Faruday Discuss. Chern. SOC., 1987,83, 125-137 Computer Simulations of Cluster-Cluster Aggregation R. Jullien,* R. Botet and P. M. Mors? Physique des Solides, Universite' Paris- Sud, Centre d 'Orsay, 91405 Orsay, France We briefly introduce a simple extension of the diff usion-limited aggregation model of Witten and Sander, the cluster-cluster aggregation model, in which all clusters diffuse and grow by sticking at their first contact. This model is particularly well adapted to describe aggregation experiments on colloids or aerosols. Three recent extensions of the model are emphasized which take into account possible restructuring, cluster polarizability or indirect aggregation by polymers. In these three cases the computer simulations are compared with experiment. Since the discovery by Forrest and Witten' of the fractal structure2 of smoke aggregates, several theoretical models3 have been built to simulate aggregation phenomena.The first was introduced by Witten and Sander.4 This model considers a particle-cluster mechanism in which individual diffusing particles stick one after another onto a single growing aggregate. This model has been recognized to be quite well appropriate to describe 'field-induced' aggregation experiments, such as filtration5 or electrodeposition,6 as well as many other experiments which are not truly aggregation, such as dielectric breakdown7 or viscous fingering.' It appears, however, that the Witten-Sander model leads to more compact structures than those observed in standard aggregation experiments such as the experiments realized with colloids or aerosols.To simulate these experiments, an alternative model, the cluster-cluster aggregation model, where the growing mechanism is governed by col- lisions between diffusing clusters, was introduced independently in the U.S.A.9 and in France." In this paper we briefly recall the definition cluster-cluster aggregation and review the different versions of the model. We then describe the intrinsic anisotropy properties of aggregates grown by such mechanism and describe more precisely recent extensions which have been motivated by experiments. These extensions consider possible readjust- ing effects, electrical polarizability effects and flocculation mechanism induced by the presence of polymers. The Cluster-Cluster Aggregation Model and Its Different Versions In its original v e r s i ~ n ~ , ' ~ the cluster-cluster model starts with a collection of equal-sized individual spherical particles randomly disposed in a box.Then these particles are allowed to undergo a diffusing motion simulated by a pure random walk (with consider- ing periodic boundary conditions at the edges of the box). When two particles come into contact, they irreversibly stick together to form a rigid dimer which is also able to diffuse in the box. This dimer can stick to other dimers or single particles etc. After each collision the two colliding clusters form a new, rigid, larger cluster. The mechanism can be pursued until a unique large aggregate remains alone in the box. t Permanent address: Instituto de Fisica, Universidade Federal do Rio Grande do Sul, 90049 Port0 Alegre, RS Brazil.125126 Computer Simulations of Cluster- Cluster Aggregation In these simulations a parameter a is naturally introduced, which characterizes how the velocity (or preferably the diffusivity) of a cluster varies as a function of its number of particles, i : vi - i". For realistic cases of sufficiently negative a values, i.e. when small clusters move faster than large clusters, and in the limiting case of very low initial particle concentration, it has been shown that the aggregates exhibit a fractal structure and their fractal dimension is independent on a in a large range of a values. This fractal dimension depends only on the dimension of space and, for pure brownian motion, it has been estimated to be D == 1.44 in d = 2 and D ==: 1.78 in d = 3 , in good agreement with the experimental values for colloids" and aerosols.' In contrast to the geometrical characteristics, all the kinetics strongly depend on a.The time dependence of the mean cluster size, as well as the entire shape of the cluster-size distribution,'* can be satisfactorily described by an old kinetic equation due to Smolu~howski,'~ at least for space dimensions larger than d = 2, where concentration fluctuations can be neg1e~ted.l~ An idealized version of the cluster-cluster model, the hierarchical model, has been i n t r ~ d u c e d ' ~ in which only clusters of rigorously the same number of particles are allowed to stick together. This version, which gives the same quantitative results but is less consuming of computer time, has permitted many systematic extensions.In par- ticular, the model has been extended to high space dimensions.16 Extensions to larger a values show that, beyond a critical value, the aggregation mechanism is dominated by particle-cluster collisions: one large cluster finally dominates by absorbing all other small This abrupt change from cluster-cluster to particle-cluster aggregation, when increasing a, can be related to the change of analytical properties in the solutions of the Smoluchowski equation." Extensions to larger con- centrations, which lead to more compact aggregates, have been introduced to simulate the sol-gel transition." Other extensions introduce a modification of the cluster trajectory.Instead of considering a brownian trajectory (the fractal dimension of which being d , = 2 ) , one can consider straight-line trajectories (d, = I).*' This ballistic model is a goad simulation of the molecular regime of aerosols. Such a modification only slightly increases the fractal dimension of the resulting aggregates. To take into account long-range attractive forces one can also consider straight-line trajectories without impact parameter: now the clusters move along a line passing through their centres of m a s 2 ' The fractal dimension is again slightly increased. Another extension of the cluster-cluster model completely eliminates the role of the trajectory: this is the chemically limited cluster-cluster aggregation This model can be built by introducing a sticking probability, and by letting this probability tending to zero.In this limit the clusters have all the time to investigate all the sticking possibilities and they finally choose one at random. This model can be considered as the cluster- cluster counterpart of the Eden particle-cluster growth model, in which particles are added at random on the surface of a growing cluster.23 The interest of such model is that it is experimentally realized in colloids when the electrostatic repulsion is not completely screened. The fractal dimension is then D==2 in d = 3 , slightly, but sig- nificantly, larger than the value D =r 1.78 obtained in the pure diffuse case (with a sticking probability equal to one). Such change of the fractal properties of aggregates has been effectively observed by Weitz et al." in their experiments on gold colloids.To summarize the influence of the nature of the trajectory on the resulting shape of the aggregates, we show in fig. 1 typical three-dimensional clusters grown using the hierarchical model. Cases ( a ) , ( c ) and ( d ) correspond to Brownian trajectories, linear trajectories with impact parameter and linear trajectories without impact parameter,R. Jullien, R. Botet and P. M. Mors 127 Fig. 1. Typical aggregates showing the influence of the nature of cluster trajectory on the shape of the resulting clusters obtained by cluster-cluster aggregation process in three dimensions: ( a ) Brownian, ( b ) chemical, ( c ) linear with impact parameter and ( d ) linear without impact parameter.To give an idea of their relative sizes, all the aggregates have the same number (4096) of equal-sized particles. Table 1. Fractal dimension of the cluster-cluster model as a function of the space dimension in the case of the Brownian, ballistic (random straight-line trajectories) and chemical versions of the model as well as in the case of linear trajectories without impact parameter" d 2 3 4 5 6 ( d , = 2) Brownian 1.44 1.78 2.05 2.27 2.6 ( d , = 1) ballistic 1.51 1.91 2.22 2.47 2.7 chemical 1.55 2.04 2.32 linear without impact parameter 1.56 2.06 2.53 2.97 3.46 a The absolute error is of the order 0.05. respectively. Case ( b ) , which corresponds to chemically limited cluster-cluster aggrega- tion, is shown for comparison. The corresponding fractal dimensions, which are numeri- cally obtained by averaging over many aggregates and by extrapolating to infinite sizes, are given in table 1.The Intrinsic Anisotropy of Cluster-Cluster Aggregates There is a great difference in shape between particle-cluster and cluster-cluster aggre- gates, and this difference is not only due to the difference in their fractal dimensions. This can be seen directly for the typical two-dimensional clusters shown in fig. 2. While particle-cluster aggregates exhibit a roughly spherical shape with a well defined centre around which the growth mechanism takes place, cluster-cluster aggregates exhibit instead an oblong, ellipsoidal, shape, in general. Since the conception of the cluster- cluster model, it has been recognized that clusters grown by this process are a n i s ~ t r o p i c .~ ~ However, it is only recently that a systematic quantitative study of these anisotropy properties has been made.25128 Computer Simulations of Cluster- Cluster Aggregation Fig. 2. A typical Witten-Sander aggregate (left) compared with a typical cluster-cluster aggregate (right) containing the same number of particles (4096), in two dimensions. Table2. Anisotropy ratio Al between the largest and the smallest eigenvalues of the radius of gyration tensor, in dimension 2, estimated from finite-size simulations in the case of three versions of the cluster-cluster model Brownian linear chemical A1 5.7 f 0.2 5.2 f 0.2 4.7 * 0.2 D 1.44 f 0.05 1.51 k0.05 1.55 f 0.05 The anisotropy can be quantitatively investigated by diagonalizing the tensor of the radii of gyration: ( ~ u b ) ’ = C ( r i u - rju>(rih - rjb)/(2N) i,j where a and b refer to the coordinates and i and j to the particles in the aggregate.The usual radius of gyration (squared) is simply the trace of the tensor: R’ = c ( R J 2 . U Using the hierarchical procedure in two dimensions, where 1000 independent clusters of 1024 particles have been built, the tensor has been diagonalized, and the largest and lowest eigenvalues and (RJ’ have been averaged over all the clusters of the same number of particles. It is found that the ratio varies very slowly with cluster size. The results for the extrapolation to infinite size are given in table 2 in the case of the Brownian, ballistic and chemical models. When comparing the three cases, one observes a systematic decrease in Al when increasing the fractal dimension.In all cases the anisotropy ratio is quite large. The same calculation performed on particle-cluster aggregates gives an extrapolated value A, =; 1. Thus particle-cluster aggregates can be considered as isotropic in the sense that the eigenvalues of the tensor are degenerate. This does not exclude other more subtle effects such as anisotropy induced by the underlying lattice26 or self-affinity proper tie^,^^ as recently found in the Witten-Sander model. The calculations have been extended to higher dimensions in the case of the cluster- cluster model with linear trajectories. The results for all the d - 1 anisotropy ratios:R. Jullien, R. Botet and P. M. Mors 129 Table 3.Anisotropy ratios Ai = (( R ( ) * ) / ( ( R,)') between the ith and the lowest eigenvalues of the radius of gyration tensor, estimated from finite size extrapolations in the case of the cluster-cluster aggregation model with linear trajectories, for space dimensions up to d = 4 . The anisotropy ratio A: for a two-dimensional projection of a three-dimensional aggregate is also given. d A3 2 5.2 + 0.02 3 10.0 f 0.3 2.5 f 0.3 4.5 f 0.3 4 14.0 f 0.5 4.2 f 0.4 2.0 f 0.3 (the eigenvalues being given in decreasing order) for space dimensions d = 2, 3 and 4, are reported in table 3. For a given d, all the eigenvalues are different. Moreover, the larger ratio A, grows rmghly linearly with d. This last result is supported by simple analytical considerations.2s For a comparison with experiment, and in particular for the analysis of two- dimensional micrographs, it is of interest to have some information on the anisotropy properties of the projection of three-dimensional clusters onto a plane.The ratio A\ for such a projection has been calculated and is also reported in table 3. A; is of the order 4.5, slightly smaller than for a pure two-dimensional process. When quantitatively comparing with experiments it is important to recall that the A, quantities are the ratios of averaged eigenvalues. The alternative quantities B, = (( Ri)2/ ( Rd)*) are 23 '/o smaller than the Ai values.28 Readjusting Effects in Cluster-Cluster Aggregation An important approximation to the original cluster-cluster model is that the clusters stay rigid along their diffusive motion and do not rearrange themselves after sticking.There is experimental evidence that, at least on a small length scale, restructuring and readjusting may occur in aggregation phenomena. An example is shown in plate 1, where one can see aggregates of wax balls floating on water surface under shear forces.29 These aggregates appear to be quite compact on a short-range scale. It is a delicate problem to simulate simply such restructuring effects. An attempt has been made in two dimension^.^' This calculation considers an off -lattice extension of the hierarchical version of the cluster-cluster model in which some readjusting is allowed after sticking. As in the original hierarchical ~ c h e m e , ' ~ one cluster, say cluster 1, stays at the centre of the coordinates while the other cluster, cluster 2, is released randomly on a large circle centred on the origin.Then cluster 2 undergoes a random walk in space until a first contact occurs. Then, as shown in fig. 3 , cluster 2 is rotated rigidly about the centre of the contacting particle on cluster 1 until a second bond is obtained. Sometimes the new contacting point is again located on the same contacting particle of cluster 2, so that another rotation can then be performed around the centre of the contacting particle on cluster 2 until a true loop is obtained. Different options can be adopted to perform these rotations. Either one considers a pure Brownian-like rotational motion or one systematically chooses the smallest possible angle.The precise choice does not affect the result too much. An important point, however, is that, in any case, one waits until the complete readjustment has been made before considering a new collision. Using this procedure, loops are systematically built at each step of the hierarchical procedure. Typical examples of small clusters ( 5 12 particles) built with and without restructuring are shown in fig. 4 (top). Here two rotations have been considered (when possible) and130 Computer Simulations of Cluster- Cluster Aggregation / W Fig. 3. Sketch of the readjusting mechanism in the cluster-cluster model. After contact, the clusters are rotated about their contacting particle until a true loop is obtained. In case ( a ) only one rotation is possible. In case ( b ) two rotations can be performed.the smallest angle has been chosen. Compactification is clearly visible and the readjusted cluster strongly resembles the clusters of plate 1. The calculation has been pursued to reach larger cluster sizes, and typical examples of larger clusters (16 384 particles) are shown in fig. 4 (bottom). Surprisingly, the restructuring effect is now less visible. This is quantitatively checked when calculating the fractal dimension, which characterizes long-distance correlations. When extrapolating to the infinite size, the fractal dimension is estimated to be D = 1.48 f 0.05, only very slightly larger than the one, D = 1.44 f 0.05, without restructuring, the difference being of the order of the error bar. A spectacular experimental confirmation of this has been recently produced by Skjelt~rp.~' His aggregation experiment with uniformly sized polystyrene microspheres confined to thin layers between solid boundaries is shown in plate 2, at two different scales.On the short scale one clearly sees readjusting and compactifications, while this is less visible on larger length scale. Moreover, the fractal dimension D = 1.49 f 0.05, estimated from the experiment is in perfect agreement with the calculations. One can, however, observe that the aggregates are slightly more compact, on the short length scale, in the experiment than in the simulation. This can only be accounted for, in any calculation, by involving more complex restructuring effects such as diffusion of individual particles on the surface and/or slight modifications of the interparticular distances, which have not been allowed in the calculations presented above.30 Such more complicated effects are hard to simulate simply.The Effect of Cluster Polarizability (Tip-to-tip Model) Some aggregation experiments in two dimensions32 or in three dimensions33 lead to a smaller fractal dimension than the one predicted by the original cluster-cluster model. This discrepancy has been recently explained by introducing polarizability effects in the existing r n ~ d e l s . ' ~ In the case of polarizable clusters, two clusters, before they collide, develop opposite electrical charges on their neighbouring tips. Then, an electrostatic attraction biases their relative diffusive motion. In the limiting case of very strongFaraday Discuss.Chem. SOC., 1987, Vol. 83 Plate 1 Plate 1. Aggregation of wax balls floating on the surface of water in the presence of shear forces (experiment carried out by Camoin and B l a n ~ ~ ~ in Marseille, France). Plate 2. Two-dimensional aggregation of 4.7 pm spheres showing compactification on a short length scale ( a ) and ramified clusters on a large length scale ( b ) (experiment carried out by Skjeltorp3' in Kjeller, Norway). R. Jullien, R. Botet and P. M. Mors (Facing p . 130)R. Jullien, R. Botet and P. M. Mors 131 * < * 80 d i a m e t e r s 60 d i a m e t e r s < * * 1100 d i a m e t e r s 800 d i a m e t e r s Fig.4. Effect of readjusting in the cluster-cluster model. Cases (a) and ( b ) correspond to small clusters (5 12 particles) with and without readjusting, respectively.Cases ( c ) and (d) correspond to larger clusters (16 384 particles). polarizability, one can imagine that the Brownian diffusion does not play any role and that these neighbouring tips effectively become the most probable sticking points. This is the spirit of a new model, the tip-to-tip model, which can be simply described in its hierarchical version. The procedure differs from the standard hierarchical procedure only in the way the collision between two clusters is simulated. As sketched in fig. 5 , a random direction in space is first chosen. Then, the two clusters are placed far apart along this direction. The two nearest particles on the first and second clusters are determined. Then the two clusters are translated so that these two particles come into contact, with their centres aligned along the chosen direction.This model gives a fractal dimension equal to 1.26 and 1.42, respectively, in d = 2 and d = 3. In its particle-cluster counterpart, this procedure leads to star-like clusters with a fractal dimension trivially equal to one. Typical two-dimensional and three-dimensional clusters obtained with this model are given in fig. 6.132 Computer Simulations of Cluster- Cluster Aggregation Fig. 5. Sketch of the sticking mechanism in the tip-to-tip model. Fig.6. Typical aggregates obtained with the tip-to-tip model. Cases ( a ) and (c) correspond to the cluster-cluster version in d = 2 and d = 3 (fractal dimensions equal to 1.26 and 1.42), respec- tively.Cases ( b ) and ( d ) correspond to the particle-cluster version.R. Jullien, R. Botet and P. M. Mors 133 I I - 3 I . 1 6 - h v U Y cl aD 5 - 0 4 - 4 - 3 - 2 log (s) Fig. 7. Experimental small angle X-ray scattering functions of aluminium hydroxyde aggregates Al(OH),, with x = 2.5 ( a ) and x = 2.6 ( b ) , plotted as a function of s = q / ( 2 ~ ) . In each case one can show the corresponding theoretical curve obtained directly from the simulation, using the tip-to-tip cluster-cluster model in case ( a ) and the ballistic cluster-cluster model in case ( b ) . The theoretical curves take care of the diffusion by individual particles and have been adjusted to the experimental curves at the inverse of the particle diameter. (-), (- - -) and (. a - - - -.) correspond to clusters with N = 64, 128, 256 particles in case ( a ) and N = 256, 512, 1024 particles in case ( b ) .The cluster-cluster tip-to-tip model has been recently used to explain small-angle X-ray scattering experiments on aluminium hydroxide aggregate^.^^ The experimental scattering functions obtained with samples of Al(OH),, with x = 2.5 and 2.6, both exhibit a linear part in their log-log plots characteristic of fractal structures, but with a smaller slope in the case x = 2.5 than in the case x = 2.6. These curves have been fitted with the theoretical expression: with where I , ( q ) is the scattering function of a single subunit, which is known from other experiments to be a spherical AlI3 small cluster of radius ro = 10 A, and where P ( r ) is a properly normalized distance distribution function, P( r ) d r being proportional to the134 Computer Simulations of' Cluster- Cluster Aggregation number of inter-particle distances located between r and r+dr.The normalization is such that P P ( r ) d r = N - 1 J where N is the number of subunits in an aggregate. P ( r ) has been directly obtained from the simulations of an N-particle aggregate, using the hierarchical procedure. The quantative fit to the experimental curve for x = 2.5, using the three-dimensional tip-to-tip model, is shown in fig. 7(a). The only adjustable parameter is the intensity at q = l / r o . The fit of the low-q part of the curve (the so-called Guinier regime) allows one to estimate the number of subunits to be N ~ 6 4 . The same kind of fit has been obtained in the case x=2.6 [fig.7 ( b ) ] , but using the usual three-dimensional cluster-cluster model (for simplicity linear trajectories have been considered but the results would not have been so different with brownian trajectories). Here the number of particles is estimated to be N = 512. Since in these fits one does not take care of any size distribution of clusters, these deduced numbers of particles must be considered as very rough averages. There are some qualitative arguments to explain why the tip-to-tip model is better adapted to the x = 2.5 case. The Al,, subunits are positively charged, and water molecules are oriented near their surfaces, producing an electrical field whose intensity is directly linked with the intensity of the charge on the subunits.For x = 2.5, this charge is quite large and one can imagine that the polarization of water molecules on the surface of the aggregates might induce a sticking mechanism like in the tip-to-tip model. However, when x increases, the charge decreases, leading back to the regular cluster-cluster mechanism for x = 2.6. Flocculation of Colloids in the Presence of Polymers The aggregation of charged colloids can be induced by the adjonction of polymers of opposite charges into the colloidal solution. This is a quite common procedure which has a lot of industrial applications. The colloidal particles, attracted by polymers, can cover their surface with a limited number of polymeric units. If the polymeric chain is sufficiently long, the same chain can be attached to several colloidal particles and, since a given particle can be covered with units coming from distinct polymeric chains, a random set of polymeric connections is progressively built, leading to an efficient aggregation mechanism between particles.A first approach to simulating this mechanism could be to extend the existing cluster-cluster model to the case of two diffusing species A (particles) and B (polymers), in which A-B connections are allowed while A-A and B-B connections are forbidden. Such a model has recently been introd~ced,'~ but with another motivation. Very interesting results, such as a dependence of the fractal dimension on the ratio of concentrations of A and B species, or saturation effects occurring when one species is in excess, are observed.However, this model, which treats the A and B species equally, as though they were both diffusing colloidal particles, is too crude a model to describe aggregation through polymers, since it does not take into account the specific properties of polymers. A more adequate simulation is in progre~s.'~ This is an iterative method in which a collection of aggregates of balls connected by polymers is built in a finite box, using a chemical-type mechanism. The first version is restricted to two dimensions. The general principles are the following. The aggregates are assumed to be rigid (the interparticle distances remain constant), while all the polymeric chains have a different random configuration at each step. Once a polymer is attached to a ball, however, this sticking remains permanent and stays at the same distance (counted along the chain)R.Jullien, R. Botet and P. M. Mom 135 Fig. 8. A typical two-dimensional aggregate of 190 particles built with a model of aggregation of balls with polymers (see text). of the polymer end. One assumes that a ball cannot be attached by more thanf contacts to the same or different polymers. The iterative process starts with a given number P of polymers and a given number B of balls. Then at a given iteration one has a collection of aggregates together with remaining single balls and/or polymers. An iteration step is performed as follows. All these objects are randomly disposed in the box under the constraint that balls cannot overlap (many trials are performed if necessary).(ii) The single polymers (if there are any) and all the polymeric chains connecting the balls in aggregates are reconstructed in a random order, using a pure random walk for the single polymers and the dangling ends, and an adequate approximately biased random walk for the segments connecting two balls. (iii) Once a polymer sticks a non-saturated ball, a contact is permanently obtained and a new cluster is built. The parameters are as follows: the number P of polymers, the number B of balls, the functionalityf of the balls, the length N of the polymers (i.e. number of monomers), and the size L of the box. A big difference from the usual cluster-cluster process is that, if there is a sufficient number of polymers, all the balls within a cluster can be saturated and the overall process can stop with more than one cluster in the box. Fig.8 gives a typical example of a final cluster (the largest) obtained in a simulation with B = 200, P = 100, f = 4, L = 100, N = 100. This cluster contains 190 balls. The theoretical Fourier transform of the positions of particles in this cluster, S ( q ) , has been calculated and reported in fig. 9. We have used the three-dimensional formula given above as if this two-dimensional cluster would be randomly disposed in space. One observes a characteristic linear qPD regime whose slope gives D = 1.7, in agreement with a direct estimation of D in real space. This is a preliminary result obtained on a unique and quite small aggregate which must be checked by averages over many samples. One can, however, already notice that this fractal dimension is larger than the one of the chemical model in two dimensions ( D = 1.55), as if the effect of polymers would be to increase the density compared to a pure chemical process.This must be understood as a preliminary conclusion, and a systematic study of the fractality and other characteris- tics of such an aggregate as a function of the parameters of the model is in progre~s.”~ This model is quite crude in its early stages, and some improvements are looked for in the future. First, it must be extended to three dimensions. Then, instead of randomly disposing the objects (which corresponds to a chemical mechanism where the objects touch many times before sticking) one can build a true diffusion. Also, instead of randomly constructing the polymers before they are irreversibly attached to the balls136 Computer Simulations of Cluster- Cluster Aggregation 0 -2 - 4 Fig.9. Theoretical curve for S ( q ) , calculated as if the cluster of fig. 8 would be randomly disposed in three-dimensional space. (which corresponds in assuming a strong short-range attraction between balls and polymers), one can envisage a biased random walk to take care of long-range attractions. Finally, one could take care of the excluded volume for polymers, and other effects, such as repulsion between balls and a possible variation of the distances between balls, can also be considered. Conclusion In conclusion, the cluster-cluster model appears to be quite appropriate to describe aggregation of colloids and aerosols. Several realistic extensions are able to reproduce quantitatively specific experimental results.In many cases a direct comparison between computer simulations and experiment has been very useful in providing a better under- standing of the underlying mechanisms of real aggregation processes. Such studies will be developed in the future. We thank M. A. V. Axelos, B. Cabane, J. P. Chevalier, M. Kolb, P. Meakin, A. T. Skjeltorp, D. Tchoubar and M. Ten& for stimulating discussions. This work has been supported by an A.T.P. and an A.R.C. from the C.N.R.S. Computer simulations have been performed at C.I.R.C.E. (Centre Inter-Rkgional de Calcul Electronique), Orsay, France. P.M.M. acknowledges financial support from C.N.Pq, Brazil. References 1 S.Forrest and T. Witten, J. Phys. A, 1979, 12, L109. 2 B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982). 3 R. Jullien and R. Botet, Aggregation and Fructal Aggregates (World Scientific, Singapore, 1986). 4 T. Witten and L. Sander, Phys. Rev. Lett., 1981, 47, 1400. 5 D. Houi and R. Lenormand, in Kinetics of Aggregation and Gelation, ed. D. Landau and F. Family 6 M. Matsushita, M. Sano, Y. Hayakawa, H. Honjo and Y . Sawada, Phys. Rev. Lett., 1985, 53, 286. 7 L. Niemeyer, L. Pietronero and H . Wiesmann, Phys. Rev. Lett., 1984, 52, 1033. 8 J. Nittmann, G. Daccord and H . 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A, 1986, 19, 2129. 35 M. A. V. Axelos, D. Tchoubar and R. Jullien, J. Phys. (Paris), 1986, 47, 1843. 36 P. Meakin and Z. Djordjevic, J. Phys. A, 1986, 19, 2137. 37 R. Botet, P. M. Mors and R. Jullien, personal communication Received 5 t h December, 1986