Absolute ion hydration free energy scale and the surface potential of water via quantum simulation Yu Shia and Thomas L. Becka,1 aDepartment of Chemistry, University of Cincinnati, Cincinnati, OH 45221 Edited by Roberto Car, Princeton University, Princeton, NJ, and approved October 26, 2020 (received for review August 13, 2020) With a goal of determining an absolute free energy scale for ion (16). In addition, establishing a single-ion hydration free energy hydration, quasi-chemical theory and ab initio quantum mechan- scale facilitates the development of more accurate force fields ical simulations are employed to obtain an accurate value for the employed in molecular simulations (27, 28). bulk hydration free energy of the Na+ ion. The free energy is We have suggested previously that there may be physical and partitioned into three parts: 1) the inner-shell or chemical con- chemical consequences to solvent-induced potential gradients tribution that includes direct interactions of the ion with nearby through an interface even though the shifts are not directly waters, 2) the packing free energy that is the work to produce a measurable (29). The consequences include nonuniform ion cavity of size λ in water, and 3) the long-range contribution that density profiles (8, 30), altered acid–base chemistry (31–34), involves all interactions outside the inner shell. The interfacial and electric fields probed by spectroscopic measurements at potential contribution to the free energy resides in the long-range droplet surfaces (35). Experimental evidence is accumulating term. By averaging cation and anion data for that contribution, that suggests the effective surface potential of water (pass- cumulant terms of all odd orders in the electrostatic potential ing from the gas phase or a hydrophobic medium into water) are removed. The computed total is then the bulk hydration free is approximately −0.4 V, in agreement with earlier estimates energy. Comparison with the experimentally derived real hydra- (29, 36, 37). tion free energy produces an effective surface potential of water For the sake of the discussion below, consider the case of in the range −0.4 to −0.5 V. The result is consistent with a variety a single spherical monovalent ion located near the center of a of experiments concerning acid–base chemistry, ion distributions large water droplet. We assume the droplet is large enough to near hydrophobic interfaces, and electric fields near the surface obtain a stable value for the average electrostatic potential in of water droplets. the region between rapidly varying zones near the ion and near CHEMISTRY the distant liquid–vapor interface, but not so large as to involve single-ion thermodynamics j hydration free energy j DFT j ab initio extraneous ions from the autoionization of water (or from other quantum molecular dynamics j surface potential sources). The excess chemical potential for the ion X is given by the Widom formula (10) on solvation phenomena are central to diverse condensed- phase physical, chemical, biological, materials, and energy- ex I µX = −kT lnhexp[−"X =kT ]i0, [1] related systems (1–6). The physical behaviors of solutions that include ions of the same charge but with different size, shape, where "X is the total interaction energy of the ion with all solvent and/or chemical makeup can vary dramatically. Well after the molecules and the zero subscript indicates the ion and waters are pioneering work on protein precipitation by Hofmeister, the uncoupled during sampling. Since the Coulomb potential is long study of these specific-ion effects has seen a major resur- ranged, any inhomogeneous charge density regions in the droplet gence in the last two decades due to the recognized relevance contribute to the chemical potential value. to problems spanning fields from fundamental colloid science The traditional expression for the excess chemical potential of and electrochemistry to biological channel proteins and battery an ion is of the form (38) materials (7). Progress has been made due to a set of experimental and Significance theoretical/computational tools that have begun to unravel the fundamental interactions leading to observed behavior. These tools include spectroscopic probes that interrogate specific ions Establishing an absolute free energy scale for single-ion at interfaces (8, 9), molecular theories that partition free ener- hydration has been an elusive goal of condensed-phase chem- gies into distinct physical parts (10–12), and quantum simulation istry for nearly a century. In this paper, a zero for the free methods that include polarization and charge transfer effects not energy scale is determined by employing quasi-chemical the- contained in simple classical models (13–15). ory and quantum mechanical simulations of the sodium ion Tied to the aim of gaining fundamental insights into specific in water. The simulation results produce an accurate estimate ion effects is the long-standing goal of establishing a single-ion of the effective surface potential of water by comparison of free energy scale for hydration (16, 17). While ion pair hydration the computed “bulk” hydration free energy with experimental thermodynamic quantities are directly measurable, estimation values for the “real” hydration free energy. This work provides of the single-ion values typically requires theoretical input (18– a firm quantitative footing for studies of interfacial specific 21). The need for theoretical input is in turn linked to the ion structure, thermodynamics, and kinetics in fields as diverse Gibbs–Guggenheim principle (22–24) that states that the elec- as colloid science, electrochemistry, biophysics, and energy trostatic potential shifts across chemically dissimilar interfaces storage materials. are not measurable. The single-ion surface potential term can- cels when composing the ion pair enthalpy or free energy (25, Author contributions: Y.S. and T.L.B. performed research and wrote the paper.y 26). Further motivations for developing quantitative theories of The authors declare no competing interest.y single-ion free energies include gaining a better understanding This article is a PNAS Direct Submission.y of ion–solvent interactions, ion exchange in resins, ion pair- Published under the PNAS license.y ing and its specificity, binding in or on proteins, distributions 1 To whom correspondence may be addressed. Email: [email protected] near interfaces, ion transfer reactions, and electrode kinetics www.pnas.org/cgi/doi/10.1073/pnas.2017214117 PNAS Latest Articles j 1 of 8 Downloaded by guest on September 28, 2021 ex ex µX = µX ,che + qφ, [2] In the present paper, the target quantity is the bulk free energy ex + µX ,b for the Na ion (right side of Eq. 6). This bulk value ex where µX ,che is a “chemical” value deep in the liquid and φ is a has been shown (36) to correspond to the Marcus value (51) contribution from surface effects. It is clear by comparing Eqs. 1 that includes no interfacial potential contribution. Toward this and 2 that Eq. 2 involves some kind of partitioning of the electro- end we utilize quasi-chemical theory and quantum mechanical static potential. The questions addressed in this paper are, what simulation methodology to partition the free energy into spa- is the physical content of φ, and what is its numerical value? tially resolved and manageable parts. Employing the difference ex ex There are multiple choices for defining the generic interfacial (µX − µX ,b )=q = φnp via Eq. 6, we are able to estimate the effec- potential φ above. First, following refs. 18 and 39 we can com- tive surface potential of water (the net potential) in the absence pute the (planar) liquid–vapor surface potential by integrating of other interfering ions. In other words, we obtain the sol- the Poisson equation through the interface, vent contribution to the net potential. The computational results and experimental values are summarized in the last five lines Z 1 of Table 1 (below). The resulting estimate of the effective sur- φsp = 4π zρ(z)dz, [3] −∞ face potential of water (−0.4 V) implies an electric field near the interface of magnitude 107 V/cm assuming an interfacial width where z is the coordinate perpendicular to the interface and of approximately 5 A.˚ ρ(z) is the charge density (due to electron and nuclear distri- This work builds upon previous efforts aimed at ab ini- butions). It is known that, for realistic charge distributions, this tio real ion hydration free energy calculations (14, 52–54), potential is large and positive (40, 41), with a value of about 4 V but differs by focusing on the bulk value. This avoids diffi- for the water liquid–vapor interface. The large magnitude arises culties related to the zero of the energy scale and finite-size from the internal molecular structure with electrons spatially effects for long-range electrostatics in periodic boundary simu- distributed around point-like nuclei (producing the quadrupo- lations. Once a value is established for one ion, free energies lar Bethe potential). The associated intrinsic chemical term for other ions can be obtained from thermodynamic measure- ex µX ,int (42) includes all local interactions of the ion with the ments. Our results resolve a recurrent debate concerning the solvent, including a large approximately canceling quadrupole surface electrostatic contribution to single-ion hydration free term. Thus, while the sum produces the “real” excess chemical energies. ex potential µX , We first outline the necessary theory for analysis of the ab ex ex µX = µX ,int + qφsp , [4] initio molecular dynamics (AIMD) simulations at the density the two contributing terms are of very large magnitude. Another functional theory (DFT) level. We then present the results with proposed partitioning of the surface potentials involves the dipo- relevant discussion and finish with our conclusions. The compu- lar surface potential for the liquid–vapor interface, as opposed tational methods are summarized following the conclusions.