Asymmetric Coulomb fluids at randomly charged dielectric interfaces: Anti-fragility,

overcharging and charge inversion

Malihe Ghodrat,1 Ali Naji,1, ∗ Haniyeh Komaie-Moghaddam,1 and Rudolf Podgornik2,3

1School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran2Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia

3Department of Physics, Faculty of Mathematics and Physics,University of Ljubljana, SI-1000 Ljubljana, Slovenia

We study the distribution of multivalent counterions next to a dielectric slab, bearing a quenched,random distribution of charges on one of its solution interfaces, with a given mean and variance,both in the absence and in the presence of a bathing monovalent salt solution. We use the previouslyderived approach based on the dressed multivalent-ion theory that combines aspects of the strong andweak coupling of multivalent and monovalent ions in a single framework. The presence of quenchedcharge disorder on the charged surface of the dielectric slab is shown to substantially increase thedensity of multivalent counterions in its vicinity. In the counterion-only model (with no monovalentsalt ions), the surface disorder generates an additional logarithmic attraction potential and thus analgebraically singular counterion density profile at the surface. This behavior persists also in thepresence of a monovalent salt bath and results in significant violation of the contact-value theorem,reflecting the anti-fragility effects of the disorder that drive the system towards a more ‘ordered’state. In the presence of an interfacial dielectric discontinuity, depleting the counterion layer at thesurface, the charge disorder still generates a much enhanced counterion density further away fromthe surface. Likewise, the charge inversion and/or overcharging of the surface occur more stronglyand at smaller bulk concentrations of multivalent counterions when the surface carries quenchedcharge disorder. Overall, the presence of quenched surface charge disorder leads to sizable effectsin the distribution of multivalent counterions in a wide range of realistic parameters and typicallywithin a distance of a few nanometers from a charged surface.

I. INTRODUCTION

Nature is not perfect and various types of disorder areubiquitous. Disorder often causes large changes in theproperties of condensed matter systems as predicted onthe basis of naive idealized models that assume perfectregularity. Electron properties in two-dimensional [1] andthree-dimensional disordered media [2, 3], crystalline lat-tices with structural defects [4, 5], spin glasses with ran-dom interactions [6, 7] and systems exhibiting criticalitymodified by the presence of disorder [8], are all instancesof pronounced disorder effects in the bulk of the materialsthat can fundamentally change the behavior of idealizedmodel systems. Apart from their fundamental impor-tance in modifying the bulk properties, disorder effectsat surfaces and interfaces are particularly important inthe context of the solid-electrolyte interphases [9, 10] rel-evant also for energy generation and storage technologies[11, 12]. The structural disorder in the charge distribu-tion and/or dielectric response spatial profile in the vicin-ity of the material interfaces couple to long-range electro-static interactions, leading effectively to long-range disor-der effects as well [13–23], that can not be understood interms of the usual assumptions of piecewise homogeneouscharge distribution and/or dielectric properties, under-pinning so much of colloid science and electrochemistry[24–29].

∗Email: a.naji@ipm.ir

The coupling between electrostatic interactions anddisorder has been already noted and discussed in otherimportant cases [30–46], including surfactant-coatedsurfaces [32–36], random polyelectrolytes and polyam-pholytes [41–44], contaminant adsorption onto macro-scopic surfaces or in amorphous films showing grainstructure after being deposited on crystalline substrates[47]. In all these cases the charge distribution oftenshows a fundamentally disordered component that oftenremains unaltered after the assembly or fabrication of thematerials, thus exhibiting a frozen, or quenched, type ofdisorder (see, e.g., Refs. [19, 20, 44–46, 48–51] for exam-ples of surfaces with annealed, disordered charge distri-butions that will not be considered in this paper). Thischarge disorder coupled to the long-range electrostaticinteractions can then leave its fingerprint also on the in-teractions between macromolecular surfaces that in theirturn can play a fundamental role in the stability of col-loidal systems [14–16].

In fact, this coupling between disorder and Coulombinteractions has been suggested to underly the anoma-lously long-ranged interactions observed in ultrahigh sen-sitivity experiments on Casimir-van der Waals interac-tions between surfaces in vacuo [52–56]. The intricateexperimental details of accurate measurements of theseinteractions can be properly accounted for only if oneconsiders also the disordered nature of charges on andwithin the interacting surfaces by invoking the so-calledpatch effect [13, 17–23, 52–58], where the disorder stems,for instance, from the adsorption of charged contami-nants and/or impurities that can give rise to monopo-

2

lar random surface charges, and/or the variation of thelocal crystallographic axes of the exposed surface of aclean polycrystalline sample and the corresponding elec-tron work function that can cause a variation of the localsurface potential. Such random distributions of surfacecharges can be measured directly by Kelvin force mi-croscopy measurements [59].

The salient features of electrostatic interactions them-selves, even for homogeneous charge distributions in theabsence of any disorder, are however quite involved (see,e.g., Refs. [13, 24–29, 60–73] and references therein).It has been recognized some time ago that electrostaticinteractions in fact come in several varieties, dependingon the strength of electrostatic coupling in the system[65–73]. In the presence of mobile monovalent coun-terions, they are standardly described by the Poisson-Boltzmann (PB) theory stemming from the mean-field,collective description of Coulomb fluids that gives rise topronounced repulsive interactions between like-chargedmacroions [13, 24–29, 61–63]. On the contrary, in thepresence of multivalent counterions, where electrostaticinteractions show basically a single-particle character,they can mediate strong attractive interactions betweenlike-charged macroions [65–73]. This attraction led to anew understanding of the theory of electrostatic inter-actions in colloidal domain based on the strong-coupling(SC) limit [71–73], devised to describe the equilibriumproperties of Coulomb fluids when charges involved be-come large. In the simple case of a counterion-only sys-tem, the transition from the mean-field PB description,dubbed also the weak-coupling (WC) limit, to the SClimit is governed by a single dimensionless electrostaticcoupling parameter, being a ratio of the Bjerrum length,which identifies Coulomb interaction between counteri-ons themselves, and the Gouy-Chapman length, whichdescribes electrostatic interaction between the counteri-ons and the charged (macroion) surfaces [67–73]. Theemerging picture of equilibrium properties of Coulombfluids has thus become much richer than conveyed formany years by the standard DLVO paradigm of colloidscience [24–29].

However, this is still not the complete story. Themost relevant case of a Coulomb fluid is in fact nota counterion-only system, but an asymmetric mixtureof multivalent ions in a bathing solution of monovalentions, a particularly relevant situation specifically in thecontext of bio-macromolecules, where multivalent ionstogether with the screening properties of the monova-lent salt are believed to play a key role in the stabilityof macromolecular aggregates such as liquid crystallinemesophases of semiflexible biopolymers [74–76] or DNAcondensates that form in the bulk [77–83] or confined intothe capsids of viruses [84–86].

In an asymmetric mixture, multivalent counterionsand monovalent ions are coupled differently to themacromolecular charges: multivalent ions strongly, whilemonovalent ions only weakly, as evidenced from their re-spective electrostatic coupling parameters. Since usu-

ally multivalent ions are present at very low concentra-tions, e.g., around just a few mM, their behavior is ex-pected to be properly described within the virial expan-sion in the powers of their fugacity (or bulk concentra-tion) [67–71, 87–89]. A dressed multivalent-ion theorythen emerges naturally within this context [69, 87–91]since the degrees of freedom due to weakly coupled mono-valent ions can be traced out from the partition func-tion, leading to an effective formalism based on screenedinteractions between the remaining dressed multivalentions and fixed macromolecular charges. The dressedmultivalent-ion theory for complicated asymmetric mix-tures of multivalent ions in a bathing solution of monova-lent ions can then seamlessly bridge between the standardWC and SC limits [69, 87–89].

These baroque features of electrostatic interactions fur-thermore give their imprint also on the effects of dis-ordered charge distribution along macromolecular inter-faces [13–16, 30–46]. While on the WC level and for ho-mogeneous planar systems the disorder effects are nonex-istent [14], they can lead to qualitative changes in thestability properties of the system once the dielectric con-trast between the solution and macromolecular interfaces(or the inhomogeneous distribution of salt ions) is takenfully into account [13, 15, 17–23, 37]. Nevertheless, it isin the SC limit that the coupling between electrostaticinteractions and the disorder in the external interfacialcharge distributions gives rise to fundamentally novel andunexpected phenomena [14, 16]. While studying the in-teraction between two disordered charged surfaces it wasnoticed [14, 16] that disorder can in fact lead to a loweringof the effective temperature of the system, engendering adistribution of the multivalent counterions between theinteracting surfaces that is characterized by less effectiveentropy. This is intuitively difficult to foresee, as onewould perhaps naively assume that thermal and exter-nally imposed charge disorder would somehow enhanceone another.

In order to properly understand and identify all salientfeatures of the coupling between quenched charge dis-order and long-range electrostatic interactions, we nowproceed to characterize more closely the consequencesof coupling between charge disorder and electrostaticallystrongly coupled multivalent counterions immersed in amonovalent salt solution bath. In particular, we willidentify the defining feature of this strongly coupled,disordered system as belonging to the anti-fragility [92]exhibited by this system. In the present context, anti-fragility simply refers to the fact that an externally im-posed, quenched charge disorder, effectively diminishesthe intrinsic thermal disorder in the system, forcing itsbehavior to be more ‘ordered’. We will show that thisbehavior stems from the compensation of the transla-tional entropy of the solution ions by the configurationalentropy due to the averaging over different realizationsof the quenched disorder. In the particular example ofthe counterion-only model (with no monovalent salt ionsand no interfacial dielectric discontinuity), we show that

3

the counterions experience an additional logarithmic at-traction towards the surface due to the presence of thesurface charge disorder in a way that their density profileexhibits an algebraically singular behavior at the surfacewith an exponent that depends on the disorder strength(variance). This behavior persists also in the presence ofa monovalent salt bath and results in significant violationof the contact-value theorem [93–96]. In the presence ofan interfacial dielectric discontinuity, depleting the coun-terion layer at the surface, the charge disorder still gen-erates a much enhanced counterion density further awayfrom the surface. Likewise, the charge inversion and/orovercharging of the surface are predicted to occur morestrongly and at smaller bulk concentrations of multiva-lent counterions when the surface carries quenched chargedisorder.The organization of the paper is as follows: In Sec-

tion II, we introduce our model and then in Section IIIpresent the theoretical formalism that will be used tostudy the distribution of multivalent ions next to ran-domly charged dielectric interface. We then proceed topresent our results in Section IV, where we discuss thecase of counterion-only systems and the effects due to thepresence of a bathing salt solution, an interfacial dielec-tric discontinuity, and also the overcharging and chargeinversion phenomena in the system. We conclude ourdiscussion in Section V.

II. THE MODEL

Our model is comprised of an infinite, planar dielectricslab of thickness b and dielectric constant ǫp, immersed ina bathing ionic solution of dielectric constant ǫm (see Fig.1). The ionic solution is assumed to consist of a mixtureof a monovalent 1:1 salt of bulk concentration n0 as wellas of a multivalent q:1 salt of bulk concentration c0. Thedielectric slab is assumed to be impermeable to mobileions with the surface normal in the direction of the z-axis,occupying the region −b ≤ z ≤ 0. The bounding surfaceof the slab at z = 0 is assumed to be charged, carry-ing a quenched spatial distribution of random monopolarcharges, ρ(r), while the other surface of the slab is un-charged. The multivalent counterions are assumed to bein contact only with the charged surface (see below). Thedisordered (random) charge distribution is described bythe Gaussian probability distribution function

P [ρ] = C exp

(

−1

2

∫

dr g−1(r) [ρ(r) − ρ0(r)]2

)

, (1)

where C is a normalization factor, ρ0(r) is the meanvalue and g(r) the variance of the spatial distribu-tion of random surface charges. It is obvious that theabove probability distribution function entails an uncor-related disorder with 〈〈(ρ(r)− ρ0(r)) (ρ(r

′)− ρ0(r′))〉〉 =

g(r)δ(r − r′), where double-brackets denote the average〈〈· · · 〉〉 =

∫

DρP [ρ](

· · ·)

.

FIG. 1. (Color online) Schematic view of an infinite, planardielectric slab of thickness b and dielectric constant ǫp im-mersed in a solution of dielectric constant ǫm, containing amixture of mono- and multivalent salts. The slab boundaryat z = 0 is randomly charged. The multivalent ions (counte-rions) with charge valency q are shown by large blue spheresand monovalent salt anions and cations are shown by smallorange and blue spheres. The multivalent counterions are incontact only with the charged surface (right).

In what follows, the general formalism is valid for anarbitrary shape of the charged boundaries, but for thelater developments in this paper we shall delimit our-selves to the specific example of a planar slab as notedabove. In this case, the mean charge distribution and itsvariance are given by

ρ0(r) = −σe0 δ(z), g(r) = ge20 δ(z), (2)

where g ≥ 0, and, without loss of generality, we assumeσ ≥ 0 and, for the multivalent counterions, q ≥ 0. Themonovalent salt is assumed to be present on both sides ofthe slab while the multivalent counterions are restrictedto the right half-space z ≥ 0. This spatial constraint canbe taken into account formally by introducing the indi-cator functions Ωc(r) = θ(z) for the multivalent counte-rions, and Ω+(r) = Ω−(r) = Ωs(r) = θ(z) + θ(b − z) forthe monovalent ions, where θ(z) is the Heaviside’s stepfunction. This constraint can be realized by enclosing theregion on the left side of the slab in a membrane imper-meable to multivalent ions; such a constraint will be rele-vant in our analysis only in the situations where the slabthickness is small relative to the salt screening length andwill otherwise have vanishing impact on the distributionof multivalent counterions next to the charged surface inthe regime of parameters that will be of interest in thispaper. For the most part, however, we shall focus on thecase of semi-infinite slabs.

III. THEORETICAL BACKGROUND

A. Field action for dressed multivalent ions

For a given realization of the fixed charge distributionρ(r), the microscopic Hamiltonian of the ionic mixture

4

described above can be written exactly in a functionalintegral form as [60–63, 71]

Z = e−1

2ln detG

∫

Dφ e−βS[φ], (3)

where β = 1/(kBT ) and φ(r) is the fluctuating (electro-static) potential and the effective “field-action” reads

S[φ] =1

2

∫

drdr′φ(r)G−1(r, r′)φ(r′) +

+ i

∫

dr ρ(r)φ(r) − kBT

∫

drV(φ(r)), (4)

where G−1(r, r′) is the operator inverse of the Coulombinteraction (or, the bare Green’s function) G(r, r′) inthe presence of dielectric boundaries satisfying the equa-tion −ǫ0∇ · ǫ(r)∇G(r, r′) = δ(r − r′). The “field self-interaction” term is given by [69, 87]

V(φ)=λcΩc(r)e−iβqe0φ +Ωs(r)

(

λ+e−iβe0φ + λ−e

iβe0φ)

,(5)

where λc and λ± are the ionic fugacities of multivalentcounterions (c) and monovalent anions (−) and cations(+), respectively. Note that, since the multivalent coun-terions result from a q:1 salt of bulk concentration c0mixed with a 1:1 salt of bulk concentration n0, we haveλc = c0, λ+ = n0 and λ− = n0 + qc0.The above field action obviously represents a highly

asymmetric Coulomb fluid especially when counterionsare multivalent, q > 1. These counterions couple stronglyto the fixed (mean) surface charge, whereas the monova-lent salt species couple weakly. This leads to a complexsituation where different components of the Coulombfluid couple differently to the same surface charges, thusmaking the analytical progress and, in particular, obtain-ing exact solutions [97, 98], very difficult.Nevertheless, progress is possible and systems of this

type can be treated using a combined weak-strong ap-proximation framework which has been discussed in a se-ries of recent works [69, 87–89]. It was shown, by employ-ing both analytical approaches as well as implicit- andexplicit-ion simulations, that in a wide range of realisticsystem parameters the monovalent ions can be treatedsafely within the Debye-Huckel (DH) framework, whilemultivalent ions can be handled by means of a standardvirial expansion scheme [67–69, 71–73] of the strong cou-pling approximation. The DH-type terms in this contextfollow by expanding the last two terms in Eq. (5) up tothe second order in φ(r) (which can be justified on a sys-tematic basis in highly asymmetric systems with q ≫ 1[87]) as

V(φ(r)) ≃ λcΩ(r) e−iβqe0φ − nbΩs(r)(βe0φ)

2/2 +O(φ3),(6)

where nb is the bulk concentration due to all monovalentions nb = λ+ + λ− = 2n0 + qc0.On the analytical level, the above procedure allows one

to trace the partition function over the degrees of freedomassociated with monovalent ions and thus one remains

with only “dressed” multivalent ions and surface chargethat then interact through a DH-type interaction kernel(or the screened Green’s function) defined via

−ǫ0∇·ǫ(r)∇G(r, r′)+ǫ0ǫ(r)κ2(r)G(r, r′) = δ(r−r′), (7)

where the Debye screening parameter κ(r) is non-zerooutside the dielectric slab and is given as κ2 = 4πℓBnb

with ℓB = e20/(4πǫ0ǫmkBT ) being the Bjerrum length.This type of methodology leads to the so-called dressed

multivalent-ion theory [69, 87–89], which is a direct gener-alization of the standard counterion-only SC theory [67–69, 71–73]. In fact, the dressed multivalent-ion theoryhas a hybrid character in that it reproduces both thecounterion-only SC theory and the DH theory as twoasymptotic limits at small and large salt screening pa-rameters, respectively.In what follows, we shall use this framework to study

the effects of surface charge disorder in the distribu-tion of multivalent counterions, which can be comparedwith previous findings in the context of non-disordered,charged dielectric surfaces [88].

B. Counterion density profile

The density profile of multivalent counterions followsstandardly from the general formalism defined by Eqs.(3) and (4) as [63, 67–69, 71, 73]

c(r; [ρ]) = λcΩ(r)〈e−iβqe0φ〉, (8)

where 〈· · · 〉 denotes the thermal (ensemble) average overthe fluctuating field φ(r). For a given (quenched) real-ization of ρ(r), this average can be calculated analyti-cally in both limits of weak and strong coupling in thecounterion-only case [63, 71] and also in the more gen-eral context of dressed multivalent ions in asymmetricCoulomb fluids [87–89]. As noted before, the effects dueto multivalent ions can be investigated by virial expand-ing the partition function in terms of their fugacity andkeeping only the leading order contributions (see Refs.[67–69, 71, 87–89] for further details). This procedureleads to a single-particle form for the density profile ofmultivalent ions as

c(r; [ρ]) = λcΩ(r) e−βu(r;[ρ]), (9)

where u(r) is the single-particle interaction energy

u(r; [ρ]) = qe0

∫

dr′G(r, r′)ρ(r′) +q2e202Gim(r, r). (10)

Here Gim(r, r) is the generalized Born energy contribu-tion that stems purely from the dielectric and/or saltpolarization effects (or the so-called “image charges”).In other words, Gim(r, r) = G(r, r) − G0(r, r), whereG0(r, r

′) is the formation (self-)energy of individual coun-terions obtained from the free-space screened Green’sfunction defined via −ǫ0ǫm(∇2−κ2)G0(r, r

′) = δ(r−r′).

5

In the present model, the fixed surface charge distribu-tion, ρ(r), has a disorder component and thus, in order toobtain the measurable density profile, one must averageover different realizations of this disorder field using theGaussian weight (1). This can be done straightforwardlyby computing

c(r) = 〈〈c(r; [ρ])〉〉 = λcΩ(r)〈〈e−βu(r;[ρ])〉〉, (11)

which then gives

c(r) = λcΩ(r) e−βu(r), (12)

where the effective single-particle interaction energy nowreads

u(r) = u0(r) + uim(r) + udis(r), (13)

with

u0(r) = qe0

∫

dr′G(r, r′)ρ0(r′), (14)

uim(r) =q2e202Gim(r, r), (15)

udis(r) = −βq2e202

∫

dr′g(r)[G(r, r′)]2. (16)

The three terms, respectively, originate from the con-tribution of the interaction of multivalent ions with themean surface charge density (first term), the contribu-tion of self-interactions (interactions with image charges,second term) and the contribution of the surface chargedisorder (third term). The latter can be viewed as aneffective surface-counterion interaction which is inducedby the quenched randomness in the surface charge. Itis proportional to the disorder variance but also showsan explicit temperature dependence. Another interest-ing point is that the disorder interaction term exhibits aquadratic dependence on the Green’s function and like-wise also on the multivalent ion charge valency q. Thesefeatures can be understood by noting that the disorderterm in fact represents the sample-to-sample fluctuations(or variance) of the single-particle interaction energy.This result follows by noting that, because u(r; [ρ]) inEq. (10) is a linear functional of the Gaussian field ρ(r),the effective single-particle interaction energy (13) can beexpressed also as

u(r) = 〈〈u(r; [ρ])〉〉 − β〈〈u2(r; [ρ])〉〉c/2, (17)

where 〈〈u2(r; [ρ])〉〉c ≡ 〈〈u2(r; [ρ])〉〉 − 〈〈u(r; [ρ])〉〉2, and wehave

〈〈u(r; [ρ])〉〉 = u0(r) + uim(r), (18)

〈〈u2(r; [ρ])〉〉c = −2kBT udis(r). (19)

C. Rescaled representation

In order to proceed, we introduce the dimensionless(rescaled) quantities

r = r/µ, κ = κµ, Ξ = q2ℓB/µ, (20)

where

µ = 1/(2πqℓBσ) (21)

is the Gouy-Chapman length and Ξ = 2πq3ℓ2Bσ isthe electrostatic coupling parameter associated with themean surface charge [71]. Analogously, one can define thedisorder coupling (or disorder strength) parameter [14]

χ = 2πq2ℓ2Bg, (22)

which is proportional to the disorder variance; its de-pendence on the counterion valency, q, is different fromthat of the mean electrostatic coupling parameter as firstnoted in Ref. [14].

IV. RESULTS

A. Counterion-only case

Let us first consider the special case of multivalentcounterions next to a charged surface in the absence ofany salt screening (κ = 0) and dielectric image chargeeffects (i.e., a dielectrically homogeneous system withǫp = ǫm). This case has been considered in a previouswork on the effective interaction between two randomlycharged surfaces [14], which however did not investigatethe distribution of counterions.In this case, the number of counterions, N , is fixed by

the mean charge on the surface through the electroneu-trality condition Nq = Sσ, where S is the surface area.The fugacity of counterions is thus given by [14]

λc =N

∫

drΩ(r) e−βu(r). (23)

The effective single-particle interaction is obtained byusing Eqs. (13)-(16) and noting that, in this case,G0(r, r

′) = 1/(4πǫ0ǫm|r − r′|). Hence, in rescaled units,we have βuim(r) = 0, and

βu0(z) = z, (24)

βudis(z) =χ

2ln z. (25)

and, therefore,

βu(z) = z +χ

2ln z. (26)

The rescaled density profile of counterions is then ob-tained using Eq. (12) as

c(z) ≡c(z)

2πℓBσ2=z−

χ

2 e−z

Γ(1− χ2 ), z ≥ 0, (27)

where Γ(·) is the Gamma function. This expression showsa standard SC exponential decay of the multivalent coun-terion density, which dominates at large separations fromthe surface and is a well-established result within the SC

6

0 1 2 30

0.4

0.8

1.2

χ = 0χ = 0.50χ = 1.00χ = 1.50χ = 1.95

z

~

~

c z~( ) 10-3 10-2 10-1 100

10-1

100

101

FIG. 2. (Color online) Rescaled density profile of multivalentcounterions next to a randomly charged surface in the absenceof salt and image charge effects for different values of thedisorder coupling parameter as shown on the graph. Insetshows the diverging behavior of the density profiles at smalldistances from the surface in the log-log scale.

context [65, 67–69, 71, 73]. But it also exhibits an al-gebraic dependence on z, which dominates at small sep-arations from the surface and thus shows that, in thepresence of surface charge disorder, the counterion den-sity diverges in the immediate vicinity of the surface.The presence of disorder thus clearly violates the contact-value theorem which was derived for uniformly chargedsurfaces [93–96]; this theorem entails a contact value ofc(0) = 1 in the whole range of coupling parameters. Notethat nevertheless the electroneutrality is exactly satisfiedas

∫∞

0dz c(z) = 1.

The behavior of the density profile, Eq. (27), is shownin Fig. 2 for a few different values of the disorder cou-pling parameter. As seen, due to the singular behaviorat the surface, the presence of charge disorder enhances(suppresses) the density of counterions at small (large)separations, in general agreement with the previous find-ings in the case of non-disordered but heterogeneouslycharged surfaces such as surfaces carrying discrete chargepatterns (see, e.g., [99–102] and references therein).The above results can be illuminated further by ana-

lyzing the averaged cumulative charge, defined as

Q(z) =1

σ

∫ z

0

dz′ [e0qc(z′) + ρ0(z

′)] , (28)

or, in rescaled units,

Q(z) = −1 +

∫ z

0

dz′ c(z′) = −Γ(1− χ

2 , z)

Γ(1− χ2 )

, (29)

where Γ(·, ·) is the incomplete Gamma function. Notethat Q(z) is normalized such that Q(0) = −1, which rep-resent the mean surface charge density at z = 0, andQ(z) → 0 for z → ∞, which reflects the global elec-troneutrality of the system. As seen in Fig. 3, nearly allof the counterions become strongly localized in the vicin-ity of the surface (i.e., Q(z) ≃ 0 for finite z > 0) as χ is

0 1 2 3 4 5

-0.8

-0.4

0

χ = 0χ = 0.50χ = 1.00χ = 1.50χ = 1.95

z~

Q z~( )

FIG. 3. (Color online) Cumulative charge next to a randomlycharged surface in the absence of salt and image charge ef-fects for different values of the disorder coupling parameteras shown on the graph.

increased. In fact, the counterion density profile tends tozero at any finite separation from the surface c(z) → 0when χ tends (from below) to the threshold value

χ∗ = 2, (30)

because the Gamma function in the denominator of thenormalization factor in Eqs. (27) and (29) goes to infin-ity.It should be emphasized that the strong accumulation

of counterions in the immediate vicinity of the surfacedoes not give rise to a renormalized mean surface chargedensity (see also Appendix C in Ref. [16]) and that thesingular behavior of counterions at the surface in thepresent context should be distinguished from the surfaceadsorption or counterion condensation phenomena [69].

B. Disorder-induced anti-fragility

The foregoing results clearly show that the excess ac-cumulation of counterions near the surface is driven bythe disorder-induced single-particle energy (25), which isattractive and depends logarithmically on the distancefrom the surface and thus generates the singular behav-ior of the counterion density profile at the surface. Thissuggests that the presence of quenched surface charge dis-order drives the system towards a state of lower thermal‘disorder’ as can be established systematically by calcu-lating the difference in the entropy of counterions in thepresence and absence of disorder, ∆S(χ) = S(χ)− S(0),i.e.

∆S(χ)

NkB= χψ

(

1−χ

2

)

+ lnΓ(

1−χ

2

)

, (31)

where ψ(·) is the digamma function. It thus follows that∆S(χ) ≤ 0. The entropy reduction is larger for largerdisorder strength, χ, and diverges as χ→ 2−.

7

In other words, the reduction in the translational en-tropy of counterions in the solution is driven by introduc-ing a finite degree of configurational entropy due to thepresence of quenched randomness in the surface chargedistribution. Formally, this latter type of entropy is gen-erated by the non-thermal (quenched) average taken overdifferent realizations of the surface charge disorder. Thissubtle interplay between the different kinds of entropyis therefore essential in generating the disorder-induced,singular behavior of counterions near the charged surface.It thus seems appropriate to refer to this type of behaviorof the counterions as anti-fragile [92], since introducingan external (quenched) disorder source effectively dimin-ishes the intrinsic thermal disorder in the system anddrives it towards a more ‘ordered’ state.

Another interesting point to be noted here is thatthe internal energy of the system also decreases due tothe presence of disorder but in such a way that leadsto a decrease in the free energy of the system as canbe seen from the free energy difference, β∆F(χ)/N =−ln Γ (1− χ/2) ≤ 0. Therefore, the system also attainsa thermodynamically more stable state, which is againa direct consequence of the singular behavior of counte-rions near the disordered surface. By contrast, one canshow that in the case of counterions next to a uniformlycharged surface which exhibits a regular potential, theattraction of counterions towards the surface leads to alarger free energy as compared with the ideal case wherethe surface is uncharged.

C. Salt image effects

We now turn to salt screening effects by assuming thatin addition to the multivalent ions (of bulk concentrationc0), we also have a finite amount of monovalent salt inthe system, giving a total bulk monovalent ion concen-tration of nb = 2n0 + qc0 as discussed before. We takea semi-infinite slab (b = ∞) impermeable to all ions andassume that the system is again dielectrically homoge-neous, i.e. ǫp = ǫm. This helps to disentangle the polar-ization effects due to the inhomogeneous distribution ofsalt ions (“salt image effects”) from those resulting fromthe inhomogeneous distribution of the dielectric constant(“dielectric image effects”).

For a semi-infinite slab, the Fourier transform of theGreen’s function can be obtained by standard methodsas

G(Q; z, z′) =1

2ǫ0ǫmγ

[

e−γ|z−z′| +∆s e−γ(z+z′)

]

. (32)

Note that we have used the lateral translational sym-metry of the Green’s function with respect to the trans-verse coordinates ρ = (x, y) and thus defined the Fourier

transform as G(r, r′) =∫

dQ(2π)2 G(Q; z, z′) e−iQ.(ρ−ρ

′) =∫

QdQ(2π) G(Q; z, z′)J0(Q|ρ−ρ

′|) since the Green’s function

10-9 10-7 10-5 10-3 10-1 101100

102

104

106

0 1 2 3 4 5

100

200

χ = 0χ = 0.10χ = 0.25χ = 0.50χ = 1.00

z~

c z~( )c0

FIG. 4. (Color online) Rescaled density profile of multivalentcounterions next to the randomly charged surface of a semi-infinite slab for Ξ = 50, κ = 0.3 and in the absence dielectricimage charge effects (ǫp = ǫm). Different curves correspond todifferent values of the disorder coupling parameter as shownon the graph. Inset shows the diverging behavior of the den-sity profiles at small distances from the surface in the log-logscale.

is only a function of Q = |Q|. In the above expressions,

∆s =γ −Q

γ +Q, γ2 = Q2 + κ2, (33)

which in fact describes the salt image effects.The contributions to the effective single-particle inter-

action energy, u = u0 + uim + udis, are now obtained, inrescaled units, as

βu0(z) = −2

κe−κz , (34)

βuim(z) =Ξ

2

∫ ∞

0

QdQ∆s

γe−2γz, (35)

βudis(z) = −χ

2

∫ ∞

0

QdQ(1 + ∆s)

2

γ2e−2γz, (36)

where γ = γµ. These equations can be used along withEq. (12) in order to calculate the density profile of coun-terions.As seen in Fig. 4, the rescaled density profile of coun-

terion shows a clear depletion effect in the absence of dis-order, which as noted above, is caused by the salt imageeffects. This depletion effect becomes weaker when thesurface is randomly charged as the counterions are againattracted more strongly to the surface in the presence ofsurface disorder. The interplay between salt image de-pletion and disorder attraction leads to a non-monotonicbehavior in the counterion density profile as the disor-der dominates at small separations while the salt imageeffects are most dominant at intermediate separations.At large distances from the surface, the behavior is dom-inated by salt (Debye) screening and we have c(z)/c0 → 1for z → ∞. Also, the large counterion density values inthe immediate vicinity of the surface, as seen in Fig. 4 forκ = 0.3, are suppressed when the bulk salt concentration

8

0 2

50

100

150 b =b = 40b = 20b = 5

χ = 1

~

~

~

~

8

0 2 4 60

20

40

χ = 0χ = 0.10χ = 0.25χ = 0.50χ = 1.00

z~

c z~( )c0

b = 5~

FIG. 5. (Color online) Same as Fig. 4 but for fixed rescaled

slab thickness b = 5 and different values of the disorder cou-pling parameter(main set) and fixed disorder coupling param-eters χ = 1 and different values of the rescaled slab thickness(inset) as shown on the graph.

is increased or when a finite dielectric jump is introducedat the surface (see Section IVD).

Note that the counterion density profiles again showa singular bahavior of the density at the surface evenin the presence of additional salt. This behavior is infact present at any finite values of χ (as may be dis-cerned more clearly from the log-log plot in the inset ofFig. 4) and coincides with the same algebraic divergence∼ z−χ/2 on approach to the surface as we found in thecounterion-only case in the previous Section. This is in-tuitively expected because salt screening effects diminishat separations much smaller than the screening lengthκ−1. Likewise, the surface disorder effects are more sig-nificant within the distances comparable with or smallerthan κ−1 and can extend to larger separations from thesurface as the bulk salt concentration is decreased.

The effects of a finite slab thickness, b, can be examinedby using the appropriate form of the Green’s functionin this case that can be obtained by means of standardmethods as

G(Q; z, z′) =1

2ǫ0ǫmγ×

×

[

e−γ|z−z′| +∆s

(

1− e−2Qb)

1−∆2s e

−2Qbe−γ(z+z′)

]

. (37)

Hence, the three different terms in the effective single-

particle energy follow in rescaled units as

βu0(z) = −2(1 + κb)

κ(2 + κb)e−κz, (38)

βuim(z) =Ξ

2

∫ ∞

0

QdQ∆s

(

1− e−2Qb)

γ(

1−∆2se

−2Qb)e−2γz, (39)

βudis(z) = −χ

2

∫ ∞

0

QdQ(1 + ∆s)

2(

1−∆se−2Qb

)2

γ2(

1−∆2se

−2Qb)2 e−2γz,

(40)

where b = b/µ. The finiteness of the slab thickness isexpected to be relevant mainly in the regime where thethickness is comparable with or smaller than the screen-ing length, i.e. κb . 1. As it can be seen directly fromEqs. (38) and (40), both the attraction experienced bymultivalent counterions due to the mean surface chargeand its disorder variance, as well as the repulsion dueto the salt image effects, become larger as the slab be-comes thicker or as the system becomes more stronglyinhomogeneous in terms of the salt distribution. Theoverall effect is such that the counterion density close tothe surface becomes smaller for smaller b as shown in Fig.5, inset (compare also Figs. 4 and 5). This behavior canbe understood also by noting that for thinner slabs thesalt ions on the left side of the slab also contribute to thescreening effects and, hence, further suppress the multi-valent counterion density on the right side. This howeverleaves the effects resulting from the surface charge disor-der qualitatively unchanged, especially at small distancesfrom the surface, where the singular behavior persists.

It is to be noted that the expressions for the effec-tive single-particle energy, Eqs. (38)-(40), can correctlyreproduce the counterion-only result, Eq. (26), whenthe limit κ → 0, which also gives ∆s → 0, is taken(in this case, the thickness b will be irrelevant). Thecounterion-only limit can not be recovered if we startwith the infinite-thickness expressions (34)-(36) (wherethe thickness of the slab is strictly set equal to infinity)and then take the limit of zero salt. The difference wouldbe in a factor 2 in the expression for u0, indicating thatthe two limits κ → 0 and b → ∞ do not commute. Al-ternatively, one can recover Eq. (26) from Eqs. (38)-(40)

by first taking the limit of a thin slab b → 0 and thenκ→ 0.

D. Dielectric image effects

So far we focused on the cases with no dielectric dis-continuity at the boundaries of the slab. The case ofa dielectrically inhomogeneous system with ǫp 6= ǫm canbe studied by simply replacing the definition of ∆s in theexpressions in the previous section (e.g., Eqs. (34)-(36))

9

2 4 6 8 100

5

10

15

χ = 0χ = 1.0χ = 2.0χ = 3.0

z~

c z

~

( )c0

κ = 0.3

~

(a)

2 4 6 8 100

10

20

κ = 0.25κ = 0.30κ = 0.35κ = 0.40

z

~

~

c z~( )c0

~~~

χ = 1

(b)

FIG. 6. (Color online) (a) Rescaled density profile of multi-valent counterions next to a semi-infinite, randomly chargeddielectric slab for Ξ = 50, κ = 0.3 and different values ofthe disorder coupling parameter as shown on the graph. Thedielectric jump parameter is ∆ = 0.95. (b) Same as panel(a) but for χ = 1 and different values of the rescaled saltscreening parameters as shown on the graph.

with a more general one, i.e.

∆s =ǫmγ − ǫpQ

ǫmγ + ǫpQ. (41)

In the absence of additional salt screening (κ = 0), the di-electric image charges lead to very strong repulsions fromthe surface when ǫp < ǫm (as is often the case for aqueoussolvents and macromolecular surfaces). This effect en-ters through the image interaction term βuim = Ξ∆/4z,where the dielectric mismatch is defined as

∆ =ǫm − ǫpǫm + ǫp

. (42)

The image interaction term diverges at the surface andthus implies a vanishing contact density c(z) → 0 for z →0, a behavior that is very distinct from that generatedmerely by salt-image depletion effects, as the latter cannot be described generally in terms of “point-like imagecharges” and generate much weaker repulsive forces onmultivalent ions than dielectric images.

In the most general case with both salt and dielec-tric image effects, the density profile of counterions canbe calculated by using Eqs. (12) and (34)-(36) with thedefinition in Eq. (41). The results are shown in Fig.6 for κ = 0.3 and ∆ = 0.95, corresponding to the di-electric jump at the water/hydrocarbon interface (withǫp = 2 and ǫm = 80). As can be clearly seen, bothadded salt and charge disorder effects become irrelevantin the small-distance regime, where the dielectric-imagerepulsions dominate and generate a wide depletion zonenear the surface. The counterion density thus again van-ishes at the surfaces and tends to the bulk value at largeseparations. Although the qualitative form of the den-sity profile remains the same in the absence and in thepresence of charge disorder (unlike what we found in theprevious Sections), the peak of the density profile be-comes more pronounced and shifts to smaller values ofthe distance from the surface as the disorder strength isincreased, see Fig. 6a. A similar effect is seen in Fig. 6b,where the salt screening parameter is decreased; in thiscase the maximum of the density profile remains nearlyunchanged while its height increases by almost an orderof magnitude when the salt screening parameter is de-creased by only a factor of 2. The disorder effects in thiscase clearly extend to much larger distances for smallervalues of κ.

E. Charge inversion and overcharging

The preceding results suggest that the charge inversionand/or overcharging of the surface, which are known tooccur with asymmetric Coulomb fluids (see, e.g., Refs.[65, 88, 103–106] and references therein), may be en-hanced when the surface carries a random charge compo-nent. This can be inferred from the averaged cumulativecharge, Q(z), being the sum of the average charges due tofixed and mobile charges within a finite distance z fromthe surface

Q(z) =1

σ

∫ z

0

dz′ [e0n+(z′)− e0n−(z

′) + e0qc(z′) + ρ0(z

′)] ,

(43)where n±(z) represent the averaged (DH) density ofmonovalent ions and c(z) that of the multivalent coun-terions. As shown in Ref. [88], the cumulative chargecan be written only in terms of the counterion densityby using the fact n+(r) − n−(r) ≃ −nb(βe0ψ(r)) whereψ(r) is the mean electrostatic potential generated by theexplicit charge densities, i.e.,

ψ(r) =

∫

dr′G(r, r′)[e0qc(r′) + ρ0(r

′)]. (44)

Hence, using the Green’s function expressions (32) for aninfinite slab, we have [88]

Q(z) = −e−κz +1

8χ2c (45)

×

∫ ∞

0

dz′[

sgn(z − z′)e−κ|z−z′| + e−κ(z+z′)]

c(z′),

10

5 10 15

-1

-0.5

0

0.5

χ = 0χ = 1χ = 2χ = 4

z~

Q z~( )

(a)

0 0.2 0.4 0.6 0.8 1

0

1

2

χ = 0χ = 4

κ~

χc~

charge inversion

0 0.2 0.4 0.6 0.8 10

2overcharging

(b)

FIG. 7. (Color online) (a) Cumulative charge of multivalentcounterions next to a semi-infinite, randomly charged dielec-tric slab for Ξ = 50, κ = 0.25, χc = 0.25, ∆ = 0.95 anddifferent values of the disorder coupling parameter as indi-cated on the graph. (b) “Phase diagram” showing the min-imal amount of multivalent counterion concentration, χc (inrescaled units), required to find the charge inversion (mainset) or overcharging (inset) effect as a function of the rescaledsalt screening parameter in the absence (χ = 0) and pres-ence of surface charge disorder (χ = 4). The region above thecurves show where charge inversion (main set) or overcharging(inset) is predicted to occur.

where c(z) ≡ c(z)/c0 and χc = χcµ with the definition

χ2c = 8πq2ℓBc0. (46)

As seen in Fig. 7a, the cumulative charge shows a pos-itive hump at intermediate separations from the surface,also known as the charge inversion effect. The charge-inversion degree is usually found to amount to a fractionof the total charge (i.e., the maximum value of Q(z) issmaller than unity) [88, 103–106]. In the presence ofcharge disorder, a substantially larger amount of multi-valent counterions are attracted towards the surface andalso a much larger charge inversion degree (correspondingto the height of the hump) is predicted to occur.There is a narrow region at small separations from the

surface where one can see a decrease in Q(z) as the dis-order strength, χ, is increased. For sufficiently large χ,it exhibits a dip with Q(z) < −1. In this region, the

cumulative charge has the same sign as the mean surfacecharge but with a larger magnitude and, therefore, rep-resents the so-called overcharging effect. Note that theovercharging effect can be present even in the absenceof disorder and depends strongly on the bulk concentra-tion of multivalent counterions, which enters in Eq. (45)through the rescaled parameter χc.In Fig. 7b, we show the minimal amount of bulk multi-

valent counterion concentration (represented by χc) thatis required to achieve the charge inversion and overcharg-ing effects for a wide range of rescaled salt screening pa-rameters. The region above the curves pertains to theparameter values where we find charge inversion or over-charging of the mean surface charge. As seen, for largersalt screening parameters, a larger bulk concentration ofmultivalent ions are required to achieve these effects, andfor a given salt screening parameter, a larger concentra-tion of multivalent counterions are required to cause over-charging of the surface than its charge inversion. Thepresence of surface charge disorder facilitates both theseeffects as they can occur for smaller threshold values ofχc, especially at intermediate to larger value of the saltscreening parameter.

V. CONCLUSIONS AND DISCUSSION

We have investigated the distribution of multivalentcounterions close to a dielectric slab bearing a quenched,random distribution of monopolar surface charges withset mean value and variance, both in the absence and inthe presence of an asymmetric Coulomb fluid, comprisedof a mixture of multivalent counterions in a bathing so-lution of monovalent ions. Asymmetric Coulomb fluidsare commonplace in many experimental examples suchas in the condensation of DNA by multivalent cations inthe bulk [77, 78, 80–83] or in viruses and virus-like nano-capsids [84–86]. Our analysis is done within the frame-work of the dressed multivalent-ion theory, which repro-duces the strong-coupling theory of multivalent counte-rions in the zero salt limit and takes into account thesurface-counterion as well as counterion-image correla-tions on the leading order and in the presence of a bathingsalt solution as discussed in detail elsewhere [87–89].In the case of counterions only, we show that a ran-

domly charged surface generates a singular density pro-file for multivalent counterions with an algebraically di-verging behavior at the surface; the latter is character-ized by an exponent which is determined by the disorderstrength (variance). Thus, multivalent counterions arepredicted to accumulate strongly in the immediate vicin-ity of a randomly charged surface in a way that violatesthe contact-value theorem, which describes the behav-ior of counterions at uniformly charged surfaces and pre-dicts a finite contact density [93–95]. This behavior stemsfrom the interplay between the translational entropy ofthe solution ions and the (non-thermal) configurationalentropy due to the averaging over different realizations of

11

Ξ = 50

κ = 0.2 0.25 0.3 0.4 χ g (e20/nm2)

n0 = 69mM 109mM 158mM 283mM c0 =1.1mM χc= 0.1 0.5 0.0157mM 98mM 147mM 272mM 7mM 0.25 1.0 0.0251mM 91mM 140mM 270mM 10mM 0.3 2.0 0.0436mM 75mM 125mM 250mM 18mM 0.4 4.0 0.08

TABLE I. Actual values of concentrations n0 and c0 that may correspond to a few typical values of the rescaled parametersκ and χc as shown in the table. We have chosen q = 4 (tetravalent counterions), ℓB = 0.7 nm and surface charge densityσ = 0.25 e0/nm

2 which give µ = 0.23 nm and Ξ = 50. We also show actual values of the disorder variance g corresponding toa few different values of the disorder coupling parameter χ (see the text for definitions).

the quenched disorder. Therefore, by introducing an ex-ternal (quenched) disorder component, we find that thesystem is driven towards a more ‘ordered’ state charac-terized by a diminished intrinsic thermal ‘disorder’ in thesystem. It thus seems appropriate to characterize this re-sponse of the system to an externally imposed quencheddisorder as the anti-fragile behavior [92] of counterionsin the presence of quenched charge disorder. It is to benoted that, in the presence of disorder, the system alsoattains a thermodynamically more stable state becausethe internal energy of the system drops in a way thatleads to a lowered free energy.

The singular behavior of multivalent counterions per-sists also when counterions are immersed into a bath ofa monovalent salt solution and there are no dielectricinhomogeneities in the system. In this case, the slab de-fines a salt-excluded zone and larger counterion densitiesare created close to the charged surface of the slab whenthe slab has a larger thickness. The interplay betweenthe disorder-induced attraction and the salt-image de-pletion effects lead to a non-monotonic density profile forcounterions close to the surface. The amount of multi-valent counterions accumulated near the surface is againstrongly enhanced when the surface is randomly charged.This holds also in the case of a finite jump in the dielec-tric constant even though dielectric image charges (unlikesalt images) create a finite counterion-depletion zone inthe immediate vicinity of the charged surface. The chargedisorder can thus make the overcharging and/or chargeinversion of the mean surface charge highly pronounced,depending on the values of the bulk concentration of mul-tivalent counterions and the salt screening parameter.

Our results are presented in terms of rescaled param-eters such as rescaled screening parameter and electro-static and disorder coupling parameters, which can bemapped to a wide range of values for counterion and saltbulk concentration, mean surface charge, counterion va-lency, etc. A few examples of the actual values for theselatter quantities (corresponding to the typical values ofthe rescaled parameters that were used to plot the fig-ures in the previous Sections) are shown in Table I. Notethat the typical values of the disorder coupling param-eter that we used in our study, e.g., χ ≃ 0 − 4, corre-spond to a relatively small degree of charge disorder on

the surface; e.g., for a mean surface charge density ofσ = 0.25 e0/nm

2, these values of χ give disorder vari-ances g ≃ 0 − 0.08 e20/nm

2. Assuming that the disorderoriginates from a quenched, random distribution of posi-tive and negative impurity charges, qie0, residing on thesurface with a surface density of ni, we find g = q2i e

20ni

[19]. Therefore, the above-mentioned values of g can beobtained by relatively small densities ni . 0.1/nm2 ofimpurity charges as compared with the mean number ofsurface charges (typically σ/e0 . 1/nm2) and can bethus easily realized in actual systems. Hence, we con-clude that the effects due to charge randomness, even atsuch small amounts, can be quite significant!

It is also worth mentioning that our key findings, suchas the singular behavior of the density profile of multiva-lent ions, remain valid even in the case of a net-neutralsurface, i.e., when the charged surface carries no meancharge density but only a finite charge disorder variance.The case of net-neutral surfaces has been studied recentlyin a series of works in the context of Casimir interactions[13, 17–23] and the role of an asymmetric Coulomb fluidin this case will be discussed elsewhere [107].

We should emphasize that our results are valid strictlyin the case of multivalent counterions, where strong elec-trostatic couplings allow for a single-particle, dressedmultivalent-ion description. In the opposite regimeof weak coupling, the disorder effects turn out to besmall and do not lead to qualitatively new featuresin the behavior of the system [14, 45, 46]. Thedressed multivalent-ion theory has been tested exten-sively against implicit- and explicit-ion simulations [69,87–91] and turns out to have a wide range of validityin the parameter space when the surfaces bear uniformcharge distributions. Similar simulations are still missingin the case of randomly charged surfaces with multiva-lent ions mostly because of a significantly large increasein the computational time, compared to the homogeneouscharge distribution case, which would be required in oderto produce reliable quenched disorder averages. Our re-sults however produce concrete predictions that can betested against simulations or even experiments. In gen-eral, we expect that the previously determined regimesof validity of the dressed multivalent-ion description [88]roughly hold also for the present case with disordered

12

surfaces. Also, in particular, while we expect that thepredicted boundaries of the parameter space pertainingto the onset of the charge inversion (overcharging) wouldbe relatively accurate, the single-particle approximationthat lies at the heart of the dressed multivalent-ion de-scription is not expected to be adequate in the regime ofparameters deeply within the region of charge inversion(overcharging) due to non-negligible many-body contri-butions [88]. These considerations remain to be assessedfurther in future simulations.Our model is based on a few simplifying assumptions

and, as such, neglects several other factors including sol-vent structure (see, e.g., Refs. [24, 29, 108–112] and ref-erences therein), the polarizability and internal structureof mobile ions (see, e.g., Refs. [113–118] and referencestherein), specific surface ion-adsorption effects [119, 120],etc. Also, we have assumed that the charge disorderis distributed according to a Gaussian weight and thatit is uncorrelated in space. Spatial correlations can beincluded in our formalism in a straightforward manner[20, 23] and will be considered in future works. But it isimportant to note that the precise statistical characteris-tics of charge disorder in real systems can be highly sam-ple and material dependent, involving also the method

of preparation, features that should all be considered ifthe theoretical findings are to be compared with exper-iments. Furthermore, annealed as opposed to quencheddisordered surfaces, containing mobile, random surfacecharges that are in thermal equilibrium with the rest ofthe system, also represent an interesting case [19, 20, 44–46, 48–50]. Charge regulating surfaces [51, 121–126] alsoconstitute another example that can be studied in thepresent context. All of these additional features we planto address in our upcoming publications.

ACKNOWLEDGMENTS

A.N. acknowledges partial support from the Royal So-ciety, the Royal Academy of Engineering, and the BritishAcademy (UK). H.K.M. acknowledges support from theSchool of Physics, Institute for Research in FundamentalSciences (IPM), where she stayed as a short-term visit-ing researcher during the completion of this work. R.P.acknowledges support from the ARRS through GrantsNo. P1-0055 and J1-4297. We thank M. Kanduc and E.Mahgerefteh for useful comments and discussions.

[1] W.R. Clarke et al., Nature Physics 4, 55 (2008).[2] P.W. Anderson, Phys. Rev. 109, 1492 (1958).[3] A. Lagendijk, B. van Tiggelen, and D.S. Wiersma,

Physics Today 62, 24 (2009).[4] N. Mermin, Rev. Mod. Phys. 51, 591 (1979).[5] H. Kleinert, Gauge Fields in Condensed Matter, Vol.

II, “Stresses and defects” (World Scientific, Singapore,1989).

[6] K. Binder, A.P. Young, Rev. Mod. Phys. 58, 801 (1986).[7] M. Mezard, G. Parisi, M.A. Virasoro, Spin glass theory

and beyond (World Scientific, Singapore, 1987).[8] A. Weinrib and B. Halperin, Phys. Rev. B 27, 413

(1983).[9] P. Verma, P. Maire, P. Novak, Electrochimica Acta 55,

6332 (2010).[10] B. Hribar-Lee, M. Luksic and V. Vlachy, Annu. Rep.

Prog. Chem., Sect. C, 107, 14 (2011).[11] B. Scrosati, J. Garche, Journal of Power Sources 195,

2419 (2010).[12] M. Eikerling, A.A. Kornyshev, and A.R. Kucernak,

Physics Today 59, 38 (2006).[13] D.S. Dean, J. Dobnikar, A. Naji, R. Podgornik (Eds.),

Electrostatics of Soft and Disordered Matter (Pan Stan-ford Publishing, Singapore, 2014).

[14] A. Naji, R. Podgornik, Phys. Rev. E 72, 041402 (2005).[15] R. Podgornik, A. Naji, Europhys. Lett. 74, 712 (2006).[16] Y.Sh. Mamasakhlisov, A. Naji, R. Podgornik, J. Stat.

Phys. 133, 659 (2008).[17] D.S. Dean, R.R. Horgan, A. Naji, R. Podgornik, Phys.

Rev. A 79, 040101(R) (2009).[18] D.S. Dean, R.R. Horgan, A. Naji, R. Podgornik, Phys.

Rev. E 81, 051117 (2010).[19] A. Naji, D.S. Dean, J. Sarabadani, R. Horgan, R. Pod-

gornik, Phys. Rev. Lett. 104, 060601 (2010).[20] J. Sarabadani, A. Naji, D.S. Dean, R.R. Horgan and R.

Podgornik, J. Chem. Phys. 133, 174702 (2010).[21] D.S. Dean, A. Naji and R. Podgornik, Phys. Rev. E 83,

011102 (2011).[22] A. Naji, J. Sarabadani, D.S. Dean and R. Podgornik,

Eur. Phys. J. E 35, 24 (2012).[23] V. Rezvani, J. Sarabadani, A. Naji and R. Podgornik,

J. Chem. Phys. 137, 114704 (2012).[24] J. Israelachvili, Intermolecular and Surface Forces (Aca-

demic Press, London, 1991).[25] E.J. Verwey, J.T.G. Overbeek, Theory of the Stability

of Lyophobic Colloids (Elsevier, Amsterdam, 1948).[26] S.A. Safran, Statistical Thermodynamics of Surfaces,

Interfaces, and Membranes (Addison-Wesley PublishingCo., Massachusetts, 1994).

[27] R.J. Hunter, Foundations of Colloidal Science, 2nd Ed.(Oxford University Press, USA, 2001).

[28] W.C.K. Poon and D. Andelman (Eds.), Soft CondensedMatter Physics in Molecular and Cell Biology (Taylor& Francis, New York, London, 2006).

[29] C. Holm, P. Kekicheff, R. Podgornik (Eds.), Electro-static Effects in Soft Matter and Biophysics (KluwerAcademic, Dordrecht, 2001).

[30] S. Manne, J.P. Cleveland, H.E. Gaub, G.D. Stucky, P.K.Hansma, Langmuir 10, 4409 (1994).

[31] S. Manne, H.E. Gaub, Science 270, 1480 (1995).[32] E.E. Meyer, Q. Lin, T. Hassenkam, E. Oroudjev, J.N.

Israelachvili, Proc. Natl. Acad. Sci. USA 102, 6839(2005).

[33] E.E. Meyer, K.J. Rosenberg, J. Israelachvili, Proc. Natl.Acad. Sci. USA 103, 15739 (2006).

[34] S. Perkin, N. Kampf, J. Klein, Phys. Rev. Lett. 96,

13

038301 (2006).[35] S. Perkin, N. Kampf, J. Klein, J. Phys. Chem. B 109,

3832 (2005).[36] G. Silbert, D. Ben-Yaakov, Y. Dror, S. Perkin, N.

Kampf and J. Klein, Phys. Rev. Lett. 109, 168305(2012).

[37] D. Ben-Yaakov, D. Andelman and H. Diamant, Phys.Rev. E 87, 022402 (2013).

[38] S. Panyukov and Y. Rabin, Phys. Rev. E 56, 7055(1997).

[39] D.B. Lukatsky, K.B. Zeldovich and E.I. Shakhnovich,Phys. Rev. Lett. 97 178101 (2006)

[40] D.B. Lukatsky and E.I. Shakhnovich, Phys. Rev. E 77,020901(R) (2008).

[41] Y. Kantor, H. Li and M. Kardar, Phys. Rev. Lett. 69,61(1992).

[42] Y. Kantor, M. Kardar, Europhys. Lett. 28, 169 (1994).[43] Y. Kantor, M. Kardar, D. Ertas, Physica A 249, 301

(1998).[44] I. Borukhov, D. Andelman, H. Orland, Eur. Phys. J. B

5, 869 (1998).[45] C. Fleck, R.R. Netz, Europhys. Lett. 70, 341 (2005).[46] C. Fleck, R.R. Netz, Eur. J. Phys. E 22, 261 (2007).[47] Z.H. Liu, N.M.D. Brown, A. McKinley, J. Phys.: Con-

dens. Matter 9, 59 (1997).[48] A. Naydenov, P.A. Pincus, and S.A. Safran, Langmuir

23, 12016 (2007).[49] R. Brewster, P.A. Pincus, and S.A. Safran, Phys. Rev.

Lett. 101, 128101 (2008).[50] Y.S. Jho, R. Brewster, S.A. Safran, and P.A. Pincus,

Langmuir 27, 4439 (2011).[51] Y.S. Velichko and M. Olvera de la Cruz, J. Chem. Phys.

124, 214705 (2006).[52] W.J. Kim, M. Brown-Hayes, D.A.R. Dalvit, J.H.

Brownell and R. Onofrio, Phys. Rev. A 78, 020101(R)(2008); 79, 026102 (2009); R.S. Decca, E. Fischbach,G.L. Klimchitskaya, D.E. Krause, D. Lopez, U. Mo-hideen and V.M. Mostepanenko, Phys. Rev. A 79,026101 (2009); S. de Man, K. Heeck and D. Iannuzzi,Phys. Rev. A 79, 024102 (2009).

[53] W.J. Kim, A.O. Sushkov, D.A.R. Dalvit and S.K. Lam-oreaux, Phys. Rev. Lett. 103, 060401 (2009).

[54] W.J. Kim, A.O. Sushkov, D.A.R. Dalvit and S.K. Lam-oreaux, Phys. Rev. A 81, 022505 (2010).

[55] W.J. Kim and U.D. Schwarz, J. Vac. Sci. Technol. B28, C4A1 (2010).

[56] D. Garcia-Sanchez, K.Y. Fong, H. Bhaskaran, S. Lam-oreaux and H.X. Tang, Phys. Rev. Lett. 109, 027202(2012).

[57] L.F. Zagonel, N. Barrett, O. Renault, A. Bailly, M.Baurer, M. Hoffmann, S.-J. Shih and D. Cockayne, Surf.Interface Anal. 40, 1709 (2008).

[58] C.C. Speake and C. Trenkel, Phys. Rev. Lett. 90,160403 (2003).

[59] H.T. Baytekin, A.Z. Patashinski, M. Branicki, B.Baytekin, S. Soh and B.A. Grzybowski, Science 333,308 (2011).

[60] S. F. Edwards, A. Lenard, J. Math. Phys. 3, 778 (1962).[61] R. Podgornik, B. Zeks, J. Chem. Soc., Faraday Trans.

II 84, 611 (1988).[62] R. Podgornik, J. Phys. A: Math. Gen. 23, 275 (1990).[63] R.R. Netz, H. Orland, Eur. Phys. J. E 1, 203 (2000).[64] R. French et al., Rev. Mod. Phys. 82, 1887 (2010).[65] A. Yu. Grosberg, T. T. Nguyen, B. I. Shklovskii, Rev.

Mod. Phys. 74, 329 (2002).[66] Y. Levin, Rep. Prog. Phys. 65, 1577 (2002).[67] H. Boroudjerdi, Y.W. Kim, A. Naji, R.R. Netz, X.

Schlagberger and A. Serr, Phys. Rep. 416, 129 (2005).[68] A. Naji, S. Jungblut, A.G. Moreira, R.R. Netz, Physica

A 352, 131 (2005).[69] A. Naji, M. Kanduc, J. Forsman and R. Podgornik, J.

Chem. Phys. 139, 150901 (2013).[70] M. Kanduc, M. Trulsson, A. Naji, Y. Burak, J. Fors-

man, and R. Podgornik, Phys. Rev. E 78, 061105(2008).

[71] R.R. Netz, Eur. Phys. J. E 5, 557 (2001).[72] A.G. Moreira, R.R. Netz, Phys. Rev. Lett 87, 078301

(2001).[73] A.G. Moreira, R.R. Netz, Eur. Phys. J. E 8, 33 (2002).[74] T.E. Angelini, H. Liang, W. Wriggers, G.C.L. Wong,

Proc. Natl. Acad. Sci. USA 100, 8634 (2003).[75] D.C. Rau, V. A. Parsegian, Biophys. J. 61, 246 (1992).[76] D.C. Rau, V. A. Parsegian, Biophys. J. 61, 260 (1992).[77] V.A. Bloomfield, Curr. Opin. Struct. Biol. 6, 334 (1996).[78] J. Pelta, D. Durand, J. Doucet, and F. Livolant, Bio-

phys. J. 71, 48 (1996).[79] J. Pelta, F. Livolant, and J.-L. Sikorav, J. Biol. Chem.

271, 5656 (1996).[80] K. Yoshikawa, Adv. Drug Deliv. Rev. 52, 235 (2001).[81] M. Takahashi, K. Yoshikawa, V.V. Vasilevskaya, A.R:.

Khokhlov, J. Phys. Chem. B 101, 9396 (1997).[82] G.E. Plum and V. A. Bloomfield, Biopolymers 27, 1045

(1988).[83] E. Raspaud, I. Chaperon, A. Leforestier, and F.

Livolant. Biophys. J. 77, 1547 (1999).[84] H.S. Savithri, S.K. Munshi, S. Suryanarayana, S. Di-

vakar and M.R.N. Murthy, J. Gen. Virol. 68, 1533(1987).

[85] M. de Frutos, S. Brasiles, P. Tavares and E. Raspaud,Eur. Phys. J. E 17, 429 (2005).

[86] A. Leforestier, A. Siber, F. Livolant and R. Podgornik,Biophys. J. 100, 2209 (2011).

[87] M. Kanduc, A. Naji, J. Forsman and R. Podgornik, J.Chem. Phys. 132, 124701 (2010).

[88] M. Kanduc, A. Naji, J. Forsman and R. Podgornik,Phys. Rev. E 84, 011502 (2011).

[89] M. Kanduc, A. Naji, J. Forsman and R. Podgornik, J.Chem. Phys. 137, 174704 (2012).

[90] L. Javidpour, A. Losdorfer Bozic, A. Naji and R. Pod-gornik, J. Chem. Phys. 139, 154709 (2013).

[91] L. Javidpour, A. Losdorfer Bozic, A. Naji and R. Pod-gornik, Soft Matter 9, 11357 (2013).

[92] N. N. Taleb, Antifragile: Things That Gain from Dis-order (Random House, 2012).

[93] D. Henderson, L. Blum, J. Chem. Phys. 75, 2025 (1981).[94] S.L. Carnie, D.Y.C. Chan, J. Chem. Phys. 74, 1293

(1981).[95] H. Wennerstrom, B. Jonsson, P. Linse, J. Chem. Phys.

76, 4665 (1982).[96] D.S. Dean, R.R. Horgan, Phys. Rev. E 68, 061106

(2003).[97] D.S. Dean, R.R. Horgan, D. Sentenac, J. Stat. Phys.

90, 899 (1998).[98] D.S. Dean, R.R. Horgan, A. Naji, R. Podgornik, J.

Chem. Phys. 130, 094504 (2009).[99] A.G. Moreira, R.R. Netz, Europhys. Lett. 57, 911

(2002).[100] D.B. Lukatsky, S.A. Safran, A.W.C. Lau and P.A. Pin-

14

cus, Europhys. Lett. 58, 785 (2002).[101] M.L. Henle, C.D. Santangelo, D.M. Patel and P.A. Pin-

cus, Europhys. Lett. 66, 284 (2004).[102] W. Pezeshkian, N. Nikoofard, D. Norouzi, F.

Mohammad-Rafiee and H. Fazli, Phys. Rev. E 85,061925 (2012).

[103] O. Lenz and C. Holm, Eur. Phys. J. E 26, 191 (2008).[104] G. I. Guerrero-Garcia, E. Gonzalez-Tovar andM. Olvera

de la Cruz, Soft Matter 6, 2056 (2010).

[105] F. Jimenez-Angeles and M. Lozada-Cassou, J. Phys.Chem. B 108, 7286 (2004).

[106] G.I. Guerrero-Garcia, E. Gonzalez-Tovar, M. Chavez-Paez and M. Lozada-Cassou, J. Chem. Phys. 132,054903 (2010).

[107] M. Ghodrat, A. Naji, H. Komaie-Moghaddam and R.Podgornik, to be submitted (2014).

[108] D. Ben-Yaakov, D. Andelman, D. Harries, R. Pod-gornik, J. Phys.: Condens. Matter 21 424106 (2009).

[109] D. Ben-Yaakov, D. Andelman, and R. Podgornik, J.Chem. Phys. 134, 074705 (2011).

[110] D. Ben-Yaakov, D. Andelman, R. Podgornik, and D.Harries, Curr. Opin. Colloid Interface Sci. 16, 542(2011).

[111] Y. Burak, D. Andelman, Phys. Rev. E 62, 5296 (2000).[112] Y. Burak, D. Andelman, J. Chem. Phys. 114, 3271

(2001).

[113] J.J. Bikerman, Phil. Mag. 33, 384 (1942).[114] M. Kanduc, A. Naji, Y. S. Jho, P.A. Pincus, R. Pod-

gornik, J. Phys.: Condens. Matter 21, 424103 (2009).[115] V. Demery, D.S. Dean, and R. Podgornik, J. Chem.

Phys. 137, 174903 (2012).[116] M.M. Hatlo, R. v. Roij, and L. Lue, Europhys. Lett. 97,

28010 (2012).[117] R.R. Netz, Eur. Phys. J. E 5, 189 (2001).[118] D. Horinek and R.R. Netz, Phys. Rev. Lett. 99, 226104

(2007).[119] O. Spalla, L. Belloni, Phys. Rev. Lett. 74, 2515 (1995).[120] J. Forsman, Langmuir 22, 2975 (2006).[121] B.W. Ninham, V.A. Parsegian, J. Theor. Biol. 31, 405

(1971).[122] D. Chan, J.W. Perram, L.R. White, T.H. Healy, J.

Chem. Soc. Faraday Trans. I 71, 1046 (1975).[123] H.H. von Grunberg, J. Colloid Interface Sci. 219, 339

(1999).[124] G.S. Longo, M. Olvera de la Cruz, I. Szleifer, Soft Mat-

ter 8 1344 (2012).[125] G.S. Longo, M. Olvera de la Cruz, I. Szleifer, ACS Nano

7, 2693 (2013).[126] N. Adzic and R. Podgornik, “Field theoretic descrip-

tion of charge-regulation interaction”, Eur. Phys. J. E(2014), in print.