史惠琦^*, 王惠明^,*,?,2)*浙江大学航空航天学院工程力学系,杭州 310027
^?浙江大学浙江省软体机器人与智能器件研究重点实验室,杭州 310027

THEORETICAL NONLINEAR ANALYSIS OF A BIOMIMETIC TUNABLE LENS DRIVEN BY DIELECTRIC ELASTOMER ¹⁾

Shi Huiqi^*, Wang Huiming^,*,?,2)*Department of Engineering Mechanics,Zhejiang University,Hangzhou 310027, China
^?Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province,Zhejiang University, Hangzhou 310027, China

通讯作者: 2) 王惠明,教授,主要研究方向：多场耦合力学, 智能软材料非线性力学. E-mail:wanghuiming@zju.edu.cn

收稿日期:2020-06-18接受日期:2020-09-7网络出版日期:2020-11-18

基金资助:

1) 国家自然科学基金资助项目.11772296

Received:2020-06-18Accepted:2020-09-7Online:2020-11-18
作者简介 About authors


摘要
介电弹性体 (dielectric elastomer) 是电活性聚合物智能材料的一种,在外加电场作用下,可产生多种形式的响应.在驱动柔性透镜的变焦方面,相对于传统的机械操控变焦方法 显示出独特的优势.针对一款在电压激励下可高效调节焦距的介电弹性体仿人眼变焦透镜,该透镜由上下两层介电弹性薄膜和固定框架构成,并在封闭腔内充入盐水,上层薄膜涂覆环形柔性电极.在电压激励下,上层膜发生变形,由于盐水的体积保持恒定,引 起下层膜随之变形,使得透镜的焦距发生改变.采用 neo-Hookean 模型,利用变分原理导出了该透镜的控制方程、边界条件和连 续条件.利用打靶法求解了该非线性问题并高效地处理了非线性问题的界面连续条件. 理论分析结果与实验结果相吻合. 利用此模型开展了广泛的参数分析,研究表明,透镜的几何形状、初始焦距、介电弹性体薄膜的预拉伸率、涂覆的电极面积、材料的剪切模量等对透镜焦距的调节性能都有重要的影响.所建立的理论分析模型可为柔性仿生透镜的设计和参数优化提供有效的分析方法.
关键词： 介电弹性体;柔性可调焦透镜;焦距;非线性

Abstract
Dielectric elastomer (DE) is a class of electroactive polymer smart materials. Under the external electric field, it can produce various forms of responses. Comparing with the traditional lens with which the focus length is manipulated by the mechanical controls, the DE soft tunable lenses exhibit the distinct advantages in the tuning way of the focal length. The DE soft tunable lenses tune the focal length by mimicking the eyeball of human beings. The lens is composed of two circular DE films which are fixed on the rigid frame. The salty water is filled in the enclosed space and forms a convex lens. The top DE film is coated by the annular compliant electrodes. Under the voltage excitation, the upper film is deformed. Accordingly, the lower film is deformed due to the incompressibility of the salt water sealed in the enclosed space. Subsequently, the focal length of the tunable lens is changed. By employing the variational principle and the neo-Hookean model, we obtain the governing equations, boundary conditions and the continuity conditions of the biomimetic lens when driven by the dielectric elastomers. The nonlinear governing equations are solved by the shooting method and the continuity conditions at the interface are treated with in an effective way. The theoretical results agree well with the experimental data. The extensive parametric analysis is carried out based on the presented model. The numerical results show that the geometrical configuration, the initial focal length, the area of the coated annular compliant electrodes, the pre-stretch of the top DE film and the shear modulus of the bottom film have significant effect on the adjusting performance of the tunable lens. The presented theoretical model provides an effective tool in designing and optimizing the biomimetic adaptive focus lens.
Keywords：dielectric elastomers;soft tunable lens;focal length;nonlinearity


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本文引用格式
史惠琦, 王惠明. 一种新型介电弹性体仿生可调焦透镜的变焦分析¹⁾. 力学学报[J], 2020, 52(6): 1719-1729 DOI:10.6052/0459-1879-20-212
Shi Huiqi, Wang Huiming. THEORETICAL NONLINEAR ANALYSIS OF A BIOMIMETIC TUNABLE LENS DRIVEN BY DIELECTRIC ELASTOMER ¹⁾. Chinese Journal of Theoretical and Applied Mechanics[J], 2020, 52(6): 1719-1729 DOI:10.6052/0459-1879-20-212

引言

随着智能手机和相机产业的迅速发展,有关可变焦透镜的研究受到广泛关注.传统可变焦透镜系统的驱动要依赖电机和齿轮等 调控装置,这会使整个系统变得笨重,导致无法在小空间内使用^[1].人们在不断寻找小型化办法的同时,也在寻找其他更为有效的驱动调焦方式^[2-12]. 介电弹性体驱动器 (dielectricelastomer actuators,DEA) 在这一领域显示出了巨大潜能.

介电弹性体薄膜在外加电场作用下能够在厚度方向产生收缩,并在面内方向扩展,从而产生较大的应变及作动力,具有超大变形、高弹性能密度、轻质量、低价格、易加工、易成形、高效率和短反应时间等特点^[13-15],已被广泛应用于各种领域,如柔性机器人^[16-18],能量采集器^[19-20]等.其中,由介电弹性体驱动的可变焦透镜引起了广泛关注^[21-23].与此同时,关于介电弹性体机电耦合行为和材料的非线性行为的理论研究也在不断深入^[24-27],对基于此驱动方式的机电系统的设计和优化提供强有力的理论支撑和有效的分析方法.

最近,Li 等^[28]设计完成了一款基于环状介电弹性体薄膜驱动的变焦透镜,结构如图1 所示,将两层圆形介电弹性体薄膜固定在框架上,上层膜涂覆了环形柔性电极,在两层膜和框架围成的空间区域内注入一定体积的盐水.当对涂覆了柔性电极的环状区域施加电压时,由于 Maxwell 应力的作用^[29],上层膜在厚度方向发生收缩变形,由于盐水体积不变的约束条件,驱动下层膜发生变形,从而导致整个透镜系统的光学性质发生改变.实验结果表明,将透镜的上下膜的半径设计成不相等,即图1(c) 中 $a \ne c$,可以获得更加优越的光学性能.

图1

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图1仿人眼介电弹性体可调焦透镜示意图(上层和下层介电弹性体薄膜分别固定于半径为 $a$ 和 $c$ 的框架上,其中上层膜涂覆了环形柔性电极)

Fig. 1Schematics of the biomimetic dielectric elastomer adaptive focus lens (The upper and lower dielectric elastomer films are fixed on frames with radius $a$ and $c$, respectively. The upper film is coated with an annular electrode)



本文建立了新近提出的一种新型高性能透镜的理论分析模型.利用上下两层膜的直径不相等的特点,可实现透镜在较大范围内快速 调焦的功能.利用所建立的理论分析模型,模拟得到的光学性质 (焦距变化率) 随外界激励 (电压) 的变化,与实验结果相吻合.在参数分析中,讨论了影响透镜焦距变化的参数,如初始焦距、透镜的几何形状、预拉伸率、电极面积和膜的剪切模量等.

1 力学模型的建立

图2给出了仿人眼介电弹性体可调焦透镜的力学模型,非线性分析模型的建立可参考文献^{[30,31,32,33,34]}. 图2(a) 表示参考状态,无外力和电压作用,薄膜上任意一点的位置用 $R$ 表示. 上层膜分为两个区域,即圆形透镜部分 ($0\leqslant R \leqslant B$) 和环形驱动部分 ($B \leqslant R \leqslant A$),初始厚度为 $H$.下层膜的半径为 $C$,初始厚度为 $H'$.图2(b) 表示预拉伸状态,上下两层膜分别被预拉伸后固定于半径为 $a$ 和 $c$ 的框架上.然后在上下两层薄膜和框架构成的封闭腔内注入体积为 $V_{0}$ 的盐水,设此时的液体压强为 $p_{0}$,见图2(c) 表示的充液状态.图2(d) 为驱动状态,在环形柔性电极的上下表面施加电压 $\varPhi$,由于 Maxwell 应力的作用,使得该部分薄膜的厚度减小,面积增大,这种机电耦合效应加上液体的耦合作用,导致所封闭腔体内液体压强由 $p_{0}$ 变为 $p$,上下两层膜产生轴对称非均匀变形,引起透镜光学性质的改变.

图2

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图2仿人眼介电弹性体可调焦透镜的力学模型(图中实心红色圆点表示处于参考状态时介电弹性体 薄膜上距离中心点为 $R$ 的点在变形各阶段所处的位置. $r$-$z$ 和 $r'$-$z'$ 分别为上层膜和下层膜在现时构形中的坐标系)

Fig. 2Mechanical model of biomimetic dielectric elastomer tunable lens (In each state, the position of a particular material particle is identified by a red dot. $r$-$z$ and $r'$-$z'$ are the coordinate systems of the upper film and the lower film in the current configuration, respectively)



采用坐标系 ($r$-$z$) 和 ($r'$-$z'$) 来分别描述上下两层膜的变形,如图2(c) 和图2(d) 所示. 利用非线性连续介质力学分析方法,考察在参考状态薄膜上距离中心点为 $R$ 的点,在变形后,处于位置 $(r(R),z(R))$,对应的参考状态距离中心点为 $R +\text{d}R$ 的点,变形后占据位置 $(r(R+ \text{d}R)$, $z(R +\text{d}R))$.两点之间初始长度为 $\text{d}R$ 的线元变形后长度为 $\lambda _{1}\text{d}R$, $\lambda_{1}$ 为径向伸长率,令 $\theta(R)$ 表示参考状态距离中心点为 $R$ 的点在变形后该点的切线方向与水平方向的夹角,则

(1)$\dfrac{\text{d}r}{\text{d}R} = \lambda_1 \cos\theta$
(2)$\dfrac{\text{d}z}{\text{d}R} = - \lambda_1 \sin\theta$
定义 $\lambda_{2}$ 为环向伸长率,有 $\lambda_{2}=r/R$.

当膜发生面外变形后,上层膜与 $r$ 轴所在水平面包围的液体体积为

(3)$V = \int_0^a 2\pi zr \text{d}r = \int_0^A 2\pi zr\dfrac{\text{d}r}{\text{d}R} \text{d}R$
为了研究施加电压后该透镜的响应,定义名义电位移 $\tilde {D}$,即某一微元的电荷总量除以该微元在未变形状态下的面积,则总电荷

(4)$Q = 2\pi \int_B^A \tilde {D}R \text{d}R$
在恒温条件下,介电弹性薄膜变形的热力学系统,可用 3 个独立的变量 $\lambda_{1}$, $\lambda_{2}$, $\tilde{D}$ 来表征,则 Helmholtz 自由能密度 $W$ 可表示为

(5)$W = W\left( {\lambda_1,\lambda_2,\tilde {D}} \right)$
当介电弹性体处于平衡态时,自由能的增大等于机械载荷和电场做的总功,即

(6)$\delta W = s_1\delta \lambda_1 + s_2\delta \lambda_2 + \tilde {E}\delta \tilde {D}$
其中,$s_{1}$ 为径向名义应力, $s_{2}$ 是环向名义应力,$\tilde {E}$ 是名义电场强度且有 $\tilde {E} = \varPhi / H$.

由式 (5) 和式 (6) 可得

(7)$s_1 = \dfrac{\partial W\left( {\lambda_1,\lambda_2,\tilde {D}} \right)}{\partial\lambda_1}$
(8)$s_2 = \dfrac{\partial W\left( {\lambda_1,\lambda_2,\tilde {D}}\right)}{\partial \lambda_2}$
(9)$\tilde {E} = \dfrac{\partial W\left( {\lambda_1,\lambda_2,\tilde {D}}\right)}{\partial \tilde {D}}$
当自由能密度函数具体给定后,由式 (7) $\sim$ 式 (9) 可以确定出材料的本构关系.

利用变分原理,可以建立外边界固定且涂覆环形电极的介电弹性体薄膜的控制方程、边界条件和连续条件 (具体推导见附录). 用上标 (1) 和 (2) 分别表示中间圆形透镜部分 $( R \leqslant B)$ 和环形驱动部分 $( B \leqslant R \leqslant A )$ 的物理量,则控制方程为

(10)$ \dfrac{\text{d}}{\text{d}R}(HRs_1 \sin\theta ) - \lambda_1\lambda_2Rp \cos\theta = 0$
(11)$\dfrac{\text{d}}{\text{d}R}(HRs_1 \cos\theta ) + \lambda_1\lambda_2Rp \sin\theta = s_2H$
由于中间圆形透镜部分和环形驱动部分的控制方程在形式上完全相同,因此在式 (10) 和式 (11) 中略去了上标 (1) 和 (2). 对于上层膜,边界条件如下：

根据对称性,在透镜部分的中心位置 ($R =0$) 处

(12)$r^{(1)}(0) = 0 , \ \ \theta ^{(1)}(0) = 0$
在膜的外边界处 ($R=A)$,有

(13)$r^{(2)}(A) = a , \ \ z^{(2)}(A) = 0$
在两部分的连接处 ($R=B)$,有连续条件

(14)$ \left.\begin{array}{l} r^{(1)}(B) = r^{(2)}(B) , \ \ z^{(1)}(B) = z^{(2)}(B) \\ \theta ^{(1)}(B) = \theta ^{(2)}(B) , \ \ s_1^{(1)} (B) = s_1^{(2)} (B) \end{array}\!\!\right\}$
对于下层膜,边界条件为

(15)$ \left.\begin{array}{l} r '(0) = 0 , \ \ \theta '(0) = 0 \\ r '(C) = c , \ \ z '(C) = 0 \end{array}\!\!\right\}$
其中 $\theta '$ 代表下层膜任意一点在变形后切线方向与坐标轴 $r'$ 的夹角.

2 本构关系

真实电场 $E$ 和真实电位移间 $D$ 存在线性关系 $D = \varepsilon E$, $\varepsilon$ 是与材料变形程度无关的介电常数. 假设介电弹性体薄膜具有体积不可压缩性质,所以厚度方向的伸长率 $\lambda_{3} =1/( \lambda_{1}\lambda_{2})$,名义电场与真实电场间的关系为 $E = \lambda_1\lambda_2\tilde{E}$,名义电位移与真实电位移的关系为 $D = \tilde {D} /(\lambda_1\lambda_2)$,因此名义电场强度与名义电位移的关系有

(16)$\tilde {E} = \dfrac{\tilde {D}}{\varepsilon }\lambda _1^{ - 2} \lambda _2^{ - 2}$
将介电弹性体的自由能函数写成^[35]

(17)$W\left( {\lambda_1,\lambda_2,\tilde {D}} \right) = W_{stretch} \left( {\lambda_1,\lambda_2} \right) + \dfrac{\tilde {D}^2}{2\varepsilon }\lambda _1^{ - 2} \lambda _2^{ - 2}$
上式右端的第一项代表聚合物材料变形引起的弹性应变能函数,第二项是电场能量密度函数. 采用neo-Hookean 模型,有

(18)$ W_{stretch} (\lambda_1,\lambda_2) = \dfrac{\mu }{2}(\lambda _1^2 + \lambda _2^2 + \lambda _1^{ - 2} \lambda _2^{ - 2} - 3)$
式中,$\mu $ 代表聚合物材料在小应变时的剪切模量. 将式 (17) 代入式 (7) 和式 (8) 得

(19)$s_1 = \mu \left( {\lambda _1 - \lambda _1^{ - 3} \lambda _2^{ - 2} } \right) -\varepsilon \tilde {E}^2\lambda _1 \lambda _2^2$
(20)$s_2 = \mu \left( {\lambda _2 - \lambda _1^{ - 2} \lambda _2^{ - 3} } \right) - \varepsilon \tilde {E}^2\lambda _1^2 \lambda _2$
根据非线性连续力学理论,可建立名义应力与真实应力间的关系为 $\sigma _1 = s_1\lambda _1 $, $\sigma _2 = s_2 \lambda _2 $,即有

(21)$\sigma _1 = \mu \left( {\lambda _1^2 - \lambda _1^{ - 2} \lambda _2^{ - 2} } \right) - \varepsilon E^2$
(22)$\sigma _2 = \mu \left( {\lambda _2^2 - \lambda _1^{ - 2} \lambda _2^{ - 2} } \right) - \varepsilon E^2$
式 (21) 和式 (22) 中的第二项代表Maxwell 应力. 对于没有涂覆电极的透镜部分和下层膜,在力学模型中只需设 $E = 0$ 或 $\tilde {E} = 0$ 即可.

3 计算实现

由式 (10) 和式 (11) 可导出以下两式

(23)$\dfrac{\text{d}\theta }{\text{d}R} = - \dfrac{s_2 }{s_1 }\dfrac{\sin\theta }{R} + \dfrac{\lambda _1 \lambda _2 }{s_1 }\dfrac{p}{H}$
(24)$\dfrac{\text{d}\lambda _1 }{\text{d}R} = \dfrac{s_2 \cos\theta - s_1 + \dfrac{\text{d}s_1 }{\text{d}\lambda _2 }(\lambda _2 - \lambda _1 \cos\theta )}{R\dfrac{\text{d}s_1 }{\text{d}\lambda _1 }}$
联立方程 (1) 和 (2) 以及方程 (23) 和 (24),得到关于 4 个函数 $r(R)$,$z(R)$, $\theta(R)$, $\lambda_{1}(R)$ 的一阶常微分方程组.

对于上层膜,中间圆形透镜部分 ($R \leqslant B$) 无电压作用,令式 (19) 和式 (20) 中的 $\tilde {E} = 0$,环形驱动部分 ($B \leqslant R \leqslant A$) 受电压激励,有 $\tilde {E} = \varPhi / H$. 求解时采用打靶法,从中心点 $R =0$ 开始,利用边界条件 (12) 和 (13),使用 Matlab 中的 fsolve 求解函数,利用介电弹性体薄膜在界面 $(R=B)$ 处的连续条件 (14),完成 4 个物理量 $r(B)$,$z(B)$, $\theta (B)$ 和 $\lambda_{1} (B)$ 在界面处的传递,可得出最终的结果. 对于下层膜,直接利用无电压作用情形的控制方程,结合边界条件 (15),采用打靶法得到最终结果.

对于图2(c) 所示的充液状态,将体积为 $V_{0}$ 的盐水注入两层膜之间的封闭的腔内,假设液体的压强,通过计算弹性场,利用式 (3),可以确定出液体初始压强 $p_{0}$. 当施加电压激励后,液体压强将发生变化,通过这种液体耦合作用,可驱动上下两层膜的变化,从而引起透镜光学性质的变化. 采用双球面厚透镜公式计算透镜的焦距^[36]

(25)$ \dfrac{1}{f} = (n - 1)\left[ {\dfrac{1}{R_1 } - \dfrac{1}{R_2 } + \dfrac{(n - 1)d}{nR_1 R_2 }} \right]$
式中,$n$ 为透镜的折射率,数值计算中取为 1.476^[36]. $R_{1}$ 和 $R_{2}$ 分别为透镜两个表面的曲率半径,$R_{1}$ 对应于近光源表面,此处为上层膜,$R_{2}$ 对应于另一表面. $d$ 为透镜的厚度,即上层膜和下层膜两中心点之间的距离. 本文在计算时使用了球面假设,当 $r(0)/a<0.3$, $| r'(0)/c | <0.3$ 时,即两层膜变形后中心高度较小时,可近似认为成两个球冠^[36].

4 结果讨论

计算用到的参数见表1.另外,电极材料为碳脂,介电常数为 4.16 $\times $ 10$^{ -11}$ F/m. 参数 $B/A$. 代表环形驱动部分的大小,值设定为 0.4. 首先将本文建立的理论模型计算结果与实验结果进行比对,根据表1 中的上下两层膜的参数和充液后的初始高度,确定出盐水体积为 1210 mm$^{3}$,初始焦距 $f_{0}=17$ mm,通过计算一系列电压驱动下透镜焦距的相对变化,将数值结果与文献^[28] 中的实验结果绘制于图3 中,两者吻合良好,表明本文的理论模型是有效的.理论模拟时考虑了拉力消失 (loss of tension) 的失效形式,即环向应力或径向应力减小到 0 的情形.当达到拉力消失点后,薄膜将会出现褶皱,影响整体系统的光学性能,将无法正常工作.文献^[28] 中的透镜系统在 5000 V 电压内工作,通过计算可知,在此电压范围内工作时,薄膜的最大电场强度远低于电击穿强度 86 MV/m^[37],因此后续的计算中可不考虑电击穿的失效形式.

Table 1
表1
表1数值模拟中用到的材料及其参数
Table 1Materials and parameters of the lens used in numerical simulation

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图3

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图3本文模型的数值模拟结果与文献^[28] 中实验结果的对比(实心三角形表示实验结果,实心圆点表示拉力消失点)

Fig. 3The comparison between the numerical results and the experimental data from Ref.^[28] (The filled triangles denote the experimental data. The solid circle corresponds to loss of tension)



图4绘制了上下两层膜在不同电压下的形状改变及内腔压强改变. 计算时使用无量纲量,电压的无量纲形式为 $\varPhi / ({H}\sqrt {\mu / \varepsilon } )$,压强的无量纲形式为 $pA /( \mu H )$.通过观察图4(a) 发现：随着电压增大,圆形透镜部分呈现出"下塌"形式,而环形驱动部分呈现出"外胀"形式,整体上层膜透镜部分形状的曲率半径由小变大,这是由于压强变小的结果,如图4(c) 所示,因此曲率减小,且上层膜与中心平面包围液体的体积增大.图4(b) 绘制出了下层膜的变形情况：随着电压变大,形状逐渐变扁,曲率减小,下层膜与中心平面包围液体的体积减小,但由于盐水体积的不可压缩特性,整个内腔中液体体积保持不变.随着电压的增大,压强下降速率增大,两层膜变化幅度也越来越大,且两层薄膜的位移方向均向对方方向移动,使得整体透镜厚度加速变薄,焦距加速增大.

图4

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图4介电弹性体薄膜在电压驱动下的构形图和液体压强的变化规律

Fig. 4The configuration of the upper and lower films and the liquid pressure as a function of voltage



图5和图6分别给出了上、下两层膜中径向、环向伸长率和真实应力以及电场强度的分布. 在上层膜中,除环向伸长率 $\lambda_{2}$ 外,其他参数均在薄膜的无电极覆盖和有电极覆盖的连接处出现突变.连接处环向伸长率的连续特性可由连续性条件 (14) 的第一式获得,径向伸长率在连接处出现了突跳的行为,这一现象可解释为：介电弹性体的电致应变特性导致其在厚度方向变薄,在面内方向扩展,这一特性明显增大了驱动部分的径向伸长率,同时挤压无电极覆盖的区域,致使该区域的径向伸长率降低,施加的电压越大,这种现象越明显.由本构方程 (21) 和 (22) 可知, $\sigma_{1}$ 和 $\sigma_{2}$ 在连接的界面处同样出现突跳的行为,其中环向应力的最小值出现在电极覆盖区域的内边界处,随着电压的增大而不断降低,此处将最先发生拉力消失的失效模式. 图5(e) 给出了无量纲电压 $\varPhi / (H\sqrt {\mu / \varepsilon} )=0$, 0.1, 0.2 和 0.25 四种情形上层膜中的电场分布,结果表明,薄膜内的真实电场强度明显低于电击穿强度.

图5

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图5上层膜发生非均匀变形时各参量的分布曲线

Fig. 5Distributions of various quantities in the upper film with inhomogeneous deformation

图6

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图6下层膜发生非均匀变形时各参量的分布曲线

Fig. 6Distributions of various quantities in the lower film with inhomogeneous deformation



对于下层膜,由于液体压力的耦合作用,所给出的各物理场 (径向和环向的伸长率和真实应力) 均从外边缘到中心处呈现出单调增大的形式,在中心点处达到最大值,图6(a) 和 图6(b) 以及图6(c) 和图6(d) 的计算结果表明,中心点处呈现等双轴拉伸状态.

接下来利用本文建立的计算模型,开展透镜光学性质的参数分析,探究影响透镜焦距改变的各种途径. 讨论了透镜的几何尺寸 (上下窗口直径)、初始焦距、涂覆电极面积、上层膜预拉伸率以及下层膜的剪切模量等因素对焦距和焦距相对变化的影响.在图7 和图8 中,只标明改变的参数,其余参数均与原始透镜的参数保持一致 (见表1),图8 中黑色曲线代表原始透镜的性能.

图7

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图7可调焦透镜光学性质的变化规律

Fig. 7Changes in optical properties of the DE soft tunable lenses

图8

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图8可调焦透镜的参数分析

Fig. 8Parametric analysis of the DE soft tunable lenses



图7给出了通过调整窗口半径 $a$ 和 $c$ 来改变透镜的焦距. 图7(a) 显示初始焦距随着透镜尺寸的整体变小而变小. 此处的算例中,控制初始充液体积相等,对于尺寸较小的透镜,初始膨胀会增大.图7(b) 显示了焦距的最大相对变化与透镜尺寸的关系.当固定 $a$ 时,随着 $c$ 增大,最大焦距相对变化率下降,当固定 $c$ 时,随着 $a$ 增大,最大焦距相对变化率增大,这是由于假设液体体积不可压缩的约束,上层膜形状的微小改变通过液体的耦合作用会引起下层膜相对大的变化,两者尺寸差距越大,此现象越明显.$a/c$ 是控制透镜性能的一个重要参数,也是采用类似结构的柔性透镜设计中一个必须考虑的问题^[36].

图8(a) 给出了不同初始焦距 $f_{0}$ 对焦距相对变化率的影响. 增大初始焦距 (增加充液的体积) 可略微提高光学可调性能,即在相同电压下可获得更大的焦距相对变化,但当液体体积过大时,系统整体质量和惯性增大,响应时间也会相应的延长,且液体本身的重量将对上下两层膜的变形造成一些干扰. 图8(b) 给出了涂覆电极面积对焦距相对变化率的影响. 结果表明,涂覆电极区域面积越大 (即 $B/A$ 的值越小),在相同电压驱动下,上层膜与中心平面包围的液体体积增大量更多,由于液体体积的不可压缩性,下层膜的"收缩"会加大,故透镜焦距的可调性能增强. 由于电极材料为黑色的碳脂,过度加大电极面积会影响透镜的透光性. 文献^[36] 中报道了一种使用透明的碳纳米管做电极材料的光学透镜,结构与本文的类似,但采用的是全涂覆,即上层膜全部被电极覆盖. 当使用全覆盖电极的结构时,透镜调节性能并非最佳,上层膜随着电压增大将不会出现图4(a) 所示的中心部分"下塌"的现象,而是呈现整体均向上鼓起,与下层膜中心位置的移动方向一致,整体的调节性能会变差^[32]. 图8(c) 给出了上层膜预拉伸率对焦距相对变化率的影响. 可观察到随着预拉伸率的增大,透镜焦距的可调节性能增强,但预拉伸率的增大,更容易引起拉伸破坏这一失效模式,这也是透镜结构设计中须考虑的问题. 图8(d) 绘制了下层膜剪切模量 $\mu _{lower}$ (即下层膜的软硬程度)对焦距相对变化率的影响. 当下层膜较硬时,充液后中心的凸起会降低,较软时情况相反. 由图3 可知随着电压增大,焦距变化率越来越快,此时下层膜的中心突起也越来越低,即"扁平"形状的透镜更易受到电压的影响.

近些年,以介电弹性体电致变形机制而设计的驱动器受到了很大的关注,各驱动器的结构形式和驱动方式各有特色.文献^[30] 报导 了一种仿人眼可变焦柔性透镜,其驱动方式是在透镜周边的 DE 膜的厚度方向施加电压进行驱动.文献^[31] 中研究的介电弹性体驱动器,其下层膜圆形 DE 膜上下表面是完全涂覆电极的.本文所研究的 DE 可调焦透镜,是在一张圆形 DE 膜的外圈涂覆环形电极,中间未涂覆电极的区域是透镜主体,通过在环形电极上施加电压进行驱动,这种结构和驱动方式使得透镜的调焦能力增强.理论分析中,在未涂覆电极的部分与涂覆电极部分的交界面处,涉及连续性条件的处理 问题.

5 结论

本文建立了一种基于介电弹性体驱动的仿人眼柔性可变焦透镜的理论分析模型并开展了参数分析. 这种透镜采用上下两层介电弹性膜固定在圆形框架结构形成封闭的腔体,并在腔体内充入一定量的盐水. 该透镜结构的特点是在上层膜的上下表面涂覆环形柔性电极. 在电极上施加电压驱动时,由于上层膜涂覆电极部分在膜的面内发生扩张而表现出外凸的效应并使得没有涂覆电极的中心区域的膜呈现出"下塌"的效应,又利用液体的耦合作用,使得下层膜向上层膜方向靠近,可实现透镜焦距的高效调节.

利用建立的理论模型,本文的模拟结果跟实验结果吻合良好. 在利用打靶法求解薄膜的物理场时,高效地处理了薄膜在涂覆电极和未涂覆电极的界面处的连续条件. 开展了影响透镜焦距变化的参数分析,如初始焦距、透镜的几何尺寸、预拉伸率、电极面积和剪切模量等因素. 此模型可为的柔性透镜设计提供理论指导,进一步的研究可开展透镜的动态性能分析.

附录

采用虚功原理推导上层膜的控制方程和连续条件. 在平衡状态下,对于微小的扰动,系统的 Helmholtz 自由能的增加应等于外界激励做的功

(A1)$ 2\pi H\left[ {\int_0^B \delta W\left( {\lambda _1^{(1)} ,\lambda _2^{(1)} } \right)R \text{d}R + \int_B^A \delta W\left( {\lambda _1^{(1)} ,\lambda _2^{(1)} ,\tilde {D}} \right)R \text{d}R } \right] = p\delta V + \varPhi \delta Q $
式中,上标 (1) 和 (2) 分别表示中间圆形透镜部分 $(0 \! \leqslant\! R \!\leqslant \!B)$和环形驱动部分 $(B \!\leqslant\! R \!\leqslant\! A)$ 的径向和环向伸长率. 由式 (1) 和式 (2),可得到径向拉伸率的变分为

(A2)$ \delta \lambda _1^{(i)} = \dfrac{\text{d}\delta r^{(i)}}{\text{d}R}\cos \theta ^{(i)} - \dfrac{\text{d}\delta z^{(i)}}{\text{d}R}\sin \theta ^{(i)} \ \ \left( {i = 1,2} \right) $
环向拉伸率变分为

(A3)$ \delta \lambda _2^{(i)} = \dfrac{\delta r^{(i)}}{R} $
将$\delta r^{(i)}$和$\delta z^{(i)}$ ($i =1, 2)$ 和 $\delta \tilde {D}$ 作为独立变量,利用式 (5) $\sim$ 式 (8) 以及式 (A2) 和式 (A3) 可得

(A4)$\begin{array}{l} \int_0^B {\delta W\left( {\lambda _1^{(1)} ,\lambda _2^{(1)} } \right)} R \text{d}R =\\ s_1^{(1)} R\cos \theta ^{(1)}\delta r^{(1)}\vert _0^B - s_1^{(1)} R\sin \theta ^{(1)}\delta z^{(1)}\vert _0^B +\\ \int_0^B {\left( {s_2^{(1)} - \dfrac{\text{d}\left( {s_1^{(1)} R\cos \theta^{(1)}} \right)}{ \text{d}R }} \right)} \delta r^{(1)} \text{d}R +\\ \int_0^B {\dfrac{\text{d}\left( {s_1^{(1)} R\sin \theta ^{(1)}} \right)}{ \text{d}R }} \delta z^{(1)} \text{d}R \end{array}$
(A5)$\begin{array}{l}\int_B^A {\delta W\left( {\lambda _1^{(1)} ,\lambda _2^{(1)} ,\tilde {D}} \right)} R \text{d}R =\\ s_1^{(2)} R\cos \theta ^{(2)}\delta r^{(2)}\vert _B^A - s_1^{(2)} R\sin\theta ^{(2)}\delta z^{(2)}\vert _B^A +\\ \int_B^A {\left( {s_2^{(2)} - \dfrac{\text{d}\left( {s_1^{(2)} R\cos \theta^{(2)}} \right)}{ \text{d}R }} \right)} \delta r^{(2)} \text{d}R +\\ \int_B^A {\dfrac{d\left( {s_1^{(2)} R\sin \theta ^{(2)}} \right)}{ \text{d}R }} \delta z^{(2)} \text{d}R + \int_B^A {\tilde {E}\delta \tilde {D}} R \text{d}R \end{array}$
液体压力做功为

(A6)$\begin{array}{l} p\delta V = 2\pi pr^{(1)}z^{(1)}\delta r^{(1)}\vert _0^B - 2\pi pr^{(2)}z^{(2)}\delta r^{(2)}\vert _B^A+\\ 2\pi p\int_0^B {\left( {r^{(1)}\dfrac{\text{d}r^{(1)}}{\text{d}R }\delta z^{(1)} - r^{(1)}\dfrac{\text{d}z^{(1)}}{ \text{d}R }\delta r^{(1)}} \right)} \text{d}R +\\ 2\pi p\int_B^A {\left( {r^{(2)}\dfrac{\text{d}r^{(2)}}{ \text{d}R }\delta z^{(2)} - r^{(2)}\dfrac{\text{d}z^{(2)}}{ \text{d}R }\delta r^{(2)}} \right)} \text{d}R \end{array}$
电场力做功为

(A7)$\varPhi \delta Q = 2\pi \int_B^A {\varPhi \delta \tilde {D}R \text{d}R} $
将式 (A4) $\sim$ 式 (A7) 代入式 (A1),并利用式 (A2) 和式 (A3),可得到如下控制方程 (其中 $i =1, 2$)

(A8)$\dfrac{\text{d}}{\text{d}R}(HRs_{1}^{(i)} \sin\theta ^{(i)}) - \lambda _{1}^{(i)} \lambda_{2}^{(i)} Rp\cos\theta ^{(i)} = 0 $
(A9)$\dfrac{\text{d}}{\text{d}R}(HRs_{1}^{(i)} \cos\theta ^{(i)}) + \lambda _{1}^{(i)} \lambda_{2}^{(i)} Rp\sin\theta ^{(i)} = s_{2}^{(i)} H $
(A10)$\tilde {E} = \varPhi / H $
和连续条件

(A11)$s_1^{(1)} \cos \theta ^{(1)} = s_1^{(2)} \cos \theta ^{(2)}$
(A12)$s_1^{(1)} \sin \theta ^{(1)} = s_1^{(2)} \sin \theta ^{(2)}$
由式 (A11) 和式 (A12) 可进一步得到

(A13)$\theta ^{(1)}(B) = \theta ^{(2)}(B) ,\quad s_1^{(1)} (B) = s_1^{(2)} (B) $

参考文献 View Option 原文顺序
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被引期刊影响因子

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IEEJ Transactions on Electrical and Electronic Engineering, 2007,2(3):232-248
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Despite its compactness, the human eye can easily focus on different distances by adjusting the shape of its lens with the help of ciliary muscles. In contrast, traditional man-made optical systems achieve focusing by physical displacement of the lenses used. But in recent years, advances in miniaturization technology have led to optical systems that no longer require complicated mechanical systems to tune and adjust optical performance. These systems have found wide use in photonics, displays and biomedical systems. They are either based on arrays of microlenses with fixed focal lengths, or use external control to adjust the microlens focal length. An intriguing example is the tunable liquid lens, where electrowetting or external pressure manipulates the shape of a liquid droplet and thereby adjusts its optical properties. Here we demonstrate a liquid lens system that allows for autonomous focusing. The central component is a stimuli-responsive hydrogel integrated into a microfluidic system and serving as the container for a liquid droplet, with the hydrogel simultaneously sensing the presence of stimuli and actuating adjustments to the shape--and hence focal length--of the droplet. By working at the micrometre scale where ionic diffusion and surface tension scale favourably, we can use pinned liquid-liquid interfaces to obtain stable devices and realize response times of ten to a few tens of seconds. The microlenses, which can have a focal length ranging from -infinity to +infinity (divergent and convergent), are also readily integrated into arrays that may find use in applications such as sensing, medical diagnostics and lab-on-a-chip technologies.

[3] Li GQ, Mathine DL, Valley P, et al. Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications
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We propose a focus-tunable double-convex (DCX) lens based on a non-ionic PVC (nPVC) gel to be used at close conjugates. The proposed lens is composed of an nPVC gel and two plates with electrodes. Each plate has a hole whose boundary and inner part are pasted with an electrode (anode) and has another ring shaped electrode (cathode) whose center point is the same as the hole's center. The gel is sandwiched between an upper plate and a lower plate, and it is bulged inward between the holes of two plates by applied pressure from the plates (double-convex lens shape). The lens's focal length changed from 3 mm to 24.5 mm with applied voltages from 0 V to 400 V. We also observed that the proposed lens's field-of-view decreased from 121.9 degrees to 41.9 degrees according to the applied voltages. The proposed lens brings additional benefit for users with higher transmittance (over 94%).

[6] 贾书海, 唐振华, 董君 等. 柔性变焦透镜发展现状
中国光学, 2015,8(4):535-547


( Jia Shuhai, Tang Zhenhua, Dong Jun, et al. Recent advances in flexible variable-focus lens
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[7] 邹异, 李华, 曹洋 等. 频率调控液体变焦透镜
苏州科技大学学报:自然科学版, 2017,34(3):45-49


( Zou Yi, Li Hua, Cao Yang, et al. Variable-focus liquid lens using frequency control
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Usually, an adaptive liquid lens only has a positive focal length, which severely limits its application in imaging and other fields. Therefore, a liquid lens consisting of polyvinyl chloride/dibutyl phthalate (PVC/DBP) gel, glycerol solution, and a glass substrate is proposed to extend the dynamic focal length range. A spherical tube is formed by the PVC/DBP gel under the effect of hydrostatic and surface tensions, which is used to restrict the glycerol solution. The PVC/DBP gel does not deform under the effect of an electric field, so the tangent line at the three-phase junction changes with the change of contact angle, which leads to an enlargement of the dynamic focal length range. At different voltage values, the proposed lens can be configured to work in three different schemes, namely, converging light, nondeflecting light, and diverging light. Here, the proposed lens has high imaging quality; the resolution is better than 114 lp/mm. A lens with a reconfigurable focal length holds great promise in diverse applications such as fluorescence detection, beam shaping, and adaptive optics.

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The holy grail of reconfigurable optics for microscopy, machine vision and other imaging technologies is a compact, in-line, low cost, refractive device that could dynamically tune optical aberrations within a range of about 2-5 wavelengths. This paper presents the first electrically reconfigurable, fully elastomeric, tuneable optical lenses with motor-less electrical controllability of astigmatism in the visible range. By applying different voltage combinations to thin dielectric elastomer actuator segments surrounding a soft silicone lens, we show that the latter can be electrically deformed either radially or along selectable directions, so as to tune defocus or astigmatism, up to about 3 wavelengths. By mounting the new lenses on a commercial camera, we demonstrate their functionality, showing how electrically reconfiguring their shape can be used to dynamically control directional blurring while taking images of different targets, so as to emphasize directional features having orthogonal spatial orientations. Results suggest that the possibility of electrically controlling aberrations inherent to these smart lenses holds promise to develop highly versatile new components for adaptive optics.

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Optical lenses with tunable focus are needed in several fields of application, such as consumer electronics, medical diagnostics and optical communications. To address this need, lenses made of smart materials able to respond to mechanical, magnetic, optical, thermal, chemical, electrical or electrochemical stimuli are intensively studied. Here, we report on an electrically tunable lens made of dielectric elastomers, an emerging class of "artificial muscle" materials for actuation. The optical device is inspired by the architecture of the crystalline lens and ciliary muscle of the human eye. It consists of a fluid-filled elastomeric lens integrated with an annular elastomeric actuator working as an artificial muscle. Upon electrical activation, the artificial muscle deforms the lens, so that a relative variation of focal length comparable to that of the human lens is demonstrated. The device combined optical performance with compact size, low weight, fast and silent operation, shock tolerance, no overheating, low power consumption, and possibility of implementation with inexpensive off-the-shelf elastomers. Results show that combing bioinspired design with the unique properties of dielectric elastomers as artificial muscle transducers has the potential to open new perspectives on tunable optics.

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AbstractIn response to a stimulus, a soft material deforms, and the deformation provides a function. We call such a material a soft active material (SAM). This review focuses on one class of soft active materials: dielectric elastomers. When a membrane of a dielectric elastomer is subject to a voltage through its thickness, the membrane reduces thickness and expands area, possibly straining over 100%. The dielectric elastomers are being developed as transducers for broad applications, including soft robots, adaptive optics, Braille displays, and electric generators. This paper reviews the theory of dielectric elastomers, developed within continuum mechanics and thermodynamics, and motivated by molecular pictures and empirical observations. The theory couples large deformation and electric potential, and describes nonlinear and nonequilibrium behavior, such as electromechanical instability and viscoelasticity. The theory enables the finite element method to simulate transducers of realistic configurations, predicts the efficiency of electromechanical energy conversion, and suggests alternative routes to achieve giant voltage-induced deformation. It is hoped that the theory will aid in the creation of materials and devices.]]>

[25] 刘彦菊, 刘立武, 孙寿华 等. 介电弹性体驱动器的稳定性分析
中国科学 E 辑: 技术科学, 2009,39:1564-1573


( Liu Yanju, Liu Liwu, Sun Shouhua, et al. Stability analysis of dielectric elastomer film actuator
Science in China (Series E), 2009,39:1564-1573 (in Chinese))


[26] Lu TQ, Ma C, Wang TJ. Mechanics of dielectric elastomer structures: A review
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( Wei Zhigang, Chen Haibo. A new elastic model for rubber-like materials
Chinese Journal of Theoretical and Applied Mechanics, 2019,51(2):473-483 (in Chinese))
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Science, 2000,287(5454):836-839
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Electrical actuators were made from films of dielectric elastomers (such as silicones) coated on both sides with compliant electrode material. When voltage was applied, the resulting electrostatic forces compressed the film in thickness and expanded it in area, producing strains up to 30 to 40%. It is now shown that prestraining the film further improves the performance of these devices. Actuated strains up to 117% were demonstrated with silicone elastomers, and up to 215% with acrylic elastomers using biaxially and uniaxially prestrained films. The strain, pressure, and response time of silicone exceeded those of natural muscle; specific energy densities greatly exceeded those of other field-actuated materials. Because the actuation mechanism is faster than in other high-strain electroactive polymers, this technology may be suitable for diverse applications.

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[36] Shian S, Diebold RM, Clarke DR. Tunable lenses using transparent dielectric elastomer actuators
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Focus tunable, adaptive lenses provide several advantages over traditional lens assemblies in terms of compactness, cost, efficiency, and flexibility. To further improve the simplicity and compact nature of adaptive lenses, we present an elastomer-liquid lens system which makes use of an inline, transparent electroactive polymer actuator. The lens requires only a minimal number of components: a frame, a passive membrane, a dielectric elastomer actuator membrane, and a clear liquid. The focal length variation was recorded to be greater than 100% with this system, responding in less than one second. Through the analysis of membrane deformation within geometrical constraints, it is shown that by selecting appropriate lens dimensions, even larger focusing dynamic ranges can be achieved.

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