Propagation of acoustic waves in nematic elastomers E. M. Terentjev, 1 I. V. Kamotski, 2 D. D. Zakharov, 2 and L. J. Fradkin 2 1 Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom 2 School of Engineering, South Bank University, London SE1 0AA, United Kingdom ~Received 26 February 2002; published 20 November 2002! We develop a theory of elastic waves in oriented monodomain nematic elastomers. The effect of soft elasticity, combined with the Leslie-Ericksen version of dissipation function, results in an unusual dispersion and anomalous anisotropy of shear acoustic waves. A characteristic time scale of nematic rotation determines the crossover frequency, below which waves of some polarizations have a very strong attenuation while others experience no dissipation at all. We study the anisotropy of low-frequency Poynting vectors and wave fronts, and discuss a ‘‘squeeze’’effect of energy transfer nonparallel to the wave vector. Based on these theoretical results, an application, the acoustic polarizer, is proposed. DOI: 10.1103/PhysRevE.66.052701 PACS number~s!: 83.80.Va, 61.41.1e, 62.30.1d Liquid crystalline elastomers ~LCE! represent an exciting physical system that combines the local orientational sym- metry breaking and the entropic rubber elasticity, producing a number of unique physical phenomena. In ordinary elastic solids, the deformations are created by relative movement of the same atoms ~or molecules! that form the bonded low- symmetry lattice. Hence, when the deformation is small, the lattice symmetry is preserved and one obtains an ordinary anisotropic elastic response. In contrast, in polymer net- works, the macroscopic elasticity arises from the entropy change of chains on relative movement of their cross-linked end points, which are relatively far apart. On a smaller length scale, a liquid crystalline order can be established within these chains. In nematic elastomers, the local director can rotate, in principle, independently of deformation of the cross-linking points. Such an internal degree of freedom within the elastic body constitutes what is known as the Cosserat medium: the relative movement of cross-linking points provides elastic strains and forces, while the director rotation causes local torques and couple stresses. Recent re- view articles summarize these ideas and report physical ef- fects, predicted theoretically and found experimentally in nematic LCE; e.g., Refs. @1,2#. A pioneering study of oscillating dynamic-mechanical properties @3# has been followed by further work on aligned monodomain nematic LCE, @4,5#, which demonstrated a dra- matic reduction of storage elastic modulus G 8 and the asso- ciated increase in the loss factor tan d in certain geometries of deformation. This effect, named the ‘‘dynamic soft elas- ticity,’’ allows one to directly probe the basic equilibrium properties of nematic rubbers and also access the new kinetic parameters—viscous coefficients and relaxation times. Here we follow the earlier theoretical work @6#, which formulated the constitutive relations of linear viscoelasticity for nematic elastomers in the hydrodynamic ~low-frequency! limit, and develop a theory of acoustic waves propagating through an elastic medium with the mobile anisotropic mi- crostructure and dissipation. Viscoelastic waves in aniso- tropic media have been thoroughly investigated over the last decade @7#. We apply a similar analysis to the nematic rub- bers and obtain unusual predictions for directions of energy propagation and conditions for anomalous dissipation. We also find the configurations of propagation and polarization, where the attenuation vanishes. This ‘‘acoustic polarization,’’ similar to the optical polarization in a birefringent medium, could lead to many new discoveries and applications. Equilibrium elastic properties of monodomain nematic rubbers are well studied, both theoretically and experimen- tally, and are described at some length in review articles. A molecular theory of ideal nematic networks @8# gives the elastic free energy density in terms of the Cauchy strain ten- sor and the the uniaxial matrices of chain step lengths before and after the director n has rotated by a certain angle during the deformation: , ij 5, ’ d ij 1@ , i 2, ’ # n i n j . One finds that, apart from the universal rubber-elastic energy scale m 5c x k B T , with c x proportional to the cross-linking density, the theory depends on a single equilibrium parameter r 5, i / , ’ , the ratio of the principal step lengths of the aniso- tropic polymer backbone @or equivalently, r 5( R i / R ’ ) 2 for the principal values of gyration radii#. This ratio is a function of the nematic order parameter. As in all polymeric materials, the bulk modulus is inde- pendent of the configurational entropy of polymer chains and mainly determined by molecular forces resisting the com- pression of a liquid, B ˜ ;10 10 J/m 3 , much greater than the typical value of rubber modulus m ;10 6 J/m 3 . In this paper, we shall explicitly implement the incompressible limit. The small-deformation limit of the elastic energy depends on « ˜ ik 5« ik 2 1 3 Tr @ « = # d ik , the traceless part of linear symmet- ric strain « ik 5 1 2 ( ] k u i 1] i u k ), with u the displacement vector, which is the only variable of classical continuum elasticity @9#. In a system with an internal orientational degree of free- dom, the antisymmetric part of strain, V5 1 2 curl u, also con- tributes to the physical response via the relative rotation, denoted here by the vector Q[V2@ n3d n# . The elastic po- tential energy density of a uniaxial incompressible solid takes the form @10# F 5C 1 ~ n• « =• n! 2 12 C 4 @ n3« = 3n# 2 14 C 5 ~@ n3« =• n#! 2 1 1 2 D 1 @ n3Q# 2 1D 2 n• « =• @ n3Q# . ~1! All constants in Eq. ~1! are of the same order of magnitude, PHYSICAL REVIEW E 66, 052701~R!~2002! 1063-651X/2002/66~5!/052701~4!/$20.00 ©2002 The American Physical Society 66 052701-1