Electrically Induced Twist in Smectic Liquid–Crystalline Elastomers

Article pubs.acs.org/JPCB

Electrically Induced Twist in Smectic Liquid−Crystalline Elastomers Christopher M. Spillmann,† Jawad Naciri,† B. R. Ratna,† Robin L. B. Selinger,‡ and Jonathan V. Selinger*,‡ †

Center for Bio/Molecular Science and Engineering, Naval Research Laboratory, Code 6900, 4555 Overlook Avenue, SW, Washington, DC 20375, United States ‡ Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, United States S Supporting Information *

ABSTRACT: As an approach for electrically controllable actuators, we prepare elastomers of chiral smectic-A liquid crystals, which have an electroclinic effect, i.e., molecular tilt induced by an applied electric field. Surprisingly, our experiments find that an electric field causes a rapid and reversible twisting of the film out of the plane, with a helical sense that depends on the sign of the field. To explain this twist, we develop a continuum elastic theory based on an asymmetry between the front and back of the film. We further present finite-element simulations, which show the dynamic shape change.

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INTRODUCTION A major goal of modern materials science is to develop smart materials that change their shape in response to external stimuli, for applications as actuators or artificial muscles. One promising approach is to use liquid-crystalline elastomers, polymer networks bonded to liquid-crystalline units, in which the orientational order controls the shape of the polymer.1−4 Elastomers of chiral smectic-A liquid crystals5−23 may be especially useful because they have an electroclinic effect, i.e., molecular tilt induced by an applied electric field.24,25 The electroclinic effect should induce an in-plane shear of the elastomer, linear in the applied field, leading to a distortion from a rectangle to a parallelogram. The purpose of this paper is to investigate the shape changes in chiral smectic-A elastomers. Instead of the expected in-plane shear, our experiments find that an electric field causes a twisting of the film out of the plane, leading to a helically curved shape. The twist is rapid and reversible, and is apparently linear in the applied field, with a helical sense that depends on the sign of the field. To explain this electrically induced twist, we develop a continuum elastic theory based on the assumption that the film has an asymmetry between front and back, which can be attributed to the preparation conditions. We further present finite-element simulations of the twisting process, which show the dynamic shape change without any mathematical assumptions of small distortions.

Figure 1. Concept for smart materials based on chiral smectic elastomers. (a) Smectic-A phase of chiral liquid crystals, showing the electroclinic effect. (b) Smectic liquid-crystalline elastomer. (c) Expected shape change in a chiral smectic elastomer under an electric field.

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RESULTS AND DISCUSSION The concept for smart materials based on chiral smectic-A elastomers is shown schematically in Figure 1. In the smectic-A phase of a pure liquid crystal, with no polymer network, the molecules form layers, and the average molecular orientation (known as the director) is perpendicular to the layers. If an © XXXX American Chemical Society

Special Issue: William M. Gelbart Festschrift Received: March 30, 2016

A

DOI: 10.1021/acs.jpcb.6b03241 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

Figure 2. Synthesis and preparation of smectic elastomer films. (a) Chemical structure of the chiral polymerizable liquid crystals and cross-linker. (b) Alignment, polymerization, and extraction of the film. (c) Coating of the film with flexible electrodes, connected to an external voltage.

Figure 3. Columns a−c: Front and side views of the elastomer film distortion as a function of applied field. The films are cut with the smectic layers perpendicular, parallel, and at a 45° angle with respect to the long edge of the film. When there is no field, the films are approximately flat, with a slight curvature as discussed in the text. When a field is applied, the first two films twist into a helical structure, while the last film kicks forward. The opposite field induces the opposite distortion. Columns d−e: Theory for the elastomer film distortion under an applied electric field.

electric field is applied, the molecules tilt perpendicular to the plane defined by the layer normal and the field (Figure 1a). The tilt direction is determined by the sign of the applied field and by the molecular chirality. The liquid-crystalline units can be

incorporated into a cross-linked polymer network, with the polymers running between and through the smectic layers (Figure 1b). In this case, the director strongly couples to the shape of the polymer network, with the chains elongating along B

DOI: 10.1021/acs.jpcb.6b03241 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B the director. Thus, one expects that an applied electric field would cause a rectangular film to deform into a parallelogram (Figure 1c). The opposite field should induce the opposite parallelogram distortion. We prepared smectic elastomer films as described previously.19 The films were composed of two chiral polymerizable liquid crystals, 70 weight % ACKN115 and 30 weight % ACBKN115, mixed with 5 mol % of the diacrylate cross-linker DACP11 (Figure 2a). The mixture was filled into an indium− tin-oxide (ITO) coated glass cell with two nylon-rubbed alignment layers, polyimide and poly(vinyl alcohol) (PVA). It was aligned by slow cooling (0.1°/min) under a square-wave electric field (±6 V/μm, 0.5 Hz). Following alignment, the field was removed and the sample was immediately photopolymerized under UV light. The cell was then soaked in water to solubilize the PVA and extract the free-standing elastomer (Figure 2b). Free-standing films were cut into rectangular pieces, with the long edge at a well-defined direction with respect to the smectic layers. Typical dimensions were 2 cm long, 0.5 cm wide, and 50 μm thick. Thin layers of silver conductive paste were applied to the front and back surfaces to provide conformal electrodes. The electrodes were thin enough to allow curvature of the film, yet thick enough to provide uniform and continuous coverage (Figure 2c). Once the films were prepared, we applied a triangle-wave electric field across the electrodes and observed the resulting deformation. Remarkably, the films did not exhibit the expected parallelogram distortion, but instead showed a more complex 3D curvature, presented in Figure 3 (columns a−c) and Supplementary Video 1. First consider row 1, observed for films cut with smectic layers perpendicular to the long edge. When an electric field is applied, the film twists into a helical shape, with a saddle-like curvature proportional to the field. Under the opposite sign of the field, the film twists with the opposite helicity, again with saddle-like curvature. The shape change is rapid; it follows the applied field up to a frequency of at least 10 Hz. When the field is removed, the film returns to its initial shape. Although the observed curvature is different from the expected parallelogram distortion, it is still potentially useful for technological applications. An electrically controlled actuator with this deformation might be used, for example, as a valve for a microfluidic system. Hence, we would like to develop a theoretical model for the smectic elastomer film under an applied field, and use this model to predict and control the shape change. In developing a model, the key issue is to explain the symmetry of the distortion. The experiment finds that one sign of the field induces a twist with one helicity, while the opposite sign of the field induces a twist with the opposite helicity. Hence, this curvature cannot be a dielectric effect, which must be even in the applied field. Rather, it must be a polar effect, linear in the applied field, like the electroclinic effect. However, in a uniform flat film, the electroclinic effect would not induce any curvature, because it respects the symmetry between the front and back surfaces of the film. Hence, the electroclinic effect must combine with some other pre-existing asymmetry between the front and back surfaces of the film. There is clear experimental evidence that the front and back surfaces are not equivalent to each other. Even when there is no applied electric field, the films are not exactly flat rectangles. Rather, they consistently show some curvature toward one side, resembling a celery stalk. This asymmetry between the two

sides is not intentional, but is a serendipitous result of the sample preparation process. It might arise from UV illumination from only one side during polymerization and cross-linking, or might be caused by a plastic deformation when the film is removed from the glass cell. Regardless of its origin, this asymmetry is present and should be considered as part of the theory. For that reason, we develop linear elasticity theory for electroclinic films that are not uniform from front to back. The simplest possible free energy that couples the nematic order tensor Qij with the three-dimensional (3D) elastic strain tensor e3D ij can be written as F=

1 3 D 3D λeii ejj + μeij3Deij3D − αΔQ ijeij3D 2

(1)

implicitly summed over i and j. Here, λ and μ are the Lamé elastic coefficients for the elastomer, and α is a coupling between strain and nematic order. Minimizing this free energy over the strain tensor gives

eij3D =

αΔQ ij 2μ

(2)

provided that this strain is compatible. We can now relate the 3D elastic strain tensor to the shape of the elastomer film. Suppose the film has internal coordinates (x,y,z), where z ranges across the narrow thickness from front to back. The midplane of the film, at z = 0, is represented by the height function h(x,y). For a point slightly off of the 2D midplane, the 3D elastic strain is given by e3D βγ = eβγ + z Kβγ, 2D where eβγ is the 2D strain tensor of the midplane, and Kβγ = −∂β ∂γh is the curvature tensor. Hence, eq 2 becomes 2D + zKβγ = eβγ

αΔQ βγ 2μ

(3)

By evaluating eq 3 at z = 0, we obtain ⎡ αΔQ βγ ⎤ 2D ⎥ eβγ =⎢ ⎣ 2μ ⎦ z = 0

(4)

This result implies that the 2D strain responds to the stimulus αΔQβγ/(2μ) at the midplane. Likewise, by evaluating the first derivative of eq 3 with respect to z, we obtain ⎡ ∂ ⎛ αΔQ ⎞⎤ βγ ⎟⎟⎥ Kβγ = ⎢ ⎜⎜ ⎢⎣ ∂z ⎝ 2μ ⎠⎥⎦ z=0

(5)

That result implies that the curvature tensor responds to the gradient of the stimulus across the thickness of the film. If any of the variables α, ΔQβγ, or μ vary as a function of z, then the film will curve. As a specific example, suppose that α′ = ∂α/∂z represents the gradient of film properties across the thickness, while ΔQβγ and μ are uniform with respect to z. In that case, the curvature tensor becomes Kβγ =

α′ΔQ βγ 2μ

(6)

Now suppose that the change in orientational order ΔQβγ arises from the electroclinic effect. When the electric field is off, the director n̂ is oriented along the layer normal x̂, parallel to the long axis of the film, as in Figure 3 (row 1). Under a nonzero electric field E, it tilts to n̂ = x̂ cos θtilt + ŷ sin θtilt, C

DOI: 10.1021/acs.jpcb.6b03241 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B where θtilt = CE is the electroclinic tilt angle and C is the electroclinic coefficient. The order tensor is related to the 3 1 director by Q βγ = S 2 nβ nγ − 2 δβγ , where S is the magnitude

(

agreement confirms that the theory captures the essential features of the experimental shape change. As a final point, we can compare our experiments on smectic elastomers with previous experiments on nematic elastomers. Several experiments have studied nematic elastomer films with twist in the director orientation between the front and back surfaces of the film.28−30 When the temperature is changed, the differential strain between the front and back surfaces leads to curved structures, which are very similar to the curved structures that we have found in smectic elastomers. In both cases, the asymmetry between front and back surfaces is essential, because it converts the anisotropic in-plane strain of liquid-crystal elastomers into curvature out of the plane. In conclusion, we have found experimentally that chiral smectic-A elastomer films exhibit a rapid and reversible twisting under an electric field, and we have developed a theory to explain the shape change. Based on this work, the elastomer films are a promising new approach for the development of smart materials.

)

of orientational order. Hence, to lowest order in E, we have 3 ΔQ 12 = ΔQ 21 = 2 SCE , with all other components zero. Putting this expression into eq 6, we find the curvature tensor K12 = K 21 = −

∂ 2h 3α′SCE = ∂x ∂y 4μ

(7)

with all other components zero. Thus, the shape of the film is h(x , y) = −

3α′SCE xy 4μ

(8)

This shape is plotted in Figure 3 (column d), and has the helical saddle-like form seen in experiment. The distortion is linearly proportional to electric field, and hence is reversed when the field changes sign. From this calculation, the twist angle from one end of the film to the other is θtwist

∂h(0, y) ∂h(L , y) 3α′SCEL = − = ∂y 4μ ∂y

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S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.6b03241. Supplementary Video 1: Experimental video of electrically induced twist in a smectic elastomer film (MOV) Supplementary Video 2: Finite element simulation of the electrically induced twist (MOV)

(9)

By comparison, the parallelogram angle (defined in Figure 1c) can be estimated as

θ=

3α0SCE 2μ

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(10)

where α0 is the coupling coefficient α(z) at the midplane. Because θtwist is proportional to the film length L, and θ is not, it is reasonable that the twist is much more conspicuous than any parallelogram distortion. One limitation of linear elastic theory is that it assumes small distortions, while the actual distortions seen in experiments can be large. Hence, we have done finite-element modeling, using a nonlinear elastodynamics approach described previously.26,27 The calculation is based on the exact Green-Lagrange strain tensor, a measure of deformation that is invariant under rotation. With this approach, the algorithm is not limited to the small-deformation limit. The finite-element result is shown in Figure 3 (column e) and Supplementary Video 2; it has the same helical saddle-like form as the linear elastic calculation and the experiment. An important feature of the theoryboth the linear and finite-element versionsis that it can predict the shapes of elastomer films that are prepared in different ways. If the film is cut with the smectic layers parallel to the long edge, the theory predicts exactly the same shape change as for perpendicular layers. We have done this experiment, and indeed it shows the same helical saddle-like deformation as in the previous case (Figure 3, row 2). By contrast, if the film is cut with the smectic layers at a 45° angle with respect to the long edge, the theory predicts a different shape change: h(x , y) =

3α′SCE 2 (x − y 2 ) 8μ

ASSOCIATED CONTENT

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We would like to thank Fangfu Ye for helpful discussions. Work at the Naval Research Laboratory was supported by the Office of Naval Research, and work at Kent State University was supported by National Science Foundation Grants DMR0605889, DMR-1106014, and DMR-1409658. Part of this work was completed at the Kavli Institute for Theoretical Physics, supported by NSF Grant PHY11-25915.

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