Gel point calculations

Compare the gel points calculated from the Carothers equation (and its modifications) with those using the statistical approach. Describe the effect of unequal functional groups reactivity (e.g., for the hydroxyl groups in glycerol) on the extent of reaction at the gel point. 

Fig. 5. Interrelation between critical concentrations of crosslink forming (a) and ring forming (a) functionalities at the gel point calculated on the basis of random flight statistics for trifunctional polycondensation (/ = 3) and different values of parameters A and 91 [Gordon and Scantleburry (72)]...

Fig. 6.16 Gel point calculation on a Bethe lattice. Each site that is already connected to the gel from its grandparent has/ - 1 possible additional connections (potential children). site is chosen as the starting point of this procedure makes no difference because they are all statistically identical for an infinite Bethe lattice. Let us assume that our starting ( parent ) site has already formed a bond with one of its neighbours (the grandparent site) as sketched in Fig. 6.16. We would like to calculate the average number of additional bonds the parent site forms with its/- 1 remaining neighbours (potential children ). The probability of each of these bonds being formed is p and is...

Compare the gel points calculated from the Carothers equation with those using the statistical approach of Floiy. 

Can the system be reacted to complete conversion without gelation If not, what is the extent of conversion of the acid functionality at the gel point calculated from (a) the Carothers equation and from the statistical approaches of (b) Flory-Stockmayer and (c) Macosko-Miller ... 

Compare the gel point calculated from the Carothers equation with that using the statistical approach of Flory for the following mixture phthalic anhydride, ethylene glycol, and glycerol in the molar ratio 1.50 0.70 0.50. 

HGURE 6 GEL POINT CALCULATION FOR A CROSSLINKED THIOL-NORBORNENE RESIN SYSTEM... 

It will be noted, on examining Fig. 5, that the cyclic portions of the polymer are incapable of cross-linking. In other words, approximately 40% of the polymer caimot undergo this reaction. This fact contributes to the deviation from the theoretical gel point calculation in the direction of gelation taking place at an unusually high degree of conversion [40]. 

The process is repeated for the "crosslinker" component (lines 1180-1950). From the calculated expectation values, the program determines the ratio of A to B groups (line 1960) and the gel point in the absence of B group self-condensation (percent reaction of A groups at gel, line 1990). The various expectation values along with the gel point and the ratio of crosslinker to polymer functionality (1/R), are printed at the terminal (lines 2000-2220). 

BELOW THE GEL POINT, THE PROGRAM CALCULATES THE WEIGHT AVERAGE" MOLECULAR WEIGHT. ABOVE THE GEL POINT, THE PROGRAM CALCULATES"... 

Dusek (1), Shy (2 and Bauer (3) give examples of modelling the structure-property relation of several networks (Tg, gel point, etc.). Examples are described of models on solvent evaporation, calculation of functionalities, molecular weight of resins (4), etc. 

The results of the experiment depicted in Fig. 62 and others similarly obtained are summarized in Table XXXI. In every case the observed gel point is reached at higher than the theoretical extent of reaction. The discrepancies between the observed and calculated ojc s appear to be due to the failure of the theory to take into account a minor degree of intramolecular condensation. Since some of the interunit linkages... 

During our early experiments on chemical gels, when first observing the intermediate state with the self-similar spectrum, Eq. 1-5, we simply called it viscoelastic transition . Then, numerous solvent extraction and swelling experiments on crosslinking samples showed that the viscoelastic transition marks the transition from a completely soluble state to an insoluble state. The sol-gel transition and the viscoelastic transition were found to be indistinguishable within the detection limit of our experiments. The most simple explanation for this observation was that both phenomena coincide, and that Eqs. 1-1 and 1-5 are indeed expressions of the LST. Modeling calculations of Winter and Cham-bon [6] also showed that Eq. 1-1 predicts an infinite viscosity (see Sect. 4) and a zero equilibrium modulus. This is consistent with what one would expect for a material at the gel point. 

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. 

In order to control the crosslinking reaction so that it can be used properly, it is important to understand the relationship between gelation and the extent of reaction. Two general approaches have been used to relate the extent of reaction at the gel point to the composition of the polymerization system—based on calculating when Xn and Xw, respectively, reach the limit of infinite size. 

The gel point is usually determined experimentally as that point in the reaction at which the reacting mixture loses fluidity as indicated by the failure of bubbles to rise in it. Experimental observations of the gel point in a number of systems have confirmed the general utility of the Carothers and statistical approaches. Thus in the reactions of glycerol (a triol) with equivalent amounts of several diacids, the gel point was observed at an extent of reaction of 0.765 [Kienle and Petke, 1940, 1941], The predicted values of pc, are 0.709 and 0.833 from Eqs. 148 (statistical) and 2-139 (Carothers), respectively. Flory [1941] studied several systems composed of diethylene glycol (/ = 2), 1,2,3-propanetricarboxylic acid (/ = 3), and either succinic or adipic acid (/ = 2) with both stoichiometric and nonstoichiometric amounts of hydroxyl and carboxyl groups. Some of the experimentally observed pc values are shown in Table 2-9 along with the corresponding theoretical values calculated by both the Carothers and statistical equations. 

As pointed out earlier, an elastomer cross-linked above its gel point will not dissolve in a solvent, but will absorb it and swell. The swelling will continue until the forces of swelling balance the retroactive forces of the extended chains of the network. The cross-link density can then be calculated from the degree of swelling using the Flory-Rehner equation ... 

Three common methods of measuring crosslinking (swelling, elastic modulus, and gel point measurements) have recently been critically appraised by Dole (14). A fourth method using a plot of sol + sol against the reciprocal dose has also been used extensively. However, Lyons (23) has pointed out that this relation, even for polyethylenes of closely random distribution, does not have the rectilinear form required by the statistical theory of crosslinking. Flory (19) pointed out many years ago that the extensibility of a crosslinked elastomer should vary as the square root of the distance between crosslinks. More recently Case (4, 5) has calculated that the extensibility of an elastomer is given by ... 

A study of benzocyclobutene polymerization kinetics and thermodynamics by differential scanning calorimetry (DSC) methods has also been reported in the literature [1]. This study examined a series of benzocyclobutene monomers containing one or two benzocyclobutene groups per molecule, both with and without reactive unsaturation. The study provided a measurement of the thermodynamics of the reaction between two benzocyclobutene groups and compared it with the thermodynamics of the reaction of a benzocyclobutene with a reactive double bond (Diels-Alder reaction). Differential scanning calorimetry was chosen for this work since it allowed for the study of the reaction mixture throughout its entire polymerization and not just prior to or after its gel point. The monomers used in this study are shown in Table 3. The polymerization exotherms were analyzed by the method of Borchardt and Daniels to obtain the reaction order n, the Arrhenius activation energy Ea and the pre-exponential factor log Z. Tables 4 and 5 show the results of these measurements and related calculations. 

The calculation of the gel point or the molecular weight distribution becomes difficult if the functional groups in monomers differ in reactivity (26, 27). Polycondensation of glycerol with symmetrical dibasic acid or of an unsymmetrical dibasic acid with symmetrical tribasic alcohol may serve as examples. The reaction probabilities of various functional group , necessary for the calculation of molecular weight distributions and gel points, have to be determined either from complete chemical analysis or from the kinetics of the respective reactions. 

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