Energy harvesting with dielectric elastomer tubes: active and (responsive materials-based) passive approaches

Based on the theoretical framework illustrated above, figure 3(a) plots the normalized external pressure $p/G$ in the absence of voltage ($\tilde{V} = 0$) as a function of the radial stretch $r_{i}/R_{i}$. As expected, the neo-Hookean model predicts a softening of the DE tube with increasing the pressure while the Yeoh model accounts for the experimentally observed stiffening and lock-up effect.

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The normalized voltage $\tilde{V}$ as a function of the radial stretch $r_{i}/R_{i}$ in the absence of pressure ($p/G = 0$) is depicted in figure 3(b). Note that the neo-Hookean model provides a response which reaches a peak voltage at $\tilde{V}\approx0.7$, beyond which no solutions can be found. This non-physical behavior is a well-known drawback of the neo-Hookean model [61, 68]. The issue can be overcome by accounting for the strain-induced stiffening with the Yeoh model. This model predicts a snap-through of the tube at a critical voltage, followed by additional deformation and strain-induced stiffening.

In order to understand the harvesting mechanisms demonstrated in the next sections, it is worth emphasizing that the deformed configuration depends on the loading direction. Upon an increase in voltage, the DE tube snaps through at $\tilde{V}\approx0.7$ such that the radial stretch increases from $r_{i}/R_{i}\approx1.3$ to $r_{i}/R_{i}\approx3.1$. Increasing the voltage results in further radial stretch. However, unloading from voltages that are above the critical value results in a reverse snap through from $r_{i}/R_{i}\approx2.3$ to $r_{i}/R_{i}\approx1.1$ at $\tilde{V}\approx0.5$, thereby accessing different deformation states.

The active energy harvesting cycle, based on the application of a mechanical pressure (as described in section 2), is shown in figure 4. First, the referential DE tube is shown. As a result of an applied inflation pressure p, the DE tube inflates (state 2). State 3 describes the configuration of the tube after it has been charged with a charge $Q = V_{3}C_{3}$, where V₃ and C₃ are the voltage and the capacitance in the charged loaded tube, respectively. Due to electrostatic forces, a further deformation takes place in the tube from state 2 to 3. Lastly, the inflation pressure is removed (i.e. the tube deflates) while the charge Q is maintained (see state 4). This leads to a decrease in the capacitance $C_{4}\lt C_{3}$ and an increase in voltage $V_{4}\gt V_{3}$.

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In the following, we investigate the energy which can be harvested from the DE tubes. To this end, we define and set the normalized inflation pressure $p/G$ and control the charging voltage $\tilde{V}_{3}$. It is emphasized that we assume that the material does not experience electrical breakdown, instabilities, or mechanical failure during the inflation and the charging process. In addition, it is worth noting that the inflation (which can be viewed as a prestretch of the material) allows for larger electrically induced deformations and thus reduces the likelihood of electrical breakdown [12, 69].

Figures 5(a) and (b) plot the normalized energy-density $e/G$ and the difference in the capacitance $C_{3}/C_{4}-1$ as a function of the normalized voltage $\tilde{V}_{3}$, respectively, for the neo-Hookean and the Yeoh materials with an inflation pressure $p/G = 0.05$. In the inflation stage, the pressure is not sufficient to observe the strain-induced stiffening of the DE and accordingly the deformed configurations of the neo-Hookean and the Yeoh materials are similar. As expected, charging the DE tube with higher voltages $\tilde{V}_{3}$ leads to an increase in the harvested energy e.

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The jump observed at $\tilde{V}_{3}\approx0.52$ is the result of the electrically induced snap through effect (illustrated in figure 3(b)). To understand the simulated trends, figure 5(b) plots the difference in the capacitance. We find that for neo-Hookean materials, once a peak voltage is reached, no other solutions can be found. However, in the Yeoh material, the quantity $C_{3}/C_{4}-1$ initially increases, reaches a peak value, and then loops back towards 0. The 'loop' stems from the electrically-induced snap through and the differences between the loading-unloading response shown in figure 3(b). The value of $C_{3}/C_{4}-1$ reaches zero at high voltages due to the stiffening effect, in which the difference between the conformations in stages 3 and 4 is small. However, we emphasize that the harvested energy depends on $V_{3}^{2}$, which is more dominant than the decrease in the capacitance ratio (see equation (1)).

The normalized energy-density $e/G$ harvested for the higher normalized pressure $p/G = 0.15$ is shown in figure 6(a). While the neo-Hookean model exhibits roughly the same trend as under lower pressures, the Yeoh model predicts a different response. Specifically, we find that the harvested energy does not monotonically increases with voltage. This behavior is explained as follows: the inflation pressure deforms the DE tube and leads to strain-induced stiffening. If the applied voltage in state 3 is in the snap through region of a DE tube that is electrically excited without mechanical loading, then the removal of the inflation pressure results in conformations which can only be accessed along the unloading path shown in figure 3(b). This effect must be considered when designing an energy harvesting DE tube.

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The difference in capacitance $C_{3}/C_{4}-1$ is shown in figure 6(b). As opposed to the case of $p/G = 0.05$, the quantity $C_{3}/C_{4}-1$ monotonically decreases with voltage according to the Yeoh model. This behavior stems from the application of a high inflation pressure, which deforms the tube into the strain-stiffening regime. This trend is also responsible for the drop $C_{3}/C_{4}-1\rightarrow0$ at high voltages, since the deformed configurations 3 and 4 are similar.

It is important to note that given the applied shear modulus G, the above-illustrated simulations predict a harvesting of energy-densities in the range of ∼10–50 mJ cm^–3 with VHB 4910. This is comparable to available experimental findings for VHB-based generators [29–32]. Specifically, with $p/G = 0.15$ an electrical energy-density of e ≈ 15 mJ cm^–3 can be harvested at $V = 1\,\mathrm{kV}$. The work of Jiang et al [32] harvested e ≈ 12 mJ cm^–3 under an equivalent voltage in a cone DE generator with $Z = 60\,\mathrm{mm}$. At a voltage $V = 1.2\,\mathrm{kV}$, a circular ring generator, as proposed by Wang et al [17], is capable of harvesting e ≈ 11 mJ cm^–3, while our design predicts e ≈ 20 mJ cm^–3. We point out that the differences stem from the charging mode—in our design the DE is charged radially while in the work of Wang et al [17] the electric field acts across the film.

The main disadvantage of the above-described DE-based energy harvesting device is the need for a constant input of active mechanical energy (in the form of work of inflation pressure). Here, we propose an alternative method to harvest energy using a passive charging mechanism. Specifically, as opposed to mechanically applying a pressure, materials that deform in response to a change in environmental conditions such as temperature, hydration, and presence of liquids, can be employed to passively charge the energy harvesting device. The advantage of this method is that an environmental source triggers the deformation and generates mechanical energy, which is stored in the DE tube. In the following, we consider polymeric gels as the passive component. The gels swell in the presence of an appropriate solvent due to gel-solvent affinity, and in turn exert inflation pressure on the DE tube. In the following we describe the concept illustrated in section 2 with respect to a passive charging mechanism.

To demonstrate this concept, consider a long cylindrical neo-Hookean dry polymeric gel that is placed in a DE tube. A thin insulating, impermeable, and highly deformable membrane is placed between the gel and the DE to prevent direct contact of the charges with the solvent permeating the gel (see state 1 in figure 7). Next, the inner part of the tube is filled with solvent and the gel swells. As a result, the gel applies a pressure that inflates the tube, thereby storing mechanical energy (see state 2 in figure 7). In state 3 of figure 7, the tube is charged with a charge $Q = V_{3}C_{3}$. It is emphasized that in order to achieve a successful energy harvesting cycle, the contact between the gel, the insulating, impermeable, and highly deformable membrane, and the DE must be maintained. Practically, this leads to the requirement that the radial deformation of a freely swollen gel must be larger than the electrically induced radial stretch. Lastly, the solvent is drained out of the tube and the gel deswells (see state 4 in figure 7). As illustrated before for a mechanically applied inflation pressure, the charge is constant and therefore the capacitance $C_{4}\lt C_{3}$ and the voltage $V_{4}\gt V_{3}$. Consequently, electrical energy can be harvested from the DE tube. From a practical viewpoint, it is emphasized that deswelling is a slow process, and therefore this mechanism is useful as a battery rather than a fast power supply source.

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In the following, we demonstrate the concept of passive charging. To this end, we summarize the governing equations and the boundary conditions of for the swelling of the constrained polymeric gels, as developed by Chester and Anand [70].

Consider a long cylindrical neo-Hookean dry polymeric gel with initial radius and length R_i and L_g , respectively. The material points are given in a polar coordinate system such that $0\unicode{x2A7D} R\unicode{x2A7D} R_{i}$, $0\unicode{x2A7D}\Theta\lt2\pi$, and $0\unicode{x2A7D} Z\unicode{x2A7D} L_{g}$. The gel is placed in a dielectric tube, which is coated with an insulating impermeable membrane. The inner part is then filled with water and the gel swells. As a result, the gel applies a pressure that leads to the dilation of the tube.

In the deformed configuration, the radius and length of the gel are r_i and l_g , respectively. The mapping of the material points is given by

$$\begin{align} r = \lambda_{g} R;\,\,\,\theta = \Theta;\,\,\,z = A_{g} Z, \end{align} \tag{ 13 }$$

where λ_g and A_g are the radial and the longitudinal stretches. Accordingly, the deformation gradient is

$$\begin{align} \mathbf{F}_{g} = \left(\begin{array}{ccc} \lambda_{g} & 0 & 0\\ 0 & \lambda_{g} & 0\\ 0 & 0 & A_{g} \end{array}\right), \end{align} \tag{ 14 }$$

where $J_{g} = \lambda_{g}^{2}A_{g}$ is the volumetric deformation.

Following the framework of Chester and Anand [70], the stress that develops in the gel is

$$\begin{align} \boldsymbol{\sigma}_{g} = \frac{1}{J_{g}}\left(G_{g}\mathbf{F}_{g}\mathbf{F}_{g}^{T}-\Pi_{g}\mathbf{I}\right), \end{align} \tag{ 15 }$$

where G_g is the shear modulus of the dry gel and $\Pi_{g}$ is a pressure-like term stemming from the solvent-polymer interactions and the mechanical interaction with the constraining tube. In addition, the chemical potential of the solvent molecules in the gel is given by

$$\begin{align} \mu = \mu^{0}+k T\left(\ln\left(1-\frac{1}{J_{g}}\right)+\frac{1}{J_{g}}+\chi\frac{1}{J_{g}^{2}}\right)+\frac{v}{J_{g}}\left(\Pi_{g}-G_{g}\right), \end{align} \tag{ 16 }$$

where k is the Boltzmann constant, T is the absolute temperature, µ⁰ is a reference chemical potential, χ is the dimensionless interaction parameter governing the gel-solvent affinity.

The boundary conditions are employed to determine the swelling-induced pressure. Specifically, the gel is traction free along the longitudinal direction such that $\sigma_{g}^{\left(zz\right)} = 0$. Along the radial direction, the gel applies a radial stress that dilates the DE tube such that $\sigma_{g}^{\left(rr\right)}\left(R = R_{i}\right) = \sigma_{rr}\left(R = R_{i}\right)$. Lastly, chemical equilibrium requires that $\mu = \mu^{0}$.

To simulate the behavior of the system, we consider hydrogels and accordingly set the temperature and the volume of a water molecule $T = 300\,\mathrm{K}$ and $v = 3\cdot10^{-29}\,\mathrm{m^{3}}$, respectively. We begin by studying the configuration of the system in state 2, which results from the application of the passive gel component due to swelling. Figures 8(a) and (b) plot the stretch $r_{i}/R_{i}$ of the gel tube as a function of the stiffness ratio $G_{g}/G$ for selected values of χ in systems with neo-Hookean and Yeoh DE tubes, respectively. We find that extremely soft gels (i.e. $G_{g}/G\ll1$) experience little swelling. This is due to the stiffness of the DE tube, which constrains the gels from dilating and expanding [71]. Stiffer gels with a shear modulus that is comparable to that of the DE exert larger swelling-induced stress on the tube and therefore experience significant radial swelling, resulting in higher mechanical energy of the DE tube. As can be seen, the radial stretch varies non-monotonically with the stiffness ratio, resulting in an optimum value $G_{g}/G$ in both neo-Hookean and Yeoh tubes that leads to maximum radial stretch. This ratio is important since it governs the capacitance of the DE tube in state 3. In addition, note that lower interaction parameters, corresponding to gels with a higher polymer-solvent affinity, result in increased swelling, as expected [16, 72, 73].

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Figures 8(c) and (d) plot the stretch $r_{i}/R_{i}$ of the gel tube as a function of the interaction parameter χ for three selected values of $G_{g}/G$ in systems with neo-Hookean and Yeoh DE tubes, respectively. The interaction parameter governs the gel-solvent affinity, and therefore lower χ values result in a larger degree of swelling. The degree of swelling decays as the interaction parameter χ increases. These plots highlight the non-monotonous trends of the degree of swelling with the ratio $G_{g}/G$. Specifically, gels with a stiffness that is comparable to that of the DE tube experience significant swelling and apply interfacial pressures which are sufficient to significantly deform the tubes. In the case of $G_{g}/G = 0.1$, the gels cannot exert sufficient interfacial pressure to stretch the DE tube. Therefore, the behavior is governed by the gel-fluid affinity—lower values of χ lead to larger solvent uptake and radial stretch. Stiff gels, as shown via $G_{g}/G = 10$, are characterized by a high chain-density which mechanically hinders swelling. Therefore, the radial stretch obtained from such gels is smaller than with $G_{g}/G = 1$.

To investigate the energy harvesting capabilities of the gel-DE system, we choose the shear moduli ratio $G_{g}/G = 1$ and the dimensionless gel-fluid interaction parameter χ = 0.2, a value that corresponds to a high gel-fluid affinity. Figures 9(a) and (b) plot the normalized energy-density $e/G$ and the difference in the capacitance $C_{3}/C_{4}-1$ as a function of the normalized voltage $\tilde{V}_{3}$ for neo-Hookean and Yeoh materials. The observed trends suggest that the inflation pressure applied by the gel, resulting in state 2, deforms the tube to the stiffening regime, as explained in relation to the above discussed case characterized by $p/G = 0.15$ in the active charging (see figures 6(a) and (b). In addition, the electrical energy-density that can be harvested using passive charging is comparable to that obtainable with active charging under the examined properties. Specifically, under the voltages $V = 1\,\mathrm{kV}$and $V = 1.2\,\mathrm{kV}$, the energy-densities e ≈ 14 mJ cm^–3 and e ≈ 18 mJ cm^–3 can be harvested. This is comparable to available experimental results. However, it is again emphasized that due to the diffusion mechanism that governs swelling, the charging is a slow process.

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