Numerical analysis of characteristics of microactuators driven by liquid crystals

Chemical Engineering Science 64 (2009) 4625 -- 4631

Contents lists available at ScienceDirect

Chemical Engineering Science journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / c e s

Numerical analysis of characteristics of microactuators driven by liquid crystals Shigeomi Chono ∗ , Tomohiro Tsuji Department of Mechanical Engineering, Kochi University of Technology, Tosayamada-cho, Kami-shi, Kochi 782-8502, Japan

A R T I C L E

I N F O

Article history: Received 5 September 2008 Received in revised form 12 March 2009 Accepted 16 March 2009 Available online 26 March 2009 This article is dedicated to the 70th birthday of Professor Morton M. Denn Keywords: Liquid crystal Complex fluids Non-Newtonian fluids Microactuator Microstructure Leslie–Ericksen theory Numerical analysis Backflow

A B S T R A C T

For the purpose of developing liquid crystalline microactuators, the transient behaviors of a nematic liquid crystal between two parallel plates have been computed for various parameters such as applied voltage, the gap between the plates, and the twist and tilt angles at the plates. The Leslie–Ericksen theory has been selected as a constitutive equation. The twist angle has an effect on the induced velocity profiles; for example, the induced flow is planar at the twist angle of 0◦ , while the flow has an out-of-plane component when the twist angle is not 0◦ . Transient behaviors of shear stress acting on the plates, the flowrate, and the maximum values of the velocity, and the tilt angle between the plates have been reported. In addition, we have investigated the effects of the applied voltage, the gap between the plates, and the tilt angle at the plates on the above-mentioned values. We can develop microactuators with arbitrary characteristics by suitably controlling the applied voltage, the size of the actuators, and the director anchoring conditions. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Liquid crystals, which consist of rod-like or disk-like molecules, are anisotropic materials whose physical properties such as dielectric constants, magnetic susceptibilities, viscosities, and elastic constants depend on the orientation state of the molecules (de Gennes and Prost, 1993; Chandrasekhar, 1992). Liquid crystalline displays are a successful example of an industrial application where these anisotropic features of liquid crystals are applied to the field of optics. In contrast, in the present study we approach the anisotropy of liquid crystals from a mechanical viewpoint. Solid mechanics and fluid dynamics are systematized disciplines that are particularly important elements in the foundations of mechanical engineering, and many useful industrial products have been developed on the basis of principles discovered in these fields. Liquid crystals should have more applications than solids and liquids because they are unique materials with reciprocal features such as solidity and fluidity. However, most studies on liquid crystals have been confined to the optical field. The authors aim to develop a discipline of the mechanics or dynamics of liquid crystals, to apply the findings to the elementary techniques in mechanics, and to develop novel microactuators that are considerably different from conventional microactuators, such as thermal, electrostatic, electromagnetic, and piezoelectric types or devices utilizing shape memory alloys. Some studies have aimed to develop artificial muscles ∗ Corresponding author. Tel.: +81 887 57 2318; fax: 81 887 57 2320. E-mail address: [email protected] (S. Chono). 0009-2509/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2009.03.024

and MEMS by using liquid crystalline polymers including elastomers and gels (Warner and Terentjev, 2003; Yu and Ikeda, 2006; Xie and Zhang, 2005; de Gennes, 1997; Wang et al., 2003; Buguin et al., 2006). Interesting applications of liquid crystals are reviewed (Woltman et al., 2007). In previous work (Chono and Tsuji, 2008), the authors showed a possibility of development of microactuators by using low-molarmass nematic liquid crystals which are more fluid-like materials; that is, computations and visualization experiments were performed to study the unsteady behavior of a nematic liquid crystal between two parallel plates. As a result, it was found that the application of an electric field to a liquid crystal induces a flow called a backflow, and that the velocity profile depends on the twist angle; that is, the profile is S-shaped for the twist angle of 0◦ and the flow is unidirectional for the twist angle of 180◦ . Furthermore, the mechanism of the flow generation was elucidated. The objective of the present study is to obtain fundamental data required to design liquid crystal microactuators. We numerically investigate the effects of the applied voltage, the cell gap, the twist angle, and the tilt angle on the induced velocity and the stress acting on the plate. Since the twist angle investigated was 0◦ and 180◦ in previous work, we paid attention to only the velocity component parallel to the anchoring direction at the plates. In this study, we show the velocity components between plates for the various twist angles. As a constitutive equation, we have used the Leslie–Ericksen theory (Ericksen, 1960, 1961; Leslie, 1968), which has been widely employed for low-molar-mass nematic liquid crystals. More complete theories have been reported; for example, the

S. Chono, T. Tsuji / Chemical Engineering Science 64 (2009) 4625 -- 4631

y

Beris–Edwards theory (Beris and Edwards, 1994) and the original Doi theory (Doi and Edwards, 1986) with the Marrucci–Greco potential (Marrucci and Greco, 1991). In the Leslie–Ericksen theory, the orientation state is described with a unit vector, while the other theories use a second order tensor, which can express the spatially dependent order parameter as well as the orientation direction. The Leslie–Ericksen theory is applicable to the rheology of low-molarmass nematic liquid crystals, where the order parameter does not vary significantly except for defect regions.

E

H

4626

2. Governing equations

x

Nematic liquid crystals have a microstructure where the positions of molecules are random like isotropic liquids but molecules are oriented in a certain direction. Hence, the nematic materials require an additional new variable, which indicates the local anisotropy. Theories which are accountable for the rheology of the nematic materials couple the molecular orientation and velocity fields. As stated before, the Leslie–Ericksen theory is the most elementary theory for nematic liquid crystals. The isothermal flow of nematic liquid crystals under an electric field is governed by the following equations:

Fig. 1. Coordinate system.

y

• Continuity equation: vj,j = 0.

θ

(1)

Here, the subscript “, j” means partial differentiation with respect to xj . • Linear momentum equation:   * vi + vj vi,j = (⊥ Ej + nk Ek nj )Ei,j − p,i + ji,j .  (2) *t In these equations, v is the velocity vector,  the fluid density, p the pressure, s the extra stress tensor, n the unit vector indicating the average direction of the liquid crystal molecules called the director, E the electric-field vector, ⊥ and  the dielectric constants perpendicular and parallel to the director, respectively,  =  − ⊥ the anisotropy of the dielectric constants. Determination of the director field n requires the angular momentum equation,   *F *F nj Ej Ei +ni −(3 − 2 )Ni − (6 −5 )Aij nj − + = 0. (3) * ni *ni,j ,j

 is required because of the constraint of the unit length of the director. A and N are the rate of strain tensor and the angular velocity of the director relative to surrounding rotation, respectively, defined as Aij = 12 (vi,j + vj,i ), Ni =

(4)

*ni + vj ni,j − ij nj , *t

(5)

where X is the vorticity tensor

ij = 12 (vi,j − vj,i ).

(6)

The extra stress tensor s is described by

ji = 1 Akl nk nl nj ni + 2 nj Ni + 3 Nj ni + 4 Aji + 5 nj nk Aki 



*F + 6 Ajk nk ni − nk,i , *nk,j

φ

z Fig. 2. Coordinates of the director.

Table 1 Computation parameters. Gap H (m) Voltage V (V) Twist angle w (deg) Tilt angle w (deg)

5–100 3–10 0, 60, 120, 180 1–40

Here, K1 , K2 , and K3 the elastic constants representing the splay, twist, and bend deformations of the director, respectively. Details of the Leslie–Ericksen theory and solutions to flow problems are discussed in de Gennes and Prost (1993), Chandrasekhar (1992), Leslie (1968), Jenkins (1978), and Rey and Denn (2002). When a uniform electric field with intensity E (=constant) is applied in the y direction, as shown in Fig. 1, the components of the velocity vector v, the director n, and the electric-field vector E are expressed as v = (u, 0, w)T ,

(9)

n = (nx , ny , nz )T ,

(10)

E = (0, E, 0)T .

(11)

(7)

where 1 –6 are the Leslie viscosities, and F is the free energy density due to spatial distortion of the director, defined as follows: F = 12 {K1 ni,i nj,j + K2 (ni,j ni,j − ni,i nj,j − ni nj nk,i nk,j ) + K3 ni nj nk,i nk,j }.

x

(8)

If an electric field is non-uniform, some secondary flow may be generated and the y-component of the velocity is non-zero. However, in the present situation, no secondary flow is anticipated, and hence we set the y-component in Eq. (9) to 0. By substituting Eqs. (9)–(11) into Eqs. (2) and (3), we obtain the following linear momentum and

S. Chono, T. Tsuji / Chemical Engineering Science 64 (2009) 4625 -- 4631

4627

Table 2 Material constants of 5CB.

1 (Pa s) 0 × 10

2 (Pa s)

−2

3 (Pa s)

−8.6 × 10

−2

4 (Pa s) −2

−0.4 × 10

8.9 × 10

5 (Pa s)

−2

6 (Pa s)

−2

5.9 × 10

−3.1 × 10

K1 (N) −2

K2 (N)

6.37 × 10

−12

 (F/m)

K3 (N)

3.81 × 10

−12

−12

8.60 × 10

15.7 × 10

⊥ (F/m) −11

5.7 × 10−11



*w *  2  *u *  2 2  *w = 1 nx ny nz + ny nz  *t *y *y *y *y

50



2

40

+n2y nz nx

30

+ 3

20

+

10

+nx nz



y µm

2

* u * w + nz *y2 *y2



4 * w * * nz + 2 ny 2 *y2 *y *t





2

* ny *ny *w * 2 − 5 * w + n2y 2ny nz − 2 *y *t *y *y *y2

3 + 6



2

2

-60

-40

-20 um µm/s

0

20

40





* u * w + n2z , *y2 *y2

(12b)



0 -80



* *u *nz *w + 2nz (nx nz ) *y *y *y *y

2

(3 − 2 )



2

+

* nx * nx * ny *u = ny 2(K3 − K2 ) − 2 *t *y *y *y



2

+ {K2 + (K3 − K2 )n2y }

50

* nx , *y2



(3 − 2 )

40

* ny *u *w = ny E2 − 3 nx + nz *t *y *y − (K3 − K2 )ny

y µm

30

(13a)

⎧  ⎨ *n 2 ⎩

x

*y





*ny + *y

2



* nz − *y

2 ⎫ ⎬ ⎭

2

20

+ {K1 + (K3 − K2 )n2y }

φw = 0 deg



φw = 60 deg

(3 − 2 )

φw = 120 deg

10

-60

-40

*ny *nz * nz *w = ny 2(K3 − K2 ) − 2 *t *y *y *y

(13b) 

2

φw = 180 deg 0 -80

* ny , *y2

+ {K2 + (K3 − K2 )n2y }

-20 wm µm/s

0

20

40

Fig. 3. Velocity profiles.

* nz . *y2

(13c)

It is noted that the conditions of */ *x = 0 and */ *z = 0 have been used because the parallel plates are infinitely long in both x and z directions. 3. Computations

angular momentum equations:

3.1. Boundary conditions





*u * 2 2 *u * *w = 1 (nx ny ) + (nx n2y nz ) *t *y *y *y *y 

+nx n2y nx 

+ 3 +

2





2

+ 



4 * u * * nx + 2 ny 2 *y2 *y *t 2

* ny * ny * u * 2 − 5 * u + n2y 2ny nx − 2 *y *t *y *y *y2

3 + 6 2 2

+n2x

2

* u * w + nz *y2 *y2



2nx

A no-slip condition for the velocity field was used at the plates. For the orientation field, when the polar coordinates of the director are expressed in terms of two angles (y) and (y), as shown in Fig. 2, we define the tilt angle w as (0) [= (H)] and the twist angle w as (H) − (0), where H is the gap between the upper and lower plates. In the present computation, we varied the tilt angle from 1◦ to 40◦ and the twist angle from 0◦ to 180◦ .





*nx *u * *w + (nx nz ) *y *y *y *y 2

3.2. Computational procedure



* u * w + nx n z , *y2 *y2

(12a)

A finite difference method and the Crank–Nicolson method were used to discretize the governing equations, Eqs. (12) and (13). On

4628

S. Chono, T. Tsuji / Chemical Engineering Science 64 (2009) 4625 -- 4631

0.25

0.15

1500 Q µm2/s

0.2

Τ Pa

2000

φw = 0 deg φw = 60 deg φw = 120 deg φw = 180 deg

0.1

1000

500

0.05 0 70

0 90

60

Θ deg

U µm/s

50 40 30

60

30

20 10

0

0 0

0.2

0.4

ts

0.6

0.8

1

0

0.2

0.4

ts

0.6

0.8

1

6

8

10

Fig. 4. Transient behaviors of typical quantities.

1.2

400

1

Tm Pa

0.8 0.6

= 0 deg = 60 deg = 120 deg = 180 deg

300 Um µm/s

φw φw φw φw

200

0.4 100 0 1.2

0 12000

1

10000

0.8

8000

Qm µm2/s

tm s

0.2

0.6 0.4 0.2

6000 4000 2000

0

0 0

2

4

6

8

10

0

2

4 V V

V V Fig. 5. Effect of applied voltage.

the basis of a preliminary computation, we selected y = H/100 and t = 10−6 s, in which y is the mesh size and t is the time step. The computational parameters are the voltage V(=EH) applied to the liquid crystal and the gap H as well as the director boundary conditions, w and w . The values of these parameters are presented in Table 1. When we investigated the effect of a parameter, we fixed the other parameters to representative values of w = 1◦ , w = 0◦ , V = 5 V, and H = 50 m. A single-step change of the voltage from 0

to V V was applied at t = 0, then we kept the value of V constant. For initial values, we set v = 0 in the entire computation domain and chose a one-dimensional equilibrium solution without an electric field for the director orientation. In this computation, we used the material constants of 4-n-pentyl-4 -cyanobiphenyl (5CB), which is a low-molar-mass nematic liquid crystal. The material constants are presented in Table 2 (Kneppe et al., 1981, 1982; Karat and Madhusudana, 1977). The density  is 1000 kg/m3 .

S. Chono, T. Tsuji / Chemical Engineering Science 64 (2009) 4625 -- 4631

6

400

4 3

= 0 deg = 60 deg = 120 deg = 180 deg

300 Um µm/s

φw φw φw φw

5

Tm Pa

4629

200

2 100 1 0 1.6

0 2000

1.4 1.2

1500 Qm µm2/s

tm s

1 0.8 0.6 0.4

1000 500

0.2 0

0 0

20

40 60 H µm

80

100

0

20

40 60 H µm

80

100

Fig. 6. Effect of gap between parallel plates.

4. Results and discussion 4.1. Velocity profiles Fig. 3 shows the induced velocity profiles for various twist angles w ; the upper figure shows the x component and the lower one shows the z component of the velocity. As explained in the previous paper (Chono and Tsuji, 2008), a single-step application of an electric field with a constant value to a liquid crystal induces a flow, which disappears rapidly owing to its viscosity (see Fig. 4). um and wm in Fig. 3 are the maximum velocities in a period from start to cessation; that is, the suffix “m” means the maximum during the time period. Both components of the velocity reach their maxima at the same time. The shapes of the velocity profiles between the two plates depend on the twist angle w . In the previous study it was shown that the profile is S-shaped at w = 0◦ , while the flow becomes unidirectional at w = 180◦ . In the present study, we paid attention to not only the x component but also the z component of the velocity, and it was found that the z component is 0 at w = 0◦ but is increased with increasing w because the director has a z component. In particular, at w = 180◦ , wm becomes an S-shaped profile, which is similar to the um profile at w = 0◦ , while um is unidirectional as mentioned above. Thus, the twist angle w has an effect on the induced velocity, particularly on the velocity profile. Hereafter, we select w as a parameter and consider the effects of the applied voltage, the gap between the plates, and the tilt angle.

4.2. Transient behaviors of physical quantities Fig. 4 shows the transient behaviors of the shear stress acting on the plates T = (2yx + 2yz )1/2 , the maximum value of the velocity vector between the plates U = (u2 + w2 )1/2 , the flowrate Q, and the maximum value of the tilt angle between the plates . The computation parameters except for the twist angle were fixed to the representative values of w = 1◦ , V = 5 V, and H = 50 m. T, U, and Q exhibit almost the same changes; they increase to a maximum

then decrease to 0 with time. At the initial stage of these changes, a time lag of about 0.2 s is discernible, which is independent of the twist angle. The electric torque acting on the director is a minimum when the angle between the electric-field vector and the director is 0◦ . The tilt angle is 1◦ for every twist angle, so that the torque is too small to rotate the director quickly and such a time lag, independent of the twist angle, occurs. On the other hand, since a liquid crystal molecule aligns parallel to the electric-field vector upon the application of a voltage,  exhibits a gradual increase, a rapid increase, and then asymptotically approaches 90◦ . Tm , Um , and Qm are the maximum values of T, U, and Q, respectively, and tm is the time at which the shear stress acting on the plate takes its maximum. Tm , Um , and Qm are measures of the output power and tm is the responsiveness of the liquid crystalline microactuators. In the following sections, we investigate the effects of the parameters V, H, and w by plotting the maximum values Tm , Um , and Qm and the time tm as functions of the parameters. It is noted that the y-position of maximum velocity shows little change with time. 4.3. Effect of applied voltage Fig. 5 shows the effect of the applied voltage V. With increasing the voltage, the maximum shear stress Tm increases and the time required to reach the peak tm decreases, as expected. The effect of the twist angle is small. At V = 10 V, tm is about 0.1 s, which is a disappointing value compared with the response in a liquid crystal display field. We discuss this further in the following section. Um and Qm increase with increasing the applied voltage. The twist angle w has a comparatively large effect, and Um and Qm increase with increasing w , because the velocity changes from an S-shaped profile to a unidirectional one. 4.4. Effect of gap between plates Tm , Um , Qm , and tm are shown in Fig. 6 as functions of the gap between the plates. The effect of H on Tm is considerable for small H, and when H < 20 m, Tm increases exponentially with decreasing

4630

S. Chono, T. Tsuji / Chemical Engineering Science 64 (2009) 4625 -- 4631

25

1 H = 5 µm

0.8

H = 10 µm H = 15 µm

0.6

Τm Pa

15

H = 20 µm

10

0.4

5

0.2

0 250

0 1

200

0.8

150

0.6

tm s

U µm/s

T Pa

20

100

0.4

50

0.2

0

0 100

90

Um µm/s

Θ deg

80 60

60 40

30

20 0

0 0

0.02

0.04

0.06 ts

0.08

0.1

0

0.12

Fig. 7. Transient behaviors for small gaps.

H. tm decreases as H decreases. We show the transient behaviors of T, U, and  for H = 5–20 m in Fig. 7, which corresponds to Fig. 4. w is fixed to 0◦ and in this case the flowrate is 0. When H is small (5 m), T, U, and  increase sharply and reach their maxima in several milliseconds. The response is good and the values of T and U are large; in particular, T is more than 20 Pa. In liquid crystal displays, the cell gap (the gap between two glass plates) has being reduced over time, and liquid crystal displays with a several micron gap are common. The realization of displays with such small gaps has made it possible to change the molecular aligning direction in a short time and thus to display animations satisfactorily. Therefore, with respect to the application of liquid crystals to actuators, a smaller gap may allow the realization of higher-performance actuators, assuming that we can overcome any difficulties in manufacturing. We return to Fig. 6. With increasing H, Um decreases and Qm remains constant. w , which has no effect on Tm and tm , affects Um and Qm , and both values take maxima at w = 180◦ . Because the electric-field strength E is expressed as E = V/H, a decrease in the gap H while keeping an applied voltage V constant (=5 V) results in an increase in E. Thus, the effects of both H and E are shown in Figs. 6 and 7. Then, to extract only the effect of H, we changed V while keeping E constant (5 V/50 m). The results for w = 0◦ are shown in Fig. 8. When H increases, Tm and Um increase and tm decreases, which means an improvement of the performance of liquid crystalline actuators. This tendency is opposite to that in

20

40

60 H µm

80

100

Fig. 8. Effect of gap with constant electric field.

Fig. 6, although the values of Tm and Um are small compared with those in Fig. 6. Tm and Um are almost 0 at H = 10 m, because at this value of H, V = 1 V, which is near the critical voltage Vc (=0.745 V) at which a Freedericksz transition occurs (Chandrasekhar, 1992); thus, no change in the molecular orientation occurs. 4.5. Effect of tilt angle Fig. 9 shows the effect of the tilt angle w on Tm , tm , Um , and Qm . The effect of w is small compared with the above-mentioned parameters, and the physical quantities do not change significantly, except for tm , which shows a slight decrease with w , meaning that the response is improved. This is because the electric-field vector and the director in the angular momentum Eq. (3) take an innerproduct form; thus, the torque acting on the director becomes large with increasing w . 5. Conclusions For the purpose of developing microactuators driven by liquid crystals, we have numerically studied the effects of the applied voltage, the gap between the two plates, and the molecular orientation at the plates on the velocity of the backflow, the flowrate, the shear stress on the plates, and the responsiveness. The induced backflow does not have an out-of-plane component when the twist angle is equal to 0◦ but has a component when the twist angle is not 0◦ .

S. Chono, T. Tsuji / Chemical Engineering Science 64 (2009) 4625 -- 4631

1.2

400

0.8 0.6

= 0 deg = 60 deg = 120 deg = 180 deg

300 Um µm/s

φw φw φw φw

1 Tm Pa

4631

0.4

200 100

0 1.2

0 12000

1

10000

0.8

8000

Qm µm2/s

tm s

0.2

0.6 0.4

6000 4000 2000

0.2

0

0 0

10

20

30

40

0

θw deg

10

20 θw deg

30

40

Fig. 9. Effect of tilt angle.

Therefore, the twist angle affects the velocity profile of the induced flow. With increasing applied voltage, the shear stress at the plates, the velocity magnitude, and the flowrate increase, and the response is improved. The gap between the two plates has a large effect, especially for a small gap; for example, when the gap is 5 m, the response is so good that the physical quantities reach their maxima in a few milliseconds, and the shear stress on the plates is more than 20 Pa; the maximum values are also large. Therefore, with respect to the application of liquid crystals to actuators, a smaller gap may allow the realization of higher-performance actuators, assuming that we can overcome any difficulties in manufacturing. However, when the electric-field strength is kept constant, actuators with a larger gap give better performance. The effect of the tilt angle is little compared with the other parameters, but the response is improved with increasing the tilt angle. It is found that the induced velocity, the flowrate, the shear stress on the plates, and the response are changed by varying the applied voltage, the gap between the plates, and the anchoring conditions at the walls. Since it is assumed that the shear stress, the velocity vector, and the flowrate are measures of the output power of the liquid crystalline microactuators, and that the time at which the shear stress takes its maximum means its responsiveness, we conclude that actuators with arbitrary characteristics can be developed by suitably controlling such parameters. Acknowledgment This work was partially supported by Japan Society for the Promotion of Science KAKENHI (17360084).

References Beris, A.N., Edwards, B.J., 1994. Thermodynamics of Flowing Systems. Oxford University Press, Oxford. Buguin, A., Li, M.H., Silberzan, P., Ladoux, B., Keller, P., 2006. Micro-actuators: when artificial muscles made of nematic liquid crystal elastomers meet soft lithography. J. Am. Chem. Soc. 128, 1088–1089. Chandrasekhar, S., 1992. Liquid Crystals. second ed.. Cambridge University Press, Cambridge. Chono, S., Tsuji, T., 2008. Proposal of mechanics of liquid crystals and development of liquid crystalline microactuators. Appl. Phys. Lett. 92, 051905. de Gennes, P.G., Prost, J., 1993. The Physics of Liquid Crystals. second ed.. Oxford University Press, Oxford. de Gennes, P.G., 1997. A semi-fast artificial muscle. C. R. Acad. Sci. Ser. IIB 324, 343–348. Doi, M., Edwards, S.F., 1986. The Theory of Polymer Dynamics. Clarendon Press, Oxford. Ericksen, J.L., 1960. Arch. Ration. Mech. Anal. 4, 231. Ericksen, J.L., 1961. Trans. Soc. Rheol. 5, 23. Jenkins, J.T., 1978. Ann. Rev. Fluid Mech. 10, 197. Karat, P.P., Madhusudana, N.V., 1977. Mol. Cryst. Liq. Cryst. 40, 239. Kneppe, H., Schneider, F., Sharma, N.K., 1981. Ber. Bunsenges. Phys. Chem. 85, 784. Kneppe, H., Schneider, F., Sharma, N.K., 1982. J. Chem. Phys. 77, 3203. Leslie, F.M., 1968. Arch. Ration. Mech. Anal. 28, 265. Marrucci, G., Greco, F., 1991. Mol. Cryst. Liq. Cryst. 206, 17. Rey, A.D., Denn, M.M., 2002. Ann. Rev. Fluid Mech. 34, 233. Wang, X., Engel, J., Liu, C., 2003. Liquid crystal polymer (LCP) for MEMS: processes and applications. J. Micromech. Microeng. 13, 628. Warner, M., Terentjev, E.M., 2003. Liquid Crystal Elastomers. Oxford University Press, Oxford. Woltman, S.J., Jay, G.D., Crawford, G.P., 2007. Liquid-crystal materials find a new order in biomedical applications. Nature Mater. 6, 929. Xie, P., Zhang, R.B., 2005. Liquid crystal elastomers, networks and gels: advanced smart materials. J. Mater. Chem. 15, 2529. Yu, Y., Ikeda, T., 2006. Soft actuators based on liquid-crystalline elastomers. Angew. Chem. Int. Edn. 45, 5416.