Eur. Phys. J. E 15, 133–140 (2004) DOI 10.1140/epje/i2004-10044-x
THE EUROPEAN PHYSICAL JOURNAL E
Monte Carlo simulation of particle aggregation and gelation: I. Growth, structure and size distribution of the clusters M. Rottereau, J.C. Gimela , T. Nicolai, and D. Durand Polym`eres Collo¨ıdes Interfaces, UMR CNRS, Universit´e du Maine, F-72085 Le Mans cedex 9, France Received 27 April 2004 / c EDP Sciences / Societ` Published online: 26 October 2004 – ° a Italiana di Fisica / Springer-Verlag 2004 Abstract. Lattice and off-lattice Monte Carlo simulations of diffusion-limited cluster aggregation and gelation were done over a broad range of concentrations. The large-scale structure and the size distribution of the clusters are characterized by a crossover at a characteristic size (m c ). For m < mc , they are the same as obtained in a dilute DLCA process and for m À mc they are the same as obtained in a static percolation process. mc is determined by the overlap of the clusters and decreases with increasing particle concentration. The growth rate of large clusters is a universal function of time reduced by the gel time. The large-scale structural and temporal properties are the same for lattice and off-lattice simulations. The average degree of connectivity per particle in the gels formed in off-lattice simulations is independent of the concentration, but its distribution depends on the concentration. PACS. 64.60.Ak Renormalization-group, fractal, and percolation studies of phase transitions – 02.70.Uu Applications of Monte Carlo methods
1 Introduction Random aggregation of particles in solution is observed for a wide variety of systems like spherical [1] and disk-like mineral particles [2], globular proteins and emulsions [3, 4]. Often the aggregation process leads to the formation of a gel and in order to understand the material properties of such gels, it is necessary to know how the structure of the system evolves during the aggregation process. Initially, the aggregation occurs by random collisions between on-average well-separated clusters. Two limiting cases have been proposed to describe this so-called flocculation regime [5, 6]: diffusion-limited cluster aggregation (DLCA) for which each collision leads to binding; and reaction-limited cluster aggregation (RLCA) for which a large number of collisions are needed before the particles bind. The properties of the aggregates that are formed in these two cases have been studied in some detail using Monte Carlo simulations [7–10]. Good agreement was found between the simulation results and experiments [8, 11]. The results can be described by kinetic equations with the appropriate kernels [9, 12]. A fundamental property of the aggregates is that they have a self-similar structure with a fractal dimension less than 3. This means that as the aggregates grow the cumulated volume fraction occupied by clusters increases and at some point they start to interpenetrate [13]. The interpenetrating aggregates continue to grow until at the gel point a
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one of the clusters spans the whole system. Close to the gel point, the degree of interpenetration is large and here the aggregation process can be described by the percolation model. It has been shown [14] that the clusters formed by a percolation process have a different fractal dimension and a different size distribution from those formed in the flocculation regime. At present, it is not known how this situation can be described in terms of kinetic equations and so far only computer simulations can be used to model the aggregation of interpenetrated clusters. After the gel point an increasing fraction of the aggregates forms a system-spanning network until finally all the particles are part of the gel. The gel retains the structure of the aggregates on a local scale and becomes homogeneous above a characteristic length scale, the so-called correlation length. Images of DLCA at different stages of the aggregation and gelation process are shown in Figure 1. Many simulations have been done to explore the flocculation and the percolation regimes, but relatively few studies have been made of the complete evolution of the aggregating particles from initial flocculation through the gel transition to the fully aggregated gels. Gonzalez et al. [11] have done lattice simulations of both DLCA and RLCA while Hasmy et al. [8] have done off-lattice simulations of DLCA. They have studied the evolution of the pair correlation function and the structure factor for a range of concentrations. Both groups found that the fractal dimension of the system evolves with the particle concentration. Rather disturbingly, the variation of the fractal
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Fig. 2. Volume fraction dependence of the average distance between the surfaces of nearest-neighboring spheres. The solid line indicates the theoretical result derived by Torquato [19].
simulations of DLCA over a wide range of concentrations. Lattice simulations cannot give a realistic picture of the local structure of the aggregates, but we have found that the large-scale structure is almost identical in lattice and off-lattice simulations. In the present paper, we will focus on the structure of the aggregates and the kinetics. In the accompanying paper [16], referred to as part II, we will discuss the pair correlation function and the structure factor of the systems as a function of time and particle concentration. We are treating only DLCA, but most of the general features of the transition between flocculation and percolation will be equally valid for RLCA even if the quantitative results will be different.
2 Simulation method Fig. 1. Example of an aggregation process for L = 200, φ = 0.005 with gel time tg = 900. The gel fraction is represented in red (only in the on-line version: www.europhysj.org).
dimension was found to be opposite whether the pair correlation function or the structure factor was used to determine it. Recently, Lattuada et al. [15] suggested that this contradiction could be resolved by taking into account the influence of the local structure. Also in our group we have done lattice simulations of DLCA [13]. The emphasis was different in that we focused on the characterization of the aggregates that were formed at different stages of the process and the way in which they fill up the space. Still many issues remain open in this area. Here we present further results of extensive lattice and off-lattice
The simulation method on a cubic lattice was explained in reference [17]. The off-lattice simulations were done in a cubic box with size L3 . N spheres with diameter d = 1 were regularly placed in the box with concentration C = N/L3 . The volume fraction in off-lattice simulations is φ = Cπ/6, while in lattice simulations, φ = C. Here, we will use C both for lattice and off-lattice simulations and φ only for off-lattice simulations. A sphere is randomly chosen and moved with step size s in a random direction. If a movement leads to overlap with another sphere it is truncated at contact. After many movements, the positions of the spheres are completely randomized. We have followed this process by measuring the average distance ∆ between nearest neighbors [18]. The value at equilibrium, ∆e , is independent of the step size if s < ∆e − 1. Figure 2 shows ∆e − 1 as a function of the volume fraction. We find good agreement with the theoretical predictions given by
M. Rottereau et al.: I. Growth, structure and size distribution of the clusters
Torquato [19] for randomly distributed hard spheres, see the solid line in fig. 2. In reference [18] we showed that the pair correlation function g(r) is independent of the step size for distance r larger than s, and well described by the Ornstein-Zernike equation with the Percus-Yevick closure, see also part II. Once the spheres are randomly positioned, the aggregation process is started. Again, if during a movement two spheres overlap, the movement is truncated at contact, but this time the spheres, that become part of larger clusters, are connected and start moving collectively. In order to mimic Brownian diffusion, clusters are chosen randomly and moved in a random direction with a probability inversely proportional to their diameter. The number of clusters, Nc , decreases during the simulation until the box contains a single cluster. The gel is defined as one or more clusters that percolate the box in at least one direction. The simulation time, tsim , is incremented by unity after Nc clusters have been chosen. The average displacement of diffusing clusters is determined by their diffusion coefficient (D): hr 2 i = 6Dt. The physical time unit is defined as the time required for an individual sphere to diffuse a distance equal to its diameter d. For a colloid with d = 1 µm in water at 293 K, this time is 0.4 s. Since d = 1, it follows that in our simulations the diffusion coefficient of the spheres is D = 1/6. The diffusion coefficient of larger clusters is inversely proportional to the average diameter of the clusters. This proportionality is correct for large clusters, but ignores the effect of asymmetry for small clusters. The physical time (t) is related to the simulation time as t = tsim s2 . Of course, the displacement is only diffusional for r À s. We have checked the effect of the step size s on the cluster aggregation process. The results, which will be reported in detail elsewhere, showed that the effect of the step size on the kinetics becomes important if s > ∆e − 1. The effect on the cluster structure becomes important at length scales smaller than s. Here, we present only results at sufficiently small s, so that the effects of the step size are negligible. Finite-size effects were investigated by varying the box size up to L = 200 for off-lattice simulations and L = 500 for lattice simulations. Unless specifically mentioned, the results shown here are independent of the box size.
3 Results and discussion 3.1 Structure of the clusters The pair correlation function of clusters formed by random cluster-cluster aggregation of particles with diameter d = 1 can be described by the following equation [20]: g(r) = r (df −3) f (r/R)
for r À d ,
(1)
where f (r/R) is a cut-off function at the radius R of the clusters that is a constant for r ¿ R and decays to zero faster than any power law for r À R. The aggregation
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number (m) and the radius of gyration (Rg ) of the clusters can be derived from g(r): m=
Z∞
4πr2 g(r)dr + 1
(2)
0
and Rg2 =
1 2m
Z∞
4πr4 g(r)dr .
(3)
0
Utilizing equation (1), it follows that m scales with Rg as follows: m = a · Rgdf for m À 1 . (4) For DLCA the fractal dimension of the aggregates is df = 1.8 if the clusters are on average still far apart, i.e. in the flocculation regime [5, 6], while in the percolation regime, where the clusters are strongly interpenetrated, df = 2.5 [14]. The cut-off function for both types of aggregates is well described by a stretched exponential: f (r/R) = k · exp[−(r/R)γ ], with γ close to 2 [21, 22] and k a constant. The transition between these two regimes is best observed when m is plotted as a function of Rg . We discussed this transition earlier for lattice simulations [13]. Here we compare these results with off-lattice simulations. Figure 3a shows m as a function of Rg obtained during the aggregation process at different volume fractions. We found that the relation between m and Rg is the same during the whole process. The expected scaling behavior is obtained in the limit of small m and small φ for the flocculation regime and in the limit of large m and large φ for the percolation regime. However, there is a broad transition regime covering about two orders of magnitude in m. If the concentration is decreased, the transition between flocculation and percolation occurs at larger m and Rg . This is expected because the flocculation regime ends when the aggregates begin to interpenetrate. We have characterized the transition by the values of mc and Rc at which the limiting slopes cross. The results at different concentrations superimpose if we normalize m by mc and Rg by Rc , see Figure 3b. The superposition demonstrates that the transition between flocculation and percolation is independent of φ and that it is not controlled by the absolute size of the clusters, but by their occupation of space, i.e. their degree of interpenetration. Within the experimental error, off-lattice simulations give the same master curve as lattice simulations, see solid line in Figure 3b. The dependence of mc on C is shown in Figure 4 and a linear least-squares fit gives mc = 3.3C −1.4 . The results obtained with off-lattice simulations are consistent with those obtained with simulations on a cubic lattice taken from reference [13], see squares in Figure 4. The relation between mc and Rc is the same as that found for lattice simulations: mc = 5.7Rc1.8 . The transition between flocculation and percolation occurs when the aggregates start to interpenetrate, i.e. when the cumulated volume fraction of the clusters is approximately unity: CRg3 /m ≈ 1, ignoring the effect of the
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Fig. 4. Concentration dependence of mc obtained from lattice and off-lattice computer simulations. The solid line has slope −1.4.
Fig. 3. (a) Relation between the aggregation number and the radius of gyration of clusters obtained from off-lattice simulations at different concentrations. The solid lines indicate the expected power law behaviour for percolating clusters, while the dashed line indicates the expected behaviour for very dilute clusters. (b) Master curve obtained from the data shown in (a) by normalizing m and Rg with a concentration-dependent characteristic aggregation number (mc ) and radius (Rc ), respectively. The solid line shows the results obtained from lattice simulations.
polydispersity. This means that mc ∝ C df /(df −3)
(5)
because mc ∝ Rcdf . Using df = 1.8, we find that equation (5) is consistent with the simulations. We showed in reference [13] that the cumulated volume fraction diverges at the gel point where the clusters are hierarchically interpenetrated.
One should be careful not to interpret the gradual change of the slope in Figure 3b as a change of the fractal dimension of the aggregates. Large aggregates formed close to the gel point are simply not characterized by a single fractal dimension over all length scales. On length scales much smaller than Rc , they have the fractal dimension of flocculating clusters (df = 1.8 for DLCA), while on much larger length scales they have the fractal dimension of percolating clusters (df = 2.5). The change of the slope that is seen in Figure 3b is thus a consequence of the transition of the structure on small length scale to that on large length scale. The superposition of the data at different concentrations shows that only the length scale Rc at which the structure changes depends on the concentration. From an experimental point of view, it is difficult to appreciate the transition shown in Figure 3a, because it is gradual. Generally, only a rather limited range of molar masses is covered in an experimental study. Over a limited range of molar masses, the data can be easily interpreted as a power law with exponents that vary with the concentration.
3.2 Connectivity of the final gel A given sphere may be connected to up to 12 other spheres. In our simulations, bonds are only formed on contact and the aggregates are rigid. Therefore, there are only binary collisions and no loops can be formed, so that the average coordination number, hzi, has to be 2 for the final system, independent of φ up to 50%. This is indeed what we have observed in the simulations. However, the distribution of coordination numbers does depend on the concentration. The concentration dependence of the fraction of spheres on a given value of z (F (z)) is shown in Figure 5.
M. Rottereau et al.: I. Growth, structure and size distribution of the clusters
Fig. 5. Fraction of particles connected to z other particles in the final gel formed in off-lattice simulations of DLCA as a function of φ.
P The fraction of branch points ( F (z) 3) and P for z ≥ P their average coordination number ( z · F (z)/ F (z) for z ≥ 3) increase with increasing volume fraction. Because hzi = 2 independent of φ, the fraction of free ends (F (1)) also increases with increasing φ.
3.3 Aggregation number distribution of the clusters The number concentration of clusters (N (m)) with aggregation number m can be derived from the kinetic equations in the flocculation regime [23, 24]. For DLCA, it has a bell-shaped form with a maximum at a characteristic aggregation number that increases with time. In the percolation regime, i.e. for m À mc , N (m) decreases as N (m) ∝ m−τ f (m/m∗ ) with τ = 2.2 and a cut-off function at a characteristic aggregation number m∗ that diverges at the gel point [14]. For the gelation process, we expect a crossover between the two distributions around m = mc . In Figure 6 we have plotted the distribution close to the gel point as a function of m/mc for different concentrations obtained from lattice simulations. The results for offlattice simulations are the same but contain more noise. In order to obtain equal area under the curves at differone needs to plot N (m)m2c /C because Rent concentrations, m2 N (m)d(log(m)) ≈ C and m is replaced by m/mc . In this representation, we obtain a universal distribution independent of the particle concentration, which might be expected from the universality of the structural transition between flocculation and percolation. We find the bell-shaped distribution for 1 ¿ m ¿ mc and the power law decay with exponent τ = 2.2 for m À mc . Again, as for the structural transition, there is a broad transition regime around mc of about 2 orders of magnitude in m.
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Fig. 6. Normalized representation of the number distribution of clusters as a function of m/mc at the gel point for different concentrations. The dashes line has slope −2.2 and indicates the expected dependence for percolation clusters.
Experimentally, using scattering techniques one determines the weight average molar mass (Mw ) as a function of the z-average radius of gyration (Rgz ). Polydispersity influences the power law relation between these two pad (3−τ ) rameters if τ > 2 as follows: Mw ∝ Rgzf . The polydispersity does not influence the dependence of Mw on Rgz in the flocculation regime, but it does influence it in the percolation regime, where τ = 2.2. The implication is that if one measures Mw as a function of Rgz for aggregates formed in DLCA up to the gel point, the power law exponent changes only weakly from 1.8 for Mw ¿ Mc to 2.0 for Mw À Mc . In the opposite case of RLCA, the fractal dimension in the flocculation regime is 2.1, so that the experimentally observed exponent varies only from 2.1 to 2.0. Considering the broad transition regime, it is nearly impossible to clearly observe the transition between flocculation and percolation by measuring Mw and Rgz . 3.4 Time dependence The time dependence of the number of aggregates (Ni ) with aggregation number i may be described in terms of kinetic equations as long as the positions of the aggregates are uncorrelated [25]: i−1 ∞ X 1X dNi Ki−j,j Ni−j Nj . Kij Ni Nj + =− dt 2 j=1 j=1
(6)
The so-called kernel, Kij , of this equation determines the sticking rate between particles with aggregation number i and j. For a DLCA process, the kernel depends on the diffusion coefficient (D) and the collision radius
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Fig. 7. Dependence of mw on t obtained from off-lattice simulations at different volume fractions indicated in the figure.
(Rcol ) of the aggregates [26, 6]. D is inversely proportional to the hydrodynamic radius Rh of the aggregates (D = kT /6πηRh , where k is the Boltzmann constant, T the absolute temperature and η the solvent viscosity). For individual spheres, Rcol and Rh are simply the particle radius. For large fractal aggregates, Rcol and Rh are proportional to m1/df , although not necessarily with the same proportionality constant [27]. The kernel Kij may be written as [25] Ki,j = 4π(Rcol,i + Rcol,j ) · (Di + Dj ) .
(7)
If it is assumed that the kernel is constant and equal to K11 then equation (7) can be solved analytically and in that case the weight average aggregation number (mw ) increases linearly with time: mw = 1 +
6 K1,1 t · φ . π
(8)
In our simulations, the particle diameter is unity, Rcol,1 = 0.5 and the diffusion coefficient of the individual spheres D1 = 1/6, so that K11 = 8π/6. Equation (8) is an exact description of the initial aggregation of spheres as long as the fraction of clusters larger than dimers is negligible. Apparently, equation (8) works even for larger mw as long as the aggregates are dilute, see the solid line in Figure 8. The time dependence of mw was obtained from offlattice simulations for a range of volume fractions (Fig. 7). mw increases sharply close to tg , the gel time. For φ > 10%, gels are formed before the monomers have diffused over their own diameter, i.e. tg < 1. To observe in detail the initial growth due to formation of dimers, we plotted in Figure 8 mw − 1 versus t · φ. The data show that for φ > 0.5%, correlations between the positions of the particles influence the concentration dependence of the aggregation process from the start. For lower concentrations, the initial growth rate is proportional to φ and is
Fig. 8. Dependence of mw − 1 on t · φ obtained from off-lattice simulations at different volume fractions indicated in the figure. In this representation, the initial increase becomes independent of φ for small φ. The solid line represents equation (8).
well described by equation (8). As the clusters grow, their cumulated volume fraction increases and they start to interpenetrate. We have seen above that the influence of interpenetration on the structure and the size distribution is a universal function of mw /mc . One might therefore expect that the evolution of mw /mc is a universal function of t/tc , where tc is the time needed to reach mw = mc . Of course, this is only true for mw À 1, so that the influence of the local structure is no longer important. Lattice simulations have shown that for mw sufficiently large, the increase of mw /mc is indeed a universal function of t/tc independent of the concentration [13]. It was shown in reference [13] that gel time (tg ) is proportional to tc , which means that mw /mc is also a universal function of t/tg , see Figure 9. Lattice and off-lattice simulations give identical results, which demonstrates that the evolution of the large-scale structure is independent of local details. We note that at low concentrations we cannot determine tg and mc directly due to finite-size effects. Therefore, we have used the power law concentration dependence of mc established at higher concentrations to estimate mc at low concentrations. The value of tg at low concentrations was estimated by superposition of the time dependence of mw /mc for large mw . Figure 9 shows that mw increases linearly with time until mw ≈ 0.01mc . For larger mw , the aggregation rate increases and mw diverges at tg . If the evolution of mw in the flocculation regime can be described by equation (8), then the concentration dependence of tc is given by tc ∝ C 3/(df −3) [13]. The gel time is proportional to tc , so that tg ∝ C 3/(df −3) . The concentration dependence of tg obtained in lattice and off-lattice simulations is compared in Figure 10. For both lattice and off-lattice simulations, we find for C < 0.1 a power law dependence that is consistent with the
M. Rottereau et al.: I. Growth, structure and size distribution of the clusters 6
10
5
10
4
10
3
10
2
10
1
10
0
tg
10
139
10
-1
lattice off-lattice
-2
10 0.001
Fig. 9. Master curves of mw /mc as a function of t/tg obtained from lattice (solid line) and off-lattice simulations.
predicted dependence using df = 1.8: tg = 0.2 × C −2.5 , see the solid line in Figure 10. At higher concentrations the gel time is faster because the volume fraction is no longer small and the initial growth is no longer in the flocculation regime. On a cubic lattice, mw at t = 0 increases with increasing concentration and diverges at the static percolation threshold Cp = 0.3116 [14]. Off-lattice, no particles are at contact in the initial state for all volume fractions below that of close packing. The aggregation process ends with the formation of a single cluster containing all particles in the system. The time needed to reach the final state is about 8 times the gel time both for lattice and off-lattice simulations. For a given system size L, the radius of gyration of the final 1 cluster increases with increasing N as Rg ∝ N 1.8 , as long as Rg ¿ L, i.e. in the flocculation regime where N ¿ mc . For N > mc , the later stage of the growth of the clusters is in the percolation regime and Rg approaches L. For a given box size, the system percolates at a critical number of particles proportional to L1.8 , and thus at a critical concentration proportional to L−1.2 . It follows that DLCA leads to gelation at any arbitrarily small concentration if the system size is sufficiently large. In practice, it means that L should be larger than Rc .
0.01
0.1
1
C
Fig. 10. Concentration dependence of the gel time obtained from lattice and off-lattice simulations. The values for C > 0.01 could be determined directly, but the data at lower concentrations were deduced from the growth rate at shorter times.
3.5 Developpement of the connectivity
Fig. 11. Time evolution of the fraction of particles connected to z other particles for off-lattice simulations at φ = 2%. The dashed line represents the gel time.
Figure 11 shows an example of the development of the connectivity in the system obtained from off-lattice simulations at 2%. Initially, dimers are formed leading to a decrease of F (0) and an increase of F (1). At later times, larger clusters are formed and particles with higher coordination numbers appear. F (0) decreases to zero when all particles are connected. F (1) passes through a maximum while the fraction of more densely connected particles increases monotonically. The variation of F (z) after
the gel point, indicated by the dashed line in Figure 11, is small because most particles are already connected at tg . The evolution of F (z) at other concentrations is similar. However, at the highest concentrations the fraction of monomers and small oligomers is not negligible at the gel point, which means that F (z) continues to evolve somewhat even after the gel point.
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In these simulations we have ignored the influence of rotational diffusion. Meakin et al. [10] have shown that for DLCA the fractal dimension is not influenced by rotation in the flocculation regime. We do not expect that the structure of interpenetrated clusters is much influenced by rotation either, because the growth in the percolation regime is caused by binding of clusters without major change in their relative positions. However, we do expect that the aggregation rate is strongly accelerated by rotation once the clusters are interpenetrating especially for t > tg .
4 Summary Diffusion-limited aggregation of particles at low volume fractions (φ < 5%) occurs initially by collisions between clusters that are on average well separated. Close to the gel point, the clusters are highly interpenetrated and the aggregation can be described as a percolation process. The transition between the flocculation regime and the percolation regime leads to a change of the structure and the size distribution of the aggregates. This rather broad transition occurs at a characteristic aggregation number mc and radius of gyration Rc . For m < 0.1mc we find the fractal dimension and the size distribution of flocculating clusters, while for m > 10mc we find the properties of percolating clusters. mc decreases with the particle number concentration as mc ∝ C df /(df −3) and mc ∝ Rcdf , with df = 1.8. Off-lattice simulations show that the degree of branching of the final gels increases with increasing volume fraction, although the average connectivity of a particle is independent of the volume fraction. mw increases initially linearly with time as long as mw < 0.01mc . Subsequently, the growth rate accelerates because the aggregates start to interpenetrate and mw diverges at tg as described by the percolation model. The gel time decreases as increasing concentration as tg ∝ C 3/(df −3) with df = 1.8. The structure and the growth of the aggregates on larger length scales is the same in lattice and off-lattice simulations.
This work has been supported in part by a grant from the Marie Curie Program of the European Union numbered MRTN-CT2003-504712.
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