Combining elasticity theory with electrostatics, we calculate the actuation response of a charged multilayer film, which is composed of consecutively stacked piezoelectric and polymeric elastomer layers, with respect to an applied voltage. Depending on the individual material properties and polarity of the applied voltage, we show that the composite multilayer can exhibit actuation properties which are considerably better than the response of the individual constituents. It is also shown that there is a window for applied voltages within which the thickness of the composite film expands, a condition that cannot be achieved in single layer films. In particular, we apply and evaluate our theory for various piezoelectric and dielectric elastomers. The results obtained will be helpful to develop tailored composite materials with specific actuation characteristics.

## I. INTRODUCTION

Active composite materials are a wide class of materials used in engineering, electronics, and transducer applications, such as hydrostatic hydrophones, acoustic sensors, medical and non-destructive testing transducers, and artificial muscles.^{1–9} The actuation of such materials is provided by an electroactive polymer (EAP) component in their structure. Generally speaking, the EAP can be classified into two main categories: ionic electroactive polymers (IAP) and field active polymers (FAP).^{3,10} IAP materials use electrically driven mass transport of ions and other charged species to obtain a shape change and vice-versa. The mostly used examples for such IAP are ionic polymers.^{3} The main disadvantages of these materials are their slow strain relaxation due to the built-up pressure gradient, the necessity of their humidification, and, as a result, a restricted temperature regime for their operation.^{3} Contrary to IAP materials, the actuation of FAP materials is based on polarization processes in the dielectrics, or, on the interaction between immobile charges and the applied field. Examples of such materials include most of dielectric elastomers^{3,10–14} and porous polypropylene (PP) film electrets.^{4,15} Because of the Coulomb nature of the interactions, the field activation in such polymers is fast, and most importantly, FAP are insensitive to temperature and humidity conditions.

Most of current FAP applications rely on using monolithic films sandwiched between metal electrodes. When polarized, a compressing Maxwellian pressure squeezes the polymer film. Thus, by changing the voltage between the electrodes, it is possible to regulate the strength of compressing force and achieve necessary deformations in the layer. Then, by assembling such monolithic Maxwell films into different structures, it is possible to enhance the actuator behavior of the multilayer. Another way to improve actuation properties of the FAP materials is based on the use of piezo-elastomer composite materials with sharp dielectric discontinuities.^{3,14,16–19} Here, the film actuation gets additional enhancement from the interface polarization under applied voltages. The actuation can be further boosted if the interface is deliberately charged. Such charging has been employed in electrets, for example, in highly insulating polymer foam films, such as biaxially stretched PP foams.^{14,20} In this case, additional charge separation appears on the opposing bubble surfaces, which creates huge permanent dipoles in the film. These dipoles strongly react to the external field generating additional intrafilm forces, which either swell or shrink the film. Assembling of charged PP foam films into a stack increases their piezoelectric coefficient from 600 pC/N to 2010 pC/N.^{15}

Corona charging has been also used for inorganic electrets like SiO_{2} and for SiO_{2}/Si_{3}N_{4} multilayers, where a positive charge *q* is deposited at a depth *d* in the electret.^{21} In order to detrap the bulk trapping sites, which are shallower in energy and have lower density than the interface trapping sites, the charged multilayer is annealed at higher temperatures. As shown in Ref. 22, the piezoelectric constant is large when the injected charges are close to interfaces rather than being in the bulk. Thin layers also inhibit partial diffusion of the interface charges into the bulk,^{23} preventing the decrease in piezoelectric reaction constants.

The morphological structure of a composite material is usually described by means of the *a-b* connectivity factor,^{1,24} where the *a* term describes the connectivity of the primary active phase, and the *b* term corresponds to the connectivity of the secondary passive phase. The simplest architecture for a piezo-polymer composite material is a layered material corresponding to a *2–2* connectivity with a flat interface between the layers. Such multilayer films have low density and fast actuation speed, and are best candidates for artificial muscle applications.^{10} Other possible architectures include a matrix elastomer polymer impregnated with piezoelectric fillers. In this case, the connectivity can be varied between *1–3* and *3–3*.^{16,25}

In this study, we present a generalized theory for the actuation properties of charged multilayer films by combining elasticity theory with a phenomenological description of electrostatics. Our research broadens the study of Tuncer *et al*.^{22} and Kacprzyk *et al*.^{30,31} to the case of a multilayer film for which we carry out a systematic analysis of the piezoelectric charge (*d*_{33}) and voltage (*g*_{33}) coefficients under applied voltages and pressures. We consider a composite multilayer with a *2–2* connectivity, where one layer is an active piezoelectric layer and the second layer is a passive dielectric elastomer layer. Our aim is to find optimal elastic and dielectric coefficients for the hybrid material at which its actuation response is superior to the response of a single layer film. We show that the response coefficients of the composite material have very rich features compared to the individual constituents of the multilayer. For example, under suitable conditions, the composite is considerably squeezed than the individual material would do. We also show that there is a threshold voltage *V*_{0} which defines a swelling window for the applied voltage −$V0<V<0$ for the multilayer, when the composite swells. For a charged multilayer under an external voltage, we derive a full expression for the coefficient *d*_{33} showing its linear dependence on *V*. Transferring the multilayer deformation to its capacity of lifting a mass, we show that considered multilayers are several times more efficient than their single layer counterparts. The rest of the paper is organized as follows. We describe our system parameters in Sec. II. The general theory of multilayer actuation is given in Sec. III. Section IV is devoted to the calculation of actuation coefficients in linear perturbation theory. We discuss our results in Sec. V and conclude in Sec. VI.

## II. SYSTEM SET-UP AND BASIC PARAMETERS

We consider a sandwich-like multilayer structure consisting of consecutively assembled dielectric layers *1* and *2* with dielectric permittivities *ϵ*_{1} and *ϵ*_{2}. The multilayer in total has *2n-1* layers: *n* hard layers *1* of thickness *d*_{1} and $(n\u22121)$ soft layers *2* of thickness *d*_{2}, see the schematic illustration in Figure 1. We assume that the layering is along the *z* direction, and along the *xy* direction, the multilayer has infinite dimensions.

For the layer *1*, we consider three different piezo-dielectric candidates: piezoelectric transducer (PZT), polyvinylidene fluoride (PVDF), and Barex (Barex 210 acrylonitrile-methyl acrylate copolymer). For the layer *2*, two different dielectric elastomers, Very High Bond acrylic foam tape (3M $VHBTM$) and SEBS (styrene-(ethylene- *co*-butylene)-styrene triblock copolymer), are considered. The dielectric constants and Young's moduli of these materials are listed in Table I. For typical materials, such as multilayer films given in Table II, we will consider the case of $\u03f51>\u03f52$ and $Y1>Y2$.

Material . | ϵ
. | Y [MPa]
. |
---|---|---|

PZT | 3800 | 50 000 |

PVDF | 12 | 3000 |

Barex | 5.2 | 65 |

VHB | 4.7 | 0.5 |

SEBS | 2.0 | 1 |

Material . | ϵ
. | Y [MPa]
. |
---|---|---|

PZT | 3800 | 50 000 |

PVDF | 12 | 3000 |

Barex | 5.2 | 65 |

VHB | 4.7 | 0.5 |

SEBS | 2.0 | 1 |

Multilayer . | $\u03f51/\u03f52$ . | $Y1/Y2$ . | V_{0} (in volts)
. |
---|---|---|---|

PZT/VHB | 810 | 10^{5} | 2.98 × 10^{3} |

PVDF/VHB | 3 | 6 × 10^{3} | 1.51 × 10^{6} |

PVDF/SEBS | 6 | 3 × 10^{3} | 1.13 × 10^{6} |

Barex/SEBS | 3 | 7 × 10^{2} | 3.53 × 10^{6} |

Multilayer . | $\u03f51/\u03f52$ . | $Y1/Y2$ . | V_{0} (in volts)
. |
---|---|---|---|

PZT/VHB | 810 | 10^{5} | 2.98 × 10^{3} |

PVDF/VHB | 3 | 6 × 10^{3} | 1.51 × 10^{6} |

PVDF/SEBS | 6 | 3 × 10^{3} | 1.13 × 10^{6} |

Barex/SEBS | 3 | 7 × 10^{2} | 3.53 × 10^{6} |

We assume that all layers *1* were initially impregnated with free ions at elevated temperatures. Then applying a strong polarizing field, the charges of opposite signs are separated towards the layer boundaries. We assume that there are strong trapping centers at the *1–2* interface which block possible ion recombination when the poling field is removed and the membrane is cooled down to room temperature. Thus, after poling, the *1–2* interfaces possess bipolar $\xb1\sigma b$ surface charge densities, and the total charge of the film is zero. The interface charges will polarize neighboring layers *1* and *2*, which will in turn generate internal electrostatic forces acting on the interfaces. We assume that the charged film has been given enough time to relax to an equilibrium state where internal forces are fully balanced by the elastic forces of the deformed layers.

When a charged and equilibrated film is put under external voltage, both layers will be polarized and subsequently deformed. We disregard any deformation related structural changes in the polymer, as well as the altering of their dielectric and mechanical properties. For simplicity, fringe effects at the layer boundaries will be considered non-important. This will avoid mechanical instability problems on the layer interfaces, like wrinkling and buckling,^{26–28} when the deformation rates of the layers do not match. For all deformations, we will consider a constant volume condition explained in Appendix A. We consider a particular case when the interface on the left corner is charged positively, and the interface on the right corner is charged negatively. The applied voltage *V* can change its polarity from positive to negative values. Note that a negative voltage *V* < 0 applied on the multilayer with positive $\sigma b>0$ on its left corner is qualitatively the same as applying a positive voltage *V* > 0 on the multilayer with negative interface $\sigma b<0$ on its left corner.

In the Sec. III, we employ linear response theory to calculate local fields in the layers, and define the interface electrostatic pressure. These parameters are then used to analyze piezoelectric actuation properties of the films.

## III. ACTUATION OF A CHARGED MULTILAYER UNDER EXTERNAL VOLTAGE *V*

The local fields *E _{i}* in the layers

*i*(

*i*= 1,2) of the film can be found from the following two equations:

- Gauss law for the electric displacement $Di=\u03f50\u03f5iEi$ across a surface carrying the free-charge surface density
*σ*:_{b}(1)$D2\u2212D1=+\sigma b.$ - Kirchhoff's voltage law for the potential drop across the film(2)$V=nE1d1+(n\u22121)E2d2=D\u03f50\u03f51nd1+D\u03f50\u03f52(n\u22121)d2.$

Thus, for the local fields, we get

and

where $A(\u03f5,n,d)=\u03f52d1n+\u03f51d2(n\u22121)$. Note that under a short circuit condition *V* = 0, fields *E*_{1} and *E*_{2} are directed in opposite directions and $E1=\u2212n\u22121nd2d1E2$. Also, a thinner layer is having a larger electric field in it.

The pressure on the interface *1–2* is defined as

Here, we assume a constant volume condition for deforming layers, which results in the doubling of the Maxwellian pressure, see Appendix A for more details. Maintaining a constant volume condition also means that lateral extensions of the layers will be different and thus cause nonlinear deformations mentioned in Sec. II. Using Eqs. (3) and (4), we arrive at

Here, the first term has a quadratic dependence on *V* and corresponds to a polarization force. The second term is a linear function of *V* and *σ _{b}*, and thus describes the Coulomb interaction between a charged interface and electrodes. The last term with no dependence on

*V*describes internal electrostatic forces between charged interfaces. This term, as already mentioned in Sec. II, is balanced by internal elastic forces of the film, and thus should be omitted from further consideration.

The sign of the pressure, $sign(p)$, depends on the competition of the first two terms in Eq. (6). Whereas the first term is always positive, the second term can be either positive or negative depending on the sign of *V*. A positive sign of pressure *p* means that the force acting on the interface *1–2*, and defined as $F\u219212=pSz\u2192/2$, is directed from the material *1* to the material *2*. Here, *S* is the interface area. The force on the *2–1* interface is $F\u219221=\u2212F\u219212$, thus under a positive pressure *p* > 0, the layer *1* swells, whereas the layer *2* shrinks. Opposite deformations for the layers will take place for a negative pressure *p* < 0.

For *V* > 0 and $V<\u2212V0$, the total pressure in Eq. (6) is positive, whereas for $\u2212V0<V<0$, it is negative. The threshold potential *V*_{0} is defined as

Maximal squeezing of the layer *1* happens at $V=\u2212V02$ under

The applied voltage *V* changes the multilayer thickness from *L* to $Lnew=L+\Delta L$, where $\Delta L=n\Delta d1+(n\u22121)\Delta d2$, and individual layer deformations are

with their sign defined as

The deformation of the boundary layers *1* facing the electrodes is different from the deformation of the inner layers *1*. This is because of the fact that boundary layers have only one interface *1–2*. On the other hand, for a multilayer with a large number of layers, a contribution from the two boundary layers to the total deformation of the film is small. Thus, for simplicity, we will assume that all layers *1*, including the boundary layers, experience the same deformation $\Delta d1$ under the applied voltage *V*.

The most interesting case here is a positive value of $\Delta L,\u2009\Delta L>0$, because a film swelling can never be achieved in a single layer film. For *p* > 0, a film swelling $\Delta L>0$ is possible when

For *p* < 0, a film swelling $\Delta L>0$ happens if

## IV. PIEZOELECTRIC VOLTAGE COEFFICIENT *g*_{33}

The material's suitability for sensing and actuation applications is usually based on the assessment of its piezoelectric coefficients *g*_{33} and *d*_{33}.^{2–6,25} The *g*_{33} coefficient defines a multilayer strain resulting from an applied voltage *V*

where the multilayer strain is

and $\Delta L$ is given by expression Eq. (11).

Because we are interested in developing composite multilayers with actuation features superior to a single layer, it is advisable to calculate the relative piezoelectric coefficient $g33R$:

where

is a piezoelectric coefficient for a single layer film of a thickness *L* made from material *2*, see Appendix B for details. Equation (16) can be written as

Similar to $\Delta L$, the most interesting case here is $\u2212g33R>0$, meaning a swelling of the multilayer. When $\u2212g33R<\u22121$, the multilayer will be shrunken more efficiently than a single layer film.

## V. ACTUATION COEFFICIENT *d*_{33}

The *d*_{33} piezoelectric coefficient measures the induced electric field at the multilayer surface generated by the applied *P*. This is a main mechanism harbored in the microphone applications.^{29} The field at the film surface *E*_{1} is related to the film surface charge density *σ _{s}*, see Eq. (19) below. Therefore, we will focus on the change $\Delta \sigma s$ instead of the change $\Delta E1$. We assume that a fixed voltage

*V*is provided from an external source, such as a capacitor, and the initial film thickness

*L*

_{0}and surface charge

*σ*

_{0}have relaxed to their equilibrium values

*L*and

*σ*as a response to

_{s}*V*. The surface charge

*σ*is defined as

_{s}Replacing *E*_{1} by Eq. (3), we get

Note that the sign of the surface charge changes from negative to positive at *V* = *V*_{1}:

where *V*_{1} is always positive provided $\sigma b>0$.

The change of the surface charge $\Delta \sigma s$ under the applied pressure *P* is associated with the layer deformations $\Delta di=diP/Yi$,

where $\Phi (V)$ is a linear function of *V*

It is evident that *d*_{33} has two separate contributions: the first contribution comes from the interface charge density *σ _{b}* and the second contribution is associated with

*V*. For a particular case of a double layer with

*V*= 0, the first term coincides with the expression given in Refs. 22, 30, and 31. Also, the first term in Eq. (23) is completely negative for $Y1>Y2$ considered in this paper. Hence, under a decompressing pressure

*P*> 0 (a pressure under which the film swells in

*z*direction, $Lnew>L$) and no external field

*V*, $\Delta \sigma s<0$, which means that any film expansion in

*z*direction in this simple case will result in the decrease in the absolute value of

*σ*. Note that $\Delta \sigma s$ is directly related to the change of the surface potential.

_{s}For the comparison of multilayer actuation with the actuation of a single layer, we calculate the relative piezoelectric coefficient

Here, the single layer coefficient $d33S$ can be derived from Eq. (24) by using $d1=0,\u2009d2=L,\u2009\u03f51=\u03f52$, and *Y*_{1} = *Y*_{2},

The final expression for $d33R$ is

where $d33I$ and $d33II$ have no dependence on *V* and *σ _{s}*,

These two terms have opposite contributions to the relative coefficient $d33R$. For example, at positive voltages *V* > 0, the $d33I$ term is negative and thus contributes to the increase in $\sigma s(P)$ when the film is expanded in *z* direction. However, for the same potential, positive $d33II$ results in the decreasing in $\sigma s(P)$ when the film is expanded in *z* direction. The third part of $d33R$,

which depends both on *σ _{s}* and

*V*, is always positive, $d33III>0$, because the following conditions $\sigma b>0$ and $Y1>Y2$ were assumed in our setup. Again, similar to the Sec. IV, the most interesting case here is when the total coefficient is positive, $\u2212d33R>0$, when the surface charge density $\sigma s(P)$ increases under a decompressing

*P*> 0, or, equivalently, when $\sigma s(P)$ decreases under a compressing

*P*< 0 (a pressure under which the film shrinks in

*z*direction, $Lnew<L$). Another interesting case is $\u2212d33R<\u22121$, which corresponds to a better actuation of the multilayer compared to the actuation of a single layer film.

## VI. RESULTS

Local electric fields in layers *1* and *2* of the PZT-VHB multilayer film according to Eqs. (3) and (4) are shown in Figure 2 as a function of the number of layers *1*. First, as it is expected, the layer *2* with a smaller dielectric constant *ϵ*_{2} has a larger electric field in it. Second, with the increase in *n*, both fields decrease, however the rescaled field $E2L/V$ is always above the single layer field $EL/V=1$. The decrease in *E _{i}* (

*i*= 1, 2) is related to the fact that the derivative $(Ei)n\u2032$ is $(EiE)n\u2032\u2248\u22121n2$, thus $Ei/E\u22481/n$.

Film strains from Eq. (15) for four different composites listed in Table II are shown in Figure 3. Our analytical results are given for $L=5000\u2009\mu $m, *σ _{b}* = 0.01 C/m

^{2}, and for three thickness ratios $d=d1/d2=0.1,1,10$. The applied voltage is varied between $\u22122V0$ and $2V0$, and the values of

*V*

_{0}are given in Table II. The multilayer has in total $2n\u22121=39$, from which

*n*= 20 are piezoelectric layers

*1*, and

*n*− 1 = 19 are elastomer layers

*2*. As it seen from Figure 3, the multilayer strain is positive in the $\u2212V0<V<0$ window. The strain has a maximal swelling at $V=\u2212V0/2$. A higher swelling is expected for larger values of

*V*

_{0}. For all other voltages

*V*, the multilayer shrinks, similar to a single layer behavior, which is shown as a black dashed line in Figure 3. If the thickness of the piezolayer becomes much larger than the thickness of the elastomer, which is the case

*d*= 10, the multilayer swelling capability decreases. This is obvious for the lines corresponding to Barex-SEBS and PVDF-VHB multilayers; however, the PVDF-SEBS film does not show exactly the same trend. The reason for that inconsistency is the nonlinear dependence of the swelling amplitude on the thickness ratio

*d*. This dependence is separately shown in Figure 4 for the PVDF-SEBS film. Both the swelling potential at $V=\u2212V0/2$ and squeezing potential at $V=V0/2$ show a maximum and minimum, respectively. Dashed line in the figure corresponds to the squeezing of a single SEBS layer of a thickness

*L*.

We compare the strains of the multilayer films and single layer elastomers for fields up to $E=50$ MV/m in Figure 5. It is evident that only the PVDF-VHB film has the greatest shrinking strain, which is about 100% at *σ _{b}* = 0.01 C/m

^{2}, and about 200% at

*σ*= 0.02 C/m

_{b}^{2}. Also, all multilayer membranes show higher strains compared to the single layer elastomers. For example, the PVDF-VHB film is more than eight times effective than its VHB single layer counterpart, and the Barex-SEBS film is about 16 times effective than its SEBS single layer counterpart when the multilayer films are charged up to $\sigma b=0.02$ C/m

^{2}.

Figure 6, where the strain Σ is plotted for different interface charges *σ _{b}*, shows that higher swellings can be achieved in strongly charged films. However, in reality, there is an upper limit for how much charge can be put on the interface. This limit depends on the Bjerrum length

*λ*of the electron at an interface, which is about

_{B}*λ*= 6 nm for a medium with

_{B}*ϵ*= 10 at room temperature

*T*= 300

*°*C. At distances smaller than

*λ*, which corresponds to the surface charge densities $\sigma b=0.004$ C/m

_{B}^{2}, the electrons strongly repel each other. This surface density limit can be increased if sufficient ion trapping centers are generated at the interfaces. In our study, we will consider a surface charge density $\sigma b=0.01$ C/m

^{2}as being an optimal value for the multilayers.

In Figure 7, we analyze the dependence of $g33R$ on the elastic and dielectric constants of the layers. This analysis is helpful for locating the domains of the $(\u03f5,Y)$ plane ($\u03f5=\u03f51/\u03f52$, and $Y=Y1/Y2$) where maximal swelling or shrinking is expected. We change *ϵ* between 1 and 20, and *Y* between 1 and 2000. We note that for $\u03f5\u226b20$, our linear theory predicts unrealistically strong membrane swellings.

Multilayer actuation under the shrinking and swelling voltages, $V=V0/2$ and $V=\u2212V0/2$, is separately shown in Figure 7 on the left and right columns, respectively. White lines on the 3D surfaces are for zero multilayer strains. For the convenience of the data presentation, we plot $\u2212g33R$ instead of $g33R$, therefore negative values of $\u2212g33R$ in Figure 7 correspond to a film shrinking, and positive values of $\u2212g33R$ correspond to a film swelling. The strength of $g33R$, except for using a *z*-direction color mapping, is additionally visualized by contouring isolines separated from each other by two units. The first pink line above the white line corresponds to $\u2212g33R=2$, and the first line below the white line corresponds to $\u2212g33R=\u22122$. 3D $g33R$ surfaces are calculated for three different thickness ratios $d=0.1$, 1, and 10. For a positive *V* and *d* = 0.1, see the top left picture in Figure 7, the film always shrinks, because the main role here is played by the elastomer layer. For *d* = 1 and low *Y*, the coefficient $\u2212g33R$ starts to show an upturn, which consequently develops to a strong swelling for *d* = 10 in accordance with Eq. (12).

The case of negative voltage $V=\u2212V0/2$ is very interesting. Here, a strong swelling is seen for all *Y* values and $\u03f5<10$ for *d* = 0.1 and *d* = 1. Comparing these two cases, we see that as the volume fraction of the material *1* increases, the film's swelling strength decreases nearly twice. A further increase in *d*, see the case *d* = 10 in Figure 7, the $\u2212g33R$ coefficient decreases again in the region $\u03f5<10$ and almost for all values of *Y*. However, in the region *Y* < 10 and $\u03f5>15$, a strong and narrow swelling strip develops. There are two interesting facts associated with this behavior. First, in this area, previous thickness factors showed weak shrinking. Second, under a positive voltage, this area also showed a strong swelling. Thus, we conclude that the best candidate for a multilayer film with $d\u22641$, which strongly shrinks under a squeezing potential $V=V0/2$, and strongly expands under a swelling potential $V=\u2212V0/2$ is a film with $Y1\u224810Y2$, and $\u03f51<10\u03f52$. Also, the best candidate for a multilayer which always swells regardless the polarity of the applied voltage is a film with *d* > 1, *Y* < 10, and $\u03f5>15$.

The case *d* = 10 has another unexpected feature, such as it has two zero $g33R$ lines for negative *V*. The consequences of this are elaborated in Figure 8, where a projection of $\u2212g33R$ on the ($\u03f5,Y$) plane is shown for *d* = 10 and for six different values of *V*: $V=\xb1V0/4,\xb1V0/2,\xb1V0$. It is evident that the change of the potential from $V0/4$ to *V*_{0} on the left column has only a quantitative impact on the film actuation: as the amplitude of the voltage increases, both the squeezing and swelling strengths of the film weaken. On the other hand, when *V* is changed from $\u2212V0/4$ to $\u2212V0$ on the right column of the figure, besides of the quantitative changes we also observe strong qualitative differences. First, at $V=\u2212V0/4$, in comparison with the case of $V=Vo/4$, see the first row of Figure 8, the swelling and shrinking regions flip their places. Second, the two zero lines at $V=\u2212V0/2$ and $V=\u2212V0$ divide the coordinate space into four distinct regions. An imaginary vertical line going from *ϵ* = 1 to *ϵ* = 20, and along the fixed $Y=102$ line will pass from a swelling area to a shrinking area. However, if we follow a horizontal line with *Y* = 5 and go along it from *ϵ* = 1 to *ϵ* = 20, the result will be opposite: we will pass from a shrinking area to a swelling area. In other words, the results of Figures 7 and 8 can serve as a powerful guide for manufacturing composite material with better and tailored properties. In experiments, a manipulation of the multilayer elastic and dielectric parameters can be achieved by using dopants and/or impregnation of the material with different fillers.

In Figure 9, we plot the relative piezoelectric coefficient $d33R$ as a function of *ϵ* and *Y*. For positive voltages *V*, see the left column on Figure 9, the value of $\Delta \sigma s$ for the multilayer is always higher than its corresponding value for a single layer. However, this behavior depends strongly on the relative thickness parameter *d*. Whereas for *d* = 0.1, a big change in the value of *σ _{s}* takes place at low

*ϵ*values, for $d\u22651$, more than 10 times stronger changes of

*σ*are expected at large

_{s}*ϵ*values. In other words, the layer thicknesses in a multilayer should be coupled to their dielectric constants for getting better sensory responses for the composite films. The swelling branch of $\u2212d33R$ at negative voltages

*V*, however, has practically no qualitative dependence on

*d*and shows both negative and positive regions. A positive $\u2212d33R$ is an indication of the fact that the surface charge

*σ*will decrease (increase) when the multilayer thickness is shrinking (expanding) under the applied pressure

_{s}*P*. We note that this trend cannot be observed in single layer elastomers. In conclusion, comparing the pictures for positive and negative voltages

*V*in Figure 9 for $d\u22651$, we see that a better candidate for a film which, when mechanically deformed under the applied pressure

*P*, strongly increases its surface charge at

*V*> 0, and strongly decreases its surface charge at

*V*< 0 is a film with $\u03f51>10\u03f52$, and practically with all possible values of

*Y*. Note that this conclusion is valid for a moderate value of the interface charge

*σ*.

_{b}Combining the analyzes of the Figures 7–9, we conclude that a charged multilayer has a versatile response which can be utilized for various sensor applications.

### A. Lifting a mass

We now estimate how much weight *M* can be lifted by a deformed multilayer under an applied voltage *V*. For this purpose, we calculate the pressure on the film surface resulting from the voltage induced strain Σ,

Here, *Y _{c}* is the elastic modulus of the composite and $\Sigma (V)$ is given by Eqs. (11) and (15). For finding the parameter

*Y*, we take into account the fact that the pressure

_{c}*P*is the same for all layers

_{s}*1*and

*2*, therefore,

Then, Eq. (31) can be rewritten as

Finally, the composite elastic modulus is

where *η _{i}* is the volume fraction of the material

*i*, and $\u2211i\eta i=1$. Once the pressure

*P*given by Eq. (31) is known, the value of the lifted mass

*M*can be determined as

where *S* is a surface area of the film, *g* is the gravity acceleration constant. The dependence of *M* on the voltage *V* and on the multilayer strain Σ is shown in Figure 10, where a solid thick line corresponds to a multilayer, and a dashed thin line corresponds to a single layer. Here, two points have to be highlighted. First, whereas the mass lifting of a single layer film is based on its shrinking deformation under the applied *V* with no dependence on the latter's polarity, the lifting capability of the multilayer is based on its shrinking under a positive *V* and on its swelling under a negative *V*. Second, a charged multilayer can lift at least four times more mass than a single layer film does. This is evident from the gradient coloring used in Figure 10: The tips of the dashed line are red, which corresponds to a lifted mass $M\u224815$ kg, whereas the highest tip of the solid line, which is dark pink, corresponds to $M\u224860$ kg. Additional inspection of the Figure 10 also reveals that the multilayer shrinking is more powerful than its swelling. The effectiveness of the membrane lifting can be further boosted if the multilayer parameters *ϵ*, *Y*, and *d* are optimized in accordance with the results of Figures 8 and 9.

## VII. CONCLUSION

Multilayer films have a great potential for being used as actuators with both swelling and shrinking capabilities. We showed that a charged multilayer has a swelling window $\u2212V0<V<0$ with a maximal swelling at $V=\u2212V0/2$. For all other voltages *V*, the multilayer shrinks like a single layer film. At the swelling $V=\u2212V0/2$ and squeezing $V=V0/2$ voltages, the film deformation has a nonlinear dependence on the ratio $d=d1/d2$ showing a maximum and minimum, respectively.

We found that a best candidate for a multilayer film with $d\u22641$, which strongly shrinks under a squeezing potential $V=V0/2$, and strongly expands under a swelling potential $V=\u2212V0/2$ is a film with $Y1\u224810Y2$ and $\u03f51<10\u03f52$. Also, a best candidate for a multilayer which always swells regardless the polarity of the applied voltage is a film with *d* > 1, *Y* < 10, and $\u03f5>15$. Note that no single layer film can swell under any applied voltage. We show that when $d>1$ and $V<V0$, two zero $g33R$ lines divide the ($\u03f5,Y$) plane into four distinct regions. Then, depending on which of the film parameters is changed by the addition of dopants, *ϵ* or *Y*, it is possible to swell the initially shrunken film, or to shrink the initially swollen film.

Our analysis of the coefficient $d33R$ reveals that the change of *σ _{s}* strongly depends on the relative thickness parameter

*d*. The larger change in

*σ*happens at low

_{s}*ϵ*for small

*d*, and at high

*ϵ*for large

*d*. In other words, the layer thicknesses in a multilayer should be coupled to their dielectric constants for getting better sensory responses for the composite films. Our results show that a better candidate for a film which, when mechanically deformed, strongly increases its surface charge at

*V*> 0, and strongly decreases its surface charge at

*V*< 0 is a film with $\u03f51>10\u03f52$, and practically with all possible values of

*Y*.

We demonstrated that a multilayer shrinking has more power to lift a weight compared to a membrane swelling. Also, the multilayer deformation is energetically stronger than a single layer deformation. For example, a charged PVDF-SEBS film can lift at least four times more weight than a single layer SEBS film.

When it comes to the role of the number of layers in the multilayer, our results show that the parameter *n* has a dual role in the multilayer response. On one hand, the mass lifting capacity of the multilayer, as shown in Figure 11, decreases as *n* increases. On the other hand, more layering stabilizes the piezoelectric response of films. In Figure 12, we show $g33R$ coefficients for two different multilayers with *n* = 2 (upper picture) and *n* = 40 (lower picture), respectively. It is evident that the case *n* = 40 has an overall better shrinking response, namely, the low *ϵ* and low *Y* regions of $\u2212g33R$ are stabilized. Also, thinner layers are better suited for avoiding charge migration problems in stacked layers under external fields.^{23}

In the framework of current analytical theory, we omitted the change of the film's lateral *x* and *y* directions for the sake of getting tractable analytical results. A full consideration of the membrane actuation^{7} must address the mismatch between the lateral stretching of the layers and associated with it the change of the interface charge densities. A modified theory should also take into account the generation of permanent dipoles and charges in the piezolayer caused by the applied voltages and pressures. Another issue concerns huge strains and associates with it the change in *ϵ* and *Y* of the multilayer. Also, high strains cannot appear instantly, and thus the theory should introduce a separation of the stress and time functions.^{7}

The results reported here allow us to propose other than simple slab geometries for exploring the dependence of Σ on the multilayer parameters. For example, it is possible to create a film which differently responds to the applied field through the variation of $d1/d2$ along the interface. In Figure 13, we show a specific case when the upper part of the film swells, whereas the bottom part of the film shrinks. Such film can be used as a building block for more complicated structures, such as for creating a gating architecture, or developing a caging geometry. Another interesting application might be core-shell colloidal particles which can be used for shape-controlled colloidal interactions.^{32} As shown schematically in Figure 14, such system, depending on where a high *ϵ* material is used, in the core or in the shell, can either elongate or shrink along the applied voltage.

It is also possible to build smart structures from the multilayer blocks *A* and *B* which differently respond to the applied field, see Figure 15. For example, a chained *A-B-A-B-A* structure, when put on a surface with a vanishing friction coefficient in one direction, will move in that direction by converting the energy of the applied *ac*-field $V=V02ei\omega t$ into a kinetic energy.

## ACKNOWLEDGMENTS

H.L. thanks the Deutsche Forschungsgemeinschaft for support of the work through the SPP 1681 on magnetic hybrid materials.

L.Z. acknowledges support by National Science Foundation through the Science and Technology Center for Layered Polymeric System (CLiPS) under Grant No. DMR-0423914 and Department of Defense (W911NF-13-1-0153). E.A. thanks Philip Taylor from Physics Department of CWRU for interesting and fruitful discussions.

### APPENDIX A: FILM DEFORMATION UNDER CONSTANT VOLUME CONDITION

Constant volume condition for a film with initial dimensions *L* and *S* and final dimensions $L+\Delta L$ and $S\u2212\Delta S$ under a decompression field is written as $LS=(L+\Delta L)(S\u2212\Delta S)$. The energy density stored in the film is defined as

where *D* and *E* are local electric displacement and fields inside the film, $q=\sigma S$ is the total charge on the plates. Total energy stored in the dielectric is

Deformations $\Delta L$ and $\Delta S$ lead to the change in the stored energy

Here, the partial derivatives of the stored energy are defined from Eq. (A2)

For the cases when the contact area of the dielectric does not change, $\Delta S=0$, which is valid for an infinite (in the *y* and *z* directions) plates, we have

and thus recover Eq. (B1)

For a finite size plate

Taking into account that in many cases, we can omit the second term $\Delta SS\u226a1$ on the right side of Eq. (A8), we end up with

or

### APPENDIX B: MAXWELLIAN STRAIN OF SINGLE LAYER FILMS

Interaction force between charged electrodes with a dielectric between them is

where surface charge density on the electrodes is $\sigma =\u03f5o\u03f5E=\u03f50\u03f5V/L$. Under the applied voltage *V*, the film shrinks by

Thus, for the strain $\Sigma =\Delta L/L$ we get

The negative sign of the strain means a shrinking of the film under the Maxwellian stress.

## References

_{33}coefficients

_{33}coefficient of cellular electret films

_{2}and Si

_{3}N

_{4}

_{33}and g

_{33}properties of 0-3 piezoelectric composites by dielectrophoresis