A correlation-hole approach to the electric double layer with counter-ions only
aa r X i v : . [ c ond - m a t . s o f t ] M a r A correlation-hole approach to the electric double layer with counter-ions only
Ivan Palaia
LPTMS, CNRS, Univ. Paris-Sud, Universit´e Paris-Saclay, 91405 Orsay, France
Martin Trulsson
Theoretical Chemistry, Lund University, Lund, Sweden
Ladislav ˇSamaj
Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia
Emmanuel Trizac
LPTMS, CNRS, Univ. Paris-Sud, Universit´e Paris-Saclay, 91405 Orsay, France (Dated: August 10, 2018)We study a classical system of identically charged counter-ions near a planar wall carrying auniform surface charge density. The equilibrium statistical mechanics of the system depends on asingle dimensionless coupling parameter. A new self-consistent theory of the correlation-hole typeis proposed which leads to a modified Poisson-Boltzmann integral equation for the density profile,convenient for analytical progress and straightforward to solve numerically. The exact densityprofiles are recovered in the limits of weak and strong couplings. In contrast to previous theoreticalattempts of the test-charge family, the density profiles fulfill the contact-value theorem at all valuesof the coupling constant, and exhibit the mean-field decay at asymptotically large distances fromthe wall, as expected. We furthermore show that the density corrections at large couplings exhibitthe proper dependence on coupling parameter and distance to the charged wall. The numericalresults for intermediate values of the coupling provide accurate density profiles which are in goodagreement with those obtained by Monte-Carlo simulations. The crossover to mean-field behaviorat large distance is studied in detail.
I. INTRODUCTION
Experiments with large macromolecules are often per-formed in water, which is a polar solvent. This is thecase for many applications using colloids, including theproteins in our bodies. This results in the release of lowvalence micro-ions into the solution, so that the colloidsacquire a surface charge density, opposite to the chargeof mobile micro-ions (coined as “counter-ions”). The to-tal surface charge can exceed thousands of elementarycharges e . In the first approximation, the curved sur-face of a macromolecule can be replaced by an infiniterectilinear plane.The charged macromolecule and the surroundingcounter-ions form a neutral electric double layer, see re-views [1–3]. In turn, the double layer is paramount inmediating the effective interactions between charged bod-ies in solution. At large enough Coulombic coupling, itis for instance known that like-charged macromoleculescan effectively attract each other in some intermediatedistance range, as was observed experimentally [4–9] andby computer simulations [10–13].In a wealth of natural or synthetic systems, micro-ions can be of both signs, with positively and negativelycharged species. In this paper, we restrict ourselves tosimplified so-called salt-free (or deionized) Coulomb sys-tems with counter-ions only. This is a convenient startingpoint for analytical progress, where detailed computersimulation results are also available [14–28]. Such modelsapply to deionized suspensions, see e.g. the experiments reported in Refs. [29–33]. In the deionized limit, systemsof counter-ions near charged surfaces have poor screeningproperties, but the standard Coulomb sum rules relatingthe one-body and two-body densities do apply [34, 35].For the system of counter-ions near a charged wall, thehigh-temperature (weak-coupling, WC) limit is describedby the Poisson-Boltzmann (PB) mean-field theory [36]and by its systematic improvement via the loop expan-sion [16, 37, 38]. The opposite strong-coupling (SC) limitwas investigated within a field-theoretical formulation ofthe model by using a renormalized expansion of virialtype [17, 18, 39–42]. In the leading SC order and inthe present planar geometry, the counter-ions effectivelybehave as non-interacting objects, as far as one is notinterested in the tail of the density profile; this fact wasconfirmed numerically in a number of numerical studies[43–48]. The first correction to the single-particle den-sity profile, calculated within a fugacity expansion with arenormalization of infrared divergences [18], is correct inits functional form, but with a wrong prefactor, depart-ing by orders of magnitude from its Monte-Carlo (MC)estimate [22]. Other SC approaches [49] emphasize thetwo-dimensional Wigner crystallization of mobile chargesat the wall surface for low temperatures. Recently [50],by a perturbative approach around the Wigner crystal,the single particle treatment was recovered in the lead-ing SC order. Moreover, the derived prefactor of the firstSC correction is in excellent agreement with MC simula-tions, also in the coupling range where no Wigner crystalis formed (strongly modulated liquid regime). Notewor-thy are also field theoretic techniques, that allow to coverthe crossover regime between WC and SC, by a propersplitting of the interactions between ions, discriminatingshort and long distances [21, 22, 25, 51].For a system of identical charges with Coulomb repul-sion, the pair correlation function is strongly depletedat small distances. This gives credit to the image of acorrelation hole around each ion in the system, an ideathat turned useful in various approaches going beyondthe PB theory [21, 22, 25, 40, 52–56]. Recently [28], fora dielectric interface, the single particle strong-couplingview was combined with the idea of the correlation hole,to obtain very accurate density profiles for strongly tomoderately coupled charged fluids. This latter contribu-tion provides the most accurate theory available so farfor these systems. We emphasize that this approach isnot self-consistent, and does not reproduce mean-field PBresults at small couplings, two key differences with thetheory to be developed below.In Ref. [20], an attempt has been made to establisha universal theory which works adequately for any valueof the coupling. Based on a mean-field treatment of theions response to the presence of a test charge, the exactdensity profile was reproduced in the limits of weak andstrong couplings. For intermediate values of the coupling,the obtained approximate density profiles agree with MCsimulations, except for two shortcomings. Firstly, thecontact theorem for the counter-ion number density atthe wall [57–59] is not satisfied. Secondly, although acrossover from exponential to algebraic decay is observedat large distances from the wall, there is an additionalprefactor to the mean-field PB solution which dependson the coupling constant. This is in contradiction withthe common expectation that mean-field should hold atlarge distances from the wall [18, 21, 22, 39, 56, 60], asthe small density of counter-ions should effectively drivethe system into the WC regime. Note that the loop cor-rections to the PB solution [16, 37, 38] are consistent withthis expectation.In this work, we propose a self-consistent theory forcounter-ions near a charged rectilinear wall, which isbased on the idea of a cylindrical correlation hole. Aswas the case in the test-charge approach of Ref. [20], theexact density profiles are recovered in the limits of weakand strong couplings. But in contrast to that theory, atall values of the coupling constant do the density profilesfulfill the contact-value theorem. Moreover, the densityprofiles are exactly of mean-field type at asymptoticallylarge distances from the wall, as expected. This allowsus to address the elusive question of the asymptotic largedistance crossover to mean-field in this geometry.The article is organized as follows. In Sec. II, we intro-duce the basic notations for the model . The correlation-hole approach is presented in Sec. III. For the sake ofanalogy and completeness, the derivation of the PB the-ory is provided as well. Analytical progress was madepossible by an original rederivation of the contact theo-rem, that does not require the explicit resolution of the (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) − q e− q e− q e e z=0 σ z FIG. 1. The electric double layer with counter-ions of valence q . The interface at z = 0 bears a surface charge σe , so thatthe system as a whole is electroneutral. theory under study. Section IV derives a number of ex-act results. The SC limit is worked out. Then, at ar-bitrary coupling parameter, the large-distance behaviorof the density profile is shown to be exactly of the PBmean-field type. In addition, we derive the subleadingcontribution to the mean-field tail. Numerical resultsfor the density profile at specific values of the couplingconstant are compared with those obtained by the test-charge method [20] and by MC simulations in Sec. V.The crossover distance from the wall to the mean-fieldalgebraic decay of the density profile is determined too.A short recapitulation and concluding remarks are givenin Sec. VI, where we present some results pertaining toan interacting two-plate system both in Monte Carlo andwithin our self-consistent scheme. II. BASIC FORMALISM
We consider the one-wall geometry pictured in Fig. 1,with positions denoted by r = ( x, y, z ). A hard wall,impenetrable to particles, is localized in the half-space { r , z < } . In the complementary half-space { r , z > } ,there are N mobile q -valent counter-ions (classical point-like particles) of charge − qe , where e is the elementarycharge. The particles are immersed in a solution withthe same dielectric constant ε as the confining wall, sothat no electrostatic image forces ensue. The infinitewall surface, localized at z = 0, carries a fixed uniformsurface-charge density σe with σ >
0. The system as awhole is electro-neutral, and the particles are in thermalequilibrium at some inverse temperature β = 1 / ( k B T ).There are two relevant length scales in the model. InGauss units, two unit charges at distance r interact bythe 3D Coulomb energy e / ( εr ); the distance at whichthis energy coincides with the thermal energy k B T is theBjerrum length ℓ B = βe ε . (1)The potential energy of an isolated counter-ion of charge − qe at distance z from the wall surface is given by E ( z ) = 2 πqe σε z ; (2)the distance at which this energy equals to the thermalenergy k B T defines the Gouy-Chapman length µ = 12 πqℓ B σ . (3)The dimensionless coupling parameter Ξ, reflecting thestrength of electrostatic correlations, is defined as theratio of the two length scales:Ξ = q ℓ B µ = 2 πq ℓ σ. (4)Denoting by h· · · i the canonical thermal average, theparticle number density at point r (with thus z ≥
0) isdefined as n ( r ) = h P Ni =1 δ ( r − r i ) i . It depends only onthe distance z from the wall, n ( r ) = n ( z ). The elec-troneutrality condition corresponds to the constraint q Z ∞ dz n ( z ) = σ. (5)The contact density of counter-ions at the wall is relatedto the surface charge density via the planar contact-valuetheorem [57–59] as follows n (0) = 2 πℓ B σ . (6)The averaged particle density will be often written in arescaled form with a dimensionless particle z -coordinateconsidered in units of the Gouy-Chapman length µ : e n ( z ) ≡ n ( µz )2 πℓ B σ . (7)In terms of e n , the electroneutrality requirement (5) andthe contact-value constraint (6) take the forms Z ∞ dz e n ( z ) = 1 (8)and e n (0) = 1 , (9)respectively. To avoid unnecessarily heavy notations, z will in the remainder refer to the rescaled distance z/µ ,whenever it appears in a expression involving the reduceddensity e n . The model is exactly solvable in two limits. In theweak-coupling limit Ξ →
0, the PB approach [36] impliesa slowly decaying particle density profile e n PB ( z ) = 1(1 + z ) . (10)In the strong-coupling limit Ξ → ∞ , the single-particlepicture of counter-ions in the linear surface charge poten-tial [17, 18, 56] leads to an exponentially decaying profile e n SC ( z ) = exp( − z ) . (11) III. THE CORRELATION-HOLE APPROACH
At any point r with z ≥
0, the relation between the(averaged) electric potential ψ and the charge distribu-tion ρ is given by the Poisson equation ∇ ψ ( r ) = − πε ρ ( r ) . (12)For the present geometry, the electrostatic potential andthe charge distribution ρ = − qen depend only on thedistance from the wall z , so that d dz ψ ( z ) = 4 πε qen ( z ) . (13)With respect to the boundary condition for the electricfield at z = 0, ddz ψ ( z ) = − πε σe (14)and the 1D relation d dz | z | δ ( z ) (15)with δ the Dirac delta distribution, the electric potentialis expressible explicitly as ψ ( z ) = − πε σez + 2 πε qe Z ∞ dz ′ ( | z − z ′ | − z ′ ) n ( z ′ ) . (16)The interpretation of this expression is transparent: inaddition to the bare plate potential (first term on therhs, linear in z ), the mobile counter-ions contribute to theelectric potential though the integral term. The potentialis determined up to an irrelevant constant; here we fixedthe “gauge” ψ (0) = 0. A. PB theory
The electrostatic energy of a counter-ion in the poten-tial ψ ( z ) is − qeψ ( z ). In the PB approach, the particledensity is related locally to the corresponding Boltzmannfactor as n PB ( z ) = n exp [ βqeψ PB ( z )] (17)where the parameter n ensures the normalization (5).In terms of the dimensionless e n (7), the self-consistentPB equation (17) is written as e n PB ( z ) = e n exp [ φ PB ( z )] , (18)where the PB reduced potential φ PB is given by φ PB ( z ) = − z + Z ∞ dz ′ ( | z − z ′ | − z ′ ) e n PB ( z ′ ) . (19)Note the gauge φ PB (0) = 0.The normalization constant e n is determined by theelectroneutrality condition (8) through e n = 1 R ∞ d z exp [ φ PB ( z )] . (20)There exists a simple way to obtain the explicit value of e n ; it will prove useful below and we thus present it in itssimplest clothing. We first differentiate the φ PB -potential(19) with respect to z : ddz φ PB ( z ) = − Z ∞ dz ′ e n PB ( z ′ )sgn( z − z ′ ) , (21)where sgn denotes the standard signum (sign) function.The integral Z ∞ dz (cid:18) dφ PB dz + 1 (cid:19) e n PB ( z ) = Z ∞ dz Z ∞ dz ′ e n PB ( z ) e n PB ( z ′ )sgn( z − z ′ ) (22)vanishes due to the anti-symmetric property of the func-tion under integration in the rhs with respect to the in-terchange transformation z ↔ z ′ . From (18) we get e n PB ( z ) dφ PB dz = d e n PB ( z ) dz . (23)Consequently, we have from (22) that Z ∞ dz ddz e n PB ( z ) = − Z ∞ dz e n PB ( z ) . (24)The density e n ( z ) vanishes as z → ∞ , so that e n PB (0) = Z ∞ dz e n PB ( z ) = 1 , (25)which is nothing but the contact-value theorem (9). Wesee that, within the PB theory, the normalization (8)automatically ensures the contact-value theorem (9), andvice versa. Under the gauge φ PB (0) = 0, the contact-value relation (25) fixes e n = 1 (26)in (18). It is easy to check that under this normalization,the PB solution (10) satisfies Eqs. (18) and (19). B. Inclusion of the correlation hole
In the single-particle SC solution (11), the only act-ing potential is due to the fixed surface-charge density;this potential is present also in the PB solution (18), butthere is an additional potential related to the mean par-ticle density profile. Thus in some sense, the SC solutionis simpler than its mean-field counterpart, since mutualcounter-ions interactions do not contribute to the lead-ing order SC response. Yet, for large Ξ, counter-ions arestrongly correlated in the ( x, y ) plane, because of theirstrong mutual repulsion; this leads to a marked correla-tion hole (“Coulomb hole”), inaccessible to other chargedparticles [18]. At smaller Ξ value, the correlation hole isless marked (in the sense that the pair correlation func-tion does not vanish for distances smaller than the holesize [18]), but a feature of depletion remains. In addition,the form of the correlation hole depends on the distancefrom the wall of the particle under consideration. It is ex-pected, in the large Ξ regime, that the correlation hole iscylindrical if the particle is close to the wall, and spher-ical for large distances from the wall (the bulk region)[20]. In this work, independently of the particle positionwith respect to the wall, we take as the correlation holean infinite cylinder, perpendicular to the wall surface,whose axis passes through this particle. The radius R ofthe cylinder is determined by the requirement that thetotal disc surface of all cylinders πR N equals the planarinterface surface, namely R = qπσ = 2 q ℓ B µ. (27)Note that, in units of the relevant Gouy-Chapman length µ , R /µ = 2 Ξ, and up to an irrelevant prefactor, similarchoices were made in [20, 22]. This means that in unitsof µ , the correlation-hole radius R vanishes in the PBlimit, while it goes to ∞ in the SC regime. Here, it canbe stressed that µ is the relevant length scale for densitygradients, both in the WC and SC regimes, as revealedby Eqs. (10) and (11).The exclusion of other particles from the cylindricalneighborhood of the given particle localized at z modifiesthe electric potential ψ ( z ) (16) to ψ ch ( z ) = ψ ( z ) − δψ ( z ),where δψ ( z ) = Z ∞ dz ′ Z R dρ πρ − qen ( z ′ ) p ( z − z ′ ) + ρ = 2 πqe Z ∞ dz ′ n ( z ′ ) × h | z − z ′ | − p R + ( z − z ′ ) i . (28)We take ψ ch ( z ) as the mean-field potential which deter-mines the counter-ion density via n ( z ) = n e βqeψ ch ( z ) .We shift the reduced correlation-hole potential φ ( z ) = βqeψ ch ( z ) by a constant to fix the gauge φ (0) = 0. Thus,the rescaled density profile e n is given by e n ( z ) = e n exp [ φ ( z )] , (29)where the reduced potential φ ( z ) = − z + Z ∞ dz ′ e n ( z ′ ) × (cid:16)p
2Ξ + ( z − z ′ ) − p
2Ξ + z ′ (cid:17) (30)satisfies the gauge φ (0) = 0 and the normalization con-stant e n is determined by the electroneutrality condition(8).The explicit value of e n can be derived in close analogywith the above PB treatment. We first differentiate the φ -potential with respect to z : ddz φ ( z ) = − Z ∞ dz ′ e n ( z ′ ) z − z ′ p
2Ξ + ( z − z ′ ) . (31)The integral Z ∞ dz (cid:18) dφdz + 1 (cid:19) e n ( z ) = Z ∞ dz Z ∞ dz ′ e n ( z ) e n ( z ′ ) × z − z ′ p
2Ξ + ( z − z ′ ) (32)vanishes due to the anti-symmetric property with respectto the interchange z ↔ z ′ of the function under integra-tion in the rhs. Then the equality Z ∞ dz d e ndz = − Z ∞ dz e n ( z ) (33)implies the contact-value theorem e n (0) = Z ∞ dz e n ( z ) = 1 . (34)We see that, as is the case within PB theory, the den-sity normalization automatically ensures the validity ofthe contact-value theorem. This is a nontrivial and exactproperty of our Coulombic system [61], that an approx-imate or phenomenological theory may violate (in thisrespect, it is thus remarkable that PB theory does fulfillthis condition). None of the theories presented in [20] or[22] do obey the contact theorem. The gauge φ (0) = 0fixes the normalization constant e n = 1. The densityprofile then takes the form e n ( z ) = exp [ φ ( z )] , (35)where the reduced potential φ ( z ) is given by (30).To summarize at this point, our key relation is (35),supplemented by the closure relation (30). The latter ex-presses the test-particle potential φ in terms of the meancounter-ion density, in a self-consistent fashion. IV. ANALYTICAL RESULTS
To begin with, it is straightforward to realize that inthe weak-coupling limit Ξ →
0, the reduced potential (30) takes the PB form (19). Due to the same normal-ization e n = 1, our correlation-hole profile (29) reducesto the PB one (18). In this section, we prove that ourcorrelation-hole theory also provides the exact densityprofiles in the strong coupling limit, where a series ex-pansion is constructed to account for corrections to SC.Then, we focus on the tail of the ionic profile, showingthat it is of mean-field type, and working out at arbitraryΞ the corresponding large- z correction to the dominanttail. All these results will be compared to numerical datain section V. A. SC limit
In the SC limit Ξ → ∞ , assuming that e n is short-ranged (e.g., decaying exponentially) and all its momentsexist, we can perform in Eq. (30) the expansion p
2Ξ + ( z − z ′ ) − p
2Ξ + z ′ ∼ ( z − z ′ ) √ − z ′ √
2Ξ (36)to obtain φ ( z ) = φ SC ( z ) = − z . Inserting this one-bodypotential due to the surface-charge density into (35) re-produces the SC solution (11).To construct an expansion around the SC limit, weanticipate the systematic 1 / √ Ξ-expansion of the densityprofile of the form e n ( z ) = e − z " ∞ X k =1 f k ( z )(2Ξ) k/ (37)with as-yet unknown functions f k ( z ). The contact theo-rem (9) fixes the values of these functions at the wall, f k (0) = 0 , (38)and the normalization (8) fixes their integrals over z , Z ∞ dzf k ( z ) = 0 . (39)Since φ ( z ) = ln e n ( z ), we have φ ( z ) = − z + ln " ∞ X k =1 f k ( z )(2Ξ) k/ . (40)Consequently, ddz φ ( z ) = − P ∞ k =1 f k ( z )(2Ξ) k/ ∞ X l =1 f ′ l ( z )(2Ξ) l/ . (41)At the same time, from (31) we get ddz φ ( z ) = − Z ∞ dz ′ e − z ′ " ∞ X k =1 f k ( z ′ )(2Ξ) k/ × ( z − z ′ ) √ " ∞ X l =1 (cid:18) − / l (cid:19) ( z − z ′ ) l (2Ξ) l . (42)Comparing the last two relations, we obtain an infiniteiterative sequence of equations which relate f ′ l ( z ) to all f k ( z ) with k ≤ l −
1. It turns out that f k ( z ) is a polyno-mial of order 2 k , the absolute term is equal trivially tozero because of the contact condition (38).The first correction to the SC profile reads as f ( z ) = z − z. (43)Writing formally the SC density profile plus the first cor-rection as e n ( z ) = e − z (cid:20) θ (cid:18) z − z (cid:19)(cid:21) , (44)we have θ = √
2Ξ = 1 . √ Ξ. This has to be comparedwith the very accurate estimate based on the Wignercrystal θ = 1 . √ Ξ [50]. A similar result θ ∝ √ Ξ wasobtained in Ref. [25]. On the other hand, the finding θ = Ξ of the renormalized virial expansion [18] fails inthe dependence on Ξ. Indeed, Monte Carlo simulationsfully corroborate the θ ∝ Ξ / scaling [50].The next expansion functions read f ( z ) = z − z z − z,f ( z ) = z − z z − z − z − z,f ( z ) = z − z
48 + z − z
24 + z − z − z, (45)etc. It is interesting that the normalization constraint(39) is automatically ensured by respecting the contactrelation (38), which can serve as a check of algebra. Notethat, at arbitrary order of the expansion around the SClimit, the density profile is decaying exponentially. B. Large-distance decay: asymptotic validity of PB
For any finite value of the coupling Ξ and at asymptot-ically large distances from the wall ( z → ∞ ), the exactdensity profile is expected to exhibit the PB power-lawbehavior (10) [18, 21, 22, 39, 56, 60], e n ( z ) ∼ /z . Itis worthwhile emphasizing that this power law behaviorimplies that the (unscaled) counter-ion density becomesindependent of the surface charge density σ , thereby re-vealing a universal behavior. An important feature ofour theory is that this asymptotic behavior indeed takesplace, at variance with the approach of Ref. [20].To prove this fact, let us first assume that at largedistances e n ( z ) ∼ z →∞ az (46)with some positive number a which might depend on Ξ.Since the positive density e n does not exhibit divergent singularities, it must be bounded from above at any point z by the function e n ( z ) ≤ A (1 + z ) , (47)where A ≥ a . For Ξ = 0 we can take A = 1, while inthe SC limit Ξ → ∞ we have A = 4 /e = 1 . . . . .The precise value of A is immaterial, as long as it isfinite. Writing − − R ∞ dz ′ e n ( z ′ ), thepotential derivative is expressed after simple algebraicmanipulations as follows ddz φ ( z ) = − Z ∞ z dz ′ e n ( z ′ ) − I ( z, Ξ) + I ( z, Ξ) , (48)where I ( z, Ξ) = Z z dz ′ e n ( z ′ ) " − z − z ′ p
2Ξ + ( z − z ′ ) ,I ( z, Ξ) = Z ∞ z dz ′ e n ( z ′ ) " − z ′ − z p
2Ξ + ( z − z ′ ) . (49)Both e n ( z ′ ) and the functions in square brackets are pos-itive. Using the inequality (47), in the large- z limit theintegrals are bounded from above by I ( z, Ξ) ≤ A Ξ + √ z + O (cid:18) z (cid:19) ,I ( z, Ξ) ≤ A √ z + O (cid:18) z (cid:19) . (50)Considering these bounds in (48), it holds that ddz φ ( z ) = − Z ∞ z dz ′ e n ( z ′ ) + O (cid:18) z (cid:19) . (51)Since φ ( z ) = ln e n ( z ), the asymptotic formula (46) im-plies that φ ′ ( z ) ∼ z →∞ − /z . Inserting this asymptoticrelation together with (46) into Eq. (51), one gets a = 1.Consequently, at any finite value of the coupling Ξ, theasymptotic large-distance behavior of the density pro-file is exactly of PB type, as was expected. This prop-erty is confirmed also by a numerical treatment of ourcorrelation-hole equations in the next section. C. Subleading asymptotic correction
It is possible to go one step further and to compute thelarge- z correction to the mean-field asymptotics (large- z analysis at fixed Ξ). We use the electroneutrality con-dition (8) to rewrite the correlation-hole relation (31) asfollows ddz φ ( z ) = − Z ∞ z dz ′ e n ( z ′ ) + I ( z, Ξ) , (52)where I ( z, Ξ) = Z ∞ dz ′ e n ( z ′ ) " z − z ′ p
2Ξ + ( z − z ′ ) − z − z ′ | z − z ′ | . (53)To proceed, we change variables z → u = (1 + z ) − , andperform a small u expansion in Eqs. (52) and (53). Usingthe fact that I ( u, Ξ) = − Ξ u + o ( u ), writing n ( u ) = u + ∆ n ( u, Ξ) and keeping in mind that ∆ n is o ( u ) butnot necessarily O ( u ), we get − ∂ ∆ n∂u + 2 u ∂ ∆ n∂u + 2Ξ u + o ( u ) = 0 , (54)from which the correction to the PB asymptotics follows: e n ( z ) ∼ z ) −
23 Ξ log(1 + z )(1 + z ) . (55)As the exact loop-derived correction, it is of order Ξ anddecays at large z like z − log( z ) [16, 37, 38]. Yet, our − / −
1. We mention herethat repeating the analysis of [16] lead us to a correctedprefactor − /
2, closer to the present − / V. NUMERICAL RESULTSA. The methods
The correlation-hole integral equation for the rescaleddensity profile e n , given by Eqs. (30) and (35), bears somesimilarities with the nonlinear PB formulation. Solvingit numerically is straightforward. In practice, an effi-cient numerical scheme was found to be the following.Rescaled distances z are first mapped onto a variable x = (1 + z ) − / , such that x ∈ [0 , φ ( x ) is then discretized on a regular grid with N points ( N up to 2 × ). We initialize the density tobe of PB form, meaning that e n ( x ) = x (which results inan improved convergence), before an iterative resolution.Convergence is typically achieved in 100 iterations if fineproperties are sought. It is important here to emphasizethat from a computational point of view, the resolutionof our self-consistent equation is significantly faster andmore convenient than the test charge approach [20], orthe theory of Santangelo [22].In parallel, we have performed a number of MonteCarlo simulations in a quasi-2D geometry. Ewald sum-mation techniques corrected for quasi-2D-dimensionalityallow to account for long-range electrostatic interactions z -0.0100.010.020.030.040.05 n ~ - n ~ PB MC simulationhybrid field theorytest-charge theorycorrelation-hole theory
Ξ = 1
FIG. 2. Deviation from the PB density profile, e n − e n PB , asa function of the dimensionless distance z for the couplingconstant Ξ = 1. Symbols correspond to the results of MonteCarlo simulation, the dashed curve is for the test-charge the-ory of Ref. [20], the dotted curve is for the approach of Ref.[22], and solid curve shows the present correlation-hole ap-proach. z n ~ - n ~ PB MC simulationhybrid field theorytest-charge theorycorrelation-hole theory
Ξ = 10
FIG. 3. Same as Fig. 2 for the coupling Ξ = 10. (see e.g. [62–64]). The Monte Carlo results provide thecorrect reference behavior of our system of point ions inthe vicinity of a charged plate.
B. Comparison to Monte Carlo results
The numerical results for the deviations from PB pro-files, e n − e n PB are presented for the coupling constantsΞ = 1, Ξ = 10 and Ξ = 100 in Figs. 2, 3 and 4 respec-tively. Our MC simulations are compared to the test-charge theory [20], to the hybrid field theory of Ref. [22]where long and short distances are treated separately andto the present correlation-hole approach. We see that forthe small value Ξ = 1, the accuracy of the test-chargeand correlation-hole theories is comparable. The hybridfield theory of Ref. [22] (which is solved at the expense z n ~ - n ~ PB MC simulationhybrid field theorytest-charge theorycorrelation-hole theory
Ξ = 100
FIG. 4. Same as Fig. 2 for Ξ = 100. of enhanced technical complexity) fares better at shortdistances, but worse for z >
2. For intermediate Ξ = 10,the accuracy of our approach is better. For relativelylarge Ξ = 100, our solid curve practically passes throughMC data. The accuracy of our results improves uponincreasing Ξ.For the tail of the ionic profile, at larger distances thanthose in the previous graphs, we see in Fig. 5 that thecorrelation-hole picture captures qualitatively the depar-ture from SC behavior, although in a distance range thatis not close enough to the charged plate. Yet, the testcharge theory fails in getting the qualitative trend. ForΞ = 100, the MC result clearly follow the exponentialprofile at e z <
10 [18], then crosses over to a longer rangedecay, following a trend that is reminiscent of that ob-served within the correlation-hole approach (same shapein the log-log plot presented). Observing properly thePB algebraic tail in 1 /z , with MC at Ξ > − , and is beyond our scope.For this reason and in order to study nevertheless thecrossover to mean-field, we will in the remainder relin-quish MC method and focus on the self-consistent treat-ment, which is considerable simpler to solve. C. Discussion of asymptotic features
We wish to investigate the behavior of ionic density atlarge distances, to first test the relevance of the correctionworked out in Eq. (55), but also to discuss the crossoverto the mean-field regime. Fig. 6 extracts the correctionto the PB profile, and compares it to the predicted func-tional form in Ξ log(1 + z ) / (1 + z ) . This is achievedthrough the computation of the following quantity: Q ( z ) = n ( z ) − (1 + z ) − − / z ) / (1 + z ) . (56)It is observed that for Ξ < Q saturates at large dis-tance close to the expected value Ξ. For Ξ = 50 (and z -8 -7 -6 -5 -4 -3 -2 -1 n ~ ( z ) e - z Ξ = 10
MC simulationtest-charge theorycorrelation-hole theory
Ξ = 100
MC simulationtest-charge theorycorrelation-hole theory
FIG. 5. Large-distance counter-ion densities for Ξ = 10 (ingreen) and Ξ = 100 (in blue). Monte Carlo data (symbols)are compared to the correlation-hole results (solid curves) andthose of the test-charge theory of Ref. [20] (dashed lines). Thedotted line is for the SC limiting behavior Ξ → ∞ . higher), the range of distances probed does not allow toreach large enough z to observe the phenomenon. z Q ( z ) Ξ = 50Ξ = 10Ξ = 5Ξ = 1Ξ = 0.5Ξ = 0.1
FIG. 6. Plot of Q ( z ) as defined in Eq. (56), vs distance to thecharged wall. Eq. (55) predicts that Q asymptotically tendsto Ξ, indicated the by horizontal dotted lines. For Ξ >
50, the large- z density profile exhibits a newproperty, that is only beginning to emerge in Fig. 5.This is illustrated in Fig. 7: the expected exponentialSC regime at short z and mean-field tail at large z areconnected by a plateau, starting at the crossover length z cross , where the density is quasi-constant. To be morespecific, the existence of a plateau followed by a z − de-cay is precisely the PB prediction, with an effective Gouy-Chapman length µ eff , and a density e n ( z ) = 1( z + µ eff ) . (57)Thus, for z < µ eff (but z > z cross ), the density profile isflat, while for z > µ eff , it decays algebraically. Keeping in z -20 -15 -10 -5 n ~ ( z ) -20 -10 Ξ = 1000Ξ = 500Ξ = 300Ξ = 200Ξ = 10Ξ = 50Ξ = 100 z cross µ eff µ eff −2 FIG. 7. Scenario for the density large-distance asymptotics.The SC limiting behavior on the left hand side is displayedwith the dashed line. The inset shows the crossover distance z cross and the effective Gouy-Chapman length µ eff for Ξ =500. mind that by its definition in Eq. (3), a Gouy-Chapmanlength scales like the inverse plate charge, it is naturalto expect µ eff to largely exceed the bare Gouy-Chapmanlength. Indeed, the PB-like profile sets in for z > z cross ,and subsumes all nonlinear screening effects at work for0 < z < z cross into an effective plate surface charge, thussignificantly smaller than σ .It can be noted that the large- z expansion of Eq. (57)yields e n ∼ /z − µ eff /z . The resulting correction tothe 1 /z tail is of smaller order than the term in log z/z stemming from Eq. (55). Hence, the value of µ eff cannotbe simply extracted from the asymptotic tail of the pro-file, but at smaller distances, where Eq. (57) is relevant[65]. The plateau seen in Fig. 7 illustrates this point: for z > z cross , Eq. (57) states that e n − / increases linearlywith distance, so that the quantity displayed in Fig. 8 of-fers a convenient measure of the effective Gouy-Chapmanlength. It can be observed in Fig. 8 that for Ξ = 10, onecannot properly extract a µ eff , which is consistent withthe data in Fig. 5 (absence of a well defined plateau).The inset of Fig. 8, where the line shown has equation y = x + 0 .
62, then indicates that µ eff changes with Ξ aslog µ eff ∼ r Ξ2 + cst . (58)This in turn sets the crossover distance to be z cross ∼ √ , (59)by equating e − z with 1 /µ at z cross . It does not comeas a surprise to recover here the value of the correlation-hole size [20, 22], see Eq. (27) which reads e R = 2 Ξ. Theeffective length µ eff diverges with Ξ, such that log µ eff islinear in √ Ξ, a conclusion also reached in [22]. Largevalues of µ eff were observed numerically as well in thecase of counter-ions around charged cylinders [66]. z √ n ~ - z
10 30 √Ξ/2 l n ( µ e ff ) Ξ = 1000Ξ = 500Ξ = 300Ξ = 200Ξ = 100Ξ = 50Ξ = 10
FIG. 8. Extraction of the effective Gouy-Chapman length µ eff , from the plot of 1 / √ e n − z , for Ξ between 10 and 1000.The plateau reached defines µ eff . The inset shows how theresulting effective length depends on the coupling parameter.The line has slope 1. Finally, we present an operational way to decide whena system with an arbitrary Ξ is in the mean-field regime.The idea is to take advantage of the fact that the stresstensor is divergence-free [67]. For mean-field theories,this yields an extended contact theorem (not only at z =0, but at any z ). In the present planar geometry, thismeans that, using dimensionless quantities p ( z ) ≡ e n ( z ) −
14 [ φ ′ ( z )] = 0 . (60)To check for that identity with numerically obtained re-sults, one could compute the correct potential φ , fromintegrating the charge density. However, keeping in mindthat we seek here a mean-field probe, it is more conve-nient to assume φ = log e n and we arrive at p ( z ) ≡ e n ( z ) −
14 [ ∂ z log e n ] = 0 . (61)Deviations of p ( z ) from 0 provide a (sufficient) condi-tion for mean-field violation. The fact that p = 0 withina mean-field treatment is a consequence of the contacttheorem, that reads p (0) = 0. It indicates that the pres-sure vanishes in our setting (single plate problem, corre-sponding to a two-plate in interaction, in the limit whereinter-plate distance is infinite). Figure 9 corroborates theexistence of a PB tail, at large enough distances. Yet, aword of caution is in order here. It can rightfully be ar-gued that a quantity such as p ( z ) may only distinguishexponential profiles from algebraic ones, but that anydensity of the type e n ∝ ( µ eff + z ) α yields p → α >
0, and not only α = 2. A possible solution would beto consider the ratio of the two terms subtracted in (61),rather than their difference; the ratio goes to a constantfor the PB behavior only ( α = 2). However, this has adrawback: it amplifies the contribution of any residual0exponential tail in the density, and requires larger dis-tances to qualify the density as PB-like. A point to keepin mind though is that our probe (61) is more interestingfor a two plate system where the real (e.g. Monte Carlo)pressure P is non-vanishing, rather than for the one platesituation. Indeed, in such a case, comparing p ( z ) to P can be viewed as signaling the mean-field regime. z / √2Ξ -0.200.20.40.6 p Ξ = 10
MC simulationcorrelation-hole
Ξ = 100
MC simulationcorrelation-hole
Ξ = 1000 correlation-hole
FIG. 9. Implementing our mean-field probe. The vanishingof the local pressure p ( z ), as defined in Eq. (61), signals thePB regime. On the x -axis, distances have been rescaled by z cross = √ < p = 0 curve,since mean-field holds at all distances. For large enough Ξ, p starts at 3/4 for small z (since e n = exp( − z ) locally holds),then reaches a minimum value close to − /
4, before vanishingon a scale z cross . Symbols are for MC, and the curves for thecorrelation-hole theory. VI. CONCLUDING REMARKS
We have studied a system of identical counter-ions neara wall carrying a uniform surface charge density, in ther-mal equilibrium. This is probably the simplest model ofthe electrical double layer, depending only on one param-eter, the coupling constant Ξ. It provides an interestingtest-bench, since both Weak Coupling (WC) and StrongCoupling (SC) limits are known.We have proposed a method which combines physicalideas from both WC and SC regimes. From the WC side,the particle density is determined by the Boltzmann fac-tor of the mean potential. From the SC side, there is acylindrical correlation hole around each particle, inacces-sible to other particles, which modifies the value of themean potential. The theory is simple by its constructionand leads to a nonlinear integral equation, similar to thePB one, which converges quickly in an iterative scheme.Remarkably, all exact constraints are respected by ourcorrelation-hole theory, for all coupling constants. Thecontact theorem for the particle density at the wall holds.The WC and SC limits are reproduced as well, and thecorrection to the SC limit Ξ → ∞ is proportional to 1 / √ Ξ, in accordance with recent approaches and MCsimulations. For large distances from the wall and atarbitrary Ξ, the algebraic mean-field density profile is re-covered. Moreover, we showed that the correspondingsubleading correction, in Ξ log z/z , is of the same formas found in a loop-wise field theoretic treatment of fluc-tuations beyond Poisson-Boltzmann [16]. Focusing onthe approach to mean-field behavior at large distances,we showed that beyond a crossover distance z cross (co-inciding with the hole size), the density takes a Poisson-Boltzmann form. This allows to define an effective Gouy-Chapman length to describe the density tail. In units ofthe bare length µ , it behaves as µ eff ∝ exp( p Ξ / p ( z ) inEq. (61), we recover the results of a direct analysis of thenumerical profiles. d -0.500.51 P ~ ( d ) d σ e − qe σ e FIG. 10. Interplate pressure versus rescaled distance, for Ξ =1, 10, 50 and 100 (from top to bottom). Monte Carlo results(symbols) are compared to the prediction of the correlation-hole theory (lines). The rescaled pressure is defined as e P = P/ ( k B T πℓ B σ ), and is measured from the contact theorem. For the sake of completeness, we also considered thesituation of two parallel uniformly charged plates (sur-face charge density σe ), at distance d , sandwiching a slabof counter-ions. There, an ambiguity arises when enforc-ing the idea of a correlation hole. Indeed, we have todistinguish between the two limits d → ∞ and d → d → ∞ , i.e. σπR ∞ = q , and by 2 σπR = q if d →
0. A possible, d -dependent interpolation formula for the correlation-holesize might be relevant, but for simplicity, we took thesame prescription as in the one-plate case, Eq. (27). Thealternative choice turned out to be slightly worse. Theequation of state of this system, as measured in MonteCarlo simulations, is reported in Fig. 10. To test ourcorrelation-hole approach (accurate at both small and1large couplings), we concentrate in Figure 10 on inter-mediate coupling strengths, where the phenomenon oflike-charge attraction sets in [3, 17, 19, 56]. We seethat the qualitative features of the pressure are well cap-tured, with an agreement that is quantitative for smalldistances, up to the range where like-charge attraction ismaximal (minimum of the pressure). The asymptotic de-cay to vanishing pressure then takes place over too largedistances, as compared to MC. The correlation-hole ideathere overestimates the SC non-mean-field features; cor-recting for this deficiency is left for future work. Yet, itis noteworthy that the present theory captures here alsoa number of exact features. Not only is the proper equa-tion of state recovered when Ξ → → ∞ , butalso, the pressure minimum arises at z ∝ Ξ / , as foundin Monte Carlo simulations [21, 50]. ACKNOWLEDGMENTS
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