Constrained Swelling of Polymer Networks: Characterization of Vapor-Deposited Cross-Linked Polymer Thin Films

In certain real situations, polymer networks cannot swell freely but are subjected to constraints by action of external forces or geometric restrictions. In this contribution, the restriction to swelling by adhesion of a cross-linked polymer film deposited on a rigid substrate is considered. Such constraints prevent swelling in plane, and the osmotic force concentrates on extension of network chains in the direction normal to the surface. We analyze existing rubber elasticity models taking into account finite extensibility of network chains and derive the elastic contribution to the chemical potential of the solvent in swollen networks. For the mixing contribution to the change of the Gibbs energy, it is more appropriate to consider the network containing cross-links, but with undeformed network chains than an un-cross-linked polymer of infinite degree of polymerization, because continuing formation of bonds within an infinite molecule further decreases the entropy. The additional term is proportional to the cycle rank of the network. The reformulation of the mixing contribution is important for consideration of cross-linking induced phase separation during network formation. The phase separation limit is due to the cross-link contribution to the mixing part of the Gibbs energy and has nothing to do with chain stretching. The swelling theories respecting finite extensibility of network chains are then applied to cross-linked poly(ethylene oxide) films obtained by plasma-assisted vapor deposition with the aim to determine their effective cross-link density.