Explicit computational model of dielectric elastomeric lenses

Yanjie Cao; Yanan Wang; Yang Liu; Yu-Xin Xie

doi:10.1364/OE.25.028710

1. Introduction

Adaptive focus lenses can be applied in a variety of instruments across a range of different applications, from consumer items to more specialized fields, such as cell phone cameras, medical endoscopes, and machine vision apparatus etc. By changing the refractive index of the optical medium or the lens curvature, adaptive lenses can replace traditional assemblies in old focus tunable lenses [1, 2]. Generally speaking, the adaptive lenses are made of different responsive materials, either for the medium or the lens itself, such as liquid crystal materials [3,4], hydrogels [5], piezoelectric materials [6] and dielectric elastomers [7] (more relevant works can be found in the references cited therein). On the other hand, the aperture variation is also a possible way to achieve an adaptive liquid-filled lens [8]. In particular, the dielectric elastomers that can deform significantly by an applied voltage or stretch provide another way to assemble an adaptive lens with fast response and relatively large focal length variations. Inspired by the human eyes, Carpi et al. [9] proposed a tunable lens where a dielectric elastomer actuated by voltage functions as the ciliary muscle and the fluid enclosed by membranes works together as the crystalline lens. However, the actuator is located outside the optical path in this kind of tunable lens configuration. Consequently, an important part of the lens lateral area is occupied by the actuation device, which then causes a reduction of the usable optical area [2]. According to the recent development of compliant, transparent, and electrically conductive electrodes, it is now possible to fabricate a tunable lens without external actuation [7, 10]. The use of transparent dielectric elastomer actuators (which is abbreviated as DEAs in this paper) for lenses has been recently demonstrated. However, since the lens consists of a buckled solid elastomer membrane, it only operates as a diverging optic [11]. Here we study the mechanical deformation for a lens induced by voltage with an integrated actuator in the optical path, and then determine the focal length change.

The lens is composed of a stiff frame, a transparent liquid of fixed volume, and two elastomeric membranes, one of which (numbered as II) is subjected to a voltage and pressure, whereas the other is only subjected to a pressure; see Fig. 1. The electroactive membrane (membrane II) is covered by transparent and flexible electrodes on both sides, which realizes the actuation system. The initial focal length of the tunable lens f₀ is determined by the refractive index of the liquid and curvature radii of the membranes, while the change of focal length is only affected by the latter factor. Such a tunable lens can exhibit both an increase and a decrease in its focal length. When a voltage is applied to membrane II, the focal length increases when the voltage ramps up, and decreases otherwise.

Fig. 1 Schematics of the tunable lens from Fig. 1 in [7] (a) Assembly of the tunable lens. (b) Electric actuated membrane, with electrodes deposited on its both sides. (c) Schematic of the lens at rest state and (d) during actuation, indicating an increase of focal length. D, R(r) and B (b) are the diameter, membrane radius of curvature at rest (actuated) state and volume of liquid between the membrane and dashed line at rest (actuated) state, respectively.

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Shian et al. [7] also wrote the MATLAB codes to calculate the maximum focal length change, using Suo’s model [12]. Their codes, however, did not give the relationship between the focal length and applied voltage. In addition, the relation between the liquid volume injected into two membranes and the initial focal length f₀ is not shown. In this paper, we present a computational model to study the performance of tunable lenses. We plot the variation of focal length with respect to the voltage and discuss the effects of liquid volume and geometry of the device.

2. Computational model

We assume that both membranes I and II are composed of incompressible hyperelastic materials. In the sequel, we use subscripts I and II to indicate parameters for the two membranes respectively. The transparent electrodes are attached on membrane II to assemble a dielectric actuated membrane. The dashed lines in Fig. 1(c) and 1(d) represent the initial positions of the two membranes. Before an actuation starts, a small amount of liquid has been injected into the space between two membranes to induce the rest state, and the deformed membranes can be viewed as part of a sphere with curvature radii denoted respectively by R_I and R_II [12]. Thus we have

R^{2} - {(R - H)}^{2} = \frac{D^{2}}{4},

where D and H are the diameter of the membrane and the distance between the center of the membrane and dashed line respectively. The subscripts I and II are omitted as equation (1) is valid for both membranes, and the same convention is adopted hereafter. The surface areas of membranes A_I and A_II are given by

A = 2 π R^{2} (1 - \frac{\sqrt{R^{2} - \frac{D^{2}}{4}}}{R}) .

The liquid volume between the membrane and dashed line are denoted respectively by B_I and B_II and the expressions are given by

B = π (R H^{2} - \frac{H^{3}}{3}) .

Then the two deformed membranes form a thick lenses with the focal length given by 1/f₀ = (n −1) [1/R_I − 1/R_II + (n − 1)(d + H_I + H_II)/(nR_IR_II)], where n and d +H_I +H_II are the refractive index and distance between centers of the two membranes respectively.

If there is no electrical actuation, the focal length remains to be f₀. Then we apply an electrical actuation on membrane II to induce a deformation and assume that the deformed state could maintain the spherical shape. Thus the longitudinal and latitudinal stretches share the same value of λ_II. We then utilize the incompressibility condition to find that the stretch along the thickness direction is $1 / λ_{II}^{2}$ . Since membrane II deforms, the liquid volume between the dashed line and membrane II becomes b_II accordingly. We mention that the liquid is assumed to be incompressible so that the liquid volume between the dashed line and membrane I becomes b_I in order to remain constant. Then we have

b_{I} + b_{II} = B_{I} + B_{II} \equiv b_{c},

where b_c is a given constant. Accordingly, it is also assumed that the deformed shape of membrane I is spherical such that it has the same longitudinal and latitudinal stretches given by λ_I. Similarly, the stretch along the thickness direction is

1 / λ_{I}^{2}

. After the deformation, the distance H, curvature radius R and surface area A become h, r and a respectively, and the new focal length is determined by 1/f = (n − 1) [1/r_I − 1/r_II + (n − 1)(d + h_I + h_II)/(nr_Ir_II)].

The expressions of λ_I and λ_II are given by

λ = \sqrt{\frac{a}{A}} .

The surface area a can be obtained by substituting R by r in equation (2). For membrane I, we assume that the Helmholtz energy function takes the form [13]

W_{I} (λ_{I}) = - \frac{μ J_{m}}{2} log (1 - \frac{λ_{I}^{- 4} + 2 λ_{I}^{2} - 3}{J_{m}}),

where μ is the ground state shear modulus and J_m denotes a material parameter. For an ideal dielectric elastomer, i.e., membrane II, the Helmholtz energy function is given by

W_{II} (λ_{II}, \tilde{D}) = - \frac{μ J_{m}}{2} log (1 - \frac{λ_{II}^{- 4} + 2 λ_{II}^{2} - 3}{J_{m}}) + \frac{{\tilde{D}}^{2}}{2 ε} λ_{II}^{4},

where D̃ and ɛ denote respectively the electric displacement and permittivity respectively. The last term in equation (7) demonstrates the fact that membrane II stores not only the strain energy but also the electrostatic energy.

The device and electric power make up a conservative system, with total energy given by

E = b_{I} W_{I} (λ_{I}) + b_{II} W_{II} (λ_{II}, \tilde{D}) - Φ Q,

where Q is the total charge on the electrodes deposited on the electroactive membrane II, and Φ is the power voltage. After a voltage is applied, the equilibrium state is determined by minimizing E subject to the constraint (4) and the stretches λ_I and λ_II are determined as a result.

Following the standard procedure of constrained minimization, we obtain the equilibrium equations

{\begin{array}{r} \frac{\partial E}{\partial λ_{I}} + η \frac{\partial L}{\partial λ_{I}} = 0, \\ \frac{\partial E}{\partial λ_{II}} + η \frac{\partial L}{\partial λ_{II}} = 0, \\ L = 0, \end{array}

where L = b_I +b_II −b_c, and η is the Lagrangian multiplier. In our calculations, we take μ = 30kPa, J_m = 120 [14], and the permittivity for VHBs 4905 and 4910 (dielectric elastomers fabricated by 3M) is ɛ = 3.98 × 10⁻¹¹F/m [15]. Once the stretches λ_I and λ_II are determined, one could utilize equation (5) to find the deformed curvature radii. Finally the ratio of f/f₀ can be obtained.

3. Results and discussions

We now compare our results with the experimental data extracted from [7]. The same parameter values and materials reported in [7] are used in our calculations. The elastomer membrane I is made of a 0.5mm thick acrylic elastomer (VHB 4905, 3M) and is bi-axially pre-stretched with a homogeneous strain of 100%, while elastomer membrane II is made of a 1mm thick acrylic elastomer (VHB 4910, 3M) and is bi-axially pre-stretched with a homogeneous strain of 300%. The refractive index of the membranes is 1.476, then we follow [7] to choose clear silicone oil whose refractive index is 1.4022 as the medium liquid. After all the geometric and material parameters are specified, we could utilize equation (9) to determine the focal length change.

We first characterize the performance of the lens in Fig. 2 by ramping up the applied voltage when the geometry and volume of liquid are fixed. A sharply rising curve indicates a big change in the focal length, especially as the actuation voltage reaches high values. From Fig. 2, it can be seen that our calculation results agree well with the experimental data, which validates the effectiveness of our computational model. Note that the red dots correspond to the case f₀ = 36mm in Fig. 4 by [7]. We use the current model to find that b_c = 160mm³. When the actuation voltage is greater than 4900V, however, the recorded images by the tunable lens in [7] are degraded significantly due to non-concentric lens deformation, thus defining the maximum usable actuation voltage. Since the focal length can be determined by the stretches, we next show the stretches λ_I and λ_II in Fig. 3 for both membranes. It is noted that the stretch for membrane I is an increasing function in terms of the applied voltage, yet the stretch for membrane II has an opposite feature. Meanwhile, we find the incremental strain caused by the voltage is relatively small.

Fig. 2 Focal length variation as function of the actuation voltage and the comparison between the experimental data and computational results. The black curve represents the computational results while the red dots indicate the experimental data extracted from the paper by Shian et al. [7], D_II/D_I = 1.6, b_c = 160mm³.

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Fig. 3 The stretches λ_I and λ_II with varying applied voltage, D_II/D_I = 1.6 and b_c = 160mm³.

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Then we plot the relation between the focal length change f/f₀ and the applied voltage V for D_II/D_I = 1.6, 1.4 and 1.3 in Fig. 4. It is found that the maximum focal length change increases as D_II/D_I increases. In the meantime, as the ratio of D_II/D_I increases, the initial focal length f₀ decreases (see Fig. 6), and a larger focal length variation can be achieved. When D_II/D_II = 1, the focal length will remain unchanged, whereas when D_II/D_I < 1, the focal length will decrease with increasing actuation voltage. On the other hand, the focal length variation is not only a function of the actuation voltage V, the geometry D_II/D_I, but also influenced by the liquid volume injected into the lens, which is another factor that can be controlled to adjust the focal length variation. We plot the relation of f/f₀ and applied voltage V for b_c = 90mm³, 160mm³, 200mm³ in Fig. 5. It can be observed that the maximum focal length change decreases slightly as the liquid volume increases significantly, which means that the liquid volume affects a little on the maximum focal length change. Note that this conclusion is obtained under the assumption that the deformed state is spherical. Additionally, the underlying assumption is that the internal liquid volume b_c should not be very large. In [7], they found that the spherical approximation is accurate for h_I/D_I and h_II/D_II are less than 0.15. If the liquid volume b_c is very large, for example, D_II/D_I = 1.3 and b_c = 260mm³, we find that h_II/D_II > 0.15, so the fundamental assumption is no longer correct. However, our model is quite accurate roughly for b_c ⩽ 210mm³. Yet the internal liquid volume b_c definitely affects the initial focal length f₀. Indeed, the geometry D_II/D_I determines the maximum focal length change and both D_II/D_I and liquid volume b_c affect the initial focal length f₀. From applicable considerations, we plot the initial focal length f₀ with varying liquid volume b_c in Fig. 6 for two geometries. In real life applications, once the parameter D_II/D_I is specified, the maximum focal length change f/f₀ is determined accordingly, one could control the liquid volume to adjust the initial focal length f₀ for certain purposes.

Fig. 4 Focal length variation as function of the actuation voltage and geometry. D_II/D_I =1.3, 1.4, 1.6; b_c = 160mm³.

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Fig. 5 Focal length variation as function of the actuation voltage and liquid volume. b_c = 90mm³, 160mm³, 200mm³; D_II/D_I = 1.6.

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Fig. 6 The initial focal length f₀ with varying liquid volume.

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4. Conclusions

To summarize, in this paper by expressing the Helmholtz energy of the two membranes on the lens and applying the minimization of energy principle under constraint, and by adopting the assumption that the focal length is only affected by the actuation voltage, lens geometry and liquid volume inside the lens, we present an explicit computational model for the optimal design of an adaptive focus lens that can be electrically controlled by tuning the voltage applied to an electroactive membrane. This model is able to describe the focal length of the lens as a function of the actuation voltage, the lens geometry as well as the liquid volume contained between two membranes. The computational results predicted by the model fit well with the reported experimental data. We also demonstrate how two parameters would affect the maximum focal length analytically. Additionally, this model has shown that the maximum focal length change also depends mainly on the geometry D_II/D_I, the volume of liquid b_c only has marginal effect, although both of them could affect the initial focal length f₀.

It is worth mentioning that for the case f₀ = 53mm, D_II/D_I = 1.6 in Fig. 4 by [7], the curve of the focal length change f/f₀ against the applied voltage V is almost the same as the case f₀ = 36mm. The latter case has been used for validating our model in Fig. 2. When f₀ = 53mm, we find that b_c is around 100mm³ according to the left figure in Fig. 6. Thus the model also indicates the same result in [7], according to the conclusion that b_c has marginal effect on f/f₀. However, if D_II/D_I = 1.3, the experimental settings by [7] are respectively f₀ = 24mm, 29mm and 52mm. We carefully calculated the corresponding deformations in the rest state and find that the deformed states are no longer spherical (one could also refer to Fig. 6 to find that the liquid volume is large for each case). Thus, our model can not be used to predict the behaviors for such lenses, i.e., the internal liquid volume is very large, as the fundamental assumption is violated. For the case that b_c is large enough, it may also affect the maximum focal length variation since the geometry of the rest state is non-spherical and a bifurcation may occur (with a large pressure) [16]. On the other hand, more factors including the gravity would affect the final focal length, which makes the problem more complicated. Moreover, this model mainly focuses on the effects of the applied voltage, membrane width ratio and liquid volume (not too large), other factors that might also contribute to the change of focal length such as the aberration by optical curvature and the shift of refractive indices due to density mode, as well as other optical characteristics affected by the shape change of membranes that might exist within lenses have not been considered. Nevertheless, the good agreement between results predicted by the model and experimental data has warranted the value of the current model. It is hoped that our work will contribute to a better design of tunable lenses hence better products for the growing market in the future.

Funding

National Natural Science Foundation of China (11172201 and 11602163); China Scholarship Council.

Acknowledgments

We thank the two anonymous reviewers for improving the original manuscript. We also acknowledge Prof. Yibin Fu from Keele University and Dr. Samuel Shian for valuable discussions.

References and links

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Explicit computational model of dielectric elastomeric lenses

Abstract

Corrections

1. Introduction

2. Computational model

3. Results and discussions

4. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (6)

Equations (9)

Optics Express