Nematic elastomers: From a microscopic model to macroscopic elasticity theory

A Landau theory is constructed for the gelation transition in cross-linked polymer systems possessing spontaneous nematic ordering, based on symmetry principles and the concept of an order parameter for the amorphous solid state. This theory is substantiated with help of a simple microscopic model of cross-linked dimers. Minimization of the Landau free energy in the presence of nematic order yields the neoclassical theory of the elasticity of nematic elastomers and, in the isotropic limit, the classical theory of isotropic elasticity. These phenomenological theories of elasticity are thereby derived from a microscopic model, and it is furthermore demonstrated that they are universal mean-field descriptions of the elasticity for all chemical gels and vulcanized media.

Figure 1

(Color online) Model of dimers cross-linked by harmonic springs.

Figure 2

(Color online) Nematic free energy as a function of $Q$ for various values of the coupling constant associated with the Maier-Saupe interaction. The transition point is denoted by $J_0^c$, whereas $J_0^{sp}$ indicates the limit of local stability of the isotropic phase.

Figure 3

A macroscopic uniform deformation, imposed after cross-linking, changes affinely the average positions of particles in the measurement state (i.e., the measurement replicas $\alpha =1,\dots ,n$), with respect to their positions in the preparation state (i.e., preparation replica $\alpha =0$).