Avoiding the pull-in instability of a dielectric elastomer film and the potential for increased actuation and energy harvesting

Abstract

Pull-in instability often occurs when a film of a dielectric elastomer is subjected to an electric field. In this work, we concoct a set of simple, experimentally implementable, conditions that render the dielectric elastomer film impervious to pull-in instability for all practical loading conditions. We show that a uniaxially pre-stretched film has a significantly large actuation stretch in the direction perpendicular to the pre-stretch and find that the maximal specific energy of a dielectric elastomer generator can be increased from 6.3 J g⁻¹ to 8.3 J g⁻¹ by avoiding the pull-in instability.

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Soft dielectrics are capable of achieving significantly large deformations and find application in humanlike robots,^1,2 stretchable electronics,³ actuators,^4–6 and energy harvesters^7–11 among others.^12,13 However, soft dielectrics under applied electric fields are also vulnerable to various types of electromechanical instabilities.^14–19 Instabilities are often thought to be detrimental to material and device functionality and often avoided by design. Recent works have, however, focused on harnessing instabilities for various applications such as artificial muscles,^20,21 dynamic surface patterning,^22,23 giant voltage-triggered deformation,^24,25 and energy harvesting.^26,27

A commonly used actuator is a dielectric thin film coated with two compliant electrodes. Upon application of a potential difference between the two electrodes, the dielectric film thins down in the thickness direction and expands laterally. When the thickness decreases to a certain threshold, the film is unable to sustain the electric field and the so-called pull-in instability occurs.^{14–16,26,28} To avoid failure and to enhance the actuation strain and the harvested electrical energy density, pull-in instability is often suppressed by using a pre-stress,²⁸ materials that exhibit load-dependent stiffening^29,30 and charge-controlled operation.^6,31 Moreover, pull-in instability can also be delayed or eliminated by pre-stretch.^32–34 In this work we analyze how properly chosen (and experimentally realizable) boundary conditions can be exploited to avoid or delay pull-in instability. We show that the pull-in instability can be summarily avoided by a judicious combination of dead-loads and controlled-displacement boundary conditions which renders the pertinent Hessian matrix of the equilibrium state positive-definite under all practical conditions. By prevention of pull-in instability and through exploitation of the competition between various pertinent factors such as electromechanical loading conditions, consideration of buckling, and pre-stretch, among others, we theoretically highlight a significantly increased ability for energy harvesting and actuation. However, realizing the theoretical maximum energy conversion experimentally has still remained elusive and the reduction of the disparity between experiments and theoretical predictions appears to be a challenging task.^7–10

Consider an elastic dielectric film with dimensions (L₁, L₂, L₃) in its undeformed state. Subjected to the voltage Φ and the dead load P₂ at a prescribed stretch λ₁, the incompressible film deforms to a homogeneous state with stretches λ₁, λ₂, and λ₃ = λ₁⁻¹λ₂⁻¹. Moreover, the film gains a magnitude of total charge Q that distributes uniformly on either side of the compliant electrodes. The homogeneously deformed film is shown in Fig. 1. We remark that the loading device actually controls the normal displacement (or the stretch λ₁) only on the two sides of the film in the X₁ direction. Since the deformation is homogeneous under these circumstances, the entire film exhibits prescribed stretch λ₁.

Fig. 1 Schematic diagram of a deformed film of a dielectric elastomer. The film is extended/compressed between two well-lubricated (with rollers), rigid plates by means of a controlled displacement (or the stretch λ₁) in the X₁ direction. A dead load P₂ is applied in the X₂ direction and a voltage Φ is applied in the thickness direction of the film that is coated with two compliant electrodes. The loads deform the film from L₁, L₂, and L₃ to λ₁L₁, λ₂L₂, and λ₃L₃, as well as induce an electric charge of magnitude Q on either electrode.

The nominal electric field and the nominal electric displacement are defined as Ẽ = Φ/L₃ and D̃ = Q/L₁L₂, respectively. The true electric field and the electric displacement are defined as E = Φ/(λ₃L₃) and D = λ₃Q/(L₁L₂). Furthermore, the nominal and true stresses from the dead load P₂ are denoted, respectively, as s₂ = P₂/(L₁L₃) and σ₂ = λ₂P₂/(L₁L₃).

The free energy of the system is^28,35

G = L₁L₂L₃W(λ₁,λ₂,D̃) − P₂λ₂L₂ − ΦQ. (1)

We remark that the stretch λ₁, the dead load P₂, and the voltage Φ in eqn (1) are prescribed parameters. In contrast to Yang et al.³⁶ and Dorfmann and Ogden³⁷ the dielectric film in this work admits only a class of homogeneous deformations. Thus, the general coordinate λ₁ has a zero variation δλ₁ = 0 for any homogeneous perturbation. When other two generalized coordinates λ₂ and D̃ vary by small amounts δλ₂ and δD̃, the free energy of the system varies by

where only the first and second variations are retained and all the high-order terms are omitted. At equilibrium, the first variation vanishes, and yields the following equilibrium equations:where the nominal stress s₂ = P₂/(L₁L₃) and the nominal electric field Ẽ = Φ/L₃ are prescribed parameters. In contrast, the nominal stress s₁ is defined as which is no longer prescribed but depends on λ₂ and D̃ as well as λ₁. Indeed, the partial derivative in eqn (4) is to be understood as the partial derivative of W with respect to λ₁ at the pre-stretch λ₁. It is exactly the coefficient of the zero δλ₁ in the first variation of the energy. Thus, it has been omitted in the variation form for simplicity. In equilibrium, from Cauchy’s stress theorem, s₁ in eqn (4) is equal to the stress generated by the force applied on the left and right surfaces, and the magnitude of the force is s₁L₂L₃.

From the principle of minimum energy, the stability of the equilibrium solution requires a positive-definite second variation in eqn (2) for arbitrary δλ₂ and δD̃, that is, the Hessian matrix

must be positive definite for the equilibrium solution.

Consider an ideal dielectric elastomer with the free energy function,^28,35

where μ is the small-strain shear modulus and ε is the permittivity. The first and second terms on the right-hand side of eqn (6) are the elastic and the dielectric energy. Then the equilibrium condition [eqn (3)] and the nominal stress s₁ in eqn (4) becomeand

Eqn (7) contains two algebraic equations with two variables λ₂ and D̃ and three prescribed parameters λ₁, s₂, and Ẽ. A quartic equation in terms of λ₂ can be obtained by eliminating D̃, such that

(μ − λ₁²εẼ²)λ₂⁴ − s₂λ₂³ − μλ₁⁻² = 0, (9)

which has only one positive real root of λ₂ if and only ifand the real root of λ₂ in eqn (9) is a real number between max {s₂(μ − λ₁²εẼ²)⁻¹, μ^1/4λ₁^−1/2(μ − λ₁²εẼ²)^−1/4} and their sum. Hence the critical nominal electric field is given bybelow which the equilibrium solution can exist. We note that under the condition [eqn (10)] with prescribed parameters λ₁ > 0, s₂ ≥ 0, and Ẽ ≥ 0, eqn (7) exhibits positive solutions for λ₂ and D̃. Otherwise, eqn (7) has no solution for realistic physical situations that admit positive λ₂ and nonnegative D̃.

The electromechanical stability of the electrostatic system directly relates to the property of the Hessian matrix [eqn (5)] that, at equilibrium [eqn (7)], is given by

The 1 × 1 principal minors of the Hessian matrix [eqn (12)] are the diagonal entries H₁₁ > 0 and H₂₂ > 0, and the only 2 × 2 principal minor is the determinant of the Hessian matrix [eqn (12)], which is always positive, namely

due to the fact that s₂ = P₂/(L₁L₃) ≥ 0. Since all principal minors are positive, the Hessian matrix H [eqn (12)] is always positive-definite. A positive-definite Hessian matrix ensures the stability of the homogeneous deformation in equilibrium, because the dielectric film in equilibrium is in a state of minimum free-energy. Thus the pull-in instability never appears in this homogeneously deformed system loaded with a prescribed stretch, a dead load and an electric voltage under the condition [eqn (10)]. We emphasize that the stability is under the condition that the film is not buckled in the X₁ direction—we will return to this point later in the communication.

To further understand the behavior of the dielectric film in equilibrium, two special cases are discussed. Case I is a dielectric film at a prescribed stretch λ₁ = 1 under an electric field and several dead loads s₂, while case II is a dielectric film at a zero dead load s₂ = 0 under an electric field and several prescribed stretches λ₁.

In Fig. 2, we plot the behavior of the dielectric film for case I, i.e. λ₁ = 1. Fig. 2(a) shows that the nominal electric field increases monotonically with the increase of the nominal electric displacement. The nominal electric field is bounded by eqn (11) and the critical nominal electric field for the nonexistence of equilibrium solutions at λ₁ = 1 is . With a constant nominal electric field but an increase of the dead load s₂, the nominal electric displacement increases in Fig. 2(a), since a dead load leads to a reduced thickness but a larger area, thus resulting in a larger capacity (and charge). Fig. 2(b) shows the relation of the nominal electric displacement and the true electric field under several dead loads s₂.

Fig. 2 Behavior of the neo-Hookean dielectric film at λ₁ = 1 under various loads, s₂/μ: (a) nominal electric displacement vs. nominal electric field, (b) nominal electric displacement vs. true electric field, (c) stretch λ₂vs. nominal electric field, (d) actuation stretch λ₂/λ_2pvs. nominal electric field, (e) true stress vs. nominal electric field, and (f) true stress vs. true electric field.

In Fig. 2(c), the variation of the stretch λ₂ is plotted. Both the nominal electric field and the dead load s₂ can increase the stretch λ₂. At the critical value , the stretch λ₂ increases to infinity and the thickness (λ₃ = λ₂⁻¹) decreases to zero, which, of course, is impossible in reality since prior to such a blowup, electric breakdown will ensue due to the large true electric field or, alternatively, mechanical rupture will take place. The actuation stretch is defined as λ₂/λ_2p, where λ_2p is the pre-stretch that exists due to the prescribed stretch λ₁ and the dead load s₂. At a prescribed stretch λ₁ = 1, the actuation stretch λ₂/λ_2p under several dead loads s₂ is shown in Fig. 2(d).

The true stress σ₁ = λ₁s₁ shown in Fig. 2(e) is directly related to electrical buckling and will be analyzed in the following. At a zero dead load in Fig. 2(e), the true stress is σ₁/μ = 1 − (1 − εẼ²/μ)^−1/2, and the nominal electric field induces a compressive state in the film, i.e. σ₁ < 0, and the magnitude |σ₁| increases monotonically with the increase of the nominal electric field. On the other hand, at a zero electric field, a dead load s₂ expands the film (λ₂ > 1) and induces a tensile state i.e. σ₁/μ = 1 − λ₂⁻² > 0 in Fig. 2(e). Interestingly, there exists a competition between the electric field and the dead load due to their opposite effects on the stress σ₁. At a low electric field, the dead load makes the dielectric film extend within the plane. When the electric field increases, the stress σ₁ gradually decreases from a tensile stress (σ₁ > 0) to a compressive one (σ₁ < 0). Without considering electric breakdown (under a high true electric field) and rupture by stretch (at a high stretch λ₂), the continually increasing |σ₁| of the compressive stress will finally make the dielectric film buckle.

Fig. 3 plots the behavior of the dielectric film for case II, i.e. s₂ = 0. We note that eqn (11) gives the limit of the nominal electric field, for example, for a prescribed stretch λ₁ = 1, while it is 0.2 for λ₁ = 5 in Fig. 3(a), (c) and (e). The increase of the stretch λ₁ in Fig. 3(a) increases the film surface area, leading to higher capacity and gain of additional charge. In other words, at a prescribed nominal electric field below , a larger stretch λ₁ corresponds to a higher nominal electric displacement. Fig. 3(b) shows the corresponding relation between the true electric field and the nominal electric displacement.

Fig. 3 Behavior of a neo-Hookean dielectric film at s₂ = 0 under various stretches λ₁: (a) nominal electric displacement vs. nominal electric field, (b) nominal electric displacement vs. true electric field, (c) stretch λ₂vs. nominal electric field, (d) actuation stretch λ₂/λ_2pvs. nominal electric field, (e) true stress vs. nominal electric field, and (f) true stress vs. true electric field.

In Fig. 3(c), the increase of the stretch λ₁ in the length direction will make the film decrease its width (or the pre-stretch λ_2p at a zero electric field), pre-stretch but an electric field, on the other hand, will tend to make the film expand in-plane and exhibit a larger stretch (λ₂). At s₂ = 0, the actuation stretch λ₂/λ_2p under several prescribed stretches λ₁ is shown in Fig. 3(d).

The true stress from eqn (7) and (8) at s₂ = 0 is obtained as σ₁/μ = λ₁s₁/μ = λ₁² − λ₁⁻¹(1 − λ₁²εẼ²/μ)^−1/2. Without the electric field, the true stress σ₁/μ is λ₁² − λ₁⁻¹. When the electric field increases from zero, for example, at a prescribed stretch λ₁ > 1 in Fig. 3(e) and (f), the true stress σ₁ will decrease from a tensile stress (σ₁ > 0) to a compressive one (σ₁ < 0).

It is well-known that a thin film subjected to a lateral compression is easy to buckle. In our model, the deformation in the X₁ direction is controlled by two well-lubricated plates, and a compressive stress (σ₁ < 0) in the film can occur under some conditions (see Fig. 2(e), f, 3e and f for example). In the following, we will discuss electric buckling of a dielectric film subjected to electromechanical loads.

The special case, loss of tension, is of interest because it is a turning point for the compression–tension behavior of the dielectric film. The compressive stress can make the film buckle and should be avoided.^5,7,36,37 When the nominal stress s₁ in eqn (8) becomes zero, it is in the so-called state of loss of tension. From the equilibrium equations [eqn (7)], the nominal electric field at loss of tension is

where due to s₂ = P₂/(L₁L₃) ≥ 0.

At the state of loss of tension, if we, for example, continue to prescribe the stretch λ₁ and the dead load s₂ but increase the nominal electric field, the stress s₁ in eqn (8) will decrease from zero to negative and the film will be in a state of compression. Inspired by Euler’s buckling of a long, slender, ideal column, we analyze here the electrical buckling of a dielectric film. Euler’s formula for the buckling of a column with two fixed end supports is

where F^c is the critical compressive force, E^e is the effective elastic modulus, I^e is the area moment of inertia of the cross section, and L₁ is the length of the column. For a film of an incompressible neo-Hookean dielectric with shear modulus μ under the small-deformation assumption, the effective modulus is E^e = 3μ and I^e = L₃³L₂/12. Thus the critical compressive stress f^c from eqn (15) is

It is assumed that the dielectric film begins to buckle when the magnitude of the compressive stress σ₁ = λ₁s₁ < 0 in eqn (8) reaches f^c in eqn (16). Together with the equilibrium equations [eqn (7)], the critical nominal electric field for the electrical buckling can be expressed as

where and f̄^c = π²/(L₁/L₃)². Compared with the length (L₁) and the width (L₂) of the film, the thickness (L₃) is often much smaller. Then the buckling stress f̄^c in eqn (16) is very small, for example, f̄^c < 0.1 for a film with an aspect ratio L₁/L₃ > 10. Therefore, the critical nominal electric field in eqn (17) for buckling is very close to that in eqn (14) for the loss of tension.

Fig. 4 shows the critical electric fields for the loss of tension [eqn (14)] and electrical buckling [eqn (17)] in which the aspect ratio is chosen to be L₁/L₃ = 10. The difference between the two critical nominal electric fields is negligible at a stretch λ₁ > 1, while it is minor at a stretch λ₁ < 1. Moreover, the critical electric field for the loss of tension is always below that of electrical buckling. Therefore, there is no electrical buckling when a loss of tension occurs in the film, but the film will buckle if the electric field increases.

Fig. 4 Comparison of the loss of tension (dashed lines) and electrical buckling (solid lines) under several dead loads s₂/μ. Effects of the stretch λ₁ on (a) the critical nominal electric field and (b) the critical true electric field.

For each curve in Fig. 4(a), there exists a peak that corresponds to the maximum of the critical nominal electric field. At that peak, any infinitesimal variation of the stretch λ₁ will make the film buckle; however, the decrease of the nominal electric field will avoid the electrical buckling. On the other hand, if a point on the buckling curve is on the left-hand side of the peak, the increase of the stretch λ₁ will avoid the electrical buckling, while a point is on the right-hand side of the peak, the electrical buckling can be avoided by a decrease of the stretch λ₁. The corresponding relation between the critical true electric field and the stretch λ₁ is shown in Fig. 4(b).

A high actuation strain for an actuator driven by an electric field is desirable. Without an electric field, the pre-stretch λ_2p due to the prescribed stretch λ₁ and the dead load s₂ can be determined by eqn (7) with Ẽ = 0. The effects of s₂ and λ₁ on the pre-stretch λ_2p are shown in Fig. 5(a). The pre-stretch is λ_2p = λ₁^−1/2 at s₂ = 0, while it is approximately equal to s₂/μ at a high dead load.

Fig. 5 Effects of the prescribed stretch λ₁ and the dead load s₂ on (a) the pre-stretch λ_2p and (b) the critical actuation stretch λ^c₂/λ_2p.

Subjected to the electric field, the dielectric film thins down and expands in the plane. When the electric field increases to the critical value in eqn (17), the electrical buckling occurs in the dielectric film with a critical stretch λ₂^c. The critical actuation stretch is defined as λ^c₂/λ_2p. Fig. 5(b) plots the effects of the stretch λ₁ and the dead load s₂ on the actuation stretch λ^c₂/λ_2p. It shows that when the dielectric film is subjected to a prescribed stretch λ₁, the actuation stretch in the direction normal to the prescribed stretch is significantly larger, especially in the case of a larger prescribed stretch (λ₁) and at a zero dead load (s₂ = 0). Hence the actuation stretch can be dramatically increased by a prescribed stretch. A similar observation has also reported before in the analysis of the electromechanical instability of a uniaxial pre-stressed dielectric film.²⁸

Inspired by the aforementioned discussion related to the avoidance of pull-in instability, we now show the possibility of increasing the capacity of the energy conversion of a dielectric elastomer generator. It is known that the usual modes of failure in a dielectric film include electrical breakdown (EB), electromechanical instability (EMI or pull-in instability), loss of tension, and rupture by stretch. The area of the cycle enclosed by these four modes of failure is exactly the maximal energy that can be converted in a dielectric film subjected to equal biaxial in-plane forces.⁷ With the same dielectric film but mechanical boundary conditions suggested in this work, the pull-in instability can be avoided and then the four modes of failure reduce to three. This reduction admits the possibility of enhanced energy conversion. In the following, we will show that not only the maximal energy of a dielectric elastomer generator but also the specific energy enclosed by a rectangular in the voltage–charge plane and the amplification of voltage (ratio of the input voltage to the output voltage) can be increased significantly.

In a previous work,⁷ the dielectric film is subjected to equal biaxial in-plane forces and voltage in the thickness direction, such that the equal nominal stresses s₁ = s₂ and the equal stretches λ₁ = λ₂ at equilibrium. It should be noted that there is no difference between the forms of the equilibrium equations in the work⁷ and in this paper, but the difference is the control parameters. Unlike the equal biaxial in-plane forces,⁷ this paper takes a prescribed stretch λ₁ and a dead load s₂ as control parameters. Since the equilibrium equations have the same forms, the equilibrium solutions of a film subjected to equal biaxial in-plane forces – as discussed by Koh et al.⁷ – can be achieved by choosing properly controlled parameters (λ₁, s₂) in this work; however, the pull-in instability will be avoided. This similarity makes feasible the direct use of their model in this work for the illustration of enhanced energy conversion by the proposed avoidance of the pull-in instability.

To make this communication self-contained, we briefly review the four modes of failure when subjected to equal nominal stresses s = s₁ = s₂ and equal stretches λ = λ₁ = λ₂ in equilibrium [eqn (7)]. First, the curve under a zero electric field (E = 0) in Fig. 6(a) is determined by eqn (7a) with D̃ = 0, while it is the origin in Fig. 6(b). Next, the electrical breakdown (EB) curve is governed by eqn (7) with Ẽ = E_EBλ⁻², where E_EB = 3 × 10⁸ V m⁻¹ is the critical true electric field when EB occurs.^7,38 Other material parameters used in the numerical calculations are μ = 10⁶ N m⁻² and ε = 3.54 × 10⁻¹¹ F m⁻¹ as well as the mass density ρ = 1000 kg m⁻³. Then, the curve of the loss of tension is the horizontal axis in Fig. 6(a), while it is represented by eqn (7) with s = 0 in Fig. 6(b). The curve of rupture by stretch in Fig. 6(a) is the vertical line λ = λ_R while in Fig. 6(b) it is determined by eqn (7b) with λ = λ_R, where λ_R ≤ 6 when the film ruptures when subjected to equal biaxial stretch.¹⁵ Here we use λ_R = 5. Last, the curve of electromechanical instability (EMI) is based on eqn (6) and (7) in the work by Koh et al.⁷

Fig. 6 A thermodynamic state of the dielectric film is represented by (a) a point in the force–displacement plane or (b) a point in the voltage–charge plane. The equilibrium curve under the zero electric field is denoted as E = 0. Four modes of failure including EB, EMI, loss of tension s = 0, and rupture by stretch λ = λ_R are also plotted.

In Fig. 6, the shaded areas enclosed by various modes of failure (also the E = 0 curve) define the maximal energy of conversion, that is, the maximal specific energy. In the work by Koh et al.⁷ four modes of failure leads to a maximal specific energy of 6.3 J g⁻¹. In contrast, the EMI (pull-in instability) is avoided in our proposed scheme and the remaining three modes of failure admit a maximal specific energy of 8.3 J g⁻¹, increasing the capacity of the dielectric elastomer generator by nearly 33%.

The aforementioned maximal-energy cycle is idealized and may be difficult to realize in practice. The rectangular⁷ and triangular³⁹ cycles are often used for energy conversion. We refer the reader to the Koh et al.^7,39 for the detailed circuit design that pumps electric charge from a low-voltage battery to a high-voltage battery. A rectangle with vertices 1–2–3–4 is plotted in Fig. 6(b), where the energy enclosed by the rectangle is called the specific energy. The electric voltage corresponding to vertices (1,2) is the input voltage Φ_in, while the electric voltage corresponding to vertices (3,4) is the output voltage Φ_out. The specific energy and the amplification of voltage for various Φ_in are plotted in Fig. 7(a) and (b). Evidently, the avoidance of pull-in instability can enhance the ability of energy conversion of a dielectric elastomer generator by increasing the specific energy and the voltage amplification.

Fig. 7 Specific energy (a) and amplification of voltage (b) for cycles represented by various rectangles in Fig. 6(b).

In summary, in this work, we propose the avoidance of the pull-in instability of a dielectric film by introducing controlled-displacement boundary conditions, which ensure that the Hessian matrix is always positive definite in equilibrium regardless of the values assigned to the prescribed stretch, the dead load, and the electric field. The limit of the nominal electric field for the existence of equilibrium solutions is presented. We also show that the critical electric field for the loss of tension is slightly below that of electrical buckling and a uniaxial pre-stretched dielectric film can exhibit a significantly larger actuation strain in the direction perpendicular to the pre-stretch. Here we should emphasize that the film needs to be highly pre-stretched, uniaxially, to avoid loss of tension (or electrical buckling) when the electric field is high and the dead load is low. Finally, we find that the maximal specific energy that can be harvested may be increased from 6.3 J g⁻¹ to 8.3 J g⁻¹ by avoiding pull-in instability.

Acknowledgements

The authors gratefully acknowledge support from NSF CMMI Grant No. 1463205 and the M. D. Anderson Professorship.

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