The totality of soft-states in a neo-classical nematic elastomer (original) (raw)
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Nonlinear elasticity, fluctuations and heterogeneity of nematic elastomers
Annals of Physics, 2008
Liquid crystal elastomers realize a fascinating new form of soft matter that is a composite of a conventional crosslinked polymer gel (rubber) and a liquid crystal. These solid liquid crystal amalgams, quite similarly to their (conventional, fluid) liquid crystal counterparts, can spontaneously partially break translational and/or orientational symmetries, accompanied by novel soft Goldstone modes. As a consequence, these materials can exhibit unconventional elasticity characterized by symmetry-enforced vanishing of some elastic moduli. Thus, a proper description of such solids requires an essential modification of the classical elasticity theory. In this work, we develop a rotationally invariant, nonlinear theory of elasticity for the nematic phase of ideal liquid crystal elastomers. We show that it is characterized by soft modes, corresponding to a combination of long wavelength shear deformations of the solid network and rotations of the nematic director field. We study thermal fluctuations of these soft modes in the presence of network heterogeneities and show that they lead to a large variety of anomalous elastic properties, such as singular length-scale dependent shear elastic moduli, a divergent elastic constant for splay distortion of the nematic director, long-scale incompressibility, universal Poisson ratios and a nonlinear stress-strain relation for arbitrary small strains. These long-scale elastic properties are universal, controlled by a nontrivial zero-temperature fixed point and constitute a qualitative breakdown of the classical elasticity theory in nematic elastomers. Thus, nematic elastomers realize a stable "critical phase", characterized by universal power-law correlations, akin to a critical point of a continuous phase transition, but extending over an entire phase.
Continuum Mechanics and Thermodynamics, 2006
We consider the bending of a nematic elastomer slab using the material model of DeSimone and Dolzmann. This model supplies a coarse description for the mechanical response associated with three distinct microstructural phases: a solid phase, a smectic phase and a liquid phase. The liquid and smectic phases permit soft modes, in which a variety of deformations are consistent with the same external tractions. Multiple-phase solutions are constructed making use of a standard analytical procedure associated with the fact that the deformation is universal in isotropic, incompressible, hyperelasticity. This requires an exhaustive accounting of the possible phase combinations consistent with the bending deformation. It is shown how an original soft mode response at low flexure angle, wherein the slab is completely in the smectic phase, can give way to conventional (non-soft) response at a threshold flexure angle associated with the nucleation of solid phase at the external boundary. Two separate phase interface fronts then travel through the slab as the flexure angle increases further, quasi-statically transforming smectic phase to solid phase.
Material instabilities in nematic elastomers
Physica D: Nonlinear Phenomena, 2000
Soft deformation paths and domain patterns in nematic elastomers are analyzed through the minimization of a nonconvex free-energy recently proposed in the literature. The free-energy density has multiple wells, and is not restricted to small deformations. The problems of calculating the quasiconvex hull of the energy wells and the quasiconvex envelope of the free-energy density are formulated and solved (the latter only in two spatial dimensions). This leads to a complete characterization of the set of soft deformations paths available to a given material, and of its effective macroscopic energy.
Soft elastic response of stretched sheets of nematic elastomers: a numerical study
Journal of the Mechanics and Physics of Solids, 2002
Stretching experiments on sheets of nematic elastomers have revealed soft deformation modes and formation of microstructure in parts of the sample. Both phenomena are manifestations of the existence of a symmetrybreaking phase transformation from a random, isotropic phase to an aligned, nematic phase. The microscopic energy proposed by Bladon, Terentjev and Warner [Phys. Rev. E 47 (1993), 3838] to model this transition delivers a continuum of symmetry-related zero-energy states, which can be combined in different ways to achieve a variety of zero-energy macroscopic deformations. We replace the microscopic energy with a macroscopic effective energy, the so-called quasiconvexification. This procedure yields a coarse-grained description of the physics of the system, with (energetically optimal) small-scale oscillations of the state variables correctly accounted for in the energetics, but averaged out in the kinematics. Knowledge of the quasiconvexified energy enables us to compute efficiently with finite elements, and to simulate numerically stretching experiments on sheets of nematic elastomers. Our numerical experiments show that up to a critical, geometry-dependent stretch, no reaction force arises. At larger stretches, a force is transmitted through parts of the sheet and, although fine phase mixtures disappear from most of the sample, microstructures survive in some pockets. We reconstruct from the computed deformation gradients a possible composition of the microstructure, thereby resolving the local orientation of the nematic director.
Nematic elastomers: From a microscopic model to macroscopic elasticity theory
Physical Review E, 2008
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.
Orientational order and finite strain in nematic elastomers
The Journal of Chemical Physics, 2005
Nematic elastomers exhibit large, spontaneous shape changes at the transition from the hightemperature isotropic phase to the low-temperature nematic phase. These finite deformations are studied here in the context of a nonlinear, properly invariant, variational theory that couples the orientational order and elastic deformation. The theory is based on the minimization of a freeenergy functional that consists of two contributions: a nematic one due to the interaction of the mesogenic units and an elastic one arising from the stretching of the cross-linked polymer chains. Suitable choices for these two contributions allow for large, reversible, spontaneous shape changes in which the elastic deformation can affect the isotropic-nematic transition temperature. The change in transition temperature as well as the magnitude of the resulting spontaneous deformation are illustrated for various parameter values. The theory includes soft elasticity as a special case but is not restricted to it.
Semisoft elastic response of nematic elastomers to complex deformations
Physical Review E, 2008
We consider a relaxed semisoft elastomer with its director oriented along the z axis that is first subjected to a large stretch in the x direction then to a slight x-z shear. We give a general argument that in any theory including director rotation, at the onset and end of the director rotation induced by these large stretches, there will be kinks in the stress-large strain curve ͑forming a stress-strain plateau͒ and zeros in the x-z shear modulus ͑C 5 ͒ associated with small shears imposed on top of the stretches. We then find the analytical forms of the C 5 -strain curves for a particular model of semisoftness ͑arising from compositional fluctuations͒ and show that it, together with the known stress-strain curve, provides the basis for a strong test of this theory. Finally, we consider the scope for other semisoft models and show that the compositional fluctuations model in fact yielded a generic form, that is, it is the most general quadratic free energy that does not explicitly include a final state direction other than the director. By introducing such additional directions, a large range of alternative models could be developed.
Linear hydrodynamics and viscoelasticity of nematic elastomers
European Physical Journal E, 2001
We develop a continuum theory of linear viscoelastic response in oriented monodomain nematic elastomers. The expression for the dissipation function is analogous to the Leslie-Ericksen version of anisotropic nematic viscosity; we propose the relations between the anisotropic rubber moduli and new viscous coefficients. A new dimensionless number is introduced, which describes the relative magnitude of viscous and rubber-elastic torques. In an elastic medium with an independently mobile internal degree of freedom, the nematic director with its own relaxation dynamics, the model shows a dramatic decrease in the dynamic modulus in certain deformation geometries. The degree to which the storage modulus does not altogether drop to zero is shown to be both dependent on frequency and to be proportional to the semi-softness, the non-ideality of a nematic network. We consider the most interesting geometry for the implementation of the theory, calculating the dynamic response to an imposed simple shear and making predictions for effective moduli and (exceptionally high) loss factors.
Anomalous Viscoelastic Response of Nematic Elastomers
Physical Review Letters, 2001
We report a combined theoretical and experimental study of linear viscoelastic response in oriented monodomain nematic elastomers. The model predicts a dramatic decrease in the dynamic modulus in certain deformation geometries in an elastic medium with an independently mobile internal degree of freedom, the nematic director with its own relaxation dynamics. Dynamic mechanical measurements on monodomain nematic elastomers confirm our predictions of dependence on shear geometry and on nematic order, and also show a very substantial mechanical loss clearly associated with director relaxation.
Free-energy density functions for nematic elastomers
Journal of the Mechanics and Physics of Solids, 2004
The recently proposed neo-classical theory for nematic elastomers is a molecular-statistical generalization of classical Gaussian network theory. The resulting free-energy density predicts the phenomenon of soft elasticity-the ability of elastomers to undergo large deformations with zero force and energy cost. The theory, however, suffers from several drawbacks: (i) extreme non-uniqueness as zero applied force corresponds to infinitely many possible deformations, (ii) insufficient moduli to model observed experimental behavior, and (iii) physically, a small, but non-zero, force must be applied. Here we propose an alternative continuum model for nematic elastomers that removes these drawbacks. Motivated by the molecular-statistical theory, we identify microstructural degrees of freedom as well as two independent strain tensors (the overall macroscopic strain plus a relative strain that indicates how the deformation of the elastomeric microstructure deviates from the macroscopic deformation) and propose expressions for the free energy as a function of the three quantities. The resulting theory provides a self-consistent bridge that connects neo-classical theory to continuum microstructural theories as well as to the classical theory of anisotropic nonlinearly elastic solids.