Networks with crosslink densities

Some further remarks concerning the Mooney-Rivlin equation are in place (14, 112). In dry rubber networks Ca in extension is often of the same order of magnitude as Cx, so that we are by no means confronted with a minor correction. In unilateral compression C2 is almost zero, and perhaps slightly negative. The constant Cx increases with the crosslinking density and with the temperature the ratio C2/C( in extension seems... 

As mentioned, the introduction of monoamine in the network induces a decrease of the crosslink density, associated with the presence of pending groups. 

Figure 17. The birefringence. An, as function of temperature for differently crosslinked networks, = weak, A = strong crosslinking density (reproduced with permission from [11]).

Fluorosilicones consist of PDMS backbones with some degree of fluoro-aliphatic side chains. The fluorinated group can be trifluoropropyl, nonafluorohexylmethyl, or fluorinated ether side group [78,28,79]. These polymers differ not only in substituent group, but also in the amount of fluoro-substitution relative to PDMS, the overall molecular weight and crosslink density, and the amount of branching. In most commercially available cases, these polymers are addition cure systems and the reactions are those discussed previously for silicone networks. 

To determine the crosslinking density from the equilibrium elastic modulus, Eq. (3.5) or some of its modifications are used. For example, this analysis has been performed for the PA Am-based hydrogels, both neutral [18] and polyelectrolyte [19,22,42,120,121]. For gels obtained by free-radical copolymerization, the network densities determined experimentally have been correlated with values calculated from the initial concentration of crosslinker. Figure 1 shows that the experimental molecular weight between crosslinks considerably exceeds the expected value in a wide range of monomer and crosslinker concentrations. These results as well as other data [19, 22, 42] point to various imperfections of the PAAm network structure. 

Returning to the evaluation of the SAH network parameters, it should be noted that the crosslinking densities obtained from the modulus and swelling data agree satisfactorily with each other [22]. Analysis of the data from Refs. [18,90] confirms this conclusion. 

Crosslinking resoles in the presence of sodium carbonate or potassium carbonate lead to preferential formation of ortho-ortho methylene linkages.63 Resole networks crosslinked under basic conditions showed that crosslink density depends on the degree of hydroxymethyl substitution, which is affected by the formaldehyde-to-phenol ratio, the reaction time, and the type and concentration of catalyst (uncatalyzed, with 2% NaOH, with 5% NaOH).64 As expected, NaOH accelerated the rates of both hydroxymethyl substitution and methylene ether formation. Significant rate increases were observed for ortho substitutions as die amount of NaOH increased. The para substitution, which does not occur in the absence of the catalyst, formed only in small amounts in the presence of NaOH. 

It is possible to calculate a number of different kinds of "effective" crosslink densities. Bauer et al have used a quantity they termed the "elastically effective crosslink density " (Cel) correlate cure with solvent resistance and other physical properties of coatings (7-10). The correlation was basically empirical. Formally, the is a calculation of the number of functional groups attached to the infinite network for which there are at least two other paths out to the network on the given polymer or crosslinker. Thus, chains with only one or two paths to the infinite network are excluded. The following expression can be written for... 

Table II shows Tgs obtained from DSC traces. (Footnotes a and b in Table II show T s values of three reference polymers two PIBs, whose Mns are similar to the Mns of MA-PIB-MA used in the network synthesis, and a PDMAAm the difference in the Tg for the Mn=4,000 and 9,300 PIBs is due to the dependence of Tg on Mn(72)). The DSC traces of the networks exhibited two Tgs, one in the range of -63 to -52 °C (PIB domains) and another in the range of 90 to 115 °C (PDMAAm domains) indicating microphase separated structures. The Tgs associated with the PIB phase in the PDMAAm-1-PIB networks were higher than those of the reference homoPIBs which may be due to PIB chain-ends embedded in the glassy PDMAAm phase restricting segmental mobility. The Tg of the PIB phase in the PDMAAm-1-PIB increases by increasing the PIB content which may be due to an increase in crosslink density. In contrast, the Tg for the PDMAAm phase in the network decreases upon increasing the PIB content. Interaction of the (-CH2-CH-) moiety of the PDMAAm with the flexible PIB and thus the formation of a more flexible structure may explain this phenomenon.

Table II shows Tg data obtained from DSC traces of the PHEMA-1 -PIB networks. The traces showed two Tgs indicating microphase separation into PHEMA and PIB domains. The presence of the PHEMA Tg at - 110°C indicates complete desilylation of all networks. The Tgs for the reference PIBs (see footnote a in Table II) are lower than the Tgs of the PIB incorporated into the network. This may be due to the flexible PIB chain-ends embedded in the glassy PHEMA matrix. The increase in the Tg of the PIB phase in the network with increasing % PIB is most likely due to an increase in crosslink density.

In order to answer these questions, the kinetic and network structure models were used in conjunction with a nonlinear least squares optimization program (SIMPLEX) to determine cure response in "optimized ovens ". Ovens were optimized in two different ways. In the first the bake time was fixed and oven air temperatures were adjusted so that the crosslink densities were as close as possible to the optimum value. In the second, oven air temperatures were varied to minimize the bake time subject to the constraint that all parts of the car be acceptably cured. Air temperatures were optimized for each of the different paints as a function of different sets of minimum and maximum heating rate constants. 

In order to enable these fluctuations to occur, the network chains are assumed to be "phantom" in nature i.e. their material properties are dismissed and they act only to exert forces on the junctions to which they are attached. With common networks having tetrafunctional junctions, the results of the two approaches differ by a factor of two. Identical results are only obtained from both theories, when the functionality is infinite. From a practical viewpoint, however, a value of about 20 for f can already be equated to infinity because crosslink densities can hardly be obtained with an accuracy better than 10%. 

Hence the measurement of this ratio for networks with different functionalities can serve to test the theory. Note that should be independent of temperature (this is confirmed by measurements) and also independent of crosslink density (however, a decrease of CJz with increasing v is found experimentally (10,34-36) 1... 

In Figure 6, these data are plotted versus the branching density z of crosslinking molecules. Gy./G is fairly independent of network microstructure. It covers a range of 0.24 to 0.32 as a result of statistical scattering, averaging to 0.28 as in case of the networks with tetrafunctional crosslinks. 

For imperfect epoxy-amine or polyoxypropylene-urethane networks (Mc=103-10 ), the front factor, A, in the rubber elasticity theories was always higher than the phantom value which may be due to a contribution by trapped entanglements. The crosslinking density of the networks was controlled by excess amine or hydroxyl groups, respectively, or by addition of monoepoxide. The reduced equilibrium moduli (equal to the concentration of elastically active network chains) of epoxy networks were the same in dry and swollen states and fitted equally well the theory with chemical contribution and A 1 or the phantom network value of A and a trapped entanglement contribution due to the similar shape of both contributions. For polyurethane networks from polyoxypro-pylene triol (M=2700), A 2 if only the chemical contribution was considered which could be explained by a trapped entanglement contribution. 

---