Gel-point Conversion

Gel-point Conversion Let us represent schematically 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)) 

Let be the event that — is the start of a finite chain then from Eq. (5.89) it follows that (Macosko and Miller, 1976)  

Roots between 0 and 1 for higher / are, however, easy to find numerically. 

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

---

In the foregoing considerations, formation of elastically inactive cycles and their effect have not been considered. For epoxy networks, the formation of EIC was very low due to the stiffness of units and could not been detected experimentally the gel point conversion did not depend on dilution in the range 0-60% solvent therefore, the wastage of bonds in EIC was neglected. For polyurethanes, the extent of cyclization was determined from the dependence on dilution of the critical molar ratio [OH] /[NCO] necessary for gelation (25) and this value was used for the statistical calculation of the fraction of EIC and its effect on Ve as described in (16). The calculation has shown that the fraction of bonds wasted in EIC was 2-2.5% and 1.5-2% for network from LHT-240 and LG-56 triols, respectively. 

One can see that for a polymer obtained from BA2 at aB = 0.95, ( Af)2 =191 while (Af)n = 19. If this polymer is crosslinked with a bifunctional crosslinking agent C2 under stoichiometric conditions, the gel point conversion is about 0.07. However, the gel point conversion is expected to be somewhat higher not only because of the lower polydispersity of the hyperbranched polymer, but also because some cyclization can occur or multiple crosslinks can be formed during crosslinking. 

In our previous article (14) the gelation in the polymerization of diallyl aromatic dicarboxylates has been experimentally examined in detail and discussed according to Gordon s theory (15). The discrepancy between actual and theoretical gel point conversion was quite large. In this connection, the gelation behavior of glycol bis(allyl phthalate)s was explored in detail from both experimental and theoretical standpoints. 

Table I summarizes the results obtained the actual gel-point conversions increased with an increase in the molecular weight of monomer. On the contrary, the discrepancy between actual and theoretical gel-point conversions was notably reduced from DAP to tetraethylene glycol bis(allyl phthalate)(n=3). This may be ascribed to the reduction of excluded volume effects on crosslinking (16,17) with the increase of the molecular weight of monomers, although the details of excluded volume effects on crosslinking will be discussed elsewhere.

Thus, in order to interpret the above correlations between mechanical properties and cocrosslinkers, the function of glycol bis(allyl phthalate)s as cocrosslinkers on the polymerization process of DAP beyond the gel-point conversion should be considered in connection with the microgel and, moreover, macrogel formation. Here it should be recalled that PEGBAP showed a rather drastic effect ... 

Among them is the gel point conversion, if multifunctional units are present, as well as accompanying divergence of viscosity, onset of equilibrium elasticity modulus, etc. By comparing the results of modeling with experiment, one can verify to what extent the chemistry is affected by physical interactions which are practically always active in polymerizations. 

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. 

Another proof against inhomogeneous cure in simple epoxy-amine and other systems has been supplied by gel point measurements. The critical conversion at the gel point (cf. Sect. 4) is a sensitive function of any inhomogeneity. For epoxy-amine systems, the gel point conversion has been found to agree well with the prediction of the theory assuming uniform distribution of reactive groups throughout the volume The deviation does not exceed 1 %. In contrast, for free-radical... 

The critical excess for amino groups is more sensitive to the substitution effect within the amino group than the gel point conversion of a stoichiometric system and it is thus more suitable for characterization of q (Fig. 6). In this way, the value of q was found to be 0.33-0.40 for the aliphatic amino group and 0.18-0.24 for the aromatic group in DDM (Refs. 16 and 18 and impublished measurements). These values are close to these obtained in model reactions of compounds of low functionality. The determination of the critical molar ratio necessary for gelation i.e. the... 

This type of microheterogeneous copolymerization of DAP with vinyl monomers having long-chain alkyl groups was applied further for the bulk copolymerization systems to obtain direct evidence to support the idea of the microheterogeneity of the systems beyond the gel-point conversion [82]. Also, the solution copolymerization of DAT was explored to demonstrate the incompatibility of the initially obtained precopolymer with a high content of comonomer units with DAT-enriched polymer chains [83]. 

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. 

Note that Eq. (5.148) gives the gel-point conversion pc of the functional group which is the limiting reactant. The conversion of the other functional group is rpc, where r is the mole ratio of the two functional groups, such that r < 1. 

Problem 5.34 Calculate the gel-point conversions for the systems cited in Problem 5.27 using the recursive approach for comparison with the corresponding values calculated according to Flory-Stockmayer theory. 

A polyesterification batch consists of tricarballylic acid (1 mol), adipic acid (4 moles), and diethylene glycol (5.5 moles). Calculate (a) the gel point conversion and (b) the weight fractions of polymeric mixture containing, respectively, 1, 3, and 5 chains per molecule at this conversion. 

Recalculate the gel point conversion of the polymerization system of Exercise 5.22 by the simpler recursive approach and compare the results. Also calculate the M of the polymer formed at 50% conversion of the A groups. Assume that the molecular weights of all the monomers in the polymerization system are equal and have a value of 100. 

An alternative expression to Eq. (7.92) for the gel point conversion can be derived in terms of the reactivity ratios of the two types of vinyl groups in A and BB. Let represent the relative reactivity of an A monomer and a B monomeric group, when reacting with a free radical of type A, and r2, the relative reactivity of a B group and an A group, when reacting with a B-type radical, i.e.,... 

---