Microscopic in situ observation of electromechanical instability in a dielectric elastomer actuator utilizing transparent carbon nanotube electrodes†

Zhen-Qiang Song*^ad, Li-Min Wang^a, Yongri Liang^a, Xiao-Dong Wang^b and Shijie Zhu*^c
aCenter for Advanced Structural Materials, State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, China. E-mail: zqsong@ysu.edu.cn
^bSchool of Civil Engineering and Mechanics, Yanshan University, China
^cDepartment of Intelligent Mechanical Engineering, Fukuoka Institute of Technology, Japan. E-mail: zhu@fit.ac.jp
^dHebei Key Lab for Optimizing Metal Product Technology and Performance, Yanshan University, China

First published on 13th August 2024

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1. Introduction

As one type of electroactive polymer, dielectric elastomers (DE) exhibit impressive performance to generate large mechanical deformation under electrical stimuli.^1–4 Consequently, it is extensively studied as a soft actuator for diverse applications.^5–8 The operational principle involves a dielectric elastomer membrane sandwiched between the compliant electrodes on both sides. When a bias voltage is applied through the thickness of the membrane, opposite charges accumulate on the two electrodes, and the electrostatic attractive force causes membrane thinning in thickness and expansion in area, during which the electrical energy is transformed into mechanical motion.^9,10 It is reported that the linear strain under electrical actuation is up to 100%,¹¹ with energy density and response speed comparable to those of natural muscles, which inspired significant development of dielectric elastomer actuators (DEAs) as artificial muscles for soft robotics.^3,9–14

Enhancing the actuation performance of dielectric elastomer actuators (DEAs) requires optimization of both the electrodes and the elastomer membranes. In particular, the dielectric permittivity and elastic modulus of the membrane are crucial factors influencing the actuation strain. Generally, the stiffness of the elastomers can be lowered via addition of plasticisers or through chemical modifications that tailor their molecular weight and cross-linking density.¹⁵ Nevertheless, it is worth noting that very soft elastomers tend to exhibit significant viscoelastic effects, which decrease the energy conversion efficiency of DEAs. A high dielectric permittivity effectively increases the capacitance of the DEA, thereby reducing the electric field intensity required to drive the actuator. High permittivity can be achieved either by chemical modification of the elastomer backbone with a high density of polar groups¹⁶ or by using a conventional strategy of elastomer composites composed of certain fillers, such as conductive particles^17,18 or ceramic fillers with high permittivity.^19,20 Note that the filler concentration should remain below the percolation threshold to avoid the formation of a conductive network through the matrix.

It is well established that the maximum actuation strain of the DEA is also strongly affected by pre-stretching of the membrane.^21–24 Optimal pre-stretching enhances actuation performance, whereas insufficient pre-stretching can render DEAs prone to electrical breakdown, severely limiting actuation deformation. This phenomenon is commonly attributed to the occurrence of electromechanical instability (EMI), or snap-through instability, which results from the positive feedback between membrane thinning and an increasing electric field.^25–32 Theoretical models predict that the EMI is characterized by the abrupt thinning and sudden area expansion of the membrane at a critical voltage. However, this particular failure feature is seldom observed in the actuation plots from numerous experimental tests conducted under low pre-stretch conditions (typically λ_pre < 2).^25,33–37 Instead, these studies often show a continuous increase in actuation strain with the driving voltage until reaching electrical breakdown. Hence, it raises the question of whether the failure of DEAs can be definitively attributed to the EMI mechanism under low pre-stretch conditions.

To elucidate the process of electromechanical instability of the dielectric elastomer actuator, certain in situ experiments were conducted at the macroscopic scale utilizing a thick (typically hundreds of micrometers) and opaque carbon grease electrode, which conceals the actual morphology of the elastomer membrane.^{25,33–35,38–42} Recently, researchers resorted to transparent and compliant electrodes to study the actuation process of DEAs at the microscopic scale, such as the liquid electrodes of NaCl solution,^43–45 silver nanowires⁴⁶ and carbon nanotube electrodes.^47,48 However, in these tests, only one side of the membrane was coated with a compliant electrode, while the other side was bonded to a rigid substrate, which contrasts with the practical operation of DEA that expands without any constraints on either side of the membrane. Therefore, the failure features observed in these studies, such as the creasing and cratering instability phenomena,^43–48 may not be directly applicable to the DEAs. Furthermore, these studies did not consider the impact of varying pre-stretch ratios on surface instability. To the best of our knowledge, the microscopic in situ observations of the electromechanical response of DEAs under unconstrained conditions are still few and far between.

An ideal electrode suitable for the in situ investigation of the DEA requires the maintenance of structural integrity with low electrical resistance, even under maximum actuation stretching. Additionally, the electrodes should be sufficiently thin and compliant to minimize the impedance to the deformation of the dielectric elastomer membrane. Consequently, conductive, flexible and transparent electrodes have been widely explored in recent years, such as carbon nanotubes,^49–54 silver nanowires (AgNW),^55–59 graphene^60,61 and electrolytic hydrogel.^62–64 These compliant electrodes are not only capable of sustaining extremely large strains to match the deformation of the DE membrane but also exhibit great optical transmittance akin to the thin metal^65–67 and traditional transparent electrodes, such as indium tin oxide (ITO).⁶⁸ Among these, single-walled carbon nanotube (SWCNT) electrodes show unique self-clearing capability when the DE membrane experiences electrical breakdown.^50,69 Therefore, the SWCNT electrode is regarded as an ideal choice for the in situ study of DEA, considering its nanometer-scale thickness, high conductivity, great compliance and transparency.

In this paper, we elaborately prepared a small active region (1 × 1 mm²) coated with a compliant and transparent carbon nanotube electrode on both sides and successively detected the electromechanical instability of DEAs under unconstrained conditions at the microscopic scale. Additionally, the impact of various pre-stretch ratios on the failure mechanism of DEA was examined, and the EMI failure of DEA was analyzed based on the thermodynamic principle and phase transition theory.

2. Experimental methods

2.1 DEA fabrication

The single-walled carbon nanotube provided by the National Institute of Advanced Industrial Science and Technology (AIST, Japan) was utilized for the electrode fabrication. 2.5 mg of SWCNT was dispersed in 45 mL of an aqueous sodium dodecyl sulfate (SDS, Sigma-Aldrich Company) surfactant solution (1% wt), and the dispersion was mixed with a cup-horn sonicator (US-50E, NISSEI Corporation) for 80 min at a power of 50 W. The solution was centrifuged at a rotation speed of 15000 rpm for 6 hours (HITACHI CF15RXII), and the upper 75% supernatant liquid was decanted. This procedure yielded a typical nanotube concentration of approximately 20–25 mg L⁻¹.^51,70,71 The uniform SWCNT film was fabricated by vacuum filtration of the solution with a Whatman Nuclepore polycarbonate hydrophilic membrane filter with 50 nm pore size and 47 mm diameter. After filtering the CNT solution, the membrane was rinsed six times with pure water to remove residual SDS. By adjusting the volume of the SWCNT dispersion used in the filtration process, SWCNT films with varying areal densities were obtained. The morphology of the SWCNT networks was characterized using a field emission scanning electronic microscope (JSM-7100F JEOL Ltd), and the electrical properties of the SWCNT film were measured with a 4-point probe resistivity measurement system (RTS-9).

The circular elastomer membrane (VHB 4905, 3M Company) with a thickness of H = 0.5 mm was equi-biaxially pre-stretched at a ratio of λ_pre = 3 (the ratio of the deformed diameter to the initial diameter). The membrane was then masked by release paper with a circular aperture of about 12 mm in diameter. The SWCNT film was transferred onto the adhesive surface of the VHB 4905 by gently stamping the filter onto the membrane. Finally, the filter was removed using a few drops of ethanol. The DE membrane was fully relaxed for 0.5 hour, and then voltage was applied at a ramping rate of 20 V s⁻¹ using a power supply (Model HAR-30P1, Matsusada Precision Inc.). A digital camera was positioned above the actuator to record the area of the electrode during the electrical actuation process. To examine the microscopic morphological evolution of the electrodes during actuation, the active region was observed in situ with an optical microscope (Keynce VHX-600E).

2.2 Transmittance measurements

The optical transmittance of the DEA coated with SWCNT electrodes at 532 nm was measured using a light source and a digital light intensity lux meter (Model SK-20LX, SATO KEIRYOKI MFG. Co., Ltd). The transmittance, T, was calculated as the ratio of the luminous flux transmitted through the DEA to the incident flux, T = I/I₀.⁷² The optical transmittance at the active region was recorded in real time as the driving voltage increased in steps of 0.1 kV.

2.3 In situ observation of DEA failure

To study the effect of the pre-stretch ratio on the actuation performance of the VHB 4905 actuator, the elastomer membrane was equi-biaxially pre-stretched to various ratios (λ_pre = 1.5, 3 and 5) before being clamped onto an annular rigid frame. The SWCNT electrodes with a width of 1 mm were oriented perpendicular to each other on the opposite sides of the membrane, with an intersecting area of 1 × 1 mm², as shown in Fig. 1, with the detailed fabrication process illustrated in Fig. S1 (ESI,† Appendix). Real-time observations of the active region were conducted using a Keyence VHX-600E microscope equipped with a VH-Z100 lens, offering magnification from 100× to 1000×. Under the multi-direction illumination variation analysis mode, the incident light reached the surface of the membrane at various angles, facilitating the imaging of the uneven surface of the dielectric elastomer caused by the voltage-driven deformation.

3. Results

3.1 SWCNT film characterization

Fig. 2(a)–(d) illustrate the morphology of the SWCNT networks on the filter membrane, employing solution volumes of 80 μL, 150 μL, 250 μL, and 600 μL, respectively. It was found that the SWCNT networks were deposited homogeneously on the filter, indicating an even dispersion of carbon nanotube bundles within the SDS solution, without any noticeable aggregation. In Fig. 2(a), when the solution volume was 80 μL, the sparsely packed nanotube network failed to completely cover the filter surface. As the SWCNT concentration increases, a more densely interconnected network of tubes with proliferating conducting channels is formed, as depicted in Fig. 2(b) and (c). Additionally, the thickness of the compact SWCNT film increases as more carbon nanotubes are deposited, as exhibited in Fig. 2(d).

The sheet resistance of the SWCNT film, measured using a collinear four-probe array, is presented as a function of the filtered solution volume in Fig. 3(a). The data reveal that the sheet resistance sharply decreases as the solution volume increases from 80 μL to 300 μL. This decline is attributed to the densification of the SWCNT network through successive deposition, which increases the concentration of the conducting nanotube–nanotube contacts,⁷¹ thereby enhancing the film's conductivity. The sheet resistance of the SWCNT film levels off when the filtered solution exceeds 300 μL. It is assumed that, at this point, a compact layer of the SWCNT network is formed, and further deposition of SWCNT primarily increases the film thickness, which accounts for the slow reduction in sheet resistance.

3.2 Transmittance results of DEA

The transmittance (T) of the DEAs coated with SWCNT electrodes of varying areal densities was measured under both open-circuit and voltage-driving conditions, as shown in Fig. 3(b) and (c), respectively. Under open-circuit conditions, the transmittance of the DEA increases as the amount of deposited SWCNT decreases. The dashed plot in Fig. 3(b) was derived by fitting the data with the Beer–Lambert law, given by A = −lgT = εcl ∝ εV, where A is the absorbance, T is the transmittance, ε is the absorptivity of the materials, c is the concentration of the species, l is the optical path length and V is the filtered volume of the SWCNT solution. Upon applying a voltage, the transmittance of DEAs with the deposited SWCNT exceeding 250 μL progressively increases, which is attributed to the gradual thinning of both the elastomer membrane and the SWCNT network during the actuation process, as indicated in Fig. 3(c). However, in the case of the DEA with 150 μL of deposited SWCNT, the transmittance shows a non-monotonic relationship with the driving voltage, which will be further elucidated in the following section.

The actuation strain of the DEAs with SWCNT electrodes is plotted as a function of applied voltage in Fig. 3(d). The relative linear actuation strain is defined as ε_l = 100% × (l − l₀)/l₀, where l is the actuated length (voltage-on) and l₀ is the unactuated length (voltage-off).¹¹ For comparison, the carbon grease electrode (MG Chemicals Co., Ltd) was chosen due to its minimal stiffening effects and high conductivity throughout the actuation process. For the DEA with 250 μL of SWCNT deposition, the electromechanical response is comparable to that of the DEA with carbon grease electrodes, exhibiting a maximum actuation strain exceeding 90%. In contrast, for the cases of 100 μL and 150 μL of SWCNT deposition, the actuation plots deviate from the behavior of the carbon grease electrode, causing reduced maximum actuation strains of 57% and 67%, respectively. When the SWCNT deposition exceeds 600 μL, the stiffening effect of the SWCNT network becomes pronounced, shifting the plots towards higher voltage levels. The stiffening effect of the thick SWCNT electrodes was confirmed through a customized test bench under equi-biaxial tensile loading, the details of which are described in Fig. S2 and S3 (ESI,† Appendix).

To clarify the real-time morphological evolution of the SWCNT electrodes during the actuation process, the active region was carefully monitored at both macroscopic and microscopic scales as the driving voltage increased. Fig. 4 displays the images of DEA with 150 μL of SWCNT. Fig. 4(a)–(c) present the overall appearance of DEA at voltages of 0 V, 2850 V and 3700 V, respectively, while Fig. 4(d)–(f) exhibit the corresponding microscopic features of the active region. Initially, before applying the voltage, the transmittance of the active region is around 84%, with the SWCNT electrode film uniformly distributed across the membrane surface, as indicated in Fig. 4(a) and (d). As the driving voltage increases to 2850 V, the active region appears frosted and hazy, as shown in Fig. 4(b), which accounts for the transmittance reduction in Fig. 3(c). Microscopically, the initially homogeneous SWCNT film splits due to excessive stretching of the DE membrane, as evidenced in Fig. 4(e). The interconnected nanotube network ruptured with numerous isolated domains exposed to air, thereby diminishing the charge storage capacity of the electrodes. As the voltage further increased, the electromechanical performance continually deteriorated as the cracking area expanded, as shown in Fig. 4(c) and (f), leading to the reduction in the actuation strain, as shown in Fig. 3(d). In contrast, for the DEA with 250 μL SWCNT deposition, the transmittance of the active region consistently increases with the driving voltage, as indicated in Fig. 5(a)–(c) at applied voltages of 0 V, 2980 V and 3500 V, respectively. Microscopic examination revealed that the SWCNT network maintained its structural integrity throughout the actuation process, as seen in Fig. 5(d)–(f). This uniform, thin and transparent SWCNT layer endows DEA with electromechanical performance on par with that of carbon grease electrodes in Fig. 3(d), making it feasible to investigate the failure mechanisms of DEAs in situ at the microscopic scale.

3.3 Electro-mechanical failure at the microscopic scale

The SWCNT electrode, fabricated with 250 μL of the solution, was employed for the in situ observation in the context of the above results. Fig. 6 shows the actuation behavior of the DEAs with an active area of 1 mm², subjected to pre-stretches of λ_pre = 1.5, 3 and 5, respectively. For λ_pre = 1.5, the membrane initially expands uniformly with an increase in the voltage, and then a localized region (denoted by arrow A) exhibits snap-through deformation, as indicated in Fig. 6(b) and (c). This domain experiences a sudden expansion at nearly constant voltage, causing abrupt thinning of the membrane, which results in electric field surging and ultimately triggers electrical breakdown. Fig. S4(a) and (b) (ESI,† Appendix) present the actuation strain plots of the local domain-A and the entire active region, respectively. For the entire active region, the actuation strain exhibits a smooth and monotonic increase as the driving voltage increases, with no discernible snap-through instability identified from the actuation strain plot, which is consistent with the experimental findings in the literature.^25,33–37 In contrast, the snap-through instability was apparent in the actuation plot for the local domain-A, as shown in Fig. S4(a) (ESI†). These observations suggest that the theoretically predicted electromechanical instability, characterized by discontinuous deformation, occurs within the micro-sized domain and may hardly be detected via conventional global observations. To confirm the reproducibility of these findings, additional tests were conducted under the same conditions, and a similar phenomenon was detected with snap-through instability occurring away from the corner of the square-shaped region, as shown in Fig. S5 (ESI,† Appendix). Therefore, the observed localized sudden thinning of the dielectric elastomer is mainly caused by EMI, rather than the stress concentration.

For the DEA under a pre-stretch of λ_pre = 3, as the actuation strain increases to approximately 110%, distinct wrinkles are formed due to the excessive planar expansion of the membrane (Fig. 6(f)), which is referred to as the loss of tension (LT) in the literature.⁴² Although the acrylic elastomer is commonly regarded as an electrical insulator, its conductivity (κ) increases with the applied electric field (E) following an exponential law, κ = κ₀exp(E/E_C), where κ₀ and E_C are material constants. Consequently, considerable charge leakage occurs when subjected to extreme stretching. As a result, the membrane elastically contracts and restores the flat morphology due to the reduction in Maxwell stress (Fig. 6(g)), followed immediately by the final electrical breakdown. The corresponding actuation strain plot is shown in Fig. S4(c) (ESI,† Appendix).

For the DEA under a λ_pre = 5, the maximum actuation strain is relatively low due to the pronounced stiffening of the acrylic elastomer under large deformation, as indicated in Fig. S4(d) (ESI,† Appendix). Despite a slight increase in the roughness of the membrane at high voltage, the overall morphology of the membrane remains flat throughout the actuation process, as shown in Fig. 6(i)–(l).

4. Discussion

4.1 Mechanisms for localized electromechanical instability

Considering a homogeneous DE membrane subjected to an external force and voltage, with the initial dimensions of L₁ (length), L₂ (width) and H (thickness), the membrane deforms to the dimensions of l₁ = λ₁L₁, l₂ = λ₂L₂ and h = λ₃H due to the combination of mechanical pre-stretch and electrical Maxwell stress. Assuming an ideal dielectric elastomer model, where the material exhibits a purely elastic response and is incompressible (λ₁λ₂λ₃ = 1), the permittivity is independent of deformation. The Helmholtz free energy density of this system can be given as a function of the stretch ratios, λ₁ and λ₂²⁷:

	(1)

where W_s is the elastically stored strain energy density for the elastomer, P₁ and P₂ are the pre-stretched forces in the plane, ε₀ is the free space permittivity, ε_r is the relative dielectric constant (typical value of 3.9 for VHB 4905²⁵) and Φ is the driving voltage. For equi-biaxial tensile loading, where P₁ = P₂ = P and λ₁ = λ₂ = λ, the thermodynamic principle dictates that an equilibrium state should minimize F, i.e., ∂F/∂λ = 0, and accordingly, the relationship between the stretch λ and the applied voltage Φ can be obtained as

	(2)

where the two terms on the left denote the pre-stretched and Maxwell stresses, respectively, and the right term is the elastic recovery stress of the DE membrane. From this point of view, eqn (2) describes the force balance within the system per se.

In this study, the Gent hyperelastic model was employed to characterize the VHB 4905 elastomer under equi-biaxial tensile loading:⁷³

	(3)

where μ represents the shear modulus in the small strain range, and J_m denotes a dimensionless parameter that defines the limiting stretch when the polymer chains approach their contour lengths under external loading. The typical values of μ = 40 kPa and J_m = 125 are selected for the commonly used VHB 4905 acrylic elastomer.^25,27

For a defect-free DEA under a pre-stretch of λ_pre = 1.5, the relationship between stretches and the applied voltage was computed based on eqn (1)–(3) and is presented in Fig. 7(a). It is observed that the stretches monotonically increase with the driving voltage at the initial stage (Φ < Φ_A), indicating that the membrane reaches an equilibrium state characterized by a specific stretch value at a given voltage. However, as the voltage increases to the values between Φ_A and Φ_E, three distinct equilibrium stretches are determined on the plot at a certain voltage. Once the driving voltage exceeds Φ_E, the equilibrium stretch abruptly jumps to E′, signifying the occurrence of electromechanical instability, given by dΦ(λ)/dλ = 0,⁷⁴ where Φ_E is the critical voltage for the onset of EMI.^33,75,76

The DEA eventually undergoes electrical breakdown under an excessively high electric field, which can be expressed as

Φ = EEBHλ−2	(4)

where E_EB is the electrical breakdown field, which is dependent on the stretch of the membrane and can be empirically calculated as E_EB = E₀λ^1.13,⁷⁴ where E₀ is the electrical breakdown field of the unstretched elastomer and is approximately 30.6 MV m⁻¹ for the VHB 4905. Consequently, the operational limit can be determined using eqn (4) and is presented on the voltage–stretch plane in Fig. 7(a). The snap-through instability results in a sharp increase in the stretch at a critical voltage Φ_E, ultimately intersecting with the electrical breakdown limit, as indicated by asterisks in Fig. 7(a). It is noteworthy that for a given pre-stretch ratio, the critical voltage for EMI increases linearly with the film thickness, H, while the critical stretch at EMI remains constant.

Although the dielectric elastomer membrane is conventionally considered to be structurally homogeneous, with uniform mechanical and electrical properties across all locations, the presence of impurities during the fabrication process is inevitable, such as microscopic air voids and conductive particles.^27,50 In this study, it is assumed that the former defect modifies the mechanical properties of the membrane by reducing the local elastic modulus, whereas the latter defect alters the electrical properties of the elastomer by increasing the local permittivity.

When considering a small domain containing microscopic air voids, it is reasonable to assume that both the small-strain shear modulus (μ) and ultimate limiting stretch (J_m) of this local region would decrease slightly. In this study, these values are assumed to be 5% of those of the defect-free matrix. To simplify the analysis, the influence of air voids on the electrical properties of the elastomer is neglected. The actuation plot of this void-containing domain was calculated and is presented in Fig. 7(a). Initially, the plot for the void-containing domain closely aligns with that of the defect-free matrix. However, when the applied voltage exceeds 4000 V, a noticeable bifurcation emerges, indicating a marginally higher actuation strain for the void-containing domain at a given voltage. Notably, the critical voltage for snap-through instability shifts to a lower level compared to that of the matrix, transitioning from Φ_E to Φ_EV, which suggests that the void-containing domain is more susceptible to sudden EMI than the defect-free matrix. This finding corroborates the phenomenon depicted in Fig. 6, wherein a burst of deformation occurs in a highly localized domain, while the matrix sustains continuous extension. Although the assumption of a 5% reduction in μ and J_m is somewhat arbitrary, it can be adjusted to other small values, such as 3%, without altering the main conclusion, as illustrated in Fig. S6 (ESI,† Appendix).

In addition to microscopic air voids, the elastomer membrane may also contain conductive particles, leading to an increase in the dielectric constant within localized regions. For simplicity, it is hypothesized that micro-sized conductive particles have a negligible impact on the mechanical properties of the elastomer. Fig. 7(b) illustrates the calculated voltage–stretch curve for the particle-containing domain, assuming a 5% increase in the relative dielectric constant vis-à-vis the surrounding matrix. The presence of conductive particles induces a bifurcation in the curves between the particle-containing domain and the surrounding matrix, akin to the effect caused by microscopic air voids, with the critical voltage for EMI shifting towards a lower value Φ_EP. In the context of the above analysis, both the microscopic air void and conductive particles contribute to EMI occurring in the defective regions before it occurs in the defect-free matrix.

A similar analysis was implemented on the DEAs with pre-stretch ratios of λ_pre = 3 and λ_pre = 5, and the voltage–stretch curves for both the defect-free matrix and defective domain are presented in Fig. 7(c) and (d), respectively. It is evident that the stretch ratio monotonically increases with the applied voltage, indicating the elimination of snap-through instability. Although there are slight disparities in the actuation strain between the defective domain and matrix, they are insufficient to trigger instability in the defective region. Thus, the employment of a high pre-stretch not only suppresses electromechanical instability but also mitigates the adverse impacts of foreign impurities on the homogeneous deformation of DEA.

Achieving precise equations to describe the electric-mechanical coupling around imperfections remains challenging due to the complex variations in the stress and electrical fields, especially under large nonlinear deformations, as illustrated by the finite element analysis (FEA) results shown in Fig. S7–S9 (ESI,† Appendix). Although the detailed distribution of the stress and electric field was not fully considered in the above analysis, the results obtained were reasonable and did not impact the main conclusion.

4.2 Electromechanical phase transition

The influence of micro-sized defects on the electromechanical instability of DEA can also be analyzed through the lens of phase transition theory. For λ_pre = 1.5, the free energy surface is calculated using eqn (1) and presented as a function of the stretch (λ) and applied voltage (Φ) in Fig. 8(a). It was found that more than one extreme value exists on the free energy surface. At low voltages, e.g., Φ_A = 3860 V, the free energy density demonstrates a single minimum as a function of stretch λ in Fig. 8(c), representing a thermodynamically stable state, as illustrated by point A1 in Fig. 8(b). As the applied voltage increases to Φ_B = 3980 V, the free energy plot exhibits two local minima (B1 and B3) and one local maximum (B2) in Fig. 8(d), which are also highlighted in the voltage–stretch curve in Fig. 8(b). The local maximum point (B2) with intermediate stretch is unstable against small perturbations, while the other two states, B1 and B3, are stable against small perturbations, which are termed as “thick-phase” with a smaller stretch and “thin-phase” with a larger stretch, respectively. As the free energy of the “thick-phase” is lower than that of the “thin-phase” at Φ_B, the former retains its stability while the latter is in a metastable state, making phase transition unlikely. As the applied voltage reaches Φ_C = 4066 V, the free energies of the “thick-phase” and “thin-phase” become equal, indicating the co-existence of two states, with Φ_C referred to as the transition voltage, as illustrated in Fig. 8(e). With further voltage increase toΦ_D, the free energy of the “thin-phase” drops below that of the “thick-phase”, implying that the “thick-to-thin” phase transition is thermodynamically possible at this stage. However, such a phase transition requires overcoming the energy barrier (ΔU), as depicted in Fig. 8(f). Hence, a carefully manufactured DE membrane, free of defects, would remain in the “thick-phase” state until it is overcharged to a voltage of Φ_E, where the energy barrier vanishes and only one minimum exists in the free energy plot, as illustrated in Fig. 8(g). At this critical point, the DEA undergoes a snap-through deformation, transitioning from the “thick-phase” to the energetically stable “thin-phase”.

At the molecular level, distinct configurations can be detected between the “thick-phase” and “thin-phase”. When a voltage is applied, the dipole moments of the polar groups tend to align with the direction of the electric field. During electromechanical instability, the abrupt thinning of the elastomer causes a local surge in the electric field, resulting in a more uniform orientation of polar groups within the “thin-phase” compared to the surrounding “thick-phase”. Additionally, the polymer chains in the “thin-phase” are stretched to a greater length due to the abrupt planar expansion, as schematically shown in Fig. S10 (ESI,† Appendix).

The essence of snap-through instability is that more than one extreme value exists on the free energy surface, resulting in two equilibrium states characterized by different stretches at a certain voltage Φ. Therefore, for a dielectric elastomer material (not limited to VHB 4905), as long as its mechanical parameter W_s and dielectric permittivity (ε_r) give rise to more than one extreme value of F in eqn (1), snap-through instability will occur during the actuation process. Analysis of the free energy surfaces for dielectric elastomer actuators (DEAs) with pre-stretch ratios of λ_pre = 3 and λ_pre = 5 revealed only a single minimum throughout the entire operational voltage range, as depicted in Fig. S11 (ESI,† Appendix), suggesting the absence of any phase transitions during the actuation process.

The above process is reminiscent of homogeneous nucleation in the water–ice phase transition during cooling, with the “thick-phase” and “thin-phase” corresponding to the water and ice phases, respectively. At a temperature of 0 °C, the free energies of water and ice are equal, analogous to the transition voltage of Φ_C in Fig. 8(e). Actually, crystallization rarely occurs spontaneously at temperatures just below 0 °C, despite ice being more energetically stable than water. This is because the liquid must overcome an energy barrier to initiate nucleation, counteracting the increase of the surface free energy due to the formation of a solid–liquid phase boundary during solidification, analogous to the behavior in Fig. 8(f). For pure impurity-free water, a higher activation free energy is required to generate a stable nucleus in the interior of the liquid in the form of atom clusters. This homogeneous nucleation mechanism allows pure water to be supercooled to a temperature well below the equilibrium solidification temperature, even below −41 °C,⁷⁷ analogous to a perfect (defect-free) DEA overcharged to the critical voltage of Φ_E before undergoing phase transition. However, for water containing a mass of impurities, the energy barrier for heterogeneous nucleation is lowered as the nuclei form on preexisting interfaces, enabling the water to readily undergo solidification at temperatures slightly below 0 °C. Similarly, the flaws in the DE membrane, such as microscale air voids and conductive particles, act as preferential nucleation sites that significantly reduce the energy barrier for the “thick-to-thin” phase transition, from ΔU to ΔU_v, as shown in Fig. 8(h). Consequently, the defective domain preferentially undergoes localized phase transition prior to the rest of the DE membrane. Particularly, when the critical point of EMI at the defective domain (Φ_EV or Φ_EP in Fig. 7(a) and (b)) decreases to the transition voltage of the matrix (Φ_C in Fig. 8(b)), the DEA will experience a phase transition (i.e., EMI) just above the transition voltage, Φ_C, analogous to the crystallization of impure water at temperatures slightly below 0 °C.

5. Conclusions

In this work, an SWCNT electrode with exceptional optical and electrical properties was fabricated, and its microscopic morphology was monitored in real time throughout the voltage-driven actuation process. Both excessively thin and overly thick SWCNT electrodes degrade the electromechanical performance of DEA, which is attributed to microscopic cracking in the thin electrodes and increased stiffness in the thick electrodes. The impact of pre-stretching on the failure mechanism of the DEA was studied in situ utilizing optimized SWCNT electrodes. Under low pre-stretch conditions, snap-through instability occurs in a highly localized domain, which is ascribed to the heterogeneity of the elastomer materials, such as microscopic air voids or conductive particles that act as nucleation sites for the new phase, thereby reducing the energy barrier for phase transition and shifting the critical voltage of snap-through instability to a lower level. Our findings clarify the discrepancy between the theoretically predicted abrupt deformation and the experimentally observed continuous expansion of DEAs under low pre-stretch conditions. This research highlights the critical role of material purity in the electromechanical response of DEAs, especially those operating under low pre-stretch conditions.

Author contributions

Zhen-Qiang Song: conceptualization, investigation, writing – original draft, and funding acquisition. Li-Min Wang: writing – review & editing. Yongri Liang: methodology and writing – review & editing. Xiao-Dong Wang: validation and formal analysis. Shijie Zhu: writing – review & editing, supervision, and funding acquisition.

Data availability

The data supporting this article have been included as part of the ESI.†

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 52201191 and the Hebei Natural Science Foundation under Grant No. E2022203108, and funded by the Science and Technology Project of Hebei Education Department under Grant No. QN2022055, and the Innovation Ability Promotion Program of Hebei under Grant No. 22567609H.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4sm00596a

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