Gelation/gels percolation theory

The Soxhlet extraction method discussed in Section 6.6 can be used to separate the sol and gel fractions of a gel in the gelation regime, allowing direct determination of the gel fraction gel- Percolation theory expects the molar mass of a network strand M to be the same as the characteristic molar mass in the sol fraction. Hence, M can be determined by the size exclusion chromatography methods of Section 6.6, applied to the sol fraction. Equation (7.93) is tested in Fig. 7.19, where the shear modulus is shown to be proportional to Pgei/M. ... 

Statistical network models were first developed by Flory (Flory and Rehner, 1943, Flory, 1953) and Stockmayer (1943, 1944), who developed a gelation theory (sometimes referred to as mean-field theory of network formation) that is used to determine the gel-point conversions in systems with relatively low crosslink densities, by the use of probability to determine network parameters. They developed their classical theory of network development by considering the build-up of thermoset networks following this random, percolation theory. 

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. 

This chapter studies the local and global structures of polymer networks. For the local structure, we focus on the internal structure of cross-Unk junctions, and study how they affect the sol-gel transition. For the global structure, we focus on the topological connectivity of the network, such as cycle ranks, elastically effective chains, etc., and study how they affect the elastic properties of the networks. We then move to the self-similarity of the structures near the gel point, and derive some important scaling laws on the basis of percolation theory. Finally, we refer to the percolation in continuum media, focusing on the coexistence of gelation and phase separation in spherical coUoid particles interacting with the adhesive square well potential. 

In these expressions, tg is the reaction time at gel point, s and t are the static scaling exponents which describe the divergence of the static viscosity, nO 6 , at trstatic elastic modulus. Go at tggelation mechanism has been discussed on the basis of several models based on the percolation theory (for review, see ref 16), that provide power laws for the divergence of the static viscosity and the elastic moduli. Characteristic values for the s, t and A exponents are predicted by each of these models (Table I). 

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. 

As discussed briefly in the previous section, gelation can generally be discussed within the framework of critical phenomena [7] by having the gel point and critical point correspond. Stauffer applied the percolation theory often used for the general theory of critical phenomena to the crosslinking reaction of polymers [8, 9],... 

The best test of the validity of percolation theory as a description of gelation is its power to predict the behavior of the properties near the gel point (Pc)> so we now examine those predictions in some detail. 

The interaction parameter B, proportional to the intrinsic viscosity, is a function of the distance to the gelation threshold. This is an evidence of the swelling of the clusters. If in the expression of B(Eqs. 8 and 16) we insert the D and o (= (3-x)/y = 0.47 + 0.04) values determined experimentally, we find D = 2.42 + 0.15 in perfect agreement with the percolation theory (D = 2.5). This fractal dimension, which corresponds to the unswollen state is larger than that measured in the swollen state (D = 1.98 + 0.03). The e independence of the ratio [ql/B, means that the fractal dimension D of clusters of polyurethane is identical in the two solvents used (THF and dioxane). Due to this swelling, exponent v linking size to the distance to the gel point cannot be directly compared with the percolation exponent value v = 0.88. 

Gel phase fraction was measured on samples (see Ref. 14 for more details) formed beyond the gel point. One can see in Fig. 8 that G is a linear function of the stoichiometric ratio. Concerning this figure two comments have to be made. The p value corresponding to the stoichiometric ratio at which the gel phase is nul (p = 0.5639) is very different from the ifferent batches of monomers were used for the preparation of the two series of samples. A small but finite gel phase was measured below p (G = 1.6 10 at p = 0.5632) this could be due to either experimental imprecision on G or the fact that this sample prepared near the gel point contains very large polymer clusters of finite size which could not pass through the membrane used for the sol extraction. This result, G P is a linear function of p with a P-exponent value deduced from x and y exponent values measured below the gel point, indicates that below and beyond the gelation threshold, the percolation theory is well adapted to describe the properties linked to connectivity of polymer clusters formed by polycondensation. 

The static theories of equilibrium gelation (percolation theory) and vulcanization (Flory-Stockmayer theory) are well known. But many of the interesting experiments on polymer sols and gels concern dynamics viscoelasticity), which are less universal... 

In Section C.I.V, it was shown that, below the gel point, the viscosity diverges as Tj K Ap" . Using the percolation theory, the exponent k = 0.7 if an analogy is made between gelation viscosity and electric conductance of a random network of superconductors and normal conductors ... 

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. 

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