A structural model for equilibrium swollen networks

– A tensile blob construction for branched structures is used to explain structural sizes larger than the strand length, as observed in neutron scattering data from equilibrium swollen networks. Under this model, equilibrium swollen networks display a base structural size, the “gel tensile blob” size, ξ, that follows the scaling relationship ξ ∼ l/(1/2−χ)P , where l is the monomer length, χ is the Flory interaction parameter and P is a power determined by the connectivity of the network and the degree of interpenetration. The gel tensile blobs compose a large-scale linear structure, whose length, L, follows the scaling relationship L ∼ QNavg, where 1/N avg = ((1/fN 2 c ) + (1/4N 2 e )), Q is the equilibrium swelling ratio, Nc is the strand length, Ne is the entanglement length and f is the functionality of the cross-links. The variation of the swelling ratio with molecular weight can now be expressed as Q ∼ N avg , which reduces to the correct expressions under the limits Ne Nc and Nc Ne. The microstructure of polymer gels and its relationship to swelling has been a source of controversy in the polymer literature [1–4]. It is well known that the microstructure of a polymer gel, in general, and the degree of topological constraint, in particular, depend on the preparation conditions, especially the polymer concentration in the solution in which the network is cross-linked [2, 4]. For networks prepared in solution with cross-links of identical functionality, the degree of topological constraint arising from trapped entanglements is expected to increase with the polymer concentration in the state of preparation and presumably reaches a maximum for networks prepared in the melt state. Trapped entanglements present in a network affect the modulus [5, 6] and possibly even the swelling behavior. The equilibrium swelling of polymer gels is often described in terms of the Flory-Rehner theory [1], in which the free energy of the gel is written as the sum of an osmotic contribution, calculated using the Flory-Huggins free energy of mixing of a polymer and a solvent; and an elastic contribution, calculated assuming affine deformation of the network strands (i.e. the chains between cross-links) which exhibit Gaussian statistics. The equilibrium swollen state is calculated by minimizing the free energy with respect to the volume fraction of the gel [1] predicting that the equilibrium swelling ratio, Q ∼ N c , Nc being the strand length. The prediction seems to be reasonably confirmed by experiments, especially for highly cross-linked networks [7]. A polymer chain in a semi-dilute solution does not follow Gaussian statistics