VOLUME 87, NUMBER 3 P H Y S I C A L R E V I E W L E T T E R S 16 JULY 2001

03830

Hydrodynamic Interactions in Quasi-Two-Dimensional Colloidal Suspensions

Jesús Santana-Solano and José Luis Arauz-LaraInstituto de Física “Manuel Sandoval Vallarta,” Universidad Autónoma de San Luis Potosí, Alvaro Obregón 64,

78000 San Luis Potosí, S.L.P., Mexico(Received 6 June 2000; published 2 July 2001)

The effects of the hydrodynamic interactions on the short-time dynamics of colloidal, hard-sphere-likeparticles confined between two parallel walls are measured by digital videomicroscopy. We find thatsuch effects can be described in terms of an effective two-dimensional hydrodynamic function H�k�, de-fined as a straightforward adaptation to two dimensions of the corresponding object describing collectivedynamics for the three-dimensional (3D) suspensions. Interestingly, the behavior of H�k� is qualitativelysimilar to the hydrodynamic function of 3D suspensions of hard spheres. We also found that for valuesof k where the static structure factor is 1, the dynamics is determined only by self-diffusion.

DOI: 10.1103/PhysRevLett.87.038302 PACS numbers: 82.70.Dd, 05.40.–a

When a colloidal particle moves in a solvent, its mo-tion perturbs the medium. This perturbation propagatesthrough, reflecting itself on the neighboring particles. Thisprocess gives rise to an indirect, long-ranged, complexcoupling of the particles’ motion, referred to as hydrody-namic interactions (HI). The understanding of the effectof the HI on the dynamic properties and on other transportproperties of suspensions has been a challenging problemfor many years [1–3]. Recently, diffusion phenomena inthree-dimensional (3D) colloidal suspensions of both hardand charged spheres have been investigated by several au-thors, and an understanding of the HI contribution to thetime and wavelength dependence of those phenomena inthese model systems is emerging [4–11]. In contrast, theunderstanding of the same phenomena as they occur in aconfining geometry (inside a pore, for instance) is still ina far less developed stage, in spite of its scientific [12–16]and technological [17] relevance. The aim of the presentwork is to study colloidal dynamics in a specific case ofconfinement, namely, the case of quasi-two-dimensionalcolloidal suspensions. Previous work has reported mea-surements of the dynamics of isolated particles [18], or onthe self-dynamics and collective dynamics of dilute sus-pensions in the intermediate-time regime where the effectsof both direct interactions (DI) and HI are already inter-mingled [12,14]. Here we shall be mainly interested inthe effects of HI. Thus, we study concentrated systemswhere such effects should be stronger, but we focus onlyon the short-time regime, where the effects of the HI canbe decoupled from those of the DI. The formal deriva-tion of a theory describing these dynamic properties in aquasi-two-dimensional geometry would have to incorpo-rate a number of rather complex effects, such as the (di-rect and hydrodynamic) interactions of the particles, notonly among themselves, but also with the confining walls.Thus, its derivation will certainly be considerably morecomplicated than in 3D. Instead of pursuing such deriva-tion, here we adopt a more practical approach; namely,we define and measure effective quantities, experimen-tally accessible, analogous to those describing the dynamic

2-1 0031-9007�01�87(3)�038302(4)$15.00

properties in bulk 3D suspensions. Interestingly, we findthat the properties thus defined exhibit a qualitative be-havior strikingly similar to their 3D counterparts. Thisindicates that, in spite of all the complexity involved, thedescription of the dynamics of quasi-two-dimensional sus-pensions can be cast in a fashion formally no more com-plicated than in the case of 3D bulk suspensions. Clearly,this observation should serve as an important guideline forthe future development of the theory of the dynamics ofconfined suspensions.

The systems studied are concentrated aqueous suspen-sions of polystyrene spheres of diameter s � 2.05 mm(size polydispersity �3%), at room temperature �27.7 60.1 ±C�. The suspensions are confined between twoglass plates separated a distance h � 2.92 mm. Polysty-rene spheres, of diameter h, scattered across the sampleserve as spacers. The sample’s area is around 1.5 cm2.Measurements at different sites, surrounded by spacerslocated outside the field of view (area �80 3 60 mm2),led to identical results. For more details on the samplepreparation see Refs. [12,19,20]. Since s is comparableto h, there is only little room for the particles to movein the perpendicular direction, and they remain mostlyin the midplane between the glass plates, forming in thisway an effective two-dimensional system [12,18,20,].Thus, we study here only the lateral motion along theplane parallel to the plates. The particles are imagedusing an optical microscope. Their motion is sufficientlyslow and the two-dimensional trajectories of the particlesin the field of view can be accurately traced by standardvideo equipment with a time resolution of 1�30 s.With our setup we measure s � 16.8 pixels. Fromthe trajectories, we obtain various physical quantitiesdescribing self-dynamics and collective dynamics alongthe plane of motion as we explain below. All the resultspresented here were obtained from the analysis of atleast 104 video frames in runs of 120 consecutive frames[12,14,20,].

Since the theory of the dynamics of quasi-two-dimensional suspensions has not yet been developed,

© 2001 The American Physical Society 038302-1

VOLUME 87, NUMBER 3 P H Y S I C A L R E V I E W L E T T E R S 16 JULY 2001

we define here effective two-dimensional (2D) quantitiesanalogous to the quantities describing colloidal dynamicsin 3D. For this purpose, let us briefly recall the definitionof those quantities. In 3D, the structural properties (staticand dynamic) are described by the dynamic structure factorF�k, t� which is the time correlation function of dn�k, t� �PN

l�1 exp�2ik ? rl�t��, the fluctuations of wavelengthl � 2p�k of the local particle concentration, i.e.,F�k, t� �

1N �

PNl,j�1 exp�2ik ? �rl�t� 2 rj�0��, where

rl�t� is the position of particle l at time t, and N is thetotal number of particles in the system. At sufficientlyshort times, F�k, t� decays as [2,3]

F�k, t� � S�k� exp�2k2H�k�t�S�k�� . (1)

S�k� � F�k, t � 0� is the static structure factor deter-mined by the DI. Equation (1) constitutes the operationaldefinition of the hydrodynamic function H�k� in termsof the measurable properties F�k, t� and S�k�. It isthis very equation that we shall adopt to operationallydefine the effective hydrodynamic function of our realquasi-two-dimensional systems, for which we shallrecord F�k, t� and S�k�, as explained below. The inverseFourier transform of F�k, t� is the Van Hove functionG�r, t� � N21�n�r0, t � 0�n�r00, t�, where n�r, t� �PN

j�1 d�r 2 rj�t�� is the local concentration of particlesat the position r and time t, with r � jr00 2 r0j. G�r, t�and F�k, t� are the most general quantities describing thedynamic properties of colloidal suspensions, in the realspace and in the reciprocal space, respectively.

In our systems, we observe the dynamics of the colloidalparticles in real space only along the plane parallel to thewalls. Thus, in our experiment, physical quantities aremeasured as if the system were strictly two dimensional.The quantities directly measured are the 2D mean squareddisplacement W �t� � ��Dr�t��2�4, the 2D short-timeself-diffusion coefficient DS

s � limt!0W�t��t, and the2D Van Hove function G�r, t�. The formal expressionsfor W �t� and G�r, t� are identical to those in 3D, buthere n�r, t� is the local 2D concentration of particles andr is the 2D vector position. One should keep in mindthat, in reality, W �t�, DS

s , and G�r, t� are effective 2Dquantities describing the dynamic properties of quasi-two-dimensional systems. As in the case of 3D, here one canwrite G�r, t� � Gs�r, t� 1 Gd�r, t�, where Gs�r, t� (theself-part) is the correlation function of a particle withitself, and Gd�r, t� (the distinct part) is the correlationfunction between different particles. At time t � 0,Gs�r, 0� � d�r� and Gd�r, 0� � n�g�r�, n� � ns2 isthe reduced concentration, with n being the averagenumber of particles per unit area, and g�r� is the ef-fective 2D pair correlation function. Figure 1(a) showsthe measured G�r, t� vs r for a system with a fractionarea fa � pn��4 � 0.38. Here one can appreciate thecontributions of both components to G�r, t� at differenttimes. At t � 0, Gd�r, t��n� is just g�r� (dots withsolid line) and the self-part is a peaked function atr � 0 (not shown). At times t . 0 (dots with dashed

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0 1 2 3r/σ

0

1

2

G(r

,t)/n

*

t=0.0 st=0.5 st=1.0 st=3.0 s

0 1 2 3r/σ

0

1

2EXPMC(2D)

2 8 14 20kσ

0

1

2

g(r)

S(k)

(a)

(b)

(c)

FIG. 1. (a) Effective 2D Van Hove function G�r, t� vs r ,at different times, of a system with area fraction fa � 0.38.(b) Measured effective 2D pair correlation function g�r� (opencircles) compared with g�r� of a strictly 2D system of harddisks at the same area fraction, obtained by Monte Carlocomputer simulations (solid line). This shows that the maininteraction between the particles in the experimental systemis given by the hard-core interaction. (c) Effective 2D staticstructure factor obtained from the measured g�r�.

lines), the self-part spreads out due to self-diffusion, andthe initial structure of the distinct part smears down dueto the loss of interparticle correlation as time evolves.For times t , 1 s, both components are clearly distin-guishable from each other, and at later times they mergetogether combining their relative contributions to G�r, t�.Of particular interest is the time at which the contributionsof Gs�r, t� and Gd�r, t� start to overlap. This time identi-fies the time span of the short-time regime. We measuredG�r, t� for a set of concentrated systems and in all of themthe merging time is between 0.5 1 s. This means that thedynamics in our systems is sufficiently slow that we canstudy the short-time regime by using the time resolutionof standard video equipment.

Figure 2 shows the measured W�t� for the system ofFig. 1 (open circles). The solid line is a straight linewith its slope (DS

s � determined by a linear regression us-ing only the five initial experimental data points (0.166 s)of W�t�. As it is seen here, the linear regime spans upto about 0.5 s. This time is approximately the same timeat which Gs�r, t� and Gd�r, t� start to overlap. For timest . 0.5 s, W�t� deviates from the linear behavior due tothe direct interaction between neighboring particles. Theclosed circles represent W�t� measured in a highly dilutesystem �fa � 2.3 3 1023�. In this system we observe,in the field of view, only 2-3 particles far apart morethan 20s. In this case, both DI and HI are negligible,and the value of DS

s contains only the effect of the HIbetween the particles and the walls. For this concentra-tion, W�t� is shown only for short times since it requiresthe analysis of a considerably larger amount of data tohave reliable values of W�t� at larger times. The dashedline in Fig. 2 is W�t� � D0t, corresponding to free dif-fusion of the same particles in 3D with D0 � kT�3phs

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VOLUME 87, NUMBER 3 P H Y S I C A L R E V I E W L E T T E R S 16 JULY 2001

0 0.2 0.4 0.6 0.8 1t [s]

0

0.006

0.012

0.018

W(t

)/σ2

DStD0t

S

FIG. 2. Effective 2D mean squared displacement W �t�. Close�fa � 2.3 3 1023� and open �fa � 0.38� circles are experi-mental data from video microscopy. The solid line is the initiallinear behavior of W �t� in the system with fa � 0.38, and thedashed line corresponds to free diffusion in 3D.

being the free diffusion coefficient. The value of the ini-tial slope of W �t� in the dilute quasi-two-dimensional sys-tem �DS

s � 8.28 3 10210 cm2�s� is about 30% of D0 �2.59 3 1029 cm2�s. This shows that the hydrodynamiccoupling of the particles with the walls has a strong effectalready on the motion of isolated particles. An additionaldecrease of DS

s is observed at larger fa as a consequenceof the HI between the particles. The friction coefficient ofa colloidal particle diffusing close to a rigid wall increasesdrastically (diverges) as the separation of the particle to thewall decreases [21]. Thus, a reduction of DS

s is expecteddue to the presence of the confining walls. However, ourmeasurements indicate that the combined effect from bothwalls is stronger than that expected from a simple linearsuperposition of the effects from each wall separately [21].We discussed this observation in Ref. [12], where we re-ported measurement of self-diffusion of 0.5 mm particlesat different separations of the walls. See also Ref. [16].

Following the relation between F�k, t� and G�r, t� in3D, here we define the effective 2D dynamic structurefactor F�k, t� as the Fourier transform of the measuredG�r, t�. As in the case of 3D, the initial slope of F�k, t�should contain the effects of the HI. Thus, following thestructure of Eq. (1), we define the effective 2D hydro-dynamic function H�k� as the initial slope of the func-tion f�k, t� � 2k22S�k� ln�F�k, t��S�k��, where S�k� �F�k, t � 0�. Figure 3 shows f�k, t� vs t, for the sys-tem of Fig. 1, at various values of k. As it is seen here,the initial time evolution of f�k, t� is indeed linear [i.e.,F�k, t� decays exponentially] for a wide range of k. Thus,the initial k-dependent slope of f�k, t� defines H�k�, thequasi-2D analog of the 3D hydrodynamic function definedby Eq. (1). Figure 4 shows H�k��DS

s , with DSs obtained

from the initial slope of W �t�, for four values of fa. Thestructure of H�k� shows that the HI contribute differentlyto the relaxation of the particle fluctuations of differentwavelengths, and that the effect is larger for higher con-

038302-3

0 0.1 0.2 0.3t [s]

0

0.002

0.004

0.006

f(k,

t) [σ

2 ]

kσ=5.7kσ=4.75kσ=10.5kσ=4.2

φa=0.38

FIG. 3. f�k, t� vs t (symbols) for various values of k. Theinitial slope (lines) defines the effective 2D hydrodynamic func-tion H�k�.

centrations. Here one can see that single particle mo-tion, characterized by DS

s which depends only on fa butnot on k, serves as a reference to collective motion de-scribed by H�k�. Deviations of H�k� from DS

s are dueto the fact that collective motion is not just a superposi-tion of the individual motion of the particles, except atvalues of k where H�k� � DS

s . Although in our case theHI between particles are combined with the hydrodynamiceffects from the walls, it is interesting to see that H�k�resembles the measured H�k� in 3D suspensions of hardspheres [4]. Although there is not a simple correspondencebetween quasi-two-dimensional and 3D systems, this ob-servation suggests that perhaps the main effect from thewalls can be accounted for by the value of DS

s [the long-klimit of H�k�].

As another important result of our work, we show ex-perimentally that particle fluctuations relax by self-diffusion, i.e., F�k, t� � Fs�k, t�, when the wavelengthof the fluctuation is such that S�k� � 1 [Fs�k, t� is the

2 6 10 14kσ

0.6

0.8

1

1.2

H(k

)/D

s

φa=0.38φa=0.32φa=0.27φa=0.23

S

FIG. 4. Effective 2D hydrodynamic function, normalized withthe 2D short-time self-diffusion coefficient, for various concen-trated systems.

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VOLUME 87, NUMBER 3 P H Y S I C A L R E V I E W L E T T E R S 16 JULY 2001

2 6 10 14kσ

−0.75

−0.25

0.25

0.75F

d(k,

t)

k where S(k)=1t=0.033 st=0.5 st=1.0 s

φa=0.38

FIG. 5. Time correlation function between different particlesFd�k, t� vs k at different times (dots with lines). Open circlesare the values ki of the wave vector where S�k� � 1. One cansee here that Fd�ki , t� � 0 for t $ 0.

self-part of F�k, t�]. This is a very important and usefulresult concerning measurements of self-diffusion by lightscattering (LS) techniques. In 3D, LS measures the fullF�k, t�. Self-diffusion is obtained at long wave vectorswhere Fd�k, t�, the distinct part of F�k, t�, vanishes sinceit is the configuration average of a phase factor oscillatingvery rapidly. Since large k implies short displacements,only short-time self-diffusion is measured in that limit.However, self-diffusion at longer times is extracted fromLS measurements by assuming that F�k, t� � Fs�k, t�for values of k where S�k� � 1. Since, by definition,F�k, 0� � S�k� and Fs�k, 0� � 1, the assumption in LSexperiments is that Fd�ki , t� � 0 for k � ki , whereS�ki� � 1. An advantage of measuring the individualtrajectories of the particles in real space is that we canmeasure both components of G�r, t� separately andsimultaneously at all times. Therefore, we can alsodetermine Fs�k, t� and Fd�k, t�, independently, by Fouriertransforming Gs�r, t� and Gd�r, t�, respectively. In Fig. 5,Fd�k, t� vs k, at different times, is plotted for the systemin Fig. 1. This shows the wavelength dependence of thetime correlation function between different particles (dotswith lines). The open circles represent the values ki whereS�ki� � 1. As it is seen here, Fd�ki , t� remains vanishingsmall for t $ 0. This proves that in our systems F�ki , t�is, in fact, Fs�ki, t�.

Here we report the first measurements of the wavelengthdependence of the HI effects on the short-time dynamicsof colloidal particles confined between two walls. Theseresults are an important step in understanding the role ofthe HI in a restricted geometry, and can serve as a guide-

038302-4

line for theoretical and computer simulation calculationsof colloidal dynamics under confinement. Interestingly,we found that the effective 2D hydrodynamic function de-fined here shows a qualitative behavior similar to H�k�measured in 3D suspensions of hard spheres. This obser-vation poses the question about the connection betweencolloidal dynamics in 3D and in confined geometries. Wealso found that at wave vectors where S�k� � 1 the par-ticle fluctuations relax only by self-diffusion. This resultis important for the interpretation of measurements of col-loidal dynamics by light scattering techniques.

This work was partially supported by the ConsejoNacional de Ciencia y Tecnología, México, Grant No.G29589E, and by the Instituto Mexicano del Petróleo,México, Grant No. FIES-98-101-I.

[1] J. F. Brady, Curr. Opin. Colloid Interface Sci. 1, 472 (1996).[2] P. N. Pusey, in Liquids, Freezing and Glass Transition,

edited by J. P. Hansen, D. Levesque, and J. Zinn-Justin(Elsevier, New York, 1991), p. 763.

[3] G. Nägele, Phys. Rep. 272, 215 (1996).[4] P. N. Segrè, O. P. Behrend, and P. N. Pusey, Phys. Rev. E

52, 5070 (1995).[5] S. Fraden and G. Maret, Phys. Rev. Lett. 65, 512 (1990).[6] X. Qiu, X. L. Wu, J. Z. Xue, D. J. Pine, D. A. Weitz, and

P. M. Chaikin, Phys. Rev. Lett. 65, 516 (1990).[7] C. W. J. Beenakker and P. Mazur, Physica (Amsterdam)

126A, 349 (1984).[8] M. Medina-Noyola, Phys. Rev. Lett. 60, 2705 (1988).[9] A. J. C. Ladd, H. Gang, J. X. Zhu, and D. A. Weitz, Phys.

Rev. Lett. 74, 318 (1995).[10] G. Nägele and P. Baur, Europhys. Lett. 38, 557 (1997).[11] W. Härtl, Ch. Beck, and R. Hempelmann, J. Chem. Phys.

110, 7070 (1999).[12] M. D. Carbajal-Tinoco and G. Cruz de León, J. L. Arauz-

Lara, Phys. Rev. E 56, 6962 (1997).[13] K. Zahn, J. M. Méndez-Alcaraz, and G. Maret, Phys. Rev.

Lett. 79, 175 (1997).[14] H. Acuña-Campa, M. D. Carbajal-Tinoco, J. L. Arauz-Lara,

and M. Medina-Noyola, Phys. Rev. Lett. 80, 5802 (1998).[15] J.-C. Meiners and S. R. Quake, Phys. Rev. Lett. 82, 2211

(1999).[16] A. H. Marcus, J. Schofield, and S. A. Rice, Phys. Rev. E

60, 5725 (1999).[17] M. Sahimi, Flow Transport in Porous Media and Fractured

Rock: From Classical Methods to Modern Approaches(VCH, Weinheim, Germany, 1995).

[18] L. P. Faucheux and A. J. Libchaber, Phys. Rev. E 49, 5158(1994).

[19] G. Cruz de León and J. L. Arauz-Lara, Phys. Rev. E 59,4203 (1999).

[20] J. Santana-Solano and J. L. Arauz-Lara (to be published).[21] G. S. Perkins and R. B. Jones, Physica (Amsterdam) 189A,

447 (1992).

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