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Journal of Sol-Gel Science and Technology 15, 129–136 (1999) c 1999 Kluwer Academic Publishers. Manufactured in The Netherlands. °

3D Monte Carlo Simulations of Diffusion Limited Cluster Aggregation up to the Sol-Gel Transition: Structure and Kinetics J.C. GIMEL, T. NICOLAI AND D. DURAND Chimie et Physique des Matériaux Polymères, UMR CNRS 6515, Université du Maine, Avenue Olivier Messiaen, F-72085 Le Mans cedex 9, France [email protected]

Received May 5, 1998; Accepted January 7, 1999

Abstract. Three-dimensional (3D) Monte Carlo simulations of diffusion limited cluster aggregation at different concentrations (φ) show a crossover from a flocculation regime at short times to a percolation regime close to the gel time (tg ). Contrary to suggestions in the literature tg is independent of the system size (L) for large L. The structural and temporal crossovers between flocculation and percolation take place at characteristic values of the cluster mass (m c ) and the time (tc ) which depend on φ. After normalisation by these characteristic values the crossovers are independent of φ except for very small clusters and at short times. The concentration dependence of m c and tc indicates that the crossover takes place at a given cumulated volume fraction of the clusters independent of φ. At low concentrations the φ-dependence of tg is determined by the cluster growth in the flocculation regime. Keywords:

simulation, percolation, aggregation, structure, kinetics

Introduction Many gels are formed by aggregation of dilute small particles, e.g., silica gels made from dilute solutions of tetramethoxysilicon in methanol [1] and protein gels made by heat-induced aggregation of globular proteins [2]. The kinetic approach proposed by Smoluchowski [3] can be used to describe the first stage of the aggregation process which is called flocculation. This approach assumes that collisions between clusters are not correlated and thus that clusters are on average widely separated. The kinetic approach does not give information about the structure of the clusters which can, however, be obtained from computer simulations [4, 5]. The final stage close to the gel point may be viewed as a connectivity transition and can be described in terms of the percolation model [6, 7]. This approach is static and assumes that the clusters are already strongly interpenetrating so that the effect of varying mobility of different clusters is negligible. At present, no theory is available to describe the crossover between these two

regimes. Computer simulation is probably the best alternative to study the complex intermediate regime. Here we report results of Monte Carlo simulations of diffusion limited cluster aggregation (DLCA) up to the gel point. In the past DLCA has been studied extensively in the flocculation regime. However, only in a few cases have simulations been continued up to the sol-gel transition. More than a decade ago Kolb and Herrmann [8, 9] did DLCA simulations in two dimensions up to the gel point. They found a fractal dimension intermediate between the values expected for flocculation and percolation. They concluded that the gel time (tg ) is infinite for macroscopic systems if the mobility decreases with increasing cluster size, which is the case for real systems. More recently, we have done 3D Monte Carlo simulations of DLCA up to the gel point [10] and shown that close to the gel point the large scale properties of the system are well described by the static percolation model. The gel time was independent of the lattice size (L) used in the simulation for L sufficiently large. tg

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Gimel, Nicolai and Durand

increased strongly with decreasing concentration (φ) following a power law dependence at small φ. A crucial parameter is the degree of overlap of the clusters which controls the transition from flocculation to percolation. Let Vi be the volume fraction occupied by the ith cluster. Vi can be defined as the volume of the smallest box containing the cluster divided by L 3 . The degree of overlap can be characterized by comparing two different volume fractions, the cumulative (Vcum ) and the effective (Veff ), defined by: Vcum =

Nc X i=1

Vi ,

Veff =

Nc [ i=1

Vi

(1)

S where the symbol means the union. Initially Vcum and Veff are the same and very small, while at the gel point Vcum diverges and Veff becomes unity. If Vcum = Veff then there is no overlap and aggregation occurs by flocculation, while if Vcum À Veff the clusters are strongly overlapping and aggregation occurs by percolation. A broad transition was observed between Vcum ≈ 0.1 and Vcum ≈ 2. The fact that Vcum becomes larger than unity close to the gel point was not appreciated in Ref. [9], and is perhaps one of the reasons that tg was found to depend on L. Subsequently, Hasmy et al. [11] included flexibility in the simulation using a fluctuating bond model. They found that including flexibility leads to a well defined critical concentration below which no gel is formed contrary to the original DLCA model for which no such critical concentration exists. Like Kolb and Herrmann, they reported that the gel time depends on the size of the lattice (L) used in the simulation unless flexibility is introduced [11, 12] and that for rigid clusters there exists no well defined gel time in the thermodynamic limit. This is in contradiction with our simulations where we found no dependence of tg on L for large L. The gel time of real systems very rarely depends on the size of the recipient. Even if such a dependence is found it could very well be a sign of specific interactions with the surface of the recipient. Another discrepancy with our work is that Hasmy et al. find a much weaker concentration dependence of the gel time [13]. Here we present further Monte Carlo simulations of DLCA up to the gel point to resolve these issues and to characterize the temporal and structural transition from flocculation to percolation. We define that the system percolates if there is a spanning cluster in any direction. We will show unambiguously that the gel time is independent of the size of the recipient once L is sufficiently large so that finite size effects become negligible. We

conclude that the concentration dependence of tg is dominated by the growth in the flocculation regime. This concentration dependence can be understood in terms of scaling arguments only if due consideration is taken of the concentration dependence of the time (t0 ) needed to reach the flocculation regime. Theory We briefly summarize predictions of various models for aggregation proposed in the literature that we need for the discussion of our results. The following function has been proposed to describe the number density of the clusters at different times in the flocculation regime [14, 15]: n(m, t) ∝ m −τ · t −ω f (m/m ∗ ) m À 1; t À t0 (2) f (x) is a cut-off function at a characteristic cluster mass m ∗ which is constant for x ¿ 1 and decreases faster than any power law for x À 1. If one assumes that m ∗ ∝ t z , then it follows from conservation of mass that ω = (2 − τ ) · z. For τP < 1, both theP number m · n(m)/ Pn(m), average cluster mass, m n = and thePweight average cluster mass, m w = m2 · ∗ n(m)/ m · n(m), are proportional to m and thus to t z . The polydispersity index (I = m w /m n ) is in that case a constant that depends on τ and f (x). The total number of clusters, Nc , decreases as: Nc ∝ t −z . Computer simulations of DLCA in the flocculation regime show that the clusters are self similar with fractal dimension d f ≈ 1.8 [4, 5]. The movement of the clusters is determined by the translational diffusion coefficient (D(m)). For dilute systems, D(m) is inversely proportional to the size of the clusters (R(m)) and therefore: D(m) ∝ m −1/d f . Using scaling arguments it can be shown that z = 1 if τ < 1 [14, 15]. Computer simulations of DLCA and experiments on DLCA of dilute colloid solutions show that Eq. (2) is a good description of the number distribution with z ≈ 1, τ = 0 and ω = 2 [14, 15]. However, at large times n(m) for small m decreases more strongly, i.e., there is a depletion effect of very small clusters. A different approach is the use of the Smoluchowski equation: 1 X ∂n(m) = K (i, j) · n(i) · n( j) ∂t 2 i+ j=m −

∞ X j=1

K (m, j) · n(m) · n( j)

(3)

3D Monte Carlo Simulations

If it is assumed that the kernel is constant Eq. (3) can be solved analytically [3, 16, 17]: µ ¶ 1 m−1 φ n(m, t) = 2 1 − mn mn

(4)

The time dependence of m n is: m n = 1 + t/t0

(5)

with t0 = 2/(φ · K (1, 1)). For t À t0 , Eq. (4) is equivalent to Eq. (2) if z = 1, τ = 0, ω = 2 and f (m/m ∗ ) = exp(−m/m ∗ ). Using τ = 0 and a simple exponential cut-off function in Eq. (2) leads to I = 2. Apparently, the Smoluchowski equation with a constant kernel gives a good description of DLCA in the flocculation regime. Of course, the Smoluchowski equation can no longer be used when the average distance between clusters is of the same order as their size, i.e., when Vcum approaches unity. Close to the gel point the percolation model [6] has been used to describe n(m) as a function of ε = ( pc − p)/ pc where p is the fraction of bonds formed and pc is the critical fraction at the gel point. Monte Carlo simulations show that self similar clusters are formed with d f ≈ 2.5. n(m) has again a power law dependence on m for m À 1 and ε ¿ 1 : n(m, t) ∝ m −τ f (m/m ∗ ), where the cut-off function has the same properties as in Eq. (2). Close to the gel point m ∗ diverges as m ∗ ∝ ε−1/σ . In three dimensions one finds τ = 2.2 and σ = 0.45. Both m w and I diverge at the gel point, but m n remains constant. In DLCA the parameter corresponding to ε is (tg − t)/tg or, as was used in Ref. [10], (1 − Veff ). For (1 − Veff ) < 0.1 the dependence of m w on (1 − Veff ) is the same for DLCA and for static percolation. Results Monte Carlo simulations were done on cubic lattices with varying size. Initially, each site is occupied with a probability φ which results in an average volume fraction of occupation φ. At each simulation step one cluster is randomly selected and moved in one out of six possible directions with probability m −1/d f , where m is the mass of the cluster defined as the number of occupied sites in the cluster. In the simulations we have used d f = 1.8 which is the fractal dimension of the clusters in the flocculation regime. Using d f = 1.8 for all clusters means that the mobility of very small clusters and of large clusters in the percolation regime is

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not exactly inversely proportional to their radius. Each time a cluster is selected the simulation time (tsim ) is incremented by 1 and the physical time (t) is incremented by 1/Nc , where Nc is the total number of clusters in the system at time t. If two monomers are nearest neighbors they are irreversibly linked and the two clusters to which these monomers belong are merged. The gel time is defined as the time when a cluster spans two opposite faces of the lattice. More details of the procedure are given in Ref. [10]. Structure Before discussing the kinetics of the aggregation process, we will show that in the simulation we recover the structural properties of flocculating clusters far from the gel point and of percolating clusters close to the gel point. In Fig. 1(a) the radius of gyration (Rg ) of the clusters is plotted as a function of their mass at different concentrations. At the highest concentration used in the simulations (φ = 0.3116) the system percolates at t = 0. The data were obtained by averaging values obtained during the whole simulation. We verified that the result is the same if the distribution is analyzed at different fixed times. In a double logarithmic representation the slope gives the fractal dimension of the clusters. The limiting slopes at small and large m are about 1.8 and 2.5 as expected for flocculating and percolating clusters, respectively. Deviations of the limiting behavior occur for very small clusters that are not yet in the flocculation regime and for very large clusters due to finite size effects. The crossover between flocculation and percolation takes place at a characteristic mass (m c ) which we define arbitrarily as the mass where the limiting slopes intersect. The transition is better visualized if m · Rg−1.8 is plotted versus m, see Fig. 1(b), because m · Rg−1.8 is constant in the flocculation regime. It is not easy to observe clearly both regimes for a single value of φ due to the limited lattice size (L = 400) used in the simulation. However, the data at different concentrations can be superimposed if m is normalized by m c , see Fig. 1(c). The superposition gives a very clear picture of the structural transition between the flocculation and the percolation regime. The crossover regime extends over three orders of magnitude of the mass. It is probable that the intermediate value for d f found by Kolb and Herrmann [8] was obtained in the crossover regime. The dependence of m c on φ is shown in Fig. 2 and will be discussed below. We note that even when the gel point is reached

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(a)

(b)

(c) Figure 1. (a) Dependence of the cluster mass on their radius of gyration at different concentrations indicated in the figure. The solid lines represent the limiting behaviors at large Rg for flocculating and percolating clusters. We used L = 400 except at φ = 0.1 for which L = 316. (b) Different representation of the data shown in Fig. 1(a). This representation shows more clearly the crossover between flocculation and percolation. (c) Superposition of the data obtained at different concentrations indicated in the figure by normalizing m with m c .

at t = 0, i.e., for static percolation, the data can be superimposed on the master curve. Of course, at this concentration we do not observe a flocculation regime as m c is small.

Kinetics Having shown that in our simulations we recover the well known limiting behavior for d f at early and late stages of the DLCA process, we will now discuss the time dependence. We can distinguish three regimes: (1) an initial period needed to reach the flocculation regime; (2) the flocculation regime characterized by

m n ∝ t, I = 2 and Vcum = Veff , (3) the percolation regime characterized by a power law divergence of I and m w with (tg − t)/tg and a logarithmic divergence of Vcum . There are three characteristic times: the time t0 needed to reach the flocculation regime, the time tc that characterizes the crossover to the percolation regime and tg the time needed to form a gel. All three characteristic times vary with φ as is illustrated in Fig. 3(a) where the time dependence of I is given for different φ. At low concentrations I stabilizes at a value close to 2 consistent with the Smoluchowski equation using a constant kernel, see above. For larger values of φ the flocculation regime is not clearly observed as regime 1 crosses over directly into regime 3.


Figure 2. Concentration dependence of the cluster mass that characterizes the transition from percolation to flocculation. The solid line represents a linear least squares fit to the data.

If the kernel in Eq. (2) is independent of φ then t0 is inversely proportional to φ. In Fig. 3(b) we plotted the same data as in Fig. 3(a) as a function of t · φ. It is clear that if t is normalized by t0 ∝ 1/φ the flocculation regime is reached at the same normalized time for all values of φ where the flocculation regime can be observed. We have seen that the structural crossover from flocculation to percolation occurs at m = m c . In Fig. 4(a) we give the time dependence of m w /m c . If the temporal crossover is a universal function of φ for t À t0 it should be possible to superimpose the data shown in Fig. 4(a) by normalizing t with tc which we arbitrarily define as the time needed to reach m w = m c . Figure 4(b) shows that the temporal crossover is indeed universal for t À t0 . For t0 ¿ t ¿ tc , m w increases linearly with t which is characteristic of the flocculation regime. At t > tc , m w increases steeply which is characteristic of the percolation regime. The small deviations close to the gel point are due to finite size effects. Like for the structure it is difficult to observe both limiting behaviors at a single concentration due to limitations of the lattice size that can be used in the simulations. The dependence of tc on φ is plotted in Fig. 5. The crossover from flocculation to percolation occurs when the clusters start to overlap, i.e., when Vcum becomes significantly larger than Veff . We have shown in Ref. [10] that the dependence of Veff on Vcum is independent of φ. One might therefore expect that tc is proportional to the time, tx , needed to reach a given value, x, of Vcum . In the flocculation regime Vcum ∝ φ · (3/d −1) and m n ∝ t/t0 which means that tx /t0 scales mn f

133

(a)

(b) Figure 3. (a) Time dependence of the polydispersity index at different concentrations as indicated in the figure. We used L = 400 except at φ = 0.08 and 0.16 for which L = 316. Vertical doted lines indicate tg for φ = 0.16, 0.08, 0.04 and 0.02. (b) Same data as in Fig. 3(b) plotted as a function of t · φ which is proportional to t/t0 .

with φ as tx ∝ φ d f /(d f −3) t0

tx > t0

(6)

Taking into account the φ-dependence of t0 (t0 ∝ φ −1 ) it follows that the time needed to reach a given value of Vcum in the flocculation regime is given by tx ∝ φ 3/(d f −3)

(7)

Figure 5 shows the time needed to reach Vcum = 0.1(t0.1 ) as a function φ. The system is in the flocculation regime up to Vcum = 0.1 [10]. At low concentrations where t0.1 À t0 we find t0.1 ∝ φ −2.45±0.05 .

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(a) Figure 5. Concentration dependence of the time (t0.1 ) needed to obtain Vcum = 0.1; the time (tc ) needed to reach m w = m c ; and the gel time (tg ).

(b) Figure 4. (a) Time dependence of the normalized weight average cluster mass at different concentrations indicated in the figure. (b) Superposition of the data shown in Fig. 4(a) obtained by normalizing t with tc .

Choosing different values of Vcum the same scaling is found as long as the system remains in the flocculation regime and tx À t0 . The scaling observed in the simulation is very close to that expected from the predicted scaling for DLCA in the flocculation regime given by Eq. (7). Comparing t0.1 with tc we find that tc has the same concentration dependence at low φ even though at time tc the system is no longer in the flocculation regime. This is a consequence of the universal temporal crossover. The third characteristic time of the simulation is the time needed to form a gel. In view of the results by Hasmy et al. [11, 12] we have done a systematic study of finite size effects on tg . Figure 6(a) shows the effect of the lattice size (L) on the average value of tg for different concentrations. We have repeated the simulations many times (at least 103 ) in order to reduce

(a)

(b) Figure 6. (a) Dependence of the gel time on the lattice size at different concentrations indicated in the figure. (b) Dependence of the probability to percolate as a function of the lattice size. Symbols are as in Fig. 6(a).


the error in the average value to less than 5% (95% confidence interval). Of course, we include in the averaging of tg only trials that lead to a gel. It is clear from Fig. 6(a) that tg is independent of L for large L. As was already shown in Ref. [10] and confirmed by this more extended study, the standard deviation around the average value of tg decreases strongly with increasing L. These results show without any ambiguity that the gel time is well defined in the thermodynamic limit for DLCA. Figure 6(b) shows the probability to percolate P as a function of L at different volume fractions. There is clearly no lower limit to the concentration for percolation. However, larger values of L are needed to observe percolation with a given probability at lower concentration. The relation between L and the concentration for which we observe for instance 50% percolation (φg ) depends on the fractal dimension of the largest clusters: φg ∝ L (d f −3) . The fractal dimension of percolating clusters is 2.5 which gives φg ∝ L −0.5 close to the dependence found in the simulation, see also Ref. [10]. Comparing Fig. 6(b) with Fig. 6(a) it appears that tg becomes independent of L once the probability to percolate is practically 1. The φ-dependence of tg is compared with that of t0.1 and tc in Fig. 5. At low concentrations where t0.1 has a power law dependence on φ the concentration dependence of tg is very close to that of tc and t0.1 demonstrating again the universality of the temporal crossover. In Ref. [10] we reported a slightly stronger φ-dependence (tg ∝ φ −2.85 ). The difference is due to the smaller lattice sizes used in our previous study yielding a small residual finite size effect. It is clear from Fig. 6(a) that large lattices are needed to determine tg at low concentrations. Discussion We have seen that the structural crossover between flocculation (d f = 1.8) and percolation (d f = 2.5) is independent of the concentration if we normalize m by a characteristic value m c . Of course, very small particles up to m ≈ 10 deviate from the universal behavior shown in Fig. 1 due to the discrete character of the simulation. Surprisingly, also at larger concentrations when the flocculation regime is not clearly seen, the way the percolation regime is approached is the same. Even in the extreme case that the percolation threshold is reached at the start of the simulation the structure of the particles is the same if we use m c ≈ 18. This observation confirms the equivalence of DLCA and static percolation already noted in

135

Ref. [10]. It is important to realize that even for static percolation the scaling behavior is only reached for large particles. To explain the φ-dependence of m c we suppose that the structural crossover occurs at a conc ). If m c is reached by flocculastant value of Vcum (Vcum (3/d −1) f c tion Vcum ∝ φ·m c so that m c ∝ φ d f /(d f −3) . In the simulation we find m c ∝ φ −1.4 , which is in good agreement with the value expected for DLCA. In Ref. [10] we showed that the cumulated volume of clusters up to a given value of m decreases with increasing time in the percolation regime. Larger upper limits of m are needed to reach the same Vcum due to depletion of very small clusters. However, depletion of small clusters does not influence the structure of the larger clusters so that m c is independent of the advancement of the aggregation. In this work we confirm our previous observation that the gel time is well defined for DLCA in the thermodynamic limit. Comparing our simulation method with that used by Hasmy et al. [10] we find that the most important difference is that Hasmy et al. used periodic boundary conditions (PBC) while we used free edges (FE). PBC have been introduced with the object to reduce finite size effects. However, Heermann and Stauffer [18] have shown in a direct comparison that FE give an at least as good, if not better, description of percolation. In the present work we do not try to correct for finite size effects, but we simply keep increasing the lattice size until the result becomes independent of L. It is not clear if the L-dependence of tg observed by Hasmy et al. is only a trivial finite size effect or also in part an artifact of the use of PBC. Kolb and Herrmann [8, 9] also used PBC in their early 2D simulations. Their conclusion that if the diffusion coefficient, D(m), increases with m no finite gel time exists, is again most likely based on finite size effects. tg is approximately proportional to tx and therefore its φ-dependence is given by Eq. (7). In the literature (e.g., Eq. (8.22) in Ref. [14] and Eq. (3) in Ref. [13]) the φ-dependence of tg has wrongly been assumed to be the same as that of tx /t0 given by Eq. (6) ignoring the φ-dependence of t0 . Hasmy and Jullien [13] studied the φ-dependence of tg in an off-lattice DLCA simulation. For a given box size they found a much weaker φdependence (tg ∝ φ −1.75 ) than reported here. Further simulations are needed to verify whether this weaker dependence is due to finite size effects or intrinsic for off-lattice simulations. We note that for these off-lattice simulations φg ∝ L −1.28 [19], implying that the fractal dimension of the largest clusters is 1.72 which is much smaller than that of percolating clusters.

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Conclusions The most important conclusions that we draw from this work are: • DLCA leads to the formation of a gel in a well defined finite time. • At early times and at low concentrations the Smoluchowski equation with constant kernel gives a good description of DLCA. • At low concentrations the gel time is determined by growth in the flocculation regime. • In the flocculation regime clusters have a fractal dimension close to 1.8. • The fractal dimension of very large clusters and the divergence of m w at tg are close to the predictions of the percolation model. • The temporal and structural crossovers from the flocculation to the percolation regime are universal functions of m/m c and t/tc . The dependence of m c and tc on the concentration is in quantitative agreement with the idea that the crossover occurs at a given value of Vcum independent of φ. References 1. J.E. Martin and K.D. Keefer, Phys. Rev. Lett. 34, 4988 (1986); J.E. Martin and J.P. Wilcoxon, Phys. Rev. A 34, 1803 (1987).

2. J.C. Gimel, D. Durand, and T. Nicolai, Macromolecules 27, 583 (1994). T. Nicolai, D. Durand, and J.C. Gimel, in Light Scattering: Principles and Development, edited by W. Brown (Clarendon Press, Oxford, 1996), p. 201; P. Aymard, D. Durand, T. Nicolai, and J.C. Gimel, Fractals 5, 23 (1997). 3. M. von Smoluchowski, Z. Phys. 17, 585 (1916); M. von Smoluchowski, Z. Phys. Chem. 92, 129 (1917). 4. P. Meakin, Phys. Rev. Lett. 51, 1119 (1983). 5. M. Kolb, R. Botet, and R. Jullien, Phys. Rev. Lett. 51, 1123 (1983). 6. D. Stauffer and A. Aharony, Introduction to Percolation Theory (Taylor & Francis, London, 1992). 7. P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979). 8. M. Kolb and H.J. Herrmann, J. Phys. A: Math. Gen. 18, L435 (1985). 9. H.J. Herrmann and M. Kolb, J. Phys. A: Math. Gen. 19, L1027 (1986). 10. J.C. Gimel, D. Durand, and T. Nicolai, Phys. Rev. B 51, 11348 (1995). 11. A. Hasmy, E. Anglaret, R. Thouy, and R. Jullien, J. Phys. I France 7, 521 (1997). 12. E. Anglaret, A. Hasmy, and R. Jullien, Phys. Rev. Lett. 75, 4059 (1995). 13. A. Hasmy and R. Jullien, J. Non-Cryst. Solids 186, 342 (1995). 14. T. Vicsek, Fractal Growth Phenomena (World scientific, Singapore, 1989), chap. 8. 15. P. Meakin, Physica Scripta 46, 295 (1992); T. Vicsek and F. Family, Phys. Rev. Lett. 52, 1669 (1984). 16. R.J. Cohen and G.B. Benedek, J. Phys. Chem. 86, 3696 (1982). 17. M.L. Broide, PhD, MIT, 1988. 18. D.W. Heermann and D. Stauffer, Z. Phys. B 40, 133 (1980). 19. A. Hasmy, E. Anglaret, M. Foret, J. Pelous, and R. Jullien, Phys. Rev. B 50, 6006 (1994).