VOLUME 81, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 3 AUGUST 1998

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Colloidal Interactions in Partially Quenched Suspensions of Charged Particles

G. Cruz de León, J. M. Saucedo-Solorio, and J. L. Arauz-LaraInstituto de Fı´sica “Manuel Sandoval Vallarta,” Universidad Autónoma de San Luis Potosı´, Alvaro Obregón 64,

78000 San Luis Potosı´, S.L.P., Mexico(Received 12 March 1998)

We report measurements of the effective pair potential between charged colloidal particles in abidimensional matrix of fixed obstacles. A binary mixture of polystyrene spheres in water is confinedbetween two glass plates. The larger particles are trapped by the plates in a disordered configuratiowith respect to which the smaller species of particles equilibrates. The structures of both the mobileand the fixed species are measured by videomicroscopy. The pair potential, obtained by deconvolutinthe structural information via the Ornstein-Zernike equation, exhibits two attractive components.[S0031-9007(98)06744-1]

PACS numbers: 82.70.Dd, 05.40.+ j, 47.55.Mh, 61.20.Gy

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Recent measurements of the direct interaction betwecharged colloidal particles in suspension, such as postyrene spheres in water, have provided evidence ofattractive effective pair potential between the particles uder conditions of confinement [1–3]. The physical properties of such systems have been extensively studied inhomogeneous three-dimensional (3D) space. In this cathe assumption of a repulsive screened Coulomb pair ptential, as the one derived by Derjaguin, Landau, Verweand Oveerbek [4], provides a functional form for the interparticle potential in terms of which the experimentaobservations have been reasonably well described [5,However, for the same kind of systems but now confinebetween two parallel glass plates, direct and indirect dterminations of the effective interparticle potential showan attractive component, rather than repulsive, at intemediate distances [1–3]. This observation is quite inteesting and raises the question about the physical origof such effective attractive interaction, and also wheththis interaction is further modified under different conditions of confinement, for instance, in an arbitrary geomtry such as a disordered porous medium. The answto these questions is quite important because the acrate determination of colloidal interactions is essentialorder to understand on a fundamental basis colloidal proerties such as their stability, structure, dynamics, thermdynamics, and so on. Here we address the latter questinamely, the question about the sensitivity of the effectivcolloidal interactions on the local environment. In thisLetter, we report indirect measurements of the effectivinteraction between charged colloidal particles in suspesion when it is permeating a bidimensional porous matriWe measure both the structure of the colloidal suspensiand the porous matrix. Then, the effective pair potentiis determined by deconvoluting the information containein the structural properties of the system, by using thmulticomponent Ornstein-Zernike (O-Z) integral equatioand a closure relation. This deconvoluting method waemployed to determine the effective interaction potenti

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between polystyrene spheres in water, confined betwtwo parallel glass plates in such a way that the systbecomes an effective two-dimensional colloidal suspesion [2]. As mentioned before, the result in that cais an attractive interaction potential between charged cloidal particles. Essentially the same potential was aobserved by different indirect [1] and direct [3] measurments on similar systems.

Let us now describe our system. It consists of a binamixture of polystyrene spheres of diameterss1 and s2,with s2 . s1, suspended in water. The suspensionconfined between two glass plates in such a way tthe separation between the inner surfaces of the placoincides with s2. Thus, the dynamics of speciesis quenched, i.e., the particles of species 2 are froin a given configuration forming in this way a twodimensional (2D) (porous) matrix of fixed particlesThe particles of species 1 then form an effective 2colloidal fluid which equilibrates in the field of thefixed particles of species 2. The particles of specieare fluorescent polystyrene spheres of a diameter0.5 mm, and the particles of species 2 are nonfluorescpolystyrene spheres of a diameter of 2mm. Under theseconfining conditions, the most important contributioto the solution’s ionic strength comes from the ionreleased from the glass plates (surface charge densit,106 mm22) [7]. Thus, the Debye length is,10 20 nm,and the effective diameters are only slightly larger ththeir physical values. Further details of the samppreparations can be found elsewhere [8,9]. Figureshows a typical image of a small area of one of osystems as it is seen through an optical microscousing both transmitted and fluorescence illumination aa 1003 objective. The differences in size and in coloof both species allow us to clearly identify the particleof the different species and to determine accurately thpositions using digital videomicroscopy. In practice, thpositions of the particles of the two species are determinseparately. For species 1, we use only fluoresce

© 1998 The American Physical Society

VOLUME 81, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 3 AUGUST 1998

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FIG. 1. Image of a small area of a sample with reducconcentrations of fixed particlesnp

2 ­ 0.25 and mobile particlesnp

1 ø 0.02. The bright and shadowed particles are the moband fixed particles, respectively. The scale of the picture60 mm along thex direction.

illumination (the fixed particles are not seen). For tlarger particles, we use only transmitted light and wfocus the top of the fixed particles so that only they aclearly imaged.

For the first part of our work, we determine the effeof the porous matrix on the structure of the colloidal fluicharacterized by the radial distribution functiong11srdof the diffusing particles along the plane parallel to tplates. We measuredg11srd for different concentrationsof fixed particles. In order to minimize the effect othe interactions between the mobile particles themselon g11srd, we kept the average particle concentrationspecies 1,n1 (; N1yA, where N1 is the average num-ber of particles of species 1 in the areaA), very low.In all our samples the dimensionless reduced concention np

1 ; n1s21 was ø0.02 [10]. The measurement o

g11srd in the field of the fixed particles constitutes, by iself, an important contribution of this work. Additionallyour model system allows us to characterize the structof the porous matrix where the colloidal suspensionabsorbed, as well as the correlation between mobile pticles and the matrix. Those quantities are described hby the 2D radial correlation function of the fixed paticlesg22srd and by the cross-correlation function (mobiparticles-matrix)g12srd, respectively. It is important tostress here that the precise characterization of the phcal properties of liquid systems absorbed in a poromedium is an interesting area of research in its own rigwhich is important in processes such as catalysis, oilcovery, and others [11,12]. Let us mention that theorecal approaches based on integral equation methods hbeen developed [13–15], as well as computer simulatmethods that have been employed [16,17], to studystructure, the thermodynamic and dynamic propertiesmobile liquids in equilibrium with a rigid matrix of obstacle particles. Thus, our systems constitute a particularperimental realization of those theoretical model syste

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Although, in our case the presence of the confining glaplates plays the role of an extra external field. In principle, if in our samples the interparticle potentials werknown, one could use those theoretical approaches anor computer simulation methods to calculate the structuof the colloidal liquid and compare with our experimentameasurements. However, the pair potentials are precisthe important quantities we would like to determine fromour experiments. Let us now discuss our results.

Figure 2 shows the radial distribution functiong11srdof the colloidal suspension for various values of the fixeparticle’s reduced concentrationnp

2 ; n2s22 , keeping con-

stant np1 ø 0.02 [10]. Here one can see the effect of

the porous matrix on the structure of the colloidal fluidIn the limit of very low concentrations of fixed parti-cles (np

2 ø 0), the system corresponds to a monodispersquasi-bidimensional suspension. Here the few large paicles present in the system serve only as spacers and allus to control accurately the distance between the plateSince the colloidal suspension is rather dilute, one couexpect the system to not be highly structured. In fact, this what we obtain, the measuredg11srd exhibits only alow first peak (solid line) at aroundr ­ 1.5s1, followedby minor oscillations that fade away rather quickly. Asdiscussed elsewhere [2], the apparent unphysical resg11srd . 0 for r , s1 is only an artifact of the quasi-bidimensionality of the system and has no relevance heAt finite values ofnp

2, g11srd reveals an increase in thestructure of the colloidal fluid. As one can see here, thheight of the first peak ofg11srd increases quite appre-ciably as np

2 increases and, interestingly, its position isalmost independent ofnp

2. One can also notice in Fig. 2that the first peak ofg11srd decays more slowly for largervalues of np

2 and that there is an abrupt change in thslope that, also interestingly, happens at the same pla(about r ­ 2.2s) in all the samples. This behavior ofg11srd is notoriously different from what is observed in

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FIG. 2. Radial distribution function of the mobile particles avarious concentrations of the fixed particles.

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VOLUME 81, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 3 AUGUST 1998

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homogeneous suspensions, i.e., when instead of increing the number of frozen particles, one variesnp

1 keepingnp

2 ø 0. Then, the system is an effective 2D system thcan be considered homogeneous along the plane palel to the plates [2,8]. There, as the particle concentrtion increases, the radial distribution function developshigher first maximum, followed by a minimum, a lowersecondary maximum, and so on. The positions of thmaxima and minima depend on concentration and theare not abrupt changes in the slope of the radial distbution function. Thus, the presence of fixed obstaclor porous matrix enhances the correlation between tmobile particles, but there is not any development of seondary maxima or minima, as is the case for homogneous systems. Here the correlations become strongerof longer range, butg11srd has basically only one peakthat decays slower for larger values ofnp

2 and exhibits amarked change in the slope. The curves ofg11srd shownin Fig. 2 represent the average of about1.4 3 104 differ-ent configurations of the mobile particles digitized at foudifferent configurations of the fixed particles as the onshown in Fig. 1. Therefore,g11srd in Fig. 2 is also an av-erage over the structure of the matrix.

The cross-correlation functions between fixed and mbile particles,g12srd, are shown in Fig. 3(a). An interest-

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FIG. 3. Cross-correlation function (a) and fixed particle’radial distribution function (b) for the systems in Fig. 2.

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ing feature to be noticed here is the existence of a lfirst maximum ofg12srd with height and position varyingvery little with np

2. Another important effect of increasingthe number of fixed particles is the development of lorange correlations between the particles of species 12, exhibited in the presence of more evident oscillationsg12srd at longer distances asnp

2 increases. The curves fog12srd in Fig. 3(a) are also the average of cross-correlatfunctions measured at four different configurations of tfixed particles as in the case ofg11srd. For the particlesof species 2, we can also determine the correspondradial distribution function,g22srd, as a practical way tocharacterize the structure of the porous matrix. Thus,Fig. 3(b) we showg22srd for the same systems. In all othese samples,g22srd has a very high first maximum closeto contact. For the larger values ofnp

2, one can observethe development of a minimum and a secondary mamum. This shape ofg22srd resembles a system of hardisks, although in our systems the first peak is higher ththat corresponding to a hard disk system at the same picle concentration. The curves ofg22srd were determinedfrom 1 2 3 103 different configurations.

The radial distribution functions of the colloidal suspension in Fig. 2 show clearly that the presence of tfixed particles increases the correlation between the mbile particles. The question is whether the observedhavior of g11srd is due only to the fact that the largparticles cage the mobile particles within the porousthat they remain close together for longer times thin the homogeneous case (np

2 ø 0), or if the presenceof the matrix also induces changes on the effectiveteraction between the mobile particles. To answer tquestion let us determine the effective pair potential btween the mobile colloidal particles. This potential cabe extracted from the detailed structural informationthe system contained ingijsrd, by deconvoluting thisinformation via the 2D Ornstein-Zernike integral eqution. For a two-component system, the O-Z equationa set of coupled integral equations for the total corretion functionshijsrd ­ gijsrd 2 1 and the direct correla-tion functionscijsrd between particles of speciesi andjsi, j ­ 1, 2d. For symmetry, one assumeshjisrd ­ hijsrdand cjisrd ­ cijsrd. Usually, the O-Z equation is employed to calculategijsrd for a given system, assumingthe pair potentialsuijsrd are known between the differenspecies. In those equations both sets of functionshijsrdand cijsrd are unknown and an additional set of reltions or closure relations is required to be able to sothe equations. The closure relations more frequently uare the mean spherical, the Percus-Yevick, and thepernetted chain (HNC) approximations [18]. Thus, if thpair potentials are known, one can use the O-Z eqtion and one of these closure relations to obtain the pcorrelation functionsgijsrd. Here we follow the inverseprocess. We Fourier transform the O-Z equation to otain a set of four coupled linear equations involvingcijskd

VOLUME 81, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 3 AUGUST 1998

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andhijskd, the Fourier transforms ofcijsrd andhijsrd, re-spectively. Since we measuredhijsrd, we solve the setof equations forcijskd, which are then transformed backto the real space to get the measured direct correlatifunctions cijsrd. The pair potential between the mobileparticlesu11srd is obtained fromc11srd using one of theclosure relations. Here we use the HNC approximatiobut the results from the other closure relations are quatatively similar. The results of this deconvoluting methodof measuring the pair potential are presented in Fig. 4.

The curve in Fig. 4(a) corresponds to the sample whenp

2 ø 0. Hereu11srd shows a well defined attractive com-ponent at about1.5s1, which is the attractive interactionobserved previously [2]. At finite values ofnp

2, u11srdalso exhibits this attractive component atr ø 1.5s1. Asone can see, its position remains basically independentnp

2 and it becomes only slightly deeper asnp2 increases.

At large concentrations of fixed particles there is an additional feature of the pair potential, namely, the attractiveness of the mobile particles becomes of larger rangOne can observe the development of a second attrative component ofu11srd at larger distances, as seen inFig. 4(c). This second component widens out towardsmaller distances and merges with the first attractive we[see Fig. 4(d)]. Thus, one could say that the effectivpair potential between the particles of the colloidal fluidexhibits two well defined attractive components under thparticular confinement studied here. The first attractivcomponent, the minimum atr ø 1.5s1, can be attributedto the effect of the confining by the glass plates. Ashown in Fig. 4 this component changes very little asnp

2

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n*

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n*

2 = 0.25 n*

2 = 0.29

(a) (b)

(c) (d)

FIG. 4. Effective pair potential between mobile particlesabsorbed in matrices of different porosities.

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is varied. The second attractive component arises asextra effect of the fixed particles, which then not onlyserve as obstacle particles reducing [19] and molding thspace available to the colloidal fluid, but their interactionwith the mobile particles introduces a strong modificatioof the effective pair potentialu11srd.

The specific features reported here, for the structure athe effective interparticle potential between the particleof a colloidal fluid permeating a matrix of obstacles, correspond to our particular system where charged polystrene spheres are used for both the mobile and the fixparticles. We expect, however, that this simple modesystem captures the general trends of those physical qutities, at least for systems of charged particles.

This work was partially supported by the ConsejoNacional de Ciencia y Tecnologı´a (CONACyT, México),Grant No. 3882E.

[1] G. M. Kepler and S. Fraden, Phys. Rev. Lett.73, 356(1994).

[2] M. D. Carbajal-Tinoco, F. Castro-Román, and J. L. ArauzLara, Phys. Rev. E53, 3745 (1996).

[3] J. C. Crocker and D. G. Grier, Phys. Rev. Lett.77, 1897(1996).

[4] W. B. Russel, D. A. Saville, and W. R. Schowalter,Col-loidal Dispersions (Cambridge University Press, Cam-bridge, England, 1989).

[5] P. N. Pusey, inLiquids, Freezing and Glass Transition,edited by J. P. Hansen, D. Levesque, and J. Zinn-Jus(Elsevier, New York, 1991), p. 763.

[6] G. Nägele, Phys. Rep.272, 215 (1996).[7] K. Iler, The Chemistry of Silica(Wiley, New York, 1979).[8] M. D. Carbajal-Tinoco, G. Cruz de León, and J. L. Arauz-

Lara, Phys. Rev. E56, 6962 (1997).[9] G. Cruz de León and J. L. Arauz-Lara (to be published).

[10] In all of the samples studied here,np1 and np

2 were keptfar from the closed packing valuenp

cp ­ 2yp

3 ø 1.15corresponding to a 2D hard disk system.

[11] J. M. Drake and J. Klafer, Phys. Today43, No. 5, 46(1990).

[12] M. Sahimi, Rev. Mod. Phys.65, 1393 (1993).[13] W. G. Madden, J. Chem. Phys.96, 5442 (1992).[14] J. A. Given and G. Stell, J. Chem. Phys.97, 4573 (1992).[15] C. Vega, R. D. Kaminsky, and P. A. Monson, J. Chem

Phys.99, 3003 (1993).[16] G. Viramontes-Gamboa, J. L. Arauz-Lara, and M. Medina

Noyola, Phys. Rev. Lett.75, 759 (1995).[17] G. Viramontes-Gamboa, M. Medina-Noyola, and J. L

Arauz-Lara, Phys. Rev. E52, 4035 (1995).[18] J. P. Hansen and I. R. McDonald,Theory of Simple Liquids

(Academic, New York, 1986), 2nd ed.[19] The large particles may form clusters with internal free

space unreachable to the mobile particles. This, howeveintroduces only a small correction to the value obtainefrom a direct excluded volume calculation [9].

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