Effect of Hydrodynamic Interactions on the Distribution of Adhering Brownian Particles

J. Bafaluy ~ ) ~ ) B. Senger, t ) J.-c. Qoegei ~ ) and P. Schaaf~ ) ~ ) t lInstitut National de la Saute et de la Recherche Medicale CJIi M-O), Centre de Recherches Odontologiques, Universite Louis Pasteur, 1 Place de l'Hopital, 67000 Strasbourg, Prance Departament de Fx'sica, Universitat Autonoma de Barcelona, 08198 Bellaterra, Barcelona, Spain Ecole Europeenne des Hautes Etudes des Industries Chimiques de Strasbourg,

The adhesion of Brownian particles on solid surfaces has attracted much interest from both theoretical and experimental points of view during the last years.One of the most popular models used to describe this appar- ently simple process is the so-called random sequential adsorption (RSA) model [1], which takes excluded vol- ume effects into account.However, it has serious limi- tations.In particular, it does not account for the trans- port of the particles to the adsorbing surface.This ef- fect is introduced in the diffusion RSA model (DRSA) [2], where the adsorbing particle is allowed to diffuse in three-dimensional space subject to hard sphere interac- tions with previously adsorbed ones.The DRSA leads to an increased adsorption probability for an incoming sphere in the close vicinity of an already attached one as compared to RSA.The DRSA distribution of adsorbed particles is thus different at a given coverage from its RSA counterpart except near saturation (jamming limit) [3].However, since the diffusion coeKcient is taken constant, the DRSA seems to model a particle as moving in "dry water" [4].This Letter is devoted to take hydrodynamic interactions into account and should thus represent a significant jump toward reality.
The effect of the hydrodynamic interaction is to in- crease the frictional force experienced by a particle when it approaches another one or a fiat surface [5].This kind of interaction between a sphere and a clean wall is well known, and its effect on the rate of adsorption has al- ready been studied [6].The main goal of this Letter is to investigate the inHuence of the hydrodynamic interac- tions between an adsorbing particle and (i) the already adsorbed ones, and (ii) the planar adsorbing surface, on the distribution of the particles on this surface.It will in particular be compared to RSA distributions in order to investigate the degree of accuracy of this simple and now well-known algorithm.
The Brownian motion of a spherical particle is com- pletely described by the friction tensor, which is in general position dependent and nonisotropic.
In particular, the normal component of the friction tensor diverges when the separation between the particle and any solid surface vanishes due to lubrication forces.As a conse- quence, contact of the particles with the adsorbing sur- face is impossible in the absence of a strong attractive force, like the van der Waals attraction.The latter be- comes strong enough at small separations to overcome the lubrication effects.In general van der Waals, hard core, and electrostatic forces also act between the parti- cles.However, in this study we will solely consider, besides the hydrodynamic interactions, the different hard core repulsions, and the van der Waals forces.For the latter we restrict ourselves to the forces acting between the moving sphere and the adsorbing plane.The sphereplane van der Waals force is derived from the corresponding potential, U, given by [7] AH 9 0 f 8 -+ + ln ." ' "") where 8 represents the shortest distance between the sphere of radius a and the plane.The Hamaker constant AH corresponds to the interaction between the particle and the surface in the presence of the fluid.It is typically of the order of 10 J.
The diffusion tensor D is related to the friction tensor R through the Einstein relation D = kTR. .Far from the surface, D becomes independent of the position of the sphere and is then given by the Stokes-Einstein relation D(oo) = IkT/6~rla, rl being the viscosity of the fluid and I the unit tensor.In general, the position dependence of the diffusion tensor may be derived as follows.Using the linear dependence of the force F and torque L with the translational and angular velocities, u and 0, respec- ).f'BD, , \, Ox~D , ~BU at+AX, , i =1,2, 3; AX, is a Gaussian random variable with zero mean and variance given by (AX, AAz) = 2D,&At All the quanti-.
ties are computed at the position of the particle at time tively, one has according to Ref.
[8] F = A. u+ B 0, L = B u+ C 0, where the second rank tensors A, B, and C are components of the resistance matrix.B repre- sents the transpose of matrix B. For an isolated sphere there is no coupling between the torque and the trans- lational motion (B = 0).However, in the presence of boundaries, e.g. , a sphere near a surface, the hydrody- namic interaction produces such a coupling.After elimi- nation of the angular velocity in the previous linear equa- This shows that the effective force acting on the particle, F,g --F -B C L, is linearly related to the velocity through the effective friction tensor . This is the friction tensor that must be used in the Einstein relation to obtain the diffu- sion tensor needed in the Smoluchowski equation which governs the diffusion of the particles.
The lubrication forces depend on the local flow of the fluid in the small regions between the moving particle and (i) the plane and (ii) the adsorbed spheres.Since these regions are well disconnected, one can assume as a first approximation that they contribute additively to the resulting force, and thus also additively to the effective friction tensor [9].The contributions from the sphere- sphere interactions to the friction tensor were calculated by using the analytical results given in Ref. [8].The sphere-plane contributions were computed according to

Refs. [10 ill
The hydrodynamic interaction increases the friction experienced by a particle that approaches a surface.This increase is larger for the motion normal to the surface than for the parallel one.Therefore, the lateral diffusion of the particle is enhanced compared to the diffusion in the direction perpendicular to the surface.A randomiza- tion of the adsorption location along the surface is then expected, rendering the distribution of the adsorbed par- ticles to look more like its RSA counterpart.
For low to intermediate coverages, interactions with only one or two adsorbed particles are relevant.Therefore, we simulated the Brownian motion of an incoming particle in the presence of, respectively, one and two ad- sorbed spheres on the surface.The center of the moving particle diffuses in-the region shown in Fig. l.Its tra- jectory was simulated by using the classical algorithm of Ermak and McCammon [12]: Given the position of the particle at time t, its displacement along the ith coordi- nate axis during a small interval of time 4t is calculated according to The value of 6 in the bulk, bo, fixes the spatial resolution of the simulation.It should be a small fraction of the radius of the particle, i.e. , bo && c.When the center of the particle approaches the exclusion surface (Fig. 1), the components of the diffusion tensor and the van der Waals potential depend strongly on the distance, s, to the surface which becomes the relevant distance.
The step of integration is then modified to become of the order of tI = s6o/a.This process is stopped at some small, but finite distance, to avoid an infinite number of elementary displacements and thus an infi.nite computer time.
Fortunately, the behavior of the particles at sufficiently small separations can be obtained using the asymptotic behavior of the diffusion tensor [D~s, D~~-(lns) ] and of the potential U s .Because we do not take attractive van der Waals forces between particles into account, the surfaces of the adsorbed spheres are perfectly reflecting boundaries.The surface of the exclusion sphere around a fixed particle is then an entrance boundary [13] that cannot be reached by the center of the diffusing particle.Spheres arriving at a small distance eq (( a will then be reflected at some position on the spherical surface at a distance 2eq from the particle.Using the expressions of the potential and of the diffusion coefIi- cients near the surface, the mean time r for this diffusion process can be computed using Eq.(5.2.160) of Ref. [13].
The lateral displacement of the particle can then be es- timated by ((Ar~~)) = (4D~~r) I'v'4era.It is taken smaller than the resolution bo, and therefore one takes er = 6o/4a (« a).Near the adsorbing surface, in con- trast, the attractive van der Waals force prevents the particle from escaping; the adsorbing surface is then an exit boundary [13].The simulation is stopped when the particle reaches a distance e~.A particle starting from a plane of height e2 eventually reaches a surface at height 2~2 with a predetermined small probability p.Using Eq. (5.2.189) of Ref.
Taking the radius of the particles c as the unit of length, and the time to = az/D(oo) as the unit of time, the problem can be cast in dimensionless form.The only dimensionless parameter is the adhesion number A~dh, measuring the relative strength of the van der Waals force with respect to the random force.If A d») 1 the tra- jectory of the particle becomes deterministic for 8 1, and a distribution similar to that obtained in ballistic deposition models has to be expected.If, on the con- trary, A dh & 1, the diffusive motion is dominant until the particle is very near to the surface; a homogeneous distribution is to be expected.For polystyrene particles in water at 300 K one has A gh = 0.4426.In the follow- ing, all the quantities will be expressed in a dimensionless way.
For the case of one adsorbed particle the simulation procedure is as follows: One particle is permanently fixed at the center of a square of side 10, with its center at height h = 1.This square is surrounded by four ver- tical walls to which periodical boundary conditions are applied.The periodicity did not influence the results.The starting position of the center of the moving parti- cles is chosen randomly on a horizontal plane at a height h above the surface.Afterwards, the values of the diffu- sion matrix and the force corresponding to the position of the particle are computed as described previously.The displacement of the particle is then determined by us- ing Eq. ( 2).It was verified on preliminary simulations that long-range hydrodynamic interactions do not mod- ify significantly the random distribution of particles if h & 5. Thus, the starting height was fixed at h = 5.Furthermore, to avoid particles escaping towards infin- ity, a sphere reaching the plane h = 8 was rejected, and a new one restarted from a random position in the plane fixed on the surface.(b) Adhesion probability along the axis perpendicular to the line of centers of two spheres fixed on the surface.Each point corresponds to one of the rectangles in (a), y = 0 being the center of the simulation area.h = 5.The mean displacement of the particle in each step is initially bo --0.1.The value of p was chosen to be 10 .Therefore, ei --0.0025 and ~2 --0.022.Once the particle touches the adsorbing plane, without overlap with the fixed sphere, the coordinates (x, y) of the con- tact point are recorded.The particle is then removed and a new one started randomly from h = 5.This procedure is repeated until a chosen number of particles, 10 in the present study, has reached the surface.At the end of the simulation stage, the surface is divided into small regions of size d.The ratio q(r) of the density of contact points (recorded previously) at a distance r from the fixed particle to the average density is plotted in Fig. 1, together with the corresponding result obtained ignoring hydro- dynamic interactions and van der Waals forces (DRSA).
It shows that the effect of the hydrodynamic interactions is to cancel out practically the nonuniformity introduced by diffusion.One recovers in this way a uniform distri- bution.This is precisely the basic hypothesis of the RSA that seems, therefore, to be a quite correct model to de- scribe the structure of the adsorbed phase at least at low coverages.
To verify this conclusion for higher coverages, we per- formed also simulations in the presence of two adsorbed particles, fixed on a rectangular 15 x 10 surface, the dis- tance between their centers being 5 [Fig.2(a)].Periodic boundary conditions were applied as previously.The re- mainder of the simulation procedure is identical to the one-sphere case.There is now a relatively small region between both spheres for the deposition of a third one, and the effects of the hydrodynamic interaction are ex- pected to be large there.Figure 2(b) shows the probabil- ity density for the adhesion of a third particle along the axis normal to the line of centers of the Axed spheres.A small depletion (of the order of 10%) seems to appear in the region nearest to the spheres.Away from that region, the distribution is uniform within statistical error.We can understand these results in the following way: The most important efFect of the hydrodynamic interactions is to diminish the motion perpendicular to the surface in comparison to the lateral motion.Therefore the lat- eral diffusion homogenizes the distribution of particles.Thus, they reach the surface in a practically homogeneous way.The inclusion of repulsive electrostatic forces between the moving particle and the adsorbing surface would even strengthen this effect: If the particles meet a repulsive barrier or a secondary minimum before reach- ing the surface, they will diffuse for a long time parallel to the surface before being adsorbed.
The main goal of this study was to introduce accurately the hydrodynamic interactions during the adhesion process of spherical particles on a solid surface.We have treated here the case in which interparticle interactions other than hydrodynamic or hard core are neglected.In this case, the particles seem to adsorb almost randomly on the surface without correlation with previously ad- sorbed spheres.This implies that, despite the complexity of the adhesion phenomenon, RSA is a suitable model to describe accurately the particle distribution on the sur- face.This result is, at Erst sight, surprising and gives a new validity to the numerous RSA studies undertaken during the last years.Even though the results reported in this Letter can only lead to this conclusion for coverages lower than or of the order of 30% (for higher coverages three-particle interactions become important [14]), it can be assumed that it remains valid up to the jamming limit.
Indeed, for high coverages, only small regions in space remain accessible to the centers of new particles.Then, during the difFusion of the particles toward the surface, a randomization of the particle in the direction parallel to the surface is likely to occur, Further studies are now un- der way to include van der Waals and double layer forces between particles in the model.

4 FIG. 1 .
FIG. 1. (a)Geometry of the simulation.The region from which the center of the diffusing particle is excluded is delim- ited by the dashed line (exclusion surface).(b) Relative dis- tribution of the points of adsorption of particles obtained by simulation of 10 trajectories with A dh = 0.4426.The results of the simulation are compared to the distribution obtained when the particles diffuse in the absence of hydrodynamic in- teractions and van der Waals forces (DRSA, continuous line).