PHYSICAL REVIEW E, VOLUME 65, 021405

Adsorption of colloidal particles on a charged surface: Cluster Monte Carlo simulations

Takamichi Terao and Tsuneyoshi NakayamaDepartment of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan

~Received 11 June 2001; revised manuscript received 11 October 2001; published 15 January 2002!

We investigate the adsorption of colloidal particles on a charged surface by the cluster Monte Carlo simu-lation. Under the primitive model of asymmetric electrolytes, the density profiles of colloidal particles at asurface are analyzed as a function of the electrostatic coupling parameter. In the strong coupling regimenumerical results show thelike-charged adsorptionbetween colloidal particles and a charged surface, whichcannot be explained by the Derjaguin-Landau-Verwey-Overbeek theory.

DOI: 10.1103/PhysRevE.65.021405 PACS number~s!: 82.70.Dd, 82.70.Kj, 61.20.Ja

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I. INTRODUCTION

Electrostatic interaction plays an important role in cloids, polyelectrolytes, and surfactant micelles@1–5#. Thelinearized screening theory of Debye and Hu¨ckel alwaysleads to a repulsive interaction between like-charged colloin an aqueous solution. In the Derjaguin-Landau-VerwOverbeek~DLVO! theory, the electrostatic part of the effetive interaction between particlesUDLVO(r ) is given by

UDLVO~r !5Z2e2

4pe S ekr m

11kr mD 2 e2kr

r, ~1!

where Z, e, k, r m , e, and r denote the surface chargecolloidal particles, the elementary charge of an electron,inverse of the Debye-Hu¨ckel screening length, the radius ocolloidal particles, the dielectric constant of the medium, athe center-to-center distance between two colloidal particrespectively. The inverse screening lengthk is given by

k254plB(j

njqj2, ~2!

wherelB5e2/4pekBT is the Bjerrum length andnj is theqj -valent ion density. In general, the effective interaction btween colloidal particles is of importance to determinephysical properties of colloidal systems in an aqueous stion @2,6–11#.

Recently, lots of works are devoted to clarify the electstatic effect in soft matters@3,12–29#. Rouzina and Bloom-field @3# and Guldbrandet al. @12# have analyzed the attraction between like-charged planer surfaces at a shseparation. Linse and Lobaskin have applied their theorya spherical geometry, and proposed a criterion that the lcharged attraction between colloidal particles can hapwhen the coupling parameterG obeysG'2 or larger, whereG is the ratio between the characteristic energy of the Clomb interaction and the thermal energykBT @13#. Here thedimensionless coupling parameterG is defined to be

G[q2lB

ac5S Z

4pqD 1/2 q2lB

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where ac and r c are the average distances between tneighboring counterions on the charged colloidal particand the radius of counterions, respectively. These stuhave shown that the effective interaction between colloiparticles becomes repulsive at weak electrostatic couplinand on the other hand, the attractive force dominatestween them at stronger electrostatic couplings@13–15,17,30–34#. At larger concentrations of multivalent ions,has been clarified that charged colloids or polyelectrolytean aqueous solution strongly bind so many oppositcharged microions that the sign of the net macroion chabecomes inverted. The charge inversion is one of the chateristic phenomena in soft matters with strong electrostcouplings and is of great interest@31,35–41#.

In this paper, we study numerically the structural formtion of colloidal particles on a charged surface by the clusMonte Carlo simulation. Using the primitive model of asymmetric electrolytes, the density profile of colloidal particlesclarified, as a function of the strength of the coupling paraeter G. The effective interaction between colloidal particland a charged surface has been studied numericallySvensson and Jo¨nsson@42#. In Ref. @42#, they have pointedout that a lot of counterions accumulate on a macroion sface and the mobility of macroions becomes very retardwhich makes it very difficult to reach a thermodynamic eqlibrium in the computer simulation. In this paper, we emplthe cluster Monte Carlo technique to treat this problem aclarify the physical properties of charged colloidal particlin an aqueous solution@14#. In the strong coupling regimeour numerical results demonstrate thelike-charged adsorp-tion between macroions and a charged surface.

This paper is organized as follows. In Sec. II, we descrthe primitive model of colloidal particles in an aqueous slution. In Sec. III, the numerical results on the density profiof colloidal particles at a charged surface are displayed. Stion IV is devoted to discussions and conclusions.

II. MODELS

We adopt the primitive model of strongly asymmetrelectrolytes to describe colloidal suspensions in an aquesolution, involving the excluded volume and the Coulominteraction of negatively charged colloidal particles acounterions @13,16,30#. Employing the primitive model,

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TAKAMICHI TERAO AND TSUNEYOSHI NAKAYAMA PHYSICAL REVIEW E 65 021405

fluctuation-induced attraction between like-charged collohas been found in computer simulations@14,18,40#. In thismodel, all particles are represented by hard spheres withferent charges and sizes. The model contains two typespherical charged particles:~i! colloidal particles~macroions!of radiusr m and charge2Ze, and~ii ! counterions of radiusr c and chargeqe, whereas the discrete structure of the svent is neglected and the solvent enters into the model vidielectric constante, which reduces the Coulomb interactioWe considerNm spherical macroions with the surface char2Ze (Z.0). In addition, counterions carrying an opposchargeqe(q.0) are fully taken into account. These macrions and counterions are dispersed in the cubic box, whthe positions of these particles are randomly moved insimulation. The linear system size of the cubic box in thex,y, andz directions is taken to beL (2L/2<x,y,z<L/2). Thepair potentialVi j (r ) between particlesi and j is given by

Vi j ~r !5H ` for r<r i1r j

qiqje2

4perfor r .r i1r j ,

~4!

whereqi andr i are the valence and the radius of a particlei,respectively. In the absence of salt, only the macroionsthe counterions are treated in Eq.~4!, and this level of de-scription is also referred to as the two-component mo@27#. The model system is described by physical parametsuch asZ, q, r m , r c , rm , rc , ands, wherer i is the numberdensity of particlesi (5m,c), and s is the surface chargedensity of a surface. The macroion chargeZ is varied from20 to 60, the counterion valenceq from 1 to 3, and theelectrostatic coupling parameterG from 0.4 ~weak couplingregime! to 3.7 ~strong coupling regime!.

In this study, the effect of a charged surface is treatedfollows: we impose periodic boundary conditions in thexandy directions, and the hard-wall condition in thez direc-tion. We place uniformly discrete negative charges on aface perpendicular to thez axis, and the other parallel surfacis set to be electrically neutral. We assume that the imcharge effect across the surfaces in thez direction, due to alower dielectric constant than the medium, is expected tosmall and can be neglected in the following simulation. Tcorresponds to the matchinge boundary condition at the surface. The counterions discharged from the negativcharged surface are included explicitly. We also considercondition of global charge neutrality such as

2rmZ1rcq1s

L50. ~5!

TABLE I. System parameters for the runsA, B, C, andD.

Run q Z Nm L ~nm! G

A 1 20 20 40.0 0.4B 1 20 44 52.0 0.4C 2 40 20 40.0 1.6D 3 60 20 40.0 3.7

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III. NUMERICAL RESULTS

The long-range nature of the Coulomb interaction is ually handled by the Ewald summation technique. Howevthe simulation becomes more difficult to treat in aquasi-two-dimensional system, because we have to take into accouthe effect of surfaces and interfaces~e.g., membranes! @43#.To treat the periodic boundary condition in a quasi-twdimensional system, we adopt the Lekner summation tenique in the following simulation@44#. In the Lekner sum-mation technique, the pair potentialV2d(x,y;z) between twoparticles in a quasi-two-dimensional system is given by

V2d~x,y;z!5qiqje

2

4pe (nx52`

`

(ny52`

`1

ur1nxLex1nyLeyu

5qiqje

2

4pe F4(n

(k

K0~2pnA$y1k%21z2!

3cos~2pnx!2 ln$cosh~2pz!

2cos~2py!%1c2G , ~6!

where the indicesnx andny run over the periodic images othe simulation box, andex , ey , K0(x), andc2 are unit vec-tors in thex andy directions, the modified Bessel functionand the constant to be added, respectively@44#. Equation~6!is rapidly convergent due to the asymptotic behavior ofmodified Bessel function, such as

K0~x!;Ap

2xexp~2x!. ~7!

For the more technical details of treating periodic imacharges, see Ref.@44#.

As explained in the preceding section, there is a difficuin performing the traditional Monte Carlo~MC! simulationof asymmetric electrolytes with strong electrostatic coplings, where the accumulation of counterions close tomacroion surface depresses the macroion mob@13,14,42#. We apply a cluster-move technique in the Msimulation to enhance the mobility of macroions, wheredefine a cluster by selecting a macroion, and surroundcounterions within a distancer cl from the center of the macroion @45#. Then we perform a trial move of the whole cluter with an acceptance probabilityPacceptsuch as

Paccept5min$1,exp~2DE/kBT!%dnnew,nold, ~8!

whereDE, nnew, andnold denote the total energy differencbetween, after, and before its trial move, the number of coterions inside the cluster (r ,r cl) after the move, and thabefore the move, respectively. The factordnnew,nold

in Eq. ~8!

is required to satisfy the detailed balance condition. Thevantage of the cluster MC method will be greater aselectrostatic coupling is increased. In the MC procedutranslational displacements in thex, y, and z directions aremade with step lengths uniformly sampled from@2D/2,D/2#, whereD is the translational displacement parameter.

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ADSORPTION OF COLLOIDAL PARTICLES ON A . . . PHYSICAL REVIEW E65 021405

the following simulations, we take the same magnitudethe translational displacement parameterD for all colloidalparticles and counterions.

At first, the profile of negatively charged colloidal paticles is studied by cluster Monte Carlo simulation in t

FIG. 1. Snapshot of colloidal particles with a planer chargsurface.~a! The electrostatic coupling parameterG is taken to beG50.4. ~b! The electrostatic coupling parameterG is taken to beG53.7.

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(N,V,T) ensemble. We start with an arbitrary particle cofiguration that does not penetrate into other particles. Tdiameter of colloidal particles is set to be 2r m54.0 nm, andthe magnitude of the surface charge on colloidal particleZis taken to beZ520, 40, and 60. It takes 104 Monte Carlosteps ~MCS! to get the system into equilibrium, and 105

MCS to take the canonical average after the equilibriumbeen reached. We also take the sample average ovesamples for each run. In the following, the temperatureT andthe relative dielectric constant of watere r([e/e0) are takento be T5300 K ande r578, respectively, wheree0 is thevacuum dielectric constant. The surface charge densitys of acharged surface is set to bes520.03 (C/m2), and the radiusof counterionsr c is taken asr c52.0 Å. Other parameters othe system are displayed in Table I.

Figure 1 shows the snapshot of colloidal particles acounterions, where the left side of the box corresponds tonegatively charged surface. Figures 1~a! and 1~b! denote theresults with the electrostatic couplingsG50.4 ~run A! and3.7 ~run D!, respectively, and we can see that these twosults are apparently different. As the electrostatic couplparameterG increases, diffusing counterions are boundmacroion surfaces. In Fig. 1~a!, there is no colloidal particlenear the charged surface, reflecting the double-layer resion between colloidal particles and the surface. In Fig. 1~b!,on the contrary, we can see that negatively charged colloparticles are adsorbed on the surface. The latter result cabe explained by the mean-field DLVO theory.

To check the finite-size effect, we study the density dtribution function r(z) of colloidal particles at a chargesurface. Figure 2 shows the normalized densities of colloparticlesr(z) with the different number of colloidal particleNm and the system sizeL. Herez is the distance between thsurface and the center of colloidal particles. The density dtribution functionr(z) is sampled by use of a uniform griof 2.5r c . Here the macroion volume densityfm is fixed tobe fm50.01. In Fig. 2, open squares and open circles rresent the calculated results withNm520 and 44, respectively. The abscissa and the ordinate denote the distancetween the surface and the center of colloidal particles,the normalized density of colloidal particles, respective

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FIG. 2. Density distribution functionr(z) of colloidal particlesat a charged surface with different number of colloidal particNm . The electrostatic coupling parameterG is taken to beG50.4.

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TAKAMICHI TERAO AND TSUNEYOSHI NAKAYAMA PHYSICAL REVIEW E 65 021405

and the statistical error is also shown by the vertical bar. Tmagnitude of the surface chargeZ and the valence of counterionsq are taken to be (Z,q)5(20,1). In Fig. 2, we havean agreement between calculated results withNm520 ~runA! and 44 macroions~run B! within the statistical error.

Figure 3 shows the density distribution functionr(z) ofcolloidal particles at a negatively charged surface wmonovalent, divalent, and trivalent counterions. Opsquares, open circles, and open triangles show the rewith the electrostatic couplingsG50.4 ~run A!, 1.6 ~run C!,and 3.7~run D!, respectively. The solid lines are only guidto the eye. The translational displacement parameterD, theacceptance ratio of the cluster movementspcl , and that ofthe counterion movementspco are shown in Table II. Theseresults indicate that the profile of the density distributifunction r(z) drastically changes atG>2. The inset showsthe effective potentialU(z) between colloidal particles andcharged surface obtained byU(z)[2kBT ln r(z). Solidcircles correspond to the result with the electrostatic cpling G53.7, and we can see that the magnitude of attracwell at z'2r m is larger thankBT. This indicates that theattractive interaction between colloidal particles andcharged surface is strong enough, compared with the thefluctuation energy (;kBT).

IV. CONCLUSIONS

In conclusion, we have investigated the adsorption pnomena of colloidal particles on a charged surface bycluster Monte Carlo simulation. From the technical pointview, it is very important to study the adsorption of fincolloidal particles to a surface. We have studied the strucand thermodynamics of colloidal suspensions on a char

FIG. 3. Density distribution functionr(z) of colloidal particlesat a charged surface with different strength of electrostatic cplings G. The inset shows the effective potentialU(z)[2kBT ln r(z) between colloidal particles and a charged surface.

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surface, where the electrostatic coupling parameterG is var-ied systematically. Using the primitive model of asymmetelectrolytes, the density profile of colloidal particles has benumerically clarified as a function of the electrostatic copling parameterG, from G50.4 ~weak coupling regime! toG53.7 ~strong coupling regime!. It has been found that thestructural formation of colloidal particles on a charged sface is sensitive toG in an aqueous solution. In the stroncoupling regime, numerical results have indicated thelike-charged adsorptionbetween colloidal particles and a chargsurface, which the DLVO theory cannot reproduce. Thefeatures have shown a good agreement with the prevstudy by Svensson and Jonsson@42#, and the importance othe Coulomb correlation and the fluctuation effect in the stem. In an aqueous solution with the strong electrostatic cpling, such like-charged adsorption will occur not onlycolloidal suspensions, but also in polyelectrolyte solutioWe have applied the cluster Monte Carlo technique to trthe difficulty, due to the accumulation of counterions closethe macroion surface@14#. It has been confirmed that thcluster MC technique is efficient to study the adsorption pnomena in an asymmetric electrolyte, especially in the cof the stronger electrostatic coupling.

It has been suggested that multivalent counterionscondensed and form a two-dimensional strongly correlaliquid at the surface of a macroion@32#. In previous studies,the counterion condensation has been studied by molecsimulations in a solution of highly charged rigid polyelectrlytes @46# and in spherical colloidal suspensions@40#. Theyhave indicated that while the agreement betweenPoisson-Boltzmann theory and simulation is excellent inmonovalent, weakly charged case, it deteriorates withcreasing the strength of the electrostatic interaction andparticular, with increasing the valence of microions. Oursults will shed light on a characteristic feature in soft mattbeyond the mean-field theory. The application of these prerties to produce various molecular assemblies is highlysirable, and these are future problems.

ACKNOWLEDGMENTS

This work was supported in part by a Grant-in-Aid frothe Japan Ministry of Education, Science, and CultureScientific Research. The authors thank the SupercompCenter, Institute of Solid State Physics, University of Tokfor the use of the facilities.

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TABLE II. Simulation parameters for each run~see the text fornotation!.

Run D ~nm! pcl pca

A 1.0 0.11 0.76B 1.0 0.11 0.78C 1.0 0.24 0.39D 0.5 0.35 0.25

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ADSORPTION OF COLLOIDAL PARTICLES ON A . . . PHYSICAL REVIEW E65 021405

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