Theory of the Increase in Rigidity of Rubber during Cure

Abstract

The rigidity of rubber, considered as a network of flexible molecules with Gaussian configuration functions, can be calculated for a particular sample if one is given a complete description of the molecular network or certain types of statistical description. In particular, it is sufficient to know the distribution of lengths and vector-mean extensions of the segments of the network, or only the latter distribution if it is of Gaussian form. The assumption that the vector-mean extensions have a Gaussian distribution corresponds to a similar postulate in the theory of Wall, and has been applied to a simplified network theory by Flory. In the complete theory it leads to calculation of the same proportionality between the rigidity of the material and the number Ga of segments, per unit volume, in the ``active'' part of the molecular network. However, consideration of the process of cure shows that this assumption cannot be expected to be correct, though it does lead to results of the right order of magnitude. An alternative approach to the problem is based on a study of the increase in rigidity of the material as cure proceeds. It uses a more realistic picture of the process of cure than those hitherto employed, and uses Gaussian distribution functions only where they can be logically justified. The final result is of the same form as that given by the previous theory, except that Ga is replaced by Ba, the number of bonds formed within the molecular network during cure. Since Ba is smaller than Ga by a factor of about two or more, the improved theory leads to the prediction of somewhat lower rigidities per bond formed. Available experimental results of Flory on Butyl rubber vulcanizates do not appear to give an adequate check on the quantitative features of the theory.

Tube Models for Rubber−Elastic Systems

“…The spring constant, k = 3kB T r 2 , is proportional to the temperature. Treating a piece of rubber as a random network of non-interacting entropic springs (the phantom model [4][5][6] ) qualitatively explains the observed behavior, including -to a first approximationthe shape of the measured stress-strain curves.…”

“…For a given connectivity the phantom model Hamiltonian for non-interacting polymer chains formally takes a simple quadratic form, [4][5][6] so that one can at least formulate theories which take the random connectivity of the networks fully into account. [11][12][13] The situation is less clear for entanglements or topological constraints, since they do not enter the Hamiltonian as such, but divide phase space into accessible and inaccessible regions.…”

Deterministic solutions for a step‐growth polymerization

“…Similarly, the molar concentration of chain links may be derived and equal (5) Solutions expressed by eqs 4 and 5 yield stochastic interpretations. Conversion terms express the likelihood (a mole fraction) that a site is reacted or not.…”

Partitioning of hexavalent chromium in temperature-sensitive, polyelectrolyte hydrogels

“…Here, L\llelastic is evaluated using a model for the elastic Helmholtz energy of a swollen gel, such as the affme or phantom models (36,37). The correction to the chemical potential arising from L\llelastic is negligible for our systems because the partial molar volumes of the ions are small.…”