Nematic rubber elasticity

Let us consider in more detail the elasticity of nematic rubbers, which is at the heart of understanding their specific properties. Consider a weakly crosslinked network with junction points sufficiently well spaced to ensure that the conformational freedom of each chain section is not restricted. We recall that for a conventional isotropic network the stress-strain relation for simple stretching (compression) of a unit cube of material can be derived as [62, 63] ... 

The difference between nematic and isotropic elastomers is simply the molecular shape anisotropy induced by the LC order, as discussed in Sect. 2. The simplest approach to nematic rubber elasticity is an extension of classical molecular mbber elasticity using the so-called neo-classical Gaussian chain model [64] see also Warner and Terentjev [4] for a detailed presentation. Imagine an elastomer formed in the isotropic phase and characterized by a scalar step length Iq. After cooling down to a monodomain nematic state, the chains obtain an anisotropic shape described by the step lengths tensor Ig. For this case the stress-strain relation can be written as ... 

Rusakov 107 108) recently proposed a simple model of a nematic network in which the chains between crosslinks are approximated by persistent threads. Orientional intermolecular interactions are taken into account using the mean field approximation and the deformation behaviour of the network is described in terms of the Gaussian statistical theory of rubber elasticity. Making use of the methods of statistical physics, the stress-strain equations of the network with its macroscopic orientation are obtained. The theory predicts a number of effects which should accompany deformation of nematic networks such as the temperature-induced orientational phase transitions. The transition is affected by the intermolecular interaction, the rigidity of macromolecules and the degree of crosslinking of the network. The transition into the liquid crystalline state is accompanied by appearence of internal stresses at constant strain or spontaneous elongation at constant force. 

J.-P. Jarry and L. Monnerie, Effects of a nematic-like interaction in rubber elasticity theory, Macromolecules, 12, 316 (1979). 

We have only dealt with the main chain nematic networks so far. Actually many liquid crystalline networks are formed by crosslinking the backbones of side chain liquid crystalline polymers. The side chain nematic polymers have three nematic phases and their backbones have either prolate or oblate conformations, depending on their phase. It is expected that the rubber elasticity of a side chain nematic polymer network is more complex. For instance, the stress-induced Ni-Nm phase transition is predicted as the network shape transforms from oblate to prolate. Liquid crystalline networks have a bright potential in industry. 

An extension of rubber elasticity (i.e. of the description of large, static and incompressible deformations) to nematic elastomers has been given in a large number of papers [52, 61-66]. Abrupt transitions between different orientations of the director under external mechanical stress have been predicted in a model without spatial nonuniformities in the strain field [52,63]. The effect of electric fields on rubber elasticity of nematics has been incorporated [65]. Finally the approach of rubber elasticity was also applied recently to smectic A [67] and to smectic C [68] elastomers. Comparisons with experiments on smectic elastomers do not appear to exist at this time. Recently a rather detailed review of the model of an-... 

Unfortunately swollen LCEs are still not robust enough for long-term operation. However, if the rod-like molecules of a conventional LCE are substituted by BC nematics, a bent-core hquid crystalline elastomer (BCLCE) can be created. BCLCEs might combine excellent flexoelectric properties with rubber elasticity. 

Studies of isotropic-nematic transitions such as are discussed by Chandrasekhar [1, Ch. 2], and other such transitions, revert to (U), dealing primarily with Fq. For nonlinear analyses of this kind, and others, it is convenient to make use of representations analogous to those used in rubber elasticity, to evade the need to calculate eigenvalues. This makes use of the fact that... 

In this section, we describe the dynamic features of EOM effects. In particular, we focus on the response times to field-on and field-off these are the rise and decay times, respectively. In the case of the EO effects of nematic liquids, the rise and decay times exhibit the characteristic dependencies on voltage, and these characteristic times reflect the elastic, viscous, and dielectric properties of the materials. They have fully characterized the dynamics of the electric-field responses of LMM-LCs in experiments and established the theoretical background [6]. The dynamic features of the EOM effects in nematic gels are expected to differ from those of the EO effects in LMM-LCs, because the gels possess rubber elasticity and also have a different origin of the memory of the initial director. In addition, the dynamic properties give important information about the applicability of EOM effects in practical applications. 

Neither does the microbrownian motion of the amorphous mesh inhibit the liquid crystal phase, nor does the positional order of the molecules interfere with the elasticity. Hence, as a hybrid material that combines LC and rubber characteristics, LCEs have unique properties in which the molecular orientation of the liquid crystal is strongly correlated with the macroscopic shape (deformation) which is unparalleled to other materials. The most prominent example in the physical properties derived from this property is the huge thermal deformation. Figure 10.1 shows an example of the thermal deformation behavior of side-chain nematic elastomers (NE) [3]. When the molecules transform from the random orientation in the isotropic phase to the macroscopic planar orientation in the nematic phase, the rubber extends in the direction of the liquid crystal orientation and increases with decreasing temperature as a result of an increase in the degree of liquid crystal orientation. This thermal deformation behavior is reversible, and LCEs can be even considered as a shape-memory material. Figure 10.1 is from a report of the early research on thermal deformation of LCEs, and a strain of about 40 % was observed [3]. It is said that LCEs show the largest thermal effect of all materials, and it has been reported that the thermal deformation reaches about 400 % in a main-chain type NE [4]. 

The notion of soft (semi-soft) elasticity was also introduced to describe the flat or nearly flat plateau with a practically zero modulus observed in the stress-strain curves when the deformation is applied to the sample perpendicularly to the initial orientation of the director (Verwey et al 1996 Warner and Terentjev 2003). The nematic extension of the Gaussian rubber elasticity (the so-called neo-classical model), which will be reviewed in the next section, allows one to make the link with the previous definition of soft elasticity by demonstrating that this modulus is C5. This model has also been used for numerical simulations of the behavior of the elastic plateau (Conti et al. 2002). 

The non-Gaussian elasticity of the NEs prepared with the two-step cross-linking process leads to the failure of the neo-classical model, which is based on Gaussian rubber elasticity, to describe some physical properties of these NEs. Typical examples are given by the stress-strain curves, in particular those with the nearly flat plateau associated with the sample twice cross-linked at high temperature. In contrast, the neo-classical model describes well the NEs prepared by a UV cross-linking of an oriented nematic polymer. For both types of NEs, the value of C5 is not small and comparable to that of the other elastic coefficients. This clearly shows that the NEs are not semi-soft materials. On the other hand, the elastic plateau of the stress-strain curve is usually called semi-soft because it refers to the noticm of ideal softness. This term has become incorrect since the elastic plateau is not due to the vicinity of a hypothetical ideal state and should be modified. 

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