Article pubs.acs.org/JPCB
Spatial-Decomposition Analysis of Energetics of Ionic Hydration George Mogami,†,‡ Makoto Suzuki,*,†,‡ and Nobuyuki Matubayasi*,§,∥,‡ †
Department of Materials Processing, Graduate School of Engineering, Tohoku University, Sendai, Miyagi 980-8579, Japan Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai, Miyagi 980-8578, Japan § Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan ∥ Elements Strategy Initiative for Catalysts and Batteries, Kyoto University, Katsura, Kyoto 615-8520, Japan ‡
S Supporting Information *
ABSTRACT: Hydration energetics is analyzed for a set of ions. The analysis is conducted on the basis of a spatial-decomposition formula for the excess partial molar energy of the solute that expresses the thermodynamic quantity as an integral over the whole space of the solute−solvent and solvent−solvent interactions conditioned by the solute−solvent distance. It is observed for all the ionic solutes treated in the present work that the ion− water interaction is favorable at the expense of the water−water interaction and that the variations of the ion−water and water−water interactions with the ion−water distance compensate against each other beyond the contact distance. The extent of spatial localization of the excess partial molar energy is then assessed by introducing a cutoff into the integral expression and examining the convergence with respect to the change in the cutoff. It is found that the excess energy is not quantitatively localized within the first and second hydration layers, while its correlations over the variation of ions are good against the first-layer contribution.
1. INTRODUCTION When dissolved into water, an ion changes the intermolecular interactions of water around it. This change is a subject of intensive studies over several decades, and such concepts as “structure making” and “structure breaking” have emerged.1−8 The water structure has been inferred from thermodynamic, transport, and spectroscopic measurements. They do not necessarily refer to a certain layer around ion, though. A thermodynamic quantity provides information integrated over the whole system, for example. A rigorous framework is thus necessary to fill the gap between an intuitive picture on local perturbation of water around ion and an experimentally accessible quantity. In previous works, the scheme of spatial decomposition was developed to rigorously bridge local correlations of molecules and macroscopic observables.9−14 A key of the development is to formulate an integral expression for the macroscopic observable in terms of molecular correlation functions over the whole space. The formulation was then employed to analyze the extent of localization of the solute’s perturbation on the water structure with respect to partial molar thermodynamic quantities for energetics and volumetrics in hydrophobic hydration.9−11 Dynamic properties were similarly treated, and statistical-mechanical frameworks were provided to discuss the ion-pair contribution to the electrical conductivity12,13 and the dipolar correlations related to the dielectric spectroscopy.14−16 In the present work, we conduct the spatial-decomposition analysis of energetics of hydration for a series of ionic solutes. We focus on the intermolecular interaction of a water molecule resolved by the ion−water distance and its connection to the (excess) partial molar energy of the ion. The analysis is based on a rigorous, statistical-mechanical formula for the partial © XXXX American Chemical Society
molar energy expressed as an integral over the whole space of the ion−water and water−water interactions conditioned by the ion−water distance. On the basis of the integral expression, the spatial extent of the ion’s effects on the intermolecular interaction of water is examined to elucidate quantitatively which layers of water around the ion are responsible for determining the partial molar energy. The analysis can be carried out in combination of molecular dynamics (MD) simulation for a solute with any form and/or charge. A series of monatomic ions with varying size and charge are investigated in this work, and polyatomic ions are a subject of subsequent work. The intermolecular interaction of a water molecule in an ionic solution consists of the ion−water and water−water interactions, and their dependencies on the ion−water distance are treated separately in our analysis. It will then be observed in section 4 that the ion−water interaction becomes favorable at the expense of the water−water interaction. This is a (partial) compensation of the two interactions and may be described more clearly by examining the water−water interaction conditioned by the ion−water pair interaction rather than by the ion−water distance. In the present work, we also develop an integral expression for the (excess) partial molar energy in terms of the water−water interaction energy conditioned by the ion−water energy. It is an “energy-decomposed” formula and satisfies the “sum rule”, as in the spatial-decomposition case, Special Issue: Bruce C. Garrett Festschrift Received: September 29, 2015 Revised: December 6, 2015
A
DOI: 10.1021/acs.jpcb.5b09481 J. Phys. Chem. B XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry B
process of solute insertion and the isochoric (constant-volume) process. References 9 and 11 then showed that the ensemble dependence of Bpure accounts for that of the excess partial molar energy and that eq 1 provides the excess partial molar energy at constant pressure when B0 is taken to be the limiting value of Bv(r), irrespective of whether Bv(r) and B0 are evaluated in NVT or NPT. See refs 9 and 11 for details. In the present work, the ions examined are monatomic and ρ(r), Bu(r), and Bv(r) are expressed as functions of the radial distance r = |r| between the ion and the oxygen site of the water molecule. Actually, r can refer to any point within solute and/or solvent. Equation 1 remains exact even when the origin of the system is set to any point within the solute and/or the position of a solvent molecule is represented by any point in it. In this sense, r (=|r|) can be rather arbitrarily chosen only on the basis of clarity and convenience, and r is taken to be the ion−oxygen distance in section 4. With this definition of r, ρ(r) is given by the product of the bulk density of solvent and the ion−oxygen radial distribution function. The extent of spatial localization of ΔE can be examined by introducing a cutoff λ into the integral of eq 1 through
that the integral over the whole energy range between ion and water provides the observable thermodynamic quantity of partial molar energy.
2. THEORETICAL BACKGROUND The thermodynamic variable treated in the present work is the excess partial molar energy. This is the change in the total energy of the system when the solute is inserted at fixed origin. We treat an aqueous solution at infinite dilution. A single solute molecule is contained in the system and is located at the origin. Let ρ(r) be the density of the solvent molecule (water) at position r; ρ(r) dr is the average number of solvent molecules found in region [r, r + dr]. It was then shown in ref 9 that the excess partial molar energy ΔE is expressed as ΔE =
⎤
⎡
∫ dr ρ(r) ⎢⎣Bu(r) + 12 Bv(r) − 12 B0⎥⎦
(1)
where Bu(r) and Bv(r) are introduced respectively as Bu (r) =
∑i ⟨uuv(i) δ(ri−r)⟩ ∑i δ(ri−r)
=
⟨uuv(i) δ(ri−r)⟩ ⟨δ(ri−r)⟩
(2)
⎡
Bv (r) =
=
∑i ∑j ≠ i uvv(i ,j) δ(ri−r)
(4)
and examining the λ dependence of ΔE(λ). The exact value is recovered in the limit of λ → ∞. When ΔE(λ) reaches the limiting value at λ corresponding to the first layer, for example, the thermodynamic quantity ΔE is spatially localized in the first layer. Similarly, the cutoff λ can be implemented to the solute− solvent and solvent−solvent (solvent reorganization) terms respectively through
∑i δ(ri−r)
∑j ≠ i uvv(i ,j) δ(ri−r) ⟨δ(ri−r)⟩
⎤
∫|r|
