Gelation cluster theory

Percolation theory describes [32] the random growth of molecular clusters on a d-dimensional lattice. It was suggested to possibly give a better description of gelation than the classical statistical methods (which in fact are equivalent to percolation on a Bethe lattice or Caley tree, Fig. 7a) since the mean-field assumptions (unlimited mobility and accessibility of all groups) are avoided [16,33]. In contrast, immobility of all clusters is implied, which is unrealistic because of the translational diffusion of small clusters. An important fundamental feature of percolation is the existence of a critical value pc of p (bond formation probability in random bond percolation) beyond which the probability of finding a percolating cluster, i.e. a cluster which spans the whole sample, is non-zero. 

All models described up to here belong to the class of equilibrium theories. They have the advantage of providing structural information on the material during the liquid-solid transition. Kinetic theories based on Smoluchowski s coagulation equation [45] have recently been applied more and more to describe the kinetics of gelation. The Smoluchowski equation describes the time evolution of the cluster size distribution N(k) ... 

Family F., and D. P. Landau, Kinetics of Aggregation and Gelation, North-Holland, Amsterdam, 1984. An excellent compendium of articles on cluster fractals—both theory and experiment. 

Theoretical and experimental treatments of gels go hand-in-hand. The former are covered first because they will help us understand gel point and other concepts. Two main theories have been used to interpret results of experimental studies on gels the classical theory based on branching models developed developed by Floiy and Stockmayer, and the percolation model credited to de Gennes. Gelation theories predict a critical point at which an infinite cluster first appears. As with other critical points, the sol-gel transition can be in general characterized in terms of a set of generally applicable (universal) critical exponents. 

It should be noted that this theory neglects loops or cyclic formations, and this affects the size distribution and other cluster properties. Some of these properties (and their relationships at gelation) are highlighted in Table 2.3 (Larson, 1999), as are their experimental values compared with the classical and three-dimensional-percolation theoretical values at gelation. 

Family [9] considered the conformations of statistical branched fractals (which simulate branched polymers) formed in equilibrium processes in terms of the Flory theory. Using this approach, he found only three different states of statistical fractals, which were called uncoiled, compensated, and collapsed states. In particular, it was found that in thermally induced phase transitions, clusters occur in the compensated state and have nearly equal fractal dimensions ( 2.5). Recall that the value df = 2.5 in polymers corresponds to the gelation point this allows gelation to be classified as a critical phenomenon. 

The first theory that attempted to derive the divergences in cluster mass and average radius accompanying gelation is that of Flory [52] and Stockmayer [53]. In their model, bonds are formed at random between adjacent nodes on an infinite Cayley tree or Bethe lattice (see Figure 47.7). The Flory-Stockmayer (FS) model is qualitatively successful because it correctly describes the emergence of an infinite cluster at some critical extent of reaction and... 

Although there are probably other universality classes, this transition was successfully modeled by bond percolation [6]. Generally, bond percolation on a lattice has each bond (line connecting two neighboring lattice sites) present randomly with probability p and absent with probability 1-p. Clusters are groups of sites connected by present bonds. For p > Pc zn infinite cluster is formed. Percolation theory (in a Bethe lattice approximation) was invented by Flory (1941) to describe gelation for three-functional polymers. 

The kind of mechanisms that lead to gelation characterised by infinite clusters are not clear. The infinite cluster contains of course a finite fraction G(t) of the total mass (M(t) + G(t) = 1). Pre-gel and post-gel states separated by a gelation transition can be analysed in terms of a kinetic equation. Sol-gel transitions are similar to phase transition phenomena. It is not surprising that scale invariance principles elaborated in the theory of phase transition can be adopted for polymer systems. Modern percolation theory (see, for example Stauffer (1979)) offer a conceptual framework to treat cluster formation. 

To derive a specific form of the equilibrium constants bi, let us introduce a simple model for the internal structure of clusters. Clusters are assumed to take a tree structure with no internal loops (Cayley tree). Cycle formation within a cluster is neglected. This is a crude approximation on the basis of the classical theory of gelation presented in Section 3.2 [5,6,7,8], but in fact it is known to work very weU at least in the pregel regime. 

The general features of structural evolution during gelation are described by percolation (or connectivity) theory, where one simply connects bonds (or fills sites) on a lattice of arbitrary dimension and coordination number (4-6). Figure 9.1 (6) illustrates a two-dimensional system at the gel point. It must be noted that gels at and just beyond the gel point usually coexist with sol clusters. These can also be seen in Figure 9.1. It is common to speak of the con-... 

The first models, describing elastic behavior of fractal structures, were used, as a rule, for simulation within the fimneworks of percolation theory [1-5], Nonhomogeneous statistical mixture of solid and liquid then only displays solid body properties (e g., not equal to zero shear modulus G), when solid component forms percolation cluster, like at gelation in pol5uner solutions. If liquid component there is replaced by vacuum, then bulk modulus. B will also be equal to zero below percolation threshold [1]. Such model gives the following relationship for elastic constants [1,3] ... 

It is clear that the exponent value is not related to the fractal dimension and to percolation theory. The structure and the connections of a gel are the result of a sequence of different processes gelation, aging and shrinkage. The a value should describe the way the clusters are connected between them and not the structure inside the clusters. 

A particularly readable review of the theory of small-angle scattering, with reference to the growth and gelation of fractal clusters, is... 

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