ALBERT

Notes: Current polymer network theories exhibit inconsistencies which show up particularly clearly when one considers the three-dimensional deformation (swelling) of networks which are prepared by crosslinking a polymer in solution. A check on the theories can only be obtained if one knows precisely the number of crosslinks in the network and if a range of deformations (range of swelling ratios) is imposed on the network. To this end a series of seven gels was prepared by crosslinking secondary cellulose acetate (D.S. = 2.42) in dioxane solution (5-10%) with dianisidine diisocyanate. The number of chemical crosslinks was determined by reacting unused isocyanate groups with C14-labeled methanol. Subsequently, changes in the degree of swelling of the gel, induced by increasing concentrations of cellulose acetate in dioxane and methyl acetate solutions around the gel, were followed by measuring accurately the changes in length of 2-cm. strips under a travelling microscope. At equilibrium swelling the activities of the solvent inside and outside the gel are equal. For the outside solutions the activities were derived from osmometry. The inside activities derive from a mixing term and an elastic deformation term. The mixing term was approximated by using the Flory-Huggins expression with an interaction parameter, as obtained from osmometry on a derivative, which was prepared by reacting cellulose acetate with an excess of diisocyanate, thus avoiding network formation. By means of the various known theoretical expressions for the elastic deformation term, the swelling data allow the calculation of the number of crosslinks in the gels. All theories lead to far fewer crosslinks than are known to be there on the basis of the chemical analysis. This has never been observed to the same extent before, but may be specific for gels obtained by crosslinking in solution. One is forced to conclude that none of the existing theories are applicable. The data can be explained, however, by postulating that the configurations of the chains between crosslinks do not follow a Gaussian distribution, but are instead given by ω(r)dr = ωx(x)-ωy(y)ωz(z) dxdydz, where ωx(x) = C|x|n exp { -bx2}. Such a non-Gaussian distribution might arise because of topological restrictions, possibly including those due to the excluded volume. For the free energy of elastic deformation we find, following James and Guth's reasoning but using the new distribution function: \documentclass{article}\pagestyle{empty}\begin{document}$$ \Delta F_{{\rm e1}} /kT{\rm = }v[(n{\rm + 1)/2][}\lambda _x ^{\rm 2} {\rm + }\lambda _y ^{\rm 2} {\rm + }\lambda _z ^{\rm 2} {\rm - 3] - }vn{\rm ln }\lambda _x \lambda _y \lambda _z$$\end{document} where v is the number of chains in the network and λx, λy, and λz are the deformation ratios with respect to the unstrained state. In the well known Gaussian theories, the term n + 1 is absent and in front of the logarithm instead of n either 0, 2/f, or 1 (f = functionality of the crosslink) is found, depending on whether the result of James and Guth, Flory and Wall, or Hermans is used, respectively.