# Anisotropic Networks, Elastomers, and Gels

**DOI:**https://doi.org/10.1007/978-3-642-27737-5_20-2

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## Keywords

Trace Formula Stripe Domain Deformation Gradient Tensor Nematic Order Selective Reflection## Definition of the Subject

Anisotropic (liquid crystalline) elastomers and gels bring together, as nowhere else, three important ideas: *orientational order* in amorphous soft materials, *responsive molecular shape*, and *quenched topological constraints*. Acting together, they create many new physical phenomena that are briefly reviewed in this article. Classical liquid crystals are typically fluids of relatively stiff rod molecules with long-range orientational order. Long polymer chains, with incorporated rigid anisotropic units, can also form orientationally ordered liquid crystalline phases. By contrast with free rigid rods, these flexible chains change their average molecular shape, from isotropic spherical to ellipsoidal, when their component rods align. Linking the polymer chains together into a network fixes their relative topology, and the polymer melt (or solution) becomes an elastic solid – an elastomer (or gel). Radically new properties arise from the ability to change average molecular shape of anisotropic polymer chains in the solid state.

In ordinary solids the deformations are created by relative movement of the same atoms (or molecules) that form the bonded lattice. Hence, when the deformation is small, the lattice symmetry is preserved and one obtains an ordinary elastic response (often anisotropic). There is a classical elastic response found in glasses as well (either isotropic or anisotropic), where in place of the crystalline lattice recording the preferred position of atoms, they are confined by constraints of local cages. Either way, the elements of the body “know” their positions, and the system responds with an increase of elastic energy when these elements are displaced. Large deformations destroy the lattice (or cage) integrity and simply break the material.

In contrast, in elastomers and gels, the macroscopic elastic response arises from the entropy change of chains on relative movement of their cross-linked end points, which are relatively far apart. Monomers remain highly mobile and thus liquid-like. What happens to chain segments (monomer moieties) on a smaller length scale is a relatively independent matter, and such weakly cross-linked network behaves as a liquid on length scales smaller than the end-point separation of strands. In particular, the liquid crystalline order can be established within these segments. The magnitude of this order can be altered, and its director can rotate, in principle, independently of deformation of the cross-linking points. Such an internal degree of freedom within, and coupled to the elastic body, provides additional local forces and torques, intricately connected in the overall macroscopic response of the body.

In the simplest and most straightforward case of nematic elastomers and gels, the uniaxial ellipsoidal average anisotropy of chain constituting the network leads to two key physical properties that make these materials so unique. If one can change the level of nematic (orientational) order, which is not difficult to achieve by, e.g., changing temperature, changes at the molecular level induce a corresponding mechanical strains: A block of rubber can contract or elongate by a large amount on heating or cooling, respectively. This process is perfectly reversible. This leads to a group of possible applications in artificial muscles and actuators, which can be driven by any stimulus that affects the local nematic order: temperature, solvent intake, irradiation, etc.

It is also possible to rotate the nematic director axis and the rubber matrix independently, although in contrast to ordinary liquid crystals, it costs energy to uniformly rotate the director within the matrix. This penalty leads to suppression of orientational fluctuations and high optical clarity of birefringent elastomers and gels. Local rotations also yield a subtle and spectacular elastic phenomenon which is called “soft elasticity.” Contrary to intuition, there is an infinite number of nontrivial mechanical deformations that can accommodate the rotation of anisotropic distribution of chains without its distortion. As a result, the entropy of the chains does not change, in spite of macroscopic deformations, and the material can be distorted without any significant energy cost! A special combination of shears and elongations/compressions is required, but it turns out not very difficult to achieve. Elastic softness, or attempts by the material to achieve it, pervades much of the elasticity of nematic elastomers and gels. For instance, another unique and spectacular property of these systems is anomalously high damping, where the mechanical energy is dissipated on these soft rotational deformation modes.

Cholesteric liquid crystals have a helical director texture. When cross-linked into elastomers or gels, this periodic helical anisotropy is made to couple to the mechanical state of the whole body. Their optical and mechanical responses to imposed stress are exceedingly rich as a result. Cholesteric elastomers are brightly colored due to selective reflection and change color as they are stretched – their photonic band structure can be powerfully manipulated by applied deformations. Such periodic (distributed feedback) photonic systems can emit laser radiation, the color of which can be continuously shifted by mechanical means across the whole visible spectrum. Further, the effect of topological imprinting can select and extract molecules of specific handedness from a mixed solvent. Cholesteric gels can act as a mechanical separator of chirality.

Smectic or lamellar elastomers and gels have plane-like, layered modulation of density in one direction (SmA) or additionally a tilt of the director away from the layer normal (SmC). Many other more complex smectic phases exist and could also be made into elastomers. Here the layers are often constrained by cross-linking density modulation not to move relative to the rubber matrix. Deformations along the layer normal can be resisted by a modulus up to 100 times greater than the normal rubber (shear) modulus of the matrix. Thus, smectic elastomers are rubbery in the two dimensions of their layer planes but respond as hard conventional solids in their third dimension. Such extreme mechanical anisotropy promises interesting applications.

^{4}times lower in modulus than conventional solid ferro- and piezoelectrics. Distortions give polarization changes comparable to those in ordinary ferroelectrics, but the response to an applied stress must necessarily much larger than in conventional materials due to the very low mechanical impedance.

It is debatable, whether an anisotropic glass could be called a liquid crystal, although nominally a glass can be made to possess orientational but not a translational order. The lack of molecular mobility makes the question of thermodynamic equilibrium very ambiguous.

## Introduction

Within the simplest affine deformation approach, one regards the change in each chain end-to-end vector * R* as

*′ =*

**R***·*

**F***, when a deformation of the overall polymer network is characterized by a deformation gradient tensor*

**R***F*

_{ ij }. In continuum elasticity, this tensor is often called

*= ∇*

**F***, a gradient of the geometrical mapping*

**f****f**from the reference to the current state of deformation. Assuming the chain connecting the two cross-links is long enough, the Gaussian approximation for the number of its conformations Z(R) gives for the free energy (per chain):

*W*

_{ch}= −

*kT*ln

*Z*(

*′) ≃ (*

**R***kT*/

**a**

^{ 2 }

**N**) [

**F**^{ T }·

*]*

**F**_{ ij }

*R*

_{ i }

*R*

_{ j }, where

*a*is the step length of the chain random walk and

*N*the number of such steps. In order to find the total elastic work function of all chains affinely deforming in the network, one needs to add the contributions

*W*

_{ch}(

*R*) with the statistical weight to find a chain with a given initial end-to-end distance

*in the system. This procedure, called the quenched averaging, produces the average \( \left\langle {R}_i{R}_j\right\rangle \simeq \frac{1}{3}{a}^2N{\delta}_{ij} \) in*

**R***W*

_{ch}. The resulting rubber-elastic free energy (per unit volume) is \( {W}_{\mathrm{el}}=\frac{1}{2}{n}_ckT\left({\boldsymbol{F}}^T:\boldsymbol{F}\right) \), with

*n*

_{ c }a number of chains per unit volume of the network. This is a remarkably robust expression, with many seemingly relevant effects, such as the fluctuation of cross-linking points, only contributing a small quantitative change in the prefactor. It is, however, incomplete since the simplified Gaussian statistics of chains does not take into account their physical volume and thus does not address the compressibility issue. The proper way of dealing with it is by adding an additional independent penalty for the volume change: \( \frac{1}{2}K{\left( \det \boldsymbol{F}-1\right)}^2 \) in elastomers, with

*K*the typical bulk modulus of the order 10 GPa in organic materials. This should be modified into \( \frac{1}{2}K{\left( \det \boldsymbol{F}-\varPhi \right)}^2 \) for gels swollen by a volume fraction Φ of solvent.

If one assumes that the rubber-elastic modulus is low, then a very reasonable approximation is to simply impose the volume conservation constraint on deformation tensor: det * F* = 1. The value of the rubber modulus is found on expanding the deformation gradient tensor in small strains, say, for a small extension,

**F**

_{ zz }= 1 +

*ε*, and obtaining \( {W}_{\mathrm{el}}\simeq \frac{3}{2}{n}_ckT{\varepsilon}^2 \). This means the extension (Young) modulus

*E*= 3

*n*

_{ c }

*kT*; the analogous construction for a small simple shear will give the shear modulus G =

*n*

_{ c }

*kT*, exactly a third of the Young modulus as required in an incompressible medium. This shear modulus

*G*, having its origin in the entropic effect of reduction of conformational freedom on polymer chain deformation, is usually so much smaller than the bulk modulus (determined by the enthalpy of compressing the dense polymer liquid) that the assumption of rubber or gel deforming at constant volume is justified. This constraint leads to the familiar rubber-elastic expression \( {W}_{\mathrm{el}}=\frac{1}{2}{n}_ckT\left({\uplambda}^2+2/\uplambda \right) \) where one has assumed that the imposed extension

*F*

_{ zz }= λ is accompanied by the symmetric contraction in both transverse directions, \( {F}_{xx}={F}_{yy}=1/\sqrt{\uplambda} \) due to the incompressibility.

*L*=

*aN*the chain contour length and \( {\ell}_{||}/{\ell}_{\perp } \) the ratio of average chain step lengths along and perpendicular to the nematic director. In the isotropic phase, one recovers \( {\ell}_{||}={\ell}_{\perp }=a \) The uniaxial anisotropy of polymer chains has a principal axis along the nematic director

*, with a prolate \( \left({\ell}_{||}/{\ell}_{\perp }>1\right) \) or oblate \( \left({\ell}_{||}/{\ell}_{\perp }<1\right) \) ellipsoidal conformation of polymer backbone. The ability of this principal axis to rotate independently under the influence of network strains makes the rubber-elastic response nonsymmetric, so that the elastic work function is (Bladon et al. 1994)*

**n****ℓ**the uniaxial matrices of chain step lengths before (

**ℓ**

_{0}) and after the deformation to the current state: \( {\ell}_{ij}={\ell}_{\perp }{\delta}_{ij}+\left[{\ell}_{||}-{\ell}_{\perp}\right]{n}_i{n}_j \) Note that the deformation gradient tensor

*does no longer enter the elastic energy in the symmetric combination*

**F**

**F**^{ T }·

*but is “sandwiched” between the matrices*

**F****ℓ**with different principal axes. This means that antisymmetric components of strain will now have a nontrivial physical effect, in contrast to isotropic rubbers and, more crucially, to elastic solids with uniaxial anisotropy. There, the anisotropy axis is immobile, and the response is anisotropic but symmetric in stress and strain. The uniqueness of nematic rubbers stems from the competing microscopic interactions and the difference in characteristic length scales: The uniaxial anisotropy is established on a small (monomer) scale of nematic coherence length, while the strains are defined (and the elastic response is arising) on a much greater length scale of polymer chain end-to-end distance (see Fig. 2) (Warner et al. 1994; Warner and Terentjev 2007).

## Nematic Elastomers: Reversible Shape Change

*n*

_{ z }= 1 and det [

*] = 1, that*

**F****ℓ**

_{0}is isotropic, \( {\ell}_{||}^{(0)}={\ell}_{\perp}^{(0)}=a \), this spontaneous extension along the director takes the especially simple form (Abramchuk and Khokhlov 1987; Warner et al. 1988):

*Q*depends on the molecular structure of the system. For weak chain anisotropy, it is certainly linear: \( {\mathbf{\ell}}_{||}/{\mathbf{\ell}}_{\perp }=1+\alpha Q \) however, there are situations (e.g., in highly anisotropic main-chain nematic polymers) when the ratio \( {\mathbf{\ell}}_{||}/{\mathbf{\ell}}_{\perp}\propto \exp \left[1/\left(1-Q\right)\right] \) takes very high values. The key relation (3) lies at the heart of nematic elastomers performing as thermal actuators or artificial muscles (Fig. 3) (Küpfer and Finkelmann 1991; Tajbakhsh and Terentjev 2001; Thomsen et al. 2001). The analogy is further enhanced by the fact that the typical stress exerted on such actuation is in the range of 10–100 kPa, the same as in human muscle (Tajbakhsh and Terentjev 2001). A large amount of work has been recently reporting on photo-actuation of nematic elastomers containing, e.g., azobenzene moieties (Fig. 4) or on the effect of adding/removing solvents (Finkelmann et al. 2001b; Hogan et al. 2002; Li et al. 2003) – in all cases the effect is due to the changing of underlying nematic order parameter

*Q*, affecting the magnitude of the ratio \( \left({\mathbf{\ell}}_{||}/{\mathbf{\ell}}_{\perp}\right) \) and thus the length of elastomer sample or the exerted force if the length is constrained (Cviklinski et al. 2002). Finally, creating an inhomogeneous internal deformation results in the bending actuation response (Fig. 5) (Camacho-Lopez et al. 2004; Tabiryan et al. 2005; Yu et al. 2003).

*M*is a number of entanglements on a typical chain of length

*N*(

**M**= 1 in the ideal Trace formula) and the notation \( \overline{\left|\dots \right|} \) denotes an orientational average of the matrix … applied to an arbitrary tube segment. The important feature of the complicated expression (4) is the “sandwiching” of the deformation gradient tensor

*between the two orientational tensors defined in the current and the reference states, respectively. The continuum model of fully nonlinear elasticity is also quite complicated in its applications, but the fundamental requirements that any elastic free energy expression must (i) be invariant with respect to body rotations of the reference state and the current state, separately, and (ii) reduce to the classical expression Tr(*

**F**

**F**^{ T }·

*) when the underlying chains become isotropic requires it to have the form*

**F***m*= 1, 2, 3. Once again, one recognizes the key symmetry of separating the two powers of deformation gradient tensor by the orientation tensors defined in the current and reference frames, respectively.

## Nematic Elastomers: Soft Elasticity and Dynamics

**ℓ**, are the reduction of the effective elastic response and the penalty on the director fluctuations. The first effect has received the name “soft elasticity” and is the result of the special symmetry of the coupling between the orientational modes and deformation gradients. It is easy to see (Olmsted 1994) that there is a whole continuous class of deformations described by the form

**V**an arbitrary unitary matrix, which leave the elastic free energy

*W*

_{el}at its constant minimum value (for an incompressible system). Remarkably, this remains true whether one examines the Trace formula (1) or any other expression above, (4) or (5), as long as they respect the correct symmetries of the two independent degrees of freedom (Golubović and Lubensky 1989; Stenull and Lubensky 2004). Figure 2c illustrates one example of such soft deformation.

This phenomenon is unique to anisotropic elastomers and gels, which have an internal microstructure (liquid crystalline order) that is capable of continuously changing its tensor structure. It can also be seen in smectic elastomers, although the layer constraints make it more complicated mathematically. Possible the closest elastic system is the famous shape-memory alloy (Bhattacharya 2003), where the martensite transition creates a crystalline unit cell with several (usually just two) equal-energy states. The ability of the lattice to switch between these states in response to certain deformation modes gives the “soft-deformation” modes to these alloys, albeit only a few discrete ones. In elastomers and gels, the “soft deformation” reflects the ability of anisotropic polymer chains to rotate their long axis to accommodate some imposed elastic deformations without changing their shape.

If one instead chooses to focus on the director modes in a nematic elastomer with a fixed (constrained) shape, the coupling between the internal mobile order parameter and the network elasticity provides a large energy penalty for uniform director rotations δn (with respect to the elastically constrained network). This penalty, which appears as a large mass term in the expression for mean-square director fluctuation, results in the suppression of such fluctuations and the related scattering of light from a nematic elastomer (Schönstein et al. 2001). In contrast to optically turbid ordinary liquid nematics, where light is scattered by long-wavelength director fluctuations, the aligned monodomain anisotropic elastomer or gel is totally transparent. However, when the elastic deformations in the network are not constrained and are free to relax, there are certain combinations of polarization and wave vectors of director fluctuations (corresponding to the soft-deformation modes), for which the “effective mass” vanishes and the fluctuation spectrum should appear as in ordinary liquid nematics (Olmsted 1994).

**n**_{0}, the director will switch and point along the axis of uniaxial extension. The early theory (ignoring the effect of soft elasticity) (Bladon et al. 1994) has predicted, and the experiment on polyacrylate LCE (Mitchell et al. 1993) confirmed that this switching may occur in an abrupt discontinuous fashion when the natural long dimension of anisotropic polymer chains can fit into the new shape of the sample, much extended in the perpendicular direction. However, the same experiment performed on a polysiloxane LCE (Kundler and Finkelmann 1995) has shown an unexpected stripe domain pattern. Further investigation has proven that the nematic director rotates continuously from

**n**_{0}toward the new perpendicular axis, over a substantial range of deformations, but the direction of this rotation alternates in semi-regular stripes of several microns’ width oriented along the stretching direction (Fig. 6a). Later the same textures have been observed by other groups and on different materials, including polyacrylates (Kundler and Finkelmann 1998; Zubarev et al. 1996), although there also have been reports confirming the single director switching mode (Roberts et al. 1997).

Theoretical description of stripe domains (Verwey et al. 1996) has been very successful and has led to several long-reaching consequences. First of all, this phenomenon has become a conclusive proof of the soft-elasticity principles. As such, the nematic elastomers have been recognized as part of a greater family of elastic systems with microstructure (shape-memory alloys being the most famous member of this family to date), all exhibiting similar phenomena. Finally, the need to understand the threshold value of extension λ_{ c } has led to deeper understanding of internal microscopic constraints in LCE and resulted in introduction of the concept of semi-softness: a class of deformations that has the “soft” symmetry, as in Eq. (6), penalized by a small but physically distinct elastic energy due to such random constraints.

_{ c }. In fact, if one uses the pure Trace formula (1), the threshold is not present at all, λ

_{ c }= 1. The backbone chain anisotropy \( {\mathbf{\ell}}_{||}/{\mathbf{\ell}}_{\perp } \), which enters the theory, is an independent experimentally accessible quantity related, e.g., to the spontaneous shape change of LCE on heating it into the isotropic phase, \( {\uplambda}_m\approx {\left({\mathbf{\ell}}_{||}/{\mathbf{\ell}}_{\perp}\right)}^{1/3} \) in Fig. 2. This allowed the data for director angle to be collapsed onto the same master curve, spanning the whole range of nonlinear deformations.

The physical reason for the stretched LCE to break into stripe domains with the opposite director rotation ±*θ*(λ) becomes clear when one recalls the idea of soft elasticity (Olmsted 1994; Warner et al. 1994). The polymer chains forming the network are anisotropic, in most cases having the average shape of uniaxial prolate ellipsoid (see Fig. 2). If the assumption of affine deformation is made, the strain applied to the whole sample is locally applied to all network strands. The origin of (entropic) rubber elasticity is the corresponding change of shape of the chains, away from their equilibrium shape frozen at network formation, which results in the reduction in entropy and rise in the elastic free energy. However, the nematic (e.g., prolate anisotropic) chains may find another way of accommodating the deformation: If the sample is stretched perpendicular to the director n (the long axis of chain gyration volume), the chains may rotate their *undeformed* ellipsoidal shapes – thus providing an extension but necessarily in combination with simple shear – and keep their entropy constant and elastic free energy zero! This, of course, is unique to nematic elastomers: Isotropic chains (with spherical shape) have to deform to accommodate any deformation. It is also important that with deformations costing no elastic energy, there is no need for the material to change its preferred nematic order parameter *Q*, so the ratio \( {\mathbf{\ell}}_{||}/{\mathbf{\ell}}_{\perp } \) remains constant for the duration of soft deformations (this is not the case when, e.g., one stretched the elastomer along the director axis). The physical explanation of stripe domains is now clear: The stretched nematic network attempts to follow the soft-deformation route to reduce its elastic energy, but this requires a global shear deformation which is prohibited by rigid clamps on two opposite ends of the sample (Fig. 6a). The resolution of this macroscopic constraint is to break into small parallel stripes, each with a completely soft deformation (and a corresponding shear) but with the sign of director rotation (and thus – the shear) alternating between the stripes. Then, there is no global shear, and the system can lower its elastic energy in the bulk, although it now has to pay the penalty for domain walls and for the nonuniform region of deformation near the clamps. The balance of gains and losses determines the domain size *d*.

_{ c }, which is necessary to overcome the barrier for creating domain walls between the “soft” stripes. However, it turns out that the numbers do not match. The energy of domain walls must be determined by the gradient Frank elasticity of the nematic (with the Frank elastic constant independently measured,

*~ 10*

**K**^{−11}N) and thus should be very small, since the characteristic length scale of nematic rubbers is \( \xi =\sqrt{K/G}\sim {10}^{-9}\mathrm{m} \). Hence, the threshold provided by domain walls alone would be vanishingly small, whereas most experiments have reported λ

_{ c }~ 1.1 or more. This mystery has led to a whole new concept of what is now called semi-softness of LCE. The idea is that due to several different microscopic mechanisms (Verwey and Warner 1997), a small addition to the classical nematic rubber-elastic free energy is breaking the symmetry required for the soft deformations:

(usually \( \alpha \ll 1 \)). The soft-elastic pathways are still representing the low-energy deformations, but the small penalty ~αG provides the threshold for stripe domain formation (λ_{c} ~ 1 + α) and also makes the slope of the stress-strain soft-elastic plateau small but nonzero (Fig. 6b). Compare this with the ordinary extension modulus of rubber before and after the region of stripe domains. Regardless of small complications of semi-soft corrections, the main advance in identifying the whole class of special low-energy soft deformations in LCE and proving their existence by direct experiment is worth noting.

The presence of the underlying nematic order, with its director capable of independently moving with respect to the matrix (Clarke et al. 2001b), leads to another remarkable feature of anisotropic elastomers and gels: the anomalous dissipation of mechanical energy (see Fig. 6c). This effect is represented in very high values of the so-called loss factor tan δ = G″/G′, representing the ratio of the imaginary and real parts of the linear modulus of response to the oscillating shear, tan *δ* > 1 in a fully elastic material is a very unusual situation, no doubt to be explored in applications in vibration and impact damping (Clarke et al. 2001a). Another aspect of high internal dissipation due to the rotating director modes is the very high amplitude and very long times of stress relaxation of stress in all liquid crystalline elastomers and gels (Clarke and Terentjev 1998).

## Swelling and Anisotropic Gels

Polymer gels are usually isotropic and swell and shrink equally in all directions. Volume phase transition phenomena of polymer gels have been extensively investigated, and it is known that an infinitesimal change in an environmental intensive variable such as solvent composition, pH, and temperature yields a discontinuous volume change for some polymer gels (Annaka and Tanaka 1992; Ilmain et al. 1991; Tanaka 1978). These volume transitions are driven by the balance between the repulsive and attractive forces acting on the network chains such as van der Waals, hydrophobic, ionic, hydrogen bonding (Shibayama and Tanaka 1993).

If gels of anisotropic molecular structure were prepared, anisotropic mechanical behavior will be expected. Additional volume transition can be triggered by nematic ordering of anisotropic molecules inside the gel (Urayama et al. 2003).

This effect adds a new driving molecular force for volume transition phenomena. Temperature-sensitive gels exhibiting volume transition at a certain temperature have attracted much attention of scientist and technologists because of their applications to drug delivery systems and sensors, originally concentrating on N-isopropylacrylamide-based isotropic systems. Spontaneously anisotropic gels undergoing a sharp and large volume change accompanied by nematic-isotropic transition are a new type of such temperature-sensitive gels, which will extend the potential applications.

*T*

_{NI}

^{Gel}(see Fig. 7a). Importantly, the volume decrease upon nematic ordering is accompanied by the shape change of a gel with a considerable degree of anisotropy, as indicated on the generic plot. The driving force behind the abrupt and pronounced shrinking of the nematic gel is the thermodynamic force (effective chemical potential) that tries to expel the “wrong” solvent and allow the network to lower its free energy of the nematic phase. Liquid crystalline networks swollen in a liquid crystalline solvent is an even richer system, expected to act as a soft actuator driven under electric field utilizing the strong dielectric anisotropy of mesogenic units controllable by external field (Kishi et al. 1997). Figure 7b shows a generic plot of the swelling ratio

*V*/

*V*

_{0}. This plot indicates that the large shrinking occurs in the window of temperatures when the solvent is still isotropic. However, the solvent is taken into the gel once again when it becomes nematic itself, since this now promotes the order in the network and lowers the joint free energy (Matsuyama and Kato 2002; Urayama et al. 2003).

^{4–6}Pa) is much greater than the energy density associated with local dielectric or diamagnetic anisotropy of the material. Essentially, the fields are too weak to rotate the director pinned to the elastic matrix. However, in highly swollen gels, the elastic modulus is much lower, and one expects (and indeed finds) an electro-optical and electromechanical response (Chang et al. 1997a; Terentjev et al. 1994; Urayama et al. 2005b). An additional interesting possibility is when a gel sample is mechanically unconstrained and therefore capable of finding a soft-deformation route in response to a dielectric torque applied to its local director. In this case the only resistance to dielectric torque would come from the weak semi-softness. Figure 8 shows what happens when a monodomain nematic gel, freely floating in a solvent, is subjected to an electric field perpendicular to its director (Urayama et al. 2005b). The director rotates, and the shape of the gel changes accordingly (see Fig. 2c) contracting in the parallel direction but remaining constant in the perpendicular direction (λ

_{⊥}= 1) as required by the soft-deformation geometry.

## Cholesteric Elastomers: Photonics

When chiral molecular moieties (or dopant) are added to nematic polymers, they form cholesteric phases. Cross-linked networks made from such polymers accordingly have a spontaneously twisted director distribution. Their director distribution being periodic with characteristic length scale in the optical range, they display their inhomogeneous modulation in brilliant colors of selective reflection. Being macroscopically non-centrosymmetric, such elastic materials possess correspondingly rare physical properties, such as the piezoelectricity.

As with nematic LCE, in order to access many unique properties of cholesteric elastomers, one needs to form well-aligned monodomain samples, which would remain ordered due to cross-linking. As with nematic systems, the only way to achieve this is to introduce the network cross-links when the desired phase is established by either external fields or boundary conditions – otherwise, a polydomain elastomer would result. As with nematics, there are certain practical uses of cholesteric polydomain state – for instance, the applications of stereoselective swelling (see below) may work even better in a system with micron-sized cholesteric domains (Courty et al. 2003a).

There are two established ways to create monodomain cholesteric elastomer films with the director in the film plane and the helical axis uniformly aligned perpendicular to the film. Each has its advantages, but also unavoidable drawbacks. Traditionally, a technique of photopolymerization (and network cross-linking when difunctional molecules are used, such as di-acrylates) was used in cells where the desired cholesteric alignment was achieved and maintained by proper boundary conditions. Many elaborate and exciting applications have been developed using this approach, e.g., (Broer et al. 1995; Hikmet and Kemperman 1998). However, it is well known that the influence of surface anchoring only extends a maximum of few microns into the bulk of a liquid crystal (unless a main-chain mesogenic polymer, with dramatically increased planar anchoring strength, is employed (Terentjev 1995)). Therefore, only very thin cholesteric LCE films can be prepared by this surface alignment and photopolymerization technique.

A different approach to preparing monodomain cholesteric elastomers has been developed by Kim and Finkelmann (2001), utilizing the two-step cross-linking principle originally used in nematic LCE. One prepares a sample of cholesteric polymer gel, cross-linked only partially, and deposits it on a substrate to deswell. Since the area in contact with the substrate remains constrained, the sample cannot change its two lateral dimensions in the plane to accommodate the loss of volume; only the thickness can change. These constraints in effect apply a large uniaxial compression, equivalent to a biaxial extension in the flat plane, which serves to align the director in the plane and the helical axis perpendicular to it. After this is achieved, the full cross-linking of the network is initiated, and the well-aligned free-standing cholesteric elastomer results. This is a delicate technique, which requires a careful balance of cross-linking/deswelling rates, solvent content, and temperature of the phases. In principle it has no significant restrictions on the resulting elastomer area or thickness. However, there is a limitation of a different nature: One cannot achieve too long cholesteric pitch, e.g., in the visible red or infrared range. The reason is the required high strength of chiral twisting at the stage of biaxial extension on deswelling: With no or weak twisting strength, the material is close to the ordinary nematic, and although the symmetric biaxial extension does confine the director to its plane, it does nothing to counter the quenched disorder of the establishing cross-links and the resulting propensity to form a two-dimensional (planar) polydomain state. Only at sufficiently high chiral twisting power the cholesteric propensity overcomes this planar disorder – and the resulting cholesteric pitch will end up relatively short. Another unavoidable drawback of this two-step cross-linking technique is the disordering effect of the portion of cross-links established at the first stage of gel formation (one has to have them to sustain elastic stress and prevent flow on deswelling), so the optical quality of samples prepared in this way is always worse than that in thin films photopolymerized in a perfect surface-aligned cholesteric state.

Another possibility is topological imprinting of helical director distribution. Being anchored to the rubber matrix due to cross-linking, the director texture can remain macroscopically chiral even when all chiral dopants are removed from the material (Courty et al. 2003b; Mao and Warner 2000). Internally stored mechanical twist can cause cholesteric elastomers to interact selectively with solvents according to its imprinted handedness. Separation of racemic mixtures by stereoselective swelling is an interesting application possibility.

The lack of centrosymmetric invariance in chiral elastic materials leads to interesting and rare physical properties (Terentjev 1993), in particular piezoelectric effect and nonlinear optical properties. Such characteristics of elastomers with a chiral smectic-C* order (the ferroelectric liquid crystalline elastomers, FLCE) have been studied with some intensity for several years now. After the permanently monodomain (fixed by cross-linking) free-standing samples of FLCE were prepared (Finkelmann et al. 1995; Gebhard and Zentel 1998), several useful experiments targeting various electromechanical and electro-optical properties, in particular the piezoelectricity and the nonlinear optical response, have been reported in recent years (Brehmer et al. 1996; Lehmann et al. 1998). Clearly, the prospect of important applications will continue driving this work. In this short review, we shall concentrate on the analogous effect in chiral nematic LCE which does not possess a spontaneous polarization.

*D*

_{∞}in a chiral material, in contrast to a very low symmetry:

*C*

_{ 2 }plus the translational effect of layers in ferroelectric smectic-C*), there is a real possibility to identify microscopic origins of piezoelectricity in amorphous polymers or indeed elastomers, if one aims to have an equilibrium effect in a stress-resistant material. The piezoelectric effect in a uniform chiral nematic LCE has been described phenomenologically (Terentjev 1993) and by a fully nonlinear microscopic theory similar in its spirit to the basic Trace formula for nematic rubber elasticity (Terentjev and Warner 1999). All experimental research so far has concentrated on the helically twisted cholesteric elastomers (Chang et al. 1997b; Meier and Finkelmann 1990; Vallerien et al. 1990). However, the cholesteric texture under the required shear deformation (Pelcovits and Meyer 1995) (Fig. 10a) will produce highly nonuniform distortions giving rise to the well-understood flexoelectric effect and masking the possible chiral piezoelectricity.

*ε*= λ − 1), is unknown in completely amorphous polymers and rubbers. Even the famous PVDF polymer-based piezoelectric has its response due to inhomogeneity of crystalline regions affected by macroscopic deformation. The molecular theory (Terentjev and Warner 1999) has examined the effect of chirality in molecular structure of chain monomers and the bias in their short-axis alignment when the chains are stretched at an angle to the average nematic director

*. If the monomers possess a transverse dipole moment, this bias leads to macroscopic polarization:*

**n***n*

_{ c }is the number of network strands per unit volume (the cross-linking density) and the parameter Δ is the measure of monomer chirality with the transverse dipole moment

*d*

_{ t }(see Terentjev and Warner 1999 for detail). Compare this with the original Trace formula of Eq. (1) to note the characteristic “sandwiching” of strain between the two different orientation tensors. This expression involves the full deformation gradient tensor

**λ**and therefore can describe large deformations of a chiral nematic rubber. When shear deformations are small, the linear approximation of (9) gives, for a symmetric shear,

*G*~ 10

^{5}Pa), the relevant coefficient

*d*= ∂

*P*/∂

*σ*= γ/

*G*is much higher than the corresponding response to stress in, for instance, quartz or PVDF. The corresponding low mechanical impedance should make the piezoelectric rubber attractive for many energy transducing applications.

## New Frontiers

When positional ordering, in the form of layering, is added to the orientational order of nematics, we have the smectic phase (Fig. 1c, d). When made into networks, they remain locally liquid-like, can suffer large extensions and have mobile internal degrees of freedom – much like with translationally homogeneous nematic elastomers and gels. This freedom, which gave spontaneous distortions and soft elasticity in nematics, is restricted in smectics by the layers to which the matrix is often strongly coupled. The mechanics of smectic elastomers (Lubensky et al. 1994; Stenull and Lubensky 2005) is decidedly more complex than that of nematic elastomers. In many cases the effect of layers means that the material has the nonlinear mechanics of a 2D rubber in the layer plane while performing as a one-dimensional crystalline solid in the third dimension (Adams and Warner 2005a) when the layer translation with respect to the rubbery matrix is penalized by a periodic potential provided by the lamellar order. Alternatively, concentrating on the layer structure, one finds the behavior radically altered from the liquid crystal case. The layers interact via the effective background provided by the elastomer network and, as a result, no longer display the celebrated Landau-Peierls loss of long-range order in one or two dimensions. The layer buckling instabilities occur when deformations along the layer normal are applied. An even more rich system is the smectic-C elastomer and its variants. This phase has the director tilted in the layer planes and a much lower symmetry, allowing for a great variety of soft-deformation modes (Adams and Warner 2005b; Stenull and Lubensky 2006). The chiral variants of smectic-C networks are ferroelectric solids: soft, greatly extensible, and noncrystalline and thus of great technological significance.

This article examines some of the recent and relevant findings about the new class of materials – liquid crystalline elastomers and gels. Nematic rubbers have already proved themselves an unusual and exciting system, with a number of unique optical and mechanical properties, indeed a real example of the Cosserat media with couple-stress elasticity. Polymer networks with cholesteric order are an equally provocative system promising new physical properties.

The basic molecular theory of nematic rubber appears to be exceptionally simple in its foundation and does not involve any model parameters apart from the backbone polymer chain anisotropy \( {\mathbf{\ell}}_{\parallel }/{\mathbf{\ell}}_{\perp } \), which can be independently measured. This is a great advantage over many other soft condensed matter systems requiring complicated, sometimes ambiguous theoretical approaches. Of course, in many real situations and materials, one finds a need to look deeper into the microscopic properties; an example of this is the “semi-softness” of nematic networks. Most of these systems are characterized by nonuniform deformations: Even in the simplest experimental setup, a large portion of the sample near the clamps is subjected to nonuniform strains and, therefore, responds with a nonuniform director field.

Looking into the future, many challenging and fundamental problems in this field are still outstanding. The field of dynamics and relaxation in rubbery networks, although not young by any means, is still not offering an unambiguous physical picture of stress relaxation. Adding the liquid crystalline order, we find an additional (director) field undergoing its own relaxation process and coupled to that of an elastic network. In the particular case of polydomain (i.e., randomly disordered in equilibrium) elastomers, we can identify a hierarchical sequence of physical processes in the underlying network (above its *T* _{ g }) and the superimposed glass-like nematic order. This leads to a particularly slow relaxation, but much remains to be done to understand the physics of such complex systems.

The general problem of dynamic mechanical properties, rheology, and relaxation in Cosserat-like incompressible solids, also characterized by the effect of soft elasticity, brings to mind a number of possible applications. An example would be the selective attenuation of certain acoustic waves, with polarization and propagation direction satisfying the condition for softness, essentially leading to an acoustic filtering system. Another example of application of soft elasticity, also related to the problem of relaxation, is the damping of shear vibrations in an engineering component when its surface is covered by a layer of nematic or smectic rubber, particularly aligned to allow the director rotation and softness.

Other very important area of applications is based on polarizational properties of materials with chirality. Most nonlinear optical applications (which have a great technological potential) deal with elastomers in the ferroelectric smectic-**C*** phase. The low symmetry (in particular, chirality) and large spontaneous polarization of **C*** smectics have a strong effect on the underlying elastic network and vice versa. This also returns back to the general problem of mechanical properties of smectic rubbers and gels. In conclusion, after the initial period of exploration and material synthesis, liquid crystalline elastomers, in all their variety, now offer themselves as an exciting ground for both fundamental research and for technology. Smectic and lamellar liquid crystalline elastomers and gels are a much more recent area of study, with much more theoretical and experimental work required to underpin their dramatically anisotropic and nonlinear mechanical properties combining a two-dimensional rubber-elastic response and a solid-like properties in the third direction.

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