Phase Diagram and Criticality of the Modified Primitive Electrolyte Model in Bulk and in Inert and Conducting Confinement

TL;DR

The study tackles vapor–liquid transitions and critical behavior of a symmetric charged Lennard–Jones fluid (mRPM) under bulk and confinement, using an extended Wang–Landau framework combined with the Constant Potential Method to access grand-canonical thermodynamics. By sampling the density of states for mixtures and projecting onto the charge-neutral diagonal, the authors obtain Landau free-energy landscapes and map coexistence, enabling precise finite-size scaling that confirms Ising universality with a bulk critical temperature $T_c = 1.598 \pm 0.001 \varepsilon/k_B$ and critical behavior consistent with $\nu \approx 0.622$. In confinement, especially with conducting boundaries, the coexistence chemical potential shifts downward and capillary condensation is enhanced, while a finite critical pore size $H_c$ emerges below which criticality vanishes; capillary critical properties scale approximately linearly with $1/H$. The work introduces a flexible, scalable approach for studying phase behavior of electrochemical fluids in porous media and electrode environments, with potential extensions to asymmetric or multivalent ions and more complex geometries.

Abstract

Ionic fluids under conductive confinement are central to technologies such as batteries, supercapacitors, and fuel cells. Their interfacial behavior governs energy storage and electrochemical processes. Despite their importance, the thermodynamics of even simple models -- such as the charged Lennard-Jones fluid -- remain underexplored in this regime. We present an extended Wang-Landau sampling approach to efficiently compute the density of states of charged mixtures with respect to the particle number. The method supports simulations in both bulk and confined geometries. Combined with the Constant Potential Method, it also enables to study effects due to confining electrodes. We employ this approach to study symmetric, binary mixtures of charged Lennard-Jones particles -- the modified Restricted Primitive Model -- in bulk, in inert confinement, and in conductive confinement at the potential of zero charge. Our results show that confinement shifts the vapor-liquid critical point to lower temperatures and higher densities compared to bulk, in line with the classical concept of capillary condensation. Importantly, conductive boundaries significantly lower the chemical potential of coexistence relative to inert confinement. These findings offer deeper insight into the phase behavior of ionic fluids in energy-relevant porous environments.

Figures (10)

• Figure 1: Schematic illustration of the sampling scheme proposed in this work. Using small WL sampling runs, which vary the number of one species while keeping the other fixed, we cover the two-dimensional phase space around the neutrality line (dashed red line). Offsets between windows can then be determined from the off-diagonals (red squares), which overlap between (multiple) windows. As an example, we highlight a sampling window in $N_+$ in bold (solid green arrow) and one in $N_-$ (dashed green arrow) with the overlap of this two being labeled. Using the overlap, the offsets between all small sampling patches can be determined. The final result of this sampling are the density of states $Q(N _{\text{tot}})$ along the diagonal, shown as blue squares.
• Figure 2: Schematic illustration of the model slit pore investigated in this work. Lennard-Jones particles of characteristic size $\sigma$ are confined between infinitely repulsive steric walls (red lines). Their positions are thus restricted to an accessible pore width region $H$ in the $z$-direction (indicated by dashed black lines). For systems of polarizable boundaries, Gaussian charge layers (thick gray lines where the Gaussians are insicated as transparent red circles) with a interlayer-separation $a$ are placed at a distance $d = 0.75 \; \sigma$ from the hard walls. Black dashes inside the electrodes indicate the underlying graphitic lattice structure. Positive and negative charged fluid particles are shown in green and blue, respectively.
• Figure 3: Illustration of the Landau free energy landscape as determined from Wang-Landau simulations at $T = 1.55\, \varepsilon / k _{\text{B}}$ at $L = 9\;\sigma$. The free energy landscape belonging to the coexistence chemical potential $\mu _{\text{coex}} = -15.866 \varepsilon$ is shown in blue, exemplary metastable states red and green. Data from WL sampling are shown shown as crosses, lines are seventh-order polynomial fits to the data.
• Figure 4: Finite size analysis for the vapor-liquid coexistence of the mRPM with $\alpha=10$. The coexistence densities determined from the extended Wang-Landau sampling scheme for are shown as round symbols, the extrapolated critical point $(\rho _{\text{c}}, T _{\text{c}})$ as a cross and the arithmetic mean of the coexisting phases is given by transparent triangles. The color code represents the simulation box size $L$.
• Figure 5: Adsorption isotherms for three pore widths and two different temperatures. We compare conducting and electrostatically inert confinement for every system. Shown are the vapor/liquid particle counts $N _{\text{v/l}}$ vs. the chemical potential. The chemical potential of coexistence $\mu _{\text{coex}}$ in each system is indicated by the dashed line. The direction of increase of control variables between panels is shown as brown arrows. Clearly, we see a shift towards higher values in $\mu _{\text{coex}}$ for a decrease in temperature and a increase in pore size. A consistent shift towards higher $\mu _{\text{coex}}$ can also be observed when going from conducting to inert confinement.