DOI: 10.1021/la903940j 6343Langmuir 2010, 26(9), 6343–6349 Published on Web 01/13/2010

pubs.acs.org/Langmuir

© 2010 American Chemical Society

Interaction between Charged Surfaces Mediated by Rodlike

Counterions: The Influence of Discrete Charge Distribution

in the Solution and on the Surfaces

John M. A. Grime,*,† Malek O. Khan,† and Klemen Bohinc‡

†Department of Physical and Analytical Chemistry, Physical Chemistry, Uppsala University,Uppsala, Sweden and ‡Faculty of Health Sciences, University of Ljubljana, Ljubljana, Slovenia

Received October 17, 2009. Revised Manuscript Received December 22, 2009

The interaction between two charged surfaces, with discrete or uniform charge distributions, embedded in a solutionof rodlike counterions has been studied. Monte Carlo simulations and density functional theory have been appliedto study the concentration profiles of counterions and the force between the charged surfaces. We show that forlow surface charge densities the repulsive force between like-charged surfaces is observed regardless of the rod length.Where high surface charge densities are present, attractive forces at surface separations related to the rod lengthare observed.

Introduction

Aqueous solutions of charged macroions1,2 play an importantrole in everyday life and have various industrial, biotechnological,andmedical applications. The presence in solution of smaller ionsmediates the electrostatic interactions between charged macro-ions; this is most commonly manifested as the so-called screeningeffect that weakens electrostatic interactions and lessens therepulsion between like-charged macroions. However, the pre-sence of multivalent ions can induce an effective attractionbetween like-charged macroions. This was first confirmed byMonte Carlo (MC) simulations made by Guldbrand et al,3 whodemonstrated that ion-ion correlations become important whenthe electrostatic interactions are large and dominate the entropiccontribution to the free energy. This can be realized for systemsfeaturing macroions with a large surface charge density, lowdielectric solvents, and low temperature or in the presence ofmultivalent ions. The effective attraction between like-chargedsurfaces is also observed when using integral equation theories,4,5

modified Poisson-Boltzmann theories,6 and field theoreticalmethods.7,8 The like-charged attraction has been observed ex-perimentally.9-12 Forweak electrostatic interactions, the effectiverepulsion between like-charged surfaces is described satisfactorilyby the mean-field Poisson-Boltzmann theory.

Natural systems do not feature truly uniform charge distribu-tion; macroions such as charged lipidmembranes, DNA, colloids,

actin molecules, proteins, viruses, and cells have discretely dis-tributed surface charges. It is of interest to study how the discretenature of the surface charge affects themacroion interactions.13-18

In earlier simulations, Khan et al18 have studied the force betweencharged surfaces with discrete charge distributions in a solution ofmultivalent ions. In theweak coupling limit, the force is dominatedby the entropic contribution, leading to an overall repulsive forcebetween like-charged surfaces, in agreement with the mean fieldPoisson-Boltzmann theory. For moderate coupling, the discretesurface charges weaken the repulsive force. In the strong couplinglimit, surfaces with a uniform charge experience an effectiveattraction between like-charged surfaces. Discrete charges on thesurfaces weaken this attractive force due to the strong correlationsbetween the discrete charges on the surface and themobile chargesbetween the surfaces.

It is necessary to move beyond simple single-point representa-tions for certain multivalent ions where the separation betweencharges is too large to be ignored. One much studied system isDNA condensation in which short, stiff multivalent oligoamines,spermine and spermidine, induce the compaction of DNA.11 Thestructure of ions also affects the stability of colloidal systems19

as well as biological macromolecules.20,21 Recently, Bohinc andco-workers22,23 showed that intraionic correlations induced bythe internal structure of multivalent ions with simple structuresenable attractive interactions between like-charged surfaces.Analysis of the molecular orientations show that at the freeenergy minimuum the counterions are oriented perpendicular to

*To whom correspondence should be addressed. E-mail: john.grime@fki.uu.se.(1) Evans, D. F; Wennerstr€om, H. The Colloidal Domain: Where Physics,

Chemistry, Biology and Technology Meet; Wiley-VCH: New York, 1994.(2) McLaughlin, S. Annu. Rev. Biophys. Biophys. Chem. 1989, 18, 113–136.(3) Guldbrand, L.; J€onsson, B.;Wennerstr€om,H.; Linse, P. J. Chem. Phys. 1984,

80, 2221–2228.(4) Kjellander, R.; Mar�celja, S. Chem. Phys. Lett. 1984, 112, 49–53.(5) Kjellander, R.; Mar�celja, S. J. Phys. Chem. B 1986, 90, 1217–1484.(6) Forsman, J. J. Phys. Chem. B 2004, 108, 9236–9245.(7) Moreira, A. G.; Netz, R. R. Phys. Rev. Lett. 2001, 87, 078301-1–078301-4.(8) Naji, A.; Arnold, A.; Netz, R. R. Europhys. Lett. 2004, 67, 130–136.(9) Khan, A.; Fontell, K.; Lindman, B. J. Colloid Interface Sci. 1984, 101,

130–136.(10) Khan, A.; Fontell, K.; Lindman, B. Colloids Surf. 1984, 11, 401–408.(11) Bloomfield, V. A. Biopolymers 1997, 44, 269–282.(12) Kjellander, R.; Mar�celja, S.; Pashley, R. M.; Quirk, J. P. J. Phys. Chem.

1988, 92, 6489–6492.

(13) Moreira, A. G.; Netz, R. R. Europhys. Lett. 2002, 57, 911–917.(14) Lukatsky, D. B.; Safran, S. A.; Lau, A. W C; Pincus, P. Europhys. Lett.

2002, 58, 785–791.(15) Foret, L.; K€uhn, R.; W€urger, A. Phys. Rev. Lett. 2002, 89, 156102.(16) White, T.; Hansen, J.-P. J. Phys.: Condens. Matter 2002, 14, 7649–7665.(17) Lukatsky, D. B.; Safran, S. A.; Lau, A. W. C; Pincus, P. Europhys. Lett.

2002, 60, 629–635.(18) Khan, M. O.; Petris, S.; Chan, D. Y. C. J. Chem. Phys. 2005, 122, 104705.(19) Linse, P.; Lobaskin, V. J. Chem. Phys. 2000, 112, 3917–3927.(20) Dai, L.; Mu, Y. G.; Nordenski€old, U. Phys. Rev. Lett. 2008, 100,

118301-1–118301-4.(21) Urbanija, J.; Bohinc, K.; Bellen, A.; Maset, S.; Igli�c, A.; Kralj-Igli�c, V.;

Kumar, P. B. S. J. Chem. Phys. 2008, 129, 1051015–1011055.(22) Bohinc, K.; Igli�c, A.; May, S. Europhys. Lett. 2004, 68, 494–500.(23) May, S.; Igli�c, A.; Re�s�ci�c, J.; Maset, S.; Bohinc, K. J. Phys. Chem. B 2008,

112, 1685–1692.

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Article Grime et al.

(and thus act to connect) the like-charged surfaces. Monte Carlosimulations have confirmed the theoretical predictions,23,24

and a systematic investigation was conducted by Kim et al.25

The latter study confirmed that in the MC weak coupling regimethe attraction occurs via bridging by counterions with a sizeabove a threshold value. In the strong MC coupling limit,the electrostatic correlations of counterions aligned parallel tothe surfaces dominate. For short rods, the depletion effect alsoplays a role.

The interaction between like-charged surfaces in a solution ofcharged polymers with variable intramolecular stiffness has alsobeen investigated via Monte Carlo simulations.26,27 Fully flexiblepolymer chains can stretch from one surface to another, andattraction can be obtained due to bridging.28 In the limit ofinfinitely stiff chains, this bridging is lost, replaced instead bya strong correlation effect at short distances.29

In this work, we combine the two distinct molecular featuresdescribed above (macromolecules with discrete surface chargesand small molecules with an internal structure) in order tofurther understand the interactions between charged species.We calculate intersurface interaction by using numerical (MC)simulations. These results are comparedwith an analytical theory.The two methods are described in the following section, followedby results and discussion. We finally conclude that for weakelectrostatic coupling a repulsive force between like-chargedsurfaces is observed. In the intermediate coupling limit, and forsufficiently long rods, an attractive force is recorded. For any rodlength in the strong coupling, the attraction at short distancesis observed.

Methods

Monte Carlo Approach. The Monte Carlo simulations wereperformed in the canonical ensemble with the standard Metro-polis formulation. The rigid rodlike counterions featured twopoint charges of valence þ1 elementary charge unit (e), and thecharged surfaces were represented as either a set of equidistantlyspaced point charges having valence -1 or a uniformly chargedplane (see Figures 1-3). An excluded region of length 0.2 nmwasplaced in front of each charged surface on the z-axis in allsimulations. We assume that there is no electric field behind eachof the two charged planar surfaces, an assumption appropriate tosituations where the dielectric constant in the macroion interior ismuch smaller than that of the aqueous region between thesurfaces.

The total energy in the system is defined as

Utotal ¼ Uww þXi

Xj 6¼i

Uiiði, jÞþXw

Uiwði,wÞþUi¥ðiÞ24

35 ð1Þ

where the direct interaction energy between the charged wallsthemselves, Uww, is ignored during Monte Carlo simulation, asthis contribution is constant; the separation D between thecharged walls does not change in the course of each simulation,and hence, no energy changes from this contribution affect theMonte Carlo procedure.

Figure 1. Schematic illustrationof two like-chargedplanar surfaceswith discrete surface charge distribution. The intervening solutioncontains multivalent rodlike ions. Each rodlike ion has two pointcharges Ze separated by a fixed distance l. The distance of closestapproach for the counterions to the charged surfaces is 0.2 nm,shown as a dashed line. The distance between the surfaces is D.

Figure 2. Counterion density adjacent to the plane of closest ap-proach (at z= 0 nm) for a simulation of discrete wall charges withσ=0.1As/m2, l=0.5 nm, andD=9nm.Concentration recordedinMC simulation is shown, alongwith order 1 and 3 polynomial fitsto the appropriate nearest concentration data points. Concentrationerror bars are plotted, but they are too small to be seen clearly.

Figure 3. Schematic illustration of two like-charged planar sur-faces with uniform surface charge density σ. The interveningsolution contains multivalent rodlike ions. Each rodlike ion hastwo point charges Ze separated by a fixed distance l. For oneparticular ion, the locations of the two charges at z and z þ s areindicated. The distance between the surfaces isD, with the plane ofclosest approach for the counterions set 0.2 nm in from eachcharged surface (shown as dashed line).

(24) Maset, S.; Re�s�ci�c, J.; May, S.; Pavli�c, J. I.; Bohinc, K. J. Phys. A 2009, 42,11815–11826.(25) Kim, Y W.; Yi, J.; Pincus, P. A. P. Phys. Rev. Lett. 2008, 101, 208305-1–

208305-4.(26) Akesson, T.; Woodward, C.; J€onsson, B. J. Chem. Phys. 1989, 91, 2461–

2469.(27) Miklavic, S. J.;Woodward, C. E.; J€onsson, B.; Akesson, T.Macromolecules

1990, 23, 4149–4157.(28) Podgornik, R.; Akesson, T.; J€onsson, B. J. Chem. Phys. 1995, 102, 9423–

9434.(29) Turesson, M.; Forsman, J.; Akesson, T. Langmuir 2006, 22, 5734–5741.

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Grime et al. Article

The direct ion-ion interactions are evaluated with the familiarCoulomb potential:

Uiiði, jÞ ¼ qiqje2

4πε0ε1rijð2Þ

Here, the charges on the ions are in elementary charge units e. Thepermittivity of the medium ε1 is expressed relative to that ofvacuum, ε0; we choose ε1 = 78.5 to approximate the permittivityof an aqueous solution at the simulated temperature (300K). TheBjerrum length, lB = e2/4πε0ε1kT, is therefore ∼0.7 nm.

The interaction between an ion and a uniformly charged wall issimply the energy between an ion and an infinite, uniformlycharged plane,

Uiwði,wÞ ¼ -qieσw

2ε0ε1jziwj ð3Þ

where |ziw| is the absolute distance between the ion and thewall onthe z-axis of the system. For the discrete systems, the uniformlycharged walls are replaced with a series of discrete charges, andhence the contribution from eq 3 is zero, with the discrete wallcharges instead considered via eq 2.

The long ranged correction to the electrostatic energy of thesystem is provided by expressing the charge distribution in thesystem as a series of infinite, uniformly charged planes afterJ€onsson et al.30 and Valleau et al.;31 the interaction of the ionswith these planes is calculated, and the finite section of the planewhich lies inside the simulation cell is removed to prevent doublecounting the direct interactions between ions:

Ui¥ðiÞ ¼Xp

Uiwði, pÞ-Uifði, pÞ ð4Þ

Here, the summation is taken over the set of charged planes usedto correct the electrostatic energy, with the surface charge densityof each plane calculated in a self-consistent manner from theaverage charge distribution in the system at that point on thez-axis. The interaction of an ion with the infinite component ofeach plane is identical to that in Uiw (eq 3), and the interactionwith the finite regionof the planewhich lies in the simulation cell isgiven by

Uifði, pÞ ¼ qieσpW

4πε0ε1f ðjzipj=WÞ ð5Þ

where the absolutedistancebetween the ionand the finite plane onthe z-axis denoted as |zip| andW is the dimensionof the simulationcell on the x- and y-axes (assumed here to be the same, so the wallplane is square inside the simulation cell). The function f(z) is31

f ðzÞ ¼ 4ln0:5þ r1

r2

� �-4z sin-1 r2

2 þ 0:5r10:5r2 þ r1r2

" #þ tan-1 1

2z-π

2

� �0@

1A

ð6ÞThe MC simulations each featured 64 counterion molecules,

with an additional 64 discrete charges per wall, arranged in an 8�8 grid for the simulations of discretely charged surfaces. Grids ofup to 12� 12 discrete wall charges were tested, and no discernibledifferences in the results were observed. Rodlike counterions wereselected at random before translation or rotation with equalprobability. Translation consisted of a random offset on eachaxis in the range [-0.5 nm, þ0.5 nm], with rotations uniformlyrandom in [-π/4:þπ/4] radians on a random axis of rotation

(generated uniformly on the surface of a unit sphere surroundingthe center of geometry of the counterion). The equilibrationperiod consisted of 106 MC steps (sufficient for the long rangedelectrostatic correction to have converged), followed by a fur-ther 2.4� 107MC steps to collect the data presented in this study.A selection of random postequilibration snapshots of the systemmay be seen in the Supporting Information.

Pressure at the charged surfaces was calculated by consideringthe electrostatic force exerted due to the location of all chargedelements of the system, in addition to the kinetic pressure due tothe presence of counterions adjacent to the charged surface. Thisincludes the pressure due to the long ranged correction to theelectrostatic interactions.31

For the case of uniformly charged surfaces, this reduces to thecontact theorem.32,33 During simulation, the counterion concen-trations were generated via histograms of resolution 50 bins pernm; the counterion concentration adjacent to the walls was fittedwith a polynomial of order 3, and the concentration was extra-polated to the plane of closest approach to determine the kineticpressure contribution. This yields a better approximation of the“true” concentration profile behavior in the vicinity of the planeof closest approach for the counterions as opposed to a linearinterpolation, for example, as demonstrated in Figure 2.

The pressure was sampled after every counterion had beenselected twice (on average) for a MC move, and then averagedover blocks of length 100 providing approximately 1.9 � 103

independent samples. Error bars were calculated for all data,but they are too small to be seen in the graphs presented.All simulations used the minimum image convention, withperiodic boundaries on all axes except the axis perpendicular tothe surface, z.

Theoretical Approach. The theoretical model consists ofplanar surfaces with uniform surface charge density σ. Eachrodlike ion carries two identical pointlike positive charges ofvalencyþ1, separated by a fixed distance l. The distance betweenthe two surfaces is D (Figure 3). The electrostatic field of thesystem varies only along the z-axis, therefore all variables dependonly on the coordinate z.

Rodlike ions are described by the particle distribution functionn(z,s) which includes the information for positional and orienta-tional degrees of freedom. If the reference charge (one of the twocharges of each ion) is located at z, then the location of the secondcharge is specified by the projection s. The concentration ofreference charges is given by the average n(z) = Æn(z,s)æ, whereÆ...æ = 1/2l

R-ll ds.

The Helmholtz free energy of the system, measured per unitarea of the surface and divided by the thermal energy unit, kT,consists of two terms: the electrostatic energy and the entropy ofthe multivalent rodlike nanoparticles,

F

AkT¼

Z ¥

-¥dz

Ψ0ðzÞ28πlB

þ nðz, sÞ ln λnðz, sÞ-1þUðz, sÞ� �� " #

Here, Ψ is the reduced electrostatic potential. Note that theconstant λ is chosen to ensure overall charge neutrality in thesystem. The function U(z,s) is the nonelectrostatic potential thatacts on the rodlike ions and ensures that they cannot penetratethrough the charged surface.

The result of the variational procedure gives the inte-gral-differential equation for the reduced electrostatic potential

Ψ00ðxÞ ¼ -8πlBZ

λ

1

2l

Z min½l,D-z�

max½-l, -z�ds e-ZðΨðzþsÞþΨðzÞÞ ð7Þ

Note that the integration limits account for the orientational

(30) J€onsson, B.; Wennerstr€om, H.; Halle, B. J. Phys. Chem. B 1980, 84, 2179–2185.(31) Valleau, J. P.; Ivkov, R.; Torrie, G.M. J. Chem. Phys. 1991, 95, 2221–2228.

(32) Henderson, D.; Blum, L. J. Chem. Phys. 1978, 69, 5441–5450.(33) Henderson, D.; Blum, L.; Lebowitz, J. L. J. Electroanal. Chem. 1979, 102,

315–319.

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Article Grime et al.

constraints imposed by the rigid surfaces. The boundary condi-tions are give at the charged surfaces

Ψ0ðz ¼ 0Þ ¼ -4πlBσ=e Ψ0ðz ¼ DÞ ¼ 4πlBσ=e ð8ÞThe integral-differential equation equation subject to theboundary conditions 8 was solved numerically.

The osmotic pressure, P, due to counterions between two like-charged surfaces can be calculated from the first derivative of thefree energy 7 with respect to the distance between the chargedsurfaces P = -∂f/A∂D:

P

kT¼ 2nð0Þ- σ22πlB

e2ð9Þ

where n(0) is the concentration of reference charges near thecharged surface.

Results and Discussion

Counterion Concentration. Figures 4-6 show the concen-tration of reference charges (defined here as the first charge in

each rodlike counterion) as a function of distance from the leftcharged surface for three different lengths of rodlike ions (l=0.5,2, and 5 nm). The distance D between the charged surfaces waschosen to be 3 nm, and three different surface charge densitieswere considered: σ=0.01, 0.025, and 0.1 As/m2.MC simulationswere performed for both discrete and uniform surface chargedistributions, whereas the density functional theory calculationswere performed only for uniformly charged surfaces.

For l= 0.5 nm and l= 2.0 nm, the concentration profiles arecomplex, with discontinuities observed. For the case of l= 5 nmno discontinuities are seen, which indicates that the discontinu-ities are a function of the rod length, specifically that whereD> lwe expect no discontinuities, as they are a product of thefrustrated geometry of the counterions. In all cases, counterionsare more highly concentrated at the discretely charged surfacescompared to the uniformly charged surfaces with identical σ, witha commensurate depletion of counterions around themidplane ofthe system.

The excellent agreement between the MC simulations withuniform charged surfaces and the density functional calculationsfor l= 5 nm suggests that, for sufficiently long rods, we expect anear perfect agreement between the two methods, demonstratingthat in the limit of long rods the interionic correlations becomenegligible for uniformly charged surfaces.24

The largest relative deviations in the behavior of the systemswith uniform and discrete surface charges are obtained for small

Figure 4. Concentration of reference charges n as a function of thedistance from the left charged surface z forD=3nm (i.e., one-halfof the symmetrical profile is shown) and σ = 0.01 As/m2. Thecircles corresponds to simulations made with uniform surfacecharge, whereas the squares represent simulations with discretesurface charge distribution. Full lines represent density functionaltheory calculations.The interval betweenplottedMCdatapoints isincreased away from the charged surface to aid the legibility of thedensity functional theory results.

Figure 5. Concentration of reference charges as a function of thedistance from the left charged surface for σ=0.025As/m2. Detailsas in Figure 4.

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Grime et al. Article

σ and l, for example, σ=0.01As/m2 and l=0.5 nm (Figure 4). Inthis limiting case, where the distance between the discrete surfacecharges becomes large compared to the rod length, the level ofsurface charge discretization is important; for this specific exam-ple, the rods of 0.5 nmare short compared to the distance betweenthe charges on the surface (4 nm), and the rods behave much likepoints in the granular distribution of charges on the surface. Suchsituations can also induce a dipolar layer.18

Osmotic Pressure. Figures 7-9 show the pressure P as afunction of the distance between equally charged surfaces forthree different lengths of rodlike ions and three different surfacecharge densities. MC simulations were performed for both dis-crete and uniform surface charge distributions, whereas thedensity functional calculations were performed only for uniformsurface charge distributions. The error bars for the pressures asrecorded in MC simulation are too small to be clearly seen in thegraphs presented here. The wall/wall pressure contributions haveno error bars, as that value is constant during simulation. Errorbars for the counterion/wall contribution to the pressure are zerofor uniformly charged surfaces (as the electric field is constant in z,and also invariant for the x and y location of a counterion charge)and ca. 3 orders of magnitude smaller than the recorded pressurecontributions for the counterion/discretely charged walls. Theerror bars on the counterion concentrations are ca. 3 orders ofmagnitude smaller than the measured concentration values, and

hence, the error bars on the kinetic pressure contribution aresmall. Nontheless, we see rather large noise in the MC data fordiscretely charged surfaces for the higher surface charge density ofσ= 0.1 As/m2; this demonstrates that although the errors on theindividual contributions to the pressure are small relative to theiroverall magnitude, the manner in which the contributions cancelone another to a large extent leads to these small errors becomingquite visible, particularly in the case of the kinetic contribu-tion from the counterion concentrations. Hence, the contacttheorem may struggle to provide good quality data where thecounterion concentrations are large and rapidly changing suchas in the cases where highly charged, discrete surfaces are present(see Figures 4-6)

For the shorter rod length of l=0.5 nm, the density functionaltheory always predicts a repulsive pressure which is monotoni-cally decreasing as D increases. By contrast, the MC simulationsof discretely charged surfaces can display sudden local drops inpressurewhere l=D for all σ. These drops can actually produce anet attractive pressure which is entirely absent from the densityfunctional theory, as in the case of σ = 0.1 As/m2 (Figure 9),before recovery to a plateau pressure for larger D. The MCsimulations of uniformly charged surfaces are generally in goodagreement with the predictions of the density functional theory,

Figure 6. Concentration of reference charges as a function of thedistance from the left charged surface for σ=0.1As/m2.Details asin Figure 4.

Figure 7. Pressure P as a function of the distance between thecharged surfaces D for σ = 0.01 As/m2. The circles correspondto simulations made with uniform surface charge, whereas thesquares represent simulations with discrete surface charge. The fulllines represent the results fromdensity functional theory. Points areconcentrated in regions of particular interest. Note that the graphsuse differing scales.

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Article Grime et al.

except the systemwith l=0.5nmandσ=0.1As/m2which displaysamarkeddecrease inpressure aroundD≈ 1nm.This is also the onlysituation where there is qualitative disagreement between the pre-dictedP fromthedensity functional theoryand the results of theMCsimulationswith uniformsurface charge.An attractive force appear-ing at small D in MC simulations of a primitive model electrolytebetween uniformly charged surfaces is well recognized,3,31 and thissituation is approximated by relatively short rodlike counterions.Such results further demonstrate the importance of counter-ion-counterion correlations, which are neglected in mean-fieldapproximations (such as the density functional theory used here)where the counterions are treated as an ideal gas. In the addition tothe usual mean-field theory (the Poisson-Boltzmann model), theorientational degrees of freedomand intraionic correlations via fixedcharge separation within each ion were introduced.

An effect due to rod length is also visible in the pressure graphsfor l=2 nm and l= 5 nm (e.g., Figures 8 and 9), where we see aslight kink in the pressure graph for D ≈ 1. For σ = 0.1 As/m2,this pressure effect is very pronounced for the longer rod lengthsand crosses over into an attractive pressure where l ≈ D beforereturning to plateau at P = 0 for larger D.

The osmotic pressure can be expressed as a derivative of freeenergy with respect to the separation between charged surfaces ofarea A:

P ¼ -∂F

A∂Dð10Þ

This implies that, if D was allowed to vary, the equilibriumdistance between the charged surfaces may be found whereP vanishes, as any deviation from that arrangement would onlyincrease the free energy of the system. The values of D for whichP= 0 may differ significantly between the density functionaltheory and theMCsimulations (e.g., l=0.5nmwithσ=0.1As/m2

in Figure 9), depending on the values of σ and l and the level ofdiscretization for the surface charge.

While the qualitative agreement between the predictions of thedensity functional theory and the results of the Monte Carlosimulations are seen to improve with increasing σ for the counter-ion concentration profiles (compareFigures 4 and 6), the oppositetrend is observed for the osmotic pressure where l is small, forexample, l = 0.5 nm; as we progressively increase σ, we see lessagreement between the pressures generated using uniform anddiscrete surface charge distributions for small D.

Conclusions

In the present work, we investigated systems composed of twoequally charged planar surfaces in an electrolyte solution composedof rigid divalent rodlike counterions. For this system,we introduceda density functional theory, and the predictions from this theorywere tested via Monte Carlo simulations. We show that, for highenough surface charge densities, attraction between both the uni-form and discretely charged surfaces may take place, and that theseparation between the surfaces where this attraction manifests isitself connected to the length of the rodlike counterions.

Figure 8. Pressure P as a function of the distance between thecharged surfaces D for σ= 0.025 As/m2. Details as in Figure 7.

Figure 9. Pressure P as a function of the distance between thecharged surfaces D for σ= 0.1 As/m2. Details as in Figure 7.

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Grime et al. Article

Acknowledgment. The authors are thankful for the highperformance computational (and storage) capacity allocatedthrough the Swedish National Infrastructure for Computing(SNIC) on resources at the National Supercomputer Centre(NSC).

Supporting InformationAvailable: Postequilibration snap-shots from the MC simulations (both uniformly and dis-cretely charged systems, rod lengths 0.5 nm and 5 nm). Thismaterial is available free of charge via the Internet at http://pubs.acs.org.