellently with TIP4P/ε(ε=

77.4 ±1.5), but underestimated with SPC/E (ε= 69.0

±1.3). Both results are in good agreement with previous

studies.3,29,64

Regarding the concentration dependence, all ion mod-

els follow the same qualitative trends for both the di-

FIG. 1. Dielectric constant (top) and molar density (bot-

tom) over true ion mole fraction of NaCl at 298 K and 1

bar. Simulation results for eight ion force fields combined

with the TIP4P/ε(left) and SPC/E (right) water models are

represented as open colored symbols as shown in the legend.

Experimental data for comparison are represented by cross

symbols5. Other experimental data6–9 are depicted by plus

symbols. The prediction for pure water is given by a dia-

mond symbol. A correlation87 to experimental density data

is shown as a black line. Statistical uncertainties are within

symbol size.

electric constant and the molar density, independent of

the employed water model. The role of the water model

becomes more significant when considering the quanti-

tative agreement with experimental data. Simulations

with TIP4P/εwater led to a higher dielectric constant

by around 8 % for all ion models at the lowest concen-

tration. This is not surprising, since water-water inter-

actions influence the dielectric constant to a greater ex-

tent in dilute solutions and the dielectric constant of pure

TIP4P/εwater is higher than that of pure SPC/E water.

The HMN-S, RH-L and JJ ion models generally led to

a higher dielectric constant than other ion force fields.

For more concentrated NaCl brines, the remaining ion

models yielded an increase of up to 10 % for the dielec-

tric constant with the TIP4P/εmodel and were thus in

better agreement with the experimental data. The molar

density predictions with all ion models were quite similar

for both water models.

The use of the TIP4P/εmodel resulted in higher di-

electric constant predictions for all ion force fields. These

results are in line with previous findings, which sug-

gest that 1-propanol89 and NaCl88 force fields combined

with TIP4P/εwater yielded dielectric constant predic-

tions that were closer to experimental data. Thus, in the

following, only TIP4P/εwas considered as water model.

FIG. 2. Dielectric constant (top) and molar density (bottom)

over true ion mole fraction of NaCl at 278 K (left) and 308 K

(right) and 1 bar. Simulation results for eight ion force fields

combined with the TIP4P/εwater model are represented as

open colored symbols as shown in the legend. Experimental

data for NaCl brine5are represented by cross symbols. The

prediction for pure water is given by a diamond symbol. A

correlation87 to experimental density data is shown as a black

line. Statistical uncertainties are within symbol size.

In order to check the experimentally observed decrease

of the dielectric constant with rising temperature, NaCl

brines were simulated for two additional temperatures

(278 K and 308 K). The simulation results are compared

with experimental data5and a correlation87 in Figure 2.

For the lower temperature, experimental density data

from Ref. 5 are shown instead of a correlation.

As in Figure 1 for aqueous NaCl solutions at 298 K,

the same qualitative trends were observed at 278 K and

308 K for all ion force fields. An increase of the temper-

ature reduced the discrepancy between simulations and

experiments for the HMN-S, RH-L and JJ force fields,

but raised deviations for the remaining ion models.

The same qualitative trends were observed for the per-

formance of the ion force fields, irrespective of the tem-

perature or the employed water model. Thus, in the fol-

lowing comprehensive study of alkali halides, only the

TIP4P/εwater model was used and only the tempera-

ture of 298 K was considered. The choice of this temper-

ature was further motivated by the fact that for many of

the investigated brines, experimental data are available

only at that temperature.

The results of a comprehensive study of eleven alkali

halides (LiCl, NaCl, KCl, RbCl, CsCl, LiI, NaI, KI, CsI,

KF, CsF) in aqueous solution are presented in Figures 3

to 5. The following discussion summarizes the findings

obtained by comparing the performance of the ion force

FIG. 3. Dielectric constant (top) and molar density (bottom)

over true ion mole fraction of alkali chloride brines at 298 K

and 1 bar. Simulation results for eight ion force fields com-

bined with the TIP4P/εwater model are represented as open

colored symbols as shown in the legend. Experimental data

for comparison for LiCl10, RbCl11, KCl12 and CsCl11 brines

are represented by cross symbols. Other experimental data

for LiCl6,9,13, RbCl9,13, KCl6,9,13,14 and CsCl12 brines are de-

picted by plus symbols. The prediction for pure water is given

by a diamond symbol. A correlation87 to experimental den-

sity data is shown as a black line. Statistical uncertainties are

within symbol size.

fields in predicting the dielectric constant and molar den-

sity of these solutions.

For nearly all considered salts, the ion force fields are

able to reproduce the experimental dielectric constant

and density trends at least qualitatively. However, strik-

ingly different results regarding the slope of the dielectric

constant decrement were observed for various force fields.

FIG. 4. Dielectric constant (top) and molar density (bottom)

over true ion mole fraction of alkali iodide brines at 298 K

and 1 bar. Simulation results for eight ion force fields com-

bined with the TIP4P/εwater model are represented as open

colored symbols as shown in the legend. Experimental data

for comparison for LiI15, NaI6,9, KI16 and CsI17 brines are

represented by cross symbols. Other experimental data for

NaI13 and KI6,13 brines are depicted by plus symbols. The

prediction for pure water is given by a diamond symbol. A

correlation87 to experimental density data is shown as a black

line. Statistical uncertainties are within symbol size.

In most cases, the dielectric constant was underpredicted

in comparison to the experimental data.

The molar density, with deviations below 1.6 % for

almost all ion models, is in good agreement with experi-

mental data. The HMN-S model set excellently repro-

duced the molar density of all potassium and cesium

halide brines (except for CsF), but yielded the overall

worst predictions for the remaining salts compared to

TABLE II. Mean relative error (MRE) between simulation

data and experimental results for dielectric constant and mo-

lar density of 11 alkali halide brines studied with different

ion force fields. All simulations were carried out with the

TIP4P/εwater model and at 298 K, except where noted oth-

erwise.

100 MRE(ε)

Model HMN Fyta JJ RH JC KBFF RDVH MP

salt

NaCl 5.7 18.9 4.7 6.1 26.8 19.9 24.9 20.5

NaCl (SPC/E) 7.2 25.1 9.7 11.6 33.1 25.8 33.5 25.5

NaCl (278 K) 6.6 13.6 8.9 4.5 20.9 14.6 20.2 16.2

NaCl (308 K) 4.2 20.6 3.2 3.9 26.9 19.9 25.7 20.8

LiCl 6.7 5.3 3.7 22.2 22.0 22.0 14.3

RbCl 22.7 24.2 15.9 12.8 13.0 12.4

KCl 11.8 19.5 15.3 17.0 21.0 16.4 18.5 20.2

CsCl 6.7 23.4 27.3 22.9 24.7 25.7 22.9 23.8

LiI 10.9 13.3 7.7 22.4 24.9 18.1 22.3

NaI 6.3 25.4 11.8 12.2 27.4 26.6 26.3 22.5

KI 4.5 33.3 9.9 18.9 27.5 17.8 25.3 25.1

CsI 3.7 12.9 18.2 16.1 21.9 19.5 19.3 19.5

KF 6.6 5.9 18.3 6.3 16.5 17.1 18.6 7.7

CsF 17.5 5.8 24.9 18.4 8.3 20.3 21.8 5.2

Average 7.6 18.6 13.8 12.4 22.5 20.2 22.2 18.3

100 MRE(ρ)

NaCl 2.3 1.1 1.8 0.3 0.2 0.6 1.3 0.2

NaCl (SPC/E) 2.2 1.0 1.8 0.4 0.2 0.6 1.8 0.3

NaCl (278 K) 1.6 0.8 1.3 0.1 0.4 0.3 1.2 0.2

NaCl (308 K) 1.6 1.5 1.2 0.2 0.2 0.9 1.2 0.6

LiCl 1.7 0.8 0.5 2.3 0.7 1.8 1.2

RbCl 5.6 0.4 0.3 1.6 2.9 0.5

KCl 0.1 0.4 1.0 0.1 0.1 0.2 0.7 0.3

CsCl 0.2 0.3 5.7 0.6 0.4 1.2 2.5 0.3

LiI 4.1 2.5 2.0 4.9 3.0 0.8 3.9

NaI 1.8 1.7 1.9 0.7 1.5 3.1 0.4 1.7

KI 0.4 1.0 4.4 1.8 2.0 3.2 1.1 2.4

CsI 0.7 0.5 4.5 1.2 0.7 2.1 0.9 0.7

KF 0.6 1.7 0.4 0.6 2.3 1.3 0.8 1.2

CsF 3.0 1.1 3.1 1.7 3.3 0.2 0.6 2.5

Average 1.6 1.0 2.6 0.8 1.3 1.4 1.3 1.1

the other force fields. The best overall agreement with

experimental density data was obtained with the RH-L

force field and the worst with the JJ force field.

The predictions of the dielectric constant generally

scatter more widely around the experimental data than

those of the solution density. For the dielectric constant,

at best average deviations of 7.6 % (HMN-S) and at

worst deviations of 22.5 % (JC-T) were observed. Di-

electric constant calculations with HMN-S were superior

to other models for all studied salts. The best repre-

sentation of the dielectric constant of NaCl brines at all

temperatures was given by the RH-L+TIP4P/εmodel

combination. The worst agreement for the dielectric con-

stant, with discrepancies to experiments of at least 20 %,

was found for the KBFF, RDVH and JC-T force fields,

but all of these models yielded much better molar den-

FIG. 5. Dielectric constant (top) and molar density (bottom)

over true ion mole fraction of alkali fluoride brines at 298 K

and 1 bar. Simulation results for eight ion force fields com-

bined with the TIP4P/εwater model are represented as open

colored symbols as shown in the legend. Experimental data18

for comparison for KF and CsF brines are represented by

cross symbols. Other experimental data for KF brine6,9,13,19

are depicted by plus symbols. The prediction for pure water is

given by a diamond symbol. A correlation87 to experimental

density data is shown as a black line. Statistical uncertainties

of the molar density are within symbol size.

sity predictions. An overview of the deviations between

simulation and experiment is provided in Table II.

In general, low concentration simulations gave bet-

ter predictions for dielectric constant and molar den-

sity, which is not surprising as the water-water interac-

tions dominate the solution behavior. By contrast, the

dielectric constant of KCl brine, calculated only up to

0.024 mol mol−1(as no reliable data for the dielectric

constant are available at higher concentrations) was in

poor agreement with experimental data.

Qualitative disagreement with experimental dielectric

constant data, as well as over- and underestimation of

density data, was found for some alkali fluoride mod-

els, cf. Figure 5. Taking into account that difficulties

in modeling alkali fluorides, especially regarding the sol-

ubility, have been reported in the literature,37–39,50 we

suggest that results for these brines should be analyzed

with care.

Interestingly, the models that performed overall best

for the dielectric constant (HMN-S, RH-L and JJ) in

general led to higher values for the dielectric constant

compared to the other models. In some cases, these ion

force fields even overestimated the experimental dielec-

tric constant for more concentrated chloride brines, e.g.

NaCl or LiCl. Predictions higher than the experimental

FIG. 6. Cation-anion radial distribution function over dis-

tance of aqueous NaCl (0.076 mol mol−1), KCl (0.024 mol

mol−1), KI (0.083 mol mol−1) and LiI (0.083 mol mol−1) so-

lutions at 298 K and 1 bar. Simulation results for eight ion

force fields combined with the TIP4P/εwater model are rep-

resented by colored lines as shown in the legend.

dielectric constant were also found for those model sets

in case of aqueous LiCl before.70

Some of the models yielded much higher predictions of

the dielectric constant than others, especially for more

concentrated brines. In order to find a possible explana-

tion, the cation-anion RDF g+,−(r) of four electrolyte

solutions (NaCl, KCl, KI and LiI) was sampled, as shown

in Figure 6.

Positions and magnitudes of the first RDF maximum,

corresponding to the contact distance and likelihood of

contact ion pairs (CIP), respectively, differ significantly

between various model sets. The lowest contact distances

were observed for the HMN-S and JC-T models and the

highest for the JJ and RDVH models. Positions of the

first RDF maximum shift to larger distances with increas-

ing cation or anion size.

Very high first peaks, suggesting excessive ion pairing,

were formed by the HMN-S model set for the studied

salt solutions in Figure 6. In addition, the ordered oscil-

lating behavior of subsequent peaks obtained with this

model set (except for LiI) indicates crystal formation in

the aqueous solution below the experimental solubility

limit. Pronounced oscillations of the RDF curves, which

suggest crystallization, were also observed in case of the

RH-L and JJ models for NaCl, as well as for the KBFF

and JJ models for KI. Models that have the lowest first

peaks for NaCl, i.e. JC-T and RDVH, accompanied by

MP and KBFF for LiI, gave a higher second than first

RDF maximum, which corresponds to a higher likelihood

of solvent-shared ion pairs (SIP) than CIP. In SIP, a sin-

gle solvent shell separates the partner ions, while in CIP

FIG. 7. Snapshots of NaCl configurations modeled by HMN-

S (top), MP (center) and RDVH (bottom) force fields in

TIP4P/εwater at 298 K, 1 bar and 0.076 mol mol−1. Vi-

olet and green spheres represent Na+and Cl−, respectively.

Water particles are left out for clarity.

there are no solvent molecules between the cations and

anions.4In nearly all other cases, higher first peaks were

followed by lower subsequent peaks, as it would be ex-

pected for a solution of a completely dissociated salt.

The magnitude of the RDF maximum can provide a

relationship with the over- and underestimation of the

dielectric constant for a given ion model. In order to

establish a connection between the present predictions

and excessive ion pairing and solubility, the number of

contact ion pairs nCIP was calculated from the cation-

anion RDF according to Eq. (5), as listed in Table III.

Based on the number of contact ion pairs, the solubil-

ity of 1:1 electrolytes in water may be roughly estimated

with the ionic pairs rule proposed by Benavides et al.52

For salts with a model solubility limit below 10 mol kg−1,

nCIP >0.5 is an indicator for a supersaturated solution,

i.e. the solubility of the model is lower than the cur-

rently investigated concentration. The rule of Benavides

et al. is, however, not applicable for salts with a model

solubility limit larger than 10 mol kg−1. Salts with an

experimental solubility limit above 10 mol kg−1, such as

LiCl, CsCl, LiI, NaI and KF, were not treated by the

ionic pairs rule. Instead, for these brines, we looked at

the snapshot of the final sampled configuration to check

visually whether the solution was supersaturated or crys-

tals were formed. All brines and ion force fields studied

in this manner at the highest concentration are listed in

Table III.

Models with nCIP >0.5 predict a higher dielectric

constant than models with nCIP <0.5, as less water

molecules are needed to form hydration shells and thus

TABLE III. Number of contact ion pairs nCIP calculated from

the cation-anion RDF of each brine and each ion force field

at 298 K, 1 bar and the highest investigated concentration.

The asterisk indicates a supersaturated solution (nCIP >0.5)

according to the ionic pairs rule of Benavides et al.52, which

can be employed for brines with a model solubility limit below

10 mol kg−1. Brines with an experimental solubility limit

above 10 mol kg−1are listed below the horizontal line. For

these brines, supersaturated solutions indicated by an asterisk

were observed in the snapshots of the final configuration for

the investigated model combination.

Model HMN Fyta JJ RH JC KBFF RDVH MP

salt

NaCl 2.97* 0.84* 3.74* 2.97* 0.14 0.69* 0.28 0.57*

NaClSPC/E 2.85* 0.85* 3.69* 2.97* 0.15 0.66* 0.26 0.57*

NaCl278K 2.69* 0.78* 3.25* 2.23* 0.10 0.62* 0.27 0.52*

NaCl308K 1.13* 0.11 1.42* 1.57* 0.09 0.12 0.24 0.39

RbCl 0.20 0.74* 0.52* 1.61* 1.28* 0.69*

KCl 3.78* 0.22 0.90* 0.55* 0.21 0.80* 0.49 0.43

KI 3.61* 0.45 4.38* 2.33* 1.16* 3.82* 1.46* 2.10*

CsI 4.30* 0.91* 2.43* 0.70* 0.97* 1.24* 1.36* 0.98*

CsF 1.63* 2.20* 0.01 1.63* 0.86* 0.57* 0.83* 1.90*

LiCl 1.36* 1.66* 1.81* 0.32 0.27 0.48 0.64

CsCl 4.26* 1.09 1.59* 1.18 0.94 0.20 1.57 0.86

LiI 0.75* 0.56 0.85* 0.03 0.04 0.25 0.08

NaI 2.65* 0.51 2.00* 1.44* 0.15 0.21 0.19 0.48

KF 3.10* 1.40* 0.14 2.71* 0.61 0.71 0.73 3.58*

the remaining ones can orient freely.

If crystallization was clearly indicated, as in Figure 6

and Table III for the NaCl models HMN-S, RH-L and

JJ, the increase of the dielectric constant was more pro-

nounced. In other words, the reduced model solubility

causes premature precipitation, which in turn increases

the dielectric constant. Striking differences for the so-

lution structure of a NaCl brine are displayed by snap-

shots of the microscopic configurations in Figure 7. We

also prepared snapshots of the final configuration of each

investigated model combination. The visual impression

that they provide was always consistent with the rule of

Benavides et al. Due to the large number of systems, we

refrained from including all these snapshots here or in

the Supplementary material.

To obtain a compact overview of the connection be-

tween the prediction for the dielectric constant and the

number of contact ion pairs, εis shown as a function of

nCIP for each brine and each ion force field at the highest

investigated concentration in Figures 8 and 9.

For most brines, a clear trend is visible in Figures 8

and 9: points are either found in the lower left or in the

upper right part of the plot. This means that all ion force

fields that yielded a comparably high dielectric constant

at high concentrations were well beyond their solubility

limit. The models that did not show this premature pre-

cipitation behavior also always predicted a lower dielec-

tric constant.

Hence, a higher number of contact ion pairs caused

FIG. 8. Dielectric constant as a function of the number of

contact ion pairs for alkali chloride brines at the highest in-

vestigated concentration. Solid symbols indicate that the so-

lution was likely supersaturated or premature crystallization

was observed. The dashed vertical line at nCIP = 0.5 indi-

cates the limit beyond which this is likely to occur according

to the rule of Benavides et al.52

an increase of the dielectric constant and thus led to a

seemingly better quantitative agreement with experimen-

tal data. The argument of a very recent study by Seal

et al.72 that stronger ion pairing leads to a reduced di-

electric constant decline at higher NaCl concentrations

is thus confirmed here. Another study of the dielectric

spectra of NaCl solutions using the GROMOS90 force

field also found an increment of the dielectric constant

due to excessive ion pairing.91

Table III clearly indicates that most of the ion force

fields showed premature precipitation for many of the

studied brines. Precipitation not only falsifies results

FIG. 9. Dielectric constant as a function of the number of con-

tact ion pairs for alkali iodide and fluoride brines at the high-

est investigated concentration. Solid symbols indicate that

the solution was likely supersaturated or premature crystal-

lization was observed. The dashed vertical line at nCIP =

0.5 indicates the limit beyond which this is likely to occur

according to the rule of Benavides et al.52

for the dielectric constant, but also for any other ther-

modynamic property. Exceptions were the JC-T and

RDVH force fields, for which only four of the investi-

gated brines were probably above the solubility limit. In

fact, Mouˇcka et al.46 tested the solubility of thirteen force

fields for NaCl brine with SPC/E water, among them all

ion force fields of this study, except for JJ and MP. They

also found that only two ion model sets (JC-SPC/E and

the medium model set of Horinek et al.) did not show

premature precipitation.

The seemingly good reproduction of both dielectric

constant and molar density with some ion models is,

therefore, probably associated with premature crystal-

lization. Hence, it may be only a coincidence of an un-

physical behavior of the respective model, which is thus

not useful.

The increase of the dielectric constant with strong ion

pairing is contrary to the expected existence of real ion

pair species in strong 1:1 electrolyte solutions. The di-

electric constant has to be very low (ε < 30) under ambi-

ent conditions for uni-univalent ion pairs to be formed.4

In fact, the strong drop of the sampled dielectric constant

of aqueous NaCl observed before at higher temperatures

was the cause of enhanced ion pairing.92

Since strong attraction of cation and anion in solu-

tion initiates excessive contact ion pairing, causes for the

specific dielectric constant behavior of some models may

eventually lie in unbalanced force field parameters. In

general, similarly sized ions (small-small and large-large)

favor ion association in water, while a combination of

small and large ions readily dissociates.93–95

In addition to that is the counterintuitive behavior of

the LJ energy parameter for ions in water, for which

lower values paradoxically lead to a softer repulsion and

stronger electrostatic attraction.41,91

Since all repulsive and dispersive interactions were de-

scribed in this study with 12 LJ parameters for like and

unlike ion and water interactions, it is not straightfor-

ward to establish a clear connection between the LJ pa-

rameters and the present results. However, in order to

relate the findings of this study to the force field param-

eters of the ions, the ratio between the cation and anion

LJ size and energy parameters for each force field is listed

in Table S.X.

Ion models that predicted a high dielectric constant,

such as HMN-S, had generally similar cation and anion

sizes and low LJ energy parameters. On the other hand,

ion models with higher LJ energies and strong deviations

to the size ratio of unity, such as JC-T, gave a lower

number of contact ion pairs. Low LJ cation diameters of

the JC-T models are an indication for an increased model

solubility96 and stronger ion solvation34 in water.

IV. CONCLUSIONS

Eleven aqueous alkali halide solutions were studied

with respect to the ion concentration dependence of the

dielectric constant and molar density under ambient con-

ditions by molecular simulation. For that purpose, eight

non-polarizable force fields for the ions and the TIP4P/ε

model for water were employed.

Almost all ion force fields predicted a decrease of the

dielectric constant with increasing salt concentration for

the considered electrolyte solutions. The quantitative

agreement with experimental data was poor in many

cases. However, the drop of the dielectric constant with

increasing temperature for NaCl brine was confirmed.

The solution density was predicted well with most ion

models. The TIP4P/εwater model somewhat improved

the predictions of the ion force fields regarding dielectric

constant and molar density compared to SPC/E.

The key finding of this study is that the generally

higher dielectric constant predictions of some ion mod-

els are associated with excessive ion pairing and/or pre-

mature crystallization. This may explain the seemingly

good dielectric constant and density agreement with ex-

periments for these models or in some cases the over-

estimation of the experimental dielectric constant. The

cause probably lies in unbalanced force field parameters

and hence reduced model solubility. These findings are

contrary to the expected formation of ion pairs with a

strong drop of the dielectric constant for 1:1 electrolyte

solutions under ambient conditions.

It is clear that further investigations of the dielec-

tric constant of aqueous electrolyte solutions require

ion models predicting solubilities close to experimental

ones. For example, the recently proposed Madrid-2019

force fields97,98 with scaled charges showed good solubil-

ity predictions. Reduced charges were used in the ex-

cellent reproduction of the experimental dielectric con-

stant for NaCl, NaBr, KCl and KBr solutions with the

TIP4P/ε88,99 water model. However, these findings were

compromised with poor solubility and activity predic-

tions of the NaCl/ε+TIP4P/εcombination.100 In the

light of the present conclusions, questions arise whether

an ion parametrization to the dielectric constant makes

sense without an adequate reproduction of the experi-

mental solubility limit.

SUPPLEMENTARY MATERIAL

See Supplementary material for tabulated numerical

simulation values and force field parameters used for the

water and ion models in this study.

ACKNOWLEDGMENTS

We gratefully acknowledge the Paderborn Center for

Parallel Computing (PC2) for the generous allocation

of computer time on the OCuLUS and Noctua clusters.

D. S. gratefully acknowledges the support by the Konrad-

Adenauer-Stiftung e.V.

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Supplementary material

Dielectric Constant and Density of Aqueous Alkali Halide Solutions by Molecular

Dynamics: A Force Field Assessment

Denis Saric,1Maximilian Kohns,2and Jadran Vrabec3, a)

1)Thermodynamics and Energy Technology, University of Paderborn, 33098 Paderborn,

Germany

2)Laboratory of Engineering Thermodynamics, Technische Universit¨at Kaiserslautern,

67633 Kaiserslautern, Germany

3)Thermodynamics and Process Engineering, Technical University Berlin, 10587 Berlin,

Germany

(Dated: 26 March 2020)

a)Electronic mail: vrabec@tu-berlin.de

TABLE S.I. Force field parameters and magnitude of the dipole moment resulting from the distri-

bution of partial charges of the TIP4P/εand SPC/E water models.a

Model σ(˚

A) ε/kB(K) qH(e) dOM (˚

A) µ(D)

TIP4P/ε3.165 93.0 0.527 0.105 2.434

SPC/E 3.166 78.2 0.424 2.351

aThe negative charge is placed on a fourth (virtual) site M in TIP4P water models so that qM=−2qH.

TABLE S.II. Lennard-Jones size and energy parameters of alkali ion force fields considered in this

work.

Li+Na+K+Rb+Cs+Ref.

Model σ(˚

A) ε/kB(K) σ(˚

A) ε/kB(K)

HMN-S 2.87 0.07 3.81 0.07 4.53 0.07 5.17 0.07 [1]

Fyta 2.58 50.3 2.69 293.5 3.49 39.1 [2,3]

JJ 2.87 0.25 4.07 0.25 5.17 0.25 5.60 0.25 6.20 0.25 [4]

RH-L 2.68 0.41 3.21 2.15 3.43 23.8 3.62 41.3 3.91 66.6 [5]

JC-T 1.44 52.4 2.18 84.8 2.83 140.7 3.05 218.1 3.36 198.6 [6]

KBFF 1.82 84.2 2.45 38.5 3.34 15.6 3.62 18 4.13 0.78 [7,8]

RDVH 1.88 200 1.89 200 2.77 200 3.26 200 3.58 200 [9,10]

MP 1.72 29 2.50 39.4 3.18 59.6 3.30 121.1 3.44 252.4 [11]

TABLE S.III. Lennard-Jones size and energy parameters of halide ion force fields considered in

this work.

F−Cl−Br−I−Ref.

Model σ(˚

A) ε/kB(K) σ(˚

A) ε/kB(K)

HMN-S 3.43 55.3 4.40 50.5 4.83 25.3 5.33 19.2 [1]

Fyta 4.16 1.80 4.40 50.35 4.63 49.6 5.83 1.80 [2,3]

JJ 3.05 357.5 4.02 357.5 4.28 357.5 4.81 357.5 [4]

RH-L 3.30 47.5 4.09 82.0 4.31 104.5 4.76 114.1 [5]

JC-T 4.52 0.79 4.92 5.87 4.93 15.3 5.26 21 [6]

KBFF 3.70 120.3 4.40 56.5 4.76 36.1 5.35 24.1 [7,8]

RDVH 3.66 200 4.41 200 4.54 200 4.78 200 [9,10]

MP 3.95 3.26 4.61 12.6 4.81 18.1 5.20 21.3 [11]

TABLE S.IV. Experimental and simulation data for dielectric constant and density of pure water

with the TIP4P/εand SPC/E water models. The numbers in parentheses denote the statistical

uncertainties of the dielectric constant in the last given digits, while those of the density are

negligibly small.

T= 278 K T= 298 K T= 308 K

ε ρ (mol l−1)ε ρ (mol l−1)ε ρ (mol l−1)

Exp.12,13 86 55.51 78.5 55.35 74.9 55.35

TIP4P/ε85 (2) 55.49 77.4 (15) 55.36 74.3 (12) 55.18

SPC/E 69.0 (13) 55.26

TABLE S.V. Experimental and simulation data for the dielectric constant εof alkali chloride brines

at 298 K, 1 bar and specified concentrations. The asterisk indicates a simulation point where a

supersaturated solution is expected according to the ionic pairs rule of Benavides et al.14 or for

which the snapshot of the final sampled configuration clearly showed the presence of a crystal

phase. The numbers in parentheses denote the statistical uncertainties in the last given digits.

salt NaCl + TIP4P/εNaCl + SPC/E

xion (mol mol−1) 0.026 0.049 0.076 0.026 0.049 0.076

Exp.13 62.2 52.4 43.5 62.2 52.4 43.5

Model

HMN-S 60.3 (13) 56.8 (12) * 54.6 (11) 51.3 (1) *

Fyta 51.9 (11) 41.3 (8) * 47.4 (9) 38.6 (7) *

JJ 56.9 (12) 52.9 (11) * 52.1 (10) 50.7 (11) *

RH-L 56.3 (12) 49.6 (11) * 50.9 (9) 49.8 (10) *

JC-T 50.3 (11) 40.1 (7) 28.9 (6) 46.4 (7) 35.7 (6) 25.2 (4)

KBFF 49.6 (9) 42.2 (9) * 47.2 (8) 37.9 (7) *

RDVH 52.4 (12) 39.2 (9) 28.7 (5) 47.9 (10) 34.7 (6) 24.3 (4)

MP 51.3 (7) 40.1 (6) * 47.1 (9) 38.4 (6) *

salt LiCl RbCl

xion (mol mol−1) 0.025 0.049 0.095 0.022 0.057 0.063

Exp.15,16 60.3 48.6 33.3 66.0 55.0 53.7

Model

HMN-S 52.7 (11) 48.2 (12) *

Fyta

JJ 56.5 (13) 50.7 (11) * 53.7 (12) 42.8 (10) 39.0 (8)

RH-L 58.8 (13) 51.0 (11) * 54.7 (12) 41.6 (8) *

JC-T 49.3 (12) 40.1 (8) 23.0 (5) 54.6 (11) 38.2 (8) *

KBFF 50.7 (10) 39.1 (7) 23.2 (5) 54.7 (12) 43.2 (9) *

RDVH 51.2 (10) 37.6 (7) 23.9 (4) 56.2 (11) 41.7 (8) *

MP 51.7 (11) 41.9 (1) 28.4 (6) 57.7 (13) 41.4 (8) *

salt KCl CsCl

xion (mol mol−1) 0.007 0.015 0.024 0.013 0.023 0.068

Exp.16,17 75.5 71.2 65.9 72.0 68.0 57.6

Model

HMN-S 66.0 (14) 64.2 (15) * 67.2 (17) 63.4 (15) *

Fyta 64.3 (15) 58.3 (12) 50.9 (10) 60.4 (13) 56.3 (12) 36.3 (9)

JJ 63.4 (13) 61.8 (13) * 56.1 (12) 52.6 (12) 36.2 (9)

RH-L 64.6 (14) 58.2 (12) * 62.8 (13) 53.6 (12) 37.6 (9)

JC-T 62.6 (14) 57.7 (12) 49.9 (10) 60.7 (14) 53.4 (12) 36.3 (8)

KBFF 64.7 (15) 59.0 (13) * 59.5 (12) 53.0 (11) 35.9 (9)

RDVH 63.9 (13) 58.1 (11) 53.2 (11) 62.0 (13) 54.3 (12) 37.6 (7)

MP 63.3 (14) 58.3 (12) 50.4 (10) 60.8 (12) 55.2 (11) 36.4 (8)

TABLE S.VI. Experimental and simulation data for the dielectric constant εof alkali iodide and

fluoride brines at 298 K, 1 bar and specified concentrations. The asterisk indicates a simulation

point where a supersaturated solution is expected according to the ionic pairs rule of Benavides et

al.14 or for which the snapshot of the final sampled configuration clearly showed the presence of

a crystal phase. The numbers in parentheses denote the statistical uncertainties in the last given

digits.

salt LiI NaI

xion (mol mol−1) 0.034 0.049 0.083 0.013 0.036 0.072

Exp.18–20 52.4 44.7 32.7 67.0 55.5 45.7

Model

HMN-S 46.6 (11) 39.9 (12) * 60.6 (13) 53.8 (11) *

Fyta 55.9 (12) 42.0 (9) 29.6 (5)

JJ 45.9 (8) 38.3 (10) * 59.7 (13) 48.4 (10) *

RH-L 47.0 (12) 42.4 (11) * 59.5 (13) 48.2 (9) *

JC-T 43.9 (12) 37.0 (8) 21.7 (4) 57.9 (13) 41.3 (8) 26.0 (5)

KBFF 43.0 (10) 34.8 (8) 21.4 (5) 56.0 (10) 42.4 (8) 27.5 (5)

RDVH 46.2 (10) 37.3 (7) 24.1 (4) 57.6 (12) 41.6 (7) 27.5 (5)

MP 44.0 (10) 36.0 (6) 22.4 (4) 58.1 (12) 44.4 (9) 30.0 (5)

salt KI CsI

xion (mol mol−1) 0.034 0.049 0.083 0.017 0.034 0.045

Exp.21,22 65.3 55.2 42.6 69.0 62.5 57

Model

HMN-S 60.3 (14) 54.4 (13) * 65.5 (15) 61.1 (14) *

Fyta 44.7 (9) 38.5 (8) 26.4 (5) 58.2 (11) 47.3 (11) *

JJ 55.1 (12) 52.9 (12) * 58.9 (14) 48.8 (11) *

RH-L 51.4 (12) 46.1 (9) * 59.3 (13) 51.1 (11) *

JC-T 46.6 (9) 40.7 (8) * 57.4 (12) 45.6 (9) *

KBFF 50.6 (11) 47.9 (10) * 56.2 (12) 49.7 (11) *

RDVH 47.4 (10) 42.4 (8) * 57.4 (13) 48.8 (10) *

MP 48.8 (11) 41.5 (7) * 57.2 (12) 48.8 (11) *

salt KF CsF

xion (mol mol−1) 0.017 0.048 0.063 0.034 0.048 0.063

Exp.23 67.9 55.4 51.1 64.6 58.8 52.7

Model

HMN-S 62.5 (17) 58.4 (17) * 53.7 (16) 48.2 (15) *

Fyta 60.5 (18) 55.8 (23) * 58.8 (20) 57.3 (22) *

JJ 58.1 (16) 46.2 (18) 38.9 (19) 48.9 (17) 46.3 (17) 37.4 (16)

RH-L 59.8 (15) 55.8 (22) * 45.8 (17) 54.3 (23) *

JC-T 60.0 (18) 48.5 (20) 38.1 (17) 58.4 (25) 54.7 (24) *

KBFF 58.4 (13) 45.1 (10) 41.5 (9) 51.8 (11) 46.6 (9) *

RDVH 56.9 (11) 45.9 (9) 39.6 (8) 51.5 (11) 45.1 (10) *

MP 61.1 (18) 64.4 (23) * 62.8 (26) 63.3 (28) *

TABLE S.VII. Correlation and simulation data for the molar density ρin mol l−1of alkali halide

brines at 298 K, 1 bar and specified concentrations. The asterisk indicates a simulation point

where a supersaturated solution is expected according to the ionic pairs rule of Benavides et al.14

or for which the snapshot of the final sampled configuration clearly showed the presence of a crystal

phase. Statistical uncertainties are negligibly small.

salt NaCl + TIP4P/εNaCl + SPC/E LiCl RbCl

xion (mol mol−1) 0.026 0.049 0.076 0.026 0.049 0.076 0.025 0.049 0.095 0.022 0.057 0.063

Corr.24 56.71 57.87 59.16 56.71 57.87 59.16 56.69 57.96 60.34 56.71 57.87 59.16

Model

HMN-S 55.90 56.01 * 55.97 56.09 * 56.05 56.70 *

Fyta 56.33 56.99 * 56.41 57.02 *

JJ 56.13 56.40 * 56.15 56.40 * 56.48 57.21 * 53.94 51.85 51.54

RH-L 56.84 57.70 * 56.90 57.60 * 56.55 57.55 * 55.72 55.83 *

JC-T 56.89 58.05 59.23 56.93 58.03 59.16 56.09 56.76 58.14 55.50 55.33 *

KBFF 56.55 57.38 * 56.58 57.34 * 57.09 58.51 60.57 55.04 54.26 *

RDVH 56.44 57.15 57.82 56.32 56.85 57.35 56.43 57.19 58.17 54.61 53.25 *

MP 56.87 57.95 * 56.96 58.00 * 56.98 57.50 58.99 55.57 55.13 *

salt KCl CsCl LiI NaI

xion (mol mol−1) 0.007 0.015 0.024 0.013 0.023 0.068 0.034 0.049 0.083 0.013 0.036 0.072

Corr.24 55.49 55.66 55.93 55.07 54.89 54.29 55.49 55.66 55.93 55.07 54.89 54.29

Model

HMN-S 55.48 55.55 * 55.16 55.01 * 53.38 52.67 * 54.77 53.72 *

Fyta 55.60 55.76 56.04 54.91 55.10 54.18 55.02 54.37 53.44

JJ 55.18 54.91 * 53.83 52.70 48.45 54.22 53.59 * 54.74 53.67 *

RH-L 55.57 55.73 * 55.07 54.78 53.36 54.43 53.96 * 55.12 54.59 *

JC-T 55.56 55.69 55.88 55.18 54.99 53.89 53.53 52.83 51.48 55.08 54.46 53.51

KBFF 55.45 55.45 * 54.93 54.56 52.75 54.33 53.86 52.73 54.75 53.63 52.04

RDVH 55.40 55.31 55.19 54.64 54.05 51.48 55.18 54.99 54.44 55.36 55.15 54.62

MP 55.75 55.63 55.76 54.91 55.10 54.18 53.99 53.37 52.02 55.03 54.34 53.34

salt KI CsI KF CsF

xion (mol mol−1) 0.034 0.049 0.083 0.017 0.034 0.045 0.017 0.048 0.063 0.034 0.048 0.063

Corr.24 54.12 53.72 53.02 53.87 52.83 52.31 56.75 59.17 60.28 56.31 56.61 56.89

Model

HMN-S 53.91 53.46 * 54.20 53.25 * 57.09 58.85 * 57.78 58.57 *

Fyta 53.95 53.34 51.99 54.10 53.17 * 57.10 60.85 * 56.77 57.39 *

JJ 52.15 50.92 * 52.19 49.69 * 56.47 58.90 60.12 54.71 54.83 55.05

RH-L 53.36 52.53 * 53.56 51.89 * 56.75 58.48 * 56.96 57.89 *

JC-T 53.28 52.42 * 53.73 52.21 * 57.10 60.46 62.83 57.81 58.79 *

KBFF 52.61 51.82 * 53.22 51.27 * 56.55 58.29 58.97 56.22 56.41 *

RDVH 53.71 52.98 * 53.66 52.07 * 56.72 58.67 59.43 56.03 56.14 *

MP 53.08 52.17 * 53.70 52.24 * 56.53 57.98 * 57.45 58.00 *

TABLE S.VIII. Experimental and simulation data for the dielectric constant εand molar density

ρof sodium chloride brines at 278 and 308 K, 1 bar and specified concentrations. The asterisk

indicates a simulation point where a supersaturated solution is expected according to the ionic pairs

rule of Benavides et al.14 or for which the snapshot of the final sampled configuration clearly showed

the presence of a crystal phase. The numbers in parentheses denote the statistical uncertainties

of the dielectric constant in the last given digits, while those of the density are negligibly small.

salt NaCl (T= 278 K) NaCl (T= 308 K)

xion (mol mol−1) 0.026 0.049 0.078 0.026 0.049 0.064

Dielectric constant εDielectric constant ε

Exp.13 66.4 52.8 40.1 59.5 50.9 46.8

Model

HMN-S 63.1 (14) 57.2 (9) * 57.9 (9) 53.8 (7) *

Fyta 56.6 (15) 46.2 (11) * 49.5 (9) 39.9 (6) 35.7 (6)

JJ 61.8 (16) 58.6 (12) * 55.8 (9) 51.0 (7) *

RH-L 60.8 (18) 52.5 (16) * 55.0 (12) 51.0 (11) *

JC-T 55.5 (15) 41.4 (14) 30.2 (12) 48.8 (10) 37.1 (10) 30.2 (10)

KBFF 56.4 (15) 45.3 (12) * 50.5 (9) 40.9 (7) 35.2 (6)

RDVH 55.7 (15) 41.5 (11) 30.8 (9) 48.9 (9) 37.1 (7) 31.8 (6)

MP 56.4 (17) 43.6 (17) * 50.5 (10) 40.6 (10) 34.1 (10)

salt NaCl (T= 278 K) NaCl (T= 308 K)

xion (mol mol−1) 0.026 0.049 0.078 0.026 0.049 0.064

ρ(mol l−1)ρ(mol l−1)

Exp.13, Corr.24 57.09 58.29 59.67 56.52 57.67 58.27

Model

HMN-S 56.19 56.35 * 55.68 55.80 *

Fyta 56.67 57.39 * 56.11 56.73 57.07

JJ 56.42 56.75 * 55.90 56.18 *

RH-L 57.17 58.27 * 56.58 57.33 *

JC-T 57.30 58.54 59.89 56.64 57.75 58.41

KBFF 56.94 57.85 * 56.31 57.11 57.54

RDVH 56.84 57.64 58.41 56.20 56.88 57.23

MP 56.96 58.00 * 56.84 57.85 58.73

TABLE S.IX. Experimental and simulation data for the dielectric constant εand molar density ρin

mol l−1of alkali halide brines at 298 K, 1 bar and the highest concentration. The asterisk indicates

a simulation point where a supersaturated solution is expected according to the ionic pairs rule

of Benavides et al.14 or for which the snapshot of the final sampled configuration clearly showed

the presence of a crystal phase. Simulation results for solutions for which this is not the case are

provided in Tables S.V to S.VIII. The numbers in parentheses denote the statistical uncertainties

of the dielectric constant in the last given digits, while those of the density are negligibly small.

salt NaCl NaCl SPC/E NaCl278K NaCl308K LiCl RbCl KCl

xion (mol mol−1) 0.076 0.076 0.078 0.064 0.095 0.063 0.024

εε εεεεε

Exp.13,15–17 43.5 43.5 40.10 46.80 33.3 53.7 65.9

Model

HMN-S 49.5 (11)* 46.4 (10)* 54.1 (7)* 52.2 (5)* 37.4 (10)* 62.1 (13)*

Fyta 35.8 (7)* 31.4 (5)* 38.1 (8)*

JJ 49.4 (11)* 47.7 (10)* 51.2 (9)* 49.9 (7)* 41.5 (11)* 53.0 (11)*

RH-L 45.4 (11)* 43.4 (9)* 47.7 (14)* 48.1 (11)* 44.3 (11)* 36.9 (8)* 55.0 (13)*

JC-T 35.4 (7)*

KBFF 34.1 (7)* 29.6 (5)* 34.4 (9)* 41.7 (8)* 53.6 (10)*

RDVH 37.3 (7)*

MP 32.3 (11)* 30.2 (5)* 33.8 (16)* 36.1 (7)*

ρρ ρρρρρ

Exp.13, Corr.24 59.16 59.16 59.67 58.27 60.34 59.16 55.93

HMN-S 56.24* 56.23* 56.66* 55.94* 58.15* 55.65*

Fyta 57.58* 57.56* 58.08*

JJ 56.83* 56.79* 57.27* 56.34* 58.41* 54.42*

RH-L 58.54* 58.50* 59.21* 57.85* 59.43* 55.83* 55.94*

JC-T 55.31*

KBFF 58.17* 58.02* 58.78* 54.15* 55.43*

RDVH 53.04*

MP 59.01* 59.01* 59.01* 55.13*

salt CsCl LiI NaI KI CsI KF CsF

xion (mol mol−1) 0.068 0.024 0.072 0.083 0.045 0.063 0.063

εε εεεεε

Exp.17–23 57.6 55.93 45.7 42.6 57 51.1 52.7

Model

HMN-S 53.5 (12)* 29.9 (9)* 45.8 (11)* 50.1 (12)* 58.2 (13)* 58.8 (18)* 51.1 (17)*

Fyta 52.2 (11)* 55.5 (26)* 53.9 (24)*

JJ 28.9 (10)* 38.8 (7)* 40.8 (9)* 46.9 (12)*

RH-L 34.0 (11)* 37.0 (7)* 37.3 (8)* 46.9 (11)* 51.2 (19)* 50.8 (25)*

JC-T 30.2 (6)* 42.4 (8)* 58.1 (29)*

KBFF 44.6 (9)* 43.1 (9)* 39.3 (8)*

RDVH 32.3 (6)* 44.1 (8)* 40.0 (8)*

MP 35.4 (7)* 43.3 (9)* 58.5 (27)* 48.9 (23)*

ρρ ρρρρρ

Corr.24 54.29 55.93 54.29 53.02 52.31 60.28 57.30

HMN-S 54.25* 51.42* 52.18* 52.66* 52.55* 60.29* 59.43*

Fyta 52.64* 62.95* 58.76*

JJ 52.02* 52.04* 48.32* 48.16*

RH-L 52.94* 53.7* 50.76* 50.87* 60.28* 58.87*

JC-T 50.63* 51.27* 60.49*

KBFF 50.36* 50.10* 56.56*

RDVH 51.34* 51.09* 56.20*

MP 50.30* 51.31* 59.50* 59.73*

TABLE S.X. Ratio between the cation and anion LJ size and energy parameters for 11 alkali halide

salts.

salt LiCl LiI NaCl NaI

Model σ+/σ−ε+/ε−σ+/σ−ε+/ε−σ+/σ−ε+/ε−σ+/σ−ε+/ε−

HMN-S 0.65 0.0015 0.54 0.0038 0.87 0.0015 0.71 0.0038

Fyta 0.59 1.0000 0.44 27.9067

JJ 0.71 0.0007 0.60 0.0007 1.01 0.0007 0.85 0.0007

RH-L 0.66 0.0050 0.56 0.0036 0.79 0.0263 0.67 0.0189

JC-T 0.29 8.9172 0.27 2.4932 0.44 14.4439 0.42 4.0385

KBFF 0.41 1.4894 0.34 3.5000 0.56 0.6809 0.46 1.6000

RDVH 0.43 1.0000 0.39 1.0000 0.43 1.0000 0.40 1.0000

MP 0.37 2.3046 0.33 1.3664 0.54 3.1279 0.48 1.8545

salt KF KCl KI CsF

Model σ+/σ−ε+/ε−σ+/σ−ε+/ε−σ+/σ−ε+/ε−σ+/σ−ε+/ε−

HMN-S 1.32 0.0013 1.03 0.0015 0.85 0.0038 1.51 0.0013

Fyta 0.65 162.6667 0.61 5.8290 0.46 162.6667 0.84 21.6667

JJ 1.70 0.0007 1.29 0.0007 1.07 0.0007 2.03 0.0007

RH-L 1.04 0.5015 0.84 0.2906 0.72 0.2090 1.18 1.4012

JC-T 0.63 177.4156 0.58 23.9648 0.54 6.7005 0.74 250.4011

KBFF 0.90 0.1300 0.76 0.2766 0.62 0.6500 1.12 0.0065

RDVH 0.76 1.0000 0.63 1.0000 0.58 1.0000 0.98 1.0000

MP 0.81 18.2985 0.69 4.7282 0.61 2.8033 0.87 77.5406

salt CsCl CsI RbCl

Model σ+/σ−ε+/ε−σ+/σ−ε+/ε−σ+/σ−ε+/ε−

HMN-S 1.18 0.0015 0.97 0.0038

Fyta 0.79 0.7764 0.60 21.6667

JJ 1.54 0.0007 1.29 0.0007 1.39 0.0007

RH-L 0.96 0.8119 0.82 0.5840 0.88 0.5040

JC-T 0.68 33.8234 0.64 9.4569 0.62 37.1435

KBFF 0.94 0.0138 0.77 0.0325 0.82 0.3191

RDVH 0.81 1.0000 0.75 1.0000 0.74 1.0000

MP 0.75 20.0360 0.66 11.8791 0.72 9.6123

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