A Review on High‐Frequency Dielectric Elastomer Actuators: Materials, Dynamics, and Applications

Dielectric elastomer actuators (DEAs) as a typical class of electroactive polymers have been developed rapidly in the last two decades due to their advantages such as large strain, high energy density, and fast response. The high‐frequency characteristics of DEAs enable them to be applied in a wider range of fields. In this review, the high‐frequency (>10 Hz) characteristics and applications of DEAs are focused on. The basic concepts and metrics for high‐frequency DEAs are first given. Next, the commercial and synthesized dielectric and electrode materials of DEAs used in high‐frequency applications are reviewed. Following materials, strategies for extending the lifetime of DEAs are introduced. Subsequently, the dynamic modeling approaches for DEAs are presented, considering damping, inertia, and the electrical loss. Building on the fundamental research, some important applications are summarized based on the high‐frequency motions of DEAs including loudspeakers, active vibration suppression, pumps/valves, haptic devices, and high‐speed mobile robots. Finally, the conclusions and future perspectives are given. This review will give readers a clearer understanding of how to design and use high‐frequency DEAs.


Introduction
A dielectric elastomer actuator (DEA) usually consists of a dielectric elastomer (DE) membrane sandwiched between two compliant electrodes. [1]Once a voltage (usually a few kilovolts) is applied across the thickness direction of the structure through the upper characteristics include 1) reducing the mechanical and electrical loss to increase bandwidth, [28,29] 2) extending the lifetime of actuators to ensure continuous functionality, [30,31] and 3) considering inertia and damping to establish dynamic models of highfrequency DEAs for performance prediction and optimization.The remaining parts of the article are organized as the following.We first give the basic concepts and metrics for high-frequency DEAs in Section 2. Section 3 reviews the commercial and synthesized dielectric and electrode materials of DEAs for highfrequency applications.In Section 4, strategies for expanding the lifetime of DEAs are introduced.In Section 5, the dynamic modeling approaches for DEAs are presented.Section 6 summarizes some important applications based on the high-frequency motions of DEAs including loudspeakers, active vibration suppression, pumps/valves, haptic devices, and high-speed mobile robots.Finally, in Section 7, we give the conclusions and future perspectives.We hope this work will give readers a clearer understanding of how to design and use high-frequency DEAs.

Basic Concepts and Metrics of High-Frequency DEAs
To understand the high-frequency DEA's characteristics, we first take a very simple example of a single-input-single-output (SISO) actuator to explain why it exhibits different responses at high frequencies compared with those at low frequencies, how dynamic characteristics are related to materials properties, as well as what metrics we can use to evaluate their highfrequency properties.
The SISO DEA (e.g., a multilayered, rolled actuator presented in our previous work [32] ) can be seen as a system with the input as the voltage V in (usually from a high-voltage generator or amplifier) and the output as the generated axial strain s z (Figure 1a).As mentioned earlier, a DEA is composed of DE films sandwiched between stretchable electrodes.Consider a simple boundary condition that a free-standing DEA film is stimulated by a sinusoidal voltage.When a voltage is applied to the two electrodes, the DEA, usually electrically modeled as a capacitor in series with a resistor (the resistance can come from the electrode resistance, the dielectric loss, or both of them), is being charged.The charging speed ( dQ dt , or current I) is determined by not only the driving voltage V in , but also the electrical impedance of the system where f is the driving voltage's oscillating frequency, R is the equivalent series resistance, and C is the ideal capacitance of the DE.The time constant of this charging process can be calculated as It indicates that the resistance and capacitance in the equivalent circuit of a DEA play important roles in its charging speed.Once the capacitance is charged, there is an electric field across the DE, and an equivalent electromechanical pressure (Maxwell stress) across the film is generated immediately where ε r is the DE's dielectric constant, ε 0 is the permittivity of vacuum, and E is the applied electric field across the capacitor.
Figure 1.Electromechanical model of a typical DEA system and evaluation methods of their dynamic properties.a) A simple model of a DEA's electric and mechanical characteristics.Reproduced with permission. [32]Copyright 2018, John Wiley and Sons.b) The step response of a series of DEAs with different dynamic performances.c) The Bode plots of a series of DEAs with different dynamic performances.
However, the deformation of the DE, which can be represented by the strain s z in the thickness direction, is not immediately produced.The mechanical model of the DE can be assumed to be a classical "mass-spring-dashpot" system, taking into consideration its inertia, elasticity, and damping, respectively.The mechanical impedance of the model is which is related to k, the spring stiffness, m, the inertia, c, the damping constant, and f, the driving frequency.Therefore, the output of the system s z has a rather complex dynamic response to p eq .The approximated settling time of the mechanical system is calculated as The natural frequency of the mechanical system is From the above equations, we could see that the damping constant determines the mechanical response speed and the stiffness of the DEA system is important for the system's natural frequency.It is worth noting DEAs with complex topologies might have more than one mode of vibration, [33][34][35] and therefore the models of DEA can be more sophisticated and more accurate, and we will discuss in detail in later sections.
To evaluate a DEA's dynamic performance, we describe a few metrics.From a time-domain aspect, response time is used to describe the speed of a DEA.Response time can be measured by stimulating a DEA with a voltage in the form of a step function, and recording the response (Figure 1b) and measuring time it takes for the DEA to reach 100% (sometimes from 10% to 90% for a damped system) of its quasistatic value.The different lines in Figure 1b show the example step response of five DEAs with different response times.Another commonly used metric is from a frequency-domain aspect, which is bandwidth.A bandwidth of a DEA is measured by first stimulating it with oscillating signals of various frequencies, and recording the amplitude of the output of each frequency, then forming a Bode plot (Figure 1c), and finding the bandwidth as the frequency at which the amplitude of output power drops to half.Another way to obtain the Bode plot of the system is to use fast Fourier transform analysis by stimulating the system randomly.The above two metrics are highly related-usually, a DEA with a shorter rise time should have a higher bandwidth, and vice versa.Another important factor to describe the dynamic response of a DEA is the quality factor, or Q factor, to describe how damped the system is (a higher Q factor means less damped).One important application of very high-Q-factor DEA is that the strain output can be exceptionally large at its resonant frequency, far exceeding its static performance.The different lines in Figure 1c show the example of Bode plots of five DEAs with different Q factors.

Material Selection and Modification for High-Frequency DEAs
The analysis in the last section shows that materials play important roles in the high-frequency performance of DEAs.In general, inappropriate DE materials may introduce mechanical loss, and inappropriate electrode materials may introduce electrical loss to a DEA running in the high-frequency domains.

Off-the-Shelf DE Materials
As most DE materials exhibit both elastic characteristics and damping characteristics, they are always seen as viscoelastic materials.The ideal DE for high-frequency uses should have high dielectric constant, high dielectric breakdown strength, low modulus, and most importantly for its dynamic response, low viscoelastic damping as seen in Equation ( 5). [36]The viscoelastic damping of a material is usually quantified using the damping coefficient, or tanδ, through dynamic mechanical analysis (DMA).A low viscoelastic damping of the DE contributes substantially to a low mechanical loss and therefore a high-speed response from the time domain and a high bandwidth from the frequency domain.As a result, low viscoelastic damping should be the prior consideration when choosing a DE material for highfrequency DEA devices.However, this principle does not mean other characteristics are not important.The dielectric constant, elastic modulus, and ultimate dielectric strength are key parameters to evaluate DEA's static performance, and in many cases, these parameters are interdependent and cannot reach their optimal values simultaneously. [27]hree types of most widely used DE materials are acrylic elastomers, polyurethanes (PUs), and silicones.The 3M VHB acrylic films are widely used to construct DEAs because of their large voltage-induced strains. [1,37]Yet due to VHB's high viscoelastic damping (tanδ > 0.5), [28,32] the composed DEAs exhibit a fastdropping trend of its strain over frequency.The bandwidth of a VHB-based DEA rarely exceeds 10 Hz; therefore, it is not suitable for applications operating at high frequencies and large strains. [38,39]However, in cases with small-strain requirements or with an out-of-plane resonance working scheme, [12,14] the actuators show remarkable vibrational motions of up to 20 kHz, but with very tiny strains.As for commercially available PUs, DE based on them can be actuated under a relatively lower electric field or produce relatively large force output, thanks to the large dielectric constants of most PU materials.This is perhaps because that besides electrostatic pressure, there is another actuation mechanism with PU named electrostriction. [40]However, due to the crystallization in their structure, PUs are usually with higher moduli, [41] and therefore in most cases generate small actuation strains. [23]In addition, PUs tend to be viscoelastic as well, and some PUs are designed to serve as dampers.As the performance of off-the-shelf acrylic and silicone-based DE improved rapidly, commercially available PUs drew less attention from researchers, whereas some turned to focus on PU-based composites or blends as DE materials. [42,43]ilicones outperform acrylics and PUs in response speed, owing to the significantly lower viscoelastic damping, [44] though the generated strains and energy density are usually much lower than acrylic DEs.Molberg et al. [45] conducted a systematic comparison of the mechanical properties of an acrylate film (VHB 4910, 3 M) and silicone (NEUKASIL RTV-23, Altropol).VHB showed a high tanδ from 0.8 to 80 Hz, exceeding 1 at 4 Hz, whereas the tanδ of silicone remained under 0.1 within the whole frequency range.In the work of Maffli et al., [8] two soft tunable lenses of identical dimensions were fabricated with silicone (CF19-2186, Nusil) and VHB, respectively.When activated, the silicone-based lens was able to change its focal length with a response time of less than 175 μs, whereas the acrylic-based lens achieved 90% of full response after 16 s.As for frequency response, as shown in Figure 2a, the silicone lens shows a bandwidth of over 1 kHz, nearly three orders of magnitude higher than that of the acrylic lens.Figure 2b shows the output optical signal in response to the input voltage, at a response time of far less than 1 ms.The results showed that the viscoelastic damping greatly limited the usage of VHB on high-frequency DEA devices, and silicones have been proved to be the superior choice among the commonly used materials.The frequently used silicones as DE materials include CF19-2186, [46] Sylgard 186, [47] Sylgard 184, [48] LSR-4305, [49] Ecoflex 00-30, [50] and Elastosil P7670. [51]

Modified and Synthesized DE Materials
Despite severe viscoelastic damping, acrylics and PUs are still appealing to build DEAs because they tend to produce large strains.Several pioneering modification methods were applied to acrylic and PU matrices, aiming to improve their response speed.One remarkable approach was optimizing the crosslinking network of acrylic polymers.Yin et al. [28] unconventionally employed a cross-linker with a large molecular weight.When the cross-linker reacted with the monomer, n-butyl acrylate (nBA), the functionality of cross-linking points was reduced to 3, and the flexible polyether diol segment in the cross-linker suppressed the strong dipole-dipole interaction between adjacent side groups of poly(nBA), as shown in Figure 2c.Besides, the butyl connected to the ester group of nBA contributed to an acceleration of polymeric segmental relaxation.As a result, the mechanical loss was greatly lowered (tanδ = 0.21 at 1 Hz, 20 °C, compared with 0.93 of VHB 4910), and the frequency response of the optimized material remained nearly flat in the 1-100 Hz frequency range.Similarly, Dong et al. [52] improved the performance of Laurel acrylate by adjusting the cross-linker.The increased cross-linker content led to more physical entanglements in the DE network, restricting slippages among the molecular chains.Thus, the internal friction under strain was reduced, and the mechanical loss decreased as a result.
Besides cross-linking adjustments, plasticizer could also be applied to enhance DE materials.Niu et al. [53] added cross-linker as well as plasticizing agent to tune the properties of the acrylic elastomer.The cross-linker flattened the mechanical loss peak throughout the working temperature range, with a decrease of up to 76% at room temperature compared with non-prestretched VHB.Meanwhile, by providing free space and improving the mobility of the chain segment, the plasticizer lowered the storage modulus and the glass transition temperature (T g , defined as the temperature when the mechanical loss factor reaches its peak), as illustrated in Figure 2d, making the loss factor comparable with silicone at 25 °C.To enhance the performance of a polyurethane acrylate polymer, Chortos et al. [54] used dioctyl phthalate (DOP) as the plasticizer to break the hydrogen bonds between chains in the polymer network, as demonstrated in Figure 2e.The original hydrogen bonds would have been breaking and forming during the actuation cycling, leading to energy dissipation.By reducing the number of hydrogen bonds, the tanδ of the plasticized dielectric matrices decreased substantially, as shown in Figure 2f, and stress relaxation was also improved.Besides, the plasticizer also exhibits certain healing capability for the DE matrix at electrical breakdown.
Some researchers sought to synthesize materials to have better control over the polymer structures.Ma et al. [55] designed and synthesized a series of triblock copolymers called SBAS, where S stood for polystyrene at both ends of the chain and BA stood for poly(n-butyl acrylate) at the middle.Reversible additionfragmentation transfer (RAFT) polymerization was applied for the synthesis of SBAS, as illustrated in Figure 2g.The fastest copolymer in this series had a response time of 0.94 ms, and both its strain and specific power at 100 Hz exceeded those of the silicone.Caspari et al. [56] synthesized and tuned a series of ethyl thioeter polysiloxanes of low molecular weight, producing lowviscoelastic-damping DE materials, with tanδ ranging from 0.005 to 0.077.Compared with a commercial PDMS film, the synthesized materials had lower moduli and similar mechanical losses, as shown in Figure 2h, thus achieving significantly enhanced strain at a relatively low electric field.Recently, Shi et al. [29] synthesized a bimodal-networked elastomer, with chains of different lengths.By tuning the hydrogen bond density in the material network, more cross-linkers were replaced by hydrogen bonds which were relatively weak and released free space for chain movements.As a result, the viscoelasticity was adjusted, and the tanδ was reduced from 0.22 to 0.1, leading to a 60% strain at 20 Hz with a multilayer DEA.In recent years, dynamic slidering cross-linkers have emerged to be promising in forming elastomers with high softness and low mechanical loss. [57,58]hintake et al. [59] reported two types of slide-ring materials, both exhibiting high dielectric strength (62.4-112.4V μm À1 ) and low mechanical loss (tanδ 0.07-0.24at 10 Hz).Bottlebrush elastomers display markedly low modulus, high elasticity, and the desired strain-stiffening characteristic. [60]Vatankhah-Varnoosfaderani et al. [61] reported a molecular design platform based on bottlebrush architecture for developing DEAs with low driving voltage and large strains.The developed materials also exhibited low hysteresis and high elasticity, which show great promise for constructing high-frequency DEAs.
Though with remarkable improvement in the last few years on the dynamic performance of DE actuators, it is worth noting that, similar to other material enhancement in the field of DEAs, improvement in some aspects was often at the cost of other electromechanical performances.Improving one specific metric may not be that difficult; it is always a challenge to optimize the modulus, viscoelastic damping, dielectric constant, electrical strength, and stability simultaneously.Copyright 2015, John Wiley and Sons.b) Optical response of the silicone lens to voltage steps.Reproduced with permission. [8]Copyright 2015, John Wiley and Sons.c) The precursor was cured to a flexible network.Reproduced with permission. [28]opyright 2021, Springer Nature.d) Storage modulus (top) and loss factor (bottom) of DEs with different plasticizer concentrations.Reproduced with permission. [53]Copyright 2013, John Wiley and Sons.e) Schematic showing how the plasticizer improved the tanδ and stress relaxation by reducing the entanglements between chains.Reproduced with permission. [54]Copyright 2020, John Wiley and Sons.f ) The tanδ as a function of frequency for cured dielectric matrices with varying plasticizer:polymer ratios.Reproduced with permission. [54]Copyright 2020, John Wiley and Sons.g) Schematic illustration of the synthesis of SBAS triblock copolymer via RAFT polymerization (top) and the microphase separation in SBAS triblock copolymer (bottom).Reproduced with permission. [55]Copyright 2017, John Wiley and Sons.h) Dynamic mechanical loss factor analysis of E3-Y, E5, and Elastosil Film.Reproduced with permission. [56]Copyright 2019, Royal Society of Chemistry.

Electrodes for High-Frequency DEAs
The ideal characteristics of the electrodes include high stretchability, low stiffness, good processibility, and high conductivity. [62]Specifically, the conductivity of the electrodes determines the electrical loss of the composed DEAs (see Equation ( 2) for the charging time constant) and should be considered with high priority for high-frequency actuations.As DEA's electrodes are usually in the form of thin and uniform sheets, sheet resistance R s is usually used to quantify its conductivity. [12]The overall resistance can then be calculated as where l and w are the length and width of the electrodes, respectively.
Common DEA electrode materials include carbon-based powder/grease/elastomer, [62] conductive-nanotube networks, [63][64][65] corrugated metallic films, [66] hydrogels with conductive ions, [12] etc.Take one of the most commonly used carbon grease-the 846 Carbon Conductive Grease from MG Chemicals-as an example, the reported volume resistance is about 117 Ω cm, and then the calculated sheet resistance of a 10 μm carbon grease film is about 100 kΩ sq À1 .Compared with this 10 μm carbon grease film, the nonstretched carbon nanotube network, silver nanowire network, and ionic hydrogel are equally or even more conductive, with sheet resistance in the range of 10 À3 -10 2 kΩ sq À1 . [12]The variation in sheet resistance may be affected by thickness, the conductivity of the material, the topology of networks, etc.An estimation of the RC constant τ should be done with Equation (2) before determining whether a specific electrode satisfies the need of a dynamic actuation.The cutoff frequency related to the electric damping can be roughly calculated as If f c is smaller than the desired working frequency, the actuator may be delayed at a low frequency simply due to the resistance of the electrode.
For DEA's electrodes, they stretch with the elastomer when actuated.Therefore, it is essential to consider how the resistance changes with strain as well.Metallic thin film electrodes exhibit excellent conductivity with no strain (sheet resistance <1 kΩ sq À1 ), [67,68] yet the conductivity would decrease rapidly as strain (>3%) is applied due to the inherent nonstretchability of metals.Carbon grease and conductive hydrogels have been reported to have excellent conductivity in response to large strains (>100%).The conductivity-stretch relationship of various potential electrodes has been studied in the literature and should be considered when large strains are required. [65,69,70]

Lifetime Extension of High-Frequency DEAs
Many applications of high-frequency DEAs such as artificial muscles, pumps, valves, and speakers require the device to stay functional over a long period.Due to the high-frequency working mode, the required life cycle of high-frequency DEAs should be especially long (ideally, >1 million cycles).DEAs are known to fail due to three major mechanisms: dielectric breakdown, electromechanical instability, and mechanical breakdown. [71]Many factors could affect these failure modes, such as fabrication methods, electrode material, mechanical and dielectric properties of DEs, working conditions (e.g., prestretch), environmental factors (e.g., medium, temperature, and humidity), etc. [72] Many efforts have been put into overcoming these failures and expanding the lifetime of DEAs.Here, we will review three common methods: utilizing the electrode, utilizing the DE, and optimizing the working conditions.

Utilizing the Electrode
It was reported that silicone-based DEAs usually fail by the dielectric breakdown at low strains (<20%). [72]The dielectric breakdown would generate permanent passages for current, making DEA unable to store charges anymore.Also, premature breakdown due to defects (e.g., air bubbles, dust, foreign particles) within the dielectrics (especially when the thin films are fabricated manually) causes an early failure for the device, and therefore short lifetime. [30]To overcome this failure, Yuan et al. [30] proposed using fault-tolerant single wall carbon nanotubes (SWCNTs) as DEA's electrodes.When a local breakdown happened, the carbon nanotube electrode was ignited by the electric arc, and burned to lose its conductivity, making the breakdown point isolated and insulated, and thus the DEA could continue working without further expanding damage. [73]In this work, although this self-clearing effect could protect DEAs from failure immediately after activation, the total working time only reached a few minutes because defects could hardly be cleared thoroughly.Later, Yuan et al. [74] found that the arc caused by uneven thickness or tiny bubbles could be suppressed by coating the electrodes with dielectric oil.SWCNT electrodes combined with dielectric oil could extend the lifetime of a VHB-based DEA to 1000 min (>16 h) of continuous actuating (not cyclic) with a relatively large linear strain (>50%).Silicone-based DEAs with SWCNTs as electrodes utilizing the same fault-tolerant characteristics achieved a life cycle of >85 000 times. [75]To further understand the fault-tolerant characteristics of SWCNTs, Jiang et al. [76] investigated different factors that might affect the self-clearing process.SWCNT electrode areal density, electrical resistance, duration of the driving voltage, and the number of layers had different yet repeatable influences on the self-clearing intensity of multilayered silicone DEAs.By using optimized combinations of the above factors, DEAs obtained 1 million life cycles with a strain of 9%.This lifetime test was carried out at 125 Hz, the resonant frequency of the DEA.In recent years, silicone-and acrylic-based DEAs utilizing SWCNT's self-clearing characteristics have been further used to expand the lifetime of DEA to facilitate more practical applications, such as haptic devices, artificial muscle, aerial vehicles, etc. [13,22,32,51,77] Due to the widespread applications, several methods to monitor the real-time states of self-clearing process in DEAs have been proposed. [76,78]esides the SWCNT electrode, there were a few other options that have been demonstrated to gain a longer lifetime for DEAs.Saint-Aubin et al. [79] tried several carbon-based electrodes and showed that silicone-based DEA with carbon grease had a limited life cycle of <100 000, whereas inkjet-printed carbon powder and carbon powder-silicone composite would extend the life cycle dramatically compared with carbon grease, as the preparation process of the formerly generated electrodes with more uniform thickness, and finally reached a life cycle of 10 million with 4% strain and 50 Hz driving frequency.
The electrodes that possess the capability of self-clearing should at least satisfy two conditions: one as being thin and the other as being combustible.The mechanism of self-clearing is still not fully understood until today, even though it has been demonstrated by many researchers as an effective way to extend the lifetime of DEAs.More in-depth works are needed in the future to better control and utilize this phenomenon to make more reliable DEAs.

Utilize the DE
DEs severely affect the failure of DEAs-all three major failure modes are closely related to the properties of the dielectrics.Particularly, the self-healing (or self-repairing) characteristics of the DE could effectively reduce or repair DEA failures and therefore extend their lifetime, especially in extreme environments.
Hunt et al. [31] proposed that selecting the two-phase mixture of open-cell silicone sponge and silicone oil as the dielectric can endow DEAs with self-healing characteristics.When a dielectric breakdown happened, the oil would flow into the breaking point, re-establishing the dielectric.50 000 cycles at 5% strain were obtained utilizing the abovementioned self-healing method, and normal DEAs without such improvement only obtained less than 1000 cycles.This achievement benefited from the mobility of liquid oil.Kellaris et al. [80] designed a hydraulically amplified self-healing electrostatic actuator (HASEL) that utilized purely liquid dielectrics, which is another type of actuator different from DEAs but with a similar electrostatic actuation principle.The HASEL actuator consisted of a chamber composed of two layers of flexible (rather than stretchable) electrodes with sealed edges, and dielectric oil that was injected inside.The HASEL actuator was reported to produce a life cycle of >150 000 with an average power density of 358 W kg À1 . [81]y optimizing the amount of cross-linker and the molecular weight of the polymer, Dünki et al. [82] obtained a silicone elastomer with an actuation strain of 20.5% at 10.8 MV m À1 and the ability to self-healing after the dielectric breakdown.Zhang et al. [83] proposed self-healing DEAs for both dielectric breakdown and mechanical damage through a thermoplastic methyl thioglycolate modified styrene-butadiene-styrene (MGSBS) elastomer.The intermolecular electrostatic interaction took place between the methyl thioglycolate modified butadiene block and the styrene block of SBS.It was shown that the dielectric strength could be recovered by up to 67% after dielectric breakdown and 39% after mechanical damage.
Brochu et al. [84] presented a dielectric all-silicone prestretchlocked interpenetrating polymer network (S-IPN) elastomer for DEAs which demonstrated improved performance in a nonprestretched state.This S-IPN elastomer was fabricated by mixing a soft room-temperature vulcanizing (RTV) silicone and a rigid high-temperature vulcanizing (HTV) silicone together, curing RTV silicone, prestaining partially cured composite, and heating to cure the HTV silicone.The tensile force in the RTV silicone balanced the compressive force in the HTV silicone, making the film into a prestretched state without external stretching.A DEA with over 30 000 life cycles and over 20% linear strain was achieved through the above process.
Yin et al. [28] designed a new polyacrylate DE with an optimized cross-linking network by rationally employing the difunctional macromolecular cross-linking agent to gain a higher life cycle and mechanical performance than VHB 4910.With 400% prestretch, the modified polyacrylate-based DEAs could reach a linear strain of 48% at 70 MV m À1 whereas the linear strain of VHB-based DEAs could only reach 15% with such a field.The dynamic response of the proposed material was over 100 Hz.Meanwhile, 50 000 life cycles were achieved under a 5 Hz driving frequency.

Optimizing the Working Conditions
Pull-in instability affects the normal functioning of DEAs at large strains (>20%).Zhao et al. [85] demonstrated that a properly designed prestretching ratio would significantly suppress this instability.Nevertheless, prestretching could reduce the reliability of DEAs due to mechanical breaks.By measuring the worst lifetime performances, Lannarelli et al. [86] reported that the prestretched DE samples have a shorter lifetime compared to nonstretched ones.Therefore, DE materials which can suppress pull-in instability that does not require prestrech would greatly expand the lifetime of DEAs, and Shi et al.'s and Chen et al.'s recent works have provided such solutions. [29,87]he working environment needs to be considered as well.For example, water vapor content is an important accelerating factor that shortens the lifetime of DEAs.The lifetime of silicone-based DEAs decreased by a factor of several to 100 times, when the water vapor content increased from 20% relative humidity (RH) to 85% RH, depending on different electrodes.Environment temperature also accelerated the failure of DEAs under a constant voltage loading test but the effect was much less major than that of humidity. [88]

Summary
The research on lifetime enhancement is summarized in Table 1.The pursuit of a long life cycle and a high strain is usually a contradiction for DEAs.First, higher strain requires a larger electric field, which increases the possibility of DEAs failing due to dielectric breakdown.Next, mechanical fatigue of both the dielectrics and the electrodes is more severe under higher strain.Therefore, we could see from Table 1 that DEAs with long lifetime are usually with small strains (<10%).However, high strain and long lifetime are both important to the practical applications of DEAs.In the future, electrodes with self-clearing capability, dielectrics with self-healing capability, and materials that are soft and with high dielectric strength as well as fatigue strength are expected to be proposed to achieve DEAs with both large strain and long lifetime simultaneously.

General Framework to Model DEAs
The early models to describe DEA's electromechanical characteristics could be traced back to the late 1990s by Pelrine et al. [1,89,90] In these models, a DEA is assumed to have uniform DE film thickness and uniform charges on the electrodes, and both the dielectric and mechanical behavior of the DE do not change with deformation.These pioneering models have been widely used to understand and predict the static behavior of DEAs.However, the nonlinearity at large strains and the viscoelasticity of the DEs were not taken into consideration, and thus they were insufficient to analyze DEAs in large-strain and high-frequency conditions.
][93] This model framework is an analytical energybased electromechanical one developed within continuum mechanics and thermodynamics, which can accommodate many important aspects that researchers are concerned with in the field of DEAs.To tackle the problem of "large-strain nonlinearity," the strain-energy function of any type of hyperelastic constitutive model can be substituted into the model to describe DE's behavior. [94,95]Many commonly used hyperelastic models have been demonstrated to be accurate in predicting the large-strain behavior of DEAs of various materials, such as Mooney-Rivlin, New-Hookean, Gent, Ogden, etc. [95][96][97][98] This model framework also successfully explained the "pull-in" instability that usually happened in DEAs and why prestretched acrylic-based DEAs could achieve large strains. [91]More impressively, the desired stressstretch curve to generate giant strains has been theoretically predicted to guide the design of materials. [92]This model framework usually considers the condition of thermodynamic equilibrium, but it can also consider the condition of nonequilibrium thermodynamics, meaning that certain type of energy loss (mechanical loss, dielectric loss, or conductive loss) occurs in the system.The solution in the condition of nonequilibrium thermodynamics requires specific models to describe the "loss."For example, a viscoelastic model of the DE containing two springs and a dashpot was used in Zhao et al.'s work and the constants were determined by fitting with the experimental data of the material's relaxation process. [93]

Model of DE's Mechanical Loss in DEAs
The time-dependent anelastic behavior of DEs, which is also named viscoelasticity, is widely considered in the modeling of the time-dependent behavior of DEAs.A viscoelastic material exhibits not only elastic characteristics (which means a resistive stress is generated in response to a strain) but also viscous characteristics (which means a resistive stress is generated in response to a strain rate), which is the main cause of DEA's mechanical loss.The simplest model of viscoelastic material includes a spring in parallel with a dashpot, which represents the linear elasticity and Newtonian viscosity, respectively (Figure 1a).The constitutive models chosen to describe viscoelasticity are important in the prediction accuracy of responses and efficiency of calculation.Commonly used linear viscoelastic models include the Maxwell model, Kevin-Vogt model, Zener model (also named Standard Linear Solid model), Burgers model, and Generalized Maxwell Model.
Various linear models have been used to describe the DEAs' viscoelastic behavior and solved analytically.Plante et al. [99] developed a viscoelasticity model based on the Bergstrom-Boyce model (similar to the Zener model) to predict the behavior of VHB 4905/4910 under a high stretch rate.The model showed good agreement with the experiment and explained how viscoelasticity affected DEAs' performance at high frequencies.Hong [100] provided a general field theory for the coupled electrostatic-viscoelastic behavior by integrating the deformable dielectrics into the framework of nonequilibrium thermodynamics.Even though the viscoelastic model used was simple, this theory could explain several basic phenomena and agreed well with the experiment.Foo et al. [101] described a method based on nonequilibrium thermodynamics to construct a model of dissipative DEAs.The elasticity part of the material was expressed using a Gent model and the viscous part was considered as a simple linear dashpot.This model could predict the dynamic response of the membrane and fitted the experiment well.Aiming to describe the behavior of a cylindrical DE more accurately, Nguyen et al. [102] combined the advantages of the Kelvin-Voigt model and the Table 1.The summary of typical methods to enhance lifetime for DEAs and their corresponding reported life cycle or life endurance under a certain strain.

Method
Linear strain Life cycle/life endurance Works Dielectric oil-coated SWCNTs as electrode and VHB as DE %50% 1000 min at a constant driving voltage (no cyclic testing) [73]   SWCNTs as electrode and silicone as DE 20% >85 000 [74]   Optimizing different factors of the self-clearing process of SWCNT and silicone as DE 9% 1 000 000 [75]   Manually applied carbon grease as electrode and silicone as DE 5% <100 000 [78]   Inkjet-printed carbon powder electrode or carbon silicone composite electrode and silicone as DE 5% 10 000 000 Open-cell silicone sponge with silicone oil as self-healing dielectric and carbon grease as electrodes 2.5% >50 000 [30]   DE: RTV silicone mixed with HTV silicone to form a prestretch state in a free-standing setting; electrode: SWCNTs >20% >30 000 [83]   %40% >500 Polyacrylate with an optimized cross-linking network as DE and CNTs as electrode Not reported >50 000 [27]  Generalize Maxwell model and proposed a nonlinear viscoelastic model.With the assist of Monte Carlo statistical simulation, they obtained a maximum prediction error of 3.762% compared with experiments.Gu et al. [103] proposed a two-part constitutive model with a nonlinear spring describing the equilibrium state and a set of nonlinear springs and linear dashpots describing the deviation from equilibrium.The work demonstrated that the increase in the number of spring-dashpot units can dramatically increase the prediction accuracy of the electromechanical nonlinear response of DEAs, but at a higher computational cost.
When the viscoelastic model or the configuration of the DEAs becomes more complex, the finite element methods are useful to solve the heavy-computation problem.Park et al. [104] developed a dynamic finite element formulation for DEs with viscoelasticity using a nonlinear viscoelasticity model.Schlogl et al. [105] derived a finite element model that extended the hyperelastic material model by viscoelastic terms using a 3D form of the Kelvin-Voigt model, and validated this model on stacked DEAs with realistic geometry.These computational frameworks may potentially be the general tools for analyzing DEAs with hyperelastic and viscoelastic characteristics.

Adding Inertia Terms to Form the Full Dynamic Model for DEAs
In cases of noncyclic actuation or low-frequency oscillation, it is reasonable to assume the system as quasistatic and ignore the inertia when modeling DEAs.However, when it comes to high-frequency cyclic actuation, either passively vibration or actively oscillation, the inertia term has to be considered.It is worth noting that the inertia of a DEA system should consider both the DE membrane and any external load that moves with the actuator. [106,107]In this part, we review methods to include the inertia in the model of DEAs to describe the full dynamics.
The methods fall into the field of continuum mechanics, where the materials are modeled as a continuous mass rather than discrete particles.One important step for a DEA's dynamic modeling is to establish the governing equations of motion (usually in the form of one or more differential equations with respect to time).For simple cases with homogeneous deformation within the DEA, one could derive the equation of motion from the linear momentum balance (or Newton's law of motions) for continuum and replace the mechanical stress term with the total stress (summation of mechanical stress and electrostatic stress).Xu et al. [108] developed a dynamic nonlinear electromechanical continuum model from the conventional equation of motion of the continuum and investigated the effect of damping on a homogeneous DEA with a sandwich structure.Based on the Euler-Bernoulli beam dynamic model, Feng et al. [109] established an analytical model for a DE-based microbeam resonator, and the approximate analytical solutions were obtained to evaluate the performance including Q-factor and frequency shift ratio.Gu et al. [110] also obtained the equation of motion of a DEA system from force balance and a feedforward controller was designed based on the dynamic model.Zhao et al. [32] derived the dynamic response of a rolled linear actuator from force balance in the axial direction.
For situations with nonhomogeneous deformation within the DEA or complex configurations of the DEA (e.g., the dielectric elastomer minimum energy structure, DEMES), the Euler-Lagrange equation is a relatively more convenient method to derive the governing equation of DEA dynamics.Actually, the Euler-Lagrange equation has been frequently used for analyzing the vibrational characteristics of elastic membranes before analyzing DEAs. [111]Xu et al. [108] and Wang et al. [112] both derived the equation of motions using the Euler-Lagrange equation on homogeneous DEAs and investigated the resonance and the effect of strain-stiffening on the vibrational state transition, respectively.The DEMES is a typical composite structure with nonlinear geometry that involves a rigid frame and soft, prestretched membrane.Zou et al. [113] and Vatanjou et al. [114] established the dynamic models of DEMES from the Lagrange equation to analyze its dynamics.Cao et al. [115] derived a mathematical model based on the Euler-Lagrange method to describe the complex response of a novel conical DE oscillator.
Another commonly used method to derive the governing equation is within the model frame of nonequilibrium thermodynamics described in Section 5.1 by further adding the work done by inertia to a DEA system.For example, Zhu et al. [116,117] modeled the nonlinear oscillation of a DE balloon and a diaphragm by assuming the variation of the free energy of the DE membrane equals the work done by the voltage, the pressure, and the inertia.Li et al. [118] developed an analytical model for the tunable DE resonator and derived the equation of motion using the virtual work method, and viscoelasticity was included.Sheng et al. [119] studied the dynamic characteristics of DEA's in-plane deformation with the consideration of viscoelasticity and the model was used to study the effects of damping and stability transition of the system.Based on the same modeling framework, Jia et al. [39] modeled the transient response of the in-plane nonuniform deformation of a circular DEA using thermodynamics and added a damping term.Huang et al. [120] established the dynamic model of a DEA with a conical shape based on the principles of nonequilibrium thermodynamics.From the last two examples, we could see the thermodynamics method is capable of handling the nonhomogenous deformation of a DEA.
The abovementioned methods to derive the equations of motion for the dynamic response of a DEA should be identical, and numerical methods (e.g., finite element methods [121] ) are frequently used to simulate the quantitative results due to the relatively complex process.One of the popular applications of the finite element method in dynamic modeling is to calculate the mode shape of an actively drive DE membrane.Goncalves et al. [122] obtained the nonlinear free vibration response under large and small amplitude with the finite element method.Chakravarty [123] also developed a finite element model to investigate the effect of pressure on the DEA's resonance frequency.Gratz-Kelly et al. [124] proposed a principle to develop multimode DEAs with linear actuation and sound generation function.By employing a coupled finite element method, Moretti et al. [125] predicted the different mode shapes of a circular out-of-plane DEA with quite consistent results with the experimental data.These works inspired works that make use of the complex structural dynamics of DEAs such as driving a robot with different harmonic frequencies to achieve various locomotion modes. [126]

Electric Modeling
In previous sections, we focus on the dynamic modeling of the electromechanical coupling and the mechanical response of DEAs.The electrical part needs modeling as well to give a full dynamic description of the system.The most widely used electric model of a DEA contains a serial resistance, a capacitance, and a parallel resistance, which represents the charging resistance, charge-storage capability, and leakage current respectively. [127]ome works have included the electric model in the dynamic modeling of DEAs. [32,101,128,129]Parallel resistance plays a key role in the electromechanical efficiency of DEAs, especially with large strains and for long duration. [101]A large serial resistance will cause a drop in the dynamic bandwidth of the DEA, and thus it also requires particular attention. [32]

Applications
][132][133] In this review, we mainly introduce applications that utilize the high-frequency characteristics of DEAs, such as loudspeakers, active vibration isolation, pumps/valves, haptic devices, and mobile robots.

Loudspeakers
Conventional loudspeakers are usually composed of magnets and conductive coils, and these rigid metal structures inevitably increase their weight and size.DEAs have the advantages of lightweight and softness and can generate sound using their highfrequency vibration, providing an alternative solution for the design of lighter and more flexible advanced audio systems.
Several studies have successfully achieved sound generation using various types of DEAs, as shown in Figure 3. Sugimoto et al. [134] designed a semicylindrical acoustic transducer using a polyurethane DE film and poly(ethylene dioxythiophene):poly(styrenesulfonate) (PEDOT:PSS) electrodes (see Figure 3a), and their research demonstrated that utilizing the change in the side length of the actuator resulted in efficient sound generation.However, this semicylindrical shape limited the size and sound pressure of the loudspeaker.Therefore, they further proposed a push-pull acoustic transducer using DEAs, [135] as illustrated in Figure 3b.The total mass of the push-pull acoustic transducer was 60 g, which was lighter than a conventional loudspeaker and this structure also contributed to suppressing the second harmonic distortion.
A typical polyhedron loudspeaker achieves nearly omnidirectional sound propagation by arranging multiple loudspeakers in all directions, and thus a polyhedron loudspeaker is unusually bulky.Hosoya et al. [136] proposed a spherical omnidirectional loudspeaker made from DEAs as shown in Figure 3c.This hemispherical loudspeaker was much small and lightweight and obtained a source omnidirectionality (0°-180°) with a frequency range of up to 16 kHz.The driving voltage of the hemispherical loudspeaker is still very high though (2 kV).They subsequently inflated the DEA like a balloon with pressured air to apply a prestretch, thereby reducing the thickness of the DEA film, [137] as illustrated in Figure 3d.In this way, the actuating voltage of the DEA loudspeaker was reduced to 800 V for maintaining the same sound pressure levels and the acoustic radiation pattern of the balloon DEA loudspeaker extended to a range of over 270°.
The diversity of DEA configurations allows the DEA-based loudspeaker to be used in some special scenarios.Keplinger et al. [12] demonstrated a transparent DE loudspeaker that produces sound over the entire audible range, i.e., 20 Hz to 20 kHz, as shown in Figure 3e.This transparent loudspeaker might be attached to a window to achieve active noise cancellation [138] without blocking the light transmission.Gratz-Kelly et al. [124] developed a cone-shaped multifunctional DEA that can produce linear actuation and sound (see Figure 3f ).Their DEA could play music while at the same time beating the tempo of the tune, simultaneously working as a loudspeaker and a metronome.

Active Vibration Isolation
The vibration isolation aims to reduce the propagation of unexpected vibrations to the sensitive structure.The general passive vibration isolation methods such as using rubber mounts or springs are more effective for suppressing high-frequency vibrations.For low-frequency vibration, active vibration isolation plays a more important role.Piezoelectric [139] and magnetostrictive [140] based actuators are commonly used in active vibration isolation.However, installing these rigid actuators on a curved surface is difficult.Soft DEAs have great potential for active vibration isolation in mechanical systems with curved surfaces.Additionally, combining their active and passive features, DEAs can realize active vibration isolation at low frequencies and passive vibration isolation at high frequencies due to the DE material's inherent damping characteristics.Also, the natural frequency of the DEAs can be tuned for semiactive vibration isolation in a wider frequency range. [141,142]arban et al. [143] fabricated a core-free rolled tubular DEA used for active vibration isolation, as shown in Figure 4a.The DE tubular actuator was able to provide isolation against harmonic vibratory disturbances with an attenuation of 66 and 23 dB, at 5 and 10 Hz respectively.The narrowband active vibration isolation was demonstrated to isolate a 250 g mass from a 2 to 8 Hz random vibration.A finite-impulse-response filter-based adaptive feedforward controller was used to produce an attenuation of 20 dB across the frequency range.Li et al. [144] presented a novel sequential deposition fabrication process for multilayer stack actuators which was used as an isolator in active vibration isolation (Figure 4b).The least mean square algorithm was used in their active vibration control system and a reduction of 34 dB was achieved at 17 Hz.Zhao et al. [145] proposed an annular DEA for active vibration isolation (Figure 4c).Filtered-x least-mean-square algorithm-based active vibration isolation experiments were carried out and the vibration at the reference harmonic frequency (7 Hz and 11 Hz) was attenuated by 57.71 and 58.38 dB, respectively.
The above works were mainly aimed at the isolation of narrowband and low-frequency vibration.Kaal et al. [146] proposed a DE stack actuator using microperforated metal electrodes, as illustrated in Figure 4d.The low electric losses of the metal electrodes [147] at high frequencies made this DEA suitable for high-frequency dynamic applications.The active vibration isolation experiment showed that the transmission curve could be kept below 0 dB with a frequency range of up to 200 Hz apart from very low frequencies and the passive resonance could be canceled out completely.Kajiwara et al. [148] studied the highfrequency active vibration control of an acrylic rectangular plate by DEA (Figure 4e).The arrangement and shape of the DEA were optimized to achieve better control performance.The results of the control experiment showed that the first four modes obtained reduced vibrations by 10.5, 7.7, 11.6, and 7.9 dB, respectively, demonstrating the efficacy of the DEA for active vibration control.

Pumps/Valves
Pumps/Valves are important components in the control of fluidic power.DEAs could help simplify the mechanical system and make pumps/valves more compact.Besides, the flow rate and pressure of a DEA-based pump/valve can be adjusted by simply controlling the amplitude of the driving voltage.On the other hand, the soft/stretchable pumps/valves based on DEAs have great application potential in soft robotic systems.
Loverich et al. [149] proposed a polymer-based micropump actuated by a DEA, as shown in Figure 5a.This pump occupying less than 10 mm 2 of the area produced a 77 μL min À1 flow rate at  [134] Copyright 2011, Acoustical Society of America.b) A push-pull acoustic transducer.Reproduced with permission. [135]Copyright 2013, Acoustical Society of America.c) A DEA-based hemispherical loudspeaker.Reproduced with permission. [136]Copyright 2015, Acoustical Society of America.d) A balloon DEA loudspeaker.Reproduced with permission. [137]Copyright 2019, Elsevier.e) A transparent loudspeaker.Reproduced with permission. [12]Copyright 2013, The American Association for the Advancement of Science.f ) A cone-DEA loudspeaker.Reproduced with permission. [124]Copyright 2022, John Wiley and Sons.
3.3 kV and 30 Hz actuation condition.The pump had a maximum thermodynamic efficiency of 0.96% and exhibited negligible performance degradation over 10 h of continuous operation.
Linnebach et al. [150] introduced a pneumatic diaphragm pump driven by a cone DEA.A demonstration device integrating the pump is shown in Figure 5b.The maximum flow rate of this diaphragm pump was 1.48 L min À1 and the maximum pressure was 19.295 kPa at an actuating frequency of 98 Hz.Cao et al. [151] also presented a pneumatic diaphragm pump driven by a magnetically coupled double-cone DEA as in Figure 5c.This pump could be almost entirely soft by using a 3D-printed soft chamber.The pump demonstrated a maximum pressure output of 3.05 kPa and a flow rate of 0.9 L min À1 .Several soft robotic prototypes, including a soft gripper, a ballooning chamber, and a suction cup driven by this pneumatic pump were demonstrated.Shi et al. [29] introduced a DEA-based roll pump that exhibited excellent performance recovery after it was bent, twisted, or hammered during operation, as shown in Figure 5d.The peak flow rate reached 20.4 mL min À1 at 2.5 kV and 10 Hz.
solutions with a flow rate of 21.5 μL min À1 and by changing the actuation frequency of the DEA, the mixing ratio of the two liquids from the inlet, as well as the flow rate, could be adjusted.
Xu et al. [153] developed an electrically driven soft valve using DEAs, as shown in Figure 5f.The loaded power density of the triple DEA designed in their research reached 290 W kg À1 .Their soft valve had a fast response time (<0.1 s), a 40 mL min À1 flow rate and could control the fluidic pressure up to kPa.Using this soft valve, multiple macroscale hydraulic soft actuators powered by a single pressure source could be controlled independently.

Haptic Devices
Haptic devices are widely used in human-machine interactions, including braille displays, virtual/augmented reality, telemanipulations, etc.[156] DEAs are promising as haptic devices because of their comparable mechanical impedance with human skin and sufficient bandwidth.
Koo et al. [157] proposed a tactile stimulating element that contained a 4 Â 5 matrix actuator array, as illustrated in Figure 6a.This haptic feedback device could be worn on a human fingertip.The maximum output displacement was 471 μm and the force-toweight ratio reached up to 6.8 N g À1 , which were sufficient for a tactile device.Phung et al. [158] presented a bidirectional haptic device by integrating an antagonistic DEA with a V-shaped electrostatic actuator, as shown in Figure 6b.Due to the introduction of the electrostatic actuator, the output displacement and blocking force of the whole device were significantly improved (680 μm of displacement and 280 mN of blocking force).They also carried out a durability test on the device, and the performance of the actuator showed almost no degradation after  [149] Copyright 2006, Royal Society of Chemistry.b) A DEA-driven pneumatic pump.Reproduced with permission. [150]Copyright 2020, IOP Publishing.c) A Magnetically coupled pump.Reproduced with permission. [151]Copyright 2019, John Wiley and Sons.d) A DE roll pump.Reproduced with permission. [29]Copyright 2022, The American Association for the Advancement of Science.e) A pumping micromixer.Reproduced with permission. [152]Copyright 2018, IOP Publishing.f ) A DEA-driven soft valve.Reproduced with permission. [153]Copyright 2021, National Academy of Sciences.
continuously running for 20 min at 4.5 kV and 1 Hz.Ji et al. [159] proposed an untethered "feel-through" haptic device using ultrathin DEAs.Most notably, they designed a wireless driver circuit (mass 1.3 g) for their "feel-through" device, as shown in Figure 6c.This novel haptic device could assist the user to identify randomly oriented black letters on the white background through tactile vibrations.
Compared with the haptic device worn on the fingertip, an arm-based wearable device can free the hands.Mun et al. [160] fabricated a wearable interface by embedding DEAs onto curvilinear surfaces such as the forearm band (see Figure 6d).The results of the perception intensity test based on 15 subjects showed that at least five levels of vibrotactile stimuli could be distinguished using the proposed haptic band.Zhao et al. [13] designed an arm-based haptic device using a 2 Â 2 array of rolled DEAs, as shown in Figure 6e.This device could be used to test human perception of simple elements of touch position and direction with a broad bandwidth (from 10 Hz to 200 Hz).However, this device was not optimized in terms of actuator structure, ergonomics, and usability of the system.Therefore, Lee et al. [161] Figure 6.Haptic devices based on DEAs.a) Fingertip tactile display.Reproduced with permission. [157]Copyright 2006, IEEE.b) Bidirectional tactile display.Reproduced with permission. [158]Copyright 2020, IOP Publishing.c) Feel-through haptics.Reproduced with permission. [159]Copyright 2020, John Wiley and Sons.d) Soft tactile interface on the arm.Reproduced with permission. [160]Copyright 2018, IEEE.e) Soft haptic communicator array.Reproduced with permission. [13]Copyright 2019, Mary Ann Liebert, Inc. f ) A wearable textile-embedded haptic display.Reproduced with permission. [161]Copyright 2021, Mary Ann Liebert, Inc. demonstrated an optimized wearable haptic device as shown in Figure 6f.Their device consisted of a 5 Â 3 array of disc-shaped DEAs with a 20 mm spacing.The mechanical impedance of the skin was considered during the design of the actuator parameters.A lumped-parameter model of the skin was developed and the human perception tests were used to refine the viability of the design.

Mobile Robots
62][163][164][165][166][167][168][169] Most DEAdriven mobile robots are based on the quasistatic deformation of the actuators to realize movements, and the speed is generally not very high.The high-frequency oscillation characteristics of DEA, especially the resonance, provide a solution for the rapid movement of mobile robots. [170]he minimum energy structure is widely adopted in designing mobile robots driven by acrylic-based DEAs.Gu et al. [171] reported a soft wall-climbing peristaltic robot with a DEA body and two electroadhesive feet as shown in Figure 7a.The robot could climb walls (made of wood, paper, and glass) at 90°with a speed of up to 0.75 body length per second at 16 Hz.This twofoot robot could also be scaled up to a four-foot one to realize spot-turning locomotion.Zhao et al. [172] reported a tethered soft robot capable of hopping-running with high speed up to 6.10 body lengths/second shown in Figure 7b.The robot showed good adaptability to the ground materials (marble, wood, rubber, sandpaper).Li et al. [173] developed a soft robot driven by DEA that mimics an inchworm (Figure 7c).On the one hand, the robot had a fast-moving speed (4 body lengths/second), and on the other hand, the robot survived a compressive weight 30 000 times its own weight during operation without mechanical fracture or electrical breakdown.
The locomotion of the robot can also be realized based on the vibration of DEAs.Tang et al. [14] introduced an ultralight, speedy locomotion cubic-shaped robot driven by a DE resonator (see Figure 7d).The foam body of the robot had anisotropic friction, and when the DE resonator vibrated back and forth, it would produce a net displacement in the forward direction.The robot could realize fast linear locomotion (up to 2.8 body lengths/second) based on the first mode resonance of the actuator and change its direction (U-turn) at the second mode resonance. [174]They also proposed a cephalopod-inspired robot based on a vibrational DEA, utilizing jet-propulsion actuation, [175] as shown in Figure 7e.The robot could be propelled via producing a jet of air on the surface of the water and when the robot sank into the water, it could also be propelled by a jet of water.
Compared with acrylic-based DEAs, silicone-based DEAs are with higher bandwidth and are more diverse in the structural design for mobile robots.Ji et al. [176] designed an insect-sized untethered and autonomous legged robot as shown in Figure 7f.This robot, DEAnsect, was driven by three low-voltage stacked DEAs operating at 450 V and more than 600 Hz.Ultralight drive electronics, including optical sensors, a microcontroller, and a battery, were designed to achieve the autonomous movement of the robot.Due to the durability of the actuators, the DEAnsect kept working even after being flattened by a fly swatter.
Tang et al. [177] proposed a DEA-driven pipeline inspection robot that could fit into pipes with subcentimeter diameters and different curvatures, as illustrated in Figure 7g.This pipeline robot achieved rapid motions horizontally and vertically (horizontal: 1.19 body lengths/second, vertical: 1.08 body lengths/second) in a subcentimeter pipe (diameter 9.8 mm).Thanks to the robot's flexible body, it was capable of moving in pipes with varying geometries (diameter-changing pipe, L-shaped pipe, S-shaped pipe, or spiral-shaped pipe).The robot also had the adaptability to traverse pipelines with different materials (glass, metal, or carbon fiber) or filled with media (air or oil).Chen et al. [22] proposed an insect-scale aerial robot powered by DEAs, as shown in Figure 7h.The multilayered DEAs used in their design had a high power density (up to 600 W kg À1 ) and high bandwidth (>400 Hz).Their work demonstrated a soft actuator with sufficient power density and bandwidth that could be used to achieve lift-off and flight control.It is worth mentioning that the DEAdriven microrobot could recover from in-flight collisions because of the actuator's high compliance and resilience.More mobile robots that can fit extreme environment driven by DEAs are expected to be developed to further leverage their capabilities such as self-healing, damage endurance, inherent adaptivity, and high power density.

Conclusions and Perspectives
In this work, we reviewed the basic concepts, materials, lifetime expansion strategies, modeling, and applications of highfrequency DEAs.Compared with quasistatic DEAs, highfrequency DEAs are less understood and developed, though great efforts and trials have been put into this field.Such a field requires researchers to be multidisciplinary, with a deep understanding on dynamics, polymers, rheology, mechanics, electronics, and structural design.Many challenges still exist that prevent the widespread use of high-frequency DEAs.
From a material aspect, off-the-shelf DE materials and stretchable electrode materials are not specialized for DEAs, whereas modified and synthesized materials are not mass-produced.Most existing DEAs are handmade in the laboratory, and the consistency of their properties, especially their high-frequency characteristics (such as response time, Q factor, resonance frequency, etc.) between different actuators are poorly controlled.The multilayer stacking technology including wet stacking methods [32,177] and dry stacking methods [29] may potentially facilitate large-scale manufacturing of DEAs.The lifetime of DEAs is still short for practical applications and the failure mode at high frequencies needs further investigation.Finally, the high-frequency actuation of the DEAs requires a higher power supply, and it may be difficult for a small high-voltage amplifier (such as EMCO series products) to provide sufficient current at high frequencies.On one hand, high-power, compact high-voltage amplifiers are in great need; on the other hand, the efficiency of high-frequency DEAs needs to be increased for future untethered applications.Copyright 2018, The American Association for the Advancement of Science.b) Hopping-running Robot.Reproduced with permission. [172]Copyright 2019, Mary Ann Liebert, Inc. c) Insect-scale Robot.Reproduced with permission. [173]Copyright 2018, Mary Ann Liebert, Inc. d) A robotic cube.Reproduced with permission. [14]Copyright 2018, Elsevier.e) Jet propulsion robot.Reproduced with permission. [175]Copyright 2020, John Wiley and Sons.f ) Autonomous untethered soft robotic insect.Reproduced with permission. [176]Copyright 2019, The American Association for the Advancement of Science.g) A pipeline inspection robot.Reproduced with permission. [177]Copyright 2022, The American Association for the Advancement of Science.h) An aerial robot.Reproduced with permission. [22]Copyright 2019, Springer Nature.

Figure 2 .
Figure 2. Different off-the-shelf and modified DE materials.a) Frequency response of silicone-based (blue triangles) and acrylic elastomer-based (red squares) tunable lenses.Reproduced with permission.[8]Copyright 2015, John Wiley and Sons.b) Optical response of the silicone lens to voltage steps.Reproduced with permission.[8]Copyright 2015, John Wiley and Sons.c) The precursor was cured to a flexible network.Reproduced with permission.[28]Copyright 2021, Springer Nature.d) Storage modulus (top) and loss factor (bottom) of DEs with different plasticizer concentrations.Reproduced with permission.[53]Copyright 2013, John Wiley and Sons.e) Schematic showing how the plasticizer improved the tanδ and stress relaxation by reducing the entanglements between chains.Reproduced with permission.[54]Copyright 2020, John Wiley and Sons.f ) The tanδ as a function of frequency for cured dielectric matrices with varying plasticizer:polymer ratios.Reproduced with permission.[54]Copyright 2020, John Wiley and Sons.g) Schematic illustration of the synthesis of SBAS triblock copolymer via RAFT polymerization (top) and the microphase separation in SBAS triblock copolymer (bottom).Reproduced with permission.[55]Copyright 2017, John Wiley and Sons.h) Dynamic mechanical loss factor analysis of E3-Y, E5, and Elastosil Film.Reproduced with permission.[56]Copyright 2019, Royal Society of Chemistry.

Figure 3 .
Figure 3. Loudspeakers based on DEAs.a) A Semicylindrical acoustic transducer.Reproduced with permission.[134]Copyright 2011, Acoustical Society of America.b) A push-pull acoustic transducer.Reproduced with permission.[135]Copyright 2013, Acoustical Society of America.c) A DEA-based hemispherical loudspeaker.Reproduced with permission.[136]Copyright 2015, Acoustical Society of America.d) A balloon DEA loudspeaker.Reproduced with permission.[137]Copyright 2019, Elsevier.e) A transparent loudspeaker.Reproduced with permission.[12]Copyright 2013, The American Association for the Advancement of Science.f ) A cone-DEA loudspeaker.Reproduced with permission.[124]Copyright 2022, John Wiley and Sons.

Figure 5 .
Figure 5. Pumps/valves driven by DEAs.a) An all-polymer micropump.Reproduced with permission.[149]Copyright 2006, Royal Society of Chemistry.b) A DEA-driven pneumatic pump.Reproduced with permission.[150]Copyright 2020, IOP Publishing.c) A Magnetically coupled pump.Reproduced with permission.[151]Copyright 2019, John Wiley and Sons.d) A DE roll pump.Reproduced with permission.[29]Copyright 2022, The American Association for the Advancement of Science.e) A pumping micromixer.Reproduced with permission.[152]Copyright 2018, IOP Publishing.f ) A DEA-driven soft valve.Reproduced with permission.[153]Copyright 2021, National Academy of Sciences.

Figure 7 .
Figure 7. DEA-driven mobile robots.a) Soft wall-climbing robot.Reproduced with permission.[171]Copyright 2018, The American Association for the Advancement of Science.b) Hopping-running Robot.Reproduced with permission.[172]Copyright 2019, Mary Ann Liebert, Inc. c) Insect-scale Robot.Reproduced with permission.[173]Copyright 2018, Mary Ann Liebert, Inc. d) A robotic cube.Reproduced with permission.[14]Copyright 2018, Elsevier.e) Jet propulsion robot.Reproduced with permission.[175]Copyright 2020, John Wiley and Sons.f ) Autonomous untethered soft robotic insect.Reproduced with permission.[176]Copyright 2019, The American Association for the Advancement of Science.g) A pipeline inspection robot.Reproduced with permission.[177]Copyright 2022, The American Association for the Advancement of Science.h) An aerial robot.Reproduced with permission.[22]Copyright 2019, Springer Nature.