Sol and gel fractions

Each site of the /-functional Bethe lattice has / possible paths to other sites (see Fig. 6.13). We define Q as the probability that a randomly selected site (A in Fig. 6.17) is nor connected to the gel through a certain randomly selected path (the path from site A to neighbouring site B in Fig. 6.17). There are immediately two possibilities. The bond between sites A and B could be unreacted with probability 1 - p, and if so this path cannot con-nect site A to the gel. The second possibility is that the bond between A and 

B is reacted with probability p. If this bond is formed, there are /- 1 remaining paths from site B that could connect to the gel. The probability that none of the/- 1 paths connect to the gel is and the probability that the bond between A and B is reacted and does not lead to the gel is pQ . Hence, there is a recurrence relation for the probability that a randomly selected site is not connected to the gel through a randomly selected potential bond (see Fig. 6.17)  

A recurrence diagram for the probability of not being connected to the gel through a given bond. 

The fraction of monomers in the sol (sol fraction) Psoi is the probability that a randomly selected site is not connected to the gel along any of its f paths (Fig. 6.18)  

There is always one solution of this equation [Eq. (6.42)] with zero gel fraction below gel point P o = 1 ) A second solution exists above the gel point for any / 3. For/— 3 this second solution of Eq. (6.51) can be easily found because it becomes a quadratic equation for 

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A simple mathematical description of the postgel stage will be presented for stepwise and free-radical chainwise polymerizations (in this case, the description will be limited to the range of low concentrations of the polyfunctional monomer leading to a homogeneous system). Calculations will be restricted to the evolution of sol and gel fractions, the mass fractions of pendant and elastic chains, and the concentration of crosslinks and EANC as a function of conversion. 

Network structures have been quantitatively determined by means of real-time XH NMR T2 relaxation experiments for several polymers [174-178]. The effect of the curing conditions on sol and gel fractions and the spatial heterogeneity of the network structure has been studied for polyethylene [174], polyacrylamide [175], PDMS [176], BR [177], epoxy resins [178] and EPDM [179]. 

Recently, a novel method for determining the microstructure of crosslinked polybutadiene in latex using solution 13C-NMR technique was reported [133]. The surfactant and polymer concentrations in the latex were adjusted to give a good signal resolution of the latex sample, as indicated by half-width of the resonance peak at 32.7 ppm. Under these conditions, the S/N ratio was almost identical to that of sample in solution, as shown in Figure 11.31. The microstructure of sol and gel fractions in a radical initiated polybutadiene, determined by this technique, was similar to that of solution measurements. 

At present, there are no physical methods to measure the concentration of amines of different types in networks and thus we cannot experimentally prove the computed values. However, the computed results seem reasonable since computer simulations give many features of the real behaviour of the systems under consideration. For example, the calculations gave kinetic curves of different reacting mixtures, sol and gel fractions, and equilibrium rubbery modulus. All results showed very good correlation with experiments 6 9,13,16,ly,31). This situation allows us to correlate the structural features of networks (for example, relative amounts of defects) obtained from computer simulations with macroscopic properties of the polymers. 

Above. . Tg,Tg must be related to the molecular weight of the sol fraction and the crosslink density of the gel fraction. If the system is considered to be a miscible binary mixture of sol and gel fractions, then... 

From these distribution functions, their moments related to sol and gel fractions and to various averages of the degree of polymerization [Eqs (6.45) and (6.46)] may be calculated. Results for functionality/= 3 are presented here. The sol fraction is defined as the fraction of all sites belonging to finite molecules... 

Calculate and plot sol and gel fractions for gelation of tetrafunctional monomers within mean-field theory, as functions of extent of reaction p. Hint Note that one of the solutions of Eq. (6.51) is Pjoi = 1 ... 

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. ... 

Instrumental analytical methods including HPLC, NMR and FT-IR have enabled the course of the reaction to be delineated by analysing the sol and gel fractions over time. In Section 1.2.1 the individual amine-epoxy reactions were presented, since the first stage of the reaction with a primary amine involves chain extension. This reaction competes with crosslinking since the reaction of the primary amine with epoxide is much faster than the reaction of the secondary amine. It is the latter reaction that results in branching of the chain and thus the formation of the first crosslinks. 

The overall crystallization kinetics of an unfractionated linear polyethylene, cross-linked by a peroxide reaction has also been studied. (95,96) A special feature of this work was the study of the separated sol and gel portions at different levels of cross-linking. The overall crystallization rates, in terms of the reciprocal of the half-time, l/ti/2, are plotted against the crystallization temperamre in Figs. 10.41 and 10.42 for a set of sol and gel fractions respectively. The gel fractions are characterized by the molecular weight between cross-links. Me, assuming ideal network formation. The sol portions are defined by their number average molecular... 

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