Cross-linking relationships

By combining random flight statistics from Chap. 1 with the statistical definition of entropy from the last section, we shall be able to develop a molecular model for the stress-strain relationship in a cross-linked network. It turns out to be more convenient to work with the ratio of stretched to unstretched lengths L/Lq than with y itself. Note the relationship between these variables ... 

Model Networks. Constmction of model networks allows development of quantitative stmcture property relationships and provide the abiUty to test the accuracy of the theories of mbber elasticity (251—254). By definition, model networks have controlled molecular weight between cross-links, controlled cross-link functionahty, and controlled molecular weight distribution of cross-linked chains. Sihcones cross-linked by either condensation or addition reactions are ideally suited for these studies because all of the above parameters can be controlled. A typical condensation-cure model network consists of an a, CO-polydimethylsiloxanediol, tetraethoxysilane (or alkyltrimethoxysilane), and a tin-cure catalyst (255). A typical addition-cure model is composed of a, ffl-vinylpolydimethylsiloxane, tetrakis(dimethylsiloxy)silane, and a platinum-cure catalyst (256—258). 

This equation is not particularly useful in practice, since it is difficult to quantify the relationship between concentration and ac tivity. The Floiy-Huggins theory does not work well with the cross-linked semi-ciystaUine polymers that comprise an important class of pervaporation membranes. Neel (in Noble and Stern, op. cit., pp. 169-176) reviews modifications of the Stefan-Maxwell approach and other equations of state appropriate for the process. 

Further improvements in the properties of PAN-based carbon fibers are likely to emerge through improved stabilization, that is, by creating the ideally cross-linked fiber. On the other hand, as purer pitch precursors become available, further improvements in mesophase pitch-based carbon fibers are likely to arise from optimized spinnerette designs and enhanced understanding of the relationship between pitch chemistry and its flow/orientation behavior. Of course, the development of new precursors offers the potential to form carbon fibers with a balance of properties ideal for a given application. 

The viscosity of most adhesives increases with time as they set by cross-linking, cooling from the melt or loss of solvent. The cross-linking of a phenolic-polyvinyl formal adhesive and of cold-setting epoxies was found by de Bruyne [41] to be represented adequately by an exponential relationship ... 

It is also possible to estimate the cross-link density from the stress-strain data, using the statistical theory of rubber-like elasticity [47,58]. For a swollen rubber the relationship is... 

It has previously been found experimentally [22], albeit for a different elastomer (a synthetic polyisoprene), that to a good degree of approximation there is a linear relationship between tan A and average inter-cross-link molecular weight, IVE, such that tan A oc Me. We therefore write ... 

The results of the application of these relationships to the cross-linking of a rectangular primary distribution consisting of equal parts by weight of every species y = l to 1000 (i.e., = 0.001 for y = l to... 

These simplified relationships offer a clearer insight into the dependence of the equilibrium swelling ratio qm on the quality of the solvent as expressed by Xh on the extent of cross-linking. Because of the nature of the approximations introduced to obtain Eqs. (40) and (40 ), their use as quantitative expressions must be limited to networks of very low degrees of cross-linking in good solvents. 

It has been shown in Chapter XI that the force of retraction in a stretched network structure depends also on the degree of cross-linking. It is possible therefore to eliminate the structure parameter ve/Vo) by combining the elasticity and the swelling equations, and thus to arrive at a relationship between the equilibrium swelling ratio and the force of retraction at an extension a (not to be confused with the swelling factor as). In this manner we obtain from Eq. (XI-44) and Eq. (39)... 

The relationship between degree of cross-linking and trapping of the entangled structure (the trapping is nearly complete in the present work). 

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