Capillary condensation and interface structure of a model colloid ...

CAPILLARY CONDENSATION AND INTERFACE ...PHYSICAL REVIEW E 68, 061404 2003A. Zero-dimensional limitIn this section we derive the zero-dimensional 0D Helmholtzfree energy for the three-component system of the AOcolloid-polymer mixture in contact with quenched hardspheres. This 0D free energy is used as an input to constructthe fundamental measure theory in the following section.Here, we give only a brief derivation, a more extensive versionwith more comments is given in Refs. 43,51. The essentialingredient is that we need to perform the so-called‘‘double average’’ which refers to the statistical average overall fluid configurations and subsequently over all matrix realizations.To that end we consider a 0D cavity which eitherdoes or does not contain a matrix particle. Hence, the 0Dpartition sum is that of a simple hard-sphere fluid,¯m1z¯m ,3FIG. 1. Sketch of the ternary mixture of mobile colloids dark,mobile polymers transparent, and immobile matrix particlesgray. The polymer coils can freely overlap. There are three modelparameters, i.e., the packing fraction of matrix particles ( m ), andtwo size ratios qR p /R c and sR m /R c , where R i is the radius ofparticles of species i. The packing fractions of colloids and polymers, c and p , respectively, are the thermodynamic parameters.N i /V, where N i is the total number of molecules of speciesi and V is the system volume. All of these components aremodeled as hard bodies, meaning that they cannot overlapbut otherwise do not interact with each other, except for thepolymer-polymer interaction, which is taken to be ideal, seeFig. 1. Consequently, when r is the mutual distance, the pairpotentials becomeu ij r if rR iR j0 if rR i R jfor i, jc,p,m except for i jp, 1and concerning the polymer-polymer interaction, this simplybecomesu pp r0 for all r. 2As all interactions are either hard core or ideal, the phasebehavior is governed by entropic packing effects and thetemperature T does not play a role. The only thermodynamicparameters are the colloid and polymer packing fractions c 4R c 3 c /3 and p 4R p 3 p /3, respectively. The remainingmodel parameters are two size ratios qR p /R c andsR m /R c and the packing fraction of matrix particles ( m4R m 3 m /3). It has to be mentioned that due to the factthat the polymers can freely overlap, the ‘‘polymer packingfraction’’ can easily be larger than one Fig. 1. The mixtureof hard spheres with these last-mentioned ideal polymersi.e., without the matrix particles is called the AO mixture15,16.III. DENSITY FUNCTIONAL THEORYwhere z¯m exp(¯m) is the fugacity of the hard spheres.Further, 1/k B T with k B Boltzmann’s constant and ¯m thechemical potential. The irrelevant prefactor scales with thevanishing volume of the cavity but has no effect to the finalfree energy and will not be discussed further. In general, weuse an overbar to refer to quantities of 0D systems. With thegrand potential, ¯mln ¯m , the average number of matrixparticles is ¯mz¯m¯m /z¯mz¯m /(1z¯m).Next, we consider the colloid-polymer mixture in contactwith the matrix in zero dimensions. If the cavity is occupiedby a matrix particle, no colloid, or polymer can be present.On the other hand, if there is no matrix particle, it can eitherbe empty, occupied by a single colloid, or an arbitrary numberof polymers. Hence,1 matrix particle in cavity¯ 4z¯cexpz¯p no matrix particle in cavity,where z¯c and z¯p are the colloid and polymer fugacities, respectively.Then, the contribution ln ¯ to the grand potentialshould contain the appropriate statistical weight for eachof the cases, i.e., z¯m /¯m for the first and 1/¯m for the second,¯ lnz¯cexpz¯p. 51z¯mAverage particle numbers are again readily obtained via ¯iz¯i¯/z¯i for ic,p not for m). The Helmholtz freeenergy can then be calculated using a standard Legendretransformation F¯¯ ic,p ¯i ln(z¯i), and we obtain forthe excess part, F¯exc F¯ ic,p ¯iln(¯i)1,F¯exc ¯c ,¯p ;¯m1¯c¯p¯mln1¯c¯m¯c1¯mln1¯m.This result can be shown to be equal from that which wouldbe obtained using the so-called ‘‘replica trick’’ 50.B. Fundamental measure theoryFMT is a nonlocal density functional theory, in which theexcess part of the three-dimensional free energy F exc is expressedas a spatial integral over the free energy density ,F exc i r dr „n i r….67061404-3

WESSELS, SCHMIDT, AND LÖWENThis free energy density in turn is assumed to depend on thefull set of weighted densities n i (r),n i r dr w i rr i r,w i i r,which are convolutions denoted with ) with the singleparticledistribution functions i (r) for species ic,p,m.The weight functions are obtained from the low-density limitwhere the virial series has to be recovered,w 3 i rR i r,w 2 i rR i r,w 1 i rR i r/4r,w 0 i rR i r/4r 2 ,iw v2 rR i rr/r,8iw v1 rR i rr/4r 2 ,with again ic,p,m being one of the three components, the Heaviside function, and the Dirac delta function. Thereare four scalar weight functions, with 3 to 0, correspondingto the volume of the particles, the surface area, the meancurvature, and the Euler characteristic, respectively, andthese are the so-called ‘‘fundamental measures’’ of thesphere. The two weights on the right-hand side of Eq. 9 arevector quantities. Often a seventh tensorial weight is used inthe context of freezing but this will not be used herei21,22,46. The dimensions of the weight functions w are(length) 3 .Then, the sole approximation made is that is taken tobe a function of the weighted densities n i (r) whereas mostgenerally one would expect this to be a functional dependence.This approximation totally sets the form of andfollowing Refs. 22,43,51 we give the expression for 1 2 3 in terms of the zero-dimensional free energyderived in the preceding section,with 3 18 1 n i 0 i n l 3 ,ic,p,m 2 n i 1 n j i j2 n v1 •n v2 ij n l 3 ,i, jc,p,mi, j,kc,p,m1 3 n 2 i n j 2 n k 2 n i j2 n v2k•n v2 ijk n l 3 , i1 ,...,i t¯j t F¯exc ¯j/¯i1 ¯ ¯it .910111213All i1 ,...,i tof which more than one indices equal p are zerodue to the form of F¯exc . Together, Eqs. 6 to 13 constitutethe excess free energy functional for this QA system.C. MinimizationHaving constructed the excess free energy, we can nowimmediately move on to the grand-canonical free energyfunctional of the colloid-polymer mixture in contact with amatrix, c r, p r; m rPHYSICAL REVIEW E 68, 061404 2003F exc c r, p r; m rk B T ic,p dr i rln„ i r i …1 ic,p dr i rV i r i .14Here, i is the ‘‘thermal volume’’ which is the product of therelevant de Broglie wavelengths of the particles of species i.Further, i is the chemical potential and V i is the externalpotential acting on component i. In this paper, we study bulkphase behavior and the free fluid-fluid interfaces, so we useV i 0. The equilibrium profiles are the ones that minimizethe functional,0 and c r p r 0.15This yields the Euler-Lagrange or stationarity equations (ic,p), i rz i expc i (1) „ j r…,16with z i 1 i exp i the fugacity of component i and theone-particle direct correlation functions given byc (1) i r F exc j r i r r. i wi n 17Obviously, the functional is not minimized with respect tothe matrix distribution m (r) as this serves as an input profile.In principle, as we are dealing with a quenched-annealedsystem in which the matrix is initially before quenching ahard-sphere fluid, m (r) should still minimize the hardspherefunctional 43,51. However, as density functionaltheory allows us to generate any distribution m (r) by applyingany suited external potential which we can then removeafter quenching, we do not need to go into the schemeof generating matrix profiles. Moreover, in the present paperwe use fluid distributions of the matrix particles which minimizeat least locally the hard-sphere functional without externalpotential for any packing fraction. This restricts thematrix distribution to be homogeneous, i.e., constant inspace, m (r)const.IV. RESULTSIn this section, we show results of the effect of the hardspherematrix on the AO colloid-polymer mixture concerningcapillary condensation in a small sample of matrix, and061404-4

CAPILLARY CONDENSATION AND INTERFACE ...PHYSICAL REVIEW E 68, 061404 2003FIG. 4. Fluid-fluid binodals of an AO colloid-polymer mixture(q0.6) inside a bulk porous matrix (s1). Tie lines connectingcoexisting state point are not drawn but run horizontal. The lowerthicker curve is the result in the absence of any matrix, m 0.For the other curves, the matrix packing fraction increases frombottom to top, m 0,0.05,0.1,0.15,0.2. The filled circles are thecritical points large, m 0). The dotted line is the Fisher-Widomline for m 0, below which the decay of correlations in the fluid isoscillatory and above which these are monotonic. The point wherethe FW line hits the binodal is marked by a large star. The FWlines for the other matrix packing fractions are not shown, only theircrossings with the binodals small stars.sitive to gravity as long as the solvent is good. When there iscoexistence inside a test tube, there is only real coexistenceat the liquid-gas interface, whereas below and above, thecolloids have slightly different chemical potentials due totheir gravitational energy. Consequently, moving upwardfrom the interface, say z, the colloid chemical potential is(m c g)z lower than at coexistence. By placing the poroussample within z* c coex /(m c g) of the interface, capillarycondensation should take place. Taking as an example, c coex 0.1 and (m c g) 1 1 m, it becomes clear that,within the context of this idealized model, values of z*0.1 m should be accessible in experiments, meaning thatcapillary condensation could in principle be observed. Usingsmaller matrix particles, with size similar to the size of thecolloids our case s1), the effect of capillary condensationbecomes much more pronounced and it should therefore beobservable in a similar way as for large matrix particles.From an experimental point of view, however, we think thatit is a much larger effort to produce such matrices with lowenough packing fractions see Fig. 4, where we discuss m0.2) in order to be penetrable to the mobile colloids andpolymers. We recall the possible realization given in Sec. Iusing laser tweezers.FIG. 5. Same as in Fig. 4, but now for s50. Matrix packingfractions increase from right to left, m 0,0.1,0.2,0.3,0.4,0.5. Inset:same curves now rescaled, i.e., p,r vs c /(1 m ).C. Phase behavior inside a bulk porous matrixWe now return to the full ternary mixture in bulk, i.e.,where in the preceding section, the matrix was only a smallsample immersed in a large system of colloid-polymer mixture,in this and the following sections we consider thecolloid-polymer mixture in a system-wide matrix. In this section,we revisit the demixing phase behavior which we needin the following sections where we study the fluid-fluid interfaceinside a matrix. Figure 4 is the phase diagram in thepolymer-reservoir representation for colloid-sized matrixparticles, s1, for various matrix densities. Increasing thematrix packing fraction, there is less volume available to thecolloids and the critical point shifts to smaller colloid packingfractions. At the same time, the porous matrix acts tokeep the mixture ‘‘mixed’’ and therefore, the critical pointshifts to higher polymer fugacities. For the case of s1, wecannot go to much higher packing fractions than m 0.2 asthen the critical fugacity shoots up dramatically to unphysicallylarge values. This may be partly due to the relativelylarge depletion shells around the matrix particles whichcause the pore sizes to become too small for the colloids andpolymers to enter the matrix. In case of large matrix particles(s50, see Fig. 5, the latter effect is negligible and the poresizes are always large enough. Consequently, only the excludedvolume remains and rescaling the binodals with (1 m ) is very effective practically mapping the binodalsonto each other, Fig. 5 inset. This rescaling is unsuccessfulfor s1 as can be directly seen from the fact that the criticalfugacities in Fig. 4 are different for each of the matrix densities.In addition, we have determined the nature of theasymptotic decay of pair correlations of the fluid inside thematrix 61. These can either be monotonic or periodic andthe corresponding regions in the phase diagram are separatedby the Fisher-Widom FW line, at which both types of decayare equally long range. This line can be determined bystudying the pole structure of the total correlation functionsh ij in Fourier space 61. In the present case of QA systems,rather than using the usual Ornstein-Zernike equations, onehas to use the replica-Ornstein-Zernike ROZ equations50. Neglecting correlations between the replicas, these areh mm rc mm r m c mm h mm r,h ij rc ij r t c it h tj r,tc,p,m23061404-7

WESSELS, SCHMIDT, AND LÖWENPHYSICAL REVIEW E 68, 061404 2003FIG. 6. Colloid density profiles (V c 4 3 R c 3 ) normal to the freefluid-fluid interface for increasing matrix packing fractions at fixedpolymer fugacity. Parameters are q0.6, s1, and p,r 1 seeFig. 4. The matrix packing fraction increases from top thick profile, m 0) to bottom: m 0,0.05,0.1,0.15,0.2. Inset: correspondingpolymer profiles ( p V p vs z/R c , with V p 4 3 R p 3 ) for the samevalues of the matrix packing fractions also increasing from top tobottom.with i, jc,p,m except i jm. Here, for ijmm, thec ij (r) 2 F exc / i (r) j (r) are the direct correlationfunctions for which we obtain analytic expressions by differentiatingEq. 7. The matrix structure is determined beforethe quench, so c mm and h mm are those of the normal hardspherefluid at density m Percus-Yevick-compressibilityclosure, see Refs. 43,51. This analysis follows closely thatof Ref. 22 in which more details are given. In view of oursubsequent interface study, we focus on the point where theFW line meets the binodal. In Fig. 4 (s1), these are denotedby stars, and we observe that the shifts due to thematrices follow the same trend as the critical points. In caseof s50, we have not determined the FW lines, but there isno reason to expect the simple rescaling of the case withoutmatrix to fail in this case. Furthermore, concerning the densityprofiles in the following section, s50), we stay wellwithin the oscillatory regime.D. Fluid-fluid profiles inside a bulk porous matrixWe have calculated density profiles at coexistence normalto the colloidal gas-liquid interface. In this case of planarinterfaces, the density distribution is only a function of onespatial coordinate z; i.e., i (r) i (z). The only dependenceon the other two degrees of freedom is in the weights andthis can be integrated out, to obtain projected weights,w˜ (z)dx dy w i i (r) see, e.g., Ref. 62. The profiles arediscretized and calculated via an iteration procedure, i.e., weinsert profiles on the right hand side of Eq. 16 and thenobtain new profiles on the left hand side, which are thenreinserted on the right. Using step functions as iterationseeds, this procedure converges in the local direction of thelowest free energy. We normalize the densities as in bulk,i.e., we plot i (z)V i so that i ()V i (I,II) i , with I and IIreferring to the coexisting phases. The zero of z is set at theFIG. 7. Same as in Fig. 6 but now for s50. Other parametersare q0.6 and p,r 1 see Fig. 5. The matrix packing fractionsincrease from top thick profile, m 0) to bottom: m0,0.1,0.2,0.3,0.4,0.5. Inset: magnification of the rescaled profilesfor the same curves, i.e., c V c /(1 m )vsz/R c where again, matrixpacking fractions increase from top to bottom.location of the interface, defined through the Gibbs dividingsurface of the colloids: dz c (z) c ()0 0 dz c (z) c ()0.In Figs. 6 and 7, we have plotted the colloid profiles normalto the interface for s1 and s50, respectively. Colloidprofiles are shown for increasing densities of the matrix atfixed fugacity, p,r 1, corresponding to the bulk binodalsin Figs. 4 and 5. For the case of s1, this means that, as thecritical point shifts to higher fugacities, the profiles are effectivelytaken at fugacities closer to the critical value. Weobserve this well-known behavior in Fig. 6; close to the criticalpoint the profiles are smoother and modulations less pronounced.Away from the critical point, the interface is sharpbut the periodic modulations due to the surface extend far inthe bulk fluid. The inset of Fig. 6 shows the correspondingpolymer profiles. In Ref. 58, the main result is that theinterface widens due to the porous medium. The same happenshere and is due to fact that one is effectively closer tothe critical point.In Fig. 7, as we saw for the bulk phase diagram, there is asimple rescaling at work and the profiles merely differ by afactor (1 m ). The inset in Fig. 7 shows the same colloidprofiles but now rescaled, and we have zoomed in on theregion close to the interface. Clearly, even the modulationsfollow the case without matrix with the same accuracy as thebulk coexistence values in the inset of Fig. 5.We have also studied the asymptotic decay of correlationswith the interface via the density profiles. These must be ofthe same nature as the decay of the direct correlations in bulkdetermined via the ROZ equations, see the preceding section,i.e., either monotonic or periodic 61. However, determiningthe crossing points of the FW line with the binodalsusing the interfacial profiles yields a systematic shift awayfrom the critical point, compared to the bulk calculation(5%). Probably, this is due to numerical limits. Close tothis crossing point both the periodic and the monotonicmodes of decay are equally strong, so only far away from the061404-8

CAPILLARY CONDENSATION AND INTERFACE ...PHYSICAL REVIEW E 68, 061404 2003FIG. 8. Fluid-fluid surface tensions vs the difference in colloidpacking fractions of the two fluid phases for q0.6, s1, andvarious values of the matrix packing fraction. The matrix packingfractions increase from right ( m 0, thick curve to left, m0,0.05,0.1,0.15,0.2. Each curve is computed from the critical(crit)point, p,r p,r where 0) until twice the critical fugacity, p,r 2 (crit) p,r .interface truly asymptotic behavior may be observed. However,there, the periodic modulations may have become toosmall to be observable. Furthermore, our numerical routinehas no real incentive to minimize the tails of the profiles asthe gain in free energy is very low.E. Fluid-fluid surface tension inside a bulk porous matrixThe presence of the matrix also affects the surface tensionbetween the colloidal liquid and gas phases. The interfacialor surface tension of planar interfaces in the grand canonicalensemble is defined throughA inh PV,24where A is the amount of surface area, inh is the grandpotential for the inhomogeneous system, and P the pressurei.e., PV is the grand potential for the homogeneous bulksystem. With our numerical scheme we calculate densityprofiles in z direction so it makes sense to write the surfacetension as an integral,with dzzP,zk B T ic,p i zln„ i z i …1 ic,p i i zk B T„n i z….2526The quantity (z) is a ‘‘local’’ grand potential density whoseaverage over space yields the actual grand potential per unitFIG. 9. Same as in Fig. 8 but now for s50. Again, matrixpacking fractions increase from right ( m 0, thick curve to left, m 0,0.1,0.2,0.3,0.4,0.5. Each curve is computed from the critical(crit)point, p,r p,r where 0) until twice the critical fugacity, p,r 2 (crit) p,r . Inset: the same but rescaled curves are shown, i.e.,R 2 c /(1 m )vs (II) c (I) c /(1 m ).of volume inh /V. In Figs. 8 and 9 we have plotted thesurface tension versus the colloidal density difference in thetwo phases for s1 and s50, respectively. In both casesthe effect of the matrix is that the surface tensions increasefaster with the difference (II) c (I) c which is of course dueto the fact that the coexistence area becomes less wide as thecoexisting packing fractions themselves become smaller. Inthe inset of Fig. 9, we show the same curves rescaled with(1 m ), and the rescaled graphs fall almost on top of theoriginal one without any matrix. Here, we note that also thesurface tension has been rescaled with (1 m ); this isneeded from Eq. 21 as the free energy density (z) needsto be rescaled as well. Again, this rescaling procedure is notsuccessful for s1.Often, the surface tension is plotted against the relativedistance to the critical point, ( p,r / (crit) p,r 1) 27. However,this does not improve the rescaling for s1 and this can beseen from the fact that the end points of the curves in Figs. 8and 9 are all at twice the critical fugacity, p,r 2 (crit) p,r , andthe surface tensions rescaled or not are at quite differentvalues at the end points.V. CONCLUSIONWe have considered the full ternary system of hardspheres and ideal polymers represented by the AO model incontact with a quenched hard-sphere fluid acting as a porousmatrix. Using a QA DFT in the spirit of Rosenfeld’s fundamentalmeasure approach, we studied capillary condensationin a small sample of matrix as well as the fluid-fluid interfaceinside a bulk matrix. The results have been presented interms of two types of matrices: i colloid-sized matrix particlessize ratio s1) being a reference system and ii matrixparticles which are much larger than the colloids sizeratio s50). The case of small matrix particles is limited torelatively low packing fractions ( m 0.2), whereas in thesecond case, much higher matrix packing fractions are acces-061404-9

WESSELS, SCHMIDT, AND LÖWENPHYSICAL REVIEW E 68, 061404 2003sible ( m 0.5), the pores of the matrix being much larger.Additionally, we have suggested that case i as well as iicould in principle be realized experimentally in 3D, i.e., usinglaser tweezers and colloidal sediments, respectively, toserve as a model porous medium for colloidal suspensions.We have shown that in the limit of infinitely large matrixparticles, the standard AO results without matrix are recoveredvia a simple rescaling. In case of s50 our bulk butalso the interface results can be mapped onto the case withoutmatrix with high accuracy. However, in the case of smallmatrix particles (s1) this mapping fails, which is due tothe more complex and smaller pore geometry on the colloidalscale.Assuming a more ‘‘experimental’’ point of view, we haveconsidered a small sample of porous matrix immersed in alarge system of colloid-polymer mixture. When the fluidfluidbinodal is approached in the colloid-poor region of thephase diagram, capillary condensation occurs in the sample.This transition appears as a capillary line in the phase diagramin system representation extending along the binodaland ending in a capillary critical point. In case of small matrixparticles, the capillary lines for various densities of thematrix are well separated from the bulk binodal but the capillarycritical points lie deep into the colloidal gas regime.Concerning the large matrix particles, these capillary criticalpoints are located closer to the bulk critical point, however,the capillary lines are also very close to the binodal. Still,using density-matched colloidal suspensions, we argue thatcapillary condensation may be observable in experiments.We have computed fluid-fluid profiles inside the porousmatrix as well as the corresponding surface tensions. For s50, these can be mapped onto the case without matrix butfor s1 the critical point shifts to higher polymer fugacities.Therefore, increasing the density of the matrix, profiles becomesmoother due to effective approach of the criticalpoint. Solving the ROZ equations, we have also determinedthe crossover between monotonic and periodic decay of paircorrelations of the mixture inside the matrix for s1. Comparingthese with the decay of the interfacial correlations wefind a small discrepancy which is probably due to numericallimits.It should be noted that we do not expect our current approachto satisfactorily describe the subtle phenomena associatedwith wetting of the curved surfaces of the matrixparticles by the colloidal liquid 60,63. Especially for s50 close to the critical point in the complete wetting regime,we can well imagine that the growths of thick films ofcolloidal liquid on the matrix spheres have a profound influenceon the occurrence and precise location of the capillarycondensation transition and on the structure of the fluid-fluidinterface inside the matrix. We do not expect this effect to beincluded in our current treatment. Note that in order to obtainthe wetting transition at a hard wall 27,28 and at a curvedsurface 63 inhomogeneous density profiles need to be calculated,which we do not do in our present method of investigationof the bulk phase behavior.We also note that the attraction between colloids, as wellas that between colloids and matrix particles which is generatedby the polymers, arises naturally from our DFT treatment.This depletion attraction, however, has a many-bodycharacter for the size ratio considered (q0.6) as multipleoverlap between one polymer and three or more colloids canoccur; for a discussion of how to obtain an effective Hamiltonianfor the colloids by integrating out the polymer degreesof freedom in the equilibrium binary AO model, see Ref.64. The presence of many-body effective interactions has aprofound influence on phase behavior inside a porous matrix.For a detailed comparison of the bulk phase behavior obtainedvia the present treatment including all polymerinducedmany-body interactions, and one, based on the ROZequations, that only retains the pairwise contribution to theeffective Hamiltonian, see Ref. 51. A more detailed studyof the present approach compared to findings for simple fluidsinteracting with pairwise attractive forces would be veryinteresting, but is beyond the scope of the present work.Concerning the fluid profiles, we have only considered ahomogeneous background of matrix particles in this paper. Itwould be interesting to use inhomogeneous matrix realizations,as, e.g., a step function of zero and nonzero matrixpacking fraction i.e., the interface of empty space and matrixor a constant matrix background in contact with a hardwall. Both types could give rise to interesting and substantiallymodified wetting behavior. Additionally, one could alsoconsider other types of matrices, e.g., quenched polymers orcombinations of quenched colloids with quenched polymers43,51. These are maybe less realistic from an experimentalpoint of view but still interesting due to the competition ofcapillary condensation with evaporation.As we have mentioned in the Introduction, there are noexperiments concerning phase behavior of colloidal suspensionsin contact with 3D porous media to our knowledge. Wehope that the accumulating results 7–12,43,51, includingthose in this paper, may encourage more experimental effortsin that direction. It is important to keep in mind that a suitableporous matrix is a compromise between length scales:large enough to allow penetration of the colloids into thevoid space, but small enough to retain significant surface andcapillary effects. In colloidal fluids in general, these lastmentionedeffects are known to be much smaller than inatomic systems, thus providing a formidable challenge toexperimentalists aiming to observe, e.g., capillary condensationof a colloidal suspension in a porous matrix.ACKNOWLEDGMENTSThe authors would like to thank D. G. A. L. Aarts foruseful discussions and R. Blaak for a critical reading of themanuscript. M.S. thanks D. L. J. Vossen for detailed explanationsof the laser-tweezer setup, T. Gisler for pointing outRefs. 7,8, and R. Evans for valuable remarks. This workwas financially supported by the SFB-TR6 program ‘‘Physicsof colloidal dispersions in external fields’’ of the DeutscheForschungsgemeinschaft DFG. The work of M.S. waspart of the research program of the Stichting voor Fun-061404-10

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