Soft Tunable Lenses Based on Zipping Electroactive Polymer Actuators

Abstract Compact and entirely soft optics with tunable and adaptive properties drive the development of life‐like soft robotic systems. Yet, existing approaches are either slow, require rigid components, or use high operating voltages of several kilovolts. Here, soft focus‐tunable lenses are introduced, which operate at practical voltages, cover a high range of adjustable focal lengths, and feature response times in the milliseconds range. The nature‐inspired design comprises a liquid‐filled elastomeric lens membrane, which is inflated by zipping electroactive polymers to tune the focal length. An analytic description of the tunable lens supports optimized designs and accurate prediction of the lens characteristics. Focal length changes between 22 and 550 mm (numerical aperture 0.14–0.005) within 260 ms, equal in performance to human eyes, are demonstrated for a lens with 3 mm aperture radius, while applying voltages below 500 V. The presented model, design rules, and fabrication methods address central challenges of soft electrostatic actuators and optical systems, and pave the way toward autonomous bio‐inspired robots and machines.


Supplementary Discussion
Total free energy of a tunable soft lens with actuator The (free) energy W of the whole tunable lens system splits into the energy of the lens membrane W M and the energy of the actuator W A .
In a quasi-static description, which we assume to suffice for moderate operating speeds, we neglect dynamic effects, such as kinetic energy and the acceleration of the dielectric liquid and the ambient medium (air in our case). Strategies for analysis of dynamic effects can be found in the literature [1,2]. The elastic energy of the membrane resides in the stretched elastomer, but it can be calculated from the integral of its pressure-volume (p-V ) characteristic, where the initial filling level V 0 defines the reference state. In this reference state, no voltage is applied, and the actuator/lens membrane system is in balance, due to equilibrium between the over-pressure and elasticity. In the actuated state, the electrodes of the actuator are connected to a constant voltage source, which then zip, and inflate the lens membrane to filling levels V > V 0 . The difference in volume ∆V = V − V 0 is called displaced volume, it is the volume exchanged between the actuator and lens membrane. We consider only reference states with convex lens shapes, hence V 0 ≥ 0, however states with V 0 < 0 can be experimentally realized and modelled within our framework. We calculate the elastic (free) energy of the incompressible lens membrane as To derive the actuator energy W A , we model the actuator as a radial-symmetric capacitor, surrounding the lens membrane. Its capacitance C(∆V ) changes, as the actuator zips. W A is dominated by the electrostatic contributions; the elastic energy of the actuator is much smaller than for the lens membrane, because the product of its material stiffness and thickness is one order of magnitude larger. The electrostatic energy of the capacitor W A ′ is: As the actuator zips, its capacitance increases, and the charges are provided by the power supply with the constant voltage U . The work of the external power supply QU should be subtracted in the energy balance, which reverses the sign of the electrostatic contribution to free energy [3,4], which then becomes: The total free energy of the system (up to an arbitrary constant) is then written as 1 The equilibrium condition is provided by the minimum of this expression with respect to the internal system variable ∆V , resulting in In the next sections, we derive expressions for p(V ) and C(∆V ) and provide an analytical model for the focus-voltage characteristics of ZEAP tunable lenses.
Inflation of a planar elastomer-membrane with prestretch Consider a thin circular elastic membrane with initial radius a and relaxed, unstretched thickness t M , which is clamped along its circumference, and is inflated by a uniform pressure p. The z-axis is the symmetry axis and r is the polar radius. In general case, the membrane can have an initial lateral equal-biaxial prestretch λ 0 . The height of the inflated membrane in z-direction is denoted as Z 2 , its area as S, and the enclosed volume as V .
Assuming that the inflated membrane is a spherical cap, these quantities can be written as Here, λ is the average stretch, based on the area change with respect to its initial value (which may includes a prestretch λ 0 ). From eq. (7) we obtain: Using the work-energy balance pdV = V M dG, where V M is the (constant) volume of the (thin, incompressible) membrane, we can relate the (over)pressure to the derivative of the elastic energy density G λ The expression (10) is rather general. 3 The p(V ) dependence can be simplified for small stretches without prestretch, where the Neo-Hookean relation holds for G and λ = 1 + δλ, δλ << 1. This case is the most relevant for elastomer lens modelling. With the shear modulus µ, the derivative of elastic energy density is written as This results in simplified expressions for the pressure, while the volume can be obtained from the eq. (8) Combining both equations results in the pressure-volume characteristics for the initial stage.
This approximations assumes small homogeneous stretch. 4 In reality, the stretch and membrane thickness vary along the membrane, and its shape deviates from a spherical cap. A more accurate fast calculation for the p-V -characteristic can still be performed numerically within the frame of linear material elasticity. 5 Numeric calculations for the inflation of a circular elastic membrane are based on a model from Pokorny et al. [5]. The applied pressure deforms the membrane such, that a point with initial coordinates (r, 0) moves to the position (r + u(r), w(r)) ( Supplementary Fig. S2). Thus, u(r) and w(r) represent the radial displacement and the deflection in z-direction. An initial prestretch λ 0 of the membrane causes a radial stretch λ r and a tangential stretch λ t in the deformed state, expressed as where u ′ (r) and w ′ (r) are the derivatives with respect to r. For a linear Saint Venant-Kirchhoff material, these stretches define the radial, tangential and initial stress, σ r , σ t and σ 0 respectively, via Young's modulus E and Poisson ratio ν.
The stresses and shape of the elastomer membrane are given by a system of ordinary differential equations with the boundary conditions For numerical calculations we use small values (e.g. 10 −4 ) for all dimensionless zero values, to avoid singularities. Solving these equations for different applied pressures gives the respective membrane shapes and enclosed volumes, resulting in the required pressure-volume characteristics, which can be used in further analysis. In particular, the paraxial focal length of the inflated membrane is calculated from the membrane shape 6 .

Capacitance of a radially-symmetric electrostatic zipping actuator
To derive the electrostatic actuator energy W A , an expression for the capacitance C is required. The actuator consists of a fixed bottom electrode with inner radius c and outer radius c + l (radial length l), and a top electrode with small slope (Supplementary Fig. S2). The initial opening angle between top and bottom electrode α 0 and is given by the actuator height h and length l.
When voltage is applied, the top electrode collapses from the outer edge, resulting in a zipped region with the radial length l 1 (and l L = l − l 1 for the unzipped region), and the opening angle changes to α. The capacitance of the actuator is the sum of the capacitance of the zipped region C 1 and of the unzipped region C L . 7 Here, S 1 is the area of the zipped region, S L the area of the unzipped region, ε F the permittivity of the plastic foil with thickness t F , and ε L the permittivity of the dielectric liquid with the radially changing thickness t L (r). The first term describes the capacitance of the zipped region C 1 and can be expressed as a function of the displaced volume.
Assuming linear thickness dependence on radius t L (r) (cone-shaped actuator), the integration in eq. (23) results in the capacitance of the unzipped region, which can be conveniently expressed via its outer radius R = c + l − l 1 The capacitance ratio C L /C 1 is large for small zipping regions with l 1 << l L . For a typical zipping regions with l 1 ∼ l L and ε L ∼ ε F this ratio is of the order of This ratio is small only asymptotically (for thin films with t F << h), and is often ∼ 1 for typical numbers. However, for the derivatives with respect to ∆V (which enter the expression (6) for the pressure), C ′ L (∆V ) is usually much smaller than C ′ 1 (∆V ) for all zipping lengths Neglecting the contribution of C ′ L (∆V ), and noting that the change in liquid volume ∆V L = −∆V , we obtain: The last expression assumes moderate volume changes, so that R is close to its initial value c+l, which results in a constant C ′ 1 (∆V ). This holds within few percent for our numbers. For large actuator sizes and small slope of the top electrode the contribution of C ′ L increases and should be included in eq. (28). Within the same "moderate changes" approximation R = c + l the derivative is modified as follows:

Displaced volume and the critical voltage
Substituting the result (28) 8 into the equality (6) together with the eq. (14) gives an analytic approximation for the equilibrium condition.
From here we obtain the dependence of the displaced volume on the applied voltage: Here, we defined the characteristic voltage U c , which combines the majority of the system properties, but is separated from the basic scaling with respect to the aperture size a and the initial volume V 0 .
Equation (31) produces negative displaced volume ∆V for small voltages, which is forbidden by system's design. This means, that zipping does not start below some threshold (critical) voltage U 0 : For small voltages U < U 0 , the equilibrium state remains completely unzipped due to the initial filling V 0 . 9

Dependence of the focal length on voltage
Now we can relate the lens volume V = ∆V + V 0 to its focal length f , and obtain its dependence on voltage. We use the thin lens expression f = ρ/(n − 1) for a plane-convex spherical cap of radius ρ = (a 2 + Z 2 )/2Z, assuming small volume (lowest order in Z/a << 1). Here, n is the refractive index of the lens (paraffin oil). This is a good approximation for our system, which results in a simple relation: 10 Together with the eq. (31), the focus-voltage relation for voltages U > U 0 becomes: For voltages below the threshold value, the focal distance remains unchanged and equal to its initial value f (U < U 0 ) = f (V 0 ) = f 0 . 8 The more accurate expression (29) with the corresponding changes was used in actual calculations, which modifies the result by a few percent. 9 The situation is similar to Peano-HASEL actuators under a constant load [6]. For U < U0 the free energy expression (5) has a minimum at V0 = 0, ∆V = 0, which has positive derivative with respect to ∆V . 10 In a more detailed approach, one can develop parametric formulas, based on the full volume and pressure expressions (8), (10) and thick lens formulas for the focal length.

Relations between a deflected laser beam and the focal length
When a thin laser beam, parallel to the optical axis, enters the lens at the radial distance d x from the axis, it is deflected by the angle β. This can be approximately treated as a single light ray. It hits the screen (positioned at a distance d s behind the lens) at the distance y from the optical axis. These quantities are related to the focal length of the lens by: When the focal length of the tunable lens changes by f 2 − f 1 = ∆f , the tangent and the spot position on the screen change accordingly: Therefore, the relative changes in tan β and spot position with respect to the reference state (1) are proportional to the relative change in focal length with respect to state (2).
Both expressions do not depend on d x , which simplifies the experimental measurements.

Cost estimation and scalability
A single small lens costs about 0.1431 e (calculation is given in Supplementary Table 3). The price is mostly determined by the PDMS elastomer, which makes 96 % of the total costs. The use of the elastomer could be readily reduced or replaced depending on the specific application or available fabrication facilities. Of course, the costs can be greatly reduced by the economies of scale as well.
The lenses are scalable without changing the fabrication methods, as the lens body is molded and the polymer foils (laser) cut to dimensions. We note however that miniaturizing the lenses to the mm scale would require a change to fabrication schemes that are assisted by high precision machines.

Optimizing homogeneous zipping
The models derived above assume axial symmetric zipping of the polymers. However, variabilities of opening angles or initial distance of the polymer lead to inhomogeneous zipping in the experimental realization. We here use two approaches to better control the start and propagation of the zipping fronts.
Welding points: Regular separated welding points at the circumference of the actuator induce starting points for zipping, due to a locally reduced separation of the electrodes (Supplementary Fig. S4). As a rule of thumb, the distance between the welding points along the circumference should be (appreciably) smaller than twice the width of the zipping annulus. Likewise, in the presence of auxiliary concentric rings (see below), it should be smaller than twice the distance between these rings. Increasing the number of welding points could lead to more homogeneous zipping, however only if the welding points are homogeneous as well. An infinite amount of points -hence a continuous circular welding-in practice still results in inhomogeneous zipping as certain regions of the welding will be (slightly) thinner and hence a (uncontrolled) starting point of a zipping front. Single points give better control over where the zipping starts.
Concentric rings: Concentrically patterned electrodes influence the zipping front propagation by inducing an anisotropy in radial direction. While the welding points represent a start of a zipping front, the concentric rings adjust the propagation speed of the zipping front. As we observed that the start of zipping fronts is much more important for homogeneous zipping than the propagation speed (at least for our lens dimensions), we disregarded the option of having concentric rings alone early in the design phase of the lenses. Future work will investigate the influence of those approaches to optimize zipping of ZEAP actuators. Non circular electrodes, such as regular polygons, might also greatly enhance the zipping behaviour. Recent work suggests that edges act as starting points for zipping as well [7].

Towards ZEAP actuators with operating voltages <50 V
The characteristic voltage in eq. (32) relates to the operating voltages of ZEAP devices. Omitting weak influence of the actuators lateral size parameters c and l eq. (32) can be simplified to The remaining quantities can be divided into actuator parameters (h, t F , ε F ) and lens membrane parameters (a, t M , µ), but all parameters influence the characteristic voltage in similar extent. A reduction of t F by a factor of 4 halves the operating voltages, as does a 4-times increase of ε F . For the current design parameters U c = 4.6 kV, 3.1 kV for the small and large actuators respectively, which results in operating voltages <500 V. Keeping the macroscopic lateral geometry unchanged, the cumulative design parameters would need to be changed by a factor 100 to achieve a reduction of operating voltages to <50 V. This is possible by using softer membranes (µ * = µ/ √ 10) and foils with reduced thickness (t * F = t F / √ 10) and higher permittivity (ε * F = 10ε F ).

Supplementary Tables
Author/ company  Comparison of lenses with large actuators featuring heat seals and no heat seals. Focal length changes require higher voltage in the initial regime (<300 V) when no heat seals are used. The focal length at higher voltages are unchanged. Note that the plateau, marking the threshold voltage, is at different initial focal length due to different initial fillings. The shape of the curves differs due to the effect of the heat seals (compare with Fig. 2g).  1 mm). b), The balance of mechanical and electrostatic forces results in the displacedvolume-voltage relations, with good agreement between the numerical (solid) and analytic (dashed) models c), Combination of actuator and lens properties results in focus-voltage characteristics. The initial filling volume determines starting (maximal) focal length and the threshold voltage for focal length tuning. Lenses with a small actuator (dashed lines) require slightly higher operating voltages due to a larger initial opening angle compared to lenses with a large actuator (solid lines). d)f ), Measured focal-length-voltage characteristics in comparison with the analytic (red dashed) and numerical (blue solid) models for lenses with different initial fillings. The experimental curves below and near the threshold voltages are influenced by effects that stem from non-uniform foil thickness due to welding spots or non-concentric zipping.