^{1,2}and Roland R. Netz

^{1,3}

### Abstract

Using molecular dynamics (MD) simulations in conjunction with the SPC/E water model, we optimize ionic force-field parameters for seven different halide and alkali ions, considering a total of eight ion-pairs. Our strategy is based on simultaneous optimizing single-ion and ion-pair properties, i.e., we first fix ion-water parameters based on single-ion solvation free energies, and in a second step determine the cation-anion interaction parameters (traditionally given by mixing or combination rules) based on the Kirkwood-Buff theory without modification of the ion-water interaction parameters. In doing so, we have introduced scaling factors for the cation-anion Lennard-Jones (LJ) interaction that quantify deviations from the standard mixing rules. For the rather size-symmetric salt solutions involving bromide and chloride ions, the standard mixing rules work fine. On the other hand, for the iodide and fluoride solutions, corresponding to the largest and smallest anion considered in this work, a rescaling of the mixing rules was necessary. For iodide, the experimental activities suggest more tightly bound ion pairing than given by the standard mixing rules, which is achieved in simulations by reducing the scaling factor of the cation-anion LJ energy. For fluoride, the situation is different and the simulations show too large attraction between fluoride and cations when compared with experimental data. For NaF, the situation can be rectified by increasing the cation-anion LJ energy. For KF, it proves necessary to increase the effective cation-anion Lennard-Jones diameter. The optimization strategy outlined in this work can be easily adapted to different kinds of ions.

The authors wish to thank I. Kalcher, S. Mamatkulov, E. Schneck, and D. Horinek for useful discussions and S. Weeresanghe for an informative correspondence. Financial support is acknowledged from the German-Israeli Foundation for Scientific Research and Development (GIF), the ‘Gender Issue Incentive Funds’ of the Cluster of Excellence in Munich, Germany, and the ‘BioFuS’ project of the Intra-European Marie-Curie Programm (call FP7-PEOPLE-2009-IEF).

I. INTRODUCTION

II. METHODS

A. Simulation details

B. Mixing rules and parameter space

C. Structural analysis

D. Kirkwood-Buff theory of solutions

III. RESULTS AND DISCUSSION

A. Structural properties and Kirkwood-Buff integrals

B. Electrolyte activity derivatives

C. A problematic case: Fluoride

IV. SUMMARY AND CONCLUSIONS

### Key Topics

- Free energy
- 16.0
- Ionic liquids
- 14.0
- Solution thermodynamics
- 13.0
- Solution processes
- 12.0
- Molecular dynamics
- 10.0

##### B01F3/00

## Figures

(a) Radial distribution functions for force field Cs9I1 at 0.31 m and four different values of the LJ energy scaling factor λ_{ɛ}. Dotted(black), dashed(red), dot-dashed(blue), and solid(green) lines correspond to values of λ_{ɛ} 0.5, 0.6, 0.8, and 1.0, respectively. The different RDFs for all components of the solution, anions, cations, and water are shown. (b) The different Kirkwood-Buff integrals for Cs9I1 at 0.31 m and four different parameters of the scaling factor λ_{ɛ} as a function of the upper integral boundary. Lines, colors and labeling are the same as in panel (a). (c) The composite Kirkwood-Buff integrals *G* _{ cc } and *G* _{ c } _{ w } as defined in Eq. (5) for Cs9I1 at 0.31 m, which enter the calculation of the activity derivative in Eq. (4a). The curves are shown again for four different values of λ_{ɛ}, with line and color coding as in panel (a). The insets zoom into the values for λ_{ɛ} = 0.6, 0.8, 1.0. All Kirkwood-Buff integrals are given in units nm^{3}.

(a) Radial distribution functions for force field Cs9I1 at 0.31 m and four different values of the LJ energy scaling factor λ_{ɛ}. Dotted(black), dashed(red), dot-dashed(blue), and solid(green) lines correspond to values of λ_{ɛ} 0.5, 0.6, 0.8, and 1.0, respectively. The different RDFs for all components of the solution, anions, cations, and water are shown. (b) The different Kirkwood-Buff integrals for Cs9I1 at 0.31 m and four different parameters of the scaling factor λ_{ɛ} as a function of the upper integral boundary. Lines, colors and labeling are the same as in panel (a). (c) The composite Kirkwood-Buff integrals *G* _{ cc } and *G* _{ c } _{ w } as defined in Eq. (5) for Cs9I1 at 0.31 m, which enter the calculation of the activity derivative in Eq. (4a). The curves are shown again for four different values of λ_{ɛ}, with line and color coding as in panel (a). The insets zoom into the values for λ_{ɛ} = 0.6, 0.8, 1.0. All Kirkwood-Buff integrals are given in units nm^{3}.

Results for the activity derivative a_{ cc } for different values of the scaling prefactor λ_{ɛ} at 0.31 m for (a) the Na^{+}-salts, (b) the K^{+}-salts, and (c) the Cs^{+}-salts, respectively. The lines are the respective experimental data as denoted by the labels, the data points denote simulation results. In all panels, the inset image shows the sizes of all ion-pairs with all cation radii increased by 0.04 nm (see text for discussion), while the inset graph shows two representative RDFs, for λ_{ɛ} = 1 (see text). In panel (a), the label KBBF denotes an alternative Kirkwood-Buff-derived force field^{42} (see text). In panel (c), results are shown for all combinations of two force field sets for Cs^{+} and I^{−}. In panel (b) the error bar is shown only for one data point for clarity. The error bar is similar for all data points.

Results for the activity derivative a_{ cc } for different values of the scaling prefactor λ_{ɛ} at 0.31 m for (a) the Na^{+}-salts, (b) the K^{+}-salts, and (c) the Cs^{+}-salts, respectively. The lines are the respective experimental data as denoted by the labels, the data points denote simulation results. In all panels, the inset image shows the sizes of all ion-pairs with all cation radii increased by 0.04 nm (see text for discussion), while the inset graph shows two representative RDFs, for λ_{ɛ} = 1 (see text). In panel (a), the label KBBF denotes an alternative Kirkwood-Buff-derived force field^{42} (see text). In panel (c), results are shown for all combinations of two force field sets for Cs^{+} and I^{−}. In panel (b) the error bar is shown only for one data point for clarity. The error bar is similar for all data points.

Activity derivatives a_{ cc } for the salts KF and NaF as a function of the LJ ion-water energy parameter ɛ_{ iO } with LJ radius σ_{ iO } constrained to the line on which the experimental ion solvation free energy is reproduced, similar to our previous study in Ref. 48. For K11F data the K11 force field is fixed and ɛ_{ FO } is varied, for KCl the Cl force field is fixed and ɛ_{ KO } is varied. The horizontal lines correspond to the experimental values for KCl and KF. The results are shown for 1.03 m (1 M) as in Ref. 48. For KF, results are also shown for 0.31 m (0.3 M). All data are given for unmodified mixing rules, i.e., λ_{ɛ} = 1, except for K11F at 1 M with ɛ_{ FO } = 0.1585 kJ/mol, where additional data for the values λ_{ɛ} = 1.2 and λ_{ɛ} = 1.5 are shown (cyan filled squares).

Activity derivatives a_{ cc } for the salts KF and NaF as a function of the LJ ion-water energy parameter ɛ_{ iO } with LJ radius σ_{ iO } constrained to the line on which the experimental ion solvation free energy is reproduced, similar to our previous study in Ref. 48. For K11F data the K11 force field is fixed and ɛ_{ FO } is varied, for KCl the Cl force field is fixed and ɛ_{ KO } is varied. The horizontal lines correspond to the experimental values for KCl and KF. The results are shown for 1.03 m (1 M) as in Ref. 48. For KF, results are also shown for 0.31 m (0.3 M). All data are given for unmodified mixing rules, i.e., λ_{ɛ} = 1, except for K11F at 1 M with ɛ_{ FO } = 0.1585 kJ/mol, where additional data for the values λ_{ɛ} = 1.2 and λ_{ɛ} = 1.5 are shown (cyan filled squares).

Activity derivatives a_{ cc } for the KF solutions as a function of the scaling factors λ_{ɛ} and λ_{σ}. Note that only one of the factors is varied, the other is fixed at unity. The line shows the respective experimental value. The concentration is 0.31 m (0.3 M).

Activity derivatives a_{ cc } for the KF solutions as a function of the scaling factors λ_{ɛ} and λ_{σ}. Note that only one of the factors is varied, the other is fixed at unity. The line shows the respective experimental value. The concentration is 0.31 m (0.3 M).

Cation-anion radial distribution functions *g* _{+−}, Lennard-Jones potentials *V* _{ LJ } , and total (Lennard-Jones plus Coulomb) potentials *V* _{ LJ } + *V* _{ C } for the salt K11F5 at 0.31 m (0.3 M). Note that *V* _{ LJ } (red dotted lines) has been magnified 10 times for clarity. The uppermost panel shows results for unmodified mixing rules with scaling factors λ_{ɛ}, λ_{σ} = 1.0. The left and right panels correspond to the modifications λ_{ɛ} > 1, and λ_{σ} > 1, respectively, where the other scaling factor is unity.

Cation-anion radial distribution functions *g* _{+−}, Lennard-Jones potentials *V* _{ LJ } , and total (Lennard-Jones plus Coulomb) potentials *V* _{ LJ } + *V* _{ C } for the salt K11F5 at 0.31 m (0.3 M). Note that *V* _{ LJ } (red dotted lines) has been magnified 10 times for clarity. The uppermost panel shows results for unmodified mixing rules with scaling factors λ_{ɛ}, λ_{σ} = 1.0. The left and right panels correspond to the modifications λ_{ɛ} > 1, and λ_{σ} > 1, respectively, where the other scaling factor is unity.

(a) Experimental activity derivatives a_{ cc } as a function of the difference *r* _{+} − *r* _{−}, where *r* _{+} and *r* _{−} denote the Pauling radii of the cation and anion, respectively. All salt solutions considered in this work are shown at concentrations of 0.31 m (0.3 M) (upper panel) and 1.03 m (1 M) (lower panel), respectively. The color-coding is for fixed cations. (b) Same as in (a), except for the color-coding which corresponds to fixed anions. (c) Experimental activity coefficients as a function of *r* _{+} − *r* _{−} for different monovalent and divalent salt solutions at 0.31 m (0.3 M) (upper panel) and 1 m (lower panel) concentration.

(a) Experimental activity derivatives a_{ cc } as a function of the difference *r* _{+} − *r* _{−}, where *r* _{+} and *r* _{−} denote the Pauling radii of the cation and anion, respectively. All salt solutions considered in this work are shown at concentrations of 0.31 m (0.3 M) (upper panel) and 1.03 m (1 M) (lower panel), respectively. The color-coding is for fixed cations. (b) Same as in (a), except for the color-coding which corresponds to fixed anions. (c) Experimental activity coefficients as a function of *r* _{+} − *r* _{−} for different monovalent and divalent salt solutions at 0.31 m (0.3 M) (upper panel) and 1 m (lower panel) concentration.

## Tables

Ion-water and bare ionic parameters for the anions and cations used in the current MD simulations. The parameters for Na^{+}, Cl^{−}, and Br^{−} are taken from Refs. 51 and 52, while the rest is taken from Ref. 48. The Pauling radii of the ions are also given.^{60}

Ion-water and bare ionic parameters for the anions and cations used in the current MD simulations. The parameters for Na^{+}, Cl^{−}, and Br^{−} are taken from Refs. 51 and 52, while the rest is taken from Ref. 48. The Pauling radii of the ions are also given.^{60}

Optimal λ_{ɛ}, λ_{σ} scaling prefactors for the cation-anion combinations studied in this work.

Optimal λ_{ɛ}, λ_{σ} scaling prefactors for the cation-anion combinations studied in this work.

Excess coordination numbers for the optimized scaling prefactors given in Table II from our simulation results, denoted as = ρ_{ c } G_{ cc } and = ρ_{ w } G_{ c } _{ w }, and experimental values as taken from the analysis in Ref. 23, denoted by and . Most data are shown for 0.31 m, numbers in parentheses correspond to solutions of 1 m concentration. The activity derivatives are computed through Eq. (4a) using the , MD data, and can be compared to the experimental data given in Refs. 23,64, denoted as ^{64} and ^{23}, respectively.

Excess coordination numbers for the optimized scaling prefactors given in Table II from our simulation results, denoted as = ρ_{ c } G_{ cc } and = ρ_{ w } G_{ c } _{ w }, and experimental values as taken from the analysis in Ref. 23, denoted by and . Most data are shown for 0.31 m, numbers in parentheses correspond to solutions of 1 m concentration. The activity derivatives are computed through Eq. (4a) using the , MD data, and can be compared to the experimental data given in Refs. 23,64, denoted as ^{64} and ^{23}, respectively.

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