1. INTRODUCTION

Tunable-focus liquid-filled lenses [1–13] have recently been demonstrated to be ideal for developing variable focus eyewear applications [14–18]. These autofocus eyeglasses aim to restore the natural accommodation of the human eye that degrades over time due to refractive errors of vision such as myopia (nearsightedness); hyperopia (farsightedness), astigmatism, and presbyopia [19–27]. A healthy human eye has the ability to change the focal length of the biological lens in the eyeball by changing the radius of curvature of the crystalline lens using muscles connected to the lens that contract and relax, thereby changing the radius of curvature of the lens. However, most people lose the ability to naturally tune the focal length of their biological lens with progression of age, with an average decrease in the accommodation range from 11 diopters at age 20 to about 1–2 diopters at age 50 [18].

We have recently demonstrated eyeglasses with automatic tunable focus implemented using liquid-filled variable focus lenses [14–18]. We demonstrated large aperture (32 mm) lightweight ($<15\mathrm{gm}$) low-power ($<20\mathrm{mW}$) tunable lenses with the ability to tune an optical power range of 5.6 diopters. A schematic of this lens is shown in Fig. 1. The working principle of the tunable-focus lens involves the use of a chamber constructed using elastomeric material that can be stretched, the chamber being filled with high-index liquid such as glycerol (refractive index $n\sim 1.47$). A transparent piston connected to the back side of this chamber is driven by three connected piezo actuators, thus increasing or reducing the chamber pressure and thereby changing the radius of curvature of the front elastic membrane. In our previous demonstrations of these tunable-focus liquid-filled membrane lenses, we used polydimethylsiloxane (PDMS) as the elastomer for the top and bottom membranes. The front membrane is $\sim 0.6\mathrm{mm}$ thick. It is well-known that elastomeric membranes that are subject to numerous flexing/relaxation cycles experience creep over time, which reduces their ability to deflect appropriately given the same amount of force [28–33]. The creep thus causes drifts in the power versus deflection characteristics, which are critical to maintain for autofocus eyeglasses applications. It is desirable to develop autofocus eyeglasses that exhibit small optical power drift rate $<120\mathrm{mD}/\mathrm{yr}$ or 0.36D over three years, for a commercially viable product lifetime. In this work, first, we have studied the creep of different materials that are potential candidates for the membranes used in tunable focus liquid lenses. For this study, we have constructed a mechanical setup that rigorously investigates the material properties that determine the creep for different materials on circular membranes. We have correlated the experimental data to the loss of diopter range over time and selected the best material suitable for our application. Experimentally realized tunable lenses using different membranes have also been compared and the data presented.

 

Fig. 1. Schematic of tunable-focus liquid-filled lenses excluding the actuators. The actuators connect to the transparent piston and are responsible for imparting the force $F_{\text{piston}}$. The change in the radius of curvature of the lens front membrane is responsible for change in the optical power of the lens.

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2. EXPERIMENTAL INVESTIGATION OF ELASTOMERIC CREEP DUE TO PERIODIC DEFLECTION

Figure 2 shows a schematic diagram of the setup constructed to periodically produce flexure in a membrane. This setup was used to test the deflection and mechanical properties of different membranes, simply by changing the membrane while keeping other components the same. Different liquids may also be used in this same setup. Because the membrane deflection is caused by fluid flow, it is effectively de-coupled from the piezo-actuator-controlled deflection mechanism used in the tunable lenses. Hence, this setup allows us to study the mechanical properties of membranes, free from the influence of piezo-actuators. We have opted to use water as the liquid in the experiments reported in this paper due to the low viscosity of water, enabling us to use high-frequency deflection cycles.

 

Fig. 2. Schematic of the experimental setup used to analyze mechanical properties of membranes for tunable-focus liquid-filled lenses subject to periodic deflections.

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A. Membrane Flexure Setup

The setup consists of an air-tight chamber (custom 3D printed), the top surface of which is closed using a tensed membrane, the mechanical properties of which are being tested. The chamber is made air-tight using strategically located fixtures, gaskets, and air-tight fittings. The chamber has only one inlet/outlet. This is connected to a liquid reservoir, through a drain valve and a fluid column, fed from the liquid reservoir using a fill pump and a pressure sensor. The fluid column stores the liquid and is kept at a higher height with respect to the chamber ($\sim 450\mathrm{mm}$), and the liquid is allowed to flow into the chamber due to this pressure difference. Because the chamber is air-tight, the liquid inflow causes the membrane to bulge upward. At this stage, the drain valve (6 V DC solenoid valve) is kept closed. Next, the drain valve is opened. The tension in the membrane releases, allowing the liquid to flow out of the chamber through the drain valve and into the liquid reservoir. Again, once the drain valve is closed, the liquid flows from the fluid column down to deflect the membrane upward. The liquid is returned from the reservoir to the fluid column using a fill pump (12 V DC peristaltic pump). The fluid column is also provided with an overflow drain that drains back to the liquid reservoir, if required. A tiny metal patch of negligible weight is placed, centered on top of the membrane. The deflection of the membrane is recorded using an optical displacement sensor (Micro-epsilon optoNCDT IL 1420 visible laser displacement sensor). A laser beam from the sensor module reflects off the top of the metal patch on top of the membrane and is detected by a photodetector integrated into the displacement sensor. The membrane position is measured by the optical displacement sensor, and the data are recorded into a computer.

It is to be noted that a large number of flex/relax cycles would be required to effectively observe any elastomeric creep in the membrane. This would require an extremely long experimental times ($\sim$ months to years). In order to reduce this time, we resorted to using an accelerated failure model to study the elastomeric creep based on the Arrhenius approach [34–36]. In this method, the chamber is heated using a silicon strip heater (30 W 120 V DC), the temperature of which is monitored using a temperature sensor (standard thermocouple) and a PID temperature controller. The entire system is fully automated using a computer control and a USB relay (Ontrak ADU208 relay box) and necessary control circuits. The heating of the chamber, the liquid inside, and, subsequently, the membrane when it is undergoing constant flex/relax cycles allowed us to observe creep and failure at an accelerated rate and thereby analyze the mechanical properties of the membranes over a short time period. Figures 3(a)–3(c) show the complete constructed setup.

 

Fig. 3. Experimental setup used to analyze mechanical properties of membranes for tunable-focus liquid-filled lenses. (a) Setup showing different components: ODS, optical displacement (laser) sensor; FP, fill pump; DV, drain valve; CC, control circuits. (b) Close-up view of a flexed membrane with a tiny metal patch and laser spot from the optical displacement sensor. (c) Close-up view of the air-tight chamber, heater, and displacement sensor assembly.

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B. Membrane Tensor Setup

In our previous work, we demonstrated liquid-filled membrane lenses using PDMS membranes. PDMS membranes inherently possess slight tension when cured. However, because this may not be repeatable every time and, also, because we aim to switch to inorganic membranes in the future, it was required to develop a mechanical setup that can introduce tension in elastomeric membranes in a repeatable manner. Figure 4 shows a schematic of such a setup. A membrane is stretched across a Teflon outer ring and held securely in place using screws. Then, three force points are introduced on this ring at equi-angular locations (120 deg apart). This outer ring with the stretched membrane is gradually lowered over an inner ring, which creates tension in the membrane. The tension can then be calculated from the values of the forces applied at the three points, recorded by three connected force gauges (Nidec-Shimpo FG-3000), and the values of the displacement for these three points as recorded by three displacement gauges (clockwise digital dial indicator displacement gauges) connected to the force gauges and in turn to the three force points. This allows us to accurately record the forces applied as well as the displacement values at the three force-point locations.

 

Fig. 4. Experimental setup used to introduce tension in elastomeric membranes in a repeatable manner. (a) Schematic showing the two-ring assembly and force points. (b) Schematic showing the introduction of tension in a membrane using this tensor setup. (c) Constructed setup showing the three force gauges and the three displacement gauges with (d) its close-up view showing a Teflon ring glued to a tensed membrane that is then used to transfer the stressed membrane to the flexure setup.

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Once tension is created in the membrane, a connecting acrylic ring is attached to preserve the tension using UV-cured glue (Norland Optical Adhesive 68). Once the glue is cured, the tension is retained in the membrane. This is then removed, reshaped, and fitted into the flexure setup discussed in the previous section as the top surface of the air-tight chamber.

3. RESULTS AND ANALYSES OF THE MECHANICAL PROPERTIES OF DIFFERENT ELASTOMERIC MEMBRANES

A. Membrane Creep Study

The periodic membrane deflection setup allows us to quantitatively measure the deflection of any membrane under controlled circumstances and study its mechanical properties. This section describes tests and results of different membranes. Figure 5(a) shows the basic measurement of the deflection of the membrane by the periodic membrane flexure setup described in Section 2.1. When the drain valve (Fig. 1) is closed, the fluid from the fluid column fills the chamber and causes the membrane to flex upward. After a specified time, the drain valve is opened, the tension in the membrane is released, and the fluid is pushed out from the chamber, causing the membrane to flex downward. The optical displacement sensor records these two positions (h1 and h2). A tiny metal patch is placed on top of the membrane, at the centermost point (here, the deflection is maximum) to facilitate the reflection of the laser spot for the optical sensor. The absolute position is measured from the plane of the sensor downward; hence, this corresponds to the plane of the sensor being at $h=0$, the upward flexed location at $h=-h_1$, and the downward flexed location at $h=-h_2$, where $h_2>h_1$. The drain valve, pump, optical sensor, etc. are controlled using the control circuit shown in Fig. 3(a) and connected to a computer control that also constantly records the displacement values from the optical sensor. Figure 5(b) shows a small data set of about 700 s extracted from a three to four day-long run. The periodicity of the flex/relax cycle can be clearly observed. The period is $\sim 30\mathrm{s}$. Here, $-h_1=-3.4\mathrm{mm}$ and $-h_2=-4.6\mathrm{mm}$ (approximate average); hence, the amplitude of deflection $\sim 1.2\mathrm{mm}$.

 

Fig. 5. (a) Schematic showing the flexing of the membrane in the periodic fluidic setup. (b) Small data set ($\sim 700\mathrm{s}$ of $3.45\times {10}^5\mathrm{s}$) of real-time measurement of the membrane position versus time.

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Following this method, we tested a number of membranes (with different tension values and at different temperatures. Then, we used the peak-to-trough deflection data at different temperatures over long time periods to estimate the accelerated failure characteristics using the Arrhenius relationships as outlined below:

The Arrhenius model is a widely accepted empirical accelerated life model based on the assumption that the degradation observed in the subject is due to physio-chemical causes. In the case of elastomeric membranes, creep is a purely physio-chemical phenomenon caused by the degradation of the chemical structure of the material due to repeated flex/relaxation cycles. The Arrhenius model [34] is widely regarded as one of the oldest and most reliable acceleration models that can predict the time-to-fail variation with temperature. Briefly, the mathematical form of this model is given by the equation (we state the mathematical expression of membrane creep deformation as follows)

However, this relationship does not allow us to measure the creep characteristics of the membranes being tested under accelerated conditions, which is required to avoid testing membranes for inordinate amounts of time (days/months), which would inevitably lead to stability issues in the experimental setup over time. Hence, we chose to use the Arrhenius model for accelerated life testing in order to calculate $S$ without having to spend an inordinate amount of time observing decrease in $h$ with respect to $t$ in room conditions.

According to the Arrhenius model for accelerated life testing, we can again express $S$ as follows:

Using Eq. (2), for two different temperatures, $T_1$ and $T_2$, and two corresponding normalized rate of drift values, $S_1$ and $S_2$, which are the slopes of the two straight-line fits through actual observed peak-to-trough membrane deflection data, $E_a$ can be calculated as follows:

First, we used this technique to analyze the membrane creep properties of a $\sim 600\mathrm{\mu m}$ thick PDMS membrane with no added tension, as was used in our previous demonstrations of liquid-filled tunable-focus lenses. Figure 6(a) shows the normalized membrane peak-to-trough deflection v/s time ($s$) for the PDMS membrane for two different temperatures (55°C and 70°C). In order to calculate the normalized peak-to-trough deflection, the absolute measurement of the membrane position (mm) is first smoothed using a Savitzky–Golay filtering method in MATLAB, with filter order = 7 and frame length = 39. Then, the peaks and troughs are extracted using peak finding algorithms; finally, the difference of these is used to calculate the amplitude, which is smoothed through an eight-sample median filter to remove noise; lastly, this value was normalized. We used a linear fitting method to fit two straight lines through the two data sets corresponding to the two different temperatures, 55°C and 70°C. It is to be noted that these temperatures correspond to the temperature at which the heater-temperature sensor pair is run; hence, the fluidic chamber is heated. However, there are some ambient temperature fluctuations in the room that cannot be avoided, and these contribute to the noise in the data seen in Fig. 6(a). Hence, it is necessary to fit a straight line through the data set. The fitting parameters and the equation of the straight-line fits are shown in Fig. 6(a). We use the slopes of the fitted straight lines in Eq. (3) to calculate $E_a$ and finally use the value of $E_a$ in Eq. (4) to calculate the value of the rate constant $A$. The values for PDMS are as follows: $E_a=2.3774\times {10}^{-20}\mathrm{kJ}/\mathrm{Mol}$ and $A=3.1206\times {10}^{-5}/\mathrm{s}$. Using these values, we can calculate the normalized rate of drift for PDMS at various temperatures. Figure 6(b) shows a plot of ${\log }_{10}S\mathrm{v}/\mathrm{s}$ temperature (°C). From this graph, we calculated that, at room temperature (27°C), $S_{27}=1.0035\times {10}^{-7}/\mathrm{s}=3.6126\times {10}^{-4}/\mathrm{hr}=3.16/\mathrm{yr}$.

 

Fig. 6. (a) Normalized membrane peak-to-trough deflection v/s time ($s$) and (b) ${\log }_{10}(S)\mathrm{v}/\mathrm{s}$ temperature (°C) for nontensed PDMS membrane creep analysis study. The values at 55°C and 70°C were measured first while the membrane creep analysis was being conducted. The value at 30°C was measured by performing an experiment after the analysis was complete. The difference between the predicted value and experimentally measured value at 30°C is 9.866%.

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Using the same principle as described above, we also tested a 50 μm thick Cosmoshine membrane (Toyobo International Industrial films transparent film Cosmoshine A4100) under applied tension. However, in the case of the Cosmoshine membrane, we used the membrane tensor setup described in Section 2.2 to apply tension to the membrane. An average of 5.23 N force for an average net displacement of 1.454 mm (recorded using the force gauges and displacement gauges, respectively) was applied to the membrane. The results are as follows: Fig. 7(a) shows the normalized membrane peak-to-trough deflection versus time ($s$) for the Cosmoshine membrane for two different temperatures (30°C and 70°C). The data processing and fitting were performed using the same method as described in the previous section. The values for Cosmoshine are as follows: $E_a=4.3234\times {10}^{-20}\mathrm{kJ}/\mathrm{Mol}$ and $A=-0.0121/\mathrm{s}$. Figure 7(b) shows a plot of ${\log }_{10}S\mathrm{v}/\mathrm{s}$ temperature (°C) for this membrane. From this graph, we calculated that, at room temperature (27°C). $S_{27}=-3.5534\times {10}^{-7}/\mathrm{s}=-0.0013/\mathrm{hr}=-11.2060/\mathrm{yr}$. Because a low rate of drift is desirable, this performance is much worse compared with that of PDMS membrane.

 

Fig. 7. (a) Normalized membrane peak-to-trough deflection v/s time ($s$) and (b) ${\log }_{10}(S)\mathrm{v}/\mathrm{s}$ temperature (°C) for tensed Cosmoshine membrane creep analysis study. The values at 30°C and 70°C were measured first while the membrane creep analysis was being conducted. The value at 55°C was measured by performing an experiment after the analysis was complete. The difference between the predicted value and experimentally measured value at 55°C is 0.27%.

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Next, we tested a 50 μm thick SKC membrane (Skyrol Polyester Film SG00L) under applied tension. First, we applied an average of 5.06 N force (comparable with that applied to the Cosmoshine membrane) for an average net displacement of 1.624 mm (recorded using the force gauges and displacement gauges, respectively). However, we were able to see only a 0.070 mm peak-to-trough deflection for this membrane, after running the deflection test for two to three days. This is insignificant when compared with $\sim 1.2\mathrm{mm}$ for PDMS and $\sim 0.5\mathrm{mm}$ for Cosmoshine. This would also not give us the desired diopter range for the tunable-focus lenses. Next, we applied an average of 0.316 N force for an average net displacement of 1.002 mm (recorded using the force gauges and displacement gauges, respectively). However, again, we were able to see only a $\sim 0.070\mathrm{mm}$ peak-to-trough deflection for this membrane, after running the deflection test for two to three days. Nevertheless, we recorded membrane peak-to-trough deflection data at two different temperatures (30°C and 50°C) and performed the similar analyses as described above. The results are as follows: Fig. 8(a) shows the normalized membrane peak-to-trough deflection v/s time ($s$) for the SKC membrane for two different temperatures (30°C and 70°C). The data processing and fitting were performed using the same method as described in the previous section. The values for SKC are as follows: $E_a=1.8424\times {10}^{-19}\mathrm{kJ}/\mathrm{Mol}$ and $A=-6.994\times {10}^{-13}/\mathrm{s}$. Figure 8(b) shows a plot of ${\log }_{10}S\mathrm{v}/\mathrm{s}$ temperature (°C) for this membrane. From this graph, we calculated that, at room temperature (27°C), $S_{27}=-3.36\times {10}^{-6}/\mathrm{s}=-0.0121/\mathrm{hr}=-106.07/\mathrm{yr}$. This performance was considerably unfavorable.

 

Fig. 8. (a) Normalized membrane peak-to-trough deflection v/s time ($s$) and (b) ${\log }_{10}(S)\mathrm{v}/\mathrm{s}$ temperature (°C) for tensed SKC membrane creep analysis study. The values at 30°C and 50°C were measured first while the membrane creep analysis was being conducted. The value at 27°C was measured by performing an experiment after the analysis was complete. The difference between the predicted value and experimentally measured value at 27°C is 21.44%.

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Last, using the same principle as described above, we also tested a 400 μm thick PDMS membrane under tension (usually, the PDMS membrane used is $\sim 600\mathrm{\mu m}$ thick but nontensed; because we applied tension, we reduced the thickness). An average of 3.1 N force for an average net displacement of 3.437 mm (recorded using force gauges and displacement gauges, respectively) was applied to the membrane. The results are as follows: Fig. 9(a) shows the normalized membrane peak-to-trough deflection v/s time ($s$) for the tensed PDMS membrane for two different temperatures (55°C and 70°C). The data processing and fitting were performed using the same method as described in the previous section. The values for tensed PDMS membrane are as follows: $E_a=1.004\times {10}^{-9}\mathrm{kJ}/\mathrm{Mol}$ and $A=-3.927\times {10}^3/\mathrm{s}$. Figure 9(b) shows a plot of ${\log }_{10}S\mathrm{v}/\mathrm{s}$ temperature (°C) for this membrane. From this graph, we calculated that, at room temperature (27°C), $S_{27}=-1.166\times {10}^{-7}/\mathrm{s}=-4.1976\times {10}^{-4}/\mathrm{hr}=-3.3771/\mathrm{yr}$. This performance was considered close to the performance of the nontensed PDMS membrane.

 

Fig. 9. (a) Normalized membrane peak-to-trough deflection v/s time ($s$) and (b) ${\log }_{10}(S)\mathrm{v}/\mathrm{s}$ temperature (°C) for tensed PDMS membrane creep analysis study. The values at 55°C and 70°C were measured first while the membrane creep analysis was being conducted. The value at 30°C was measured by performing an experiment after the analysis was complete. The difference between the predicted value and experimentally measured value at 30°C is 6.68%.

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It can be seen in Figs. 6–9 that the measured deflection values show significant fluctuations around the fitted linear trend. We attribute this fluctuation to the change in surrounding temperature in the laboratory. However, it can be seen that, for higher values of the temperature of the experimental setup, the variation is low, indicating that, when the temperature of the setup is significantly higher than ambient temperature, ambient temperature fluctuations have a lower effect on the variation in the data. At the moment, we are simply fitting the linear model through the average fluctuations and trying to observe the membrane creep, empirically.

Last, the optical power drift is directly related to the normalized rate of deflection drift [15] because the relation between the power of the lens and the height of the bulged membrane is

B. Comparison of Tunable Lens Performance

Finally, we constructed two tunable-focus liquid-filled lenses using the methods discussed above and elsewhere [15,16] using PDMS and Cosmoshine as two representative membranes amongst the ones studied here. The characterization of the performance of the lenses was done following experimental setup and techniques discuss previously [14]. Figure 10 shows the comparison of the performance of the two membranes in their respective tunable lenses, by demonstrating the variation of the lens optical power as a function of the actuator voltage. The range of powers over which a PDMS lens was successfully tuned was $-2.56\mathrm{D}$ to $+3.01\mathrm{D}$ and that for a Cosmoshine lens was 1.88 D to 3.07 D for a voltage range of $-250\mathrm{V}$ to $+250\mathrm{V}$. It can be clearly seen that, owing to its high value of Young’s modulus, rigidity, and creep characteristic, the lens made using Cosmoshine membrane performs much worse compared with the one made using PDMS membrane.

 

Fig. 10. Comparison of performance of two membranes PDMS and Cosmoshine in tunable-focus liquid-filled lenses. Tunable-focus liquid-filled lenses fabricated using (a) PDM membrane and (b) Cosmoshine membrane. (c) Lens optical power (at the lens center) as a function of voltage for the two lenses shown in (a) and (b).

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We can see that, based on this careful study and analyses, it is difficult or near impossible to meet the creep rates for an elastic membrane liquid lens for the polymers tested.

4. CONCLUSIONS

In this work, we presented the analyses of the elastomeric creep property of different materials that are potential candidates for the membranes used in the tunable-focus liquid-filled lenses for our autofocus eyeglasses. We presented theoretical studies and experimental results to compare the performance of different elastomeric membranes.