(327c) Structure of Equilibrium-Swollen Gels

The molecular structure of polymer gels has been of longstanding interest since a reasonable model for the structure should precede the prediction of thermodynamic and mechanical behavior. Flory-Rehner theory [1] and the c* theorem of de Gennes [2] link structure to the average mesh size determined by the molecular weight of the crosslinked chains. From these models the structure of the gel observed in neutron scattering should be similar to that of a semi-dilute polymer solution with the scattering displaying no structure at sizes above the mesh size. However, neutron scattering from networks display structure at sizes much larger than the mesh size [3-5] making these models inconsistent with experimental evidence. This excess scattering at low-q has been associated with fractal aggregates [6,7], percolation clusters [8], and quenched heterogeneities [9]. We have proposed [10,11] a structural model based on a generalized tensile-blob construction adapted to networks. The tensile-blob model was originated by Pincus for linear chains [12]. The gel model considers two fundamental size scales associated with a swollen network, the gel tensile-blob size, ξ, and a large-scale length, L, that is much larger than the mesh size. The effect of trapped entanglements and interpenetration of network chains is included in the model.

The tensile blob size is determined from a balance between thermal energy and the free energy of the network structure [10,11],

(1)

where the power P is determined from the networks internal dimension, reflecting the connectivity of the chains in the network, c, and a dimension associated with the packing or interpenetration of the network, α [10,11],

(2)

where d is the spatial dimension. c is 1 for linear chains and is df for regular objects (fully branched objects of dimension df). The mass-fractal dimension of the tensile blob is given by,

(3)

L reflects a balance of the elasticity of a randomly arranged network structure including the effect of crosslinks and chain entanglements and the osmotic swelling of the gel. From this balance it is determined that,

(4)

where Q is the swelling ratio (1/φ) and,

(5)

where f is the average functionality of the crosslink sites and Ne is the entanglement molecular weight.