Role of Multipoles in Counterion-Mediated Interactions between Charged Surfaces: Strong and Weak Coupling
aa r X i v : . [ c ond - m a t . s o f t ] M a y Role of Multipoles in Counterion-Mediated Interactions between Charged Surfaces:Strong and Weak Coupling
M. Kanduˇc, A. Naji,
2, 3, 4
Y.S. Jho,
5, 6
P.A. Pincus,
5, 6 and R. Podgornik Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia Department of Physics, Department of Chemistry and Biochemistry,& Materials Research Laboratory, University of California, Santa Barbara, CA 93106, USA Kavli Institute of Theoretical Physics, University of California, Santa Barbara, CA 93106, USA School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran Materials Research Laboratory, University of California, Santa Barbara, CA 93106, USA Dept. of Physics, Korea Advanced Institute of Science and Technology, Yuseong-Gu, Daejeon, Korea 305-701 Institute of Biophysics, Medical Faculty and Department of Physics,Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
We present general arguments for the importance, or lack thereof, of the structure in the chargedistribution of counterions for counterion-mediated interactions between bounding symmetricallycharged surfaces. We show that on the mean field or weak coupling level, the charge quadrupolecontributes the lowest order modification to the contact value theorem and thus to the intersur-face electrostatic interactions. The image effects are non-existent on the mean-field level even withmultipoles. On the strong coupling level the quadrupoles and higher order multipoles contributeadditional terms to the interaction free energy only in the presence of dielectric inhomogeneities.Without them, the monopole is the only multipole that contributes to the strong coupling electro-statics. We explore the consequences of these statements in all their generality.
I. INTRODUCTION
The assembly of colloidal building blocks with designerengineered size, shape, and chemical anisotropy seems tobe the next step in the fundamental and applied colloidscience and science of soft materials, vigorously pursuedby many researchers [1, 2]. Moving beyond the tradi-tional systems implies the creation of designer colloidalbuilding blocks such as colloidal molecules, janus spheres,and other patchy particle motifs that have very differ-ent propensities for self-assembly [3]. In order to real-ize this goal one needs detailed control over various as-pects of colloid geometry at the nanoscale ( e.g. , aspect ra-tio, faceting, branching, roughness) and microscale ( e.g. ,chemical ordering, shape gradients, unary and binary col-loidal “molecules”). The ability to spatially modify thesurface structure of colloids with designed chemical het-erogeneity, e.g. , to form a patchy surface structure, seemsto be becoming a realistic goal. Fabrication of stableanisotropic microcapsules was recently accomplished bythe layer-by-layer polyelectrolyte adsorption techniquecombined with particle lithography technique to pro-duce anisotropic polymer microcapsules with a singlenanoscale patch [4]. Granick and co-workers [2] recentlyreported a highly scalable synthetic pathway for creatingbipolar janus spheres, i.e. , particles that consist of op-positely charged hemispheres. Such colloidal “animals”which are probably the simplest example of patchy col-loids, exhibit orientation-dependent interactions that gotogether with localized patches of like/unlike charges orhydrophobic/hydrophilic regions. This heterogeneous in-teraction landscape promotes the formation of larger col-loidal molecules and clusters that are themselves patchy.These advances in the nano- and microscopic tailor-ing of (charged) colloids motivated various approaches to generalizations of the existing theories of electrostaticinteractions in charged colloids by explicitly includingthe structure of the counterions as embodied by theirmultipolar moments. The inclusion of structured coun-terions of a dipolar [5, 6] and quadrupolar [7, 8] typeinto the theory of electrostatic colloidal interactions hasbrought fourth some of the salient features of the coun-terion structure effects, which on face value appear tobe quite distinct from the standard Poisson-Boltzmannframework. In view of these advances in the study ofinteractions between charged colloids it thus seems ap-propriate to explore the ramifications of the emergingparadigm if applied to these more complicated structuredcolloidal molecules. Especially the non-spherically sym-metric charge distribution of microscopically tailored col-loidal particles might lead to some unexpected propertiesof electrostatic interactions in this type of systems. Withthis in mind we thus embark on a thorough examina-tion of the consequences of multipolar charge distributionof mobile counterions that mediate interactions betweencharged (planar) macroions.
II. MODEL
In what follows we will consider a system of fixedmacroions of surface charge distribution ρ ( r ) in an aque-ous solution, described as a dielectric continuum witha dielectric constant ε and temperature T , containing N neutralizing counterions. Counterions are assumed tobe pointlike particles but they do posses a rigid internalstructure described by a charge distribution ˆ ρ ( r ; R i , ω i )that we assume can be written as a standard multipolar a-a ' ' z l FIG. 1: Geometry of the system. Two charged walls with dif-ferent dielectric constants and structured counterions in be-tween. Counterions are treated as pointlike particles but withinternal uniaxial structure that can be viewed, for example,as uniformly charged rods. expansion [9]ˆ ρ ( r ; R i , ω i ) = e qδ ( r − R i ) − p ( n i · ∇ ) δ ( r − R i ) ++ t ( n i · ∇ ) δ ( r − R i ) + · · · , (1)for the i -th counterion located at R i with ω i being theorientational variables specifying the angular dependenceof the counterion charge density. The monopolar mo-ment of each counterion is e q , where q is the chargevalency and e the elementary charge, p = p n is thedipolar moment and Q = t n ⊗ n the quadrupolarmoment with director unit vector n . We must empha-size here that the quadrupolar expansion in (1) is notgeneral but adequately describes only a uniaxial counte-rion, e.g. , a charged particle of rod-like structure, Fig. 1.One possible implementation of counterions possesingjust monopolar and quadrupolar moment is a uniformlycharged rod with charge e q and length l . Counterion’squadrupolar moment is then t = e ql /
24. Anotherpossibility is a negative charge ( − e ) in the center andtwo positive charges (+ e ) located at both ends of therod leading to monopolar and quadrupolar moments of e q = 2 e − e and t = e l , respectively.Counterions interact via a Coulomb interaction poten-tial u ( r , r ′ ) so that the interaction energy of two givencounterions i and j at set positions and orientational con-figurations is obtained by integrating the Coulomb inter-action over their internal orientational degrees of freedomas ZZ d r ′ d r ˆ ρ ( r ; R i , ω i ) u ( r , r ′ )ˆ ρ ( r ′ ; R j , ω j ) . (2)Although our formalism in this section is in general ap-plicable to macroions of arbitrary shape and charge dis-tribution, we shall primarily focus on the case of twocharged planar surfaces located at z = ± a with thecharge distribution ρ ( r ) = σδ ( a − z ) + σδ ( a + z ) . (3)We also consider a dielectric inhomogeneity between thebounding surfaces and the ionic solution, such that the ionic solution is described with dielectric constant ε andthe bounding surfaces with ε ′ . In this geometry theCoulomb interaction potential is composed of the directinteraction u ( r , r ′ ) = (4 πεε | r − r ′ | ) − and the (electro-static) image interaction u im ( r , r ′ ), so that u ( r , r ′ ) = u ( r , r ′ ) + u im ( r , r ′ ) . (4)The Green’s function in planar geometry can be ex-pressed as a sum of image charge contributions, viz. u ( r , r ′ ) = 14 πεε " X n even ∆ | n | | r ′ − r − na ˆ k | ++ X n odd ∆ | n | | r ′ − r + (2 na − z ′ ) ˆ k | . (5)In both sums the index n runs through negative and pos-itive integer values. The term corresponding to n = 0represents the direct Green’s function of the particle in aspace of a uniform dielectric constant, namely u ( r , r ′ ).The unit vector ˆ k = (0 , ,
1) points in the + z direction.The relative dielectric jump at the two bounding surfacesis quantified as ∆ = ( ε − ε ′ ) / ( ε + ε ′ ). For most relevantsituations the outer permittivity is smaller than the innerone, ε ′ < ε , so that the dielectric jump is positive, ∆ > ε ′ → ε .The canonical partition function Z N of this Coulombfluid composed of N charged particles is given by Z N = 1 N ! Z d R · · · d R N d ω · · · d ω N × (6) × exp (cid:16) − βU [ R i , ω i ; R j , ω j ] (cid:17) , (7)with the configurationally dependent ionic interaction en-ergy given by U [ R i , ω i ; R j , ω j ] =12 X i = j ZZ d r d r ′ ˆ ρ ( r ; R i , ω i ) u ( r , r ′ ) ˆ ρ ( r ′ ; R j , ω j ) ++ X i ZZ d r d r ′ ˆ ρ ( r ; R i , ω i ) u ( r , r ′ ) ρ ( r ′ ) ++ 12 ZZ d r d r ′ ρ ( r ) u ( r , r ′ ) ρ ( r ′ ) , (8)where β = 1 /k B T . The three terms in U [ R i , ω i ; R j , ω j ]correspond to direct electrostatic interaction betweencounterions, electrostatic interactions between counteri-ons and fixed charges and between fixed charges on thewalls themselves, respectively. We will furthermore makethe standard assumption that the system is overall elec-troneutral, implying that the mobile countercharge ex-actly compensates the charge on the surfaces.We now proceed in the Netz [10] fashion and performthe Hubbard-Stratonovitch transformation of the parti-tion function [11], where the configurational integral overcounterion positions is transformed into a functional in-tegral over a fluctuating auxiliary electrostatic potential φ ( r ). In this way the grand-canonical partition functionis obtained straightforwardly in the form [10] Z = ∞ X N =0 λ N Z N = C Z D [ φ ( r )] e − βH [ φ ( r )] , (9)where λ is the bare fugacity which is the exponentialof the chemical potential. The prefactor C above is thefunctional determinant of the inverse Coulomb kernel u − ( r , r ′ ) while the “action” of the functional integralis given by H [ φ ] = Z d r (cid:20) ε ( r ) ε ( ∇ φ ) − i ρ ( r ) φ ( r ) −− λ ′ β Z d ω Ω( r ) exp (cid:18) i β Z d r ′ ˆ ρ ( r ′ ; r , ω ) φ ( r ′ ) (cid:19)(cid:21) . (10)Here we have introduced the renormalized fugacity as λ ′ = λ exp( u ( r , r )), where u ( r , r ) is the electrostaticdirect self-energy of a single counterions with multipolarcharge distribution. Ω( r ) is the geometric characteris-tic function of the counterions, being equal to unity inthe slab between the bounding surfaces and zero other-wise. The partition function in the above form can notbe evaluated explicitly except in the one-dimensional case[12] where the partition function evaluation is reduced toa solution of a Schr¨odinger-like equation. Neverthelessthe field-theoretical representation of the grand canon-ical partition function allows one to use quite powerfulanalytical approaches that eventually lead to an explicitevaluation of the partition function in two well definedand complementary limits [10, 13, 14] that we shall ad-dress later in this paper. These limits retain the relevancealso in the case of structured counterions. III. DIMENSIONLESS REPRESENTATION
One may obtain a dimensionless representation for thepresent system by rescaling all length scales with a givencharacteristic length scale. Recall that the character-istic distance at which two unit charges interact withthermal energy k B T is known as the Bjerrum length ℓ B = e / (4 πεε k B T ) (in water at room temperature, thevalue is ℓ B ≈ . q then the aforementioned distance scales as q ℓ B .Similarly, the distance at which a counterion interactswith a macromolecular surface of surface charge density σ with an energy k B T is called the Gouy-Chapman length,defined as µ = e / (2 πqℓ B σ ). A competition between ion-ion and ion-surface interactions can thus be quantified bythe ratio (Ξ) of these characteristic lengths, that isΞ = q ℓ B /µ = 2 πq ℓ σ/e , which is referred to as the electrostatic coupling param-eter. In what follows, we may rescale the length scales withthe Gouy-Chapman length, i.e. , r → r /µ ; hence, thesurface separation will be rescaled as ˜ D = D/µ or therescaled half-distance as ˜ a = a/µ . Other dimensionlessquantities can be defined as follows. The dimensionlessmultipolar moments are defined as p = p /e qµ and t = t /e qµ , (11)and the dimensionless pressure as˜ P = βP πℓ B ( σ/e ) . (12)For simple structureless counterions the functional in-tegral (9) can be rescaled [10] yielding Z → C Z D [ φ ( r )] e − H ′ [ φ ( r )] / Ξ , with dimensionless H ′ [ φ ( r )]. As will be shown later anidentical representation can be obtained also for struc-tured counterions, described on the level of a multipolarexpansion of their charge density. IV. STRONG AND WEAK COUPLINGDICHOTOMY
In the absence of a general approach that wouldcover thoroughly all the regions of the parameter spaceone has to take recourse to various partial formula-tions that take into account only this or that facet ofthe problem [15]. The traditional approach to theseone-component Coulomb fluids has been the mean-fieldPoisson-Boltzmann (PB) formalism applicable at weaksurface charges, low counterion valency and high tem-perature [13, 16, 17]. The limitations of this approachbecome practically important in highly-charged systemswhere counterion-mediated interactions between chargedbodies start to deviate substantially from the mean-fieldaccepted wisdom [15]. One of the most important re-cent advances in this field has been the systematizationof these non-PB effects based on the notions of weak and strong coupling approximations. They are based on thefield-theoretical representation of grand canonical parti-tion function (10), whose behavior depends on a single di-mensionless coupling parameter Ξ [10] (see below). Theweak-coupling (WC) limit of Ξ → → ∞ is diamet-rically opposite and corresponds to a single particle de-scription. It has been pioneered by Rouzina and Bloom-field [21], elaborated later by Shklovskii et al. [22, 23, 24]and Levin et al. [25], and eventually brought into a finalform by Netz et al. [10, 13, 15, 26, 27, 28].These two limits are distinguished by the pertainingvalues of the coupling parameter Ξ which can be intro-duced in the following way.The regime of Ξ ≪ µ is much largerthan the separation between two neighbouring counte-rions in solution and thus the counterion layer behavesbasically as a three-dimensional gas. Each counterionin this case interacts with many others and the collec-tive mean-field approach of the Poisson-Boltzmann typeis completely justified.On the other hand in the strong coupling regime Ξ ≫
1, which is true for high valency of counterions and/orhighly charged surfaces. In this case the mean distancebetween counterions, a ⊥ ∼ p e q/σ , is much larger thanthe layer width ( i.e. , a ⊥ /µ ∼ √ Ξ ≫ via computer simulations. As will become clearin what follows the WC and SC limits remain valid alsofor structured counterions described with a multipolarcharge distribution. V. MEAN-FIELD LIMIT
The mean-field limit, Ξ →
0, is defined via the saddle-point configuration of the Hamiltonian (10) [20] and isvalid for a weakly charged system. This leads to the fol-lowing generalization of the Poisson-Boltzmann equation εε ∇ ψ ( r ) = − λ ′ ZZ d r ′ d ω ˆ ρ ( r ; r ′ , ω ) Ω( r ′ ) ×× exp (cid:20) − β Z d r ′′ ˆ ρ ( r ′′ ; r ′ , ω ) ψ ( r ′′ ) (cid:21) (13)for the real-valued potential field ψ = − i φ . Note thatthe dielectric discontinuity at the boundaries is irrele-vant within the mean-field theory [29] since in the planargeometry this is effectively one-dimensional theory. Tak-ing now the counterion density function as a sum of themonopolar, dipolar and quadrupolar terms, where themean-field depends only on the transverse coordinate z ,the above equation can be written in dimensionless form as ψ ′′ = − Z +1 − d x Ω( x, z ) (cid:0) u ( z ) − pxu ′ ( z ) + tx u ′′ ( z ) (cid:1) , ∝ ρ ( z ) + ρ ( z ) + ρ ( z ) , (14)where we have defined dimensionless potential ψ = βe qψ with corresponding derivatives ψ ′ = βe qµψ ′ and ψ ′′ = βe qµ ψ ′′ and dimensionless multipolar moments p and t . We defined the orientational variable x = cos θ ,where θ is the angle between the z -axis, assume to coin-cide with the normal to the bounding surfaces, the direc-tor of the uni-axial counterion is n and the integral overthis variable gives the orientational average. We havealso introduced u ( z ) = C exp (cid:0) − ψ − pxψ ′ − tx ψ ′′ (cid:1) , which is obviously the local orientationally dependentnumber density of the counterions and ρ ( z ) is the corre-sponding orientationally averaged charge density of thecounterions. In the case of higher multipoles the num-ber density and the charge density of the counterionsare not proportional. Expressions ρ i ( z ) are simply theorientationally averaged multipolar charge densities, i.e. , i = 1 for monopolar charge, i = 2 for dipolar charge etc.,that are simply proportional to the three terms in theintegrand of (14). The corresponding prefactor is simplyobtained by appropriate normalisation of ρ to satisfyelectro-neutrality condition of the system.The above PB equation has to be supplemented by anappropriate boundary condition at z = ± a correspondingto the electroneutrality of the system by taking into ac-count surface charges, σ . The constant C is set by theseboundary conditions, as is the case in the standard PBtheory. In dimensionless units these boundary conditionsread ψ ′ ( ± a ) = ∓
2. Obviously in the case of counterionswith only monopolar charge distribution the above setof equations reduces to the standard Poisson-Boltzmanntheory.The characteristic function Ω( x, z ) for the parallelplane geometry simply excludes the counterion config-urations that would penetrate the bounding walls anddepends on the geometric form of the counterions. Sincein what follows we will not be interested in steric effects,we will assume that all counterions are pointlike, and soΩ( x, z ) = 1. Steric effects have been studied elsewhere[6, 7, 8].Because of the similarity with the standard PB equa-tion for monopolar charges one is led to believe that thepressure in an inhomogeneous system of multipolar coun-tercharge can be derived in the same way as in the stan-dard PB theory, via the s.c. contact value theorem [30].Indeed this can be proven exactly, namely the mean-fieldpressure can be obtained from the first integral of the PBequation (14). After some manipulations its first integralis obtained in the form˜ P = − ψ ′ + Z − d x (cid:2) u + pxuψ ′ + tx ( uψ ′′ − u ′ ψ ′ ) (cid:3) . (15)Here ˜ P is the dimensionless equilibrium pressure in thesystem, defined as (12). The r.h.s. of the above identitycan be calculated at any arbitrary point | z | < a as itsvalue is independent of z . The choice of the actual pointis governed by the symmetry of the system, as is usualalso in the standard PB theory. Since we delve only onthe symmetrical solution, with inversion symmetry cen-tered on z = 0, the derivative of the mean potential atthe mid-point must vanish, i.e. , ψ ′ ( z = 0) = 0. Thus inthis case ˜ P = Z − d x u (0) (cid:0) − tx | ψ ′′ (0) | (cid:1) . (16)While the first - van’t Hoff - term corresponds to repulsiveinteractions of the standard PB type, the second term inthis particular geometry and symmetry entails attractiveinteractions between the bounding surfaces. Note that ψ ′′ < i.e. , an extended and flexible coun-terion, one can derive a similar type of pressure formula[31], with a negative term that contributes an attractivepart to the force equilibrium. This attractive part in thecase of polyelectrolytes is due to polyelectrolyte bridg-ing interactions [32] that stem from the connectivity andflexibility of the polyelectrolyte chain.Does equation (16) have a similar physical content?This interpretation certainly does not seem likely forpoint counterions, for which the above pressure formulawas derived. The structure of the first integral of the PBequation (16) seems to be saying that apart from the idealcontribution to the equilibrium pressure, a term propor-tional to the midpoint number density, one also findsan electrostatic contribution that is due to the interac-tion between the monopolar and the quadrupolar part ofthe counterion charge density across the midplane and isproportional to the square of the midpoint density, see(14). The attractive part thus does not look like bridgingwhich has its origin in the connectivity and flexibility ofthe polyelectrolyte chain, but more like a virial expansionin terms of the multipolar interactions. This is of courseonly true for point-like counterions. For extended coun-terions a bridging interpretation would be more appropri-ate as was already clear in the early studies of structuredcounterions [34]. A. First order in t The above PB equation is a fourth order highly non-linear integro-differential equation and therefore difficultto handle with conventional numerical procedures. Sincehere we are not particularly interested in the weak cou-pling results, we want only to show that the effect ofquadrupolar moments, t , on the interaction pressure isto induce a small attractive contribution. In order toshow this we expand (14) to the first order in t and put p = 0, which then reads ψ ′′ = − C e − ψ (1 + tψ ′ − tψ ′′ ) . (17)This equation can be solved much more easily with con-ventional numerical methods. Although we must beaware that it is valid only for small t . The correspond-ing formula for interaction pressure in the 1st order in t expansion reads ˜ P = C e − ψ (0) . (18)Due to a convenient cancellation the formula contains noexplicit t -dependence, but the pressure still depends on t implicitly, due to the t dependent potential ψ . As canbe shown numerically for small t the correction to thestandard PB pressure depends linearly on t . B. Numerical results in the mean-field limit
Here we are not interested in the details of the mean-field results–they were analyzed in detail before [6, 7, 8]–but list them for completeness anyhow. In what followswe delimit ourselves to counterions that posses monopo-lar and quadrupolar charge, so we do not take into ac-count any dipoles ( p = 0).The density profile on Fig. 2(a) corresponds tomonopolar density ρ ( z ), (14). Counterions are localizedpreferentially in the vicinity of both surfaces due to mu-tual repulsion that prevents them from being localized atthe center. What we then observe is that with increaseof the quadrupolar moment, t , counterions concentrateat surfaces even more. This can be easily explained. Thepotential energy of every quadrupolar particle in electro-static potential is βW = tψ ′′ cos θ. (19)Since the second derivative of the mean-field potential ψ ′′ in the symmetric case considered here is a concave func-tion of the coordinate z , this means that the quadrupolarforce, F = − ∂W/∂z , acts away from the center towardboth surfaces.The insight into the orientation of quadrupolar mo-ments can be obtained through the average of the squareof variable x , namely (cid:10) x (cid:11) , (cid:10) x (cid:11) = R x u ( x )d x R u ( x )d x . (20)It is instructive to define an orientational order parameter S = (cid:0) (cid:10) x (cid:11) − (cid:1) . (21)This order parameter can have values in the rangefrom − / (cid:10) x (cid:11) = 1 /
3, and the orientational order parameter van-ishes, S = 0. In the completely ordered case, S = 1 and -1 -0.5 0 0.5 1 z/a ρ t = 0.0t = 0.1t = 0.2t = 0.3 (a) -1 -0.5 0 0.5 1 z/a S t = 0.1t = 0.2t = 0.3 (b) a~ P~ t = 0.0t = 0.1t = 0.2t = 0.3 (c)FIG. 2: Mean-field results for various t in first-order approximation, (17). (a) Counterion density profile, (14). By increasing t in the mean-field limit counterions are depleted from the center toward both surfaces. All the densities are rescaled tocorrespond to the same area under the curve. (b) Order parameter profile, evaluated from (20) and (21). (c) Dimensionlesspressure-distance curves obtained from the first-order approximation of (17) and (18). Quadrupolar contributions are attractive. (cid:10) x (cid:11) = 1, so that all quadrupolar moments are paral-lel with the z -axis and perpendicular to the walls, θ = 0.The other extremum, S = − /
2, corresponds to (cid:10) x (cid:11) = 0so that all quadrupolar moments are perpendicular to z -axis and parallel with the walls, θ = π/ t and is larger at both surfacesthan at the center. The order parameter S > z -axis. According to (19) the electrostatic en-ergy can be minimized by increasing cos θ , since ψ ′′ < (cid:10) cos θ (cid:11) really is preferably increased above the valueof 1 / t > intra-counterion corre-lations (entering here via the rigid structure assumed forcounterions) that lead to such attractive contributions.At this juncture it does not make much sense to us incontinuing the multipolar expansion to yet higher orders,that would become relevant for even more aspherical andelongated charge distributions that would obviously en-tail also some molecular flexibility. In that case one coulduse the well developed theory of polyelectrolyte mediatedinteractions with much better confidence [35]. VI. STRONG COUPLING LIMIT
In the strong coupling limit, Ξ ≫
1, the system ishighly charged and counterions are highly correlated.The partition function (9) can be approximated by thetwo lowest order terms in the virial expansion with re-spect to the (renormalized) fugacity [10] Z = Z (0)SC + λ ′ Z (1)SC + O ( λ ′ ) . (22) The second term corresponds to a one-particle partitionfunction as the SC limit is effectively a single particletheory. Z (0)SC and Z (1)SC are then given by Z (0)SC = exp (cid:20) − β Z d r d r ′ ρ ( r ) u ( r , r ′ ) ρ ( r ′ ) (cid:21) , (23)which is the exponential of the interaction between baresurface charges, and Z (1)SC Z (0)SC = ZZ d R d ω exp (cid:20) − β ZZ d r d r ′ ˆ ρ ( r ; R , ω ) u ( r , r ′ ) ρ ( r ′ ) −− β ZZ d r d r ′ ˆ ρ ( r ; R , ω ) u ( r , r ′ )ˆ ρ ( r ′ ; R , ω ) (cid:21) , (24)which is the partition function of all possible (single)counterion configurations. The interaction potential inthis part of the partition function is composed of the di-rect and image electrostatic interactions (4).In the case of two charged surfaces with uniformlysmeared surface charge density (3), the surface chargeelectrostatic potential does not depend on the z -coordinate; it is spatially homogeneous and is given by Z d r ′ u ( r , r ′ ) ρ ( r ′ ) = − σaεε . (25)The corresponding potential energy of a counterion inthis surface electrostatic potential is given by ZZ d r d r ′ ˆ ρ ( r ; R , ω ) u ( r , r ′ ) ρ ( r ′ ) = − σaεε e q. (26)Since all the terms in density operator (1), except thefirst one, depend on the gradients, i.e. , spatial deriva-tives, the counterion energy in a homogeneous externalelectrostatic potential depends only on the first, monopo-lar term (26). That means that higher multipoles do notinteract directly with planar surface charge, but we mustemphasize that this is not the case in inhomogeneous po-tential at curved surfaces, e.g. , at cylindrical or sphericalsurfaces or inhomogeneously charged surfaces.As for the self-energy contribution, the second term inthe exponent of (24), it only picks up terms from the z -dependent parts of the image self-interaction, u im . Thedirect self-interaction, u , does not depend on coordi-nates so it can be discarded. The self-image energy isthen ZZ d r d r ′ ˆ ρ ( r ; R , ω ) u im ( r , r ′ )ˆ ρ ( r ′ ; R , ω ) = X i =1 w i ( z, cos θ ) . (27)Self-image contributions are among monopolar, dipolarand quadrupolar moments of the counterions interactingwith their own electrostatic images of monopolar, dipolarand quadrupolar moments. Summing up all these con-tributions we remain with nine terms of which only sixare different. Writing them up in extenso we obtain w ( z, cos θ ) = q e u im ( R , R ) ,w ( z, cos θ ) = qe p ( n · ∇ ) u im ( r , R ) (cid:12)(cid:12) r = R ,w ( z, cos θ ) = qe t ( n · ∇ ) u im ( r , R ) (cid:12)(cid:12) r = R ,w ( z, cos θ ) = w ( z, cos θ ) ,w ( z, cos θ ) = p ( n · ∇ )( n · ∇ ′ ) u im ( r , r ′ ) (cid:12)(cid:12) r ′ = r = R ,w ( z, cos θ ) = p t ( n · ∇ )( n · ∇ ′ ) u im ( r , r ′ ) (cid:12)(cid:12) r ′ = r = R ,w ( z, cos θ ) = w ( z, cos θ ) ,w ( z, cos θ ) = w ( z, cos θ ) ,w ( z, cos θ ) = t ( n · ∇ ) ( n · ∇ ′ ) u im ( r , r ′ ) (cid:12)(cid:12) r ′ = r = R . (28)Here, the first expression, w , corresponds to the inter- action between a monopole and its own monopole im-age, therefore it is proportional to q . The second term, w , corresponds to the monopole-dipole image interac-tion which is obviously the same as the dipole-monopoleimage interaction, w , and so on for all the higher orderterms.These image self-interaction terms can be expressed ina dimensionless form as˜ w i (˜ z, cos θ ) = β w i ( z, cos θ ) . (29)The above expressions were derived for the planar geom-etry. Similar expressions but with different image poten-tials can be derived also in other geometries with dielec-tric inhomogeneities. Note that if there is no dielectricdiscontinuity, so that u im ( r , r ′ ) = 0, they are all identi-cally zero, w i =0!Thus one can conclude at this point that in the dielec-trically homogeneous case the higher order multipoles arecompletely irrelevant on the SC level. Only the monopo-lar term survives in the partition function. This is prob-ably one of the most important conclusions of this work,so let us reiterate it: in a dielectrically homogeneous casein plan-parallel geometry the structure of the counterionsas codified by their multipolar moments plays absolutelyno role in the strong coupling limit!For our case of two charged planar surfaces we eval-uate the above expressions by using the Green’s func-tion (5), which was derived for planar geometry witha step-function dielectric profile at the two boundaries z = ± a . We get the following somewhat cumbersomeexpressions for the dimensionless self-image interactionsdefined above˜ w (˜ z, x ) = 12 Ξ X n odd n ˜ a ∆ n ( n ˜ a ) − ˜ z − Ξ4˜ a ln(1 − ∆ ) , (30)˜ w (˜ z, x ) = 12 Ξ p x ˜ z X n odd n ˜ a ∆ n [( n ˜ a ) − ˜ z ] , ˜ w (˜ z, x ) = −
18 Ξ t (1 − x ) ( X n even ∆ n ( n ˜ a ) + X n odd n ˜ a ( n ˜ a ) + 3˜ z [( n ˜ a ) − ˜ z ] ∆ n ) , ˜ w (˜ z, x ) = 18 Ξ p ( (1 − x ) X n even ∆ n ( n ˜ a ) + (1 + x ) X n odd n ˜ a ( n ˜ a ) + 3˜ z [( n ˜ a ) − ˜ z ] ∆ n ) , ˜ w (˜ z, x ) = 34 Ξ pt (1 + x ) x ˜ z X n odd n ˜ a ( n ˜ a ) + ˜ z [( n ˜ a ) − ˜ z ] ∆ n , ˜ w (˜ z, x ) = 332 Ξ t ( (3 − x + 35 x ) X n even ∆ n ( n ˜ a ) + (3 + 2 x + 3 x ) X n odd n ˜ a ( n ˜ a ) + 10( n ˜ a ˜ z ) + 5˜ z [( n ˜ a ) − ˜ z ] ∆ n ) . All indices n here run only through positive values. Note again that self-image contributions, ˜ w i , vanish when∆ = 0. Taking all this into account, the strong cou-pling interaction free energy, βF = − k B T ln Z (1)SC , i.e. ,the part of the free energy that depends on the inter-surface separation, can be written in dimensionless formas ˜ F / ˜ A = 2˜ a − Z ˜ a − ˜ a d˜ z Z − d x exp (cid:16) − X i =1 ˜ w i (˜ z, x ) (cid:17) . (31)Just as in the WC case we have again assumed thatthe counterions are point-like particles so that the char-acteristic function Ω does not depend on their coordinate˜ z and therefore the integration goes from − ˜ a to ˜ a . Thismeans that we also disregard the possible entropic ef-fects due to the finite size and anisotropy in the shape ofcounterions. These entropic contributions to the parti-tion function are relevant only for intersurface separationon the order of the size of the counterion or smaller. Inthat regime of separations other, much stronger effectswould come into play and compete with ionic finite sizeeffects, thus these type of effects are not the focus of thispaper.The corresponding dimensionless pressure is simply ob-tained by taking the derivative of the free energy (31)with respect to wall separation, ˜ D = 2˜ a ,˜ P = − ∂ ˜ F / ˜ A ∂ ˜ a . (32)It is again obvious from here that in the dielectricallyhomogeneous case, i.e. , the case with no electrostatic im-ages, where w i (˜ z, x ) = 0, the strong coupling limit isgiven exactly by the monopolar term (the first term in(24)). Thus without the images we remain with the sameform of the strong coupling interaction free energy as inthe case of monopolar point charges. It is given by˜ F / ˜ A = 2˜ a − a, (33)with the corresponding pressure˜ P = 1˜ a − . The higher order multipoles thus make no direct con-tribution to the strong coupling interaction free energyor forces between the bounding charged surfaces withoutdielectric inhomogeneities.This is a very powerful result whose ramifications infact impose rather stringent limits on the significance ofthe multipolar expansion of counterionic charge. It ap-pears that for highly charged counterions, that are infact the only ones where the multipolar expansion reallymakes sense, most of the electrostatics is properly cap-tured by the monopolar term. Higher order multipolessimply do not contribute to the pressure in the systemin the SC limit. Of course all of this is valid in the limitof homogeneous dielectric properties for planar surfaceswithout any image effects. The counterion density profile can be extracted from(31) as an integrand in the second term,˜ ρ (˜ z ) ∝ Z − d x exp (cid:16) − X i =1 ˜ w i (˜ z, x ) (cid:17) . (34)In the case without the dielectric mismatch the densityprofile is simply homogeneous, ρ ( z ) = const. Dielectricimages induce an additional repulsion between the coun-terions and the surface charges pushing them towardsthe midplane region between the two charged boundingsurfaces.Orientational order of quadrupoles can be inspected via averaged (cid:10) x (cid:11) which is here defined as (cid:10) x (cid:11) = R x exp (cid:16) − P i =1 ˜ w i (˜ z, x ) (cid:17) d x R exp (cid:16) − P i =1 ˜ w i (˜ z, x ) (cid:17) d x . (35)We can now use the same definition of order parameter S as in WC, (21). A. Numerical results in the SC limit
We now present some numerical results for the SClimit. Without any dielectric mismatches the density pro-file of the monopolar counterions is homogeneous, i.e. , itdoes not depend on z , which is a well known result [10].In the case of multipoles with dielectric images this is nolonger true, but the counterion density does remain aneven function of z .On Fig. 3 we show the numerical results for couplingparameter strength Ξ = 50, dielectric jump ∆ = 0 . a = 1. Contrary to the WC situation, here counte-rions are localized in the vicinity of the midpoint be-tween the two bounding surfaces due to image repul-sion. Even without quadrupolar contributions ( t = 0) themonopolar interaction, w , is repulsive and long ranged, w ∼ z − [29], compressing the counterions to the mid-point.When the quadrupolar parameter t , is increased a bitthe midpoint peak first starts to widen. This is due to theattractive monopole-quadrupole image contribution, rep-resented by the w and w terms which are proportionalto t . As t increases further (approximately for t > a / w , which is proportional with t , takesover and confines counterions to the center even more sothat the density peak sharpens up.For the value ∆Ξ large enough, so that density pro-file becomes bell-shaped, the SC density profile can beapproximated by a gaussian curve with a variance, i.e. ,peak width, of the form δ ˜ z ≃ a ˜ a − a t + t . (36) -1 -0.5 0 0.5 1 z/a ρ∼ t = 0.0t = 0.2t = 0.4t = 1.0 (a) -1 -0.5 0 0.5 z/a -0.5-0.48-0.46-0.44-0.42 S t = 0.2t = 0.4t = 1.0 (b) a~ -50510 ∆ P~ t = 0.2t = 0.4t = 1.0 (c)FIG. 3: Strong coupling results for Ξ = 50 and ∆ = 0 .
95 and various t . (a) Counterion density profile. (b) Order parameterprofile. Counterions are less ordered at the center. Order is increased with increasing t . (c) Quadrupolar pressure contribution,∆ ˜ P = ˜ P t − ˜ P t =0 . Contribution is repulsive at small separations and becomes attractive at larger separations. At very largeseparations it goes to zero. As can be seen, small t widens the density peak, but fur-ther increasing t over the value t > a /
45 the peak nar-rows. Also by increasing parameters Ξ, ∆ the midpointdensity peak narrows. This also means that the repul-sive pressure between the walls increases. As the densitypeak narrows and the density of the counterions at themidpoint plane increases, the counterions become moreordered in the quasi two-dimensional midpoint layer, im-plying also that the correlations are stronger and there-fore the SC limit should a fortiori be even more appro-priate.Fig. 3(b) shows also the orientational order parameterfor the SC limit. Contrary to the WC results, here theorientational order parameter is negative, implying thatquadrupoles are preferably aligned perpendicular to z -axis. The parameter reaches its extremum − / z -axis and parallel to the walls. This is due to the strongquadrupole-quadrupole image repulsion, w , which hasits minimum at x = 0 and is not a purely steric effect asin simulations of long charged stiff rods [36].Since monopole-monopole image interaction, w , hasthe longest range, we can expect that for large separa-tions between the walls, 2˜ a , this interaction will dominateand higher multipoles can be again neglected. Fig. 3(c)shows the pressure difference ∆ ˜ P , which is the differ-ence between the pressure with finite t and the pres-sure without the quadrupolar contributions, t = 0. Asexpected this difference indeed goes to zero at largeseparations. At moderate separations the quadrupo-lar contribution becomes negative, which is caused bymonopole-quadrupole image attractions, w and w , thatare shorter ranged than w . At even smaller separa-tions quadrupole-quadrupole image repulsion, w , be-comes dominant, since it has the shortest range andvaries approximately as ∼ a − . The larger the t thesmaller is thus the maximal quadrupolar attraction atlarger distance since it is overwhelmed by repulsive con-tributions. VII. MULTIPOLES, CORRELATIONS,BRIDGING AND ALL THAT JAZZ
There appear some similarities between the polyelec-trolyte bridging interaction [32, 33] and the attractionseen with strongly charged counterions. Recent simula-tions and density functional results [36] in fact do accen-tuate a close connection between polyelectrolyte bridg-ing and strong-coupling electrostatics, a line of reasoningthat we will explore here in finer detail.Turesson et al. [36] indeed find out that for flexiblepolyelectrolytes bridging attraction appears to be thedominant source of attractive interaction. On the otherhand, for stiff charged rods the strong adsorption to thebounding charged walls prevents bridges to form. In-stead a very strong correlation attraction is apparent atshorter separations. Flexibility thus extends the rangeof attractive interactions as their nature changes fromshort range correlation attraction to longer ranged poly-electrolyte bridging. Similar conclusions have been alsoreached in [37].Here we will formulate the exact correspondence be-tween the intuitive notion of “bridging” in the case ofmultivalent counterions and the definition of “bridginginteraction” in the case of charged polymers. Let usanalyse first the partition function for two disparatesystems: a counterion Coulomb fluid confined betweentwo strongly charged walls, and a flexible polyelectrolytechain confined in the same geometry with weakly chargedwalls. For point counterions between two charged sur-faces the grand canonical partition function in the SClimit [38] can be written to the first order in fugacity as Z = Z (0)SC + λ ′ Z (1)SC + . . . , (37)where Z (0)SC = C Z D [ φ ( r )] exp (cid:18) − βε Z ε ( r )( ∇ φ ( r )) d r ++ i β Z ρ ( r ) φ ( r )d r (cid:19) , (38)0and Z (1)SC = C Z d R Ω( R ) Z D [ φ ( r )] exp (cid:18) − βε Z ε ( r )( ∇ φ ( r )) d r ++ i β Z ρ ( r ) φ ( r )d r + i βqe Z ρ ( r − R ) φ ( r )d r (cid:19) . (39)The above partition function is of course exactly the sameas the one in (22) after one evaluates the functional in-tegrals indicated above explicitly. Here ρ ( r ) is againthe external fixed charge on the two bounding surfaceslocated at z = ± a and the shorthand ρ ( r − R ) is intro-duced for ρ ( r − R ) = δ ( r − R ).On the other hand the partition function for a flexiblepolyelectrolyte chain is given by [39] Z P = C Z D [ φ ( r )] exp (cid:18) − βε Z ε ( r )( ∇ φ ( r )) d r + (40)+ i β Z ρ ( r ) φ ( r )d r + ln ZZ G φ ( r , r ′ ; N ) d r d r ′ (cid:19) . where G φ ( r , r ′ ) is the Green function of a polyelectrolytechain in external electrostatic field φ given by [40] G φ ( r , r ′ ; N ) = Z r ′ = R ( N ) r = R (0) D [ R ( n )] × (41) × exp (cid:18) − ℓ Z N (cid:18) d R ( n )d n (cid:19) d n + i βqe Z N φ ( R ( n ))d n (cid:19) , where R ( n ) is the coordinate of a polyelectrolytesegment n and the boundary condition is set aslim N → G φ ( r , r ′ ; N ) = δ ( r − r ′ ). Representing theGreen function via a corresponding Edwards equa-tion one can derive that the polyelectrolyte monomerdensity ρ φ ( r ; N ) is given by a functional derivativeof ln RR G φ ( r , r ′ ; N ) d r d r ′ with respect to the fluctuat-ing electrostatic potential φ ( r ) [40]. The polyelec-trolyte monomer density has to satisfy the conditionlim N → ρ φ =0 ( r ; N ) = δ ( r ) . In the planar geometry with two bounding surfaces,one can center the monomer density at the origin of thecoordinate system, whose z axis is directed along thesurface normal and has the origin at the midplane. De-noting R = (0 , ,
0) one can thus write ρ φ =0 ( r ; N ) = ρ φ =0 ( r − R ; N ). In the limit of small electrostatic po-tentials, the lowest order ( i.e. , the first-order) solution ofthe polyelectrolyte Green function can be written as [40]ln ZZ G φ ( r , r ′ ; N ) d r d r ′ ∼ = i βqe Z ρ φ =0 ( r − R ; N ) φ ( r )d r . (42)By construction this solution clearly corresponds to aweak coupling of the polyelectrolyte to the external fieldgenerated by the fixed charges at the boundary of thesystem. For large values of this charge higher orders inthe expansion (42) would certainly have to be taken intoaccount. In the case of weak coupling the partition function ofa polyelectrolyte chain is then given approximately by Z P ∼ = Z (0)WC [ ρ φ =0 ( r − R ; N )] = (43)= C Z D [ φ ( r )] exp (cid:18) − βε Z ε ( r )( ∇ φ ( r )) d r ++ i β Z ρ ( r ) φ ( r )d r + i βqe Z ρ φ =0 ( r − R ; N ) φ ( r )d r (cid:19) . If we now compare the expressions (39) and (43) we canestablish the following identity Z (1)SC = lim N → Z d R Ω( R ) Z (0)WC [ ρ φ =0 ( r − R ; N )] , (44)with the simple (first order SC) counterion partition func-tion on the l.h.s. and the polyelectrolyte (zeroth orderWC) partition function on the r.h.s. This “duality” re-lation connects the partition function of a strongly cou-pled point counterion system with the partition functionof a weakly coupled flexible polyelectrolyte system. Since Z (0)WC for finite N describes polyelectrolyte bridging inter-actions it is clear that the above formula establishes a for-mal connection between strong coupling point counterionattractive interactions and weak coupling polyelectrolytebridging interactions. VIII. CONCLUSIONS
Using field-theoretic methods we derived a descriptionof the counterion-mediated electrostatic interaction whenthe counterions posses internal degrees of freedom such asa rotational axis. We concentrate on the symmetricallycharged planar surfaces with a dielectric mismatch onboth sides and counterions with quadrupolar moments inbetween. We analyse this system in two different regimes,namely in the weak coupling (mean-field) and the strongcoupling limit. The former one describes the case of lowsurface charge and low counterion valency, whereas thelatter one describes high surface charge and high counte-rion valency.In the mean-field limit, we derived the Poisson-Boltzmann equation for counterion density in the casethat the counterions posses dipolar as well as quadrupo-lar moments. Since the equation is highly non-linear,we solved it numerically only in the first order approx-imation, which should be valid for small quadrupolarmoments, t . As is already known the dielectric mis-matches play no role in the mean-field theory and havetherefore no effect on quadrupoles [29]. One could addthose corrections in by hand [41], but such an approachdoes not strictly corresponds to the mean-field analysis.Quadrupolar interaction affects the counterion densitydistribution as well as interactions between the chargedsurfaces. Counterions are depleted from the central re-gion and concentrated in the vicinity of both surfaces.The orientation of quadrupoles is preferentially parallel1to the z -axis as the moment t is increased. The highestalignment is reached right next to both bounding sur-faces. The quadrupolar contribution to the interactionpressure is attractive.In the strong coupling limit higher order multipolesalso play an important role in the intersurface interac-tions but only for a dielectrically inhomogeneous case.Without any dielectric discontinuities in the system thehigher multipoles simply do not matter in plan-parallelcase and the monopolar part of the partition functioncaptures all the strong coupling effects. This is a com-pletely general result for plan-parallel surfaces and shouldbe of some importance in assessing the electrostaticallymediated forces between strongly charged planar sur-faces. As already mentioned the multipoles can haveeffects near curved surfaces even without dielectric dis-continuities. If compared to the case of monopolar coun-terions, higher multipoles interact only via the orien-tationally dependent part of the image self-interaction.In our analysis we focused on the counterions withmonopolar and quadrupolar moments, so that the in-teraction is composed of three contributions. These arethe monopole-monopole image interaction, which is longranged and repulsive, the monopole-quadrupole image aswell as quadrupole-monopole image interactions that areshorter range and attractive. Finally, very short rangedquadrupole-quadrupole image contribution is repulsiveand plays a dominant role at very small distances.We are at present involved in extensive Monte Carlo simulations in order to explore the validity of the for-mal developments described above. Preliminary resultscompletely vindicate the theoretical analysis. IX. ACKNOWLEDGEMENT
R.P. would like to acknowledge the financial supportby the Slovenian Research Agency under contract Nr.P1-0055 (Biophysics of Polymers, Membranes, Gels, Col-loids and Cells). M.K. would like to acknowledge the fi-nancial support by the Slovenian Research Agency underthe young researcher grant. A.N. would like to acknowl-edge financial support by the Institute for Research inFundamental Sciences (IPM), Tehran. This research wassupported in part by the National Science Foundationunder Grant No. PHY05-51164 (while at the KITP pro-gram
The theory and practice of fluctuation induced in-teractions , UCSB, 2008). Y.S.J. is grateful to M.W. Kimfor useful discussions. Y.S.J. and P.A.P. were supportedin part by the National Science Foundation (GrantsDMR-0503347, DMR-0710521) and MRSEC NSF DMR-0520415. Y.S.J. and P.A.P. were supported by a Ko-rea Science and Engineering Foundation (KOSEF) grantfunded by the Korean Government (MEST) (grant code:R33-2008-000-10163-0) and the Brain Korea 21 projectsby the Korean Government. [1] van Blaaderen A 2006
Nature
NanoLett. Nature Materials Soft Matter Phys.Rev. Lett. J. Phys. A: Math. Theor. Eur. Phys. Lett. Phys. Rev. Lett.
Classical Electrodynamics , The Advanced Book Program,Perseus Books Group, (1998)[10] Netz R R 2001
Eur. Phys. J. E Eur. Phys. J. E J. Chem. Phys. J.Chem. Phys.
Physica A
J. Chem.Phys. Phys. Rep.
Biopolymers J. Phys. Soc.Japan Phys. Rev. E Phys. Rev. E , 011920[20] Podgornik R and ˇZekˇs B 1988 J. Chem. Soc., FaradayTrans 2 J. Phys. A J. Phys. Chem.
Rev.Mod. Phys. J.Chem. Phys.
Phys. Rev. E Rep. Prog. Phys. Macromolecules Macromolecules Eur. Phys. Lett. Eur. Phys. J E Phys. Rev. Lett.
Soft condensed mat-ter physics in molecular and cell biology , (Taylor & Fran-cis). [31] Podgornik R 1992 J. Phys. Chem. Curr. Op. Coll. and In-terf. Sci. J. Pol. Sci. Part B: Polymer Physics J. Chem. Phys. J. Polymer Sci. Part B: PolymerPhysics Issue 19 3539[36] Turesson M, Forsman J and ˚Akesson T 2006
Langmuir
131 (2005).[37] DeRouchey J, Netz R R and R¨adler J O 2005
Eur. Phys.J. [38] Netz R R 2001
Eur. Phys. J. E J. Phys. Chem. The theory of polymerdynamics , Oxford University Press, USA[41] Onuki A 2006
Phys. Rev. E73