The effect of nonlinear material viscosity on the energy harvesting performance of dielectric elastomer generators

Abstract

Dielectric elastomer generators are capable of converting mechanical energy from a variety of sources into electrical energy. The energy harvesting performance depends on the interplay between electromechanical coupling, material viscosity, and multiple failure modes. Experiments also suggest that the material viscosity of dielectric elastomers is deformation-dependent, which makes the prediction of the performance of dielectric elastomer generators more challenging. By adopting the coupled field theory, finite-deformation viscoelasticity theory, and the theory for polymer dynamics, this work investigates the harvested energy and conversion efficiency of dielectric elastomer generators from theoretical perspective. By comparing the simulation results from the nonlinear viscosity model to the experimental data and the simulation results from the linear viscosity model, we further examine the possible factors that may strongly influence the performance of dielectric elastomer generators. It is found that dielectric elastomer generators exhibit higher harvested energy when nonlinear material viscosity is considered. Moreover, by selecting a higher voltage of the power supply for the generator, the conversion efficiency of dielectric elastomer generators can be greatly improved. The theoretical framework in this study is expected to offer some new insights into optimizing the design of dielectric elastomer generators and thus improving their performance.

Keywords

Dielectric elastomers nonlinear viscosity energy harvesters

1. Introduction

Dielectric elastomers (DEs) have drawn much attention in the field of transduction technology recently due to their high energy density, light weight, flexibility, and large deformation capability (Suo, 2010). DEs are particularly promising candidates for harvesting energy from various sources including human motions, ocean waves, and wind (Kornbluh et al., 2012; Suo, 2010). Most dielectric elastomer generators (DEGs) can be envisaged as a DE membrane coated with compliant electrodes on its surfaces, which functions as variable capacitor to collect and transfer electrical charges during an energy harvesting process. First, the DE membrane is connected to a power source and stretched by external mechanical forces, which increases the capacitance of the DE and forces the charges to flow from the power supply to the compliant electrodes on the DE surfaces. Then, the DE is disconnected from the power supply and allowed to shrink back to its original shape by releasing the applied mechanical forces. During this step, the voltage on the DE increases due to the decrease of its capacitance, which forces the charge to flow from the DE capacitor to the storage. Through this electromechanical cycle, mechanical energy is converted into electrical energy. The amount of energy transferred depends on the change of the capacitance of the DE membrane.

Based on this energy harvesting mechanism, DE generators with various configurations and harvesting schemes have been developed in the literature. One of the pioneering studies was conducted by Pelrine et al. (2001), in which they designed a plate DEG with a constant voltage scheme. The prototype developed by using acrylic elastomers (VHB 4905 by 3M) could achieve energy density up to 400 J/kg (Pelrine et al., 2001). With VHB 4905, McKay et al. (2010) developed a self-priming DEG system in order to retain the charge losses and experimentally demonstrate that the energy density ranged from 2.8 to 12.6 J/kg and the harvesting efficiency was up to 84%. Chiba et al. (2008) built a DEG with VHB 4905 aiming to collect energy from ocean waves, which was capable of generating 50 W with wave height of 0.5 m. Liu et al. (2010) designed stacking energy harvesters with silicon elastomers to collect energy from water with energy density of 3.6 J/kg. Koh et al. (2009) theoretically analyzed the maximal energy that can be converted by a DEG with the consideration of various failure modes, including electrical breakdown, electromechanical instability, loss-of-tension (tensile stress becomes 0), and rupture. Recently, more efforts have been devoted to improving the energy density or harvested efficiency of DEGs. For example, Huang et al. (2013) boosted the energy density of DEGs based on VHB 4905 to 560 J/kg by adopting equi-biaxial loading that maximizes the change of the capacitance. A triangular energy harvesting scheme was proposed by Shian et al. (2014) to optimize the electromechanical harvesting cycle of the DEG designed by Huang et al. (2013), which demonstrated that the energy density could be up to 780 J/kg. However, it should be mentioned that the energy harvesting performance of the DEGs, including energy density and conversion efficiency, is quite scattered in the literature. The currently obtained maximum energy density in the literature is still much lower than the theoretical prediction (i.e. 1700 J/kg) (Koh et al., 2011).

One of the main factors that affect the harvesting performance of DEGs is the material viscoelasticity, which could cause high energy dissipation and loss-of-tension of DE membrane (Fan et al., 2018; Huang et al., 2013; Shian et al., 2014). Recently, based on the fully coupled field theory for DEs by Suo et al. (2008) and the finite-deformation viscoelasticity theory by Reese and Govindjee (1998), Hong (2011) has proposed a novel constitutive model capable of capturing the finite-deformation viscoelastic response of DEs under electromechanical coupling. By using Hong’s (2011) model, the energy harvesting performance of DEGs with the constant voltage harvesting and the triangular scheme has been investigated. For example, Li et al. (2012) have analyzed the inhomogeneous viscoelastic deformation of DEGs and the effect of rapid loading and unloading on the performance of DEGs. Foo et al. (2012b) studied dissipative process and current leakage in a harvesting cycle of DEGs. Zhou et al. (2015) investigated the energy density and the conversion efficiency of DEGs with their fatigue life taken into account. Chen et al. (2016) investigated the effect of temperature on the dissipative process of DEGs. Zhou et al. (2017) developed a theoretical framework for analyzing the energy harvesting process of DEGs and proposed methods to improve the energy harvesting performance. Fan et al. (2018) investigated the effect of loss-of-tension on energy harvesting performance of DEGs. Those studies have demonstrated the effect of material viscosity on the energy harvesting performance of DEGs.

It should be mentioned that most existing studies about DEGs assume linear material viscosity within the framework of finite-deformation viscoelasticity. In other words, the material viscosity is assumed as a constant in these studies. However, according to the theory of polymer dynamics (Doi and Edwards, 1986), the viscosity of the polymer chains in elastomers should be nonlinear (deformation-dependent), especially when they undergo large deformation. This argument has also been proven by experiments in the literature (Hossain et al., 2012; Wang et al., 2016). Although elastomers may exhibit more or less viscosity depending on their type, it is still essential to investigate the nonlinear viscous effect on their applications that require large deformation, such as DEGs. On the contrary, the effect of such nonlinear viscosity on the energy harvesting performance of DEGs has not been investigated thus far. Recently, Zhou et al. (2018) has developed a constitutive law that incorporates the nonlinear material viscosity into the finite-deformation viscoelasticity framework of elastomers, which is expected to better capture the time-dependent and rate-dependent deformation of DEs. In this work, the modeling framework by Zhou et al. (2018) will be employed to study the energy harvesting performance of DEGs with the triangular harvesting scheme. Particularly, the influence of the deformation-dependent viscosity on the energy density and conversion efficiency of the DEGs will be explored.

2. Model and formulation

Figure 1 sketches the deformation process of the embedded DE membrane in the generator, which is covered by compliant electrodes on its top and bottom surfaces. The undeformed state is denoted by in-plane length L and thickness H as shown in Figure 1(a). When the membrane is subjected to a voltage $Φ$ between the electrodes and equi-biaxial stretching force P, it deforms to the current state with length l and thickness h. Assuming homogeneous deformation and material incompressibility, the stretch ratios in-plane and along the thickness direction are defined as λ = l / L and λ_h = h / H with $λ_{h} = 1 / λ^{2}$ . The applied voltage across the DE membrane induces the charges accumulated on the two electrodes, that is

\pm Q = C_{D E} Φ

(1)

where $C_{DE} = ε ε_{0} λ^{4} L^{2} / H$ is the capacitance of the DE membrane capacitor with $ε_{0}$ being the dielectric permittivity of the vacuum and $ε$ being the relative dielectric constant of the DE. It is evident that the change of the capacitance is realized through the continuous deformation of the membrane during the energy harvesting cycle, which governs the energy conversion of the DEG.

Under the homogeneous deformation condition, the deformation gradient of the current state with respect to the undeformed state can be expressed in terms of the stretch ratios as $F = [\begin{matrix} λ & 0 & 0 \\ 0 & λ & 0 \\ 0 & 0 & λ_{h} \end{matrix}]$ . As commonly treated in the literature (Foo et al., 2012a; Hong, 2011; Zhou et al., 2014), the deformation gradient F of the viscoelastic DE membrane is decomposed into an elastic component and an inelastic component in a multiplicative format, which gives

F = F^{e} F^{i}

(2)

where $F^{i} = (\begin{matrix} λ^{i} & 0 & 0 \\ 0 & λ^{i} & 0 \\ 0 & 0 & {λ_{h}}^{i} \end{matrix})$ and $F^{e} = (\begin{matrix} λ^{e} & 0 & 0 \\ 0 & λ^{e} & 0 \\ 0 & 0 & λ_{h}^{e} \end{matrix})$ with superscripts “e” and “i” represents the elastic and inelastic parts, respectively. The material incompressibility of the DE ensures $λ_{h} = 1 / λ^{2}$ , $λ_{h}^{e} = 1 / (λ^{e})^{2}$ , and $λ_{h}^{i} = 1 / (λ^{i})^{2}$ .

Due to the work done by the tensile forces P and voltage $Φ$ , the variation of Helmholtz free energy density W is expressed as

L^{2} H \frac{\partial W}{\partial λ} δ λ = 2 PL δ λ + Φ δ Q

(3)

Following Hong’s (2011) work, the total Helmholtz free energy W of the deformed DE membrane consists of two parts, that is, $W = W^{EQ} (λ, Φ) + W^{NEQ} (λ^{e})$ . W^EQ is the equilibrium Helmholtz free energy, which is associated with both the total deformation and applied electric voltage $Φ$ ; $W^{NEQ}$ is the non-equilibrium Helmholtz free energy, which is only related to the elastic deformation. It is worth noticing that the electric field is assumed to be in equilibrium as the electric field always reaches the equilibrium state much faster than the mechanical deformation (Hong, 2011). According to the work of Huang and Suo (2012), the equilibrium Helmholtz free energy density $W^{EQ}$ takes the form of

W^{EQ} = W_{s} (λ) + \frac{ε ε_{0}}{2} {(\frac{Φ}{H})}^{2} λ^{4}

(4)

where $W_{s} (λ)$ is the strain energy density function of the elastomer, and the second term is the free energy associated with the electric polarization. The finite deformation of elastomers has been well studied in the literature with hyperelastic constitutive models. Here we adopt the Gent (1996) model for case study to define the strain energy density function for both the equilibrium and the non-equilibrium parts, that is

W_{s} = - \frac{G^{EQ} J_{lim}^{EQ}}{2} \ln (1 - \frac{2 λ^{2} + λ^{- 4} - 3}{J_{lim}})

(5)

W^{NEQ} = - \frac{G^{NEQ} J_{lim}^{NEQ}}{2} \ln (1 - \frac{2 {(λ^{e})}^{2} + {(λ^{e})}^{- 4} - 3}{J_{lim}})

(6)

where the material constants $J_{lim}^{EQ}$ and $J_{lim}^{NEQ}$ are related to the total stretching limit and the stretching limit of the elastic component of the DE chains, respectively; $G^{EQ}$ is the equilibrium shear modulus and $G^{NEQ}$ is the non-equilibrium shear modulus.

Considering that δλ is any arbitrary small variation, substituting (4) to (6) into (3) leads to

\begin{matrix} \frac{P}{GLH} = \frac{χ J_{lim}^{EQ} (λ - λ^{- 5})}{J_{lim}^{EQ} - 2 λ^{2} - λ^{- 4} + 3} \\ + \frac{(1 - χ) J_{lim}^{NEQ} (λ {(λ^{i})}^{- 2} - λ^{- 5} {(λ^{i})}^{4})}{J_{lim}^{NEQ} - 2 {(\frac{λ}{λ^{i}})}^{2} - {(\frac{λ}{λ^{i}})}^{- 4} + 3} - \frac{ε ε_{0}}{G} {(\frac{Φ}{H})}^{2} λ^{3} \end{matrix}

(7)

where a new parameterχ = G^EQ / G $(0 \leq χ \leq 1)$ with G = G^EQ + G^NEQ is introduced to indicate the fraction of the time-independent polymer networks related to the total deformation (Bergström and Boyce, 1998). For limiting cases, $χ = 1$ represents a pure elastic medium, while $χ = 0$ is for a Newtonian fluid.

Since the free energy of the system never increases, the inelastic stretch ratio $λ^{i}$ in equation (7) must satisfy the thermodynamic evolution law (Hong, 2011; Reese and Govindjee, 1998), that is

- \frac{1}{2} F \frac{d [{(C^{i})}^{- 1}]}{dt} F^{T} {(b^{e})}^{- 1} = γ^{- 1} : τ_{NEQ}

(8)

where $C^{i} = (F^{i})^{T} F^{i}$ , $(b^{e})^{- 1} = (F^{e})^{T} F^{e}$ , $τ_{NEQ} = 2 F^{e} (\partial W^{NEQ} / \partial C^{e}) (F^{e})^{T}$ , $C^{e} = (F^{e})^{T} F^{e}$ , and $γ^{- 1}$ is an isotropic rank-four mobility tensor. According to the work of Reese and Govindjee (1998), $γ^{- 1}$ takes the form of

γ^{- 1} = \frac{1}{2 η} (I^{4} - \frac{1}{3} I \otimes I)

(9)

where $I^{4} = \frac{1}{2} (δ_{ik} δ_{jl} + δ_{il} δ_{jk})$ is the fourth-order symmetric identity tensor, I is the second-order identity tensor, and $η$ represents the shear viscosity. According to this thermodynamic evolution law, the time-dependent inelastic stretch ratios are expressed as

\begin{matrix} \frac{d λ^{i}}{dt} = \frac{G^{NEQ} J_{lim}^{NEQ} λ^{i}}{6 η [J_{lim} - 2 {(\frac{λ}{λ^{i}})}^{2} - {(\frac{λ}{λ^{i}})}^{- 4} + 3]} \\ [{(\frac{λ}{λ^{i}})}^{2} - {(\frac{λ}{λ^{i}})}^{- 4}] \end{matrix}

(10)

In order to demonstrate the effect of current leakage on harvesting efficiency, we calculate the charge Q on DE described as

\frac{dQ}{dt} = i - i_{leak}

(11)

where i is the current through the conducting wire attached to the compliant electrodes on DE, and the leakage current $i_{leak}$ is governed by empirical exponential function (Di Lillo et al., 2011; Foo et al., 2012b) as

i_{leak} = σ_{0} e^{(E / E_{0})} E L^{2} λ^{2}

(12)

Here $σ_{0}$ is the conductivity of DE under low electric field, $E_{0}$ is an empirical material constant, and the electric field is governed by $E = Φ λ^{2} / H$ . The time-dependent quantities $P, i, i_{leak}, λ^{i}, Q, and Φ$ can be obtained by solving a set of differential-algebraic equations, that is, equations (1), (7), and (10)–(12), when initial conditions are given.

It should be noted that $η$ is the viscosity of the material in the current state, which depends on the deformation of the DE and should be constitutively prescribed to obtain the inelastic stretch ratios as shown in equation (10). Here we adopt the theory of polymer dynamics by Doi and Edwards (1986) to determine the deformation-dependent viscosity η. The details of this theory are well elaborated in the literature (Doi and Edwards, 1986; Zhou et al., 2018). The basic idea of this theory is that the motion of a polymer chain is confined by the surrounding polymer chains. This topological constraint can be modeled by a tube-like region with the tube axis C-D defined as the primitive chain of the polymer chain (Figure 2(a)). The viscosity of elastomers originates from the diffusion of the polymer chains in the material. A polymer chain can wiggle within the tube and reptate along the tube. The tube diameter is a function of the primitive length L of the polymer chain and the end-to-end vector R of the primitive chain (Figure 2(b)), that is, $〈 a 〉 = 〈 R^{2} 〉 / 〈 L 〉$ with 〈*〉 being the expectation operator of a parameter. It is easily perceived that both L and R vary with the macroscopic deformation of the polymer, particularly when the DE undergoes large deformation. Therefore, the expectation of L and R is a function of the deformation gradient F and the statistical distribution function $f_{0} (R)$ of the end-to-end vector, that is, $〈 L 〉 = \int (F \cdot u / 4 π) L d^{2} u$ and $〈 R^{2} 〉 = \int {| F \cdot R |}^{2} f_{0} (R) d^{3} R$ , with $u$ being the unit tangent vector at a contour position of the polymer chain in the current state. The nonlinear viscosity was given by Doi and Edwards (1986) as

\frac{η}{η_{0}} = \frac{a_{0}^{2}}{{〈 a 〉}^{2}} = {\frac{{〈 R^{2} 〉}_{0} \frac{| F \cdot u |}{4 π} d^{2} u}{\int {| F \cdot R |}^{2} f_{0} (R) d^{3} R}}^{2}

(13)

where $a_{0}$ is the tube diameter and

η_{0} = \frac{1}{12} \frac{ζ N^{3} b_{0}^{4} G_{0}}{k_{b} {Ta}_{0}^{2}}

is the viscosity in reference state. The subscript “0” of the quantities refers to the reference state.

3. Energy harvesting cycle of DEGs

The energy harvesting performance of the DEG with the triangular energy harvesting scheme (Shian et al., 2014) has been investigated by Zhou et al. (2017) using a linear viscosity model. However, as mentioned in the Introduction section, elastomers exhibit nonlinear viscosity when they undergo large deformation. Therefore, the nonlinear viscosity model presented above is used to examine the viscoelastic effect on the energy harvesting performance of the DEG in this section. In general, the achievable electrical energy of a DEG is limited by the electromechanical integrity of the DE membrane, that is, rupture and electrical breakdown. Therefore, a maximum stretch ratio λ_max is first prescribed in order to prevent the DE from rupture. Meanwhile, a minimum stretch ratio λ_min is usually prescribed as greater than 1 for the DE since it takes a longer time for the DE to recover back to the undeformed state during an energy harvesting cycle. When the applied voltage exceeds the breakdown voltage Φ_B of the DE, the energy harvesting process will also fail due to short-circuiting (Liu et al., 2009, 2012). The electrical breakdown voltage follows the rule of $Φ_{B} = E_{B} (1) H λ^{x - 2}$ according to experiments (Huang et al., 2012), where E_B(1) = 30 MV/m for the dielectric strength of the undeformed DE and x = 1.13 are determined through experimental data fitting. To compare our results with the experiment by Shian et al. (2014), acrylic elastomer VHB 4905 is selected as the material for the simulation. The corresponding geometric and material parameters are set as $L = 3.5 cm$ , $H = 0.5 mm$ , $λ_{min} = 2$ , $λ_{max} = 5.5$ , $ρ = 960 kg / m^{3}$ , $χ = 0.5$ , $ε = 4$ , $ε_{0} = 8.85 \times 10^{- 12} F / m$ , $σ_{0} = 3.23 \times 10^{- 14} S / m$ , E₀ = 40 MV/m, $G = 350 kPa$ , $J_{lim}^{EQ} = 110$ , $J_{lim}^{NEQ} = 55$ , and the relaxation time $τ_{0} = η_{0} / G^{NEQ} = 3$ in the reference state (Foo et al., 2012b; Huang et al., 2013; Shian et al., 2014; Zhou et al., 2017).

In order to demonstrate the achievable harvested energy during the DEG harvesting cycle, Figure 3(a) depicts the electrical breakdown, $λ_{max}$ and $λ_{min}$ curves of the embedded DE membrane on the charge-voltage work-conjugate plane following the work of Koh et al. (2011). Theoretically, the maximum achievable energy during a harvesting cycle of the DEG is the area enclosed by these three curves. Hence, a desirable harvesting cycle should be designed to cover this area as much as possible. Shian et al. (2014) have proposed a triangular harvesting cycle denoted by A-B-C in Figure 3(a), which appears to be capable of achieving as much energy as possible when the voltage at point B and the slope of BC are appropriately selected. The implementation of this harvesting scheme is relatively simple and could be realized by the electric circuit as shown in Figure 3(b). However, due to the material viscoelasticity of the DE, it requires extra time for the DEG to finish such a cycle A-B-C, which may lead to high energy dissipation and low efficiency. This triangular cycle was modified to A₁-B₁-C₁-D₁ in the experiment of Shian et al. (2014). In order to further improve the energy harvesting performance of the generator, Zhou et al. (2017) has proposed a harvesting cycle as A₂-B₂-C₂-D₂ to improve the conversion efficiency of the DEG. For cycle A₂-B₂-C₂-D₂, loss-of-tension of the DE membrane is avoided by changing the timing of the electromechanical loading and unloading. However, the nonlinear viscosity of the polymer chains is not considered in both works, which may significantly affect the harvesting process. Therefore, a harvesting cycle denoted by A₃-B₃-C₃-D₃ with the consideration of material nonlinear viscosity is proposed in this work. The harvesting cycle proposed here follows the same principle as in the work of Zhou et al. (2017) and adopts the same electric circuit (Figure 3(b)) proposed by Shian et al. (2014). As shown in Figure 3, the energy harvesting cycle A₃-B₃-C₃-D₃ is described as follows. First, with both switches 1 and 2 open, the DE membrane is stretched to the prescribed maximum stretch ratio λ_max = 5.5 by an equi-biaxial force P (denoted by point A₃ in the Figure 3(a)). Then switch 1 is closed, and charges flow from the power supply to the DE capacitor. During this charging process as identified by A₃-B₃, the voltage on the DE keeps rising until it reaches the level of the power supply, which is set as Φ_L = 3000 V for example. Starting from state B₃, switch 1 is re-opened and the equi-biaxial force P is decreased to allow the DE to shrink back. During this process, the capacitance of the DE decreases, leading to the increase of the voltage on the DE. Once the voltage on the DE is higher than that on the transfer capacitor, charges accumulated on the DE capacitor flow to the transfer capacitor through the diode in the circuit. The slope of line BC or B₃C₃ is governed by the capacitance of the transfer capacitor C_p. To maximize the harvested energy, it is desirable to adjust C_p to make line B₃C₃ tangential to the electrical breakdown curve. Moreover, the DE membrane is expected to shrink back to the prescribed minimum stretch, that is, state C. However, it is found in the simulation that the loss-of-tension of the DE membrane occurs during the shrinking process, which was also reported in the experiment by Shian et al. (2014) and the simulation by Zhou et al. (2017). In the experiment by Shian et al. (2014), the loss-of-tension was not avoided and the state C₁ is reached by the servomotor position. In the simulation, before loss-of-tension (P = 0) occurs, a state C₃ that maximizes the enclosed area is selected. The selection of C₃ is realized with the simulation program by scanning all the eligible points on line B₃C₃. State C₃ is also the onset of the discharging process (closing switch 2), which allows the charges on both the DE and the transfer capacitors to flow to the harvesting circuit. This process continues until the DE membrane reaches the loss-of-tension at state D₃, then the DE is immediately stretched again and switch 2 is re-opened. Once the DE is stretched back to λ_max, that is, to state A₃, the harvesting cycle is completed and will repeat. Due to the material viscosity, the harvested cycle of the DEG may vary with time. However, as the harvesting operation of the DEG continues, the cycle will eventually become steady after a few cycles. It should be noted that all cycles presented in Figure 3(a) are steady harvesting cycles. Since the cycles in Figure 3(a) are not the first cycle, A, A₁, A₂, and A₃ do not start from the origin as there are residual charges from the previous cycle. The harvested energy is calculated by the area enclosed by A₃-B₃-C₃-D₃ in our simulation as 0.5285 J and the energy density is determined as 873.73 J/kg. Without considering the deformation-dependent viscosity, the harvested energy and the energy density were determined as 0.48 J and 794 J/kg, respectively (Zhou et al., 2017). It is found that the energy harvested through cycle A₃-B₃-C₃-D₃ is more than the other two cycles A₂-B₂-C₂-D₂ and A₁-B₁-C₁-D₁ (0.47 J in the experiment by Shian et al., 2014).

Figure 4 shows the force–stretch curve for both harvesting cycles A₂-B₂-C₂-D₂ and A₃-B₃-C₃-D₃. Based on the experimental results of Shian et al. (2014), both the stretching and the shrinking rates are set as $d λ / dt = 2.8 s^{- 1}$ . Meanwhile, the voltage of the DE is ramped up at the rate of r = dΦ / dt = 6000 V/s during the charging process (from state A₂ (or A₃) to state B₂ (or B₃)). It is found that it requires higher mechanical force to stretch the DE when the nonlinear material viscosity is considered. However, the electromechanical cycle for the nonlinear model reaches the steady state faster than the linear model. The mechanical work done by the equi-biaxial force P can be calculated as two times of the area enclosed by these curves, which are calculated as 0.8076 and 0.7072 J for the nonlinear viscosity model and the linear viscosity model, respectively. Correspondingly, the energy conversion efficiency for these two cycles A₂-B₂-C₂-D₂ and A₃-B₃-C₃-D₃ are determined as 67% and 65.4%, respectively, which is much higher than 30% obtained in the experiment (Shian et al., 2014). The main reason for the efficiency improvement is that the loss-of-tension of the DE membrane is avoided in the simulation, which prevents a large amount of energy dissipation during the inelastic deformation of the DE. This is realized through shortening the discharging process, that is, shortening the time from state C₂ (or C₃) to state D₂ (or D₃). It should also be mentioned that the possible energy dissipation through friction, plastic deformation, and electric circuit will certainly contribute to the lower energy conversion efficiency in the experiment. In addition, the effect of the electrodes and the drift of the dielectric permittivity are not considered in the simulation, which could also have an impact on the harvesting performance (Vu-Cong et al., 2013).

Although the discharging process is shortened in the simulation process, its effect on the harvested energy of DEG is negligible since most of the charges have been transferred out of the DE at state D₂ (or D₃). This is evidenced from Figure 5(a), which shows the variation of the electric charge throughout the whole energy harvesting cycle. It is demonstrated that at the end of the discharging, there are little residual charges left. The change of the electrical voltage during the energy harvesting process is shown in Figure 5(b), which follows $Φ = Φ_{C} + (Φ_{L} - Φ_{C}) (1 - e^{- t / 0.45})$ during the discharging process (state C₂ (or C₃)-D₂ (or D₃) with $Φ_{C}$ being the voltage level at state C₂ (or C₃). At state A₂ (or A₃) the voltage of the DE is non-zero since there are still some charges left on the DE capacitor from the previous harvesting cycle. In comparison of the nonlinear and linear models, it is found that the nonlinear model exhibits lower residual voltage and charge than the linear model. In other words, the voltage and charge at D₃ are lower than that at D₂. It should be mentioned that the current leakage (Figure 5(c)) is calculated as $7.7 \times 10^{- 6} C$ during the whole harvesting cycle, which is negligible by comparing with the peak value of the charges on the DE as shown in Figure 5(a). This conclusion is in agreement with both the simulation (Zhou et al., 2017) and the experiment (Shian et al., 2014).

From the above analysis, it can be concluded that the harvesting performance of the DEG, including the harvested energy and the conversion efficiency, is mainly governed by the voltage level of the power supply and the stretching state or the timing for closing switch 2, that is, the position of B₂ (or B₃) and C₂ (or C₃) on the energy harvesting curve. Therefore, a suitable combination of states B₃ and C₃ may help improve the energy harvesting performance of the DEG, either the harvested energy, or the efficiency, or both. Figure 6 shows the variation of the harvested energy (Figure 6(a)) and the efficiency (Figure 6(b)) of the DEG with the stretch ratio at state C₃ $(λ_{C_{3}})$ and the voltage level of the power source $(Φ_{L})$ when the nonlinear material viscosity is considered. It is observed that both the harvested energy and the conversion efficiency increase when a higher power supply voltage $Φ_{L}$ and a lower stretch ratio $λ_{C_{3}}$ are selected. This trend is different from the one predicted by the linear viscosity model (Zhou et al., 2017), which indicates the significance of considering nonlinear material viscosity in the modeling framework of the DE. Within the given range of the power supply voltage and the stretching ratio $λ_{C_{3}}$ at C₃, the maximum energy that can be harvested is 0.6 J and the highest conversion efficiency is 69%. It is thus concluded that choosing a higher power supply voltage and smaller $λ_{C_{3}}$ (or delay of closing switch 2) is an effective approach to improve both the harvested energy and the efficiency of DEG. However, it should be ensured that the voltage level of the power source is lower than the electrical breakdown voltage of DE at B₃, and the loss-of-tension does not occur at state C₃.

With the theoretical framework developed in the current work, we also evaluate the energy harvesting performance of DEGs with different types of polymer networks. Figure 7 depicts the maximum harvested energy and the maximum efficiency of a DEG as a function of χ when the power supply voltage level Φ_L is set as 3000 V. The results from the linear viscosity model by Zhou et al. (2017) are also plotted for comparison. It is observed that as χ increases, the maximum harvested energy of the DEG increases until approaching a constant when χ reaches a critical value for both models. This is expected since DE membranes with χ higher than the critical value, they are able to shrink back to the prescribed minimum stretch ratio λ_min = 2 without loss-of-tension, that is, both C₂ and C₃ in the simulated harvesting cycles coincide with the originally proposed state C in the triangular scheme A-B-C. It is thus concluded that the ideal triangular harvesting scheme could be realized when the fraction of the time-independent polymer networks in the material is higher than this critical value. For the limiting case of a purely elastic solid with χ = 1, the maximum harvested energy is determined as 0.62 J when Φ_L = 3000 V, which is quite close to the maximum achievable energy 0.74 J of the DEG as calculated by the area enclosed by the electrical breakdown, λ_max and λ_min curves in Figure 3(a). It is also found that the critical value of the fraction of time-independent polymer networks is different for the two models, for example, $χ = 0.45$ for the nonlinear model and $χ = 0.6$ for the linear model. On the contrary, the change of the maximum efficiency exhibits non-monotonic behavior, which attributes to the combined effect of the material viscoelasticity on the harvested energy and the mechanical work consumed during the harvesting cycle. This curve exhibits similar trend for both the linear and nonlinear viscosity models, while the discrepancy between the linear and nonlinear models indicates that neglecting the nonlinear behavior of the material viscosity may lead to significant error in analyzing the harvesting performance of DEGs.

4. Conclusion

Based on the finite-deformation viscoelasticity theory and the theory of polymer dynamics, a theoretical framework with the consideration of material nonlinear viscosity is developed to comprehensively evaluate the energy harvesting performance of DEGs. Simulation results show that using a higher voltage power supply for energy harvesting is an effective way to improve both the harvested energy and the conversion efficiency of the DEG. Meanwhile, avoiding loss-of-tension of the DE membrane by shortening the discharging process in the energy harvesting cycle can significantly increase the conversion efficiency. This work also theoretically proves that the ideal triangular energy harvesting scheme could be realized by using DEs with higher fraction of time-independent polymer networks. Comparison between the linear and nonlinear models strengthens the significance of considering nonlinear material viscosity in modeling DEs. This work aims to provide an increased understanding on how the deformation-dependent material viscosity affects the energy harvesting performance of DEGs and is expected to provide optimization guidance for further experimental works.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

ORCID iD

Liying Jiang

References

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