Modified Poisson Boltzmann Research Articles

Elucidating the Mechanism of Electro-Adsorption on Electrically Conductive Ultrafiltration Membranes via Modified Poisson-Boltzmann Equation.

Electrically conductive membranes (ECMs) were prepared by coating porous ethylenediamine-modified polyacrylonitrile (PAN-EDA) UF membranes with an ultrathin layer of platinum (Pt) nanoparticles through magnetron sputtering. These ECMs were used in electrofiltration to study the removal of brilliant blue dye from an aqueous solution under positive electrical potentials (0-2.5 V). Negative electrical potentials (-1.0--2.5 V) were also investigated to regenerate the membrane by desorbing the dye from the ECM surface. At +0 V, the EC PAN-EDA membrane adsorbed the dye due to its intrinsic positive charge. Application of -2.0 V resulted in a maximum of 39% desorption of the dye. A modified Poisson-Boltzmann (MPB) model showed that -2.0 V created a repulsive force within the first 24 nm of the membrane matrix, which had a minimal effect on dye ions adsorbed deeper within the membrane, thus limiting the electro-desorption efficiency to 39%. Moreover, increasing positive potentials from +0.5 V to +2.5 V led to increased dye electro-adsorption by 9.5 times, from 132 mg/m2 to 1112 mg/m2 at pH 8 (equivalent to the membrane's isoelectric point). The MBP simulations demonstrated that increasing electro-adsorption loadings are related to increasing attractive force, indicating electro-adsorption induced by attractive force is the dominant mechanism and the role of other mechanisms (e.g., electrochemical oxidation) is excluded. At pH 5, electro-adsorption further increased to 1390 mg/m2, likely due to the additional positive charge of the membrane (zeta potential = 9.2 mV) compared to pH 8. At pH 8, complete desorption of the dye from the ECM surface was achieved with a significant repulsive force at -2.0 V. However, as pH decreased from 8 to 5, the desorption efficiency decreased by 3.9% due to the membrane's positive charge. These findings help elucidate the mechanisms of electro-adsorption and desorption on ECMs using dye as a model for organic compounds like humic acids.

In situ polymerization of a poly(3,4-ethylenedioxythiophene):poly(sodium–styrennesulfonate) conductive layer for electro-assisted ultrafiltration and fouling mitigation

Electrification can enhance the antifouling performance of ultrafiltration membranes and improve the sustainability of membrane technologies. The design of high-performance conductive membrane materials remains one of the core technologies in electro-assisted ultrafiltration. Here, conductive PVDF/PEDOT:PSS composite membranes were prepared via in situ chemical oxidation polymerization on a PVDF surface. The PVDF/PEDOT:PSS membranes were subsequently used for electro-filtration to study the effect of the applied potential on humic acid fouling. When a voltage of 3 V was applied, the PVDF/PEDOT:PSS20 membrane was continuously polluted for 5 cycles. The results showed that after simple hydraulic cleaning, the membrane flux recovery rate reached 91 %. Compared with that of the PVDF membrane, the irreversible fouling of the PVDF/PEDOT:PSS20 membrane decreased by 65 %. The modified Poisson Boltzmann equation indicates that applying cathodic potential on the surface of PEDOT:PSS conductive layer increases the cumulative concentration of counter ions on the membrane surface and the thickness of the charged layer, thereby promoting the migration of peak forces on the membrane surface to the bulk solution, changing the fouling behaviour of HA from cake filtration to standard blocking.

Dataset for quantum-mechanical exploration of conformers and solvent effects in large drug-like molecules

We here introduce the Aquamarine (AQM) dataset, an extensive quantum-mechanical (QM) dataset that contains the structural and electronic information of 59,783 low-and high-energy conformers of 1,653 molecules with a total number of atoms ranging from 2 to 92 (mean: 50.9), and containing up to 54 (mean: 28.2) non-hydrogen atoms. To gain insights into the solvent effects as well as collective dispersion interactions for drug-like molecules, we have performed QM calculations supplemented with a treatment of many-body dispersion (MBD) interactions of structures and properties in the gas phase and implicit water. Thus, AQM contains over 40 global and local physicochemical properties (including ground-state and response properties) per conformer computed at the tightly converged PBE0+MBD level of theory for gas-phase molecules, whereas PBE0+MBD with the modified Poisson-Boltzmann (MPB) model of water was used for solvated molecules. By addressing both molecule-solvent and dispersion interactions, AQM dataset can serve as a challenging benchmark for state-of-the-art machine learning methods for property modeling and de novo generation of large (solvated) molecules with pharmaceutical and biological relevance.

Modeling the Environment-Dependent Kinetics of Oxygen Reduction Reaction – a Continuum Model for Electric Double Layer

For proton-exchange-membrane fuel cells (PEMFCs) to achieve broad commercialization, improved energy-conversion efficiency with minimal Pt-based electrocatalyst is required [1]. Because the sluggish rate of oxygen reduction reaction (ORR) limits the efficiency of PEMFCs [2], the efficiency improvement requires a better understanding of ORR kinetics and mechanism to design better catalyst. To understand the ORR mechanism, theoretical and experimental analyses have been conducted. While previous studies reasonably explained the catalyst-dependent activity on single crystal catalysts in 0.1 M perchloric acid solution [3, 4], the explicit effect of electrolyte and related microenvironments [5, 6] is not thoroughly understood. The change in the electrolyte alters the electric-double-layer (EDL) structure and thus the local microenvironment at the electrode/electrolyte interface. Thus, the structure of the EDL should be carefully analyzed to uncover the electrolyte-dependent reaction kinetics. In this talk, we propose a multiscale continuum model to predict the EDL structure and examine the effect of perchloric acid concentration on ORR activity on Pt (111). The model includes Density Potential Functional Theory (DPFT) for electron density and Modified Poisson Boltzmann equation for species’ density and electric potential [7]. Also, the interaction between adsorbents and electric field is taken into account by minimizing the grand potential. After model validation with experimentally measured double-layer capacity data as a function of applied potential and concentration, the effect of the perchloric acid concentration (0.02 M – 0.2 M) on ORR activity is analyzed and discussed. It is shown that the model reproduces the specific activity obtained in the experiments when assuming the oxygen adsorption is limiting the rate, which can be attributed to the large energetic barrier for solvent reorganization [8]. Then, extension of the model to PEMFC ionomer electrolytes will be introduced. Overall, the model framework and findings provide insights into the ORR mechanism and guidance on how to tailor catalyst materials for increased PEMFC performance. Acknowledgements: This work was partially supported by the Hydrogen and Fuel Cell Technologies Office (HFTO), Office of Energy Efficiency and Renewable Energy, U.S. Department of Energy (DOE) through the Million Mile Fuel Cell Truck (M2FCT) consortia, technology managers G. Kleen and D. Papageorgopoulos, under contract number DE-AC02- 05CH11231, as well as by a CRADA with Toyota Central R&D Labs., Inc.. Reference [1] D. A. Cullen, K. C. Neyerlin, R. K. Ahluwalia, R. Mukundan, K. L. More, R. L. Borup, A. Z. Weber, D. J. Myers and A. Kusoglu, Nature Energy, 6, 462 (2021).[2] I. E. L. Stephens, A. S. Bondarenko, U. Grønbjerg, J. Rossmeisl and I. Chorkendorff, Energy & Environmental Science, 5 (2012).[3] M. Shibata, M. Inaba, K. Shinozaki, K. Kodama and R. Jinnouchi, Journal of The Electrochemical Society, 169 (2022).[4] H. A. Hansen, V. Viswanathan and J. K. Nørskov, The Journal of Physical Chemistry C, 118, 6706 (2014).[5] K. Kodama, K. Motobayashi, A. Shinohara, N. Hasegawa, K. Kudo, R. Jinnouchi, M. Osawa and Y. Morimoto, ACS Catalysis, 8, 694 (2017).[6] M. Luo and M. T. M. Koper, Nature Catalysis, 5, 615 (2022).[7] J. Huang, J Chem Theory Comput, 19, 1003 (2023).[8] D. T. Limmer, A. P. Willard, P. Madden and D. Chandler, Proceedings of the National Academy of Sciences, 110, 4200 (2013).

Rheological impact on thermofluidic transport characteristics of generalized Maxwell fluids through a soft nanopore

The time-dependent electroosmotic flow (EOF) and heat transfer characteristic of a generalized Maxwell fluid through the polyelectrolyte layer (PEL) grafted nanopore are investigated while considering different permittivity between the PEL and electrolyte solution. The ion partitioning effects arise due to the different permittivity among these regions. Taking the ion partitioning effects, the analytic solution for the induced potential is established within and outside the PEL from the modified Poisson–Boltzmann equation assuming the Debye–Hückel approximation for a low surface charge. The Cauchy momentum equation with a suitable constitutive equation for fractional Maxwell fluids is derived, and the corresponding analytic solution is presented to provide the axial fluid flow distribution in the full domain. The energy fluxes that have major contributions to the energy equation mainly depend on axial conduction, convection due to electrolyte transport, and Joule heating effects for the external electric field. The analytical solutions of the energy equation for hydro-dynamically fully developed flow with constant thermophysical properties are presented to provide the temperature distribution considering constant heat flux at the nanopore wall. The influence of several important factors for characterizing heat transfer behavior is investigated in the present study. The maximum fluid velocity occurs when the permittivity between the PEL and electrolyte region is the same. The increasing values of fluid velocity imply higher convective heat transfer and make the Nusselt number higher. This study makes a conscious effort toward highlighting the modality controlling the heat transfer characteristics for the ion partitioning effects.

Influence of concentration and temperature dependent dielectric constants on the thermodynamics of electrolytes

The symmetric Poisson–Boltzmann theory, the modified Poisson–Boltzmann theory, and the mean spherical approximation are utilized to calculate the osmotic coefficient, and the individual and mean activity coefficients of model KCl, LiCl, NaCl, and NaClO4 electrolytes all dissociated in water where the dielectric constant depends on concentration. The theories are also employed to compute the same thermodynamic quantities for HCl and NaCl solutions where the dielectric constant is now temperature dependent. In each situation the theoretical predictions are compared with the corresponding exact benchmark Monte Carlo simulation results. The mean spherical approximation and the modified Poisson–Boltzmann theory reproduce the simulation data for osmotic coefficient quantitatively and that for the mean activity almost quantitatively. The individual activities are accurately represented by the modified Poisson–Boltzmann theory but only qualitatively by the symmetric Poisson–Boltzmann and the mean spherical approximation. In general, the symmetric Poisson–Boltzmann theory shows good agreement with the simulations at low concentrations but deviates at higher concentrations.

Charge asymmetric electrolytes around a rigid cylindrical polyelectrolyte: A generalization of the capacitive compactness

The surface mean electrostatic potential and capacitive compactness of the electrical double layer surrounding an infinite rigid cylindrical polyelectrolyte are analysed for equally-sized -1:z+ aqueous electrolytes. Monte Carlo simulations, the non-linear Poisson–Boltzman equation, the modified Poisson–Boltzmann theory, and the hypernetted chain/mean spherical approximation integral equation are used to examine the role of multivalent cations (coions) when the properties of monovalent anions (counterions) are fixed. A non-zero mean electrostatic potential in the neighbourhood of an uncharged polyelectrolyte is predicted by the simulations and the modified Poisson–Boltzmann theory in the presence of multivalent cations. The concept of capacitive compactness is generalised to overcome the divergence found in the classical definition, when a non-zero potential at the point of zero charge is present. With a highly electrified colloidal surface an inversion is observed, as a function of colloidal charge, in the precedence of the mean electrostatic potential near the polyelectrolyte’s surface and of the capacitive compactness between -1:+1 and -1:z+ electrolytes.

Coupling Effects of Ionic Surfactants and Electrolytes on the Stability of Bulk Nanobubbles.

As interest in the extensive application of bulk nanobubbles increases, it is becoming progressively important to understand the key factors affecting their anomalous stability. The scientific intrigue over nanobubbles originates from the discrepancy between the Epstein–Plesset prediction and experimental observations. Herein, the coupling effects of ionic surfactants and electrolytes on the stability of bulk nanobubbles is studied. Experimental results show that ionic surfactants not only reduce the surface tension but also promote the accumulation of net charges, which facilitate the nucleation and stabilization of bulk nanobubbles. The addition of an electrolyte in a surfactant solution further results in a decrease in the zeta potential and the number concentration of nanobubbles due to the ion shielding effect, essentially colloidal stability. An adsorption model for the coexistence of ionic surfactants and electrolytes in solution, that specifically considers the effect of the adsorption layer thickness within the framework of the modified Poisson–Boltzmann equation, is developed. A quantitative agreement between the predicted and experimental surface tension is found in a wide range of bulk concentrations. The spatial distribution of the surface potential, surfactant ions and counterions in the vicinity of the interface of bulk nanobubbles are described. Our study intrinsically paves a route to investigate the stability of bulk nanobubbles.

Effect of ion partitioning on an oscillatory electro-osmotic flow on solute transport process of fractional Jeffrey fluid through polyelectrolyte-coated nanopore with reversible wall reaction

The ion-partitioning effects on solute transport phenomena of time-periodic electro-osmotic flow in fractional Jeffrey fluid are investigated through a polyelectrolyte layer (PEL)-coated conical nanopore within a reactive wall whose ends are connected with two large reservoirs. By considering the ion-partitioning effects, analytical solutions for the induced potential and the axial velocity are presented, respectively, from the modified Poisson–Boltzmann equation and the Cauchy momentum equation with the proper constitutive equation of the fractional Jeffrey fluid model in the exterior and interior of the PEL. The analytic solution of the convection–diffusion for solute transport is established in the entire domain. The influence of the oscillating Reynolds number Rew, permittivity ratio εr between two mediums, relaxation time λ1ω, retardation time λ2ω, phase partitioning coefficient σp, PEL fixed charge density qfix, Debye–Hückel parameter κa, and softness parameter λs are investigated in this study. Asymptotic solution for the axial velocity was also presented for low-oscillating Reynolds numbers and validated. The maximum axial velocity occurs when the permittivity between the PEL and electrolyte is the same for all models. The volumetric flow rate decreases with the increase in the PEL thickness, positive PEL charge density, and softness parameter in our study. The volume flow rate of the Newtonian fluid increased 24.07% for Maxwell fluid (λ1ω=5, α = 1) and 11.56% for Jeffrey fluid (λ1ω=5, λ1ω=1, α = 1, and β=0.5), when κa=25, Rew = 10, qfix = 5, d = 0.2, εr=0.6, and λs=1.0. The mass transport rate increases with relaxation time, tidal displacement, and permittivity ratio between these layers.

Ionic-size dependent electroosmotic flow in ion-selective biomimetic nanochannels

Electrokinetic phenomena, especially electroosmosis in ion-selective environments, play a key role in many systems, from ion-selective nanopores to cellular processes. In this paper, the impact of ionic size on the electroosmotic flow through an ion-selective soft slit nanochannel is analytically studied. Meanwhile, the modified Poisson–Boltzmann and the modified Navier–Stokes equations were used for modeling the electrostatics and the electrohydrodynamics of the problem, respectively, and the derived equations were solved by linearizing method. The results reveal the importance of considering the effect of ionic size in the calculation, as the steric effects, especially at high charge densities of polyelectrolytes (PELs), dramatically alter both the ions arrangement and the electric potential; and amplify the electroosmotic flow. Considering Debye-Huckel parameters of 4 and 10 for the electrolyte layer and the PEL, respectively, we demonstrate that the dimensionless electroosmotic velocity in a soft nanochannel having a dimensionless soft layer thickness of 0.2, from 3.2 by ignoring the steric effect, can reach the value of 6 by considering the steric effect of ν=0.3.

Modified Poisson–Boltzmann equations and macroscopic forces in inhomogeneous ionic fluids

We propose a field-theoretical approach based on the thermodynamic perturbation theory and within it derive a grand thermodynamic potential of the inhomogeneous ionic fluid as a functional of electrostatic potential for an arbitrary reference fluid system. We obtain a modified Poisson–Boltzmann (PB) equation as the Euler–Lagrange equation for the obtained functional. Applying Noether’s theorem to this functional, we derive a general mean-field expression for the stress tensor consistent with the respective modified PB equation. We derive a general expression for the macroscopic force acting on the dielectric or conductive body immersed in an ionic fluid. In particular, we derive a general mean-field expression for the disjoining pressure of an ionic fluid in a slit pore. We apply the developed formalism to describe three ionic fluid models of practical importance: nonpolarizable models (including the well-known PB and Poisson–Fermi equations), polarizable models (ions carry nonzero permanent dipole or static polarizability), and models of ion-dipole mixtures (including the well-known PB–Langevin equation). For these models, we obtain modified PB equations and respective stress tensors, which could be valuable for different applications, where it is necessary to estimate the macroscopic forces acting on the dielectric or conductive bodies (electrodes, colloids, membranes, etc) together with the local electrostatic potential (field) and ionic concentrations.

Analytical Solution of Modified Poisson–Boltzmann Equation and Application to Cylindrical Nanopore Supercapacitor Energy Storage

Analytical Solution of Modified Poisson–Boltzmann Equation and Application to Cylindrical Nanopore Supercapacitor Energy Storage

Linear-scaling selected inversion based on hierarchical interpolative factorization for self Green's function for modified Poisson-Boltzmann equation in two dimensions

This paper studies an efficient numerical method for solving modified Poisson-Boltzmann (MPB) equations with the self Green's function as a state equation to describe electrostatic correlations in ionic systems. Previously, the most expensive point of the MPB solver is the evaluation of Green's function. The evaluation of Green's function requires solving high-dimensional partial differential equations, which is the computational bottleneck for solving MPB equations. Numerically, the MPB solver only requires the evaluation of Green's function as the diagonal part of the inverse of the discrete elliptic differential operator of the Debye-Hückel equation. Therefore, we develop a fast algorithm by a coupling of the selected inversion and hierarchical interpolative factorization. By the interpolative factorization, our new selected inverse algorithm achieves linear scaling to compute the diagonal of the inverse of this discrete operator. The accuracy and efficiency of the proposed algorithm will be demonstrated by extensive numerical results for solving MPB equations.

Applicability of Different Double-Layer Models for the Performance Assessment of the Capacitive Energy Extraction Based on Double Layer Expansion (CDLE) Technique

Capacitive energy extraction based on double layer expansion (CDLE) is a renewable method of harvesting energy from the salinity difference between seawater and freshwater. It is based on the change in properties of the electric double layer (EDL) formed at the electrode surface when the concentration of the solution is changed. Many theoretical models have been developed to describe the structural and thermodynamic properties of the EDL at equilibrium, e.g., the Gouy–Chapman–Stern (GCS), Modified Poisson–Boltzmann–Stern (MPBS), modified Donnan (mD) and improved modified Donnan (i-mD) models. To evaluate the applicability of these models, especially the rationality and the physical interpretation of the parameters that were used in these models, a series of single-pass and full-cycle experiments were performed. The experimental results were compared with the numerical simulations of different EDL models. The analysis suggested that, with optimized parameters, all the EDL models we examined can well explain the equilibrium charge–voltage relation of the single-pass experiment. The GCS and MPBS models involve, however, the use of physically unreasonable parameter values. By comparison, the i-mD model is the most recommended one because of its accuracy in the results and the meaning of the parameters. Nonetheless, the i-mD model alone failed to simulate the energy production of the full-cycle CDLE experiments. Future research regarding the i-mD model is required to understand the process of the CDLE technique better.

Osmotic Transport at the Aqueous Graphene and hBN Interfaces: Scaling Laws from a Unified, First-Principles Description.

Osmotic transport in nanoconfined aqueous electrolytes provides alternative venues for water desalination and "blue energy" harvesting. The osmotic response of nanofluidic systems is controlled by the interfacial structure of water and electrolyte solutions in the so-called electrical double layer (EDL), but a molecular-level picture of the EDL is to a large extent still lacking. Particularly, the role of the electronic structure has not been considered in the description of electrolyte/surface interactions. Here, we report enhanced sampling simulations based on ab initio molecular dynamics, aiming at unravelling the free energy of prototypical ions adsorbed at the aqueous graphene and hBN interfaces, and its consequences on nanofluidic osmotic transport. Specifically, we predicted the zeta potential, the diffusio-osmotic mobility, and the diffusio-osmotic conductivity for a wide range of salt concentrations from the ab initio water and ion spatial distributions through an analytical framework based on Stokes equation and a modified Poisson-Boltzmann equation. We observed concentration-dependent scaling laws, together with dramatic differences in osmotic transport between the two interfaces, including diffusio-osmotic flow and current reversal on hBN but not on graphene. We could rationalize the results for the three osmotic responses with a simple model based on characteristic length scales for ion and water adsorption at the surface, which are quite different on graphene and on hBN. Our work provides fundamental insights into the structure and osmotic transport of aqueous electrolytes on 2D materials and explores alternative pathways for efficient water desalination and osmotic energy conversion.

On the modified Poisson–Boltzmann closure for primitive model electrolytes at high concentration

The modified Poisson-Boltzmann closure is applied to the Kirkwood hierarchy of integral equations to investigate high concentration primitive model electrolytes. Two approximations are considered in the two sphere fluctuation potential problem. The derived damped oscillatory mean electrostatic potentials suggest that this closure should be of use in providing a basis for understanding the large experimental decay lengths found at high electrolyte concentrations.

Steric and Slippage Effects on Mass Transport by Using an Oscillatory Electroosmotic Flow of Power-Law Fluids.

In this paper, the combined effect of the fluid rheology, finite-sized ions, and slippage toward augmenting a non-reacting solute’s mass transport due to an oscillatory electroosmotic flow (OEOF) is determined. Bikerman’s model is used to include the finite-sized ions (steric effects) in the original Poisson-Boltzmann (PB) equation. The volume fraction of ions quantifies the steric effects in the modified Poisson-Boltzmann (MPB) equation to predict the electrical potential and the ion concentration close to the charged microchannel walls. The hydrodynamics is affected by slippage, in which the slip length was used as an index for wall hydrophobicity. A conventional finite difference scheme was used to solve the momentum and species transport equations in the lubrication limit together with the MPB equation. The results suggest that the combined slippage and steric effects promote the best conditions to enhance the mass transport of species in about 90% compared with no steric effect with proper choices of the Debye length, Navier length, steric factor, Womersley number, and the tidal displacement.

The surface potential explains ion specific bubble coalescence inhibition

HypothesisSome ions can prevent bubbles from coalescing in water. The Gibbs-Marangoni pressure has been proposed as an explanation of this phenomenon. This repulsive pressure occurs during thin film drainage whenever surface enhanced or surface depleted solutes are present. However, bubble coalescence inhibition is known to depend on which particular combination of ions are present in a peculiar and unexplained way. This dependence can be explained by the electrostatic surface potential created by the distribution of ions at the interface, which will alter the natural surface propensity of the ions and hence the Gibbs-Marangoni pressure. CalculationsA generalised form of the Gibbs-Marangoni pressure is derived for a mixture of solutes and the modified Poisson–Boltzmann equation is used to calculate this pressure for five different electrolyte solutions made up of four different ions. FindingsCombining ions with differing surface propensities, i.e., one enhanced and one depleted, creates a significant electrostatic surface potential which dampens the natural surface propensity of these ions, resulting in a reduced Gibbs-Marangoni pressure, which allows bubble coalescence. This mechanism explains why the ability of electrolytes to inhibit bubble coalescence is correlated with surface tension for pure electrolytes but not for mixed electrolytes.

From Surface Tension to Molecular Distribution: Modeling Surfactant Adsorption at the Air–Water Interface

Surfactants are centrally important in many scientific and engineering fields and are used for many purposes such as foaming agents and detergents. However, many challenges remain in providing a comprehensive understanding of their behavior. Here, we provide a brief historical overview of the study of surfactant adsorption at the air-water interface, followed by a discussion of some recent advances in this area from our group. The main focus is on incorporating an accurate description of the adsorption layer thickness of surfactant at the air-water interface. Surfactants have a wide distribution at the air-water interface, which can have a significant effect on important properties such as the surface excess, surface tension, and surface potential. We have developed a modified Poisson-Boltzmann (MPB) model to describe this effect, which we outline here. We also address the remaining challenges and future research directions in this area. We believe that experimental techniques, modeling, and simulation should be combined to form a holistic picture of surfactant adsorption at the air-water interface.

On the equivalence of self-consistent equations for nonuniform liquids: a unified description of the various modifications

A variety of self-consistent (SC) equations have been proposed for non-uniform states of liquid particles under external fields, including adsorbed states at solid substrates and confined states in pores. External fields represent not only confining geometries but also fixed solutes. We consider SC equations ranging from the modified Poisson–Boltzmann equations for the Coulomb potential to the hydrostatic linear response equation for the equilibrium density distribution of Lennard-Jones fluids. Here, we present a unified equation that explains the apparent diversity of previous forms and proves the equivalence of various SC equations. This unified description of SC equations is obtained from a hybrid method combining the conventional density functional theory and statistical field theory. The Gaussian approximation of density fluctuations around a mean-field distribution is performed based on the developed hybrid framework, allowing us to derive a novel form of the grand-potential density functional that provides the unified SC equation for equilibrium density.