PhysRevE.96.032701.pdf

PHYSICAL REVIEW E

, 032701 (2017)

Electroclinic effect in chiral smectic-

A

liquid crystal elastomers

Noy Cohen

and Kaushik Bhattacharya

Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA

(Received 30 June 2017; published 18 September 2017)

Chiral smectic-

liquid crystal elastomers are rubbery materials composed of a lamellar arrangement of

liquid crystalline mesogens. It has been shown experimentally that these materials shear when subjected to an

electric field due to the electrically induced tilt of the director. Experiments have also shown that shearing a

chiral smectic-

elastomer gives rise to a polarization. Roughly, the shear force tilts the directors which, in

turn, induce electric dipoles. This paper builds on previous works and models the electromechanical response of

smectic-

elastomers using free energy contributions that are associated with the lamellar structure, the relative

tilt between the director and the layer normal, and the coupling between the director and the electric field. To

illustrate the merit of the proposed model, two cases are considered—a deformation induced polarization and

an electrically induced deformation. The predictions according to these two models qualitatively agree with

experimental findings. Finally, a cylinder composed of helical smectic layers is also considered. It is shown that

the electromechanical response varies as a function of the helix angle.

DOI:

10.1103/PhysRevE.96.032701

I. INTRODUCTION

Smectic-

liquid crystal elastomers are rubbery materials

composed of mesogens that are uniformly oriented and

organized in a layered structure. In their pioneering work,

Garoff and Meyer [

] found that low molecular weight

smectic-

∗

liquid crystals comprising chiral molecules exhibit

an electroclinic effect, i.e., there is a direct coupling between

the molecular tilt and an applied electric field. In smectic

elastomers comprising chiral mesogens, an electric field acting

along the layer plane induces a shear deformation. This

phenomenon directly emanates from the chirality of the meso-

gens. Specifically, the electric field couples to the transverse

dipole of the chiral mesogens and induces a tilt between

the mesogens and the layer normal. As a consequence, the

elastomer shears and contracts along the direction of the layer

normal. Typically, such elastomers are incompressible and

subsequently this deformation is accompanied by an expansion

along the transverse directions. Another consequence of the

chirality of the mesogens was demonstrated by Lehmann

et al

[

], where it was found that flipping the direction of the electric

field reverses the direction of the mechanical tilt.

Several experimental works were carried out to study the

response of a smectic-

elastomer subjected to an electric

field. Spillmann, Ratna, and Naciri [

] mounted a thin sheet

of elastomer in a tension clamp with the smectic layers

either perpendicular or parallel to the tension clamps and

applied an electric field normal to the sheet. An electric

field

7MV

m was applied to a film the layers of

which are parallel to the clamps, and a contraction of

∼

was measured. Next, the sample was placed such that the

layers are perpendicular to the clamps. Upon application of

the electric field, an expansion of

∼

2% was measured.

Hiraoka

et al.

[

] prepared a smectic elastomer film with

an initial thickness

600

m and applied an electric field

1MV

M. They measured an electrically induced tilt

angle of

∼

◦

a contraction along the layer normal of

∼

2%,

noyco@caltech.edu

and a shear angle of

∼

◦

. A recent experiment by Spillmann

et al

] demonstrated that smectic elastomers can also twist

in response to an electric field.

Previous works have also investigated the purely me-

chanical behavior of smectic elastomers. The experimental

work of Nishikawa, Finkelmann, and Brand [

], Nishikawa

and Finkelmann [

], and Spillmann, Ratna, and Naciri [

]

demonstrated that the mechanical modulus of smectic-

elastomers parallel to the layers normal is significantly higher

than that along the layer plane. This anisotropy emanates from

the lamellar microstructure of the elastomer. Interestingly, the

experiments of Nishikawa and Finkelmann [

] revealed that

the elastomer becomes significantly softer above a threshold

∼

3% strain along the layers normal.

On the theoretical side, Adams and Warner [

] developed

an energy-based model for smectic-

elastomers which as-

sumes that the layer normal and the director are locked

and may only rotate in response to an external force. This

assumption prevents a tilt and a smectic-

like ordering. The

work of Stenull and Lubensky [

] argued that the director is

not rigidly coupled to the layer normal and, as a consequence, a

shear deformation can rotate the smectic layers and the director

with respect to the layer normal. The later works of Adams

et al

] and Stenull

et al

] provided another model

that accounts for the stiffness of the smectic layers and the

resistance to the tilt by adding appropriate contributions to the

free energy.

From an electric viewpoint, the work of Corbett and Warner

[

] theoretically examined the electromechanical behavior of

a nematic elastomer subjected to electric fields. In this paper, it

was shown that the mechanical and electrical anisotropies can

cause a thin film to expand in the direction of the electric field

rather than contract. As a result of the Poisson effect, the film

shrinks along the two directions in the plane of the electrodes.

Selinger

et al

] found that under high electric fields the

state of uniform induced tilt can become unstable and lead to

chiral modulations. In their work, they modeled and simulated

this phenomenon.

The aim of this paper is to model the coupled behavior

of smectic-

elastomers under finite deformations resulting

2470-0045/2017/96(3)/032701(7)

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, 032701 (2017)

from the electroclinic effect. To this end, we characterize the

chirality of the mesogens by a chirality vector which couples

to the tilt of the director. The proposed model, presented in

Sec.

, is based on the minimization of the free energy of the

smectic elastomer. To demonstrate the behavior predicted by

the model, two cases are examined in Sec.

III

—an electrically

induced deformation and a deformation induced polarization.

In the first case, an electric field is applied to the smectic-

elastomer and the resulting deformation is determined through

the minimization of the free energy. Subjected to this loading,

the tilt angle is larger than the shear angle. This trend is in

agreement with the experimental findings of Hiraoka

et al

[

]. We also show that the orientation of the chirality of the

mesogens can either enhance or diminish the polarization.

In the second case, we follow the experimental work of

Kramer and Finkelmann [

] and consider the response of a

smectic-

elastomer subjected to a simple shear deformation.

The proposed model predicts that the prescribed deformation

rotates the director with respect to the layer normal and

therefore induces a tilt of the mesogens. Subsequently, the

inherent chirality gives rise to spontaneous dipoles which, in

turn, induce a polarization in the elastomer. In accordance

with experimental findings, the proposed model predicts that

the shear angle is larger than the tilt angle. A cylindrical setup

of a smectic-

elastomer with helical layers is considered

in Sec.

. The electromechanical response of an elastomer

with various pitches and helix angles is computed. The main

conclusions are summarized in Sec.

II. A COUPLED ELECTROMECHANICAL MODEL

Consider a smectic-

elastomer that deforms from a refer-

ence to a current configuration due to a coupled electromechan-

ical loading. The material points are denoted

and current po-

sitions are denoted as

(

), so that the deformation gradient is

defined as

=∇

(

), where

∇

denotes the gradient operation

with respect to the referential coordinate system. The smectic

order is characterized by two order parameters—the director

and the chiral vector. These are denoted

and

in the ref-

erence configuration and

and

in the current configuration.

Note that bold letters with hats on top denote unit vectors. The

chirality vector is perpendicular to the director such that

0. The layer normal at the current configuration is given as

−

(1)

where

is the magnitude of the chirality vector

. Note that

gives

and characterizes smectic-

while

gives rise to smectic-

. This smectic order gives rise to a spon-

taneous elongation along the director given by the step-length

tensor

−

[

(

−

⊗

], where

is a parameter

related to the strength of the ordering [

]. It also gives rise to

a spontaneous polarization

, where

is a parameter.

We assume that the free energy-density function of an

elastomer subjected to a uniform electric field by surface

electrodes can be viewed as the sum of five contributions:

(2)

The first contribution is the classical elastic free energy of an

anisotropic elastomer [

Tr(

−

)

(3)

where

is the shear modulus and

−

[

(

−

⊗

] is the step-length tensor before the

deformation. Here,

is the identity tensor.

The second contribution to the free energy in Eq. (

)

concerns the change in the layer thickness:

−

(4)

where

is a measure of the stiffness of the layers. This is

identical to the free energy contribution in the works of Adams

and Warner [

] and Adams

et al

], but written in terms

of the magnitude of the physical chirality vector

. Above,

and

−

denote the transpose and inverse of

, respectively.

This formulation is advantageous because it is indifferent to

the sign of the director

and therefore satisfies the objectivity

requirements associated with the energy-density function.

The third term in Eq. (

) concerns the change in chirality,

or equivalently the tilt (or rotation) of the director with respect

to the layer normal:

(5)

where

is the resistance of the mesogens to tilt. This is again

the same as Adams and Warner [

] and Adams

et al

], but

written in terms of the magnitude of the chirality vector. Note

that Eq. (

) assumes that the ground state is smectic-

,butit

can be modified if the ground state is smectic-

The fourth term in Eq. (

) pertains to the energetic

contribution of the electrical sources to the total energy (see,

for example, Shu and Bhattacharya [

]). Specifically,

=−

−

(

−

)

(

−

)

(6)

where

is the polarization,

is the susceptibility, and

the permittivity in vacuum.

To determine the polarization and the deformation under

a prescribed coupled electromechanical loading, the energy-

density function is minimized with respect to

, and the

chirality vector

. By computing

∂W

∂

0, we find that

(7)

The last term in Eq. (

) concerns the external mechanical

loading:

=−

(8)

where

is the first Piola-Kirchhoff stress tensor.

For convenience, we normalize the energy-density function

and substitute Eq. (

) into Eq. (

) to obtain

Tr(

−

)

−

)

−

(9)

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PHYSICAL REVIEW E

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where

and

are the normalized layer stiffness and

the normalized resistance to tilt, respectively.

The constants in the proposed model can be determined

through experiments. The order of magnitude of the shear

modulus and the layer stiffness are

∼

1–1 MPa and

∼

10 MPa, respectively [

]. Fitting to experimental findings

resulted in a resistance to tilt parameter

∼

1–1 MPa

[

]. The anisotropy parameter

has been measured

between 1

05 and 60 in elastomers composed of prolate chains

[

]. Lastly, we can deduce the order of magnitude of the

chirality related parameter from experimental findings. It is

found that

∼

−

–10

−

III. THE ELECTROMECHANICAL RESPONSE

OF SMECTIC-

A

ELASTOMERS

To examine the predictions of the proposed model, consider

a thin film of a smectic-

elastomer with an initial thickness

We prescribe a global coordinate system

{

}

such that at

the reference configuration the director and the layer normal

are along the

direction, i.e.,

and

(see Fig.

The faces of the film are covered with flexible electrodes such

that an applied voltage induces an electric field

where

is the stretch along the thickness of the sheet.

Mechanically, an applied stress

shears the film.

The elastomer experiences the general deformation gradient

⎛

⎝

⎞

⎠

(10)

where

tan

, and

is the shear

angle (see Fig.

). Due to the incompressibility constraint,

. As a result of an electromechanical loading, the

director rotates such that at the current configuration

sin

cos

. The chirality vector is

sin (

)

, where

is the tilt angle between

and

, in accordance to Eq. (

Note that in accordance with its definition, the chirality vector

is zero when the layer normal is parallel to the director. By

making use of Eq. (

), we find that the layer normal at the

current configuration

With the above definitions, the normalized energy-density

function in Eq. (

) can be written as

(

,λ

,θ

)

−

cos(2

)

FIG. 1. A schematic of the electromechanical response of a

smectic-

elastomer. The chirality vector

points out of the plane.

−

rλ

sin(2

)

cos

−

sin

−

cV λ

sin

−

sλ

(11)

where

μt

and

μt

)

. We also normalize the

polarization [Eq. (

)]:

μt

(12)

where

μt

χε

. The constants

and

have units of voltage

and are related via

−

μt

In order to demonstrate the merit of the model, we adopt

the typical values

−

, and

The constants

7 kV and

15 kV correspond to a

film with a thickness

1 mm and a susceptibility

A. Voltage induced deformation

Due to the electroclinic effect, smectic-

elastomers

deform in response to an applied electric field [

The aim of this section is to examine the voltage induced

deformations according to the proposed model. To this end,

we take

0 and minimize the energy-density function

(

,λ

,θ

)[Eq.(

)] with respect to

, and

obtain the equilibrium state. This leads to nonlinear equations

which can be solved numerically. We can also gain insight by

simplifying the expressions for small voltages and retaining

only the leading order terms in

. We obtain

≈

ξV,

(13)

≈

−

ξV,

(14)

≈

(

(

)

−

)

(15)

≈

(

)

−

(16)

cξ

(17)

where

is the initial slope between the tilt angle and

the applied voltage. This quantity is related to the electroclinic

coefficient

∂θ

∂E

. Specifically, in the examined parallel plate

geometry

≈

. Subjected to relatively low voltages, Köhler

et al

], Spillmann, Ratna, and Naciri [

], and Hiraoka

et al

[

] measured an electroclinic coefficient

045 m

MV,

006 m

MV, and

07 m

MV, respectively. These

findings correspond to tilt angles in the range 4–10

◦

. Since

the thickness of the smectic-

elastomers is around

∼

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FIG. 2. (a) The shear angle

and (b) the tilt angle

as a

function of the applied voltage

. The inset plots depict the predicted

response subjected to high voltages. The continuous and the dashed

curves correspond to the exact and the approximated predictions,

respectively.

01–1 mm, we approximate the order of magnitude of the

initial slope

∼

01–1 kV

−

The parameters

and

can be determined from a curve fit

and

as a function of the voltage via Eqs. (

) and (

)

in the low voltage regime. The quadratic dependency of

and

on the applied voltage stems from the initial microstructural

alignment of the director and the layer normal in the smectic-

elastomers.

Figures

2(a)

and

2(b)

depict the shear angle

and the tilt

angle

as a function of the applied voltage. The insets plot

the electromechanical response of the smectic-

elastomers

under high voltages. The continuous and the dashed curves

correspond to the exact predictions and the approximations

[Eqs. (

) and (

)], respectively. A close inspection of the

insets shows that the approximations hold for voltages up to

≈

5 kV. It is noted that the application of the electric field

leads to a tilt angle that is larger than the shear angle.

To explain this, note that Eqs. (

) and (

) reveal that in the

range of small voltages the ratio between the shear and the tilt

angles is

−

. Interestingly, in smectic-

elastomers with

1 the ratio

1. This implies that the electrically induced

deformation is characterized by a tilt angle

that is greater

than the shear angle

. This observation is supported by the

experimental findings of Hiraoka

et al

], where the measured

FIG. 3.

as a function of the applied voltage

. The inset

depicts the predicted response as a result of high voltages. The

continuous and the dashed curves correspond to the polarization

[Eq. (

)] with

−

and 0 V

−

, respectively.

ratio

≈

02 led to the anisotropy parameter

02. It

should also be noted that the obtained value of the anisotropy

parameter is very low.

The predicted polarization is plotted as a function of the

voltage in Fig.

. As before, the inset plots the predicted

polarization as a result of high voltages. The continuous and

the dashed curves correspond to the polarization with

−

and 0 V

−

, respectively. It is shown that the electrically

induced tilt leads to an enhancement of the polarization of

the elastomer by as much as 1.8 times. We also find that

reversing the electric field flips the direction of the polarization,

as seen in the experimental findings of Hiraoka

et al

]. As

previously discussed, this effect emanates from the chirality

of the mesogens. We emphasize that since the direction of the

chirality parameter

is linearly coupled to the polarization, the

spontaneous dipoles in a smectic-

elastomer with

→−

decrease the polarization.

B. Deformation induced polarization

Next, we follow the experiment of Kramer and Finkelmann

[

] and the theoretical work of Stenull

et al

] and apply

a shear stress

along the

direction which deforms the

elastomer such that

1, corresponding to simple

shear. By setting

0, the normalized energy density is

(

,λ

,θ

)

(18)

To determine the equilibrium state, we minimize Eq. (

) with

respect to the shear

and the tilt angle

Figures

4(a)

and

4(b)

depict the shear angle

and the tilt

angle

as a function of

, respectively. We emphasize that the

nonlinear dependence of the shear angle

on the shear force

corresponds to a linear dependence between the shear

and

. It is demonstrated that the mesogens tilt in the direction

of the external shear force, as expected, and negating the force

simply flips the direction of the tilt angle

Interestingly, as opposed to the previous case the proposed

model predicts that under the examined deformation the shear

angle is always greater than the tilt angle. Furthermore,

since

1 is fixed and layer rotation is not permitted, the

tilt of the mesogens is not accompanied by a reduction in

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FIG. 4. (a) The shear angle

and (b) the tilt angle

as a function

of the normalized nominal stress

the layer thickness. These observations are in accord with the

experimental work of Kramer and Finkelmann [

], where the

elastomer was subjected to a pure shear deformation such that

the maximum imposed shear angle was

◦

An examination of Fig.

4(b)

reveals that the dependence

between the tilt of the mesogens and the applied shear force

can be divided into two regimes. Initially, a small increase

in the applied force leads to a large variation in the tilt angle.

Beyond a threshold

≈

5, the cost of further tilting the director

becomes very high. Specifically, a large increase in the shear

force is required to attain a small increment in the tilt of the

mesogens. Practically, elastomers fail at a shear angle

◦

and may not reach the second regime.

From Eq. (

) we deduce that the polarization in the absence

of an electric field is directed along the chirality parameter

which is perpendicular to the plane spanned by

and

Figure

plots the normalized polarization component

as a function of

It is demonstrated that the direction

of the applied shear force determines the direction of the

polarization. Two reasons lead to this behavior—the first is

the antisymmetric dependency of the tilt angle on the applied

force. As discussed in relation to Fig.

4(b)

, the directions of

the shear force and the tilt are linearly dependent. Second, the

observed trend is enforced by the direct coupling between the

director and the chirality parameter

. This coupling is defined

such that the magnitude of the polarization is proportional

to sin (

). We emphasize that the chirality of the mesogens

determines the direction of the polarization. Specifically, a

FIG. 5.

as a function of the normalized nominal stress

shear force along the

direction induces a polarization along

the

−

direction in elastomers characterized by a flipped

chirality vector (i.e.,

→−

IV. THE ELECTRICALLY INDUCED DEFORMATION

OF A CYLINDER WITH HELICAL LAYERS

To illustrate the capabilities of the electroclinic effect,

we explore the electromechanical response of a cylindrical

smectic-

elastomer with helical layers. Consider a hollow

thin-walled cylinder with an initial radius

, a wall thickness

, and a height

. The cylinder is made of a thin incompress-

ible smectic-

elastomer film with helical layers. We prescribe

a global cylindrical coordinate system

{

}

such that at

the reference configuration the director and the layer normal

are

sin(

)

cos(

)

(19)

where

is the initial helix angle (see Fig.

). The positions

of the referential material points are denoted by

{

R, ,Z

}

.The

inner and outer faces of the cylindrical film are covered with

flexible electrodes such that an applied voltage induces a radial

electric field

≈

. In addition, an internal pressure

exerted on the inner surface. As a result, the cylinder deforms

,

FIG. 6. A schematic of the electromechanical response of a

cylinder with helical layers.

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FIG. 7. (a) The twist

and (b) the change in the enclosed volume

as a function of the applied voltage for various helix angles at

such that at the current deformation

λR

β, θ

δZ, z

(20)

where

{

r,θ ,z

}

denote the material points at the current config-

uration. Accordingly, the deformation gradient in cylindrical

coordinates is

⎛

⎜

⎝

∂r

∂R

∂θ

∂

∂θ

∂Z

∂z

∂Z

⎞

⎟

⎠

⎛

⎜

⎝

δr

⎞

⎟

⎠

(21)

At the current configuration, the chirality vector is

sin

(

)

(22)

and the director is

sin(

)

cos(

)

(23)

where

is the angle between the director

and the layer

normal

and

is the helix angle at the current configuration.

By employing Eq. (

), we find that

sin(

)

cos(

)

(24)

The layer normal at the current configuration is directly

dependent on the deformation gradient via [

]

−

(25)

and therefore one finds that the helix angle at the current

configuration is

arctan

sin

(

)

λr

[

cos

(

)

−

Rδ

sin

(

)

]

(26)

The energy-density function can be written as

(

)

(

λ,β,φ,δ

)

(

λ,β,φ,δ

)

−

(27)

where

(

λ,β,φ,δ

) is computed with Eq. (

is the pressure

inside the cylinder, and (

)

accounts for the change in

the enclosed volume of the cylinder. The electromechanical

response of the cylinder is determined by minimizing the

energy

(

)

(

λ,β,φ,δ

) with respect to

, and

Figures

7(a)

and

7(b)

depict the twist

and the change in the

enclosed volume

πr

πR

, where

is the current height of

the cylinder, as a function of the applied voltage, respectively,

for various initial helix angles at

0. It is shown that

elastomers with cylindrical layers (or helical layers with an

angle

◦

) experience twist with a minor increase in the

enclosed volume. As the helix angle increases from

0to

◦

, the twist

decreases while the enclosed volume increases.

Interestingly, the twist angle in the smectic-

elastomers with a

helix angle

◦

is extremely small. The proposed model

also predicts that such an elastomer experiences the largest

volumetric expansion. The trend is reversed as the helix angle

increases from

45 to 90

◦

. Specifically, the cylinder twists

in the other direction and the enclosed volume decreases.

We highlight that the volumetric expansion of the smectic-

elastomers is due to the radial expansion and the longitudinal

compression resulting from the amplification of the voltage.

Additionally, it is important to note that, due to the low voltages

that are practically applied in the actuation of these materials,

the twist and the volumetric expansions are small.

Next, we study the effects of pressure on the actuation of

smectic-

elastomers with helical layers at an angle

◦

FIG. 8. (a) The twist

and (b) the change in the enclosed volume

as a function of the applied voltage for a cylinder comprising helical

layers with a

◦

angle for various pressures.

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Figures

8(a)

and

8(b)

plot the twist

and the change in the

enclosed volume

as a function of the applied voltage,

respectively, for a cylinder subjected to various pressures.

Figure

8(a)

reveals that the pressure causes the elastomer to

twist. However, the increase in voltage does not significantly

change the twist angle.

As shown in Fig.

8(b)

, the enclosed volume increases

with pressure. This stems from the radial expansion and the

longitudinal elongation of the cylinder in response to the

internal pressure. As the voltage is amplified, a linear increase

in the enclosed volume is observed. Interestingly, we find

that the linear dependence between

and

is independent

of the pressure in the examined range of voltages. It is

again noted that, even under an applied pressure, an increase

in voltage leads to a radial expansion and a longitudinal

compression.

V. CONCLUDING REMARKS

This paper proposes a new model that captures the behavior

of smectic-

elastomers subjected to an electromechanical

loading. Specifically, the energetic formulation of Adams

et al

[

] is rewritten in terms of a new variable

, that describes

the chirality of the mesogens composing the elastomer, and

an additional term is added to account for the effects of an

electric stimulus. The current configuration of the elastomer is

determined by minimizing the energy.

To demonstrate the merit of the model, two simple cases

are presented. In the first case, an electric field is applied

to the smectic-

elastomer. In accordance with experimental

findings, the proposed model predicts a shear deformation of

the sample. In the second case, a mechanical shear deformation

is imposed. As a consequence, the mesogens tilt and their

inherent chirality gives rise to a polarization. The predictions

qualitatively capture several experimental observations.

The potential of these elastomers is exhibited in Sec.

where a thin-wall cylindrical configuration of a smectic-

film with helical layers is considered. Upon application of an

electric field, we show that the pitch and the helix angle control

the macroscopic behavior. Specifically, the enclosed volume

and the twist depend on the formation of the lamellar structure.

It is also shown that the application of internal pressure in the

tube can effectively change the macroscopic response.

ACKNOWLEDGMENT

We are grateful for the financial support of the US Air

Force Office of Scientific Research through the MURI Grant

No. FA9550-16-1-0566.

[1] S. Garoff and R. B. Meyer, Electroclinic Effect at the

−

Phase Change in a Chiral Smectic Liquid Crystal,

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