VOLUME 86, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 FEBRUARY 2001

Colloidal Aggregation with Sedimentation: Computer Simulations

Agustín E. GonzálezCentro de Ciencias Físicas, Universidad Nacional Autónoma de México, A. Postal 48-3, Cuernavaca, Morelos 62251, México

(Received 24 July 2000)

A computer model for colloidal aggregation is presented that considers both the Brownian motion andthe gravitational drift experienced by the colloidal particles and clusters. It is shown that the aggregationcrosses over from diffusion-limited aggregation to another type with a higher cluster fractal dimension,a speeding up followed by a slowing down of the aggregation rate, an algebraically decaying cluster sizedistribution, and a higher concentration required for gelation. Although these findings are in accordancewith the experimental results, some interpretations are different.

DOI: 10.1103/PhysRevLett.86.1243 PACS numbers: 61.43.Hv, 05.10.Ln, 82.70.Dd, 82.70.Gg

The sedimentation experienced by big clusters is therule rather than the exception in colloidal aggregation. Toeliminate the action of the gravitational field, experimen-talists need to resort to a number of tricks, such as flippingthe sample very often during the aggregation, consideringa suspension fluid whose density closely matches that ofthe colloidal particles, etc. It looks therefore necessaryto advance our knowledge of colloidal aggregation in thepresence of gravity, in order to have a thorough under-standing of all the processes occurring in real aggregatingsystems. Although there have been a considerable numberof studies of colloidal aggregation in the absence of sedi-mentation [1], very few works [2–5] have been devoted todescribe the process, taking into account the coupling andinterplay that occurs between aggregation and sedimenta-tion and considering, furthermore, the subsequent deposi-tion with the consequent mass depletion in the bulk of thesample. In the most recent studies along that line, Allainet al. [4,5] performed experiments in which they found anincrease in the fractal dimension of big settling clusters,attributing this increase to a restructuring mechanism dueto the hydrodynamic stresses felt by the different parts ofthe cluster. They also found a much higher concentrationrequired to have gelation, by several orders of magnitude.

In this Letter, the results of a Monte Carlo aggregationmodel are presented, that considers both the Brownianmotion as well as the sedimentation experienced by theparticles and clusters. The deposition is considered in anaverage way, as will be explained below, due to compu-tational restrictions. Among the results, for intermediatestrengths of the downward drift, a higher fractal dimensionis also obtained at about the Allain et al. value. Simi-larly, we also get a much higher initial concentrationrequired to have gelation. The model is, however, capableof proportioning additional quantities not obtained byAllain et al., such as the number of clusters as a functionof time, Nc�t�, the weight-average cluster size as a func-tion of time, Sw�t�, and the cluster size distribution fordifferent times, Ns�t�, all of them calculated for clusters inthe bulk. With their help, it is shown that the aggregationspeeds up and eventually slows down, as the mass inthe bulk becomes scarce. The cluster size distribution

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obtained becomes algebraically decaying, reflecting thebroad range in cluster sizes. For much higher strengths ofthe drift the fractal dimension diminishes again, revealingthe anisotropy of the clusters, becoming more elongatedalong the vertical direction. Given that this is a rigidaggregation model, we should point out that it is notnecessary to invoke the restructuring of the clusters inorder to achieve the higher values of the fractal dimension,but only the sweeping of the small clusters and particlesby the big drifting clusters, getting into the holes of thebig ones and making them more compact.

The sedimentation velocity ys experienced by a clusterof N spherical particles of radius a and mass m0 is

ys �m0�1 2 r�r0�gN

f�

m0�1 2 r�r0�gkBT

DN , (1)

where r0 is the density of the particles, r is that of thesuspension fluid, f is the friction coefficient of the clus-ter, D��1�Rg� is its diffusion coefficient, Rg is its ra-dius of gyration, and T is the temperature. Let t0 bethe time for which the cluster diffuses a particle diameter�2a�, that is, t0 � 2a2�D. During the same time, thecluster drifts a distance ds � yst0 � PeNd, where Pe �m0�1 2 r�r0�ga�kBT is the Peclet number of the col-loidal particles in that fluid and d is the diameter of theparticles. It is found that Pe is of the order of unity if theparticles are 1 mm in diameter, 1 2 r�r0 is less than butof the order of unity, and T is room temperature. However,if the diameter is 0.1 mm such quantity is of the order of1024, while if the diameter is 10 mm, Pe goes as high as104. Therefore, 1 mm marks the transition between diffu-sive and drifting behavior for individual particles.

A previous effort to elucidate the experimental results[4,5] was made by González and Leyvraz [6]. In thatmodel, the particles and clusters were allowed to dif-fuse— and to settle according to Eq. (1)—on a three-dimensional lattice with periodic boundary conditionson the three spatial directions. A hint of the crossoverto another regime with an increasing fractal dimensionwas obtained; however, it was not possible to obtain themuch higher initial particle concentration required to havegelation, because there was no deposition in the model.

2001 The American Physical Society 1243

VOLUME 86, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 FEBRUARY 2001

Another consequence of this lack of deposition was thatthe dynamical quantities Nc�t�, Sw�t�, and Ns�t� for clus-ters in the bulk were incorrectly calculated. At this pointwe should recognize that, in order to study sedimentationfor not very high Peclet numbers, as Allain et al. did[4,5], large box sizes have to be used that would allowthe growth of clusters big enough, capable enough tostart drifting downwards. If, in addition, the height of thesample has to be about 20 times its lateral dimensions [5],in order to follow the drift of the big clusters, only a hugelattice would permit us to study the sedimentation withdeposition. This exceeds by far our available computermemory. Here, a cubic lattice of size L was consideredinstead, with periodic boundary conditions on the threespatial directions, which represents in an average way anycubic slice of the prism of height H (� 20L in Ref. [5]).Whenever a cluster moves on the z direction of a latticespacing d, a quantity Zs is updated that proportions thetotal distance in units of d which the cluster has moveddownwards (taking into account the movement of all itsancestors; i.e., the largest of the two quantities Zs of the“parent” clusters is inherited by the “child” cluster after amerging). If the movement is downwards, the cluster istaken out of the cubic box (which means that it has beendeposited on the bottom) with probability 1��H 2 Zs�.Thus, the bigger clusters with the larger Zs are taken outof the sample with higher probability.

Let tmin be the time taken by the most mobile clusterof the sample (with the smallest Rg or largest D) to dif-fuse one lattice spacing. At each Monte Carlo step Dt�#tmin�, in which a cluster is picked in a cyclic way, thecluster is moved downwards a distance ysDt (smaller thanor equal to a lattice spacing; if this sedimentation step doesnot lead to a downward movement by a lattice spacing, theamount moved is stored to be added at the next sedimen-tation step of the same cluster). The cluster is then movedone lattice spacing d on a random direction with proba-bility �Dt�tmin� �D�Dmax�. After each of those moves, ifit occurred downwards by a lattice spacing, the cluster istaken out of the sample with probability 1��H 2 Zs�, asmentioned. If the cluster is moved but not taken out, wecheck for overlapping with other clusters, in which casethe moving cluster is taken back to its original positionand the overlapping clusters are merged.

The simulations were stopped just before gelation forthe gelling systems; for the nongelling simulations, theywere stopped after all the clusters had been deposited.One volume fraction f � 0.01 was considered, for whichfive values of Pe were chosen: Pe � 1028 [diffusion-limited colloidal aggregation (DLCA) simulation], 1023,1022, 1021, and 100. For the first three Peclet numbers,the system was gelling, while for the last Peclet num-ber it was nongelling; for Pe � 0.1, some of the simu-lations percolated the box while the others did not. Formost of the computations, ten simulations of size L � 270and H � 10L were done for each Peclet number, except

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when trying to calculate the critical concentration to havegelation, in which case L and f were varied also.

Figure 1a, for Pe � 1028, shows a log-log plot of theaverage radius of gyration Rg versus the number of par-ticles N of all the clusters formed in the ten simulations,while in Figs. 1b and 1c we show the corresponding plotsfor Pe � 1022 and Pe � 100, respectively. The Rg was

0.1

1

10

100

1 10 100 1000 10000

Rg

N

a

a

(a)

D = 1.882fa

P = 10e-8

1

10

100

1 10 100 1000 10000

Rg

N

a

a

b

b

(b)

D = 1.868fa

D = 2.270fb

P = 0.01e

0.1

1

10

100

1 10 100 1000 10000

Rg

N

b

b

(c)

P = 1.0e

D = 2.131fb

FIG. 1. A log-log plot of the average radius of gyrationas a function of the number of particles of the clusters for(a) Pe � 1028, (b) Pe � 1022, and (c) Pe � 100.

VOLUME 86, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 FEBRUARY 2001

averaged over all N lying on segments of constant sizeon the logarithmic scale. In Ref. [6] we failed to do theaverage and were unable to locate the breaking point thatwe are about to see. For Pe � 1028, after a curvaturedescribing the corrections to scaling for small N , a veryneat straight line is clearly seen between the arrows la-beled “a,” whose inverse slope proportions the fractal di-mension. The value obtained for Dfa �� 1.882� agreesquite well with the standard DLCA value on cubic lattices[7] at the corresponding concentration �Df � 1.887�. Be-yond the second arrow, the fractal dimension starts increas-ing again, probably trying to reach the value for percolatingclusters. In Fig. 1b, for Pe � 0.01, we now clearly see abreaking point at about N � 200. On the left and rightof the breaking point, there are well-defined straight linesbetween the arrows labeled “a” and “b” that give the twofractal dimensions: the small clusters Dfa � 1.868 andthe big clusters Dfb � 2.270. The first value also agreeswell with the standard DLCA value, while the second isclose to the Allain et al. [4,5] value (2.2–2.3). For thePe � 0.001 and Pe � 0.1 cases (not shown), the breakingpoints occur at about N � 1000 and 50, respectively, withthe small clusters’ fractal dimension being Dfa � 1.872and 1.887, while the big clusters fractal dimension takesthe values Dfb � 2.326 and 2.224. Finally, in Fig. 1cfor Pe � 1.0, the breaking point disappears and the curvecrosses over directly from the corrections to the scalingpart to the big cluster Dfb . Between the arrows labeled “b”we now see a straight line that defines the fractal dimen-sion Dfb � 2.131. The decrease in the big clusters’ fractaldimension as we increase the Peclet number to very highvalues is probably related to the anisotropy of the clusters,becoming more elongated along the z direction. A cal-culation was made of the following quantities: �Rgx�Rg�,�Rgy�Rg�, and �Rgz�Rg�, where the average runs over allthe clusters lying between the arrows “b” that define thebig clusters, except for Pe � 1028 for which it was madeover all the clusters lying between the arrows labeled “a” inFig. 1a. The results for the different Peclet numbers were:Pe � 1028: 0.555, 0.553, and 0.553; Pe � 0.001: 0.570,0.571, and 0.546; Pe � 0.01: 0.552, 0.555, and 0.578;Pe � 0.1: 0.513, 0.505, and 0.657; Pe � 1.0: 0.387,0.386, and 0.815. It should be noted that Allain et al. neverconsidered such high values for the Peclet number, work-ing generally at about Pe � 1025 [5].

To obtain the critical concentration to have gelation, f�,the following procedure was used: For a given Peclet num-ber and a given L, about six values of f were consideredat about (above and below) f�. For each of those f val-ues, 20 simulations were run and a record was made ofthe number of simulations that percolated. The probabilityto percolate for a given f was defined as the number ofsimulations that percolated divided by 20. Next we fittedthe function P � �tanha�f 2 b� 1 tanh�ab���1 1

tanh�ab�� to the six points defining the probability topercolate as a function of f. The critical concentration f�

FIG. 2. The critical concentration to have gelation f� as afunction of 1�L, for the Peclet numbers Pe � 1028, 1023, 1022,1021, and 100.

was defined as that value of f for which the function Phad the value of 1�2. Figure 2 shows the plots of f� as afunction of 1�L for the different Peclet numbers studied.The following values of L were considered: 120, 150, 180,210, 240, and 270. For each Peclet number, the points lieon a straight line and it is possible to get an extrapolationto the macroscopic system. The first thing to note fromthe graph is that, for the DLCA simulations, the f� for themacroscopic system is essentially zero (the small negativevalue obtained in the extrapolation comes from the statisti-cal uncertainties). This should be the case for a gellingsystem made of fractal aggregating entities (clusters).However, for the other Peclet numbers the extrapolationdoes not go to zero, the depletion of mass in the bulkduring the aggregation being responsible for this phe-nomenon. Although it was not possible to consider thePeclet number used by Allain et al., because we wouldneed a very large box in order to start seeing the sedi-mentation of very large clusters, it is estimated from thegraph that the critical f� for Pe � 1025 is of the order ofmagnitude of 1023. This is the same order of magnitudeof the value obtained by Allain et al. [4].

Figures 3a and 3b show log-log plots for Nc and Sw ,respectively, both as a function of time. The symbolscorrespond to the average over the ten simulations madefor each Peclet number, except for Pe � 0.1 for which thecrosses correspond to the average over the six gelling simu-lations, while the triangles are for the four nongelling simu-lations. However, in the range of time for which both thegelling and the nongelling simulations were taking place,the curves coincide for both Nc�t� and Sw�t�. For Pe �1028 we can see the establishment of a straight line forboth Nc�t� and Sw�t�, defining the exponents z0 and z [7]in DLCA. As the Peclet number increases the aggregationspeeds up, in the sense that for a given time there are lessnumbers of clusters or the average cluster size is higher.However, for those simulations that did not gel with Pe �0.1 and for Pe � 1.0, the speeding up of the aggregation

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1

10

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1000

10000

100000

1000000

10 100 1000 10000

Nc

t

(a)

1

10

100

1000

10000

10 100 1000 10000

Sw

t

(b)

FIG. 3. (a) The average number of clusters Nc as a functionof time for the Peclet numbers Pe � 1028 (diamonds), Pe �1023 (pluses), Pe � 1022 (squares), Pe � 1021 (crosses forpercolating and triangles for nonpercolating systems), and Pe �100 (asterisks). (b) The weight-average cluster size Sw as afunction of time for the same Peclet numbers as in (a), identifiedwith the same symbols.

is followed by a slowing down, coming from the depletionof mass in the bulk. Another consequence of this lack ofmass is the fact that the average cluster size first increasesand then decreases to zero. As the Peclet number becomesmuch higher, the maximum average cluster size becomessmaller, indicating that medium size clusters are rapidlysettling to the bottom.

A calculation was done for each Peclet number of thescaled cluster size distribution f � S2

w�t�Ns�t��N0�t�,

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where N0�t� is the number of particles in the bulk at timet and was plotted as a function of x � s�Sw�t�. For theDLCA case �Pe � 1028�, the data collapse was good andthe master curve is bell-shaped [8] with some asymmetry,as it should be for a not very dilute system [7]. Forthe Pe � 0.001, 0.01, and 0.1 cases, the collapse of thedata was not good, probably due to the fact that we haveNs�t� data coming from two different processes. One isthe usual DLCA process of diffusion and aggregation,for the early times and small clusters, and the other isthe process of aggregation due to the sweeping of smallclusters and particles by the big drifting clusters, whichoccurs at later times. However, for the Pe � 1.0 case,the data collapse is established again, but now with analgebraically decaying f as a function of x. This showsthat for Pe � 1.0 the DLCA process no longer exists,everything comes from the sweeping mechanism process.The algebraic decay indicates the broad width of thecluster size distribution, coming from the coexistence ofvery big settling clusters, which are becoming bigger andbigger at a high pace, together with the still existent smallclusters and particles. All this and more will be describedin a forthcoming, longer publication.

This study was partially supported by CONACYT(México) Grants No. 3165-PE and No. 34343-E. I thankH. Larralde for a helpful conversation.

[1] For recent reviews, see R. Hidalgo-Alvarez, A. Martin,A. Fernández, D. Bastos, F. Martínez, and F. J. de lasNieves, Adv. Colloid Interface Sci. 67, 1 (1996); P. Meakin,J. Sol-Gel Sci. Technol. 15, 97 (1999).

[2] A. S. Michaels and J. C. Bolger, IEC Fundam. 1, 24 (1962).[3] M. Couch, Ph.D. thesis, Cambridge University, Cambridge,

UK, 1993.[4] C. Allain, M. Cloitre, and M. Wafra, Phys. Rev. Lett. 74,

1478 (1995).[5] C. Allain, M. Cloitre, and F. Parisse, J. Colloid Interface

Sci. 178, 411 (1996).[6] A. E. González and F. Leyvraz, in Proceedings of the Sta-

tistical Mechanics in Physics and Biology Symposium ofthe MRS 1996 Fall Meeting, edited by D. Wirtz and T. C.Halsey (Materials Research Society, Pittsburgh, 1997).

[7] M. Lach-hab, A. E. González, and E. Blaisten-Barojas,Phys. Rev. E 54, 5456 (1996).

[8] M. L. Broide and R. J. Cohen, Phys. Rev. Lett. 64, 2026(1990).