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Models for irreversible gel formation
ELSEVIER
Polymer Gels and Networks 4 (19%) 375-382 Copyright 0 1996 Published by Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0966.7822/96/$15.00 PII: SO%&7822(%)00025-l
Models for Irreversible
Gel Formation
M. Kolb Laboratoire Chimie ThCorique, Ecole Normale Supkrieure, 69364 Lyon, France and Institut de Recherches sur la Catalyse, 2, Ave. A.Einstein, 69626 Villeurbanne Cedex, France
ABSTRACT Many of the physical and chemical properties of gels are determined by the porosity of the underlying molecular network. The build-up of an infinite network-and even its existence-often depends on irreversible aggregation processes during the earliest stages of structure formation. The system gels only once the primary structures fill space. Aggregation and percolation models each describe one limiting situation of such a sol-gel process. The important intermediate stages cannot be treated with the same degree of generality. New results for this crossover regime are presented: it is shown that there is a smooth change of the structural properties and the cluster size distribution when passing from the aggregation to the gelation stage. The measured fractal dimension of the clusters changes continuously as one approaches the gelation transition. Copyright 0 1996 Published by Elsevier Science Ltd.
1 BASIC
MODELS
Irreversible aggregation describes the formation of elementary building blocks in one type of gel formation. Examples of such processes are the gelation of silica and more generally colloidal gels.‘-’ This class of models starts from the assumption that clusters are always perfectly rigid, that they move according to a diffusion law with a cluster size dependent diffusion coefficient, and that aggregates grow by forming permanent bonds via a strong short-range interaction between elementary blocks.‘-” One distinguishes two limiting cases: diffusion limited or reaction limited aggregation, depending on whether bonds are formed at the first contact or whether a potential barrier imposes many collisions before a bond can 375
376
M. Kolh
form. The first process is also called rapid aggregation, with the latter known as slow aggregation. The main characteristics of the structures built in this way are the following: individual clusters are fractal objects with a fractal dimension of D = 1.40-1.45 in two-dimensions and D = 1.75-1.80 in three-dimensions for fast, diffusion limited aggregation and D = 1.55-1.60, respectively, D = 2.0-2.1 for slow, reaction limited aggregation. In the former case the cluster size distribution is mono-disperse, which means that all clusters have roughly the same size. This holds for the physically relevant case when large clusters diffuse more slowly than small ones. For reaction limited aggregation the cluster size distribution is weakly poly-disperse, which means that it decays with a power law with an exponent less than two. These properties are valid irrespective of the details of the system under study (lattice structure, rotational motion of the aggregates, details of the interaction), but they do depend on the spatial dimension, the short-range nature of the interaction and the rigidity of the bonds of the aggregates. The rigidity of the clusters is of course a key assumption for the applicability of these models. The cluster diffusivity only has a marginal influence on the scaling properties, provided that large clusters move more slowly than small ones. Furthermore, aggregation models assume that the initial density is sufficiently low for cluster-cluster correlations to be neglected. However, because of the fractal nature of the aggregates the cluster density one gradually leaves the invariably increases with time. Therefore range of validity of the models. At the same time it is exactly this increase of the effective density that eventually makes gelation possible. Hence in the context of gelation one is particularly interested in this change of the aggregation process as the density grows. The resulting network may differ considerably from those predicted by percolation theory.“’ Within the framework of irreversible growth, the simplest possibility to obtain a space-filling network is to run the aggregation process beyond the low density limit. We will present numerical results on this changeover process. If one continues to aggregate beyond the point where the clusters fill space, the growth could continue in two possible ways. If the clusters are to stay fractal, they would interpenetrate each other more and more. This is an unlikely scenario as it would mean that neighbouring aggregates have an ever larger common interface and still manage to avoid touching each other. The other possibility is that clusters become non-fractal. The study of a model that starts out at very high density clearly shows that this is what happens. “.I2 The simulations show that the clusters are dense, but
Models for irreversible gel formation
377
have a fractally rough surface. The surface roughness depends on how mobile small clusters are in comparison with large clusters. Other modifications may of course also change the structure and the physical properties of a system that is about to gel. Depending on the type of bonding, one must relax the assumption of irreversible bonds. This leads, in its idealized form, to aggregation with random bond breaking. Because of the finite lifetime of the bonds the aggregation process reaches a steady state when the growth induced by the cluster collisions equals the fragmentation due to bond breaking. The clusters also restructure because of fragmentation and their fractal properties differ from the irreversible growth models, even at low densities. At very high densities the model closely resembles bond percolation. A second modification that needs to be included in a more realistic description of an aggregation-gelation process is the elasticity of large structures. Experimentally there is always a characteristic size beyond which the clusters are no longer rigid. Ideas from the scaling theory for polymers may then be applied, at least well beyond the gel point.’
2 SIMULATIONS
AT
CROSSOVER
CONCENTRATION
The results that we wish to discuss here are the change of the cluster structure as a consequence of the ever-increasing density. The model is standard cluster-cluster aggregation beyond the low density range: on a lattice single particles clusters are placed randomly with an initial density p. << 1. The clusters diffuse by jumping randomly to neighbouring sites. If a jump of a cluster leads to an overlap of one or more of its particles with other particles, the move is not accepted. Instead a permanent, rigid bond is formed between a pair of colliding particles. For several colliding pairs, one is chosen at random. The diffusion coefficient depends on the cluster size: the jump probability has scaling form, D = Y, where s is the number of particles in the cluster and (Y< 0, such that larger clusters diffuse more slowly than small ones. A physically reasonable value in three dimensions is (Y= -0.5. For reaction limited aggregation, a bond between neighbouring particles only forms after a large number of overlap attempts or after particles have spent much time on neighbouring sites. For the last model the choice of (Y has no influence on the aggregate structure and the size distribution. The simulations were performed at several intermediate initial concentrations, to show that the results can be fitted to a crossover scaling form.
37x
M. Kolh
3 RESULTS
AND
DISCUSSION
In Fig. 1 we show a snapshot of a typical situation of two dimensional growth for (Y= -0.5 and at an intermediate effective concentration where clusters are visibly highly correlated. Internally they appear homogeneous, and one notices the pronounced void spaces between the clusters. This may be compared with the experimental finding that during high density irreversible aggregation a characteristic length sets the scale for the growth, as for phase separation phenomena.” Another interesting observation comes from the mass vs radius relation
Fig. 1. Two-dimensional diffusive cluster aggregation at intermediate effective conccntralion. The correlation between clusters is clearly visible: empty “corridors” separate neighboring clusters.
Models for irreversible gel formation
379 1
:
1
10000 D
1000 100 10 1
r
031
O,l 1
10
100
1000
10000
R Fig. 2. Average cluster mass as a function of the average cluster radius, plotted logarithmically. The effective slope gradually changes from the value D = 1.43 for cluster aggregation at low concentration to the value D = 2 for dense structures. Also shown is the exponent characterizing the minimum (“chemical”) distance which is important for the mechanical properties of the structure. The dotted line is the probability p that the largest cluster spans the system.
as one enters the high-density regime. In Fig. 2 this is shown for two-dimensional aggregation, up to the finite size gel point (defined as the time when the largest cluster spans the lattice; this is not a true gel point). The effectively measured fractal dimension D steadily increases from the low density value (lower straight line) to the Euclidian value at high density (upper line). The change takes place over three decades in R; the slowness of the variation suggests that intermediate dimensions measured in experiments may correspond to an effective exponent in the crossover domain. The figure also shows the chemical distance which measures the tortuosity of the network. This slope barely changes, as the two limiting exponent values at low and at high density are almost identical. This exponent determines the electrical and mechanical properties of gels well beyond the gel point through the the properties of the elementary (fractal) blocks. Direct evidence of a scaling crossover behaviour is the particle density within the clusters (defined as the total number of particles divided by the volume of all the clusters, IZ,Rid, where R, is the radius of gyration and d the spatial dimension) as a function of the average radius. A scaling form implies that the reduced density pnorm (normalized by the initial density
380
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Fig. 3. Average density inside the clusters, normalized by the total initial density, plotted against the average cluster radius, normalized by the radius at which the clusters fill space, for different initial concentrations between 0.004 and 1.0. All curves superpose, indicating that a scaling crossover describes the intermediate concentration regime. The slope on the left corresponds to the co-dimension D-d of the low density cluster fractal dimension.
pO) as a function of the reduced cluster radius R,,,, (normalized by the radius where the clusters just fill all of space, R* = p6/("p")) should be for initial densities for independent of po. Figure 3 plots this function p. = 0.004-l. The data collapse confirms that a scaling crossover describes the change from low to high density aggregation. At low density the approach to the scaling function is indistinguishable, at the highest densities the scaling curve is approached differently; at the same time the scaling range shrinks. It is more difficult to see the crossover in the cluster size distribution as the range of concentrations required to measure it has to be larger than for the average cluster density. The scaled distribution at low and at very high concentrations nevertheless gives one an idea of the change. They are shown in Fig. 4, in three dimensions and for cx = 0, where the distribution is weakly poly-disperse at all densities. What is remarkable about the crossover towards high density-and what is of immediate relevance for gelation-is that the low density regime remains valid up to concentrations where the cluster radii are considerably larger than the width of the space separating neighbouring clusters.
381
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Fig. 4. Change of the normalized cluster size distribution with increasing effective concentration in three dimensions. In (a), for a low initial concentration of 0.01 the slope is close to T = 2, where one enters the strongly poly-disperse regime (percolation). In (b) the same type of scaling but with a lower value of 5 is observed for a very high initial concentration.
382
M. Kolb
REFERENCES 1. Carpineti, M. & Giglio, M., Phys. Rev. Left., 68 (1992) 3327. 2. Dietler, G., Aubert, C. & Cannell, D., Phys. Rev. Lett., 57 (1986) 3117. 3. Anglaret, E., Hasmy, A., Courtens, E., Pelous, J. & Vacher, R., Investigation of dynamical scaling in a mutually self-similar series of base-catalyzed aerogels. Europhys. Lett., 28 (1994) 591 and references therein. 4. Salome, L., Structure of percolating carbon black fractal aggregates dispersed in a polymer. J. Phys. Fr., 2 (1993) 1647. 5. Zrinyi, M., Kabai-Faix, M., Fuhos, S. & Horkay, F., Sedimentation and gelation of flocculated iron(II1) hydroxide gels. Langmuir, 9 (1993) 71. 6. Shih, W. -H., Shih, W. Y., Kim, S. -I., Liu, J. & Aksay, 1. A., Scaling behavior of the elastic properties of colloidal gels. Phys. Rev. A, 42 (1990) 4772. 7. Meakin, P., Formation of fractal clusters and networks by irreversible diffusion-limited aggregation. Phys. Rev. Left., 51 (1983) 1119. 8. Kolb, M., Botet, R. & Jullien, R., Scaling of kinetically growing clusters. Phys. Rev. Lett., 51 (1983) 1123. 9. Kolb, M. & Jullien, R., Chemically limited versus diffusion limited aggregation. J. Phys. Lett. (Paris), 45 (1984) L977. 10. Stauffer, D., Introduction to Percolation Theory. Taylor and Francis, London, 1985. 11. Kolb, M. & Herrmann, H. J., The sol-gel transition modeled by irreversible aggregation of clusters. J. Phys. A, 18 (1985) L435. 12. Kolb, M. & Herrmann, H., Surface fractals in irreversible aggregation. Phys. Rev. Lett., 59 (1987) 454. 13. Spinodal type dynamics in fractal aggregation of colloidal clusters.
Polymer Gels and Networks 4 (19%) 375-382 Copyright 0 1996 Published by Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0966.7822/96/$15.00 PII: SO%&7822(%)00025-l
Models for Irreversible
Gel Formation
M. Kolb Laboratoire Chimie ThCorique, Ecole Normale Supkrieure, 69364 Lyon, France and Institut de Recherches sur la Catalyse, 2, Ave. A.Einstein, 69626 Villeurbanne Cedex, France
ABSTRACT Many of the physical and chemical properties of gels are determined by the porosity of the underlying molecular network. The build-up of an infinite network-and even its existence-often depends on irreversible aggregation processes during the earliest stages of structure formation. The system gels only once the primary structures fill space. Aggregation and percolation models each describe one limiting situation of such a sol-gel process. The important intermediate stages cannot be treated with the same degree of generality. New results for this crossover regime are presented: it is shown that there is a smooth change of the structural properties and the cluster size distribution when passing from the aggregation to the gelation stage. The measured fractal dimension of the clusters changes continuously as one approaches the gelation transition. Copyright 0 1996 Published by Elsevier Science Ltd.
1 BASIC
MODELS
Irreversible aggregation describes the formation of elementary building blocks in one type of gel formation. Examples of such processes are the gelation of silica and more generally colloidal gels.‘-’ This class of models starts from the assumption that clusters are always perfectly rigid, that they move according to a diffusion law with a cluster size dependent diffusion coefficient, and that aggregates grow by forming permanent bonds via a strong short-range interaction between elementary blocks.‘-” One distinguishes two limiting cases: diffusion limited or reaction limited aggregation, depending on whether bonds are formed at the first contact or whether a potential barrier imposes many collisions before a bond can 375
376
M. Kolh
form. The first process is also called rapid aggregation, with the latter known as slow aggregation. The main characteristics of the structures built in this way are the following: individual clusters are fractal objects with a fractal dimension of D = 1.40-1.45 in two-dimensions and D = 1.75-1.80 in three-dimensions for fast, diffusion limited aggregation and D = 1.55-1.60, respectively, D = 2.0-2.1 for slow, reaction limited aggregation. In the former case the cluster size distribution is mono-disperse, which means that all clusters have roughly the same size. This holds for the physically relevant case when large clusters diffuse more slowly than small ones. For reaction limited aggregation the cluster size distribution is weakly poly-disperse, which means that it decays with a power law with an exponent less than two. These properties are valid irrespective of the details of the system under study (lattice structure, rotational motion of the aggregates, details of the interaction), but they do depend on the spatial dimension, the short-range nature of the interaction and the rigidity of the bonds of the aggregates. The rigidity of the clusters is of course a key assumption for the applicability of these models. The cluster diffusivity only has a marginal influence on the scaling properties, provided that large clusters move more slowly than small ones. Furthermore, aggregation models assume that the initial density is sufficiently low for cluster-cluster correlations to be neglected. However, because of the fractal nature of the aggregates the cluster density one gradually leaves the invariably increases with time. Therefore range of validity of the models. At the same time it is exactly this increase of the effective density that eventually makes gelation possible. Hence in the context of gelation one is particularly interested in this change of the aggregation process as the density grows. The resulting network may differ considerably from those predicted by percolation theory.“’ Within the framework of irreversible growth, the simplest possibility to obtain a space-filling network is to run the aggregation process beyond the low density limit. We will present numerical results on this changeover process. If one continues to aggregate beyond the point where the clusters fill space, the growth could continue in two possible ways. If the clusters are to stay fractal, they would interpenetrate each other more and more. This is an unlikely scenario as it would mean that neighbouring aggregates have an ever larger common interface and still manage to avoid touching each other. The other possibility is that clusters become non-fractal. The study of a model that starts out at very high density clearly shows that this is what happens. “.I2 The simulations show that the clusters are dense, but
Models for irreversible gel formation
377
have a fractally rough surface. The surface roughness depends on how mobile small clusters are in comparison with large clusters. Other modifications may of course also change the structure and the physical properties of a system that is about to gel. Depending on the type of bonding, one must relax the assumption of irreversible bonds. This leads, in its idealized form, to aggregation with random bond breaking. Because of the finite lifetime of the bonds the aggregation process reaches a steady state when the growth induced by the cluster collisions equals the fragmentation due to bond breaking. The clusters also restructure because of fragmentation and their fractal properties differ from the irreversible growth models, even at low densities. At very high densities the model closely resembles bond percolation. A second modification that needs to be included in a more realistic description of an aggregation-gelation process is the elasticity of large structures. Experimentally there is always a characteristic size beyond which the clusters are no longer rigid. Ideas from the scaling theory for polymers may then be applied, at least well beyond the gel point.’
2 SIMULATIONS
AT
CROSSOVER
CONCENTRATION
The results that we wish to discuss here are the change of the cluster structure as a consequence of the ever-increasing density. The model is standard cluster-cluster aggregation beyond the low density range: on a lattice single particles clusters are placed randomly with an initial density p. << 1. The clusters diffuse by jumping randomly to neighbouring sites. If a jump of a cluster leads to an overlap of one or more of its particles with other particles, the move is not accepted. Instead a permanent, rigid bond is formed between a pair of colliding particles. For several colliding pairs, one is chosen at random. The diffusion coefficient depends on the cluster size: the jump probability has scaling form, D = Y, where s is the number of particles in the cluster and (Y< 0, such that larger clusters diffuse more slowly than small ones. A physically reasonable value in three dimensions is (Y= -0.5. For reaction limited aggregation, a bond between neighbouring particles only forms after a large number of overlap attempts or after particles have spent much time on neighbouring sites. For the last model the choice of (Y has no influence on the aggregate structure and the size distribution. The simulations were performed at several intermediate initial concentrations, to show that the results can be fitted to a crossover scaling form.
37x
M. Kolh
3 RESULTS
AND
DISCUSSION
In Fig. 1 we show a snapshot of a typical situation of two dimensional growth for (Y= -0.5 and at an intermediate effective concentration where clusters are visibly highly correlated. Internally they appear homogeneous, and one notices the pronounced void spaces between the clusters. This may be compared with the experimental finding that during high density irreversible aggregation a characteristic length sets the scale for the growth, as for phase separation phenomena.” Another interesting observation comes from the mass vs radius relation
Fig. 1. Two-dimensional diffusive cluster aggregation at intermediate effective conccntralion. The correlation between clusters is clearly visible: empty “corridors” separate neighboring clusters.
Models for irreversible gel formation
379 1
:
1
10000 D
1000 100 10 1
r
031
O,l 1
10
100
1000
10000
R Fig. 2. Average cluster mass as a function of the average cluster radius, plotted logarithmically. The effective slope gradually changes from the value D = 1.43 for cluster aggregation at low concentration to the value D = 2 for dense structures. Also shown is the exponent characterizing the minimum (“chemical”) distance which is important for the mechanical properties of the structure. The dotted line is the probability p that the largest cluster spans the system.
as one enters the high-density regime. In Fig. 2 this is shown for two-dimensional aggregation, up to the finite size gel point (defined as the time when the largest cluster spans the lattice; this is not a true gel point). The effectively measured fractal dimension D steadily increases from the low density value (lower straight line) to the Euclidian value at high density (upper line). The change takes place over three decades in R; the slowness of the variation suggests that intermediate dimensions measured in experiments may correspond to an effective exponent in the crossover domain. The figure also shows the chemical distance which measures the tortuosity of the network. This slope barely changes, as the two limiting exponent values at low and at high density are almost identical. This exponent determines the electrical and mechanical properties of gels well beyond the gel point through the the properties of the elementary (fractal) blocks. Direct evidence of a scaling crossover behaviour is the particle density within the clusters (defined as the total number of particles divided by the volume of all the clusters, IZ,Rid, where R, is the radius of gyration and d the spatial dimension) as a function of the average radius. A scaling form implies that the reduced density pnorm (normalized by the initial density
380
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Fig. 3. Average density inside the clusters, normalized by the total initial density, plotted against the average cluster radius, normalized by the radius at which the clusters fill space, for different initial concentrations between 0.004 and 1.0. All curves superpose, indicating that a scaling crossover describes the intermediate concentration regime. The slope on the left corresponds to the co-dimension D-d of the low density cluster fractal dimension.
pO) as a function of the reduced cluster radius R,,,, (normalized by the radius where the clusters just fill all of space, R* = p6/("p")) should be for initial densities for independent of po. Figure 3 plots this function p. = 0.004-l. The data collapse confirms that a scaling crossover describes the change from low to high density aggregation. At low density the approach to the scaling function is indistinguishable, at the highest densities the scaling curve is approached differently; at the same time the scaling range shrinks. It is more difficult to see the crossover in the cluster size distribution as the range of concentrations required to measure it has to be larger than for the average cluster density. The scaled distribution at low and at very high concentrations nevertheless gives one an idea of the change. They are shown in Fig. 4, in three dimensions and for cx = 0, where the distribution is weakly poly-disperse at all densities. What is remarkable about the crossover towards high density-and what is of immediate relevance for gelation-is that the low density regime remains valid up to concentrations where the cluster radii are considerably larger than the width of the space separating neighbouring clusters.
381
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Fig. 4. Change of the normalized cluster size distribution with increasing effective concentration in three dimensions. In (a), for a low initial concentration of 0.01 the slope is close to T = 2, where one enters the strongly poly-disperse regime (percolation). In (b) the same type of scaling but with a lower value of 5 is observed for a very high initial concentration.
382
M. Kolb
REFERENCES 1. Carpineti, M. & Giglio, M., Phys. Rev. Left., 68 (1992) 3327. 2. Dietler, G., Aubert, C. & Cannell, D., Phys. Rev. Lett., 57 (1986) 3117. 3. Anglaret, E., Hasmy, A., Courtens, E., Pelous, J. & Vacher, R., Investigation of dynamical scaling in a mutually self-similar series of base-catalyzed aerogels. Europhys. Lett., 28 (1994) 591 and references therein. 4. Salome, L., Structure of percolating carbon black fractal aggregates dispersed in a polymer. J. Phys. Fr., 2 (1993) 1647. 5. Zrinyi, M., Kabai-Faix, M., Fuhos, S. & Horkay, F., Sedimentation and gelation of flocculated iron(II1) hydroxide gels. Langmuir, 9 (1993) 71. 6. Shih, W. -H., Shih, W. Y., Kim, S. -I., Liu, J. & Aksay, 1. A., Scaling behavior of the elastic properties of colloidal gels. Phys. Rev. A, 42 (1990) 4772. 7. Meakin, P., Formation of fractal clusters and networks by irreversible diffusion-limited aggregation. Phys. Rev. Left., 51 (1983) 1119. 8. Kolb, M., Botet, R. & Jullien, R., Scaling of kinetically growing clusters. Phys. Rev. Lett., 51 (1983) 1123. 9. Kolb, M. & Jullien, R., Chemically limited versus diffusion limited aggregation. J. Phys. Lett. (Paris), 45 (1984) L977. 10. Stauffer, D., Introduction to Percolation Theory. Taylor and Francis, London, 1985. 11. Kolb, M. & Herrmann, H. J., The sol-gel transition modeled by irreversible aggregation of clusters. J. Phys. A, 18 (1985) L435. 12. Kolb, M. & Herrmann, H., Surface fractals in irreversible aggregation. Phys. Rev. Lett., 59 (1987) 454. 13. Spinodal type dynamics in fractal aggregation of colloidal clusters.