Static Dielectric Permittivity Profiles and Coarse-graining Approaches for Water in Graphene Slit Pores

TL;DR

This work addresses how to correctly predict the dielectric response of water confined between graphene walls by clarifying boundary conditions and nonlocal effects. It derives and validates spatially resolved dielectric profiles ε_parallel(z) and ε_perp(z) from both equilibrium fluctuations and external-field perturbations under planar confinement, and assesses boundary-condition impacts. Applying multiple water models, it shows bulk-like dielectric behavior down to about 1 nm, with the observed capacitance reductions explained by interfacial shifts and the Stern layer through an exact effective-medium mapping to an equivalent circuit. The results bridge microscopic dielectric profiles and macroscopic observables, reinforce universal polarization behavior across water models, and provide open-source tools (MAICoS) to enable reproducible analysis of confined fluid dielectrics.

Abstract

The dielectric response of nano-confined fluids is crucial across technologies and biological systems, yet its calculation and interpretation from molecular simulations are often muddled by unclear boundary conditions. We re-derive the Green--Kubo relation for the spatially resolved linear dielectric response of fluids in planar confinement, explicitly accounting for boundary conditions and showing that equilibrium-derived profiles agree with those obtained from external fields. We identify common misconceptions in the literature and outline how microscopic dielectric behavior can be coarse-grained to connect with experimental observables. Simulations show that water retains a bulk-like dielectric response down to $\sim 1\,\mathrm{nm}$ confinement. The reduced \emph{effective} dielectric response that governs capacitance arises from the placement of the dielectric interface. Using effective-medium theory, we demonstrate that long-range reductions reported in experiments and theory are consistent with bulk-like behavior beyond about $1\,\mathrm{nm}$ from the surface. The effective response naturally maps onto an interfacial capacitance, and the dielectric properties of simulated water are robust across simulation setups and water models, reflecting universal polarization correlations.

Figures (12)

• Figure 1: Schematic illustration of typically employed boundary conditions for dielectric properties of a bulk fluid in simulations and theory. (a) Liquid droplet immersed in vacuum and (b) in a dielectric continuum with the same properties as the fluid. (c) Periodic boundary conditions. For methods such as the Ewald approach to solve the Coulomb sum, a dielectric boundary condition at infinity, $\varepsilon_\infty$, needs to be specified.
• Figure 2: Snapshot of the simulation system and representative observables discussed in this work. Data shown are obtained using the TIP4P/$\varepsilon$ model for water confined between insulating graphene sheets separated by $34\angstrom$. The position $z = 0\angstrom$ denotes the location of the carbon atoms. (a) Simulation snapshot showing carbon as gray, oxygen as red and hydrogen as white spheres. (b) Number density profile $n(z)$ of water molecules. The dashed line shows the bulk density $n_\mathrm{bulk}=0.033\angstrom^{-3}$ and the dotted line the location of the Gibbs dividing surface. (c) Parallel dielectric profile. The dashed line shows the dielectric constant $\varepsilon = 79$ of the water model at $300K$. (d) Inverse perpendicular dielectric profile. The dashed line shows the corresponding bulk value $\varepsilon^{-1} = 1/79$. In (c) and (d), the blue lines show results obtained from equilibrium fluctuations of the dipole density [\ref{['staerk24c_dielectric-planar:eq:par_fluct_diss', 'staerk24c_dielectric-planar:eq:perp_fluct_diss']}] and the red via the "direct" route [\ref{['staerk24c_dielectric-planar:eq:lin_res_par', 'staerk24c_dielectric-planar:eq:lin_res_perp']}] as explained in the text, using $E_\parallel = 0.05V\per nm$ and $D_\perp/\varepsilon_0 = 0.2V\per nm$, respectively. The dotted lines in (c) and (d) denote the corresponding locations of the dielectric dividing surfaces. The inset in (d) also shows the mean over the profiles for $z > 12\angstrom$ as round markers with corresponding error bars.
• Figure 3: Schematic illustration of the virtual cutting method to determine $m_\parallel(z)$. The highlighted box in the middle represents the primary simulation domain. The blue shaded area contains the atom's charges $\rho_{\text{cut}}$, over which the integral is performed in order to arrive at the surface charge density along the green line as a function of the $z$-coordinate. Multiple cuts (different positions of the cutting plane $a_\text{cut}$) are averaged over to obtain an estimate of the parallel polarization density $m_\parallel(z)$.
• Figure 4: Schematic illustration of the different lengths involved and the step profile approach. (a) The simulation box size $L_\mathrm{sim}$ can in general be larger than the surface atom separation $L$, which is typically taken also as plate distance when describing a capacitor. The water slab thickness $L_\mathrm{w}$ follows from the Gibbs dividing surface, cf. \ref{['staerk24c_dielectric-planar:eq:gibbs_length']}. (b) For coarse-graining a step profile of effective dielectric thickness $L_\alpha^\mathrm{eff}$ and effective dielectric constant $\varepsilon_\alpha^\mathrm{eff}$ is employed. The depletion layer thickness $\delta$ can be decomposed into contributions from dielectric interfacial shift $\delta_\alpha^\mathrm{w}$ and from the Stern layer thickness $\delta_\alpha^\mathrm{S}$, as explained in the text and \ref{['staerk24c_dielectric-planar:tab:shifts']}.
• Figure 5: Dielectric profiles at the graphene/water interface for different treatments of the boundary conditions. Labels denote the effective simulation box size $L_\text{sim} = x \cdot L$ for simulation systems that where calculated with 3d periodic boundary conditions. The data labeled with "2d" is obtained from two-dimensional boundary conditions. Data are shown for (a) the parallel dielectric profile $\varepsilon_\parallel (z)$ (curves overlap) and (b) the perpendicular inverse dielectric profile $\varepsilon_\perp^{-1}(z)$.