Generalized constitutive equation for polymeric liquid crystals Part 1. Model formulation using the Hamiltonian (poisson bracket) formulation

Abstract The Hamiltonian formulation of equations in continuum mechanics through Poisson brackets, developed in Z.R. Iwinski and L.A. Turski, Lett. Appl. Eng. Sci., 4 (1976), 179–191, P.J. Morrison and J.M. Greene, Phys. Rev. Lett., 45 (1980) 790–794, I.E. Dzyaloshinskii and G.E. Volovick, Ann. Phys., 125 (1980) 67–97, D.D. Holm, J.E. Marsden, T. Ratiu and A. Weinstein, Phys. Rep., 123 (1985) 1–116, M. Grmela, Phys. Lett. A, 130 (1988) 81–86, and A.N. Beris and B.J. Edwards, J. Rheol., 34 (1990) 55–78, for a class of incompressible fluids, is used here in order to generate a constitutive equation for the stress and the order parametr tensor for a polymeric liquid crystal. A free energy expression, of the type used by Doi in his theory for concentrated solutions of rigid rods, is used in addition to the Frank elasticity expression employed in the Leslie—Ericksen—Parodi (LEP) theory to model the effect of spatial gradients in the liquid crystalline microstructure. For homogeneous systems, the analysis leads to a model which is equivalent to a generalization of Doi theory out to fourth-order terms in S . Truncating this model at second-order terms gives the Doi equations exactly. To evaluate the expanded model, results for steady simple shear and extensional flows are compared against the Doi model predictions. The constitutive equation resulting from the expanded model is compared against the LEP constitutive equation and the parameters between the two are correlated. The additional stress terms for non-homogeneous systems reduce to a recently presented (B.J. Edwards and A.N. Beris, J. Rheol., 33 (1989) 1189–1193; M. Grmela, Phys. Lett. A, 137 (1989) 342–348) generalization of the Ericksen stress expression in terms of the second-order parameter tensor. The model presented is a generalization and extension of the order-parameter-based theory of Doi which allows a greater flexibility in describing the rheological properties of polymeric liquid crystalline systems

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