Physica A 388 (2009) 1359–1366

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Physica A

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Influence of surface affinity of planar walls on effective interactionbetween the wall and colloidsLing Zhou a,b, Yue Jiang a, You-qi Wang a, Yu-qiang Ma a,∗a National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, Chinab School of Science, Nantong University, Nantong 226007, China

a r t i c l e i n f o

Article history:Received 5 October 2008Received in revised form 18 November2008Available online 25 December 2008

Keywords:Effective interactionColloidsSurface affinityDensity functional theory

a b s t r a c t

Using density functional theory, we investigate the effective interaction between a bigcolloid immersed in a sea of small colloids and a wall which has different affinity to thesmall colloids. Steele 10-4-3 potential is introduced to mimic both short-range repulsiveand long-range attractive interactions between the wall and the small colloids. It is foundthat the surface affinity of thewall has a significant influence on the effective interaction. Inthe short-range repulsive case, the repulsion greatly enhances the big colloid-wall effectiveattraction, which sensitively depends on the concentration of small colloids, and is notsensitive to the repulsive strength. In the long-range attractive case, both the concentrationof small colloids and the attractive strength have great effect on the effective interaction,and with an increase of the attractive strength, a strong repulsion may be induced whenthe big colloid is close to the wall. In low density limit of small colloids, the present resultsagree well with those of the Asakura and Oosawa(AO) approximation.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Recently,many theoreticalmethods [1–9] and experimental techniques [10–19] have been developed to detect depletioninteractions. Such a fundamental physical behavior has a lot of potential applications in industry and biological systems[20–22]. For example, it can be used to promote the aggregation of red blood cells [23], separation of different size anddifferent packing fractions of oil-in-water emulsion droplets [24], crystallization of protein solutions [25,26], and eventhe formation of a variety of ordered structures of colloids [27,28]. It has been widely accepted that the excluded volumeinteraction between particles of different sizes can induce an entropic attraction between two big particles or betweena big particle and a hard planar wall, if only hard-core interactions are considered [9]. Now the depleting agent can besimple fluids [29], colloids [9], polymer molecules [30], micelles [31], hemoglubines [23], and globular proteins [25]. Theshape can be spherical [32], rod-like [33,34], or elliptical [35], etc. However, except for the differences in the componentand the shape, the influences of the surface affinity can not be neglected. Solid surfaces are often decorated with chemicallydistinct ligands that display preferential wetting interactions between the surfaces and the host fluid. In particular, chemicalsurface modification for biocompatibility has many useful applications such as biosensors. In recent years, more and moresoft interactions have also been concerned, which are more close to realistic circumstances. Several experiments havemeasured the interactions between cleaved or chemically modified mica surfaces that are immersed in polar liquids [36],nonpolar liquids [37], and polymer melts [38]. On the contrary, relatively little theoretical work has been done on thisaspect until recently [39]. In the present study, we focus on the effective interaction between a big colloid immersed in

∗ Corresponding author. Tel.: +86 25 83592900; fax: +86 2583595535.E-mail address:myqiang@nju.edu.cn (Y.-q. Ma).

0378-4371/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2008.12.032

1360 L. Zhou et al. / Physica A 388 (2009) 1359–1366

a sea of small colloids and a wall which has different affinity to the small colloids. The main aim of the present work isto apply density functional theory to systemically investigate how effective interactions depend on the surface affinity ofthe wall. By controlling the energy parameter of the surface, we successfully realize the effective attraction or repulsionbetween the big colloid and the wall. The results are consistent with the experiments done by Bechinger et al. [40] whomeasured the effective potential of a colloidal polystyrene sphere in front of a flat glass surface in the nonionic polymericfluids. We also notice that Louis et al. [39] once used Yukawa pair potentials to mimic repulsive and attractive interaction.Our results qualitatively agree with their results. Moreover, we further perform an extensive exploration in a wide range ofthe interaction strength and the concentration of small colloids, and observe the oscillation of the effective interaction andthe transition of the depletion force from attraction to repulsion with the increase of the attractive strength, which has notyet be predicted before.

2. Model and method

We consider a binary mixture composed of big colloids with diameter σb and small colloids with diameter σs. The wallis placed in xy plane, and z axis is vertical to the wall and directs to the liquid. The origin of the coordinate is chosen at thesurface of the wall. The colloid-colloid interaction is described by a simple excluded-volume pair potential

uij(r) =∞, r < (σi + σj)/2,0, otherwise, (1)

where the subscripts i or j (=s and b) denote small and big colloids, respectively, and r is the center-to-center distance.With the change of the surface affinity of the wall, the wall-small colloid and thewall-big colloid interactions will change

accordingly. The latter contains the direct colloid-wall interaction and the effective interaction induced by small colloidparticles. However, the real interaction between big colloids and thewall has no influence on the calculation of small colloid-mediated effective interaction which is performed under the diluted limiting of big colloids, and for simplicity, the externalpotential exerted by the wall on big colloids is given by

V extb (z) =∞, z < σb/2,0, otherwise. (2)

In order to apply a unifying potential to express both short-range repulsive and long-range attractive interactions,we choosethe external potential of the wall acting on small colloids as follows

V exts (z) =

∞, z ≤ 0,u(z)− u(Rcut), 0 < z ≤ Rcut ,0, z > Rcut .

(3)

Here, Rcut is the cutoff distance, and u(z) is taken as the Steele 10-4-3 potential [41]:

u(z) = εw

[25

(σwz

)10−

(σwz

)4−

σ 4w

3∆(z + 0.61∆)3

], (4)

with the chosen parameters σw = 0.903σs, εw = 12.96ε, and∆ = 0.8044σs. The consequence of selecting such a potentialis that ε becomes the only adjustable parameter that reflects the strength of small colloid-wall interactions (here ε is inunits of kBT ). Different cutoff distance Rcut is taken to imitate two types of wall potentials. For the repulsive surface, we takeRcut = 0.76σs, and for the attractive surface, we take Rcut = 20σs. Fig. 1 illustrates the distributions of the repulsive andattractive potentials respectively. For the sake of clarity, we use case 1 and case 2 to represent the repulsive and attractivecases, respectively. Here, the Steele 10-4-3 potential is introduced instead of Yukawapotential [39] to simulatemore realisticsystem, and we can successfully realize the attraction and repulsion of the surface by adjusting the cutoff distance Rcut .In the framework of density functional theory, the grand potential of the system can be written as a functional of local

densities ρi(r):

Ω[ρi(r)] = F [ρi(r)] +∑i=s,b

∫d3rρi(r)(V exti (r)− µi), (5)

where µi is the chemical potential of the two species, and V exti (r) is the external potential. F is the Helmholtz free energywhich includes an ideal gas part and an excess part.

F = Fid + Fex. (6)

The ideal gas part is directly written as:

βFid =∑i=s,b

∫d3rρi(r)(ln(λ3i ρi(r))− 1) (7)

where λi = (2π h/mkBT )1/2 is the thermal wavelength for species i.

L. Zhou et al. / Physica A 388 (2009) 1359–1366 1361

Fig. 1. The Steele 10-4-3 potential as a function of the distance z with the cutoff distance Rcut = 0.76σs . The inset shows the corresponding potential withthe cutoff distance Rcut = 20σs .

For excess free energy functional part, we use the Fundamental Measure Theory(FMT) of Rosenfeld [42,43], and obtain

βFex =∫d3rΦ(nα), (8)

where β = (kBT )−1, Φ is the excess free energy density, and nα denote weighted densities which are evaluated via spatialconvolutions

nα(r) =∑i=s,b

∫d3r ′ρi(r′)wiα(r− r′), (9)

where the subscripts α (=0, 1, 2, 3, v1, v2) represents the index of six weight functionswiα(r) [42,43].There are several expressions for excess free energy density Φ [43], and here we choose the White-Bear (WB)

version [44]:

Φ(nα) = −n0 ln(1− n3)+n1n2 − nv1 · nv2

1− n3+ (n32 − 3n2nv2 · nv2)

n3 + (1− n3)2 ln(1− n3)36πn23(1− n3)2

. (10)

Eq. (10) is based on Mansoori–Carnahan–Starling–Leland (MCSL) state equation instead of the Percus–Yevick (PY)compressibility equation, and gives the improvement compared with the original FMT [45].The final equilibrium density profiles of the mixtures is determined by the minimization of grand potential Eq. (5) with

Eqs. (6)–(10).Further, we hope to calculate the effective potential between a big colloid and the wall. A diluted limit of big colloids is

taken, i.e., ρb → 0 or equivalently µb →−∞. In this case, the weighted densities reduce to

ndilα (r) =∫d3r ′ρs(r′)wsα(r− r′), (11)

which depend only on the density profile ρs(r) of small colloids.In the diluted limit of big colloids, the effective potential W (r) can be easily expressed as the difference between the

one-particle direct correlation function C (1)b in the bulk(r→∞) and at the position r [9]:

βW (r) = limρb→0[C (1)b (r→∞)− C

(1)b (r)]. (12)

It is well known that the Rosenfeld density functional theory we use is accurate for describing the inhomogeneous fluid.So we choose relatively lower packing fractions of the fluid to ensure that the phase transitions (interfacial or bulk) are notoccurred.Using the definition of the one-body direct correlation function given by density functional theory,

C (1)b (r) = −βδFexδρb(r)

, (13)

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Fig. 2. Comparison of the density profiles of small colloids in the cases that the small colloids-wall interaction is repulsive (case 1) and attractive (case 2).In both cases the size ratio of big and small colloids is σb:σs = 5:1, the bulk packing fractions of them are ηs = 0.2 and ηb = 10−4 , and the interactionstrength is ε = 0.8.

we obtain

βW (r) =∑α

∫d3r ′Ψα(r′; ndilα )w

bα(r− r′), (14)

where

Ψα(r′; ndilα ) = β(∂Φ(ndilα )∂ndilα

)r′− β

(∂Φ(ndilα )∂ndilα

)∞

. (15)

A direct calculation of the effective potential can be implemented through Eq. (14), based on the one-particle directcorrelation function of big colloids in a sea of small colloids.

3. Results and discussion

We first compare the density profiles of small colloids in the cases 1 and 2. The size ratio between big and small colloidsis fixed to be σb : σs = 5 : 1 in all following calculations. Due to the translational invariance in the xy plane, the calculationscan be carried out along z axis. Fig. 2 gives the density profiles in both cases with ε = 0.8. The bulk packing fractions of smalland big colloids are ηs = 0.2 and ηb = 10−4. Here, ηb is sufficiently small, ensuring that the density profile of small colloidsis indistinguishable from that of a pure fluid at the same ηs. We find that in the case 2, a large number of small particlesare attracted by the wall and a dense layer of small particles is formed near the wall, while the case is opposite when thefluids contact a repulsive wall. Notice that in the case 1, though the potential of the wall is repulsive, there still exists a weakadsorption near the wall due to the hard-sphere repulsive property of small particles. The present result also shows thatEq. (3) can be reasonably used to describe the repulsive and attractive interactions of the wall.We now turn to the calculation of effective interactions. In the case 1, Fig. 3(a) shows the depletion potential W (h)

between a big colloid and a repulsive wall as a function of the distance h between the big colloid surface and the wall at afixed packing fraction ηs = 0.3with ε = 0.05 (dotted line), 0.2 (dot-dashed line), and 0.8 (solid line), respectively. The insetshows the corresponding profiles of the effective force F(h). F(h) is calculated through the definition F(h) = −∂W (h)/∂(h),which arises from the force that the small particles exert on the big colloid. Fig. 3(b) gives the corresponding density profilesof small colloids. For the sake of comparison, we also give the hard wall case(dashed line) with the same ηs. Comparedwith the hard-wall case, we find that the shapes of the curves are similar, namely the effective potential and force oscillatearound zero, exhibiting the attractive and repulsive effects. However, the contact value of the effective potential and forceat h = 0 and the position of the maximum are changed dramatically. When the big colloid is very close to the wall, wecan find an enhanced attraction, and the position of effective potential barrier shifts to the relatively larger value of h. Therepulsion between the small colloids and the wall pushes the small colloids into the bulk and generates a large depletionlayer. Therefore, the contact of the big colloid and thewall will providemore free volume to the small colloids, leading to theenhancement of depletion attractive potential and force. Such an enhanced attraction has been observed between a colloidsphere and a flat glass surface in polymer coils using total internal reflection microscopy [40]. Interestingly, we further findthat the repulsive strength ε has very little influence on depletion interaction. It can be seen from Fig. 3(a) that under thevarying values of ε, even the curves in the enlarged profiles are nearly indistinguishable. The reason is that the interactionhere is a short-range repulsive one. Seen from Fig. 3(b), once the small particles are pushed away from the wall, the density

L. Zhou et al. / Physica A 388 (2009) 1359–1366 1363

Fig. 3. (a) Effective potential between a big colloid and a repulsive wall at a fixed bulk packing fraction ηs = 0.3 with the repulsive strength ε = 0.05(dotted line), 0.2 (dot-dashed line), and 0.8 (solid line). Dashed line is the hard wall case with the same bulk packing fraction for comparison. The insetshows the corresponding profiles of effective force. (b) The corresponding density profiles of small colloids.

Fig. 4. (a)Effective potential between a big colloid and a repulsive wall for a fixed repulsive strength ε = 0.8with different bulk packing fractions ηs = 0.1(dotted line), 0.2 (dot-dashed line), and 0.3 (solid line). The inset shows the corresponding profiles of effective force. (b) The corresponding density profilesof small colloids.

profile of the small colloidswill not change greatly by increasing the interaction strength, and accordingly effective potentialand force keep almost unchanged. However, the concentration of small colloids has great influence on the effective potentialand force. Fig. 4(a) shows the effective potentialW (h) between a big particle and a repulsive wall at different bulk packingfractions ηs = 0.1 (dotted line), 0.2 (dot-dashed line), and 0.3 (solid line) with a fixed interaction strength ε = 0.8. The insetshows the corresponding effective force F(h) as a function of h. The density profiles of small colloids are shown in Fig. 4(b).Here, we choose the bulk packing fraction ηs from 0.1 to 0.3, which can guarantee that the system does not freeze. Withthe increase of the concentration of the small colloids, we find that the oscillation of both the effective potential and forcebecomes more pronounced, especially the effective potential and force at the contact point showmore negative values andthemaximumof them shifts to a smaller value of h. This can be explained by the packing effect of small colloids. Actually, theeffective interaction is closely related to the equilibrium density profiles of small colloids: the increase of the concentrationof the small colloids leads to a distinct layer structure of density profile, which directly results in the oscillation of theeffective potential and force.For the case 2, we emphasize the effect of the different attractive strengths ε. First we consider the weak attractive case.

Fig. 5(a) shows the effective potential W (h) between a big particle and an attractive wall with ε = 0.2 and ηs = 0.1(dotted line), 0.2 (dot-dashed line), and 0.3 (solid line). The inset plots the corresponding effective force F(h). Fig. 5(b) plotsthe corresponding density profiles of small colloids. Compared with the case 1, the curves of the effective potential havea pronounced shift. The most significant difference is that W (h) ≥ 0 at all distances. The height of the potential barrier

1364 L. Zhou et al. / Physica A 388 (2009) 1359–1366

Fig. 5. (a) Effective potential between a big colloid and an attractive wall at ε = 0.2 with ηs = 0.1 (dotted line), 0.2 (dot-dashed line), and 0.3 (solid line).The inset shows the corresponding profiles of effective force. (b) The corresponding density profiles of small colloids.

Fig. 6. Same as in Fig. 5 except for ε = 0.8 with ηs = 0.05 (dotted line), 0.1 (dot-dashed line), and 0.2 (solid line).

increases and shifts towards left (the smaller values of h). At the contact point (h = 0), the depletion force is still attractivewhich is similar to the case 1, but the magnitude is greatly decreased.We further increase the attractive strength ε, and Fig. 6(a) shows the effective potentialW (h) between a big particle and

an attractive wall with ε = 0.8 and ηs = 0.05 (dotted line), 0.1 (dot-dashed line), and 0.2 (solid line). The inset shows thecorresponding effective force F(h). It is observed that a strong repulsive interaction is induced at a small separation. Underthe influence of the dense layer of small particles near the wall, the big colloid is difficult to approach the wall, and thereforefeels an effective repulsion. The potential barrier is no longer existent and the effective potential reaches itsmaximum at thecontact point. At a small distance h, the effective potential decreases monotonously. Of course, at intermediate separations,the oscillatory behavior is still observed due to the packing efficiency. Here, we should point out that in the present case,we do not consider the case that ηs = 0.3, due to the fact that strong attractive small colloids-wall interaction can cause thedense adsorption of small colloids near the wall and they may crystallize into a solid, which is not concerned in the presentwork. Additionally, we can see from Figs. 5 and 6 that the concentration of the small colloids has an obvious influence on theeffective interaction, and the effective potential and force are increased by the increase of the small colloids in both weakand strong attractive cases. The results are also attributed to the packing effect as those in the case 1.Finally, we discuss the case that the small colloids are in the low density limit. The excess free energy functional has the

following form [9]

limµi→−∞

βFex = −12

∑i,j

∫d3r

∫d3r ′ρi(r)ρj(r′)fij(r− r′). (16)

L. Zhou et al. / Physica A 388 (2009) 1359–1366 1365

Fig. 7. Comparison of effective potential by DFT and AO approximation in (a) repulsive and (b) attractive cases with ε = 0.2 and ηs = 0.001. The insetshows the corresponding profile of the force. Solid line denotes the result of DFT, whereas open circle denotes the result of AO approximation.

where fij is the Mayer bond between colloids of species i and j. According to the definition given in Eq. (13), the one-bodydirect correlation function can be written as

C (1)b (r) =∫d3r ′ρs(r′)fbs(r− r′). (17)

Substituting Eq. (17) into Eq. (14), we have

βW (r) = −∫d3r ′(ρs(r′)− ρs(∞))fbs(r− r′). (18)

Note that in low density limit, the density profile of small colloids reduces to the density profile of an ideal gas in the externalpotential V exts (r),

ρs(r) = ρs(∞) exp[−βV exts (r′)], (19)

then we can simplify effective potential as

βW (r) = −ρs(∞)∫d3r ′(exp[−βV exts (r

′)] − 1)fbs(r− r′). (20)

This is the well-known AO depletion potential, where

fbs(r− r′) = −Θ((σb + σs)/2− |r− r′|), (21)

which is the Mayer bond between the big and small colloid.We make a comparison of the effective potential and force by DFT and AO approximation in the cases 1 and 2, as shown

in Fig. 7. Obviously, both results agree very well. The reason can resort to the fact that the excess free energy functional forthe FMT is constructed by interpolating between the low density limit described by the pair excluded volume and the idealliquid asymptotic limit characterized by one-particle geometries. Therefore, it is reasonable that when the fluid is close tothe ideal gas case, the two excess free energy expressions described by Eqs. (8) and (16) become equivalent. This is alsohelpful to verify the validity of the model studied here.

4. Conclusions

We apply density functional theory to examine the influences of the surface affinity on effective interactions. When thewall is repulsive to the small colloids, the effective attractionwill be enhanced by depleted small colloids near thewall. In theshort-range repulsive case, the effective interaction is sensitive to the concentration of small colloids, but is not sensitive tothe repulsive strength of the wall. When the wall is attractive to the small colloids, both the concentration of small colloidsand the interaction strength have great influences on the effective interaction. With the increase of the attractive strength,the transition of the effective force from attraction to repulsion can be observed. In low density limit of small colloids, thepresent results are consistentwith those of AOapproximation.We sufficiently consider the influences of the different surfaceaffinity on the effective interactions except excluded volume effects, which can provide a guide to realize the control of themicrostructures.

Acknowledgements

This work was supported by the National Basic Research Program of China, No. 2007CB925101, and the National NaturalScience Foundation of China, Nos. 10574061 and 20674037.

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References

[1] S. Asakura, F. Oosawa, J. Chem. Phys. 22 (1954) 1255.[2] A. Vrij, Pure Appl. Chem. 48 (1976) 471.[3] P. Bryk, Langmuir 22 (2006) 3214.[4] P. Attard, J.L. Parker, J. Phys. Chem. 96 (1992) 5086.[5] M. Kinoshita, T. Oguni, J. Chem. Phys. 116 (2002) 3493.[6] S. Amokrane, J. Chem. Phys. 108 (1998) 7459.[7] H. Shinto, M. Miyahara, K. Higashitani, J. Colloid Interface Sci. 209 (1999) 79.[8] A. Milchev, A. Bhattacharya, J. Chem. Phys. 117 (2002) 5415.[9] R. Roth, R. Evans, S. Dietrich, Phys. Rev. E 62 (2000) 5360.[10] D. Rudhardt, C. Bechinger, P. Leiderer, J. Phys.: Condens. Matter 11 (1999) 10073.[11] A. Sharma, S.N. Tan, J.Y. Walz, J. Colloid Interface Sci. 191 (1997) 236.[12] P.C. Odiachi Jr., D.C. Prieve, Colloids Surf. A 146 (1999) 315.[13] A.J. Milling, K. Kendall, Langmuir 16 (2000) 5106.[14] A.J. Milling, J. Phys. Chem. 100 (1996) 8986.[15] S. Biggs, J.L. Burns, Y.-d. Yan, G.J. Jameson, P. Jenkins, Langmuir 16 (2000) 9242.[16] J.C. Crocker, J.A. Matteo, A.D. Dinsmore, A.G. Yodh, Phys. Rev. Lett. 82 (1999) 4352.[17] P.K. Thwar, D. Velegol, Langmuir 18 (2002) 7328.[18] E.S. Pagac, R.D. Tilton, D.C. Prieve, Langmuir 14 (1998) 5106.[19] L. Helden, R. Roth, G.H. Koenderink, P. Leiderer, C. Bechinger, Phys. Rev. Lett. 90 (2003) 048301.[20] J.A. Lewis, J. Am. Ceram. Soc. 83 (2000) 3241.[21] M. Fuchs, K.S. Schweizer, J. Phys.: Condens. Matter 14 (2002) R239.[22] J. Janzen, D.E. Brooks, Lin. Hemorheol. 9 (1989) 695.[23] G.H. Koenderink, G.A. Vliegenthart, S.G.J.M. Kluijtmans, A. van Blaaderen, A.P. Philipse, H.N.W. Lekkerkerker, Langmuir 15 (1999) 4693.[24] M. Piech, P. Weronski, X. Wu, J.Y. Walz, J. Colloid Interface Sci. 247 (2002) 327.[25] A.M. Kulkarni, A.P. Chattarjee, K.S. Schweizer, C.F. Zukoski, Phys. Rev. Lett. 83 (1999) 4554.[26] A.M. Kulkarni, A.P. Chatterjee, K.S. Schweizer, C.F. Zukoski, J. Chem. Phys. 113 (2000) 9863.[27] J.C. Loudet, P. Barois, P. Poulin, Nature 407 (2000) 611.[28] M. Adams, Z. Dogic, S.L. Keller, S. Fraden, Nature 393 (1998) 349.[29] Y. Qin, K.A. Fichthorn, J. Chem. Phys. 119 (2003) 9745.[30] N. Patel, S.A. Egorov, J. Chem. Phys. 121 (2004) 4987.[31] R. Tuinier, E. ten Grotenhuis, C. Holt, P.A. Timmins, C.G. de Kruif, Phys. Rev. E 60 (1999) 848.[32] S.A. Egorov, Phys. Rev. E 70 (2004) 031402.[33] Y.-L. Chen, K.S. Schweizer, J. Chem. Phys 117 (2002) 1351.[34] P.G. Bolhuis, A. Stroobants, D. Frenkel, H.N.W. Lekkerkerker, J. Chem. Phys 107 (1997) 1551.[35] W. Li, H.R. Ma, J. Chem. Phys. 119 (2003) 585.[36] H.K. Christenson, R.G. Horn, Chem. Phys. Lett. 98 (1983) 45.[37] H.K. Christenson, J. Phys. Chem. 90 (1986) 4.[38] R.G. Horn, S.J. Hirz, G. Hadziioannou, C.W. Frank, J.M. Catala, J. Chem. Phys. 90 (1989) 6767.[39] A.A. Louis, E. Allahyarov, H. Löwen, R. Roth, Phys. Rev. E. 65 (2002) 061407.[40] C. Bechinger, D. Rudhardt, P. Leiderer, R. Roth, S. Dietrich, Phys. Rev. Lett. 83 (1999) 3960.[41] Y. Tang, J. Wu, J. Chem. Phys. 119 (2003) 7388.[42] Y. Rosenfeld, Phys. Rev. Lett. 63 (1989) 980.[43] R. Roth, S. Dietrich, Phys. Rev. E 62 (2000) 6926.[44] R. Roth, R. Evans, A. Lang, G. Kahl, J. Phys.: Condens. Matter 14 (2002) 12063.[45] Y. Rosenfeld, Phys. Rev. E 50 (1994) R3318.