^{1}and Thomas L. Beck

^{2}

### Abstract

The Quasichemical Theory (QCT) involves a length-scale organization of solvation thermodynamics. The QCT has been employed in studies of solutes ranging in size from small molecules and ions to proteins. There are three contributions to the QCT free energy: (1) an inner-shell term that includes the direct solute-solvent chemical interactions, (2) an outer-shell packing term that is the free energy to dig out a cavity in the solvent, and (3) an outer-shell long-ranged term that includes all interactions of the solute with the solvent conditional on an empty inner shell. The present study utilizes a regularizing generalization of the QCT and classical simulations to compute these three contributions to the ion hydration free energy out to large cavity radii for eight ions in the alkali halide series. The accuracy of simple approximations for the long-ranged term is also examined. The inner-shell contribution exhibits ion specificity for cavity sizes less than 4–5 Å, followed by a common length scale of 6.15 Å at which its value equals the bulk hydration free energy for all eight of the ions. The 6.15 Å length scale is closely approximated by the distance at which the revised scaled-particle theory packing contribution matches the magnitude of a simple Born estimate for the long-ranged term.

We would like to thank Lawrence Pratt, Hank Ashbaugh, Dilip Asthagiri, Chris Mundy, Greg Schenter, Shawn Kathmann, Marcel Baer, and David Rogers for helpful discussions. This research was supported by the National Science Foundation (Grant No. CHE-1011746) and a generous grant of computing time at the Ohio Supercomputer Center.

I. INTRODUCTION

II. QUASICHEMICAL THEORY

III. INTERFACIAL POTENTIALS

IV. CAVITY GROWTH FOR THE INNER-SHELL AND PACKING TERMS

V. APPROXIMATIONS FOR THE LONG-RANGED TERM

VI. SIMULATION METHODS

VII. RESULTS

VIII. CONCLUSIONS

### Key Topics

- Free energy
- 70.0
- Cavitation
- 23.0
- Electrostatics
- 12.0
- Water energy interactions
- 12.0
- Solvents
- 10.0

## Figures

The expansion process employed to obtain the packing part of the free energy. The expanding potentials (labelled ɛ on the left axis) are shown for 5 evenly spaced values of γ (that controls the cavity size during the expansion process, see Sec. IV below) from left to right, together with the corresponding cavity/water-oxygen radial distribution functions (rdf's, right axis). The data is for a λ value of 6.6 Å. The γ = 1 WCA potential is the rightmost (black) curve.

The expansion process employed to obtain the packing part of the free energy. The expanding potentials (labelled ɛ on the left axis) are shown for 5 evenly spaced values of γ (that controls the cavity size during the expansion process, see Sec. IV below) from left to right, together with the corresponding cavity/water-oxygen radial distribution functions (rdf's, right axis). The data is for a λ value of 6.6 Å. The γ = 1 WCA potential is the rightmost (black) curve.

The TI paths for growing cavities (a) in the pure solvent (b) around the Na+ and Cl− ions, when λ = 6.6 Å. In (a), the integrand (squares) and the cumulative integral (smooth solid curve) for the packing contribution are presented. In (b), the ion-specific curves for the inner-shell contribution for the two ions are presented. The cumulative integrals show that the growing-in of the cavities is a smooth process for the chosen potential.

The TI paths for growing cavities (a) in the pure solvent (b) around the Na+ and Cl− ions, when λ = 6.6 Å. In (a), the integrand (squares) and the cumulative integral (smooth solid curve) for the packing contribution are presented. In (b), the ion-specific curves for the inner-shell contribution for the two ions are presented. The cumulative integrals show that the growing-in of the cavities is a smooth process for the chosen potential.

Solute/water-oxygen rdfs (left axis) and hydration numbers (right axis) for all ions studied. The data for cations is shown in (a), while the data for anions is shown in (b).

Solute/water-oxygen rdfs (left axis) and hydration numbers (right axis) for all ions studied. The data for cations is shown in (a), while the data for anions is shown in (b).

Free energy contributions for the Cl− ion vs. λ. Upward triangles are the packing term, and the line through the data is the SPT fit. Open circles are the inner-shell term, while open blue downward triangles are the long-ranged term computed with the midpoint-rule approximation. Blue squares are the sum of the inner-shell and packing terms, illustrating the cancellation point at 3.48 Å. The horizontal black line at −87.6 kcal/mol is the exact numerical intrinsic free energy, , while the blue x's are the computed totals (using the midpoint-rule approximation for the long-ranged term).

Free energy contributions for the Cl− ion vs. λ. Upward triangles are the packing term, and the line through the data is the SPT fit. Open circles are the inner-shell term, while open blue downward triangles are the long-ranged term computed with the midpoint-rule approximation. Blue squares are the sum of the inner-shell and packing terms, illustrating the cancellation point at 3.48 Å. The horizontal black line at −87.6 kcal/mol is the exact numerical intrinsic free energy, , while the blue x's are the computed totals (using the midpoint-rule approximation for the long-ranged term).

Free energy data for cations, showing the inner-shell and total free energy data. The circles are for the inner-shell terms, and the lines through the data are fit to the larger cavity data (with a consistent slope of roughly −20.3). The horizontal lines are the exact results (intrinsic free energy): −61.8, −72.4, −88.8, and −113.9 kcal/mol for the Cs+, K+, Na+, and Li+ ions, respectively. The x's are the numerical results computed using the midpoint-rule approximation for the long-ranged term. The squares locate the length at which the inner-shell and packing terms cancel.

Free energy data for cations, showing the inner-shell and total free energy data. The circles are for the inner-shell terms, and the lines through the data are fit to the larger cavity data (with a consistent slope of roughly −20.3). The horizontal lines are the exact results (intrinsic free energy): −61.8, −72.4, −88.8, and −113.9 kcal/mol for the Cs+, K+, Na+, and Li+ ions, respectively. The x's are the numerical results computed using the midpoint-rule approximation for the long-ranged term. The squares locate the length at which the inner-shell and packing terms cancel.

Free energy data for anions, showing the inner-shell and total free energy data. The circles are for the inner-shell terms, and the lines through the data are fit to the larger cavity data (with a consistent slope of roughly −21.9). The horizontal lines are the exact results (intrinsic free energy): −72.2, −81.0, −87.6, and −117.2 kcal/mol for the I−, Br−, Cl−, and F− ions, respectively. The x's are the numerical results computed using the midpoint-rule approximation for the long-ranged term. The squares locate the length at which the inner-shell and packing terms cancel.

Free energy data for anions, showing the inner-shell and total free energy data. The circles are for the inner-shell terms, and the lines through the data are fit to the larger cavity data (with a consistent slope of roughly −21.9). The horizontal lines are the exact results (intrinsic free energy): −72.2, −81.0, −87.6, and −117.2 kcal/mol for the I−, Br−, Cl−, and F− ions, respectively. The x's are the numerical results computed using the midpoint-rule approximation for the long-ranged term. The squares locate the length at which the inner-shell and packing terms cancel.

Data for the long-ranged contribution for the Na+ (top) and Cl− (bottom) ions. The open black circles are for the uncoupled sampling fluctuation term, while the open black squares are for the coupled sampling fluctuation term (right axis). The green x's are for the long-ranged term computed with the midpoint rule (left axis). The blue +'s are for the trapezoid-rule approximation (average of two mean-field bounds), and the red stars are for the average of the two Gaussian approximations.

Data for the long-ranged contribution for the Na+ (top) and Cl− (bottom) ions. The open black circles are for the uncoupled sampling fluctuation term, while the open black squares are for the coupled sampling fluctuation term (right axis). The green x's are for the long-ranged term computed with the midpoint rule (left axis). The blue +'s are for the trapezoid-rule approximation (average of two mean-field bounds), and the red stars are for the average of the two Gaussian approximations.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content