Commentary on “ Mechanical properties of mono - domain side chain nematic elastomers ” by P . Martinoty

We discuss the rheology experiments on nematic elastomers by Martinoty et al. in the light of theoretical models for the long-wavelength low-frequency dynamics of these materials. We review these theories and discuss how they can be modified to provide a phenomenological description of the nonhydrodynamic frequency regime probed in the experiments. Moreover, we review the concepts of soft and semi-soft elasticity and comment on their implications for the experiments. PACS. 83.80.Va Elastomeric polymers – 61.30.v Liquid crystals – 83.10.Nn Polymer dynamics Nematic elastomers are crosslinked rubbers with the uniaxial symmetry of a nematic liquid crystal [1]. One could argue, therefore, that they are simply uniaxial rubbers. A nematic elastomer can, however, be distinguished from a simple uniaxial elastic medium by the fact that, at least in an idealized limit, it can form via a spontaneous phase transition from an isotropic phase. Since this phase transition breaks the continuous rotational symmetry of the isotropic phase, it has an associated Goldstone mode whose manifestation is the vanishing of the elastic modulus C5 measuring the elastic energy of strains in planes containing the anisotropy axis [2,3]. Thus, the ideal nematic elastomer has a “soft” elasticity compared to a traditional uniaxial elastic medium in which C5 is nonzero. In practice, ideal monodomain nematic elastomers do not form in the absence of some aligning field, which is usually produced by weakly crosslinking a sample in the isotropic phase, stretching to produce a uniaxial configuration, and then crosslinking again [4]. The resultant material is weakly anisotropic in the high-temperature paranematic phase and more anisotropic in the low-temperature nematic phase. Since the low-temperature phase arises from an already aligned phase and does not break rotational symmetry, it does not exhibit ideal soft elasticity. Rather, it exhibits “semi-soft” elasticity in which C5 is small but nonzero. Thus, for small strains, a semi-soft nematic elastomer is truly a uniaxial solid. At higher strains, however, it exhibits properties, such as a nearly constant stress for increasing strain, characteristic of a soft nematic elastomer [5]. The static elastic properties and phase transitions of both soft and semi-soft nematic elastomers are reasonably well understood [1,2,6,7,8], provided the effects of random stresses [2,9] can be ignored. They can be described either in terms of models involving strain only or in terms of models with coupling between a traditional Maier-Saupe-de-Gennes nematic symmetric-traceless order parameter Qij and strain. Considerable progress has also been made toward understanding the dynamics of nematic elastomers [10,11,12] again in the limit in which random stresses and fields can be ignored. As in a standard elastic medium, purely hydrodynamical equations, which describe all modes with frequencies ω smaller than the smallest characteristic inverse decay time τ when wavenumber q tends to zero, involve only the displacement field u and not the nematic director n, which relaxes to the local stain in a nonhydrodynamic time τn. Even though the nematic director is not strictly speaking a hydrodynamic variable in either soft or semi-soft nematic elastomers, it is of some interest to develop phenomenological equations describing both nematic director and elastic displacement. Two distinct derivations [1,11, 12] lead to identical sets of coupled director-displacement equations, which we will refer to as the NED (nematicelastomer dynamic) equations, that reduce to the rigorous hydrodynamic equations in terms of displacement only at frequencies ωτn ≪ 1 and to the standard equations of nematohydrodynamic when elastic rigidity is turned off. Rheological measurements of the complex modulus [13] G(ω) = G(ω)− iG(ω) (1) provide a useful experimental probe of the dynamical properties of viscoelastic systems. They typically probe a wide range of frequencies that stretch from the low-frequency hydrodynamic regime to well above it. Thus, interpreting these experiments requires theoretical models of dynamics beyond hydrodynamics. A nematic elastomer is characterized in general by relaxation times associated with director relaxation and with other modes, which we will simply refer to as elastomer modes. In their simplest form, the NED equations assume the director relaxation time τn is the 2 O. Stenull and T. C. Lubensky: Commentary on nematic elastomers longest non-hydrodynamic relaxation time, and that it is well separated from the longest elastomer time τE . They describe dynamics for both ωτn ≪ 1 and ωτn > 1 and provide non-trivial predictions about G(ω), in particular the appearance of a plateau in G(ω) and an associated dip in G(ω) for τ n < ω < τ −1 E . Experiments [14,15,16, 17,18] to date do not exhibit the plateau predicted by the NED equations. This has led Martinoty et al. to question their validity. Here we will review both the hydrodynamical and NED equations for nematic elastomers and discuss how they can be modified to provide a phenomenological description of the non-hydrodynamic frequency regime probed in rheology experiments. Moreover, we review the concepts of soft and semi-soft elasticity and comment on their implications for rheology experiments. We begin with a review of the dynamics of traditional isotropic solids, of which rubber is a particular example. The elastic free energy of an isotropic solid is characterized by a bulk modulus B and a shear modulus μ, and dissipation is characterized by a bulk viscosity ζ and a shear viscosity η, which are both frequency independent at low frequencies. The complex shear modulus in the hydrodynamic limit is simply