Segmental Orientation in Deformed Networks. 2. Molecular Theory for Biaxial Deformation

The Nagai formulation of segmental orientation in stretched polymer chains is applied to segmental orientation in amorphous elastomeric networks under biaxial deformation. The orientation of a reference vector, m, rigidly affied to a chain of the biaxially drawn network is investigated. The orientation of m is characterized by two orientation functions, S, and S,, where x and y are the laboratory- fmed axes along which the macroscopic biaxial deformation is prescribed. Expressions for S, and S, are obtained in the form of a series expansion, grouped into terms depending on their order with respect to lln, and on the magnitude of applied deformation. Expressions including terms up to the fourth power of lln (n = number of chain repeat units) and eighth power of the extension ratios are derived. Configurational averages appearing in the coefficients of the expression for the orientation functions are obtained by Monte Carlo simulation for a poly(ethy1ene)-like model chain with 100 bonds. Different levels of approximation for S, and S,, comprising terms linear and second order in l/n, are compared. The limits of validity of the expressions are discussed in relation to finite chain extensibility. Results of calculations show that the segmental orientation S, is strongly affected by finite extensions imposed along the transverse direction, y. When the strains are made infinitesimally small, however, S, is demonstrated to be uncoupled from the strain applied along the y-direction.