## Abstract

Symmetry arguments are advanced that, although ideal chiral nematic elastomers cannot show strain-induced electrical polarization, non-ideal ones can. Phenomenological arguments are then presented, which predict a simple and universal form for the direction and strain dependence of the polarization. A microscopic minimal model is also developed, which predicts the same form. Finally, an example of a polarization–strain curve is calculated for a typical experimental geometry. In this geometry, the polarization is exactly zero at both small and large strains, but pronounced for a large set of intermediate strains corresponding to the strains that cause incremental rotation of the nematic director.

## 1. Introduction

Nematic elastomers are remarkable materials that combine rubber elasticity and liquid crystalline order. Similar to rubbers, they are made by chemically cross-linking melts of polymer chains, so, as in rubber, their constituent chains are locally liquid-like and are in constant thermal motion exploring different conformations. Unlike a conventional rubber, the polymer chains incorporate some rigid-rod components, either as a constituent of, or hanging pendant-like from, the main chain. Below a certain temperature, these rigid rods align to form a nematic liquid crystal phase inside the elastomer. Importantly, because the elastomer is locally liquid-like, this nematic direction is not frozen into the elastomer but rather can rotate to any orientation. These materials exhibit many features of conventional elastomers and liquid crystals (long-range orientational order, glass transitions, nematic–isotropic transitions, birefringence, large-strain elasticity, etc.) and some striking new phenomena including ‘soft’ elasticity, where some deformations can occur (almost) without energy penalty, and very large spontaneous distortions on heating and cooling. These properties and many others are reviewed in Warner & Terentjev (2007). The work in this paper predicts another phenomenon, namely electrical polarization, that does not occur in conventional nematics or conventional elastomers.

In the theoretical study of nematic elastomers, an important distinction is drawn between ideal and non-ideal (or semi-soft) samples. In an ideal nematic elastomer, the director can rotate through the polymer network without energy penalty, leading to genuinely zero-energy modes of deformation. In non-ideal elastomers, it is energetically slightly favourable for the director to align along a certain direction, so the deformations associated with rotating the director away from this direction are of low energy rather than truly soft. Real nematic elastomers are always to some extent non-ideal.

Two previous papers have looked at polarizations in nematic elastomers. Warner & Terentjev (1999; and also Terentjev & Warner 1999) developed a microscopic theory of Gaussian chains of chiral rod-like monomers to model chiral ideal elastomers. It was found that the polarization always relaxed out of the system by the rotation of the nematic director. Here we argue that this is because ideal elastomers have an isotropic reference state rather than because of the details of the model. The microscopic model developed in this paper will be a non-ideal counterpart of the model in this work. Adams (2004) explored several mechanisms to circumvent this relaxation including non-equilibrium dynamics, smectic ordering and a very specific (and rather artificial) mechanism for semi-softness. Here we argue that in fact all semi-soft chiral nematic elastomers, and hence all real-life chiral nematic elastomers, will show polarizations that will not relax to zero.

Rubbers are much softer than conventional piezoelectric materials such as quartz, so they will exhibit polarizations at much higher strains but much lower stresses. Their low mechanical impedance will also help them couple to liquid and gaseous systems rather more efficiently than ceramic and crystalline transducers. These properties may eventually lead to applications as sensors in high-strain low-stress environments.

The content of this paper will be presented in four sections. In §2, an overview of the theory of nematic elastomers will be given, and then in §3, symmetry arguments will be used to show that ideal elastomers cannot show electrical polarization, but non-ideal ones can. These will then be developed to predict the full form of the polarization as a function of strain. In §4, a microscopic minimal model of chiral non-ideal elastomers—Gaussian chains with main-chain chiral liquid crystal rods and compositional fluctuations—will be constructed and shown to exhibit the form of strain-induced polarization predicted by the phenomenological analysis. Finally, in §5, an example of a polarization–strain curve will be calculated for a specific geometry, namely stretching a nematic elastomer perpendicular to its director.

## 2. Nematic elastomers

A successful microscopic model of nematic elastomers has been developed, which treats the polymers as Gaussian chains whose conformations are biased in the direction of the nematic director. This bias is introduced via the ‘step-length tensor’ , where is the identity matrix; is the nematic director; and *r* is a (scalar) measure of the anisotropy of the conformation distribution. The probability of a free polymer strand of length *L* adopting a conformation that results in a spanning vector ** R** is then (Warner & Terentjev 1996)(2.1)a Gaussian distribution completely characterized by its second moment, the ellipsoidal tensor . If the elastomer is prepared by introducing

*n*cross links (per unit volume) between the chains while the nematic director is along , and then a deformation gradient is applied and the nematic director rotates to , the free energy is given by (Warner & Terentjev 1996)(2.2)However, the final-state director, , is not set experimentally but is free to rotate to whichever direction gives the lowest free energy, so the observed free energy function is actually(2.3)This theory has been used to successfully describe and predict many behaviours of nematic elastomers, including spontaneous deformations on heating and cooling, rotation of the director as a function of strain and complicated textured deformations. This idealized theory predicts a class of zero-energy deformations (Olmsted 1994; Warner

*et al*. 1994)—any deformation of the form , where is a rotation, returns the same free energy as the elastomer at rest—there is a large class of non-trivial deformations that do not cost energy. Conceptually, this is because rotating the nematic order causes the polymer conformations to be biased in a different direction so it results in a macroscopic change in the shape of the elastomer.

Golubovic & Lubensky (1989) showed that soft deformations can be understood as a consequence of the existence of an isotropic reference state. If an elastomer is prepared in a high-temperature isotropic state and then cooled inducing nematic order, then a direction must be chosen for the nematic order to align along. The elastomer will spontaneously stretch in this direction. However, since any other direction could have been chosen, there are many equivalent states each with a different deformation with respect to the original isotropic state. Provided the nematic direction is free to rotate, deformations that map between these states must be soft. The existence of an isotropic reference state (and hence the existence of soft modes) within the ideal model can be demonstrated by substituting and observing that all dependence on cancels.

Although the ideal theory has been qualitatively very successful, real elastomers are not ideal and do not show perfect soft elasticity. Rather they are nearly ideal in the sense that the director can rotate but retains an energetic preference to be oriented along its original direction. One way this idea can be incorporated into the microscopic model, developed by Verwey & Warner (1997), is by assuming that, rather than each strand coupling to the nematic order to the same degree, each does so with a different strength and hence has a step-length tensor characterized by a different value for *r*, a mechanism known as compositional fluctuations. In a melt, all strands can achieve their preferred degree of anisotropy but, once they are cross-linked, no deformation preserves the anisotropy of all the strands so no deformations are truly soft, rather they are semi-soft as they show many features of soft deformations but do cost some energy. Including this in the model results in an additional term in the free energy(2.4)where the tensors have the average anisotropy and *α*, the measure of non-ideality, is given by . Although this specific mechanism is unlikely to be the real explanation for non-ideality, it produces very good predictions for stress–strain and strain–rotation curves (Kupfer & Finkelmann 1994; Finkelmann *et al*. 1997). This is because the resulting form of the free energy is in fact generic—it is the most general admissible quadratic free energy involving only , and (Biggins *et al*. 2008).

## 3. Phenomenological approach

Several authors have used a phenomenological approach to study strain-induced polarization in nematic elastomers (Terentjev 1993) and cholesteric elastomers (Brand 1989; Pelcovits & Meyer 1995). This work differs from all the above in that it describes strain using the full (nonlinear) deformation gradient tensor, and works directly with the polarization pseudovector rather than the free energy.

### (a) Symmetry arguments for the existence of a polarization

Ideal nematic elastomers have an isotopic reference state that is found by applying the deformation to the relaxed state, in effect compressing it along the nematic director so that the polymers follow an isotropic conformation distribution. An isotropic state clearly cannot show any polarization since every direction is equal, so no one direction can be distinguished for the polarization to point along. If a deformation applied to this state (breaking the isotropy) were to cause a strain-induced polarization ** p**, we would like to be able to write

**as a function of so that we could predict what polarization a given deformation would cause. Unfortunately, there is no function that can map deformations onto vectors. Intuitively, this is because a deformation from an isotropic state is completely characterized by three double-headed perpendicular vectors (the principal stretches), and even with a handedness and hence the ability to take cross products, this does not uniquely define any single-headed axial or polar vectors. We can easily prove this more formally by considering (without loss of generality) a diagonal deformation from the isotropic state. If the function**

*p**f*maps onto a polarization

**,(3.1)then, for any rotation it must satisfy(3.2)because the reference state is isotropic. Similarly(3.3)because a final-state rotation should rotate a final-state vector. If is a**

*p**π*rotation about one of the principal directions of , then but(3.4)which is only possible if

**is along the axis of . However, we could have chosen the axis of to be any of the three perpendicular principal stretches of , and**

*p***cannot be parallel to all of them, so the function**

*p**f*and hence an electrical polarization cannot exist.

The above argument relies crucially on the existence of an isotropic reference state, equation (3.2), which is the hallmark of an ideal elastomer. However, non-ideal elastomers do not have an isotropic reference state so the above reasoning does not apply. Indeed it is easy to see that a non-ideal elastomer does have low enough symmetry to exhibit polarization. Consider a strip of non-ideal elastomer prepared with its director, *n*_{0}, parallel to the *z*-axis, which is then subject to pure shear causing the director to rotate to ** n** as shown in figure 1. Since the elastomer is non-ideal, the energy of the elastomer increases. In this state, the director ‘knows’ about a definite rotational sense—the direction in which it would rotate if the elastomer were allowed to relax—so if the system also has a handedness, caused by the chiral nature of the nematic rods, a right-hand rule can be applied to the director and its turning sense to define a single-headed pseudovector

**along which a polarization could lie. This argument fails if the elastomer is ideal because the elastomer can relax to a zero energy state with director**

*v***, so relaxation does not define a rotational sense. There is also no rotational sense before any strain is imposed, so the relaxed state is not polarized. More subtly, there is no rotational sense after some very large strains have been imposed—if a large**

*n**x*stretch had been imposed, the director would lie along the

*x*-axis so, on relaxation, it could rotate back clockwise or anticlockwise to relax to

*z*. However, the reasoning does apply to all states where the director has started rotating and applying further deformation causes further rotation.

### (b) Strain dependence of the polarization

Any polarization in a semi-soft chiral nematic elastomer must be built out of the following ingredients, , , and *ϵ*_{ijk}. To be properly rotationally invariant, the polarization must be constructed by contracting these objects correctly, which is by only contracting final-state indices (the *i* on and the *i* on *n*_{i}) with other final-state indices, and likewise for reference-state indices (the *j* on and the *j* on *n*_{0j}). Finally, since and are nematic directors, they are quadrupolar (double-headed) vectors so they must appear only in even numbers so that the polarization is invariant under the transformations and . At zeroth order in , the only permissible term is(3.5)which is zero because contracting a symmetric tensor with *ϵ*_{ijk} gives zero. Similarly at first order in , the only possible term, , is also zero. At second order in , there are five permissible terms that are listed below(3.6)The first and second of these give zero. The third and fifth are parallel because both and are symmetric tensors. Therefore, we have only two potentially independent vectors left,(3.7)However, the mechanical free energy, equation (2.4), is both experimentally known to be a good model and theoretically shown to be a generic form for a non-ideal elastomer (Biggins *et al*. 2008). In practice, is not a variable that can be set experimentally; rather will always rotate to the direction that gives the lowest energy. The terms in the free energy (equation (2.4)) involving are(3.8)The matrix pre-multiplying is clearly symmetric and hence, in some frame, diagonal. Therefore, the energy is minimized when lies along the eigenvector of this matrix with the smallest eigenvalue, giving the relaxation condition that(3.9)for the smallest eigenvalue *A*. Contracting this relaxation condition with *ϵ*_{ijk}, we see that both the above terms (equation (3.7)) are also parallel, so the only admissible vector at quadratic order in is(3.10)However, elasticity in nematic elastomers is typically large strain, so there is little reason to truncate this series at quadratic order. At higher orders, there are other admissible vectors that can be constructed. However, rubber elasticity, particularly Gaussian rubber elasticity, is extremely well described by just the low-order terms even at large strain, motivating the consideration of just the lowest order terms.

## 4. Microscopic minimal model

The phenomenological analysis above suggests a simple and universal form for the direction and strain dependence of polarization in chiral nematic elastomers—equation (3.10). In §4*a*, we develop a microscopic minimal model that also predicts the same form. The use of this is threefold, it lends weight to the phenomenological analysis, yields an overall magnitude for the polarization (which tends to zero as the elastomer becomes ideal) and illustrates a possible microscopic mechanism for the effect. The model developed is a non-ideal counterpart of that studied by Warner & Terentjev (1999).

### (a) Setting up the model

The model is based on two different rigid-rod monomers. The first, species 1, we model as a rigid rod of length *a*. The second, species 2, we model as an L-shaped molecule with dimensions *a* and *b* and an electrical dipole ** d** assigned using a ‘right-hand rule’ (figure 2). The two monomers are polymerized to make random copolymers, which we model as freely jointed (Gaussian) chains of

*N*monomers, with binomially distributed compositions between species 1 and 2. The rigid-rod nature of the monomers allows them to form a joint nematic phase, which results in the polymers having anisotropic conformation distributions. However, the two monomers couple to the nematic order with different strengths, so a polymer made entirely of species 1 would have step-length tensor with anisotropy

*r*

_{1}and one made of species 2 would have step-length tensor with anisotropy

*r*

_{2}. For simplicity, we assume that the constants of proportionality for both these tensors are the same. The random copolymers have different overall anisotropies depending on their composition. The polymers are cross-linked in the nematic state to form a chiral non-ideal nematic liquid crystal elastomer, which, we will show, exhibits strain-induced polarization.

This model captures the two essential ingredients for a nematic elastomer to show electrical polarization, non-ideality (introduced via compositional fluctuations) and chirality, introduced by the right-hand rule used to assign the electrical dipoles to the L-shaped monomers.

The assumptions that both species of monomer have the same length, and that both step-length tensors have the same constants of proportionality (i.e. the same components perpendicular to ), are somewhat unrealistic. However, these assumptions significantly simplify the algebra. The model can be solved without these assumptions in the same manner as outlined below, with the end result being simply that the coefficient of polarization is slightly altered.

### (b) Polymers with homogeneous composition

If a polymer is constructed out of freely jointed monomers of species 2, numbered by *α*=1, …, *N*, and each monomer has an end-to-end vector (figure 2) then the total end-to-end vector for the chain is(4.1)and, defining the binormal of the chain to be(4.2)the electrical polarization of the chain is(4.3)If the monomers in the chain are freely jointed, their orientations are independent, and the probability of the chain being in a configuration with a given ** R** and

**is (Warner & Terentjev 1999)(4.4)where, if the nematic director is**

*V***, then and . This model can give a non-zero binormal for a chain if the spanning vector for the chain is not along the liquid crystal director. An intuitive sense of this can be developed by considering a two-dimensional system with perfect nematic order, so all long sides of all the Ls align perfectly along the nematic director,**

*n***. This means each monomer can contribute one of four vectors to the chain path, shown in figure 3.**

*z*If the overall conformation has a significant component along *z*, then there must be many more monomers in the first and third orientations than the second and fourth. If the conformation also includes a component along *x*, then there must be more monomers in the first orientation than the third, so an overall binormal is developed.

The equivalent result for a chain made entirely of species 1, and thus with no binormal, is simply(4.5)where .

### (c) Random copolymers

If a chain has *N*_{1} monomers of species 1 and *N*_{2}=*N*−*N*_{1} monomers of species 2, since the chain is freely jointed, the order in which the monomers are arranged does not affect the conformation probability distribution. Therefore, the chain is equivalent to a chain of *N*_{1} monomers of species 1 connected to a chain of *N*_{2} monomers of species 2, so the new chains distribution is simply(4.6)

This distribution can be calculated explicitly (see the appendix A for details) by completing the square in the exponent. The resulting distribution is(4.7)where *L*=*Na* is the total contour length of the chain; *r* is the length-weighted average of *r*_{1} and *r*_{2}, i.e.(4.8)and . We see that the distribution for the copolymer is identical to that for a homogeneous chain but with modified coefficients.

### (d) Polarization of a strand of copolymer

The polarization of a strand is given as *d*〈** V**〉. This is straightforward to calculate since(4.9)Evaluating this is a simple case of doing a Gaussian integral (using equation (4.7) as

*P*(

**) which has ) and gives(4.10)If the polymer strand is cross-linked into a nematic elastomer with nematic director and span vector**

*V*

*R*_{0}between its cross-linked ends, and then a deformation is applied to the system, causing the span to change to and the director to change to , the new binormal will be(4.11)Averaging this expression over all possible initial spanning vectors using gives(4.12)which gives a polarization per strand of(4.13)

### (e) Mechanical response of a semi-soft nematic elastomer

The arguments of §3 show that a chiral non-ideal elastomer can develop a polarization under deformation. We thus consider a random copolymer strand cross-linked into a nematic elastomer network. The mechanical part of its free energy is simply(4.14)which is the same as for a non-chiral strand in a nematic elastomer. However, in this case, the anisotropy, *r*, is a function of the individual strand's composition, so averaging across all the strands in the elastomer gives(4.15)where and . This is the usual non-ideal free energy for a nematic elastomer, so it will also follow the mechanical relaxation condition given in equation (3.9).

### (f) Polarization of a semi-soft chiral nematic elastomer

The polarization per strand given in (4.13) takes the form(4.16)where the polarizability, *p*(*r*), is a coefficient that depends on the composition of the strand, and hence its anisotropy ratio. To find the behaviour of the whole elastomer, we must average this result over all compositions (all strands in the elastomer). Introducing polarization-weighted averages as(4.17)we get the strand-averaged polarization(4.18)The first and last of these terms evaluate to zero since because they consist of a symmetric tensor contracted with *ϵ*_{ijk}. The middle two terms are exactly the two terms related by the mechanical relaxation condition (3.9). Since is also a symmetric tensor, we can use the relaxation condition to eliminate one of the two terms, giving(4.19)This expression for the polarization has a simple vector part but quite a complex coefficient. Note that the coefficient gives zero for an ideal elastomer because then all strands have the same value of *r*, so both types of average give the same result. They also give the same result (and hence the coefficient is zero) if all strands have the same polarizability.

### (g) Evaluating the coefficient for the random copolymer model

In the analysis of a random copolymer strand, we predicted the anisotropy of the strand, *r*, and the polarizability of the strand, *p*(*r*), in terms of its composition. Assuming that the strands for the elastomer were generated by randomly polymerizing a mixture of the two monomer units, a reasonable model for the compositional distribution of the chains is that all the chains have *N* monomers, of which *N*_{1} are of species 1, and *N*_{1} is distributed binomially with probability *q*,(4.20)The effective anisotropy of the chain is given by(4.21)Defining *Δ*=*N*_{1}−*Nq*, in a large *N* limit, we expect to be small, so we can expand the coefficient as a Taylor series in this quantity and then average and keep the leading term. We can easily use this method to evaluate 〈1/*r*〉(4.22)substituting *N*_{2} for *N* and *N*_{1} for *q* and *Δ* gives(4.23)(4.24)Expanding in *Δ*/*N* to second order,(4.25)Finally, taking the average over *Δ*, we see that to leading order(4.26)so the semi-soft parameter *α* is given by(4.27)The full coefficient in equation (4.19) can be expanded out in a similar manner to give(4.28)which is non-zero, so the model does predict a polarization. At first sight, it may appear small because it contains a factor of 1/*N*. However, so does the measure of non-ideality, *α*. In the compositional fluctuations model, non-ideality goes to zero as the chain length becomes infinite because the variance of the chain anisotropy vanishes. Therefore, this coefficient is only small because our model is in fact almost ideal. Taking this at face value and replacing the small (1/*N*) factor by *mα*, we can (very crudely) estimate the magnitude of the polarization for a real non-ideal elastomer as , where ** P** is the total polarization per unit volume;

*n*is the number of strands per unit volume; and we have assumed that the factors involving just

*r*,

*q*and are of order unity. Substituting reasonable values (

*n*∼10

^{26},

*b*/

*a*∼0.1,

*α*∼0.1 and

*d*∼

*e*×1 Å), we get |

**|∼10**

*P*^{−5}cm

^{−2}. This is of a similar magnitude to mechanically induced polarizations induced in quartz (|

**|∼10**

*P*^{−4}cm

^{−2}at 0.2% strain), but at much higher strain ( versus 0.2%) and much lower stress (10

^{5}versus 10

^{8}Pa).

## 5. Example of a polarization–strain curve

If a non-ideal nematic elastomer is prepared with its director oriented along the *z*-axis, and is then stretched by a factor of *λ* along the *x*-axis, the elastomer passes through three regimes. Verwey & Warner (1997) showed that there is a threshold deformation(5.1)for extensions with *λ*≤*λ*_{1} the deformation is simply that expected from a classical rubber, , and the director does not rotate. However, in the second regime, , the energy of the elastomer is significantly reduced if the deformation includes *λ*_{xz} shear and the director rotates to *θ* with the *z*-axis (figure 4). More precisely,(5.2)where the shear is given by(5.3)

In the third regime, , director rotation is complete so the director lies along the *x*-axis and the elastomer again deforms classically,(5.4)These results were derived in by Verwey & Warner (1997) and are discussed in Warner & Terentjev (2007).

In the first and third regions, the vector part of the polarization gives zero. This is simply because the deformation is symmetric and is along one of its eigendirections, so(5.5)However, in the second region, is not symmetric and the polarization is non-zero. Calculating the polarization is now a simple matter of substituting the expression for into (3.10) to get(5.6)A plot of the polarization function is shown below in figure 5. The form of the polarization is rather unusual, with a very steep profile and non-continuous changes of gradient at the onset and end of director rotation. In reality, if a sample is stretched in this manner without applying the energy-lowering shear with the clamps, then the elastomer will split into stripes (see Conti *et al*. (2002) for theory or Finkelmann *et al*. (1997) for experiment) each of which undergo opposite shear so that the average shear of the sample is zero. These stripes will have opposite polarization. This is shown in figure 6. This type of behaviour can be eliminated by simultaneously imposing a stretch and a sympathetic shear. This may facilitate experimental observation and possible applications. The imposition of a macroscopic shear also allows the sign of the polarization to be determined by the sign of the shear.

## 6. Conclusions

Semi-soft chiral nematic elastomers have low enough symmetry to exhibit strain-induced polarization. Such polarizations are expected to be caused by any deformation that causes incomplete director rotation, which means that more of the same deformation would lead to more director rotation and vice versa. Rotation of the director through the elastomer is at the root of the phenomena because, as the director rotates away from its preferred orientation, it can distinguish between rotations back to or further away from the preferred direction. The introduction of this rotational sense into the elastomer lowers the symmetry enough for a polarization to form.

On phenomenological grounds, the form of the polarization is expected to be , where is the deformation tensor and is the final-state liquid crystal director. A microscopic minimal model using compositional fluctuations to model non-ideality and L-shaped liquid crystal mesogens to incorporate chirality also predicts a strain-induced polarization of this form. The microscopic model suggests that polarizations of the order of |** P**|∼10

^{−5}cm

^{−2}could be achievable, which is comparable with the polarizations observed in quartz, but is predicted to occur at much higher strain and much lower stress.

## Acknowledgments

I would like to thank Mark Warner, James Adams and Eugene Terentjev for their helpful discussions.

### Appendix A. Conformation distribution of the copolymer

We need to calculate *P*(** R**,

**) given by equation (4.6). To complete the square in the exponent, we substitute . The result of this substitution can be simplified by noting that(A1)where is the step-length tensor with average anisotropy**

*V**r*given by(A2)This is easy to show by direct multiplication of the tensors since and are co-diagonal. The substitution gives the new exponent(A3)Having completed the square, we can now treat

*R*_{2}as a Gaussian-distributed variable with second moment proportional to so that the integral can be written as an expectation,(A4)Substituting for

*R*_{2}into the expectation part of this expression using(A5)we see that it contains terms of zeroth, linear and quadratic order in

*R*_{2}. The linear terms give zero because the first moment of a Gaussian is zero. The quadratic terms are also zero because the second moment tensor is coaxial with so their product is symmetric and gives zero when contracted with

*ϵ*

_{ijk}. This leaves only the zero-order term which, just taking the part inside the expectation, is(A6)Again, because the matrices are co-diagonal, this can be multiplied out in a diagonal frame to show that it is equal to(A7)Substituting this for the expectation in equation (A 4), we get the probability distribution stated in equation (4.7).

## Footnotes

- Received November 4, 2008.
- Accepted December 22, 2008.

- © 2009 The Royal Society