In this Letter, we analyze the problem of monolayer formation onto a flat surface by deposition of hard particles. Our detailed computer simulations of ballistic deposition of hard disks show significant deviations from the classical analytical solution obtained by J. Talbot et al. [Phys. Rev. Lett. 68, 958 (1992)]. These deviations are due to cooperative adsorption induced by particles trapped above the adsorbed layer. We show that not only the adsorption kinetics but also the jamming (saturation) coverage of the surface depends on the volume fraction of the suspension, a prediction which is completely new. These new cooperative effects cannot be neglected even in the case of very diluted suspensions

Jordi Faraudo

M. Elimelech

P. Carl

P. Schaaf

R. Jullien

4 pages, 4 figures.-- PACS nrs.: 68.43.Mn, 82.70.Dd.In this Letter, we analyze the problem of monolayer formation onto a flat surface by deposition of hard particles. Our detailed computer simulations of ballistic deposition of hard disks show significant deviations from the classical analytical solution obtained by J. Talbot et al. [Phys. Rev. Lett. 68, 958 (1992)]. These deviations are due to cooperative adsorption induced by particles trapped above the adsorbed layer. We show that not only the adsorption kinetics but also the jamming (saturation) coverage of the surface depends on the volume fraction of the suspension, a prediction which is completely new. These new cooperative effects cannot be neglected even in the case of very diluted suspensions.The author acknowledges financial support from the Spanish government (MCyT) under GrantNo. BFM2000-0351-C03-01 and from the Generalitat of Catalonia (Direcció general de recerca) under Grant No. 2001-SGR-00186.Peer reviewe

Faraudo, Jordi

Digital.CSIC

VOLUME 89, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 30 DECEMBER 2002New Cooperative Effects in Ballistic Deposition of Hard DisksJordi Faraudo*Department of Chemistry, Imperial College of Science, Technology and Medicine,Exhibition Road, London SW7 2AY, United Kingdom(Received 7 October 2002; published 20 December 2002)276104-1In this Letter, we analyze the problem of monolayer formation onto a flat surface by deposition ofhard particles. Our detailed computer simulations of ballistic deposition of hard disks show significantdeviations from the classical analytical solution obtained by J. Talbot et al. [Phys. Rev. Lett. 68, 958(1992)]. These deviations are due to cooperative adsorption induced by particles trapped above theadsorbed layer. We show that not only the adsorption kinetics but also the jamming (saturation)coverage of the surface depends on the volume fraction of the suspension, a prediction which iscompletely new. These new cooperative effects cannot be neglected even in the case of very dilutedsuspensions.DOI: 10.1103/PhysRevLett.89.276104 PACS numbers: 68.43.Mn, 82.70.Ddto the study of the deposition kinetics as well as thestructure of the adsorbed monolayer both from the ex-the saturation coverage has also been obtained analyti-cally (1  0:808 65). The success of the BD model inThe physics of particle deposition is a field with manychallenging and interesting problems from a theoreticalviewpoint and also with a recognized importance in theunderstanding of many chemical, biological, and indus-trial processes. These processes include filtration, mineralprocessing, the fouling of heat exchanger surfaces, andthrombus formation associated with artificial organs, tomention only a few [1]. In this Letter, we are interested inone of such problems, namely, the irreversible formationof monolayers by deposition of large colloidal particlesfrom fluid suspensions onto solid surfaces. In a typicalexperiment, one considers a diluted suspension of colloi-dal particles, which sediment due to the effect of gravity[2,3]. We are interested in situations in which the densitydifference between the solvent and the particles is large,so that the effect of diffusion in the motion of the par-ticles can be neglected. In many experimental conditions[2–5] (for example, in the case of latexes, erythrocytes,etc.), the deposited particles are strongly bound to theadsorbing surface, and neither desorption nor surfacediffusion are observed. Therefore, the particles in contactwith the surface form a layer of irreversibly adsorbedparticles in a kinetically frustrated state. Other particlescan deposit over this layer by sedimentation, but they arenot bound to the substrate and can easily be removed.Hence, only the particles in the layer in contact with thesurface can be considered adsorbed. The macroscopicstate of the adsorbed layer is typically characterized bythe coverage of the surface  which is the fraction of thearea A occupied by particles. The coverage  increasesmonotonically with time, and the area of the surfaceavailable for adsorption decreases due to the hard coreinteraction between particles (the so-called blocking ef-fect). The adsorption process may continue until no fur-ther particles can adsorb onto the surface and a jammingconfiguration is obtained. Much effort has been devoted0031-9007=02=89(27)=276104(4)$20.00perimental and theoretical points of view (see Refs. [4,5]for a review).In order to be more specific, let us consider the classicalproblem of deposition of spherical particles onto a planarsurface. From the theoretical point of view, it is a stan-dard practice to analyze this deposition process by usingthe so-called correlated sequential adsorption model, alsocalled the ballistic deposition (BD) model [6]. In thismodel, the actual deposition process is replaced by somesimple geometrical rules for the filling of the adsorbingsurface. These rules are as follows: (i) the particles aredeposited sequentially on the surface, (ii) for each depo-sition trial, a starting position is chosen randomly overthe adsorption plane. The particle then follows a verticaltrajectory until it reaches the adsorption plane or contactsa previously deposited particle. In this latter case, itfollows the path of steepest descent until it reaches astable position. (iii) If this position lies on the adsorbingsurface, the particle is irreversibly adsorbed. Otherwise,the particle is removed from the system and a new trial isstarted. At this point, it is important to recall that multi-layer formation is not allowed in the BD model: alltrapped particles are simply rejected (eliminated fromthe system) during the adsorption process. This rule istypically justified by assuming that the formation of theadsorbed layer is much more rapid than the growth oflayers of trapped particles. Computer simulations showthat the saturation coverage (jamming limit coverage) isgiven by 1  0:6105 [4]. The kinetics of the depositionprocess and the structure of the adsorbed monolayer havebeen analyzed extensively by simulations [4]. In addition,it is possible to derive approximate analytical expressionsfor the process kinetics valid at the limiting situations oflow coverages or high coverages [7]. Also, the simplifiedsituation of ballistic deposition of hard disks onto a linehas been solved exactly by Talbot et al. [8]. In this case, 2002 The American Physical Society 276104-1VOLUME 89, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 30 DECEMBER 2002explaining experimental data [2,4] has motivated theintroduction of many improvements in the model in orderto account for fine details neglected in the original BDmodel [9,10]. In the case of BD of hard disks, the effect ofhydrodynamic interactions [9] and external fields [10] hasbeen analyzed.Despite its success, the basic rules of the BD model arequestionable. In particular, the rejection rule in the BDmodel is not realistic: in an actual experiment, trappedparticles are not withdrawn from the system. In fact,these trapped particles can help new incoming particlesto be adsorbed. This produces a cooperative effect whichis neglected in the classical BD model. In order to illus-trate the limitations of the classical BD model, let usconsider the situation illustrated in Fig. 1. In order tosimplify the geometry, we consider the adsorption ofhard disks onto a line. In Fig. 1 (left), we consider aconfiguration of two adsorbed disks and two free disks(labeled as 1 and 2) which sediment with constant veloc-ity. After a given time, these incoming particles collidewith the adsorbed ones. In the classical BD model, both 1and 2 are rejected from the simulation and the coverage ofthe line is unchanged. However, one expects that the finalconfiguration of particles is that illustrated in Fig. 1(right): particle 1 is trapped in a stable mechanical posi-tion and particle 2 is adsorbed after rolling over particle 1.This kind of cooperative effects and other more complexeffects (involving more particles) are unavoidable in ex-perimental situations and are ignored by the BD model inits present formulation. Usually, it is assumed that if thebulk concentration of particles is small (as it is in typicalexperiments), this kind of effects can be neglected andthe usual BD is valid. However, it is not clear how thesecooperative mechanisms can be suppressed by decreasingthe volume fraction. In fact, there is no theoretical studyin the literature supporting this strong hypothesis. Ourobjective in this work is to elucidate under which con-ditions this hypothesis can be accepted and which kind ofdeviations from the BD model appear under reasonableconditions. This objective will be accomplished by com-paring computer simulations with the known results ofthe BD model.FIG. 1. Example of cooperative effects in adsorption. Left:initial configuration with two adsorbed particles and twoincoming particles (labeled as 1 and 2). Right: disk 1 isdeposited over the previously adsorbed particles and disk 2 isadsorbed at z  0 after rolling over disk 1.276104-2Detailed computer simulations of the deposition pro-cess without the simplifying hypothesis of BD are pro-hibitive from the computational point of view. In order toperform detailed simulations of deposition using a rea-sonable amount of computing time, we will consider thedeposition of hard disks instead of hard spheres. Atthis point, we recall that this is a typical procedure inadsorption-deposition studies: in the two dimensionalsystem it is easier to perform simulations and to obtainanalytical results [4,8–10]. The results obtained in thetwo dimensional system are very useful to guide theanalysis of the much more complicated three dimensionalsystem. Of course, this procedure is reasonable if no newadsorption mechanisms appear in the three dimensionalproblem, being the 3D and 2D processes essentiallydriven by the same mechanisms. We will comment onthe validity of our results in the 3D case later on.Now, let us describe in detail our model and the simu-lation algorithm. Let Lx be the length of the adsorbingsurface, Lz the maximum height of the adsorption cell,and NB the initial number of disks in the suspension. Theinitial concentration and volume fraction are given, re-spectively, by cB  NB=LzLx and B  R2cB, whereR is the radius of the disks. During the whole adsorptionprocess, no particles are added or subtracted from thesystem (closed cell conditions). All particles move withthe same constant vertical sedimentation velocity vS andinteract between them with hard core interactions. Hence,in the suspension there is a vertical flux of particlestowards z  0 given by J0  vScB. At z  0 there is anirreversibly adsorbing surface: when the center of a par-ticle reaches z  0, it is immobilized. As time passes, oneobserves the formation of an adsorbed layer at z  0.Also, particles trapped into stable mechanical positionsover other deposited particles can be observed. In adsorp-tion processes, it is customary to introduce a dimension-less time defined as   2RJ0t  2RcBvSt. The kineticsof the layer appearing at z  0 is typically characterizedby the coverage defined by  Na 2RLx ; (1)where Na is the number of particles adsorbed at z  0at time . In addition, there are Nm particles trappedinto stable mechanical positions over previously depos-ited particles.The simulation algorithm is as follows. First, an initialconfiguration of nonoverlapping hard disks is generated,with volume fraction B. We use dimensionless units, inwhich distances are measured in units of the particlediameter (x0  x2R ; z0  z2R ) and time is measured in unitsof . At each time step , each free particle is displaceda distance z0 given byz0   vS2Rt   4B: (2)276104-2VOLUME 89, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 30 DECEMBER 2002After each move of a particle, a test is performed in orderto prevent unphysical overlaps with the other particles ofthe system. If an overlap is found, the currently movingparticle is displaced along the center-to-center line untilthe overlap disappears. Also, a test is performed in orderto check if the particle is still free, or is adsorbed ortrapped into a position of stable mechanical equilibrium.We have performed simulations for different initial bulkvolume fractions in the range 104–101. In all simula-tions, NB  300, the size of the adsorbing line wasLx=2R  100 and z0  0:01. The desired initial volumefraction is obtained by adjusting the vertical dimensionLz=2R. We have stopped the simulations at   3:0. Atthis time, in most simulations the adsorbing surface hasreached the saturation or jammed state (no more particlescan be adsorbed at z  0), so the saturation coverage canbe computed. In the remaining simulations, some smallgaps with room for only one particle remain. Hence, thesaturation coverage can be predicted easily by adding toNa one particle per remaining gap. The time evolution ofthe coverage, averaged over 10 000 simulations, is shownin Fig. 2 for B  7:854 104 and B  0:1571. Incontrast to the BD model, the observed adsorption ki-netics does depend on the bulk concentration. The devia-tions appear at adsorption times  	 0:5. At intermediatevalues of , the coverage obtained in simulations is largerFIG. 2. Adsorption kinetics from simulations with B 7:854 104 (circles) and B  0:1571 (diamonds). The solidline corresponds to the classical BD model and the dashed lineto   . Inset: Adsorption kinetics near the jamming state.276104-3than the coverage predicted by the BD model. This resultagrees with the initial expectations based on the mecha-nism shown in Fig. 1, which predicts an enhancement ofadsorption. It is important to note that even for the small-est value analyzed of the volume fraction, deviationsfrom the BD model are still present. The obtained valuesof the saturation coverage also differ from the BD pre-dictions. We have obtained 1  0:8121 0:0002 forB  7:854 104 and 1  0:7904 0:0002 for B 0:1571. In order to investigate the evolution of the satu-ration coverage with the volume fraction, we have per-formed additional simulations with different values ofB. In most of the cases we have performed 103 simu-lations (instead of 104) for each volume fraction in orderto save computer time. The results are shown in Fig. 3 (theerror bars represent 95% confidence intervals). The moststriking feature of these results is that 1 is approxi-mately constant up to B 	 102 with a saturation cover-age slightly larger than the BD predictions. For largervolume fractions, the saturation coverage decreases as thebulk volume fraction increases. All these differencesbetween the BD model and the simulations can be attrib-uted only to the interaction of new incoming particleswith particles deposited at layers above z  0. Hence, itseems convenient to monitor the growth of multilayersduring the simulations. Nearly all the particles depositedin layers over z  0 can be found on the second and thirdFIG. 3. Saturation coverage observed in simulations as afunction of the initial volume fraction. The dashed line corre-sponds to the BD model saturation coverage.276104-3θ/θ0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0% of deposited particles in each layer0102030405060708090100layer 1layer 2layer 3∞FIG. 4. Fraction of particles deposited in layers 1, 2, and 3 asa function of the relative coverage of the surface. The diamondscorrespond to simulations with B  7:854 104 and thelines correspond to simulations with B  0:1571.VOLUME 89, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 30 DECEMBER 2002layers (particles over the third layer were rarely observedexcept at stages very close to saturation).We show in Fig. 4the fraction of all the deposited particles that can befound in the first layer (z  0), the second layer (particlestrapped onto particles adsorbed at z  0), and third layer(particles trapped onto particles at the second layer).From this figure it is clear that the number of particlesat secondary layers is small only at the initial stage of theadsorption process. The presence of particles deposited insecondary layers is clear at intermediate coverages, andas the first layer approaches its saturation coverage, thenumber of particles depositing in multilayers increasesdramatically. It is important to note that the fraction ofparticles deposited in secondary layers does not stronglydiffer between the cases B  7:854 104 and B 0:1571. This is consistent with the fact that the initialdilution of the suspension does not prevent the mecha-nism of secondary layer formation. Hence, the coopera-tive effects induced by these trapped particles cannot beneglected even for very diluted suspensions.In conclusion, we have shown for the first time that theformation of secondary layers cannot be neglected inballistic deposition/adsorption studies. We have foundthat both the kinetics of adsorption and the saturationcoverage depend on the volume fraction even for verydiluted suspensions. This dependence, absent in the clas-sical BD model, is a consequence of the interactionbetween incoming bulk particles and particles depositedat layers above the adsorbed layer (z  0). This result hasimportant implications in adsorption/deposition studies,where it is usually assumed that the value of the volumefraction is not an important factor determining adsorp-tion kinetics and jamming coverage provided that it issmall. At this point, the obvious question is if this kind ofeffects, predicted for the 2D system, are observable in276104-4actual 3D systems. In a recent paper by Carl et al. [2],experimental results for latexes and erythrocytes arecompared with BD predictions and Brownian dynamicssimulations. In these simulations, the trapped particlesare not removed so multilayer formation is possible. InFig. 4 of this reference (corresponding to the heavierparticles analyzed) a deviation from BD is observedboth from experiments and simulations. Qualitatively,the observed behavior is very close to our Fig. 2. Hence,it is very reasonable to infer that the cooperative effectsdescribed in the present paper are responsible for thedifferences observed between experiments (or simula-tions) and the BD model. Of course, more detailed studiesin 3D models are needed to confirm this hypothesis. Butfrom the present work it seems clear that in any detaileddeposition model, the effect of multilayer formation can-not be neglected as is done in classical models. Anothervery interesting prediction arising for the present work isthe dependence of the saturation coverage on the volumefraction. We encourage experimentalists to undertake thenecessary experiments to check if this effect is indeedobserved in real systems.The author acknowledges financial support from theSpanish government (MCyT) under Grant No. BFM2000-0351-C03-01 and from the Generalitat of Catalonia(Direccio´ general de recerca) under Grant No. 2001-SGR-00186. Useful comments by B. Senger, R. Dominguez-Cascante, and J. Bafaluy are also acknowledged.*On leave from Departament de Fı´sica, UniversitatAuto`noma de Barcelona, E-08193 Bellaterra,Barcelona, Spain.Electronic adress: jordi.faraudo@uab.es[1] M. Elimelech, J. Gregory, X. Jia, and R. Williams,Particle Deposition and Aggregation (Butterworth-Heinemann Ltd., Oxford, 1995), 1st ed.[2] P. Carl, M. Schmittbuhl, P. Schaaf, J.-F. Stoltz, J.-C.Voegel, and B. Senger, Physica (Amsterdam) 298A, 198(2001).[3] P. Wojtaszczyk, E. K. Mann, B. Senger, J. C. Voegel, andP. Schaaf, J. Chem. Phys. 103, 8285 (1995).[4] P. Schaaf et al., Ann. Phys. (Paris) 23, 1 (1998); B. Sengeret al., Colloids Surf. A 165, 255 (2000).[5] Z. Adamczyk et al., Adv. Colloid Interface Sci. 48, 151(1994).[6] A. P. Thompson and E. D. Glandt, Phys. Rev. A 46, 4639(1992); R. Jullien and P. Meakin, J. Phys. A 25, L189(1992).[7] H. S. Choi, J. Talbot, G. Tarjus, and P. Viot, J. Chem. Phys.99, 9296 (1993).[8] J. Talbot and S. M. Ricci, Phys. Rev. Lett. 68, 958 (1992).[9] I. Pagonabarraga and M. Rubı´, Phys. Rev. Lett. 73, 114(1994).[10] I. Pagonabarraga, J. Bafaluy, and M. Rubı´, Phys. Rev.Lett. 75, 461 (1995).276104-4

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