Network formation and properties: Rate theory description of effects of ring formation on elastic shear modulus of RA₂ + RB₃ networks

The rate theory of RA 2 +RB 3 polymerization has been developed to enable the concentrations of the smallest inelastic loops and the resulting reductions in elastic shear modulus at complete reaction to be evaluated. Reduction in modulus is expressed as M c /M° c , the effective molar mass of chains between elastically active junction points (M c ) relative to that for the perfect network for given reactants (M° c ). Calculations have been carried out for stoichiometric reaction mixtures and the concentration of loops is characterized by λ, a ring‐forming parameter. λ = P ab /C Ao , where C Ao is the initial concentration of A‐groups and P ab = (3/(2πvb 2 )) 3/2 /N Av , with v the number of bonds in the chain forming the smallest loop, b the effective bond length of the chain, and N Av Avogadro's number. P ab is equal to the mutual concentration, assuming Gaussian statistics, of the A‐ and B‐groups at the two ends of the linear sub‐chain of v bonds. Loop formation increases as λ increases, that is, as the initial concentration of reactive groups (C Ao ) decreases, or, at constant C Ao , as the molar mass of reactants (v) or their chain stiffness (b) decreases. Comparison with existing experimental data on two series of polyurethane networks formed from hexamethylene diisocyanate and two polyoxypropylene triols at different initial dilutions shows that the values of P ab decrease with increases in initial dilution and molar mass of triol. The decrease with molar mass is entirely accounted for by changes in v and the variation with dilution shows that the approximation of counting only smallest loops improves as dilution increases.