Graph-like state of matter: 12. Static and dynamic entanglements

Abstract

It is shown that the contribution of chain entanglements to the equilibrium shear modulus, recently deduced theoretically and measured experimentally, cancel almost quantitatively the downward adjustment in the front-factor g of the equation of state used in such theories. This downward revision of g is contested theoretically on algebraic grounds, and the classical value of unity is recovered as a result. This classical value is supported by reinterpreting literature data on silicone rubbers by Valles and Macosko, and by new measurements (which confirm less accurate ones obtained earlier by a more difficult method) on a well-tried polyester system near its gel point. There are indications that the contributions of entanglements to the equilibrium shear modulus are all too easily overestimated by theory, and the technologist is advised to prefer the remarkably simple formulae of the graph-like-state theory of phantom chains. Dynamic entanglements can readily explain qualitatively the anelastic effects here reported, viz. frequency shifts in the plot of the real part of the dynamic shear modulus against conversion in the range α α c ⩽ 1.01 (where α c is the critical conversion at the gel point). Here too it seems as if graph-like-state theory will ultimately account for the observations, but at least certain eigenvalue problems will have to be solved first. An intriguing observation, that at the gel point of the polyester system the spectrum of relaxation frequencies approaches a block distribution, so that every finite value is equally likely, may have general validity.