Statistics of Network Formation

Theoretically, if each molecule in a polymer sample were to be linked to two of its neighbors, a single highly branched molecule would form that would encompass the whole sample. In practice, due to the statistical distribution of chain lengths and the random incorporation of crosslinks, the situation is far more complex. 

If we assume a random distribution of crosslinking events, the average number of crosslinks along the length of a chain is proportional to its length. Thus, we vould expect a chain with 

000 atoms in the backbone to have twice the number of crosslinks as a chain with only 

For a given initial molecular weight distribution, the weight fraction of chains incorporated into a network increases as the number of crosslinking events increases. Similarly, the number of crosslinking events required to create a network decreases as the average molecular weight increases. These relationships are illustrated schematically in Fig, 5.16. 

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The kinetics of free radical linear polymerizations has been thoroughly studied [74] and the relationships between molecular weight distribution and polymerization conditions are well known. Gels are made by incorporating a small fraction of bi-functional or multifunctional monomers that becomes part of more than one kinetic chain so that a network forms. The statistics of network formation are also well known. 

Several theories of network formation have been developed in the past half century, including statistic [50,64-66,138-144] and kinetic ones [100,145-153], and... 

It is important to note that this statistical calculation is only vaHd as long as the kinetics of network formation is totally controlled by the reactivity between the precursor monomers. With the formation of an infinite network at gelation and corresponding increase in viscosity, the reaction is slowed down considerably. Consequently, Eq. (15) is only valid prior to gelation. 

The classical approach to crosslinking statistics, as especially developed by Flory and Stockmayer, is capable of treating the simpler type of network formation processes, but becomes mathematically intractable in the more complicated cases (unequal reactivities, cyclization). Of the various recently proposed alternatives, the link probability generating function approach of Gordon et al., which is based upon the theory of... 

The description of a network structure is based on such parameters as chemical crosslink density and functionality, average chain length between crosslinks and length distribution of these chains, concentration of elastically active chains and structural defects like unreacted ends and elastically inactive cycles. However, many properties of a network depend not only on the above-mentioned characteristics but also on the order of the chemical crosslink connection — the network topology. So, the complete description of a network structure should include all these parameters. It is difficult to measure many of these characteristics experimentally and we must have an appropriate theory which could describe all these structural parameters on the basis of a physical model of network formation. At present, there are only two types of theoretical approaches which can describe the growth of network structures up to late post-gel stages of cure. One is based on tree-like models as developed by Dusek7 I0-26,1 The other uses computer-simulation of network structure on a lattice this model was developed by Topolkaraev, Berlin, Oshmyan 9,3l) (a review of the theoretical models may be found in Ref.7) and in this volume by Dusek). Both approaches are statistical and correlate well with experiments 6,7 9 10 13,26,31). They differ mainly mathematically. However, each of them emphasizes some different details of a network structure. 

Although the major interest in experimental and theoretical studies of network formation has been devoted to elastomer networks, the epoxy resins keep apparently first place among typical thermosets. Almost exclusively, the statistical theory based on the tree-like model has been used. The problem of curing was first attacked by Japanese authors (Yamabe and Fukui, Kakurai and Noguchi, Tanaka and Kakiuchi) who used the combinatorial approach of Flory and Stockmayer. Their work has been reviewed in Chapter IV of May s and Tanaka s monograph Their experimental studies included molecular weights and gel points. However, their conclusions were somewhat invalidated by the fact that the assumed reaction schemes were too simplified or even incorrect. It is to be stressed, however, that Yamabe and Fukui were the first who took into account the initiated mechanism of polymerization of epoxy groups (polyetherification). They used, however, the statistical treatment which is incorrect as was shown in Section 3.3. 

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. 

Macosko and Miller (1976) and Scranton and Peppas (1990) also developed a recursive statistical theory of network formation whereby polymer structures evolve through the probability of bond formation between monomer units this theory includes substitution effects of adjacent monomer groups. These statistical models have been used successfully in step-growth polymerizations of amine-cured epoxies (Dusek, 1986a) and urethanes (Dusek et al, 1990). This method enables calculation of the molar mass and mechanical properties, but appears to predict heterogeneous and chain-growth polymerization poorly. 

With the advancement of a curing reaction, the of the resin will increase, but the goal is to quantitatively predict the of a resin as a function of cure conversion. Several models have been proposed to correlate the with the conversion or extent of curing (a). With the increase in conversion, the concentration of reactive functionalities decreases, and crosslinks or junction points are formed, leading to the departure from Gaussian behaviour. Steric hindrance affects chain conformation at high crosslink densities. The models are based on the statistical description of network formation and calculation of the concentration of jrmction points of different functionalities as a function of conversion. However, one issue that complicates the calculation and which is not fully resolved is whether to consider all the junction points or only those which are elastically effective. 

In order to obtain the expression of network formation, we need to consider the probability of continuing paths from a randomly chosen group to a statistically equivalent group for example, a path from Bi to B in Figure 13. Let the fractions of A and B groups be Xa and Xb respectively, then the probability of continuing path, yean be given as below ... 

The elastic properties of rubbers depend on the network molecular structure and on the microscopic response of this structure to a macroscopic constraint. Studies of network formation by computer simulation,sol-gel statistics or determination of network structure by extraction and random degradation may help to analyze results of mechanical testing. Generally, the probability parameters involved in the statistical theories of network formation are determined by comparison of theoretical predictions with experimental values of the sol fraction. ... 

O Saito. Statistical aspects of infinite network formations. Polym Eng Sci 19 234-235, 1979. 

Despite its simplicity, the statistical method has been quite successful in predicting the effect of various chemical variables on network formation (cf. e.g. [29, 30, 34-37]). Since the internal structure of the gel can be characterized to a certain degree by the statistical method (e.g. average size of dangling chains and weight fraction of material in them), these methods offer a basis for correlations between structure and viscoelastic properties. 

Figure 5.9 Difference in concepts of statistical (A) and kinetic (B) network formation theories...

Chapter II describes the various kinds of network structures which may exist before or arise during its formation. Of these, the various network defects resulting from the crosslinking statistics, have received far more attention in the literature than the effects of inhomogeneous network formation and syneresis (separation in a gel + diluent phase). This is reflected in this review (Chapter II, section 2) although a special emphasis is also laid on the latter aspect (Chapter II, section 3 and 4). 

Valuable information on whether the structure is homogeneous or inhomogeneous can also be obtained by analyzing the network formation process. A shift of experimental and estimated statistical parameters (Mw, gel point conversion, sol fraction, etc.) will be observed if inhomogeneities are formed as a result of the crosslinking process. 

So far the micro-mechanical origin of the Mullins effect is not totally understood [26, 36, 61]. Beside the action of the entropy elastic polymer network that is quite well understood on a molecular-statistical basis [24, 62], the impact of filler particles on stress-strain properties is of high importance. On the one hand the addition of hard filler particles leads to a stiffening of the rubber matrix that can be described by a hydrodynamic strain amplification factor [22, 63-65]. On the other, the constraints introduced into the system by filler-polymer bonds result in a decreased network entropy. Accordingly, the free energy that equals the negative entropy times the temperature increases linear with the effective number of network junctions [64-67]. A further effect is obtained from the formation of filler clusters or a... 

The contributions of Vulpiani s group and of Kaneko deal with reactions at the macroscopic level. The contribution of Vulpiani s group discusses asymptotic analyses to macroscopic reactions involving flows, by presenting the mechanism of front formation in reactive systems. The contribution of Kaneko deals with the network of reactions within a cell, and it discusses the possibility of evolution and differentiation in terms of that network. In particular, he points out that molecules that exist only in small numbers can play the role of a switch in the network, and that these molecules control evolutionary processes of the network. This point demonstrates a limitation of the conventional statistical quantities such as density, which are obtained by coarse-graining microscopic quantities. In other words, new concepts will be required which go beyond the hierarchy in the levels of description such as micro and macro. 

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