Post-gel Properties

Gel-point Conversion Let us represent schematieally the stepwise homopolymerization of A/ by 

Assume that polymerization has proceeded to an extent that some fraction p of the A s has reacted. Picking an A group at random (say, A in Eq. (5.150)) we ask What is the probability 

Physically, = 1 signi es that the system has not yet gelled. In order to derive post-gel 

Roots between 0 and 1 for higher / are, however, easy to nd munerically. 

Problem 5.36 Starting with Eq. (5.153) for the simple Af (J 2) homopolymerization, derive a relation between the critical extent of reaction pc for gelation and the monomer functionality /. 

The recursive method of Macosko and Miller (1976) has been described earlier for calculating molecular weight averages up to the gel point in nonlinear polymerization. A similar recursive method (Langley, 1968 Langley and Polman-teer, 1974) can also be used beyond the gel point, particularly for calculating weight-fraction solubles (sol) and crosslink density. To illustrate the principles, we consider first the simple homopolymerization, that is, reaction between similar /-functional monomers A/ and then a more common stepwise copolymerization, such as reaction of A/ with B2. 

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The theories of Miller and Macosko are used to derive expressions for pre-gel and post-gel properties of a crosslinking mixture when two crosslinking reactions occur. The mixture consists of a polymer and a crosslinker, each with reactive functional groups. Both the polymer and crosslinker can be either collections of oligomeric species or random copolymers with arbitrary ratios of M /Mj. The two independent crosslinking reactions are the condensation of a functional group on the polymer with one on the crosslinker, and the self-condensation of functional groups on the crosslinker. 

In this paper a simple BASIC program for calculating pre-gel and post-gel properties of thermoset coatings has been presented. The program is based on the work of Miller and Macosko and has been extended to incorporate two independent crosslinking reactions. In... 

Small variations in the conversion of A in the first stage (but a complete conversion of h in stage 2) have a very significant impact on the characteristics of P2 and on the pre- and post-gel properties in stage 3. This is exemplified in Figures 12-14. 

Michov, BM, Radically Simplifying the Henry Eunction, Electrophoresis 9, 199, 1988. Miller, DR Macosko, CW, A New Derivation of Post Gel Properties of Network Polymers, Macromolecules 9, 206, 1976. 

DR Miller, CW Macosko. A new derivation of post gel properties of network polymers. Macromolecules 9 206-211, 1976. 

The same remarks apply to the kinetic methods and Smoluchoswki-type approach. It is possible to calculate general post gel properties from the Smoluchowski equation, but only for simple kernels Kij, which are often too simple for the real world for each model one obtains differences in the behaviour. Ilavsky and Dusek gave an example for the calculation of the trapping factor and modulus from the kinetic method by a stepwise polyaddition involving cyclization to first order. Comparison with experiments shows strong deviations and a realistic picture of gelation-structure-elasticity relationships seems to be a still open question. ... 

The post-gel Miller-Macosko derivation determines network properties by first calculating the probability that looking out from a A group is a finite chain, P(F ° ). This probability is equal to the probability that A has not reacted (1-a) plus the probability that A has reacted times the probability that looking in to a B group is finite ... 

As might be expected, these differences in the pre-gel properties are also reflected in the post-gel regime. The sol fraction varies more smoothly with conversion for the branched prepolymer compositions, cf. Figures 3 and 6. But because the gel point is at (so... 

It is shown that model, end-linked networks cannot be perfect networks. Simply from the mechanism of formation, post-gel intramolecular reaction must occur and some of this leads to the formation of inelastic loops. Data on the small-strain, shear moduli of trifunctional and tetrafunctional polyurethane networks from polyols of various molar masses, and the extents of reaction at gelation occurring during their formation are considered in more detail than hitherto. The networks, prepared in bulk and at various dilutions in solvent, show extents of reaction at gelation which indicate pre-gel intramolecular reaction and small-strain moduli which are lower than those expected for perfect network structures. From the systematic variations of moduli and gel points with dilution of preparation, it is deduced that the networks follow affine behaviour at small strains and that even in the limit of no pre-gel intramolecular reaction, the occurrence of post-gel intramolecular reaction means that network defects still occur. In addition, from the variation of defects with polyol molar mass it is demonstrated that defects will still persist in the limit of infinite molar mass. In this limit, theoretical arguments are used to define the minimal significant structures which must be considered for the definition of the properties and structures of real networks. 

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. 

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