On the equivalence of self-consistent equations for nonuniform liquids: a unified description of the various modifications
aa r X i v : . [ c ond - m a t . s o f t ] F e b On the equivalence of self-consistent equations fornonuniform liquids: a unified description of thevarious modifications
Hiroshi Frusawa † Laboratory of Statistical Physics, Kochi University of Technology, Tosa-Yamada,Kochi 782-8502, Japan.
Abstract.
A variety of self-consistent (SC) equations have been proposed for non-uniform states of liquid particles under external fields, including adsorbed states atsolid substrates and confined states in pores. External fields represent not onlyconfining geometries but also fixed solutes. We consider SC equations ranging from themodified Poisson-Boltzmann equations for the Coulomb potential to the hydrostaticlinear response equation for the equilibrium density distribution of Lennard-Jonesfluids. Here, we present a unified equation that explains the apparent diversity ofprevious forms and proves the equivalence of various SC equations. This unifieddescription of SC equations is obtained from a hybrid method combining theconventional density functional theory and statistical field theory. The Gaussianapproximation of density fluctuations around a mean-field distribution is performedbased on the developed hybrid framework, allowing us to derive a novel form of thegrand-potential density functional that provides the unified SC equation for equilibriumdensity.
1. Introduction
Many studies on dense uniform liquids have demonstrated the dominant role of short-ranged harshly repulsive intermolecular forces in determining structural correlations,which provides the insight that attractive intermolecular forces essentially cancel dueto the symmetric configurations of molecules in uniform liquids [1-3]. In contrast,translational invariance of molecular arrangements is broken in non-uniform liquids [4-12]. Accordingly, both attractive and repulsive forces can, for instance, significantlyinfluence the structure of a Lennard-Jones (LJ) fluid next to a hard wall because thevector sums of the long-range attractive forces do not cancel each other.There are several situations in which such non-uniform states of liquid moleculesemerge. Inhomogeneities occur [4-12]: (i) at interfaces between liquid-gas, liquid-liquid,and liquid-crystal phases; (ii) in the adsorption of liquids at solid substrates or walls; (iii)for fluids in confining geometries, such as slits and pores; and (iv) in the sedimentationequilibrium of colloidal particles under gravity. Theoretically, a non-uniform state can † e-mail: [email protected] n the equivalence of self-consistent equations ρ eq ( r ) in equilibrium [4, 5, 14, 41-47]. This type of SC theory canbe traced back to the van der Waals interface theory [5, 42]. The essential ingredientof the classical van der Waals theory is that gas-liquid coexistence can be attributed tointermolecular competition between short-range repulsions and long-range attractions,thereby providing a qualitatively accurate description of slowly varying interfaces. Arecent SC equation modified the van der Waals equation to quantitatively describe localinterfacial structures on the molecular scale [5]. n the equivalence of self-consistent equations ρ eq ( r ) can be obtained from a novel form of the grand-potential density functional. Theunified equation corresponds to an extension from the equation based on the Percus’test particle method [47, 53] for uniform liquids to that for non-uniform liquids. It willalso be demonstrated that the single unified equation for ρ eq ( r ) can yield a variety ofexisting SC equations [4, 5, 14-34, 41-47].In the remainder of this paper, we first present a review of previous SC equations [4,5, 14-34, 41-47] and our main results related to the unification discussed above: sections2 and 3 provide the formal background and essential results for demonstrating theequivalence between previous SC equations. In Sections 4 and 5, comparisons are madebetween previous SC equations and the obtained forms derived from a single unifiedequation for ρ eq ( r ): we investigate the SC equations for non-Coulombic liquids [4, 5, 14,41-47] in Section 4, whereas Section 5 presents a unified equation for ρ eq ( r ) expressed bythe Coulomb potential for verifying the equivalence of modified PB equations [14-34].After demonstrating the relevance of our hybrid framework [48, 49] for SC equations,Section 6 outlines the derivation scheme for the grand-potential density functional, akey functional to obtain the unified SC equation. Summary and conclusions are givenin Section 7. In the appendices, full details of the formulations regarding not only thepresent results but also the previous modified PB equations can be found.
2. Formal background
In non-uniform liquids, there are two types of interaction potentials: one is theinteraction potential between liquid particles, which produces intermolecular forces in n the equivalence of self-consistent equations v ( r ), as a function of the intermolecular distance | r | ≡ r , includes thehard-sphere (HS) potential, LJ potential, and Coulomb potential. The latter externalpotential J ( r ), which is uniquely determined by the location r of particles, is causednot only by the existence of a hard wall, but also by long-range repulsive or attractiveparticle-wall interactions, such as Coulomb and LJ interactions. It should be noted thatnot only v ( r ) and J ( r ) but also the other energetic quantities, appearing in this paper,are defined in the k B T -unit.Let us introduce the liquid systems through explaining the potential splitting intotwo parts: v ( r ) = v ( r ) + v ( r ) , (1) J ( r ) = J ( r ) + J ( r ) . (2)In general, v and J denote harshly repulsive parts and v and J slowly varying parts.In a dense and uniform fluid, the subscript ”0” indicates that the potentials are those ina reference system that has been found sufficient for determining the essential structuralproperties [1-3]. As the first approximation in treating J ( r ), we have estimated from J an effective radius R of fixed object, such as a fixed solute, that excludes constituentliquid particles (see eq. (10) for the definition of the effective radius R ) [5, 42].We investigate the SC equations for HS fluids, attractive particle systems, andpoint-charge systems as listed in Table 1. Correspondingly, we give concrete forms ofthe interaction potential splitting ( v = v + v ) for the HS fluids, LJ ones and point-charge systems. The so-called WCA separation, proposed by Weeks, Chandler andAndersen, applies to attractive particle systems including the LJ fluids [1-3].First, in the HS fluids, v is identified with the full potential v HS of HS interactions(i.e., v = v HS ), and it follows that v ≡
0. Next, the WCA separation for the LJ fluidscreates the WCA potential v WCA ( r ) arising from the LJ potential v LJ truncated at theminimum as follows: v ( r ) = v WCA ( r ) ≡ v LJ ( r ) + ǫ LJ if r ≤ r m r > r m , (3)and v ( r ) = − ǫ LJ if r ≤ r m v LJ ( r ) if r > r m . (4)In eqs. (3) and (4), r m and ǫ LJ denote the separation and depth of potential minimum,respectively. Equation (3) is a typical form of the WCA potential in attractive particlesystems. n the equivalence of self-consistent equations Table 1.
A list of previous self-consistent equations derived from a unified equation(25) [EL = Euler-Lagrange; RY = Ramakrishnan-Yussouff; HLR = hydrostatic linearresponse; PB = Poisson-Boltzmann].Constituents Self-consistentlydetermined variables Original equation Modified equationHard spheres Equilibrium density EL equation ofthe RY functional HLR equation(eq. (14))Attractive particles Equilibrium densityReference density ditto — ditto Mean-field equation(eq. (9))Point charges Coulomb potential PB equation Finite-spread PB equation(eqs. (15), (A.17) and (A.18))Point charges Coulomb potential PB equation Higher-order PB equation(eqs. (17) and (A.20))
Lastly, the point-charge systems have the Coulomb interaction potential that isalways repulsive: v ( r ) = v el ( r ) ≡ z l B r , (5)where we consider particles carrying charges of + ze and the Bjerrum length l B is definedas the distance at which two unit charges have interaction energy k B T . Despite the long-range nature, there is no characteristic length for the potential splitting; nevertheless,previous studies have proposed the potential splitting [14, 20, 21] such that v el ( r ) = v ( r ) + v ( r ) , (6) v ( r ) = z l B erfc( r/ξ ) r , (7) v ( r ) = z l B erf( r/ξ ) r , (8)which has been demonstrated to be useful in finding the non-uniform Coulomb potentialby adjusting the characteristic length ξ . General forms of v ( r ) in point-charge systemsare given in Appendix A.2 where it is clarified that this type of potential splitting inthe Coulomb potential is equivalent to the introduction of charge smearing model (orOnsager charge smearing) for strongly-coupled Coulomb systems [14, 20-24, 35-40]. We outline previous SC equations [4, 5, 14-34, 42-47] in the above three systems, whichthis paper aims to derive in a unified manner. Table 1 lists the previous SC equations forthese systems. In Table 1, the SC equations are classified into two groups: the former n the equivalence of self-consistent equations
Conventional mean-field equation for the reference density ρ ref ( r ) [4, 5, 14, 42-47]. — The conventional mean-field equation is used when considering a slowly varyinginteraction potential v ( r ) in a non-uniform liquid with the external potential J ( r )applied. The solution of the mean-field equation has been referred to as the referencedensity ρ ref ( r ). In the outside of the excluded region with the effective radius of R [5, 42], which is defined below in eq. (10), ρ ref ( r ) satisfies ρ ref ( r ) = e µ − J ( r ) − R d r ′ v ( r − r ′ ) ρ ref ( r ′ ) , (9)where µ denotes the chemical potential in the k B T -unit. It is found from eq. (9) that thespatial dependence of ρ ref ( r ) for HS fluids arises only from J ( r ) because of v ( r ) ≡ J ( r ) to R is R = Z ∞ dr (cid:8) − e − J ( r ) (cid:9) , (10)which is a measure of the region w that excludes the liquid particles due to the harshlyrepulsive field J ( r ). Considering that the reference density ρ ref ( r ) necessarily vanishesinside the region w due to the presence of J ( r ), the reference density is expressed by ρ ref ( r ) = r ∈ w,ρ e − ∆ φ ( r ) ; r / ∈ w, (11)where the reference density ρ ≡ ρ ref ( r ) at a specified position of r is treatedseparately, and the shifted chemical potential µ ref , defined by ρ ≡ e µ ref = e µ − φ ( r ) , (12) φ ( r ) = J ( r ) + Z d r ′ v ( r − r ′ ) ρ ref ( r ′ ) , (13)is utilized for obtaining the equilibrium density, according to the hydrostaticapproximation [4, 5, 43-45]. Accordingly, the reference density at other locations isgiven by eq. (11) using ρ and the external potential difference ∆ φ ( r ) ≡ φ ( r ) − φ ( r )from the external field given by eq. (13).Previous studies based on the hydrostatic approximation have demonstrated thatthe mean-field like treatment works well for the purpose of obtaining the equilibriumdensity distribution of a dense non-uniform liquid with the help of the above referencedensity ρ . The SC equation for the equilibrium density ρ eq ( r ) [4, 5, 42-47]. — In attractiveparticle systems as well as in HS fluids, the equilibrium density ρ eq ( r ) is obtained fromevaluating the deviation from the reference density ρ at a specified position of r . The n the equivalence of self-consistent equations r ρ eq ( r ) = ρ (cid:20) Z d r ′ c ( r − r ′ ) (cid:8) ρ eq ( r ′ ) − ρ (cid:9)(cid:21) , (14)using the direct correlation function (DCF) c ( r ) for a uniform system of liquid particlesinteracting via the harshly repulsive potential v ( r ). The DCF depends on the density; ineq. (14), however, we use the DCF at a uniform density of ρ . The same equation (14)applies to both of attractive particle systems and HS fluids because the HLR equation[4, 5, 14, 42-47] is formulated for v ( r ), irrespective of the systems considered. Equation(14) is reduced to the closure in the hypernetted chain (HNC) approximation [1] whenthe density difference ρ eq ( r ) − ρ is related to the total correlation function (TCF) h ( r )in the v –interacting systems: ρ eq ( r ) − ρ = ρ h ( r − r ), according to the Percus’test particle method [47, 53]. In eq. (14), we take into account the non-uniformityby incorporating the contribution of slowly varying potential v ( r ) into the mean-fieldsolution ρ as given by eq. (9). Modified PB equations for the Coulomb potential Ψ( r ) [14-30]. — Lastly, we mentiontwo types of modified PB equations (see Appendix A for the details). Both modificationsare relevant to describe the charge-charge correlations between charged particles in theone component plasma (OCP) at strong coupling, concentrated electrolytes, and roomtemperature ionic liquids. It has also been found that these two modifications explainthe deviation from the PB solution for counterion distribution over a wide range of fromthe intermediate to strong coupling [14-30].One type is the finite-spread PB equation [14, 20-24] based on the charge smearingmodel. In the uniform OCP, a smeared charge distribution is simply characterized bythe mean distance a between adjacent charges and has been found useful to representthe strong repulsive correlations between the charges; for non-uniform systems, a issupposed to represent the mean separation in the ground state (see Appendix A.1 forthe details of the definition). In the local molecular field theory [14, 20, 21], the Gaussiancharge smearing model provides ∇ Ψ( r ) = − πl B e µ − J ( r ) − Ψ( r ) (15) O ( r ) = 1 { π ( a/m ) } / Z d r ′ exp ( − (cid:18) m | r − r ′ | a (cid:19) ) O ( r ′ ) , (16)where a constant m is an adjusting parameter (see also Appendix A.3); for instance, m = 1 / . m = 1 / . ∇ (cid:26) − a ∇ + a ∇ (cid:27) Ψ( r ) = − πl B z e µ − J ( r ) − Ψ( r ) ; (17) n the equivalence of self-consistent equations ∇ ∇ –term,has been referred to as the Bazant-Storey-Kornyshev (BSK) equation [16-18] in binarysystems such as concentrated electrolytes and room temperature ionic liquids.
3. Central results for proving the equivalence of previous self-consistentequations e Ω[ ρ ref ]As detailed in Section 6, the density-functional integral representation of the grandpotential is a powerful tool to evaluate fluctuating density fields around the referencedensity ρ ref with the help of the conventional DFT [6-12]. The hybrid method [48, 49]combining the DFT [6-12] and statistical field theory [15, 23, 31-34, 36, 50-52] allowsus to obtain the grand-potential density functional e Ω[ ρ ref ] into which the additionalcontribution due to density fluctuations is incorporated. The resulting form is e Ω[ ρ ref ] = f Ω [ ρ ref ] + U [ ρ ref ] , (18) f Ω [ ρ ref ] = F ex0 [ ρ ] − Z Z d r d r ′ ∆ ρ ref ( r ) h ( r − r ′ )∆ ρ ref ( r ′ )+ Z d r [ ρ ref ( r ) ln ρ ref ( r ) − ρ ref ( r ) − ρ ref ( r ) µ ] , (19) U [ ρ ref ] = 12 Z Z d r d r ′ ρ ref ( r ) v att ( r − r ′ ) ρ ref ( r ′ ) + Z d r ρ ref ( r ) J ( r ) . (20)In eq. (19), F ex0 [ ρ ] represents the excess part of the intrinsic Helmholtz free energy,∆ ρ ref ( r ) = ρ ref ( r ) − ρ and the reference TCF h ( r ) is related to the reference DCF c ( r ) through the Ornstein-Zernike equation given below. The subscript ”0” in the TCFhave double meanings as well as that in the DCF: the TCF is not only the function ata uniform density, which is equal to a reference density ρ = ρ ref ( r ) at r , but isalso the function in repulsive v –interaction systems. The subscript ”1” on the righthand side (rhs) of eq. (20) is altered to the subscript ”att”, clarifying that the potentialsplitting given by eq. (1) applies only to the attractive particle systems in our results : v = v + v att .Equation (19) indicates that the repulsive part of the grand potential f Ω iscomposed of two parts: the interaction energy term and the ideal-gas contribution.The second term on the rhs of eq. (19), representing the interaction energy, is expressedusing the TCF instead of the DCF, other than the conventional DFT [6-12]. In eq. (20),on the other hand, the interaction energy due to the slowly varying attractive part v att ( r )of interaction potential is evaluated in the mean-field approximation. Such treatment isapparently a crude estimation but has been found to be relevant for evaluating the long-range contribution to e Ω in non-uniform liquids [14, 46, 47]; actually, it is quite difficultto find density-density correlations by explicitly considering the non-uniformity. n the equivalence of self-consistent equations As seen from the expression (20), we regard only the contribution due to the long-rangeattractive interactions as the additional part of the grand-potential density functional e Ω[ ρ ref ]. Correspondingly, in this study, v ( r ) reads for the following three systems: v ( r ) = v WCA ( r ) (Attractive particles) v HS ( r ) (Hard spheres) v el ( r ) (Point charges) . (21)In what follows, the same rule of the subscript ”0” as above applies to the subscript”0” in the TCF ( h ( r )) and DCF ( c ( r )). Equation (9) with the above expression(21) implies that the spatial dependence of ρ ref ( r ) is determined solely by the externalpotential J ( r ) for both of the HS fluids and point-charge systems because of v att ( r ) ≡ J is treated as a variable field J , we can differentiatethe grand potential with respect to J , thereby generating the equilibrium density ρ eq ( r )with the help of a variable reference density ρ J ref ( r ) ≡ e µ −J ( r ) − R d r ′ v att ( r − r ′ ) ρ ref ( r ′ ) as follows: ρ eq ( r ) = δ e Ω[ ρ J ref ] δ J ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J = J = δρ J ref ( r ) δ J ( r ) (cid:12)(cid:12)(cid:12)(cid:12) J = J δ e Ω[ ρ J ref ] δρ J ref ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ J ref = ρ ref = − ρ ref ( r ) δ e Ω[ ρ J ref ] δρ J ref ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ J ref = ρ ref , (22)where use has been made of the following relation in the last line of the above equation: δρ J ref ( r ) δ J ( r ) (cid:12)(cid:12)(cid:12)(cid:12) J = J = δδ J n e µ −J ( r ) − R d r ′ v att ( r − r ′ ) ρ ref ( r ′ ) o(cid:12)(cid:12)(cid:12)(cid:12) J = J = − e µ − J ( r ) − R d r ′ v att ( r − r ′ ) ρ ref ( r ′ ) . (23)It also follows from eqs. (18) to (20) that the functional differentiation of e Ω[ ρ ref ] withrespect to ρ ref ( r ) yields δ e Ω[ ρ J ref ] δρ J ref ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ J ref = ρ ref = − Z d r ′ h ( r − r ′ )∆ ρ ref ( r ′ )+ Z d r ′ v att ( r − r ′ ) ρ ref ( r ′ ) + J ( r ) − µ + ln ρ ref ( r ) + ρ ref ( r ) δ J δρ J ref (cid:12)(cid:12)(cid:12)(cid:12) ρ J ref = ρ ref = − Z d r ′ h ( r − r ′ )∆ ρ ref ( r ′ ) − , (24) n the equivalence of self-consistent equations ρ eq ( r ): ρ eq ( r ) = ρ ref ( r ) (cid:26) Z d r ′ h ( r − r ′ )∆ ρ ref ( r ′ ) (cid:27) , (25)which is our main result. Plugging the Ornstein-Zernike equation into eq. (25), theHLR equation (14) [4, 5, 14, 42-47] will be obtained in the next section.
4. Comparison with the HLR equation (14) for the equilibrium density ρ eq ( r )We use the Ornstein-Zernike equation [1], h ( r − r ′ ) = c ( r − r ′ ) + Z d r ” c ( r − r ”) ρ h ( r ” − r ′ ) , (26)for the reference density ρ given by eq. (12) because both of the DCF and TCF, c ( r )and h ( r ), are defined for uniform liquids with the density of ρ . Plugging eq. (26)into eq. (25) at r , we have ρ eq ( r ) − ρ ρ = Z d r ′ h ( r − r ′ )∆ ρ ref ( r ′ )= Z d r ′ c ( r − r ′ )∆ ρ ref ( r ′ ) + Z d r ” c ( r − r ”) ρ Z d r ′ h ( r ” − r ′ )∆ ρ ref ( r ′ ) . (27)The main result (25) is again inserted into ρ R d r ′ h ( r ” − r ′ )∆ ρ ref ( r ′ ) of the aboveexpression with the relation ρ ref ( r ”) = ρ + ∆ ρ ref ( r ”), yielding ρ Z d r ′ h ( r ” − r ′ )∆ ρ ref ( r ′ ) = ρ (cid:26) ρ eq ( r ”) − ρ ref ( r ”) ρ + ∆ ρ ref ( r ”) (cid:27) ≈ ρ eq ( r ”) − ρ ref ( r ”) − { ρ eq ( r ”) − ρ ref ( r ”) } ∆ ρ ref ( r ”) ρ ≈ ρ eq ( r ”) − ρ ref ( r ”) . (28)It will be seen in Section 6 and Appendix B.2 that the above approximation used ineq. (28) is consistent with the approximation in deriving the grand-potential densityfunctional e Ω[ ρ ref ] given by eqs. (18) to (20). Thus, we find that e Ω[ ρ ref ] leads to theHLR equation (14) via eq. (25) [4, 5, 14, 42-47]: it follows from eq. (28) that eq. (27)reduces to ρ eq ( r ) − ρ ρ ≈ Z d r ′ c ( r − r ′ )∆ ρ ref ( r ′ ) + Z d r ” c ( r − r ”) { ρ eq ( r ”) − ρ ref ( r ”) } = Z d r ′ c ( r − r ′ ) (cid:8) ρ eq ( r ′ ) − ρ (cid:9) , (29)which is equal to the HLR equation (14). n the equivalence of self-consistent equations
5. Comparison with modified PB equations for the Coulomb potential Ψ( r )Before comparing the unified equation (25) for ρ eq ( r ) with the modified PB equationssuch as eqs. (15) to (17), we would like to see that the essential equation (25) impliesthe Boltzmann distribution of ρ eq ( r ). Actually, in the approximation of 1 + x ≈ e x , eq.(25) reads for point-charge systems ρ eq ( r ) ≈ ρ ref ( r ) e − Ψ( r ) , (30)Ψ( r ) = − Z d r ′ h el ( r − r ′ )∆ ρ ref ( r ′ ) , (31)where h el ( r ) denotes the electrostatic TCF of point charges interacting via v el ( r ) withoutthe potential splitting. This section investigates the modified PB equations, or the SCequations for the Coulomb potential Ψ( r ) created by Coulomb interactions betweenpoint charges. We consider two types of the modified PB equations: the finite-spreadPB equations [14, 20-24] based on the charge smearing model and the higher-orderPB equations [15-18, 25-30]. We show below that both forms can be derived from thePoisson equation for Ψ( r ) defined by eq. (31) using the electrostatic TCF h el ( r ). Ψ( r ) and the DCF c el ( r )Plugging the Ornstein-Zernike equation (26) into eq. (31) as before, we haveΨ( r ) = − Z d r ′ c el ( r − r ′ )∆ ρ ref ( r ′ ) − Z Z d r ′ d r ” c el ( r − r ”) ρ ref ( r ”) h el ( r ” − r ′ )∆ ρ ref ( r ′ )= − Z d r ′ c el ( r − r ′ ) ρ ref ( r ′ ) − Z d r ” c el ( r − r ”) ρ ref ( r ”)Ψ( r ”) − Ψ c . (32)In the last line of eq. (32), the constant potential Ψ c is related to the isothermalcompressibility κ T as Ψ c = ( k B T ρ κ T ) − − − ρ Z d r c el ( r ) = 1 k B T ρ κ T . (33)The resulting expressions of the Poisson equation for the above potential Ψ vary,according to the treatments of the Laplace equation for the electrostatic DCF c el ( r ).To see the different expressions of the Laplace equation, it is convenient to introducethe Fourier-transform of c el ( r ). In general, c el ( k ) can be written as c el ( k ) = − z l B f ( k ) k , (34)where f ( k ) take various forms depending on the approximations adopted: f ( k ) = e − ( ka )24 m (HNC [38]) (cid:8) ka j ( ka ) (cid:9) (Soft MSA [22 , , ka ) / (Bessel smoothed model [35]) , (35) n the equivalence of self-consistent equations k = | k | and a is the mean separation between adjacent charges as mentionedbefore eq. (15) (see also Appendix A.1 for the detailed definition of a ). In eq. (35),we use m = 1 .
08 according to the HNC approximation [38], the soft MSA [22, 39,40] signifies the mean spherical approximation (MSA) [1, 41] for soft spheres, andthe smeared interaction potential is identified with the DCF in the Bessel smoothedmodel [35] that supposes a smeared charge distribution expressed by the modified Besselfunction K (see Appendix A.2 for the details). In the real-space representation of the electrostatic DCF given by eq. (34), we have −∇ Z d r ′ c el ( r − r ′ ) ρ ref ( r ′ ) = Z Z d s d r ′ ∇ v el ( r − s ) f ( s − r ′ ) ρ ref ( r ′ )= − πz l B Z d s δ ( r − s ) h ρ ref ( s ) i = − πz l B h ρ ref ( r ) i , (36)where we have introduced the smeared density, h ρ ref ( r ) i , defined by h ρ ref ( s ) i = Z d r ′ f ( s − r ′ ) ρ ref ( r ′ )= Z d r ′ f ( s − r ′ ) e µ − J ( r ′ ) . (37)From eqs. (32) and (36), we find ∇ Ψ( r ) = − πz l B h ρ ref ( r ) i + 4 πz l B h ρ ref ( r ) i Ψ( r ) (38) ≈ − πz l B h ρ eq ( r ) i , (39)where the approximation (30) has been applied to the above second line: h ρ eq ( r ) i = (cid:10) ρ ref ( r ) e − Ψ( r ) (cid:11) = Z d s f ( r − s ) ρ ref ( s ) e − Ψ( s ) ; (40)see also Appendix A.3 and A.5 for the details. It is found from eqs. (36) and (37) thatthe introduction of smeared charge model is compatible with the use of the DCF in theliquid-state theory, as has been verified for the strongly coupled OCP (see Appendix A.5for the details) [37, 39, 40]. Equations (37), (39) and (40) are reduced to eqs. (15) and(16) in the HNC approximation that provides f ( r − s ) = { π ( a/m ) } − / e − ( m | r − s | /a ) ;however, a discrepancy exists between the values of m : m = 1 .
08 for the HNC [38],whereas the local molecular field theory [14, 20, 21] set that m = 1 / . ≈ .
67 for acounterion system as mentioned after eq. (16). Considering that the effective meandistance between point charges is diminished to a/m by the factor 1 /m in the Gaussiancharge smearing model, it is suggested from the fitting result ( m ≈ .
67) of the local n the equivalence of self-consistent equations a/m should be shorter than that ofthe HNC approximation of the uniform OCP model for a quantitative description ofinhomogeneous counterion distribution in perpendicular direction to oppositely chargedsurface.
The higher-order PB equations add higher-order gradient terms to the Laplacianappearing in the usual Poisson equation (see also Appendix A.4 for the details). TheFourier-space representation of eqs. (39) and (40) is − k Ψ( k ) = − πl B z f ( k ) ρ ref ( − k ) e − Ψ( k ) . (41)While f ( k ) ρ ( − k ) can be interpreted as a smeared density when leaving f ( k ) on the righthand side, the higher-order PB equations are generated by investigating the contributionof f − ( k ) on the left hand side. In other words, this subsection focuses on how to handle k /f ( k ).In the long-wave approximation valid for ka ≪
1, we have1 f ( k ) ≈ ( ka ) m (HNC)1 + ( ka ) (Soft MSA)1 + ( ka ) (Bessel) , (42)In the above results, m = 1 .
08 as before, whereas the above approximate form for thesoft MSA [37, 39, 40] is derived as follows: we have used the approximation that (cid:26) ka j ( ka ) (cid:27) − ≈ ka ) , (43)expanding the Bessel function j ( x ) as j ( x ) = sin x − x cos xx ≈ x (cid:18) − x (cid:19) , (44)where use has been made of the approximation,sin x − x cos x ≈ x − x , (45)based on the expansions such that sin x ≈ x − x / x /
120 and cos x ≈ − x / x / − k f ( k ) ≈ − (cid:8) γa ) k (cid:9) k , (46) n the equivalence of self-consistent equations γ : γ = √ m (HNC) √ (Soft MSA) (Bessel) . (47)It is found from eqs. (41), (46) and (47) that eqs. (39) and (40) can be reduced to (cid:8) − ( γa ) ∇ (cid:9) ∇ Ψ( r ) = − πl B z ρ ref ( r ) e − Ψ( r ) , (48)using eq. (47) as the values of γ . Thus, the higher-order PB equation of the BSK form[15-18] has been verified.To validate another type of the higher-order PB equation given by eq. (17), weneed to perform the expansion of f − ( k ) up to the order k . In the HNC approximation[38], for example, it is straightforward to show that1 f ( k ) = e ( ka )24 m ≈ m ( ka ) + 132 m ( ka ) , (49)yielding ∇ (cid:18) − a m ∇ + a m ∇ (cid:19) Ψ( r ) = − πl B z e µ − J ( r ) − Ψ( r ) , (50)with m = 1 .
08. In the parentheses on the left hand side of eq. (50), the coefficientof ∇ –term is close to that of of eq. (17), and yet the coefficient of the ∇ –term issomewhat smaller than the previous factor in eq. (17).
6. Derivation scheme of the grand-potential density functional given byeqs. (18) to (20)
In this section, we outline how to derive eqs. (18) to (20) starting with the densityfunctional integral representation of the grand potential. More details of the followingformulations are given in Appendix B.
Let Ω[ v, J ] be the grand potential of non-uniform liquids where constituent particles inan external potential J ( r ) interact via an interaction potential v ( r ). Once the densityfunctional form of Ω[ v, J ] is found for attractive particle systems, other grand-potentialdensity functionals for only repulsive systems (i.e., hard sphere fluids and point-chargesystems) is obtained by setting that v att ( r ) ≡ v with either hard sphereor Coulomb interaction potential (i.e., v HS or v el ). n the equivalence of self-consistent equations e − Ω[ v, J ] = Z Dρ e − Ω ∗ [ ρ ] . (51)Here the conditional grand potential Ω ∗ [ ρ ] of a given density field ρ ( r ) is an extensionof the grand-potential density functional formulated in the conventional DFT [6-12]:Ω ∗ [ ρ ] = Ω V [ ρ ] + ∆Ω , (52)where Ω V [ ρ ] denotes the variational grand potential, which reduces to the equilibriumgrand potential Ω when ρ = ρ eq , and ∆Ω corresponds to the deviation free energyfrom the equilibrium grand potential due to the imposition of a given density field ρ ( r ),instead of ρ eq ( r ). In this subsection, resulting forms of both Ω V [ ρ ] and ∆Ω[ ρ ] are onlypresented (see Appendix B.1 for the detailed derivation).The variational grand potential Ω V [ ρ ] is given byΩ V [ ρ ] = F [ ρ ] − Z d r ρ ( r ) µ + U [ ρ ] , (53) U [ ρ ] = 12 Z Z d r d r ′ ρ ( r ) v att ( r − r ′ ) ρ ( r ′ ) + Z d r ρ ( r ) J ( r ) . (54)As expected from the subscript ”0”, the first term F [ ρ ] on the rhs of eq. (53) representsthe intrinsic Helmholtz free energy of the only repulsive systems where particles in theabsence of external field J ( r ) ≡ v ( r ). We cansee from eq. (54) that the attractive interaction energy is evaluated in the mean-fieldapproximation. The second term on the rhs of eq. (54), which expresses the one-bodypotential energy, is coupled only with the slowly-varying external field J because theconfigurational integration range is set outside the excluded region defined by eqs. (10)and (11).Following the functional integral formulation combined with the conventional DFT,the first approximation of the additional contribution ∆Ω provides the logarithmic formin the Gaussian approximation of the fluctuating one-body potential field, dual to agiven density field ρ ( r ) (see Appendix B.2 for the derivation):∆Ω = 12 ln (cid:12)(cid:12) det G ( r ; ρ ) (cid:12)(cid:12) (55) G ( r ; ρ ) = ρ (cid:8) δ ( r ) + ρ h ( r ) (cid:9) , (56)where we have neglected variation of the density-density correlation function from G ( r ; ρ ) for a uniform density ρ , which is consistent with the other approximationsof F [ ρ ] described below. ρ ref as the saddle-point density Starting with the above density functional integral form (51), we would like to derive themean-field density ρ ref ( r ) where the attractive part v att ( r ) of the interaction potential n the equivalence of self-consistent equations F [ ρ ] into the ideal gas and excess contributions, F id0 [ ρ ] and F ex0 [ ρ ]: F [ ρ ] = F id0 [ ρ ] + F ex0 [ ρ ] , (57)so that we may rewrite, for later convenience, the conditional grand potential Ω ∗ [ ρ ]given by eqs. (52) to (56) asΩ ∗ [ ρ ] = F ex0 [ ρ ] + Ω ∗ att [ ρ ] + ∆Ω , (58)Ω ∗ att [ ρ ] = F id0 [ ρ ] + U [ ρ ] − Z d r ρ ( r ) µ. (59)Accordingly, eq. (51) can read e − Ω[ v, J ] = e − ∆Ω Z Dρ e − Ω ∗ att [ ρ ] (cid:18) R Dρ e −F ex0 [ ρ ] − Ω ∗ att [ ρ ] R Dρ e − Ω ∗ att [ ρ ] (cid:19) , (60)using eqs. (58) and (59). This form (60) reveals that the mean-field approximationof the attractive interaction energy becomes equivalent to the use of the saddle-pointdensity determined by δ Ω ∗ att [ ρ ] δρ (cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ ref = 0 (61)because combination of eqs. (59) and (61) yieldsln ρ ref ( r ) + Z d r ′ v att ( r − r ′ ) ρ ref ( r ′ ) + J ( r ) − µ = 0 , (62)which is identical to the previous result (9). Following the previous treatment, we approximate the functional integral (60) by e − Ω[ v, J ] ≈ e − e Ω[ ρ ref ] ≡ e − Ω ∗ att [ ρ ref ] − ∆Ω (cid:18) R Dn e −F [ ρ ref + n ] − U [ ρ ref + n ] e − Ω ∗ att [ ρ ref ] (cid:19) , (63)extracting only the saddle-point path for the attractive part of the grand potential. Theapproximate form (63) of Ω[ v, J ] has been referred to as the grand-potential densityfunctional which is denoted by e Ω[ ρ ref ] in eq. (18). We expand F [ ρ ref + n ] + U [ ρ ref + n ]around the reference energy up to the quadratic term with respect to fluctuating densityfield n ( r ), yielding (see Appendix B.2 for the derivation) F [ ρ ref + n ] + U [ ρ ref + n ] − Ω ∗ att [ ρ ref ] ≈ F ex0 [ ρ ref ] − Z Z d r d r ′ n ( r ) c ( r − r ′ )∆ ρ ref ( r ′ )+ 12 Z Z d r d r ′ n ( r ) G − ( r − r ′ ; ρ ) n ( r ′ ) , (64) n the equivalence of self-consistent equations F [ ρ ref ] for the v -interaction systemsare of the Ramakrishnan-Yussouff functional form [1, 6, 7]: F [ ρ ref ] = F [ ρ ] − Z Z d r d r ′ ∆ ρ ref ( r ) c ( r − r ′ )∆ ρ ref ( r ′ ) + Z d r ρ ref ( r ) ln (cid:26) ρ ref ( r ) ρ (cid:27) , (65)where the same approximation used in eqs. (28), (55), and (56) has been applied to thelast line of eq. (64) (see also eq. (B.13)).Accordingly, eq. (63) is further reduced to e − e Ω[ ρ ref ] = e −F ex0 [ ρ ref ] − Ω ∗ att [ ρ ref ] − ∆Ω (cid:18)Z Dn e − ∆ F [ n ] (cid:19) (66)∆ F [ n ] = 12 Z Z d r d r ′ n ( r ) G − ( r − r ′ ; ρ ) n ( r ′ ) − Z Z d r d r ′ n ( r ) c ( r − r ′ )∆ ρ ref ( r ′ ) , (67)indicating that the remaining task is to perform the functional integral over thefluctuating n -field. As proved in Appendix B.3, eq. (67) reads∆ F [ n ] = 12 Z Z d r d r ′ e n ( r ) G − ( r − r ′ ; ρ ) e n ( r ′ ) − Z Z d r d r ′ ∆ ρ ref ( r ) { h ( r − r ′ ) − c ( r − r ′ ) } ∆ ρ ref ( r ′ ) , (68)due to the shift of fluctuating field from n to e n ( r ) = n ( r ) − Z Z d r ′ d r ” G ( r − r ′ ; ρ ) c ( r ′ − r ”)∆ ρ ref ( r ”)= n ( r ) − Z Z d r ′ d r ” ρ (cid:8) δ ( r − r ′ ) + ρ h ( r − r ′ )) (cid:9) c ( r ′ − r ”)∆ ρ ref ( r ”)= n ( r ) − ρ Z d r ” h ( r − r ”)∆ ρ ref ( r ”) ≈ n ( r ) − { ρ eq ( r ) − ρ ref ( r ) } , (69)where the last approximation is the same as eq. (28). It follows from eqs. (66) and (68)that the Gaussian integration over the e n –field leads to e − e Ω[ ρ ref ] = e −F ex0 [ ρ ref ] − Ω ∗ att [ ρ ref ] − ∆Ω+ RR d r d r ′ ∆ ρ ref ( r ) { h ( r − r ′ ) − c ( r − r ′ ) } ∆ ρ ref ( r ′ ) × (cid:26)Z D e n e − RR d r d r ′ e n ( r ) G − ( r − r ′ ) e n ( r ′ ) (cid:27) = e −F ex0 [ ρ ref ] − Ω ∗ att [ ρ ref ]+ RR d r d r ′ ∆ ρ ref ( r ) { h ( r − r ′ ) − c ( r − r ′ ) } ∆ ρ ref ( r ′ ) , (70)where the logarithmic term ∆Ω given by eq. (55) is canceled by the result of Gaussianintegration over the e n –field in the underlined term. Combining eqs. (59), (65) and (70),we obtain eqs. (18) to (20), the central results for e Ω[ ρ ref ]. n the equivalence of self-consistent equations Figure 1.
A schematic flow chart of proving the equivalence of previous SC equations[4, 5, 14-34, 42-47] based on the hybrid method [48, 49] that incorporates the densityfunctional theory (DFT) [6-12] into the statistical field theory [15, 23, 31-34, 36, 50-52].The saddle-point equation (9), or eq. (61), determines the reference density ρ ref ( r ).The Gaussian approximation of density fluctuations around ρ ref ( r ) yields the grand-potential density functional e Ω[ ρ ref ] given by eqs. (18) to (20). The equilibrium density ρ eq ( r ) is obtained from e Ω[ ρ ref ] using the relation (22). The single unified equation (25)for ρ eq ( r ) produces a variety of previous SC equations, including the hydrostatic linearresponse (HLR) equation (14) [4, 5, 14, 42-47], the finite-spread Poisson-Boltzmann(PB) equation (15) (or eqs. (A.15) to (A.18)) [14, 20-24], and the higher-order PBequation (17) (or eq. (A.20)) [15-18, 25-30].
7. Summary and conclusions
The flow chart depicted in Fig. 1 outlines the unified description of previous SCequations covering the mean-field equation (9), the HLR equation (14) [4, 5, 14, 42-47], the finite-spread PB equation [14, 20-24] given by eq. (15), and the higher-orderPB equation [15-18, 25-30] expressed by eq. (17). The first step is that the hybridmethod [48, 49], as a kind of thermodynamic perturbation theory, yields the grand-potential density functional e Ω[ ρ ref ] of the reference density ρ ref ( r ) as a solution of thesaddle-point equation (61), or eq. (9). Next, a single unified equation (25) for ρ eq ( r ) isobtained from e Ω[ ρ ref ] using the relation (22). The final step is to demonstrate that theunified equation (25) is reduced to the three types of SC equations [4, 5, 14-34, 42-47].Following the process in Fig. 1, we can divide the formulations presented thus farinto the following four parts: (i) the mean-field equation (9) for the reference density ρ ref ( r ), (ii) the grand-potential density functional e Ω[ ρ ref ] given by eqs. (18) to (20), (iii)the HLR equation (14) for the equilibrium density ρ eq ( r ) [4, 5, 14, 42-47], and (iv) themodified PB equations (39), (40), (48), and (50) [14-30]. A detailed summary of eachpart is provided below. (i) The mean-field equation (9) for the reference density ρ ref ( r ).— The proposedformulation clarifies that the HLR theory [4, 5, 14, 42-47] proceeds in the opposite n the equivalence of self-consistent equations v ( r ) ≡
0. It is the aim of the saddle-pointequation (61) for ρ ref ( r ) to determine a reference state that reflects the entire profile ofthe non-uniform density distribution using a simple and precise method. Along with theHLR treatment, we investigated a perturbative contribution to the reference system fromenergetic aspects when recovering the repulsive part v ( r ) of the interaction potential. (ii) The grand-potential density functional e Ω[ ρ ref ].— The hybrid method leads toa novel density-functional form of the grand potential given by eqs. (18) to (20),thereby demonstrating that the same density-functional form validates the previousSC equations [4, 5, 14-30, 42-47]. It has been shown that there are two main features ofthe density-functional form. One is that e Ω[ ρ ref ] is a functional of the reference densityevaluated in the mean-field approximation, instead of the equilibrium density ρ eq ( r ) usedin the conventional DFT. The other is that the TCF, specifically − h ( r ), is treatedas effective interaction potential instead of the DCF. The TCF, which is generatedfrom the DCF due to the Ornstein-Zernike equation, emerges as a result of consideringfluctuations around ρ ref ( r ) when repulsive interactions are inserted into the systems asdescribed above. In more detail, the second term on the rhs of eq. (64), which is thecoupling term for the fluctuating density n ( r ) with the reference density ρ ref ( r ), plays akey role in generating the effective interaction energy expressed by the TCF. The non-vanishing coupling term reveals that the proposed formulation follows the core conceptof the HLR theory [4, 5, 14, 42-47]. (iii) The HLR equation (14) for the equilibrium density ρ eq ( r ).— The differencebetween the HLR method and our theory is that we treat the slowly varying partof the repulsive interaction potential, including the long-range part of the Coulombinteraction potential, as a perturbative potential added in the reference system, eventhough the other slowly varying parts are evaluated in the mean-field approximationaccording to the previous SC equations reviewed in Section 2. The local molecular fieldtheory [14, 20, 21] starts with the a priori separation of v el ( r ) into harshly repulsiveand slowly varying parts, thus providing the finite-spread PB equation [14, 20-24] or theaforementioned mean-field equation based on a Gaussian charge smearing model withthe harshly repulsive part of v el ( r ) to be cut-off [14, 20-24, 35-40]. However, our goalis to clarify the underlying mechanisms that generate a variety of SC equations from aunified perspective. Therefore, we investigated the theoretical background of why theGaussian charge smearing model is relevant for describing charge-charge correlations.As a result, we verified the HLR equation [4, 5, 14, 42-47] for any repulsiveinteraction system using a unified framework based on the recently developed densityfunctional integral representation [48, 49]. The field-theoretic formulation of the HLRequation also demonstrates the benefits of the hydrostatic approximation [4, 5, 14, 42- n the equivalence of self-consistent equations ρ , instead of the bulk density,as another reference system for evaluating the grand-potential density functional ofnon-uniform liquids. (iv) A set of modified PB equations, eqs. (39), (40), (48) and (50) .— In termsof the Coulomb potential Ψ( r ) defined as eq. (31) using the TCF, we can see thatthe HLR equation (14), expressed by the TCF, forms the basis of validating theBoltzmann distribution (30) for ρ eq ( r ) in both the finite-spread PB equation [14, 20-24]and higher-order PB equation [15-18, 25-30]. These modifications of the PB equationare ascribed to the Poisson equation (36) of the electrostatic DCF c el ( r ). Furthermore,the Ornstein-Zernike equation yields the SC equation (32) for Ψ( r ) with the use of c el ( r ). Combining these relations, eqs. (32) and (36), we obtain the linearized SCequation (38) for Ψ( r ), which transforms to alternative representations, the finite andhigher-order PB equations, depending on how the weight function f ( r ) is approximatedand/or interpreted. We stress again that these modifications of the conventional PBequation emerge due to the electrostatic DCF, and that the electrostatic DCF naturallyappears when calculating ρ eq ( r ) from the grand-potential density functional e Ω[ ρ ref ].In conclusion, we have proved the equivalence of previous SC equations [4, 5, 14-34,42-47] by demonstrating that the apparent diversity of SC equations can be explainedby the identical form of the grand-potential density functional given by eqs. (18) to(20). This theoretical finding implies that not only improvements to the previous formsbut also derivations of any new equations can be performed in a systematic manner,by either exploring a new system to which our formulations may apply or consideringhigher-order terms beyond the Gaussian approximation. Appendix A. Summary of previous results related to modified PBequations for point-charge systems
Appendix A.1. Coupling constant in non-uniform systems
Let a and Γ be the mean distance between point charges and the coupling constantrepresenting the strength of Coulomb interactions, respectively. As is well-known, Γ inthe uniform OCP is given by Γ = z l B a , (A.1)when the valence of each charge is z . In non-uniform point-charge systems, however,the distance between charges cannot be uniquely determined. Correspondingly, it is nottrivial to define Γ, unlike the conventional uniform OCP. Two types of coupling constantshave been defined so far for the typical non-uniform systems of small counterions thatare dissociated from oppositely charged macroions, or macroscopic objects includingcharged plates: while one uses the Gouy-Chapman length as a reference length a , theother definition of Γ is based on an analogy with the two-dimensional OCP. n the equivalence of self-consistent equations b betweencounterions on the surface is defined by πb σ = z, (A.2)using the number density σ per area of surface charges. Obviously, b is the minimumof counterion-counterion separation: b ≤ a . The second definition of Γ in counterionsystems therefore implies that eq. (A.1) leads to the unique definition of Γ for non-uniform systems as long as a is identified with b . That is, we set that a = b not onlyin eq. (A.1) but also in the main text. Consequently, the relation Γ ≫ Appendix A.2. Two types of weighted densities
There are two methods to formulate charge smearing models [14, 20-24, 35-40] althoughnot described explicitly. To see this, we first investigate the relationship between threedensity functional forms of the slowly varying part of the Coulomb interaction energy U el1 that is associated with the slowly varying part of the Coulomb interaction potentialsuch as eq. (8).The first form expresses U el1 as the sum of Coulomb interactions between smearedcharges with the interaction potential being given by the original interaction potential v el ( r ): U el1 [ ρ ] = z l B Z Z d r d r ′ ρ ( r ) v el ( r − r ′ ) ρ ( r ′ ) , (A.3) ρ ( r ) = Z d s ρ ( s ) ω ( s − r ) , (A.4)where ρ ( r ) is smeared due to the weighted function ω ( s − r ); for instance, we have ω ( r ) = { πa / (2 m ) } / e − ( mra ) (Gaussian) πa Θ( a − r ) (Homogeneous) πa (cid:16) K (2 r/a ) r (cid:17) (Bessel) , (A.5)which are the models that consider the radial distribution given by the modified Besselfunction K [26] as well as the Gaussian and homogeneous charge distributions.Equations (A.3) and (A.4) naturally lead to the second description of U el1 where theslowly varying interaction potential v ( r ) is introduced without smearing the density: U el1 [ ρ ] = z l B Z Z d s d s ′ ρ ( s ) v el1 ( s − s ′ ) ρ ( s ′ ) . (A.6) n the equivalence of self-consistent equations v el1 ( s − s ′ ) = Z Z d r d r ′ ω ( s − r ) v el ( r − r ′ ) ω ( r ′ − s ′ ) , (A.7)which also reads v el1 ( s − s ′ ) = Z Z d r ′ d r ” v el ( s − r ”) ω ( r ” − r ′ ) ω ( r ′ − s ′ ) (A.8)due to the coordinate change such that r ” = s − r + r ′ .Equations (A.6) and (A.8) thus generate the third form: U el1 [ ρ ] = z l B Z Z d r d r ” ρ ( r ) v el ( r − r ”) ρ ( r ”) , (A.9) ρ ( r ”) = Z Z d r ′ d s ′ ω ( r ” − r ′ ) ω ( r ′ − s ′ ) ρ ( s ′ ) . (A.10)Equation (A.10) represents the second type of the weighted density used in the maintext; as will be seen below, the expression (A.10) yields the same form as the distributionfunction given in eq. (16). Appendix A.3. Finite-spread PB equations [14, 20-24]
There are two kinds of expressions for the Coulomb potentials created by counterion-counterion interactions in correspondence with the above two definitions of smeareddensity, i.e., ρ ref ( r ) and ρ ref ( r ). While a finite-spread PB theory [22] uses the formerdefinition, ψ ( r ) = Z d r ′ v el ( r − r ′ ) ρ ref ( r ′ ) , (A.11)the local molecular field theory [14, 20, 21], another finite-spread PB theory, adopts thelatter: Ψ( r ) = Z d r ′ v el ( r − r ′ ) ρ ref ( r ′ ) (A.12)= Z d r ′ v el1 ( r − r ′ ) ρ ref ( r ′ ) . (A.13)Both potentials are related to each other as follows:Ψ( r ) = Z d r ′ ω ( r − r ′ ) ψ ( r ′ ) , (A.14)indicating that Ψ( r ) corresponds to the weighted potential of ψ ( r ).We can see from eq. (A.13), the key representation, that the reference density givenby eq. (9) obeys the Boltzmann distribution such that ρ ref ( r ) = e µ − J ( r ) − R d r ′ ω ( r − r ′ ) ψ ( r ′ ) (A.15)= e µ − J ( r ) − Ψ( r ) . (A.16) n the equivalence of self-consistent equations ψ ( r ) and Ψ( r ) yield ∇ ψ ( r ) = − πz l B Z d r ′ ω ( r − r ′ ) ρ ref ( r ′ ) , (A.17) ∇ Ψ( r ) = − πz l B Z Z d r ′ d r ” ω ( r − r ′ ) ω ( r ′ − r ”) ρ ref ( r ”) , (A.18)respectively. Combining eqs. (A.15) and (A.17), we obtain an expression of the finite-spread PB equation. As mentioned at the end of Appendix A.3, on the other hand, eq.(A.18) reads ∇ Ψ( r ) = − πz l B (cid:26) { πa /m } Z Z d r ′ d r ” e − | m ( r − r ′ ) /a | e − | m ( r ′ − r ”) /a | ρ ref ( r ”) (cid:27) = − πz l B (cid:26) { πa /m } / Z d r ” e −| m ( r − r ”) /a | ρ ref ( r ”) (cid:27) (A.19)for Gaussian charge smearing model. Combining eqs. (A.16) and (A.19), we verifyanother form of the finite-spread PB equation given by eqs. (15) and (16) used in thelocal molecular field theory [14, 20, 21]. Appendix A.4. Higher-order PB equations [15-18, 25-30]
One type of higher-order PB equations is truncated at the biharmonic ∇ ∇ -term, whichcan be written as (cid:8) ( γa ) ∇ − (cid:9) ∇ Ψ( r ) = 4 πl B z ρ ref ( r ) e − Ψ( r ) , (A.20)where γ takes the value of either Γ or ∼ [16-18, 25, 28]. This type of equation is thesame as the BSK equation in binary systems as mentioned in the main text. The BSKequation [16-18] has been found to be a powerful tool in investigating the structural anddynamical properties of concentrated electrolytes and room temperature ionic liquidswhere a is identified with the diameter of ions.Meanwhile, the other type has been presented in eq (17) which has the ∇ ∇ -termas the gradient term of the highest order. Incidentally, in eq. (17), the coefficients ofthe ∇ ∇ –and ∇ ∇ –terms use a previous result of σ , given that ∆ = 0 in eq. (26) ofRef [26]. Appendix A.5. Correspondence between the electrostatic DCF c el ( r ) and the slowlyvarying part v el1 ( r ) of the Coulomb interaction potential The electrostatic DCF relevant in the strong coupling regime of the OCP can be regardedas v el1 ( r ) when considering specific charge smearing models. Comparison between eqs.(34) and (A.8) implies that the equality, f ( r − r ′ ) = Z d r ” ω ( r − r ”) ω ( r ” − r ′ ) , (A.21) n the equivalence of self-consistent equations f ( r ) and ω ( r ) for the strongly coupled OCP. More specifically, f ( r ) in theHNC approximation [38] is expressed by ω ( r ) of the Gaussian charge smearing model,whereas f ( r ) in the soft MSA [37, 39, 40] by ω ( r ) of the homogeneous charge smearingmodel (see also eqs. (35) and (A.5)). Appendix B. Details of Section 6
Appendix B.1. Density functional form of the conditional grand potential Ω ∗ [ ρ ] :derivation of eqs. (51) to (54) [48, 49] The configurational representation Ω[ v, J ] is e − Ω[ v, J ] = Tr exp ( − X i We express ∆ F [ n ] given by eq. (67) as∆ F [ n ] = 12 Z Z d r d r ′ n ( r ) G − ( r − r ′ ; ρ ) n ( r ′ ) − Z Z d r d r ′ n ( r ) c ( r − r ′ )∆ ρ ref ( r ′ ) , = 12 Z d n ( ) "Z d G − ( − ) (cid:26) n ( ) − Z Z d d G ( − ) c ( − )∆ ρ ref ( ) (cid:27) , (B.14)where G ( r − r ′ ; ρ ) has been abbreviated as G ( r − r ′ ) in the last equality. Theabove underlined terms in the last line of eq. (B.14) implies the necessity to shift the n the equivalence of self-consistent equations n ( r )to e n ( r ) defined by eq. (69) for performing the Gaussianintegration over the fluctuating density field.To see this, we rewrite the first term on the rhs of eq. (68) using the n –field asfollows:12 Z Z d d e n ( ) G − ( − ) e n ( )= 12 Z Z d d n ( ) G − ( − ) n ( ) − Z Z Z Z d d d d G − ( − ) × { n ( ) G ( − ) c ( − )∆ ρ ref ( ) + n ( ) G ( − ) c ( − )∆ ρ ref ( ) } + 12 Z Z Z d d d G − ( − ) × (cid:20)Z d G ( − ) c ( − )∆ ρ ref ( ) (cid:21) (cid:20)Z d G ( − ) c ( − )∆ ρ ref ( ) (cid:21) = 12 Z Z d r d r ′ n ( r ) G − ( r − r ′ ) n ( r ′ ) − Z Z d r d r ′ n ( r ) c ( r − r ′ )∆ ρ ref ( r ′ )+ 12 Z Z Z Z d d d d ∆ ρ ref ( ) c ( − ) G ( − ) c ( − )∆ ρ ref ( )= ∆ F [ n ] + 12 Z Z d r d r ′ ∆ ρ ref ( r ) { h ( r − r ′ ) − c ( r − r ′ ) } ∆ ρ ref ( r ′ ) . (B.15)In obtaining the above last line of eq. (B.15), use has been made of the followingderivation due to the Ornstein-Zernike equation (26): Z Z d d c ( − ) G ( − ) c ( − ) = Z Z d d c ( − ) (cid:8) δ ( − ) + ρ h ( − ) (cid:9) ρ c ( − )= Z d h ( − ) ρ c ( − )= h ( − ) − c ( − ) . (B.16)Combination of eqs. (B.14) and (B.15) proves eq. (68). References [1] Hansen J P and Mcdonald I R 2013 Theory of Simple Liquids (Academic Press, London), andreferences therein.[2] Widom B 1967 Science Science Physica A Ann. Rev. Phys. Chem. Adv. Phys. Fundamentals of inhomogeneous fluids (Marcel Dekker, New York) 85-175[8] Likos C N 2001 Phys. Rep. J. Phys.: Condens. Matt. n the equivalence of self-consistent equations [10] Wu J and Li Z 2007 Annu. Rev. Phys. Chem. J. Mol. Liq. J. Phys.: Condens. Mat. Nature P. Natl. Acad. Sci. USA J. Chem. Phys Phys. Rev. Lett. Langmuir Phys. Rev.Lett. J. Phys. Chem. B J. Phys. Chem. B P. Natl. Acad. Sci. USA Adv. Chem. Phys. Phys. Rev. E Phys. Rev. E Phys. Rev. E Soft Matter Europhys. Lett. Phys. Rev. E J. Phys. Chem. C Phys. Rev. E Eur. Phys. J. E J. Stat. Mech.: Theory E. P05033[33] Buyukdagli S and Blossey R 2014 J. Chem. Phys. J. Phys.: Condens. Mat. J. Chem. Phys Condens. Matt. Phys. J. Chem. Phys. J. Chem. Phys. J. Chem. Phys. Phys. Rev. E Phys. Rev. E J. Phys. Chem. B Phys. Rev. Lett. Phys. Rev. Lett. J. Chem. Phys. J. Chem. Phys J. Chem. Phys J. Phys. A: Math. Theor. Entropy Phys. Rev. E Physica A Phys. Chem. Chem. Phys. Phys. Rev. Lett.8