Counterion condensation and fluctuation-induced attraction

PHYSICAL REVIEW E 66, 041501 2002Counterion condensation and fluctuation-induced attractionA. W. C. Lau 1 and P. Pincus 21 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 191042 Materials Research Laboratory, University of California, Santa Barbara, California 93106-9530Received 24 March 2002; published 7 October 2002We consider an overall neutral system consisting of two similarly charged plates and their oppositelycharged counterions and analyze the electrostatic interaction between the two surfaces beyond the mean-fieldPoisson-Boltzmann approximation. Our physical picture is based on the fluctuation-driven counterion condensationmodel, in which a fraction of the counterions is allowed to ‘‘condense’’ onto the charged plates. Inaddition, an expression for the pressure is derived, which includes fluctuation contributions of the wholesystem. We find that for sufficiently high surface charges, the distance at which the attraction, arising fromcharge fluctuations, starts to dominate can be large compared to the Gouy-Chapmann length. We also demonstratethat depending on the valency, the system may exhibit a first-order binding transition at short distances.DOI: 10.1103/PhysRevE.66.041501I. INTRODUCTIONCorrelation effects may play an important role in controllingthe structure and phase behavior of highly charged macroionsin solutions 1. The macroions may be charged membranes,stiff polyelectrolytes such as DNA, or chargedcolloidal particles. Recently, these effects have attracted agreat deal of attention, since they may drastically alter thestandard mean-field Poisson-Boltzmann PB picture 2–6.For example, one surprising phenomenon is the attractionbetween two highly charged macroions, as observed in experiments7–9 and in simulations 10–12. This attractionis not contained in the mean-field PB theory, even for anidealized system of two charged planar surfaces. Indeed, ithas been proven that PB theory predicts only repulsion betweenlike-charged macroions 13.Very recently, another interesting effect that is not capturedwithin the PB theory is predicted, namely, thefluctuation-driven counterion condensation 14. For a systemconsisting of a single charged surface and its oppositelycharged counterions, Netz and Orland 5 showed that asimple perturbative expansion about the mean-field PB solutionbreaks down for sufficiently high surface charge. Thus,in this limit, fluctuation and correlation corrections can becomeso large that the solution to the PB equation is nolonger a good approximation. To circumvent this difficulty, atwo-fluid model was proposed in Ref. 14, in which thecounterions are divided into a free and a condensed fraction.The free counterions have the usual three-dimensional 3Dmean-field spatial distribution, while the condensed counterionsare confined to move only on the charged surface andthus effectively reduce its surface charge density. The numberof condensed counterions is determined self-consistently,by minimizing the total free energy which includes fluctuationcontributions. This theory predicts that if surface chargedensity of the plate is sufficiently high, a large fraction ofcounterions is ‘‘condensed’’ via a phase transition, similar tothe liquid-gas transition with a line of first-order phase transitionsterminating at the critical point. Furthermore, the valenceof the counterions plays a crucial role in determiningthe nature of the condensation transition. The physicalmechanism leading to this counterion condensation is thePACS numbers: 82.70.y, 61.20.Qgadditional binding arising from 2D charge fluctuations,which dominate the system at high surface charge. In thispaper, we extend this condensation picture to a system of twocharged surfaces with their neutralizing counterions, and tostudy the electrostatic interaction between them.Previous theoretical approaches to the problem of the attractionin charged surfaces include both numerical and analyticalmethods that go beyond the mean-field PB theory.Gulbrand et al. 10 provided the first convincing demonstrationfor the attraction between highly charged walls usingMonte Carlo simulations. In particular, they showed that fordivalent counterions, the pressure between charged walls becomesnegative for distances less than 10 Å; hence, the existenceof short-ranged attraction. Subsequently, there havebeen a number of numerical studies based on the hypernettedchain integral equation 15 and the local density-functionaltheory 16, as well as analytic perturbative expansionaround the PB solution 17 that demonstrates attraction.More recently, motivated by the problems of DNA condensationand membrane adhesion, two distinct approacheshave been proposed to account for the attraction arising fromcorrelations 2–4. The first approach is based on ‘‘structural’’correlations first proposed by Rouzina and Bloomfield2; the attraction comes from the ground state configurationof the ‘‘condensed’’ counterions. This theory predicts astrong short-ranged attraction, with the characteristic lengthset by the lattice constant, typically of the order of few angstroms.In the other approach, based on charge fluctuations,the counterion fluctuations are approximated by the 2DDebye-Hückel theory, which predicts a long-ranged attractionwhich scales with the interplanar distance as d 3 . Note,however, that the mean-field PB repulsion which scales liked 2 always dominates the attraction for large distances, andthus, the range of the attraction in this picture is still short,typically of the order of 10 Å 18,19. Despite the fact thatsome conceptual issues have been resolved concerning thecrossover of the attractions from long ranged to short ranged20,21, there remain some interesting problems to be understood.In particular, some experimental observations in planarsurfaces 8 and in charged colloidal suspensions 9 aswell as computer simulations 12 provide evidence for along-ranged attraction, typically of the order of microns,1063-651X/2002/664/04150114/$20.0066 041501-1©2002 The American Physical Society

A. W. C. LAU AND P. PINCUS PHYSICAL REVIEW E 66, 041501 2002whereas the two mechanisms mentioned above give onlyshort-ranged attraction. In this paper, we show that thecharge-fluctuation approach, together with the counterioncondensation mechanism 14, can induce long-ranged attractionsfor sufficiently high surface charge. We note thatother mechanisms based on hydrodynamic interactions 22,depletion effects 23, and an exact calculation for the 2Dplasma model 24 have been proposed recently to accountfor the long-ranged attractions.In particular, we study the interaction between twocharged surfaces separated by a distance d, with counterionsdistributed both inside and outside of the gap. This boundarycondition, as opposed to all of the counterions confined betweenthe gap, is more appropriate in general, since systemsare not closed and often the counterions are in equilibriumwith a ‘‘bath’’ in surface forces experiments. In the spirit ofthe ‘‘two-fluid’’ model proposed in Ref. 14, we divide thecounterions into a ‘‘condensed’’ and a ‘‘free’’ fraction. Thecondensed counterions are allowed to move only on thecharged surfaces, while the free counterions distribute in thespace inside and outside the gap. The surface density of thecondensed counterion n c on each plate is determined byminimizing the total free energy, which includes fluctuationcontributions. Furthermore, an expression for the fluctuationpressure is derived, which includes fluctuation contributionsfrom the condensed and ‘‘free’’ counterions, and their couplings.We find that the counterion condensation can occureither by increasing surface charge density at a fixed distanceor by decreasing the separation between plates. For low surfacecharge, the counterion condensation proceeds continuouslyas a function of distance with the fraction of counterioncondensed being small but finite, and the total pressure ofthe system remains repulsive.For higher surface charges, the qualitative behavior of thecounterion condensation transition depends critically on thevalence Z of the counterions. For Z2, the counterion condensationproceeds continuously as a function of distance.However, for Z2, the behavior of the system is qualitativelydifferent, similar to an isolated charged plate 14. Inthis case, the counterion condensation occurs via a first-orderphase transition as a function of distance. Remarkably, wefind that for trivalent (Z3) counterions, there is a widerange in the surface density, in which the first-order counterioncondensation spontaneously takes the system from a repulsiveregime to an attractive regime at short distances, resultingin a first-order binding transition. For high surfacecharge, counterion condensation again proceeds continuouslyeven for Z2, but with a significant number of condensedcounterions. Thus, in this regime, the mean-field repulsion issubstantially reduced and the long-ranged charge-fluctuationattraction dominates the system even for large distances.Note, however, that the mean-field repulsion will eventuallydominate as d→. We emphasize that all these features, inparticular the special role of the valence, deviate significantlyfrom the PB mean-field predictions.This paper is organized as follows. In Sec. II, we brieflyrecapitulate qualitatively the mechanism which drives thecounterion condensation. In Sec. III, we present in detail thetwo-fluid model and derive a general expression for the totalfree energy of the counterions. In Sec. IV, we apply thisformalism to study the interaction of similarly charged surfaces.A detailed discussion of our results is presented inSec. V.II. COUNTERION CONDENSATION:QUALITATIVE ARGUMENTIn this section, we recapitulate the essential physics of thecondensation transition presented in Ref. 14. Recall that fora single plate of charge density (x)en 0 (z) immersed inan aqueous solution of dielectric constant , containing oppositelycharged Ze pointlike counterions of valence Z onboth sides of the plate, PB theory predicts that the counteriondistribution 251 0 z2Z 2 l B z 2decays to zero algebraically with a characteristic length 1/(l B Zn 0 ), where l B e 2 /k B T7 Å is the Bjerrumlength in water at room temperature, k B is the Boltzmannconstant, and T is the temperature. This Gouy-Chapmanlength defines a sheath near the charged surface withinwhich most of the counterions are confined. Typically, it ison the order of few angstroms for a moderate charge densityof n 0 1/100 Å 2 . Note that since scales inversely withn 0 and linearly with T, at sufficiently high density or lowtemperature, the counterion distribution is essentially two dimensional.In fact, in the limit T→0, we havelim 0 zdz lim 2 n 0 T→0 T→02Z n 0Z ,where is an arbitrarily small but fixed positive value of z,i.e., the counterion profile 0 (z) reduces to a surface densitycoating the charged plane with a density of n 0 /Z. Therefore,according to PB theory, all of the counterions collapse ontothe charged plane at zero temperature. However, for highlycharged surfaces Z 2 l B , the fluctuation corrections becomeso large that the solution to the PB equation is nolonger valid 5. To capture this regime in the spirit of the‘‘two-fluid’’ model 14, we divide the counterions into a‘‘free’’ and a condensate fraction. The ‘‘free’’ counterionshave the usual PB 3D spatial distribution, while the ‘‘condensed’’counterions are confined to move only on thecharged plane, as shown in Fig. 1. The free energy per unitarea for the condensed counterions with an average surfacedensity n c can be written as 26 f 2D n c n c lnn c a 2 1 1 2 d2 q2 2 ln 1 1q D 1q D ,where 1 k B T, a is the molecular size of the counterions,and D 1/(2Z 2 n c ) is the 2D screening length. The firstterm in Eq. 3 is the entropy and the second term arises fromthe 2D fluctuations. Note that the latter term is logarithmi-123041501-2

COUNTERION CONDENSATION AND FLUCTUATION- ...FIG. 1. The geometry of the problem.cally divergent, which may be regularized by a microscopiccutoff a, yielding f 2D (n c )1/(8 D 2 )ln(2 D /a).In addition, the condensate partially neutralizes the chargedplane, effectively reducing its surface charge density fromen 0 to en R en 0 Zen c . Thus, motivated by PB theory, thefree counterions can be modeled as an ideal gas confined to aslab of thickness R 1/(l B Zn R ) with a 3D concentrationof cn R /(Z R ). The fluctuation free energy in this casemay be estimated using the 3D Debye-Hückel theory: f s 3 /(12) 27, where s 2 4Z 2 l B c is the inversesquare of the 3D screening length. The free energy per unitarea of the free counterions is then approximately given by f 3D n c c R lnc a 3 1 s 312 R .All the qualitative results, including the nature of the condensationtransition, follow straightforwardly from the analysisof the total free energy: f (n c ) f 2D (n c ) f 3D (n c ); minimizingf (n c ) to find the fraction of condensed counterions,n c , we obtainln1 g 4 2 3g g1g ln1, 5where the three dimensionless parameters: the order parameterZn c /n 0 , the coupling constant gZ 2 l B /, where is the bare Gouy-Chapmann length, and the reduced temperaturea/Z 2 l B , completely determine the equilibriumstate of the system. It is easy to derive the asymptotic solutionsof the last equation corresponding to the free, 1 1,and condensed, 2 1, state of the counterions: 1g exp1 4 3 g, and 2 1 exp(1) 1/2 (g/) (g1)/2 ,respectively. For weak couplings g1, 1 is the only consistentsolution. Thus, there are almost no condensed counterions1. This is not surprising since PB theory is a4PHYSICAL REVIEW E 66, 041501 2002weak-coupling theory which becomes exact in the limit T→. However, for high surface charge g1, where correlationeffects becomes important, the behavior of dependscrucially on . In particular, for c 0.038, 1 and 2 areboth consistent solutions corresponding to the two minima off, and thus a first-order transition takes place when f ( 1 ) f ( 2 ), in which a large fraction of conterions is condensed.This occurs at a particular value of the bare surfacecharge density such that gg 0 (). For an estimate, we take0.02 divalent counterions at room temperature and obtaing 0 1.7, corresponding to c e/10 nm 2 . However,for c the behavior of is completely different; in thisregime, there is no phase transition and the condensationoccurs continuously. Thus, the condensation transition issimilar to the liquid-gas transition, which has a line of firstordertransitions terminating at the critical point where asecond-order transition occurs. If one takes l B 10 Å, e.g.,room temperature, and a1 Å, it follows from the definitionof that there is a critical value of counterion valence Z ca/(l B c )1.62, below which no first-order condensationtransition is possible. Therefore, divalent counterions behavequalitatively differently from monovalent counterions atroom temperature.Clearly, this condensation picture may also be crucial tounderstanding the attraction between two similarly chargedplates, separated by a distance d. Recall that the total pressureof this system comprises the mean-field repulsion andthe correlated fluctuation attraction 4. The repulsion comessolely from the ideal gas entropy and it is proportional tothe concentration at the midplane: 0 (d)k B T 0 (0)8k B T/( B 2 R ) for d R 28, where B 4Z 2 l B . Thefluctuation-induced attraction is (d) 0 k B T/d 3 for d D , where 0 0.048 4. Clearly, when a large fractionof the counterions is ‘‘condensed,’’ the mean-field repulsionis greatly reduced. Therefore, the attraction arising from correlatedfluctuations of the ‘‘condensed’’ counterions canovercome the mean-field repulsion even for large distances.Using the estimates in the last paragraph above, we find thatfor divalent counterions and surface charge density of aboutone unit charge per 7 nm 2 , the total pressure becomesattractive at about d10 nm; hence a long-ranged attraction.Of course, this estimate should be supplemented by a moreprecise calculation for the system of two charged plates,which is done below.III. COUNTERION FREE ENERGY IN THE‘‘TWO-FLUID’’ MODELConsider an overall neutral system consisting of counterionsand two charged surfaces separated by a distance dimmersed in an aqueous solution. The surface charged densityon each plate is 0 en 0 . We model the aqueous solutionwith a uniform dielectric constant . This simplificationallows us to study fluctuation and correlation effects analytically.In the spirit of the ‘‘two-fluid’’ model proposed in Ref.14, we divide the counterions into a ‘‘condensed’’ and a041501-3

COUNTERION CONDENSATION AND FLUCTUATION- ...d d2 q2 2qe 2qd 1q D 2 1 .30This expression is exactly the pressure derived in Ref. 4,arising from 2D fluctuations of the counterions. It scales like(d)1/d 3 for large distances.The second term in Eq. 29 may be interpreted as thecoupling between the counterions near the surfaces and thosein the bulk. This can be seen by considering the asymptoticbehavior of the pressure for large d in the no condensatelimit 0, i.e., the fluctuation corrections to the PB pressure.In this case, in addition to the usual d 3 scaling lawarising from counterion fluctuations near the surfaces, thesecond term in Eq. 29 contributes a term, which scales asd 3 ln(d/) in the large d limit. Therefore, the pressured 1 d 3 1 d 3 lnd/31contains a logarithmic term, which dominates the d 3 termfor large distances. This term has been obtained by severalauthors previously 17,18 and, in particular, Ref. 18 showsthat this term arises physically from the coupling betweencounterions near the surfaces and those in the bulk. Therefore,Eq. 29 recovers the PB limit 0 and the 2D limit1 as special cases. Although the fluctuation corrections tothe PB (0) pressure have been considered previously, westress that Eq. 29 is a generalization which allows for counterioncondensation and may apply to other physical situations,such as ions absorption.Combining with the mean-field pressure, we obtain thetotal pressure tot d 22 B1 B8 R1b 2 I 2 I 3 2 d R1b 2 d2 q qM 2 q2 2 1M 2 q .32The behavior of the total pressure depends on the couplingconstant gZ 2 l B / and the fraction of condensed counterionsZn c /n 0 . For g1 and 1, the fluctuation correctionsare small and the total pressure tot (d) is controlled bythe mean-field repulsion. However, for 1 the mean-fieldrepulsion is greatly reduced and the fluctuation attraction canovercome the repulsion at finite distances. Furthermore, forg1 the short distance behavior is highly sensitive to .Even a very small number of condensed counterions wouldturn the total pressure, otherwise repulsive for 0, intoattractive for short distances. For g1, the fluctuation attractionbecomes dominant at short distances even whenthere is no condensate, and the effect of finite is to push theattractive region out to a larger length scale. Hence, if thereis sufficient number of condensed counterions, the pressureis attractive even for large distances. Our next task is todetermine the fraction of condensed counterions as determinedby the minimum of the total free energy.B. Equilibrium propertiesThe equilibrium state of the system is determined byminimizing the total free energy with respect to the orderparameter . Therefore, we need to evaluate the derivative ofthe total free energy Eq. 16 with respect to . Let us firstconsider the mean-field contribution. Explicitly differentiatingEq. 24, we obtain f 0 2n 0Z ln2/a 2n 0Z 2n 0Z ln 1 2 2n 0Z ln1b2 , 33where 4Z/( B n 0 ) is the ‘‘bare’’ Gouy-Chapman length.To obtain the fluctuation contributions, we again make use ofthe exact identity: ln det Xˆ Tr Xˆ 1 Xˆ to evaluate the derivativeof the fluctuation free energy in Eq. 25, F 12 B d 3 xG 3D x,x 2 Dzd/2 2 e (z). 34This expression can be explicitly evaluated using similartechniques outlined in the Appendix and the result is givenby1 F 41 1A 22 I 1 1b22I 3 2b2 I 2 I 3 2 d R1b 2 ,35where I 2 and I 3 are given in Eqs. A30 and A31, respectively,and I 1 is defined byI 1 d/ R d2 qPHYSICAL REVIEW E 66, 041501 20024 RGd/21Lq, 362 2 2qwhere L(q) and G(d/2) are defined by Eqs. A18 andA19, respectively, in the Appendix. Note that I 1 d/ R islogarithmically divergent see Appendix, Eq. A34, asinthe case for 2D Debye-Huckel theory, which may be regularizedby a microscopic cutoff, chosen to be the size of thecounterion a. Finally, using Eqs. 33 and 35, the root ofthe free energy F()/0 can be determined numerically.For example, the case of an isolated charged plate can beobtained by taking the limit d→ in Eqs. 33 and 35,which leads to the following transcendental equation:1lng/2ln 12xgcxdx 121x0 1x1x1x 0,37041501-7

A. W. C. LAU AND P. PINCUS PHYSICAL REVIEW E 66, 041501 2002FIG. 2. The fraction of condensed counterions Zn c /n 0 as afunction of gZ 2 l B / for different values of a/Z 2 l B . At lowsurface charge g1, the counterion distribution is well describedby PB theory since 1. However, at high surface charge, correlationeffects lead to a large fraction of condensed counterions. Thecondensation is first order for c and continuous for c . Thecritical point is at c 0.017, g c 1.23, and c 0.43.where x c 2/(1)g is the microscopic cutoff. Theanalysis of this equation gives all the features mentioned inSec. II. As a consistency check, it can be verified that in thelimit d→0, Eq. 35 gives the fluctuation free energy for anisolated charged surface but with twice of the surface chargedensity 2en 0 .V. RESULTS AND DISCUSSIONLet us first discuss the behavior of the order parameter at a fixed separation d between the charged surfaces, as summarizedin Fig. 2. The behavior of as a function g at a finitedistance d is qualitatively identical to the case of infiniteseparation. For weak couplings g1, there is a small butfinite number of condensed counterions but the total pressureremains repulsive. For sufficiently high g1, the condensationproceeds continuously for c and via a first-orderphase transition for c at a particular value of the couplingconstant g 0 (d,). We note that in this regime, thenumber of counterion condensation becomes significant.This implies that the mean-field repulsion is drastically reducedand the correlated attractions can overcome the repulsionat a finite distance. For 0.02, roughly correspondingto divalent counterions at room temperature, we find that theonset of the attraction occurs at g1.6 or surface charge ofabout one charge per 10 nm 2 at a distance d1.540 Å.These numbers are order of magnitude consistent with computersimulations 10.Next, we discuss counterion condensation and the totalpressure of system as a function of distance. Note that thisscenario is more physically relevant, since surface force experimentsusually vary the distance between charged surfacesrather than changing their surface charge densities. Forlow surface charges g1, as shown in Fig. 3, the counterioncondensation is continuous as a function of separation andFIG. 3. The pressure profile for monovalent (0.1) and divalent(0.02) counterions in the case of low surface charges g.the fraction of condensed counterions remains small but finite.We note that generally increases as the distance of twosurfaces decreases, but remains fairly constant up to . Thisis not surprising since there is an entropy loss of the freecounterions due to confinement. However, the pressure remainsrepulsive and shows little difference from the PB pressureprofile, as expected.For sufficiently high coupling g1, we have several interestingregimes depending on the reduced temperature see Fig. 4. For c , the counterions condense continuouslyas the separation d decreases and the pressure of thesystem remains repulsive down to very short distances, asshown in Fig. 5, where we have plotted the pressure profilefor monovalent counterions (0.1) for different values ofg. Note also that there is still a repulsive barrier, which decreaseswith increasing g, while the range of the attraction isshifted to larger separations. For g1.2, corresponding to asurface charge density of about one charge per 300 Å 2 ,the total pressure becomes attractive at about d10 Å. Itshould be noted that in real experimental settings, otherFIG. 4. The fraction of condensed counterions as a function ofdistance.041501-8

COUNTERION CONDENSATION AND FLUCTUATION- ...PHYSICAL REVIEW E 66, 041501 2002FIG. 5. The pressure profile for monovalent counterions in thecase of moderate coupling g1.strong repulsive force, such as hardcore or hydration force,that we have not taken into account in our model, may becomeimportant and may overwhelm this correlated attractionat length scale less than 20 Å 1. This may explainwhy attraction is difficult to observe experimentally formonovalent counterions. Moreover, the pressure profile forlarge separations is similar to that of the PB theory, exceptwith a renormalized or effective surface charged density. Indeed,it is known experimentally that in order to fit experimentaldata to the PB theory, it is necessary to use an effectivesurface charge, which is always lower than the actualsurface charge density 1. Therefore, this counterion condensationpicture provides a possible scenario in which thisphenomenon can be accounted for theoretically, without invokingcharge regulation mechanism.However, for c the behavior of the order parameter and the total pressure of the system is qualitatively differentsee Fig. 4. We find that there is a range in the couplingconstant, g ()/2gg (), in which the order parameterdisplays a finite jump at a particular separation d 0 (g,), andthe counterion condensation is first order as a function of theseparation d. Here, g () denotes the coupling constant atwhich the first-order counterion occurs at infinite separation,i.e., an isolated charged plate see Sec. II and Ref. 14.Thus, in the limit g→g (), we must have d 0 (g,)→,since the system is composed of two isolated charged plates.However, we have d 0 (g,)→0 asg→g ()/2, because thislimit corresponds to a single charged plate with twice of thesurface charge density, i.e., 2en 0 . This striking behaviorof the order parameter has interesting implications for theinteraction for the system. Indeed, for sufficiently short distances,we find that the first-order counterion condensationspontaneously can take the system from the repulsive to theattractive regime, resulting in a first-order binding transition.This is illustrated in Fig. 6 for the case of trivalent counterionsat room temperature 0.01 at g1.3, corresponding toa surface density of one charge per 70 nm 2 . For trivalentcounterions, the first-order counterion condensation occursin the range of 0.9g1.8 14. The correspondingpressure profile is plotted in Fig. 7, which shows that theFIG. 6. First-order binding transition: for the case of trivalentcounterion (0.01), the number of condensed counterions ()exhibits a discontinuous jump at a particular distance.binding transition occurs at about d10 Å. We note that aninteresting consequence of this first-order binding transitionis the existence of metastable states, which may have importantmanifestations in surface force experiments. It is easy toimagine that the system can be trapped in different metastablestates, and therefore, hysteresis may occur as the twosurfaces are pushing in and pulling out again. Indeed, there issome experimental support for this behavior for multivalentcounterions in similar systems 31. It is important to emphasizethat this interesting behavior is not included in themean-field PB theory. Note also that this first-order bindingtransition can only take place at short distances. This is becaused 0 (g,) generally increases with increasing g, andeventually when g is near g (), the condensation occurswithin the repulsive regime and the binding transition becomescontinuous. Thus, direct experimental observation ofthe first-order binding transition may prove subtle.Finally, For gg () and c , the condensation againbecomes continuous. This is because the first-order phasetransition has already occurred at infinite separation, inwhich 1. In this regime, the length scale at which theFIG. 7. The pressure profile for the first-order binding transition.041501-9

A. W. C. LAU AND P. PINCUS PHYSICAL REVIEW E 66, 041501 2002counterion distribution is strongly modified if discreteness istaken into account. In particular, the counterions tend to bemore ‘‘localized’’ near the charged surface. It remains to beseen how this affects the condensation picture presented inthis paper; it is possible that this effect may smooth out thefirst-order transition. However, we believe that a rapid variationof the condensation reflecting the first-order transitionshould remain.ACKNOWLEDGMENTSFIG. 8. The pressure profile for high surface charges. Note thatthe distance at which the pressure turns attractive can be large comparedto the Gouy-Chapmann length .attraction starts to overcome the repulsion can be quite largesee Fig. 8. In the case of divalent counterions, we find thatthe onset of attraction occurs at d100Å. Clearly, the higherthe surface charge or g, the longer is the range of the attraction;therefore, together with the mechanism of fluctuationdrivencounterion condensation, the correlated attraction mayexplain the long-ranged attractions observed experimentally.Moreover, we note that there is a qualitative change in theshape of the pressure: the repulsive barrier disappears. Thismay mark an onset of the aggregation and has importantexperimental manifestations on the phase behavior of themacroions.In summary, by incorporating the condensation driven byfluctuations, we show that the net pressure between twosimilarly charged surfaces becomes negative, hence attractions,at a length scale much longer than the Gouy-Chapmann length. We also predict several distinct behaviorsof the system, depending on the valence of the counterions,that deviates significantly from the classical theory of thedouble-layer interactions. While our calculation is based onthe Gaussian fluctuation theory which may break down for avery high surface charge density, a complementary treatmentis considered by Shklovskii 6 in this regime, where thecondensed counterions are assumed to form a 2D stronglycorrelated liquid. That theory predicts a strongly reduced surfacecharge and exponentially large renormalized Gouy-Chapmann length, qualitatively similar to our results that forhigh surface charge most of the counterions are condensed.Moreover, it was shown in Ref. 20 that by perturbingaround the low temperature Wigner crystal ground state, thelong-ranged attraction persists to be operative, independentof the ground state. Thus, at large distances, we believe thatour picture should capture the interaction of two similarlycharged surfaces in the regime between where PB theory isappropriate low surface charge and the strong couplinglimit 6,32.However, there remain fundamental issues to be addressedin the future. For example, in real systems, thecharged surfaces are often characterized by discrete surfacecharge distribution. In recent studies 33, it is shown that theWe wish to acknowledge J.-F. Joanny, D.B. Lukatsky,T.C. Lubensky, S.A. Safran, and P. Sen for important discussions.A.W.C.L. would like to thank Professor P.-G. deGennes for fruitful discussions and for a wonderful hospitalityat Collège de France, Paris where most of this work iscompleted. P.P. acknowledges support from grants MRL-NSF-DMR-0080034 and NSF-DMR-9972246. A.W.C.L. acknowledgessupport from NIH under Grant No. HL67286.APPENDIX: DERIVATION OF THEFLUCTUATION PRESSUREIn this appendix, we present in detail the derivation of thepressure arising from fluctuations. The derivative of the fluctuationfree energy with respect to the distance d is given inEq. 27,whereFd 12 B d 3 xG 3D x,x d 2 Dzd/2 2 e (z), A1 2 e (z) 2 2 sec 2 zz2zd/2 2 ˜ z,as defined in Eq. 26, R /(1b 2 ), and G 3D (x,x) is theGreen’s function defined as the inverse operator of Kˆ 3D andsatisfies Eq. 28, which in Fourier space can be written as 2z 2 q 2 2 D B zz.zd/2 2 e (z)G 3D z,z;qThe Green’s function can be solved by standard technique34; first, we note that the homogeneous solutions are givenbyfor zd/2 andh z;qe qz 1 q tanz ,qz h 1 z;qe 1qzd/2,A2A3041501-10

COUNTERION CONDENSATION AND FLUCTUATION- ...for zd/2. We have two cases to consider, zd/2 andzd/2. In the former case, we split the space into fourregions: zd/2, d/2zz, zzd/2, and zd/2and writePHYSICAL REVIEW E 66, 041501 2002The coefficients A(z)•••F(z) are determined by the followingboundary conditions:G 3D d/2,z;qG 3D d/2,z;q,A8G z,z;qAzh z;q for zd/2, A4G z,z;qBzh z;qCzh z;qfor z G 3D z,z;q zd/2 z G 3D z,z;q zd/22/ D G 3D d/2,z;q,A9d/2zz,G z,z;qDzh z;qEzh z;qzzd/2,G z,z;qFzh z;q for zd/2.A5forA6A7G 3D z,z;qG 3D z,z;q, A10 z G 3D z,z;q zz z G 3D z,z;q zz B .A11After some algebra, we obtain specificallyEz B h zMqh z2q 1 2 /q 2 1M 2 q ,DzMqEz,A12A13e qd 1b 2 2 1b 2 q R 1b 2 q R Mq1b 2 2 1q R q R 1b 2 q R 1b 2 q R ,A14where R / D 2/(1). Therefore, the Green’s function G 3D (x,x) for zd/2 is explicitly given byG 3D x,x d2 q B2 2 2q 1M 2 q1M 2 q 2 sec 2 z 1M 2 qq 21 2 /q 2 1M 2 A15q2Mq1M q 12 /q 2 tan 2 zcosh2qz2/q tanzsinh2qz . A162 1 2 /q 2Note that the Green’s function is symmetric with respect to z, as expected from the symmetry of the problem. Similarcalculation can be done for the case zd/2 and the result isG 3D x,x d2 q B2 2 2q 1 1Lqe2q(zd/2) 1qzd/2 2, A17q 2 zd/2 2where L(q) is given byandLq eqd Gd/2h d/2 h d/2e qd, A182 h d/2Gd/2 h d/2Mqh d/2h d/2Mqh d/2 2qG1 2 /q 2 1M 2 q 3D d/2,d/2;q.BA19041501-11

A. W. C. LAU AND P. PINCUS PHYSICAL REVIEW E 66, 041501 2002Returning to the expression in Eq. A1, we note that it canbe separated into three parts,1A1A1AF A 2d B D d2 q dzG3D z,z;q2 2 0 d zd/2,F B 1 d B d2 q dzG3D z,z;qz2 2 0 d 22 sec 2 z,F C 1 d B d2 q dzG3D z,z;q˜ z2 2 0 d2zd/2 2,A20A21A22where we have used the fact that the integrand is symmetricwith respect to z. Note also that there should also be twoterms containing (z)/d and ˜ (z)/d in F B /dand F C /d, respectively; however, they cancel identicallywhen they are added together.Let us first discuss Eq. A20; using the identity(/d)(zd/2) 1 2 (/z)(zd/2), and integrating bypart, it can be transformed into1AF A 1d B D d2 q G 3D z,z;q.2 2 zzd/2 A23Using the boundary condition in Eq. A9, z G 3D z,z;q zd/2 z G 3D z,z;q zd/2 2/ D G 3D d/2,d/2;q,and the explicit expression for the Green’s function given inEq. A16, we obtain after some algebra1AF A 1d B D d2 q B2 2 2q2 D M 2 q1M 2 qq D 1b 2 2 b 2 q R q D 1b 2 2 1b 2 q R 1b 2 q R q Db 2 1b 2 21M 2 q 1b 2 2 1q R q R 1b 2 q R 1b 2 q R d2 q2 2qM 2 q1M 2 q .A24The next term, Eq. A21, can be shown to be1AF B 82d B R1b 2 b2 d R1b 2 d2 q2 20d/2dxG3D x,x;qsec 2 x1x tan x.A25In evaluating the x integral, we note that there is a nontrivialintegral which involves the last term inside the bracket ofGreen’s function in Eq. A16; it readsd˜ Q dx sec 2 x1x tan x12/k 2 tan 2 x0cosh kx4/k tan x sinh kx,where k2q/ and d˜ d/2. Note that none of these integralscan be expressed in terms of elementary functions, butintegrating by parts several times, one can show that theintegral Q can be expressed in closed form with the help ofthe relation 2b tan(d/2)1b 2 ,byQ 1b2 2 1b 2 16bq R 2coshqd b1b2 4q R 2 coshqd 1b2 2 b4q R 2 1b2 1b 2 8bq R 2 d R1b 2 2 d R1b 2 coshqd b coshqdq R 2 2 d R1b 2 sinhqd b1b2 2q R sinhqd.Substituting this result back into Eq. A25 and rearrangingterms, we obtain041501-12

COUNTERION CONDENSATION AND FLUCTUATION- ...1AF Bd 1b2 22 B R d2 q B2 2 1 B R24b 2 1b 2 2 d R1b 2 2q Gd/22Jd/2 d2 q2 2 B2q Gd/2Jd/2 21b2 Jd/22q R 2 1 2 /q 2 2 2Mqsinhqdq R 1 2 /q 2 1M q,2where J(d/2) is defined by the expressionJd/2 1 2 1M 2 q 2Mqcosh qd1M 2 q. A261M 2 qFinally, we turn to the last term in Eq. A1, Eq. A22. Withthe help of the integral G3D z,z;qdzd/2 zd/2 B 1b 2 23 22q 2 R 1 2 Gd/2 1 2 1 2 Lq ,Eq. A22 can be written asA271AF Cd 1b2 22 B R d2 q 2 2qB Gd/21Lq2 24b2 1b 2 d2 q2 2 B2q2 d 1b 2 R Gd/21Lq2, A28which can be combined with the expression for F B /dabove note that G(d/2) cancels nicely to yield1AF BC 1b2 22d B RPHYSICAL REVIEW E 66, 041501 2002 4b2 1b 2 B4 RI 3 1 B R22 d R1b 2 B4 RI 2 I 3 ,A29where we have defined the following dimensionless integrals:I 2 d2 q2 2 4 R2q 21b2 Jd/2q R 2 1 2 /q 2 2 2Mqsinhqdq R 1 2 /q 2 1M q,2A30I 3 d2 q2 2 4 R2q 1 2 1 2 LqJd/2 . A31With some straightforward but tedious algebra, they can becast into a more explicit form,I 2 d/ R 0dx2x1b 2 x 2 1b 2 x1b 2 xb 2 4b 2 x 2 4b 2 x 2 1M 2 x1b 2 2 1xx1b 2 x1b 2 x0dx2xM 2 x1b 2 x1b 2 x1b 2 xb 2 x4b 2 x 2 4b 2 x 2 1M 2 x1b 2 2 1b 2 x1b 2 xA32andI 3 d/ R 0dx2xb 21M 2 x1b 2 2 1xx1b 2 x1b 2 x 2xM 2 xxb 2 xdx0 1M 2 x1b 2 2 1b 2 x1b 2 x ,A33041501-13

A. W. C. LAU AND P. PINCUS PHYSICAL REVIEW E 66, 041501 2002where we have made a change of the integration variable xq R . Now, observe that the first term in Eq. A29 cancelsprecisely the first term in Eq. A24. Therefore, combining the two expressions, we obtain Eq. 29 for the fluctuation pressure.Similarly, I 1 d/ R defined in Eq. 36 can be expressed asI 1 d/ R d2 q 4 Rq R 1b 2 2 21b 2 q R 2 2 2q 1b 2 2 1q R q R 1b 2 q R 1b 2 q R d2 q 4 R q R M 2 q1b2 2 2q2 2 21b 2 q R 1M 2 q 1b 2 2 1q R q R 1b 2 q R 1b 2 q R 1b2 2 2q R 1b 2 2q R 2 21b 2 q R 1b 2 2 1b 2 q R 1b 2 q R . A34Note that the first term in this expression is logarithmically divergent.1 J.N. Israelachvili, Intermolecular and Surface Forces AcademicPress, San Diego, 1992; W.M. Gelbart, R.F. Bruinsma,P.A. Pincus, and V.A. Parsegian, Phys. Today 539, 382000.2 I. Rouzina and V.A. Bloomfield, J. Phys. Chem. 100, 99771996; J. Arenzon, J.F. Stilck, and Y. Levin, Eur. Phys. J. B12, 791999; B.I. Shklovskii, Phys. Rev. E 60, 5802 1999.3 B.-Y Ha and A.J. Liu, Phys. Rev. Lett. 79, 1289 1997; 81,1011 1998; Phys. Rev. E 58, 6281 1998; 60, 803 1999.4 P. Pincus and S.A. Safran, Europhys. Lett. 42, 103 1998.5 R. Netz and H. Orland, Eur. Phys. J. E 1, 203 2000.6 V.I. Perel and B.I. Shklovskii, Physica A 274, 446 1999; T.T.Nguyen, A.Yu. Grosberg, and B.I. Shklovskii, Phys. Rev. Lett.85, 1568 2000.7 R. Kjellander, S. Marcelja, and J.P. Quirk, J. Colloid InterfaceSci. 126, 194 1988; H. Wennerstrom, A. Khan, and B. Lindman,Adv. Colloid Interface Sci. 34, 433 1991; V.A. Bloomfield,Biopolymers 31, 1471 1991; R. Podgornik, D. Rau, andV.A. Parsegian, Biophys. J. 66, 962 1994.8 Patrick Kekicheff and Olivier Spalla, Phys. Rev. Lett. 75, 18511995.9 A.E. Larsen and D.G. Grier, Nature London 385, 230 1997.10 L. Guldbrand, B. Jönsson, H. Wennerström, and P. Linse, J.Chem. Phys. 80, 2221 1984.11 M.J. Stevens and K. Kremer, J. Chem. Phys. 103, 1669 1995;N. Gro”nbech-Jensen, R.J. Mashl, R.F. Bruinsma, and W.M.Gelbart, Phys. Rev. Lett. 78, 2477 1997; N. Gro”nbech-Jensen, K.M. Beardmore, and P. Pincus, Physica A 261, 741998.12 R. Messina, C. Holm, and K. Kremer, Phys. Rev. Lett. 85, 8722000; Europhys. Lett. 51, 461 2000.13 J.C. Neu, Phys. Rev. Lett. 82, 1072 1999; J.E. Sader and D.Y.Chan, J. Colloid Interface Sci. 213, 268 1999.14 A.W.C. Lau, D.B. Lukatsky, P. Pincus, and S.A. Safran, Phys.Rev. E 65, 051502 2002.15 R. Kjellander and S. Marcelja, Chem. Phys. Lett. 112, 491984; J. Phys. Chem. 90, 1230 1986.16 M.J. Stevens and M.O. Robbins, Europhys. Lett. 12, 811990; A. Diehl, M.N. Tamashiro, M.C. Barbosa, and Y.Levin, Physica A 274, 433 1999.17 Phil Attard, Roland Kjellander, and D. John Mitchell, Chem.Phys. Lett. 139, 219 1987; P. Attard, D.J. Mitchell, and B.W.Ninham, J. Chem. Phys. 88, 4987 1988; R. Podgornik, J.Phys. A 23, 275 1990.18 D.B. Lukatsky and S.A. Safran, Phys. Rev. E 60, 5848 1999.19 A. Gopinathan, T. Zhou, S.N. Coppersmith, L.P. Kadanoff, andD.G. Grier, Europhys. Lett. 57, 451 2002.20 A.W.C. Lau, Dov Levine, and P. Pincus, Phys. Rev. Lett. 84,4116 2000; A.W.C. Lau, P. Pincus, Dov Levine, and HerbFertig, Phys. Rev. E 63, 051604 2001.21 B.-Y Ha, Phys. Rev. E 64, 031507 2002.22 T.M. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 49762000.23 M.N. Tamashiro and P. Pincus, Phys. Rev. E 60, 6549 1999.24 Ning Ma, S.M. Girvin, and R. Rajaramann, Phys. Rev. E 63,021402 2001.25 S.A. Safran, Statistical Thermodynamics of Surfaces, Interfaces,and Membranes Addison-Wesley, Reading, MA, 1994.26 Alexander L. Fetter, Phys. Rev. B 10, 3739 1974; H. Totsuji,J. Phys. Soc. Jpn. 40, 857 1976; E.S. Velazquez and L. Blum,Physica A 244, 453 1997; A.W.C. Lau and P. Pincus, Phys.Rev. Lett. 81, 1338 1998.27 L.D. Landau and E.M. Lifshitz, Statistical Physics, 3rd ed.Pergamon, New York, 1980, rev. and enl. by E.M. Lifshitzand L. P. Pitaevskii.28 Note that this is the case where the counterions can besqueezed out of the surfaces, where the density at the midplanefor d, 0 (0)1/(l B 2 ), instead of 0 (0)1/(l B d) wherecounterions are confined between two charged hard walls.29 J. Hubbard, Phys. Rev. Lett. 3, 771959; R.L. Stratonovitch,Dokl. Akad. Nauk. UzSSR 115, 1907 1957.30 S. Samuel, Phys. Rev. D 18, 1916 1978; see also Ref. 5.31 Uri Raviv private communications; unpublished.32 André G. Moreira, and R. Netz, Phys. Rev. Lett. 87, 0783012001.33 D.B. Lukatsky, S.A. Safran, A.W.C. Lau, and P. Pincus, Europhys.Lett. 58, 785 2002; André G. Moreira and R. Netz,ibid. 57, 9112002.34 G. Arfken, Mathematical Methods for Physicists AcademicPress, San Diego, 1996.041501-14