Bridging 3D Confinement and 2D Correlations in Counterion Layers at Charged Interfaces: An Extended Percus Relation from First Principles

We develop a first-principles theory that bridges three-dimensional (3D) confinement and two-dimensional (2D) in-plane correlations in counterion layers at oppositely charged interfaces. The system is controlled by two independent coupling constants. While a 3D parameter (γ) for perpendicular localization varies with the strength and direction of the applied electric field, a 2D parameter (Γ) for lateral correlations depends solely on system-specific conditions. This independence allows for strongly coupled yet noncrystalline liquid states. Our theoretical framework is based on a hybrid of density functional and statistical field theory, thereby yielding an extended Percus relation that, unlike its conventional counterpart for uniform 2D liquids, is valid for the spatially inhomogeneous density profiles. This extension is critical, as it establishes a direct connection between the 3D confinement and the resulting 2D in-plane structure. Numerical investigations of this relation reveal key in-plane structural features in the strong 3D coupling limit (γ→∞): a geometric length scale, the minimal inter-particle separation (dmin), governs both the first peak of the radial distribution function and the wavelength (λ) of its oscillatory tail. These findings clarify that in-plane order in these strongly coupled counterion liquids is determined by a geometric constraint rather than any crystalline symmetry.