# Multiscale Methods Framework with the 3D-RISM-KH Molecular Solvation Theory for Supramolecular Structures, Nanomaterials, and Biomolecules: Where Are We Going?

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

_{γ}g

_{γ}(

**r**) of finding interaction site γ of solvent molecules at 3D space position

**r**around a solute molecule or arbitrary shape, with the average solvent site number density ρ

_{γ}in the solution bulk and the normalized solvent site density distribution g

_{γ}(

**r**). This 3D solvent site distribution function g

_{γ}(

**r**) assumes the values g

_{γ}(

**r**) > 1 or g

_{γ}(

**r**) < 1 in the areas of density enhancement or depletion, respectively. At a distance from the solute, it levels out to g

_{γ}→1. The 3D solvent site distribution functions around the solute molecule are obtained from the 3D-RISM integral equation

_{γ}(

**r**) = g

_{γ}(

**r**) − 1 is the 3D total correlation function of solvent interaction site γ, c

_{γ}(

**r**) ~ −u

_{γ}(

**r**)/(k

_{B}T) is the 3D direct correlation function which has the asymptotics of the solute–solvent site interaction potential u

_{γ}(

**r**), the site-site susceptibility of pure solvent χ

_{αγ}(r) is an input to the 3D-RISM theory, indices α and γ enumerate all interaction sites on all sorts of solvent species, T is temperature, and k

_{B}is the Boltzmann constant [14,15,16,17,18,19,20,21,22]. A closure relation connecting the 3D total and direct correlation functions complements the 3D-RISM integral of Equation (1), providing a computational handle to integrate the infinite chain of diagrams. The exact closure can be formally expressed as a series in terms of multiple integrals of the combinations of the total correlation functions, which is cumbersome, and in practice, is replaced with tenable approximations. For instance, the KH closure approximation accounts, in a consistent manner, for both electrostatic and non-polar features (i.e., associative and steric effects) of solvation in simple and complex liquids, and has the form:

_{γ}(

**r**) is the 3D interaction potential between the whole solute and solvent site γ specified by the molecular force field. The 3D-KH closure in Equation (2) couples, in a nontrivial way, the so-called mean spherical approximation (MSA) applied to spatial regions of solvent density enrichment g

_{γ}(

**r**) > 1 and the hypernetted chain (HNC) one applied to the regions of density depletion g

_{γ}(

**r**) < 1. The 3D solvent site distribution function and its first derivative are continuous at the joint boundary g

_{γ}(

**r**) = 1 by construct. The 3D-RISM-HNC theory is known to overestimate solvation structures in strongly associated systems, thus imparting numerical instability and also diverging for strongly charged systems like electrolytes in solution. On the other hand, the 3D-RISM-KH theory can handle all such systems with ease. There are several other closure relations which have been developed over the years but they are aimed at specific applications, lacking the generality of the KH closure [46,47,48,49]. As a critical drawback, KH closure underestimates the height of strong associative peaks. On the other side, it widens the peak to some extent, and so provides the correct thermodynamics and solvation structure [50,51]. The site–site susceptibility of solvent breaks up into the intra- and intermolecular terms,

_{αγ}(r) represents the geometry of solvent molecules. For rigid species with site separations l

_{αγ}, it has the form, ${\omega}_{\alpha \gamma}(r)=\delta (r-{l}_{\alpha \gamma})/(4\pi {l}_{\alpha \gamma}^{2})$ specified in the reciprocal k-space in terms of the zeroth-order spherical Bessel function j

_{0}(x) as

_{αγ}(r) is the intermolecular radial total correlation function between solvent interaction sites α and γ. The solvent site–site total correlation functions h

_{αγ}(r) in Equations (1) and (3) are obtained in advance from the dielectrically consistent RISM theory [52] coupled with the KH closure (DRISM-KH approach). It is applied to the bulk solution of a given composition, including polar solvent, co-solvent, electrolyte, and ligands at a given concentration. The DRISM integral equation has the form [52]

_{αγ}(r) is the site–site direct correlation function of a bulk solvent, and both the intramolecular site–site correlation functions ${\tilde{\omega}}_{\alpha \gamma}(r)$ and the total site–site correlation functions ${\tilde{h}}_{\alpha \gamma}(r)$ are renormalized via an analytical dielectric bridge correction, ensuring all inter- and intra-species interactions in liquid state,

_{0}(x) and j

_{1}(x) are the zeroth- and first-order spherical Bessel functions over the positions of each atom

**r**

_{α}= (x

_{α}, y

_{α}, z

_{α}) with a partial site charge q

_{α}of site α on species s with respect to its molecular origin, where both sites α and γ are on the same species s, and its dipole moment ${d}_{s}={\displaystyle {\sum}_{\alpha \in s}{q}_{\alpha}{r}_{\alpha}}$ is oriented along the z-axis, ${d}_{s}=(0,0,{d}_{s})$. The renormalized dielectric correction (8) is nonzero only for polar solvent species of sorbed electrolyte solution which possess a dipole moment and are responsible for the dielectric response in the DRISM approach. The value of the envelope function h

_{c}(k) at k = 0 determines the dielectric constant of the solution, and is assumed in the ae smooth non-oscillatory form, quickly falling off at wavevectors k larger than those corresponding to the characteristic size l of liquid molecules and hence, to the dielectric constant (ε) of the solvent,

_{γ}(

**r**) in Equation (13a) can be taken as the 3D-solvation free energy density arising due to all solvent–solute interactions. The solvation free energy of the solute molecule Δμ is obtained by summation of the partial contributions over all solvent sites integrated over the whole volume. Other thermodynamic quantities are derived from the solvation free energy (13) via differentiation. This includes the solvation chemical potential which is decomposed into the energetic and entropic components at a constant volume,

^{vv}gives the energy of solvent reorganization around the solute. The partial molar volume of the solute macromolecule is obtained from the Kirkwood–Buff theory [53] extended to the 3D-RISM formalism as [54,55]

_{T}is the isothermal compressibility of bulk solvent obtained in terms of the site–site direct correlation functions of the bulk solvent as

## 3. Electrical Double Layer in Nanoporous Materials

_{1}(species 1), sorbed in a porous matrix of “quenched” particles with a spatial distribution corresponding to an equilibrium ensemble with temperature T

_{0}(species 0). The mean free energy of the sorbed annealed fluid is obtained as a statistical average of the free energy with the canonical partition function Z

_{1}(

**q**

_{0}) of the fluid sorbed in the matrix with a particular spatial configuration of quenched particles

**q**

_{0}over the ensemble of all realizations of matrix configurations

**q**

_{0},

^{s}as follows: $\mathrm{ln}{Z}_{1}=\underset{s\to 0}{\mathrm{lim}}\mathrm{d}{Z}^{s}/\mathrm{d}s$. The statistical average of the moments is given by the equilibrium canonical partition function of a fully annealed (s + 1)-component liquid mixture of matrix species 0 and s equivalent replicas of fluid species 1 not interacting with each other. The average free energy of the annealed fluid is obtained, assuming no replica symmetry breaking in the analytical continuation of the annealed replicated free energy A

_{rep}(s),

_{ex}on the other electrode of the supercapacitor cause separation of electrolyte cations and anions that diffuse across the electrode separator to the electric double layers in their nanopores. The diffusion occurs until the ionic concentration bias in each electrode satisfies the condition of electroneutrality in the whole system,

_{ex}is distributed among charges ${q}_{c}^{0}$ on each sort of nanosphere, provided the electrostatic potential is the same inside all carbon nanospheres,

_{exEDL}of the external EDL at the macroscopic surface of the electrode to the solution bulk outside the electrode. The “zero” potential level is therefore shifted from the potential in vacuum by the external EDL ${\varphi}_{\mathrm{ex}\mathrm{EDL}}$.

_{s}, the chemical potential comprises the ideal gas term of spheres with molecular weight m

_{s}, $\Delta {\mu}_{s}^{\mathrm{id}}={k}_{\mathrm{B}}T\mathrm{ln}({\rho}_{s}{\Lambda}_{s})$, where ${\Lambda}_{s}=\sqrt{{h}^{2}/(2\pi {m}_{s}{k}_{\mathrm{B}}T)}$ is the de Broglie thermal wave length, the excess chemical potential Δµ

_{s}for liquid–liquid and liquid–matrix interactions in the nanopores, and the electrostatic potential of species charges q

_{s}in the average electrostatic field between the electrodes,

_{s}, with the replica DRISM-KH-VM integral Equations (23)–(25) being converged at each outer loop of iterations for ρ

_{s}. This is followed by calculating the potential changes ${\varphi}_{\mathrm{av}}^{\mathrm{I},\mathrm{II}}-{\varphi}_{\mathrm{c}}^{\mathrm{I},\mathrm{II}}$ at converged ρ

_{s}by solving the Poisson equation (30).

## 4. (Macro)Molecular Simulations with the 3D-RISM-KH Theory

^{+}and Cl

^{−}ions at infinite dilution presents the solvation structure of the ions correctly with bridging water molecule(s) between the ions embedded in a weak second solvation shell. The features of the H-site of the water molecule show interaction with the chloride ion as well as with the outer solvation shell while positioned away from the sodium ion. This corresponds to water dipole-like water oriented around a cation and hydrogen-bonded to an anion. Water molecules form a bridge of strongly associated water molecules located in a ring between the ions, a situation known as contact ion pair arrangement. At a high salt concentration, many salt bridges form in addition to water hydrogen bonding, and like ion pairs stabilize in both contact ion pair (CIP) and solvent-separated ion pair (SSIP) arrangements. In particular, the Cl

^{−}–Cl

^{−}ion pair at a high concentration in water is bridged by several Na

^{+}ions and waters, and forms a cluster.

_{C}inside the nanoporous electrode averaged over the nanoporous carbon material. It is obtained from the Poisson equation (Equation (30)) with the charge density taken from Figure 2. Inside the conducting carbon nanoparticle, $r<{R}_{c}^{0}$, the electrostatic potential levels out. The external charge of carbon nanoparticles drives the electrostatic potential run in the first and second solvation shells near the nanoparticle surface. Because of the steric constraints, there are no solutions charges in the Stern layer, and only the Coulomb potential causes the slope of the potential curves for $r>{R}_{c}^{0}$ near the carbon nanoparticle surface. Next comes the surface dipole appearing due to water hydrogens located closer to the nanoparticle surface than water oxygens, which causes the potential drop, and due to OH

^{−}ions located closer to the surface than Li

^{+}ions, which causes the subsequent potential rise. The peaks of the electrostatic potential in the first and second solvation layers constitute the outer Helmholtz layer. Due to the electric charge of carbon nanoparticles, the electrostatic potential oscillates with a period of 12 Å, which is close to the size of carbon nanoparticles, slowly diminishing with distance. That includes both the nanoparticle diffuse layer and EDL statistical average around other nanoparticles closely packed in the nanoporous carbon. The outer Helmholtz layer and further oscillations almost entirely cancel out the potential change of the Stern layer. These oscillations come mainly from the ionic cloud of the two solvation shells that screen the external charge of the carbon nanoparticle, with OH

^{−}ions for positive and K

^{+}ions for negative external charge. The latest model picks up chemical potential changes for both K

^{+}and OH

^{−}ions upon the introduction of the EDL in the computation.

^{+}and K

^{+}cations as well as OH

^{−}anions at the nanoporous carbon electrode surface. The dipole electric field formed by the solution at the electrode macroscopic boundary due to the chemical equilibrium conditions (37) shifts the bulk potential level q

_{ex}ϕ

_{C}in the nanopores. The difference between the “interior” part of the of ions in the nanoporous electrodes ${k}_{\mathrm{B}}T\mathrm{ln}({\rho}_{s}{\Lambda}_{s})+\Delta {\mu}_{s}$ is counterbalanced by an additional EDL appearing at the macroscopic boundaries of the two electrodes. This brings about a major portion of the supercapacitor voltage $U({q}_{\mathrm{ex}})$ caused by the chemical equilibrium between the solutions inside the nanoporous electrodes and the bulk solution outside them [75]. For nanoporous carbon in an ambient aqueous solution of KOH electrolyte at a concentration of 1M, it was found that the electrochemical mechanism of the supercapacitor is driven mostly by the Nernst–Planck equation, determining the chemical balance of sorbed ions in the whole electrode rather than just the EDL potential change at a planar electrode [35,36,37]. The purification efficiency of a nanoporous electrosorption cell (39) is determined with the same molecular forces [69]. The sorption capacity and specific capacitance are an interplay of the EDL potential change in the Stern layer at the nanoparticles’ surface and the Gouy–Chapman layer statistically averaged over the nanopores, the osmotic term coming from the difference of the ionic concentrations in the nanopores and in the bulk electrolyte solution, and the solvation chemical potentials of ions in the nanopores. Chemical specificities of ions, solvent, surface functional groups, and steric effects for sorbed ions strongly affect their solvation chemical potentials in nanopores. Note that the specific capacitance is strongly affected by enlarged effective sizes of sorbed ions, with strong implications on supercapacitor devices. A major factor affecting the specific capacitance is that two extra EDLs at the macroscopic boundaries of the nanoporous electrodes offset the differences of their chemical potentials from the solution bulk. This substantially changes the supercapacitor voltage.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Solvation structure of the aqueous solution of KOH electrolyte sorbed in the nanoporous carbon electrode. Specific charge of the nanoporous electrode: q

_{ex}= 0 (solid black lines); q

_{ex}= +80 [C/cm

^{3}] (long-dashed red lines); q

_{ex}= −80 [C/cm

^{3}] (short-dashed blue lines); RDFs of water O and H sites, and of K

^{+}and OH

^{−}ions around carbon nanoparticles.

**Figure 3.**Statistically averaged electrostatic potential ϕ

_{0}(r) around nanoparticles in carbon nanoporous electrode, with respect to the “zero” level ϕ

_{C}. The sorbed solution is in equilibrium with the bulk ambient aqueous solution of KOH electrolyte at concentration 120 ppm. Specific charge of the nanoporous electrode: q

_{ex}= 0 (black line); q

_{ex}= +16 [C/cm

^{3}] (yellow line); q

_{ex}= +80 [C/cm

^{3}] (red line); q

_{ex}= −16 [C/cm

^{3}] (green line); q

_{ex}= −80 [C/cm

^{3}] (blue line). Inset: statistical–mechanical average (red circle and distance vector) around a carbon nanoparticle (red ball) over carbon nanoparticles (black balls) and nanopores (white voids).

**Figure 4.**Illustration of Li

^{+}and K

^{−}cations and OH

^{−}anions in aqueous solution in a nanoporous carbon electrode.

**Figure 5.**Folding miniprotein 1L2Y by OIN quasidynamics steered by 3D-RISM-KH mean solvation forces extrapolated with GSFE [38]. α-Helices (purple), β-Hairpins (blue), Loops & Turns (grey) © 2015 American Chemical Society.

**Figure 6.**Tertiary structure of miniprotein 1L2Y from NMR experiment (

**left**) and from OIN/3D-RISM-KH/GSFE quasidynamics simulation for 25 ns with outer time step of 1 ps (

**right**) [38]. α-Helices (purple), β-Hairpins (blue), Loops & Turns (grey) © 2015 American Chemical Society.

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**MDPI and ACS Style**

Roy, D.; Kovalenko, A.
Multiscale Methods Framework with the 3D-RISM-KH Molecular Solvation Theory for Supramolecular Structures, Nanomaterials, and Biomolecules: Where Are We Going? *Thermo* **2023**, *3*, 375-395.
https://doi.org/10.3390/thermo3030023

**AMA Style**

Roy D, Kovalenko A.
Multiscale Methods Framework with the 3D-RISM-KH Molecular Solvation Theory for Supramolecular Structures, Nanomaterials, and Biomolecules: Where Are We Going? *Thermo*. 2023; 3(3):375-395.
https://doi.org/10.3390/thermo3030023

**Chicago/Turabian Style**

Roy, Dipankar, and Andriy Kovalenko.
2023. "Multiscale Methods Framework with the 3D-RISM-KH Molecular Solvation Theory for Supramolecular Structures, Nanomaterials, and Biomolecules: Where Are We Going?" *Thermo* 3, no. 3: 375-395.
https://doi.org/10.3390/thermo3030023