Inverse Flexoelectret Effect: Bending Dielectrics by a Uniform Electric Field

It is highly desirable to discover an electromechanical coupling that allows a dielectric material to generate curvature in response to a uniform electric field, which would help to simplify device design and reduce interface failure in designing actuators. Flexoelectricity, a two-way coupling between polarization and strain gradient, is a good candidate. However, its applications are usually limited to the nanoscale due to its inherent size dependence. Here, an inverse flexoelectret effect in silicone elastomers is introduced to overcome this limitation. On the basis of this idea, a flexing actuator that can generate large curvature at the millimeter length scale is fabricated and shown to have excellent actuation performance, comparable with that of current nanoscale flexoelectric actuators. Theoretical analysis indicates that the phenomenon originates from the interplay of electrets and Maxwell stress. This work opens an avenue for applying macroscopic flexoelectricity in actuators and flexible electronics.

Figure 1

The mechanism of the inverse flexoelectret effect. (a) An external electric field (yellow arrows) generated by an external voltage (yellow spheres) is capable of bending a dielectric material via intrinsic flexoelectricity, which is significant at the nanoscale but decreases dramatically with size increase. (b) The effective flexoelectricity is enhanced in an elastomer with a layer of embedded negative charges (red spheres with minus signs). The embedded charges induce positive charges on the surface electrodes (red spheres with plus signs) and a consequent built-in electric field (red arrows). Under an external voltage, the external charges (yellow spheres) and inductive charges add to each other on the upper electrode while canceling on the other one. (c) Thus, the upper and lower parts experience a different true electric field and thus undergo consequent different stretching stress via the Maxwell stress effect, leading to bending deformation.

Figure 2

Fabrication of the elastomer (control group) and flexoelectrets (experimental group).

Figure 3

Stress-strain curve of the elastomer for measuring Young’s modulus.

Figure 4

Experimental setup for measuring the voltage-induced oscillations. The inset shows an enlarged view of the cantilever.

Figure 5

Experimental results. (a) Experimental setup consisting of a clamping rigid body, a high-voltage supply, and a laser displacement sensor. (b) Applied voltage (solid blue line) and induced displacement of cantilever end. Black and red circles represent the elastomer without charge and the flexing actuator with a charge density of −0.088 mC/m², respectively. (c) Curvature as a function of the applied electric field for a flexing actuator (−0.088 mC/m²). (d) Applied voltage (solid blue line) and induced first-harmonic displacement of the flexing-actuator cantilever end. The charge density of the middle layer is positive (0.086 mC/m²). (e) Curvature as a function of the magnitude of charge density under a sinusoidal of 1 kV. All results in this figure are acquired for the sample with a size of 10 × 7 × 0.76 mm³ at a sinusoidal excitation of 1 Hz at room temperature.

Figure 6

Finite-element calculations. (a) The FE model with a charged layer in the middle plane. (b)–(d) The deformation and the distribution of Maxwell stress $p_2$ of the model (X₁-X₂ cross section) with a negatively charged layer (b), a positively charged layer (c), and without a charged layer (d) at excitation of ±1 kV. The color and arrows represent the magnitude and direction of Maxwell stress $p_2$, respectively.

Figure 7

Electrostatic model and comparison between experimental, theoretical and simulation results. (a) Theoretical model of a flexoelectret. The red spheres with minus signs represent the embedded negative charges with a density of q. The red spheres with plus signs represent the inductive positive charges on electrodes. The black arrows represent the electric field. (b) Distribution of the electric field (absolute value) along the thickness direction as a function of the applied voltage. The lines with circles and triangles represent the electric field above and below the middle plane, respectively. (c) The electric field inside the model when the applied voltage is positive. The yellow spheres represent the applied voltage. (d) Distribution of Maxwell stress p₂ along the thickness direction as a function of the applied voltage. (e),(f) Curvature as a function of charge density (e) and thickness (f) and comparison of experimental, theoretical, and simulation results. To investigate only the influence of a thickness change, we divide the curvature $\kappa$ by the charge q because it is difficult to control the charge density accurately.

Figure 8

The effect of the material’s asymmetry. (a) Output curvature and (b) bending asymmetry as a function of the relative thickness of PTFE. In our experiments, α is around 0.011–0.017, which is nearly the case without considering the PTFE layer. The material properties used in this figure are h = 0.76 mm, q = −0.88 mC/m², V = ±1.5 kV, ε = 2.85 ε₀, ε′ = 2 ε₀, Y = 56.79 kPa, and Y′ = 0.4 GPa.

Figure 9

Comparison of the bending performance between flexoelectrets and typical flexoelectrics. (a) The ratios of curvature to electric field as a function of thickness are compared for flexoelectrets (this work) and flexoelectric ${\mathrm{Sr}\mathrm{Ti}\mathrm{O}}_3$ [32, 35] and ${\mathrm{Ba}\mathrm{Ti}\mathrm{O}}_3$ [30]. (b) The evolution of bending performance under a fixed applied electric field (yellow arrows) when the size is scaled up.

Figure 10

Theoretical laminated model with a PTFE layer. The red spheres with minus signs represent the embedded negative charges with a density of q.