# Theory of dielectric elastomers capable of giant deformation of actuation

Entry by Jianyu Li, AP225, Fall, 2010.

## Contents

## Summary

A dielectric elastomer is deformed under a electric field, reducing thickness and expanding area. Its mechanism is mainly divided into two modes: for the stiff dielectric, the deformation is limited by electrical breakdown; for a compliant dielectric, electromechanical instability occurs. Various strains of actuation are reported, ranging from <1% to >100%, but what is the fundamental limit? It is what this paper reports, giant deformation of actuation beyond 100% is possible in theory.

## Deformation Mode

Type I: a stiff dielectric limited by electrical breakdown when a path of electrical conduction is formed inside.

Type II: a compliant dielectric limited by electromechanical instability when the thickness is too thin.

Type III; suggested in this paper, the dielectric averts electromechanical instability, obtains a giant deformation.

The paper further reports several designs of type III material, for example, biological tissues such as skins and vascular walls which is consisted of stiffer fibers in a compliant matrix. For a synthetic elastomer, adjusting the cross-links density doesn't work. The stress-stretch curve is available through using polymers with side chains, swelling the network with a solvent, forming interpenetrating networks, etc. Schematic plots of their structure are showed in Fig.1.

## Principle of Dielectric Elastomer Transducers

This paper use the model of ideal dielectric elastomer. Start with some basic equations: the electric field is <math>E=\lambda^2\Phi/H</math>; the actuation stress is <math>\sigma=\epsilon E^2</math>; combine them, we can derive the following equation:

<math>\Phi=H\lambda^{-2}\sqrt{\sigma(\lambda)/\epsilon}</math>

where H is thickness, <math>\Phi</math> the voltage, <math>\lambda</math> the stretch, E the electric field and <math>\sigma</math> is the Maxwell stress, <math>\epsilon</math> is the permittivity of the elastomer. The formula can perfectly describe the change of the voltage as a function of stretch.

Consider the limit of deformation, <math>\phi_B(\lambda)</math> is introduced here, which equals to <math>E_B*H*\lambda^{-2}</math>.Obviously, the breakdown voltage proportional to the inverse square of the stretch, as illustrated in Fig.2. Then, the paper claims to differentiate three types of dielectric dependent on the intersect of <math>\Phi(\lambda)</math> and <math>\phi_B(\lambda)</math>.For type I,electrical breakdown occurs before electromechanical instability, leading to small deformation; For type II, dielectric breaks down during thinning. Type III dielectric is capable of keeping a stable state before breakdown, as illustrated in Fig. 2(d).

## Application

Apply the theory to experimental observation, the stress-stretch curves for VHB and VHB-TMPPMA interpenetration networks, as showed in Fig. 3(a).

According to theory, VHB is a type II dielectric, while VHB-TMPPMA is a type III dielectric which maximum stretch is about 2. The authors also consider a network of polymers swollen with a solvent, as showed in Fig. 1(e). First assume the elastomer is incompressible, <math>\lambda_1\lambda_2\lambda_3=1</math>; if the swelling volume is <math>\alpha^3</math>, so the stretch of each chain in the presence of the solvent is represented by

<math>\Lambda=\alpha(\lambda^2_1+\lambda^2_2+\lambda^2_3)^{1/2}/\sqrt{3}</math>.

The stretch is related to the normalized force <math>\zeta</math> by,

<math>\Lambda=\sqrt{n}(\zeta/tanh\zeta-1)/\zeta</math>.

The free energy is expressed as

<math>W=kT/\nu\alpha^3(\zeta/tanh\zeta+log(\zeta/sinh\zeta))</math>.

The stress-stretch curve is followed

<math>\sigma(\lambda)=\frac{kT\zeta(\lambda^2-\lambda^{-4})}{3\alpha\nu\sqrt{n}\Lambda}</math>.

Fig.4 plots the curve of <math>\phi(V)</math> and <math>\lambda</math>, using the values <math>E_B=2*10^8 V/m, T=300K, \nu=10^{-27}m^3, \epsilon=4*10^-11F/m and H=1mm</math>. For <math>\alpha=1</math>, no swelling, the elastomer belongs to type II. When the volume swells to five times, it becomes type III, the stretch could be 5.5 when <math>n=500</math>.

## Conclusion

Simple theoretical analysis shows that giant deformation of actuation is available, the dielectric could eliminate the snap-through electromechanical instbility before breakdonw. It is valuable for searching dielectric elastomer capable of giant deformation of actuation.