short-time dynamics of colloidal suspensions in confined...
TRANSCRIPT
PHYSICAL REVIEW E APRIL 1999VOLUME 59, NUMBER 4
Short-time dynamics of colloidal suspensions in confined geometries
I. Pagonabarraga,1,* M. H. J. Hagen,1,† C. P. Lowe,1,2 and D. Frenkel11FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands2Computational Physics, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
~Received 29 September 1998!
We analyze the short-time dynamical behavior of a colloidal suspension in a confined geometry. We analyzethe relevant dynamical response of the solvent, and derive the temporal behavior of the velocity autocorrelationfunction, which exhibits an asymptotic negative algebraic decay. We are able to compare quantitatively withtheoretical expressions, and analyze the effects of confinement on the diffusive behavior of the suspension.@S1063-651X~99!09204-1#
PACS number~s!: 82.70.Dd, 05.40.2a, 66.20.1d, 83.20.Jp
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I. INTRODUCTION
The dynamics of fluids in confined geometries is a subof interest, both for its relevance in chemical engineering aenvironmental science, and because of the new phenomthat arise as a consequence of the competition betweendynamical intrinsic features of the fluid and the geometrirestrictions introduced by the walls. Such an interaction mgive rise to a qualitatively different behavior from those pdicted in unbounded systems. For example, it is well knothat in equilibrium new thermodynamic phases may beduced by the geometrical constraints. Here we will analthe effects on the dynamics of suspended particles suspeas a consequence of the modification of the solvent hyddynamics.
In a previous paper@1#, we showed how the presenceconstraining walls modifies the dynamics of colloidal paticles. We indicated there that the effect of the surfacesthe dynamics of the fluid depends on the specific momenexchange mechanisms at the interfaces, and not only ongeometrical constraints. For example, for slip boundary cditions we saw that the fluid behaved as an effective medof dimensionalityd* equal to the dimensions of the systethat were unconstrained by the presence of the boundaOn the other hand, for stick surfaces, the propagationsound modes in the fluid was modified. Depending ongeometry of the constraining medium, the effective souvelocity may become imaginary, developing into a diffusisound wave. This effect, in turn, qualitatively changesshort-time dynamical behavior of a suspended particle.asymptotic decay is controlled by the diffusive sound morather than by vorticity diffusion, as happens in unboundfluids @2#.
The effect that the coupling of a fluid to an elastic mathas on the dynamical properties of the fluid has been conered previously following the model introduced by Biot@3#.According to this model, the dynamics of the solid and flu
*Present address: Department of Physics and Astronomy, Unsity of Edinburgh James Clerck Maxwell Building, The KingBuildings, Mayfield Road, Edinburgh, EH9 3JZ U.K.
†Present address: Unilever Research, Port Sunlight LaboraQuarry Road East, Bebington, Wirral L63 3JW, U.K.
PRE 591063-651X/99/59~4!/4458~12!/$15.00
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phases are treated on the same footing, and all the dissiparises due to the momentum exchange between the fluidthe solid when they are in relative motion. This dissipationincluded as a frictional force acting on the fluid, with a frition coefficient that is either left undetermined or is computed assuming that the solid frame is a porous mediumthe fluid is filling it. One of the predictions of Biot’s theory ithat this coupling induces two sound modes in the fluid, oof which has a velocity smaller than the corresponding sovelocities in the two media. Subsequently, it was pointedthat the velocity of the slow sound could give rise to a dfusive sound mode@4#. Analogous ideas were used to anlyze the dynamics of a thin fluid layer adsorbed on a sosubstrate@5#, as well as the dynamics of gels@6#, where itwas argued that, due to the characteristic small sizes ofpores, such a diffusive mode could play an important rolethe low frequency response of the gel.
In this paper we will restrict ourselves to a simple geoetry, namely, a fluid between plane walls, where an exhydrodynamic calculation can be carried out without makany assumption about the force that the solid exerts onfluid. It then becomes apparent that it is only the role of tboundary conditions that determines the hydrodynammodes of the fluid, and from which the diffusive sound mocan be completely characterized. Although a careful hyddynamic analysis of the modes of a fluid layer confinedtween two parallel plates has previously been carried out@7#,it was restricted to high frequencies. We will concentratethe low frequency regime, and in particular we will considthe behavior of the velocity autocorrelation functio~VACF!, defined as
Cv~ t 
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PRE 59 4459SHORT-TIME DYNAMICS OF COLLOIDAL . . .
the short-time regime for the colloidal dynamics, a time scin which colloid displacements can be neglected, butwhich the solvent hydrodynamics has time to develop.
In Sec. II we will theoretically consider the VACF ofparticle suspended in a fluid. To this end, we will determthe relevant fluid modes characteristic of the confined svent. In Sec. III we present the simulation results, and copare quantitatively with the theoretical predictions. We loin detail at the behavior of the fluid, and we study both ttranslational and rotational motion of a suspended particleSec. IV we use the results of the preceding sections to sthe effect of confinement on the diffusion of colloidal supensions. We end with a discussion of our results. In Appdix A we give some of the quantities used in Sec. II, andAppendix B we present an intuitive derivation of the resuof Sec. II.
II. THEORETICAL ANALYSIS
In order to study the VACF, we start by analyzing thvelocity generated by a localized force perturbation in a psolvent in the presence of walls, which will show clearly tconnection between the dynamics of the suspended parand the relevant hydrodynamic modes of the solvent.will consider a fluid confined between two parallel platelocated aty56L/2. For simplicity, we restrict ourselves tthe two-dimensional~2D! situation, in which the confiningwalls are slits, but it is straightforward to generalize the cculation to higher dimensions.
Since the force pertubation is weak, the fluid densityr
and velocity fieldsvW , obey the linearized continuity anNavier-Stokes equations, respectively,
]r
]t1r¹•vW 50, ~2!
]vW
]t1
cs2
r¹r2n¹2vW 2G¹¹•vW 5
1
rFW , ~3!
wherecs is the speed of sound, andn and G are the kine-matic shear and bulk viscosities, respectively. The fluidinitially at rest, perturbed only by the external weak forceFWapplied in the center of the slit and pointing along the dirtion of the slit,x̂, FW 5 f 0x̂d(t)d(y2y0). In Eqs.~2! and~3!we have disregarded thermal effects, assuming the procebe isothermal, but its generalization to adiabatic proceswill not modify the features presented in this paper.
Since the fluid is confined between two walls, in orderstudy the dynamics of the flow generated by a given forcis necessary to specify the boundary conditions at the wWe will assume that stick boundary conditions are satisat the boundaries, which is the usual situation for solid intfaces. In this case, the velocity of the fluid is equal to thathe wall, namely,vW (y56L/2)50.
In order to solve Eqs.~2! and ~3!, we Fourier transformthem. Defining
vW ~v!5E2`
`
eivtvW ~ t !dt ~4!
en
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lee,
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and
vW ~kW !5E2`
`
eikW•rWvW ~rW !drW,
we can rewrite Eqs.~2! and~3! as a set of ordinary differential equations
d
dtV5D̃V1 fW , ~5!
where the vectorsV and fW are defined as
V5S k̂x•vW ,] k̂x•vW
]y,vy ,
]vy
]yD , ~6!
fW5S 0,1
rk̂x• x̂ f 0,0,0D d~ t !d~y2y0!. ~7!
The hydrodynamic matrixD̃, which is written down inAppendix A, has, as eigenvalues,
l125kx
21iv
n,
l225kx
22v2
cs21 i ~n1G!v
. ~8!
From Eq. ~5! we can express the velocity field at anpoint in the fluid as
vW ~y,kW x ,v!5M•D~y!•CW 1E2L/2
y
dy8M•D~y2y8!• fW~y8!,
~9!
with D being the diagonalized hydrodynamic matrix andMthe corresponding eigenvector matrix. Explicit expressiofor both of them are given in Appendix A. Finally,CW is avector of constants that should be specified by imposingboundary conditions in Eq.~9!. Although the derivation sofar has been focused on a localized force, we view it a~small! colloidal particle. In the limit of a vanishing radiusthe identification will be exact.
It is well established that the dynamics of suspended pticles can be understood in terms of couplings betweenparticle variables and the hydrodynamic modes@8#. Since weconsider times at which the displacement of the colloid cbe neglected, we can take advantage of Onsager’s regrehypothesis@9#, and can relate the decay of the velocity of tfluid at pointy0 ~where the force has been applied! with theVACF of a particle placed at that position. The velocitythe fluid at the point where we have initially applied thforce is given by
vx~y5y0 ,kx ,v!5c1el1y01c2e2l1y01c3
ikx
l2el2y0
2ikx
l2e2l2y0, ~10!
ary
4460 PRE 59PAGONABARRAGA, HAGEN, LOWE, AND FRENKEL
and the constantsc1 , . . . ,c4 are the components of the vectorCW . Substituting their appropriate expressions for stick boundconditions, one obtains
vx~y0 ,kx ,v!5f 0
l2n~l122kx
2!
H~kx ,l1 ,l2 ,L,y0!
$22kx2l1l2@12cosh~Ll1!cosh~Ll2!#1~kx
41l12l2
2!sinh~Ll1!sinh~Ll2!%, ~11!
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where we have introduced the functionH(kx ,l1 ,l2 ,L,y0),which is written down explicitly in Appendix A. If we nowtransform back to real space and time, we find
vx~x50, y5y0 ,t !5E2`
` dv
2pe2 ivtE dkx
2pvx~y0 ,kx ,v!.
~12!
This equation contains the complete information ondynamics of the particle. We will now focus on itasymptotic behavior, in the situation of strong confinemei.e., l1L!1 andl2L!1. The opposite limit has been examined in the literature@7#, in the context of interfacial hydrodynamic modes. In that case, the observed modes differ fthe usual bulk ones in the sense that, although qualitativthey are not modified, the damping increases due to the pence of the solid walls.
From the structure of Eq.~11!, one can see that the dynamics of the particle is determined by the relevant hyddynamic modes. The expression of the denominator shthat different dynamical behaviors will exist at different timscales. For example, the confinement will induce resonanof the propagating modes at the time scale in which sopropagates the width of the system. At low frequencies, hever, a diffusive mode develops and controls the asymptrelaxation of fluid perturbations. In this time regime, Eq.~11!reduces to
vx~y5y0 ,kx ,v!5
i f 0cs8kx
8L9F12S 2y0
L D 2G2
21233n6~l12kx2!v2~v2vd!
~13!
where vd5 i (cs2L2/12n)kx
2 , which is a diffusive mode.Therefore, density perturbations in the tube will decay difsively, and we can define an effective diffusion coefficiefor sound,D* , as
D* 5cs
2L2
12n. ~14!
As already pointed out, sound diffusion correspondsthe relevant low frequency dynamical response of the fluidthe tube. In this case we have derived an expression fordiffusion coefficient, which comes from the exact hydrodnamic analysis, and which presents a simple dependencthe geometry considered. It clearly shows that sound dision arises purely from the dissipation at the walls, withomaking any assumption about the momentum exchanNevertheless, the expression we obtain coincides withpredictions made on the basis of permeability@4#, which inthis case isL2/12. The previous analysis is also easily e
e
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-s
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-t
onhe-on-te.e
-
tended to explore the effect of different kinds of boundaconditions on the hydrodynamic response of the fluid. Fexample, for slip boundary conditions, the hydrodynammodes correspond to those of a fluid with an effectivemensionalityd* 5D21 @1#. In this case sound is alwayspropagative mode.
From Eqs.~12! and ~13!, we can derive the asymptotidecay of the velocity of a suspended particle,
vx~y5y0 , r x50,t !523A3
32Ap
f 0
csAnt3/2F12S 2y0
L D 2G2
1OS 1
t5/2D . ~15!
The asymptotic behavior is algebraic, and the velocreverses its direction during its decay. Also for unboundfluids an algebraic long-time tail is observed@2#. In this case,the slow decay is due to the coupling of the particle velocto vorticity diffusion. The velocity does not change sign, athe predicted exponent is different. In fact, on the basisvorticity, one would have expected an exponential decaythe velocity of the suspended particle in the present situa@10#. As we will show in Sec. III, vorticity cannot develop along times due to the interaction with the walls. It was thfact which led previous authors to predict an exponendecay for the VACF of a particle between walls@10#. Ourresult shows that the particle velocity couples to the slowdecaying mode in the fluid. The solid walls qualitativemodify the low frequency modes with respect to those chacteristics of an unbounded fluid, and the coupling todiffusing compressible mode leads to a qualitatively differedecay for the VACF.
We have restricted our analysis to a two-dimensiofluid, but it can be easily generalized to higher dimensioIn all cases a diffusive sound mode is present, and, accingly, a power law characterizes the long-time tail of tVACF of a suspended particle. The value of the exponcan be easily related to the number of dimensions of the flthat are not constrained,d* @1#. In Appendix B we present amore intuitive derivation of the negative algebraic long-timtail. It gives a simple physical picture of the role of the waon the development of sound modes, and predicts the cosponding exponent of the long-time tail for any dimensio
So far we have presented a hydrodynamic analysis. Ifcolloidal particle is diffusing, the amplitude of the long-timtail will be modified. It is easy to account for it by makin
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PRE 59 4461SHORT-TIME DYNAMICS OF COLLOIDAL . . .
use of mode-coupling theory@11#. In this case the algebraidecay is not modified, and the amplitude of the long-timewill depend both onn and the diffusion coefficient of theparticle in the usual way. Finally, we have always assumthat the solid surfaces are completely rigid, which ensurecomplete time scale separation between the propagatiothe fluid and wall modes. If the surfaces are deformabsound modes can partially propagate through the walls, leing also to a decrease in the amplitude of the long-time t
III. COMPUTER SIMULATIONS
We have performed computer simulations to gain a beunderstanding of the dynamical effects induced by the cfining walls, using the lattice Boltzmann model to simulatefluid @12#. This method is a preaveraged version of a lattigas cellular automaton model of a fluid. The basic dynamquantity is the fraction of particles moving in a given diretion at a certain lattice node. With this technique it is easysimulate the dynamics of colloidal particles. These are induced as surfaces where the collision rules of the populatin the neighboring nodes are modified in order to ensureappropriate boundary conditions. Therefore, it is also simto introduce bounding walls for the fluid. We use an impliupdating scheme for the moving boundaries@13# to avoidinstabilities, and to allow us to simulate buoyant particlwhile the original scheme could only deal with heavy pticles @12#. We have taken the lattice spacing as the unitlength, and the time step as the unit of time. In these unthe speed of sound of the fluid iscs51/A2, and the kine-matic viscosity used isn5 1
2 unless otherwise stated.In order to compute the VACF we should, in principl
perform equilibrium averages for the velocities of the pticles, taking into account that they move in a fluctuatifluid. Since we are interested in the short-time dynamicsfar as their displacements can be neglected, we can makof an Onsager regression hypothesis@9# and study the decayof their velocities from an initial perturbation. In this caswe can then disregard the fluctuations of the fluid, withcorresponding improvement in the simulation performan@14#. In all the simulations we will consider times such thsound has not had time to travel the length of the waTherefore, the system can be regarded as infinite, and wnot have to consider finite-size effects.
Throughout this section we will constrain ourselves tosimplest geometry. In this sense, the confining walls willeither planes~in 2D slits! or a cylindrical tube. When welook at particle dynamics, they will always be spheres~disksin two-dimensional!.
A. Diffusive sound modes
The simplest way to show the diffusive character of soumodes in a confined geometry is by analyzing the tempdecay of a localized density perturbation in the absenceany solid particle. Initially, the density of the fluid is homogeneous except for a node located in the center of a tubecompute then the moments of the density distribution alothe direction of the tube, averaging across the transverserection, i.e.,^x(t)n&5*2`
` dx*2L/2L/2 (r(x,y,t)2r0)ndy, with
n51,2, . . . ,wherer0 is the equilibrium density andL is the
il
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rn-
-ic
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,-fs,
-
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eet.do
ee
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width of the tube. We have analyzed the behavior ofdiffusion coefficient obtained from the second moment ofdensity distribution, i.e.D* 5 1
2 (d/dt)^x(t)2& at long times,as a function of the viscosity and the radius of the tube.Fig. 1 we display the diffusion coefficient measured asfunction of the inverse of the shear viscosity. The depdence of the diffusion coefficient of sound on the viscospredicted in Eq.~14! is clearly recovered. We have analogously checked its dependence on the width of the slit. Bscalings also allow us to deduce that the diffusion coefficimeasured in the simulations behaves as
Dsim* 5c2~L221!
12n1
1
2, ~16!
which agrees quantitatively with Eq.~14!, except for the fac-tors 1
2 and 1, which are due to lattice artifacts.In order to understand how the diffusive regime sett
down, in Fig. 2 we show the second and fourth momentsthe distribution as a function of time, normalized by the coresponding values for Gaussian diffusion, i.e.,^x2(t)&52D* t and ^x4(t)&512(D* t)2. We use as a value forD*the one given in Eq.~16!. At long times, both moments approach 1, which means that the diffusive regime settlesThe diffusive regime is reached on a time scale in whmomentum can diffuse the width of the tube,tn5L2/n. Onecan also see that the second moment approaches the Gian behavior faster than the fourth moment. This effectreadily seen by looking at the second cumulanta2(t)[^x4(t)&/„3^x2(t)&2
…21, which characterizes the nonGaussianity of the diffusion process, which is also displayin Fig. 2. One can see that it is a slowly decaying function.fact, a study of its asymptotic behavior shows that it vanisas a2(t);1/t, which implies than, even when the diffusivregime is achieved, still diffusion will be non-Gaussian unlonger times.
Finally, in Fig. 3 we display the time evolution of thdensity profile generated by the initially localized densperturbation. At short times two peaks start to displaceopposite directions at the speed of sound. As time proce
FIG. 1. The effective diffusion coefficient of density perturbtions D* as a function of 1/n. Results were obtained in a twodimensional slit of widthL59. The points denote the simulatioresults, and the line is a guide to the eye.D* andn are expressed inlattice units.
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4462 PRE 59PAGONABARRAGA, HAGEN, LOWE, AND FRENKEL
these peaks are progressively damped. On the other handregion between the two peaks does not relax to the equrium density. The inhomogeneity generated in this regiona result of the successive collisions of the initial density pturbation with the solid wall, develops subsequently intoGaussian and decays diffusively. It is on this time scale tthe dynamics is controlled by the diffusion coefficientD*derived in Sec. II.
FIG. 2. Second and fourth moments, and second cumulanta2 ofof the density distribution as a function of time for an initiallocalized density perturbation in a 2D slit of widthL510 and ki-nematic viscosityn50.5. The moments are normalized by the coresponding moment for Gaussian diffusion. Dimensional quantiare expressed in lattice units.
FIG. 3. Time evolution of an initial localized density perturbtion in a two-dimensional slit of widthL510 and kinematic viscosity n50.5. The density perturbation isdr50.1r0 , r0 being theequilibrium density. Density profiles at timest50.1, 0.15, 0.2, 0.25,0.3, and 0.35 correspond to continuous lines, and times 0.5 andto dashed lines. The lower the curve in the center, the largertime it corresponds to. Time is expressed in units of the diffustime tn , and distance is expressed in lattice units.
theb-s-
aat
B. Velocity correlation function for a particle
We will start by analyzing the VACF of a sphere of radiur 52.5 in the center of a cylinder of radiusR54.5, as shownin Fig. 4. The initial velocity of the particle is parallel to thaxis of the cylinder. The decay coincides with the one otained for an unbounded fluid until the initial velocity peturbation has reflected back from the wall on the particThis deviation will scale linearly with the tube width, because it is controlled by sound propagation. At longer timsuperposition of sound reflections modifies the decay ofvelocity which becomes negative, exhibiting a minimuSuch a minimum appears, roughly, at a timetc5L/cs , whenmomentum has propagated the tube width. It is, subquently, at timestn when momentum has diffused the tubwidth that the decay becomes algebraic. As shown ininset of Fig. 4, the exponent of the algebraic decay is2 3
2 inthis case, which coincides with the theoretical predictiona 2D fluid of the preceding section.
In order to test the prediction for the amplitude of thlong-time tail, we have studied the asymptotic decay olocalized velocity perturbation, and of particles of radiusr50.5 and 2.5, all of them located at the center of a twdimensional slit of width 16. In Fig. 5 we compare the amplitude of the algebraic tail predicted by Eq.~15! with thesimulation results in all three cases. Due to the discretenof the lattice, there exists an uncertainty about the aclocation of the solid boundaries@12#. Both for a point forceand for the particle of radius 0.5 this indeterminacy is neggible. For a particle of radius 2.5 we have fitted the curmultiplying the distance from the center of the particle to twall by a factor 1.1. This assumes a 10% error in the lotion of the interfaces, leading to errors in the distance thatsmaller than one lattice spacing, and which are in agreemwith previous estimates of the uncertainty in the particle-wseparations@15#.
We have focused on motions of the particles parallelthe directions that are not confined. If the particle initiamoves perpendicularly to the walls, an exponential decaobtained as could be expected, since in this case the deperturbation generated by the motion of the particle can
s
.5e
e
FIG. 4. Normalized VACF of a colloidal particle with radiusr52.5 in the center of a cylindrical tube with radiusR54.5, bothlengths expressed in lattice units.
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PRE 59 4463SHORT-TIME DYNAMICS OF COLLOIDAL . . .
propagate through the system. Therefore, even if in a sussion particles move at random, the long-time behavior wbe controlled by the components parallel to the walls.
One can also analyze the behavior of the angular veloautocorrelation function~AVACF!. The decay is always exponential as long as the particle is in equidistant fromwalls. This is due to the fact that the angular velocity donot couple to any compressible mode, and its decay is ctrolled by vorticity. However, when it is not equidistant, theit will induce a translational velocity, and therefore it widecay algebraically at long times, with the same exponeas those corresponding to the VACF. Again, if slip boundconditions are considered the situation is qualitatively diffent. In this case the fluid has an effective dimensiond*5D21, vorticity can diffuse in this effective medium, anthe AVACF decays algebraically with the correspondinggebraic power, without changing its direction of motion duing its decay@16#.
If we would have considered the VACF of a particle inclosed container, one expects that eventually the decaybe exponential. For a roughly isotropic container, in whall its dimensions are of the same magnitude, either thecay of the VACF is purely exponential or vorticity developwhen the system is large enough, showing a later crossto the final exponential decay@17#. If we consider an elon-gated axisymmetric container, even if vorticity cannot dfuse, if the long lengthL uu and the short oneL' are related insuch a way that momentum can diffuse the short distabefore travelling the long one, i.e.,L uu /cs@L'
2 /n, then diffu-sion of sound alongL uu will have time to develop. In thiscase, for timesL uu /cs.t.L'
2 /n the decay of the VACF of aparticle will be algebraic, and only at later times it will become exponential. In this case, the algebraic decay is noasymptotic behavior of the short-time dynamics, as seeFig. 6.
FIG. 5. Amplitude of the long-time tail measured in the simlation Asim relative to the theoretical predictionAth given in Eq.~15!as a function of the distance from the center of the disturbancfrom the particle centery to the center of the two-dimensional tubyc , expressed in lattice units. The tube width isL516, and theviscosityn5
12 .
n-ll
ty
esn-
tsy-
--
ill
e-
er
e
hein
C. Hydrodynamic fields
We will now look in detail at the flow fields that argenerated in a fluid confined between two parallel platesresponse of an initially moving sphere in an otherwise qescent fluid. We will consider a particle of radiusr 52.5lattice units, placed in the center of a tube of radiusR58.5,in a fluid of shear viscosityn5 1
6 . We focus on the featureof the flow, vorticity, and density fields for a particle diffusing along the axis of the tube, during the initial decay of tVACF, i.e., for times at which the asymptotic decay has nbeen reached yet.
The total simulation time corresponds tot/tn'1. Theflow, vorticity, and density fields depend on the three spacoordinates. The fields are examined in thez50 plane. Thevelocity field and density are directly calculated in thlattice-Boltzmann scheme. The vorticity,v, at a lattice point(x,y), which is a local quantity is given by
v~x,y!51
2„vy~x11,y!2vx~x,y11!2vy~x21,y!
1vx~x,y21!…. ~17!
This is a two-dimensional discretization ofv5 12 ¹3v, the
three-dimensional definition of vorticity. The time evolutioof the various fields is shown in Figs. 7 and 8.
Initially, the velocity field contains the vortex pair thatresponsible for the positive long-time tail in the VACF for aunbounded fluid, although as soon as the perturbationduced by the particle reaches the walls vortices creepinthe wall of the tube are also seen. On the other hand,motion of the particle produces an increase of the densitthe front, and a decrease in the back, as expected. In Figis already seen that the two vortices cannot grow diffusiveand that meanwhile the vortices that were creeping alongwall are already outside the picture. The density plot incates that in front of the particle there is a sizable denincrease. At later times, when the particle velocity reverdirection, a clear change in the qualitative behavior of
or
FIG. 6. Decay of the VACF for a disk of radiusr 52.5 in thecenter of a box of sidesLy59 andLx5199. The coefficienta usedto show the long-time exponential decay has been fitted. We huseda50.006. Lengths are expressed in lattice units.
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tionon.
4464 PRE 59PAGONABARRAGA, HAGEN, LOWE, AND FRENKEL
flow field is observed. The two vortices have been damout, which shows that the perturbation induced by vorticdecays exponentially due to the presence of the walls@10#,and over a significant area the velocity in thex direction isopposed to the direction at which the particle was initiamoving. This feature becomes more salient at later timescan be seen in Fig. 8, which corresponds to the minimumthe VACF. Due to the back flow around the particle, tvorticity near the colloidal particle has almost disappearAt later times the negative flow field keeps growing in sizat the end spanning the total plotted region. The velocfield resembles the Poiseuille flow field for these timwhich is the parabolic steady state flow profile for flow intube, with a pressure drop, and the density field becomore one dimensional.
FIG. 7. Flow field, vorticity, and density, from top to bottomrespectively (t/tn50.47). The velocity field, which is at the top, iscaled withs5Max vx in the x direction, whereas for they direc-tion 4s is used to scale the arrows. For the sake of clarity oneof the velocity vectors in both directions are omitted from the pture. The vorticity field is obtained by applying Eq.~17! to thevelocity field, and is shown in the middle. It is scaled such thatisovorticity lines are shown, at heights which vary linearly betwethe maximum and minimum vorticities. The largest vorticity is dpicted by the darkest color. The same procedure was performethe density, the bottom picture.
FIG. 8. t/tn50.78. For caption, see Fig. 7.
d
asf
.,y,
es
IV. DIFFUSION COEFFICIENTS
The self-diffusion of the suspended particle can be relato the integral of the VACF by means of a Green-Kubo epression. We can then obtain macroscopic information abthe behavior of the suspension, from the knowledge gaiin the analysis of its macroscopic behavior.
In Fig. 9 we show the values for the diffusion coefficiefor a sphere of radius 2.5 at the center of a cylindrical tubevariable radius, normalized by the value of the diffusion cefficient in the absence of the tube. We also plot the predtion of the center-line approximation@18#, calculated assuming a purely incompressible fluid, i.e., neglecting soueffects. We obtain a perfect agreement with the theoretpredictions. This can be understood, because the algebdecay we have described in the preceding sections afrom the coupling to the compressible modes of the fluand these vanish in the stationary limit. These modes doaffect the integral of the VACF since their amplitude isfact proportional to the frequency itself. The diffusion coeficient is determined by the decay of transverse velocity pturbations@10# which, in a confined system, is exponentiaSo, while compressibility effects dominate the long-time dnamics, they still do not contribute to the diffusion coefcient. However, it is important to resolve all the time scain order to obtain the proper diffusion coefficient from thintegral of the VACF. In Fig. 10, we show the time depedent diffusion coefficient, which is the integral of the VACfrom 0 to timet. One can see that it exhibits a maximum.we would have assumed an exponential decay for the VAwe would have obtained an overestimate for the value ofdiffusion coefficient.
Due to the flexibility of the lattice Boltzmann method tdeal with complex geometries, we can use the techniquemeans to derive profiles for the diffusion coefficient of supended particles in general geometries. In Fig. 11, we shthe values of the diffusion coefficient for a spherical particof radius 2.5 in a cylinder of radius 9. As we move ofcenter, the three components of the diffusion coefficientnot equal. In particular, the component of the diffusion pallel to the tube (Dx in the figure!, exhibits a maximum as
lf-
nn
for
FIG. 9. Diffusion coefficient of a particle with radiusr in thecenter of a cylindrical tube with radiusR, normalized with respectto its value at the center of the tube. The points denote simularesults, and the line corresponds to the center-line approximati
ol-erheu
llew
ete
hebnthe
that
elyect
beor-ion
in aob-in-
r a
-itdlts
nte
nt
PRE 59 4465SHORT-TIME DYNAMICS OF COLLOIDAL . . .
we move toward the plate. A perturbative calculation@19#implies the existence of a maximum in the parallel compnent of the diffusion for the motion inside a cylinder, athough we do not know of any explicit calculation of thvalue of the maximum. The same behavior persists fotwo-dimensional fluid, as shown in Fig. 12. However, tdiffusion coefficient parallel to the the walls is a monotonofunction for a three-dimensional fluid between two paraplates. In Fig. 13 we show the corresponding profile, andcompare it with the value for the density predicted theorcally if we neglect the hydrodynamic interactions betwethe two walls. In this case, the diffusion coefficient of tsphere is the sum of the diffusion coefficients inducedeach wall independently, corrected by the value at the ceof the layer. For the diffusion coefficient of a sphere in t
FIG. 10. Dependence of the diffusion coefficient with time foparticle of radiusr 52.5 in a slit of widthLy57. tw5(L/2)2/n isthe time needed the vorticity to diffuse the width of the tube.
FIG. 11. Profile of the diffusion coefficient of a particle of radius 2.5 in a tube of width 9 lattice spacings, as a function ofposition off-center.x is the direction of the axis of the cylinder, anwe move off-center in thez direction.D refers to the average locadiffusion coefficient defined as the mean of its three componen
-
a
slei-n
yer
presence of a single wall a theoretical expression existsis sufficiently accurate except close to contact@20#. The plotshows that the agreement is very good, even for a relativnarrow fluid layer as the one considered, in which the aspratio isl[2r /Ly51/3, with r being the radius of the colloidand Ly the width of the tube. The deviations that canexpected close to contact, where lubrication will be imptant, cannot be resolved in a lattice Boltzmann calculatwith the small sphere considered in these simulations.
Diffusion coefficient for a confined suspension
We have also considered the VACF for a suspensionslit. The time dependence observed is analogous to thetained in the preceding sections for a single particle. By
s
.
FIG. 12. Profile for the component of the diffusion coefficieparallel to the walls in a two-dimensional slit for two different tubwidths,Ly . The particle has always radiusr 52.5 lattice units.
FIG. 13. Profile for the component of the diffusion coefficieparallel to the walls for a sphere of radiusr 52.5 confined betweentwo plane walls at located at a distanceLy515 from each other.Lengths are expressed in lattice units.
r an
en-lee
r a
neo
ononpaen
slvn
caco
itaizlidktlsniah
al
lowto
de-y
uid,eionpic-ityin
al-ot
lls,ns., anf
d
heon.ntonalic-cals.er
dif-ibitx-
ouswithn,thelleon-fu-
as
ethe
o-ncl
4466 PRE 59PAGONABARRAGA, HAGEN, LOWE, AND FRENKEL
tegrating, we obtain the short-time diffusion coefficient focolloidal suspension, and we analyze its behavior as a fution of the aspect ratiol. It is clear that the wider the tubthe larger the diffusion coefficient, since the mobility is hidered due to the hydrodynamic interaction of the particwith the walls. This is clearly depicted in Fig. 14, where wshow the decay of the value of the diffusion coefficient foparticle equidistributed in the tube, as a function ofl. How-ever, if we normalize the diffusion coefficients of the suspesion by the average mobility at zero volume fraction, ththe curves for the diffusion coefficients of suspensions cresponding to differentl look much more similar to eachother, as shown in Fig. 15. In fact, a significant deviatifrom the general behavior is only observed for strong cfinement. As soon as roughly two layers of suspendedticles fit between the walls, their relative motion becomdominant, and the walls can be seen as if they provide othe mean friction felt by the colloidal particles.
V. CONCLUSIONS
In this paper we have analyzed the short-time dynamicconfined colloidal suspensions. We have restricted ourseto the simplest geometry, namely, a tube, where exact alytical results can be derived for point particles and onefocus on the basic dynamical features of the short-timeloidal dynamics.
We have shown that the presence of stick walls qualtively modifies the hydrodynamic modes that characterthe response of the fluid. The coupling of the fluid to a soelastic medium~which is a typical example where sticboundary conditions are fulfilled! induces the developmenof a diffusive sound mode at low frequencies. We have ashown that this diffusive mode is initially non-Gaussian, ait is only on longer time scales that it becomes GaussThis diffusive mode decays slower than vorticity, whiccannot propagate diffusively due to the presence of the w
FIG. 14. Linear-log plot of the component of the diffusion cefficient parallel to the walls of a two-dimensional tube as a fution of the width of the tube,l, for an equidistributed particle. In alcases particles have a radiusr 52.5. The diffusion coefficient andthe radius are expressed in lattice units.
c-
s
-nr-
-r-sly
ofesa-nl-
-e
odn.
ls,
and it becomes the leading hydrodynamic response atfrequencies. The dynamics of the suspension will couplethe longest-lived solvent mode. Therefore, the long-timecay of the velocity autocorrelation function is controlled bsound, contrary to what happens in an unbounded flwhere it is controlled by vorticity. The diffusive decay of thinitial density perturbation induces a change in the directof the suspended objects, as was clearly seen from thetures of the hydrodynamic fields. Also, the angular velocautocorrelation function differs from the usual behaviorunbounded fluids. Since its decay is due to vorticity, itways exhibit an exponential decay, insofar as it is ncoupled with the velocity. Due to the presence of the wathis is an unrealistic requirement for colloidal suspensioAs soon as the particles are not equidistant from the wallsinitial angular velocity will induce a translational velocity othe particle@21#. This coupling term will decay slower anwill control the subsequent decay of the AVACF.
From the analysis of the VACF we have extracted tvalues for the diffusion coefficient of a confined suspensiIn order to obtain proper values, it is necessary to take iaccount the algebraic decay of the VACF, although the fivalues for the diffusion coefficients agree with the predtions for incompressible fluids. We have looked at the lovalues of the diffusion coefficient for different geometrieDue to the presence of the walls the diffusion is no longisotropic, and we have focused on the component of thefusion parallel to the walls. We have seen that it may exha maximum, although it is not a generic feature. For eample, in the case of two parallel plates it has a monotonbehavior, and we have been able to compare our valuesthe predictions obtained from the additivity assumptiowhich neglects the hydrodynamic interactions betweentwo walls. This approximation is seen to work quite weeven for quite narrow tubes. The flexibility of the latticBoltzmann method in dealing with general boundaries cverts it into a useful technique to obtain maps of the difsion coefficient in general geometries, that can be used
FIG. 15. Normalized diffusion coefficient as a function of thvolume fraction for a confined suspension, as a function ofaspect ratio. In all cases the particles have a radiusr 52.5 latticeunits.
-
ar
ungr-d
ora
anntisie
pledamultivdsyhe
aeepe
n-ter
ionso-
cdse
deen
Ml
ofionnt
PRE 59 4467SHORT-TIME DYNAMICS OF COLLOIDAL . . .
inputs for other simulations techniques that work in the pticle diffusion time scale, such as Brownian dynamics@22#.
The development of the diffusive sound mode can bederstood on a simple physical basis. For time and lenscales in which the fluid is interacting with the solid intefaces, momentum is no longer a conserved variable. Unthese conditions, only mass is conserved, and has therefdiffusive character. This interpretation provides also an anogy with a Lorentz gas. This model is known to exhibitalgebraic velocity decay at long times with an expone2D/211 @23# for a D-dimensional system, and whichprecisely the one we have obtained from solving the NavStokes equations, takingD as the effective dimension inwhich the density is the only conserved variable. This simconnection explains the robustness of the results presentthis paper. We have studied the decay of the VACF for pticles moving in a porous medium built up with fixed randoparticles, and also the decay of a particle moving in a regarray of fixed rectangles, obtaining in all cases a negaalgebraic decay, with an exponent that can be understoothe number of unconstrained dimensions. In these twotems, the solid walls do not completely restrict any of tspatial dimensions, and we find, consistently,d* 5D, D be-ing the dimensionality of the system. This also implies ththe same kind of dynamical behavior we have described hshould apply to the dynamics of particles embedded in gor in suspended membranes, where in the latter case thesibility of developing diffusive sound modes has also bepointed out@24#.
lo
-
-th
ere al-
r-
ein
r-
areass-
trelsos-n
Finally, if we would have considered that the fluid is cotained between slip walls, then the fluid would develop afa transient into an effective fluid of dimensiond* . Thisclearly shows that the dynamical behavior of the suspensin confined geometries will be sensitive to the specific mmentum fluxes that occur at the boundaries.
ACKNOWLEDGMENTS
The work of the FOM Institute is part of the scientifiprogram of the FOM and is supported by the NederlanOrganisatie voor Wetenschappelijk Onderzoek~NWO!.Computer time on the CRAY-C98/4256 at SARA was maavailable by the Stichting Nationale Computer Faciliteit~Foundation for National Computing Facilities!. I.P. ac-knowledges the FOM for financial support, and the FOInstitute for its hospitality. We thank E. Trizac for a criticareading of the manuscript.
APPENDIX A: HYDRODYNAMIC EIGENVECTORS
In this appendix we write down expressions for somethe quantities used in Sec. II during the theoretical derivatof the VACF. All the symbols appearing in the subsequeexpressions are defined in Sec. II.
The hydrodynamic matrix introduced in Eq.~5! is
D̃5S 0 1 0 0
1
nF iv2 ics
2kx2
v1~n1G!kx
2G 0 0 2cs
2kx1 ikxGn
nv
0 0 0 1
0cs
2kx1 ivGkx
ic22~n1G!v2
v~ iv1nkx2!
ics22~n1G!v
0D , ~A1!
an
which after diagonalizing gives
D5S l1 0 0 0
0 2l1 0 0
0 0 l2 0
0 0 0 2l2
D ~A2!
The eigenvector matrixM corresponding toD, and whichis needed to derive the appropriate expression for the veity field from Eq. ~9!, has the form
c-
M5S i
kx
i
kx
ikx
l22
ikx
l22
2il1
kx
il1
kx2
ikx
l2
ikx
l2
21
l1
1
l12
1
l2
1
l2
1 1 1 1
D . ~A3!
Finally, the functionH(kx ,l1 ,l2 ,L,y0) introduced in Eq.~11! and which specifies the velocity at the point whereinitial perturbation is applied in the fluid is given by
4468 PRE 59PAGONABARRAGA, HAGEN, LOWE, AND FRENKEL
H~kx ,l1 ,l2 ,L,y0!52kx6 sinh~Ll1!@cosh~2y0l2!2cosh~Ll2!#1l1
3l23 sinh~Ll2!@cosh~2y0l1!2cosh~Ll1!#
24kx2l1l2 sinhF ~2y02L !l2
2 GcoshF ~2y02L !l1
2 G~l1l21kx2!24kx
2l1l2~l1l22kx2!
3sinhF ~2y01L !l2
2 GcoshF ~2y01L !l1
2 G1kx2l1
2l22@3 sinh~Ll1!cosh~Ll2!
22 sinh~2y0l1!cosh~Ll1!#2kx4l1l2@2 sinh~Ll2!cosh~Ll1!1sinh~Ll2!cosh~2y0l1!#. ~A4!
illpea
istoona
thintht t
ic
ic
om
sinfu
vel-he
e
we
,s
n-
ndIf
p-s
n-
APPENDIX B: INTUITIVE DERIVATION
Although the analysis of Sec. II is rigorous, here we wpresent an alternative derivation that provides a simphysical understanding of the role of the walls in the devopment of the sound diffusing mode. We will considerD-dimensional fluid confined between parallel walls a dtanceL apart in they direction. The basic idea is that, duethe geometrical constraints imposed by the tube, at ltimes the transversal component of the velocity field hasmost relaxed, and therefore,] uuv uu@]yvy , where] uu meansthe derivative with respect to the components parallel towalls. Then, although the velocity can be thought as havone dimension less, it still keeps the dependence in allspatial coordinates, because at all times it has to vanish awalls. At long times what happens is that they dependence isimportant only close to the boundary, and the existencesuch boundary layers will dominate the hydrodynammodes.
If we introduce the previous assumption from Eqs.~2! and~3!, we end up with
]r
]t1r0¹ uu•vW uu50,
~B1!
]v uu
]t1a¹ uur2G¹ uu
2v uu2n]2v uu
]y25FW .
The hydrodynamic modes that determine the dynamresponse of the fluid are then
v i5 ink2, ~B2!
v65i
2~nk21Gkuu
2!61
2A2~nk21Gkuu!
214cs2kuu
2,
~B3!
which correspond to one incompressible mode and two cpressible modes. The wave vectorkW is defined such that itscomponent parallel to the wall is given by the vectorkW uu , andits component perpendicular to the wallky52pn/L is dis-cretized and chosen from the appropriate sine or cotransform to ensure that stick boundary conditions arefilled. The integern will therefore not contain the value 0.
lel-
-
gl-
egehe
of
al
-
el-
The fact that, even in the hydrodynamic limit, the wavectors in they direction do not vanish implies that the reevant modes will depend on the width of the tube. In thydrodynamic regime, which corresponds to the limitkuu→0, the compressible modes read
v15 i ~nk21Gkuu2!1O~kuu
4!, ~B4!
v25 i2cs
2
nky2
kuu21O~kuu
4!. ~B5!
Therefore, in this regime we can identify an effectivsound diffusion coefficient
D̂52cs
2
nky2
, ~B6!
which has the same functional dependence as the onehave rigorously derived@Eq. ~14!#. Incidentally, if we sumover all the perpendicular wave vectors, we recover
(n51
`
D̂5cs
2L2
2p2n(n51
`1
n25
cs2L2
12n, ~B7!
which is precisely Eq.~14!. In terms of the previous modesthe velocity induced by the applied the force will decay a
vW ~ t !52 i ~1WW 2 k̂uuk̂uu!•FW
v2v i1
2 iv k̂uuk̂uu•FW
~v2v1!~v2v2!. ~B8!
Both contributions to the velocity dynamics from the icompressible mode andv1 will decay exponentially in time,becauseky cannot tend to zero. However, sincev2 is diffu-sive, it will induce an algebraic decay of the velocity, athis will become the dominant contribution at long times.we transform Eq.~B8! back to time and space, we asymtotically obtainv(t);2t2(D11)/2. Since in this case there ionly one constrained direction,d* 5D21, which implies
v~ t !;21
t ~d* /2!11. ~B9!
which for D52 agrees with the results of Sec. II, and geeralizes it for a general dimension.
ke
ys
kel,
-
PRE 59 4469SHORT-TIME DYNAMICS OF COLLOIDAL . . .
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