D De Tommasi

G Puglisi

G Saccomandi

G Zurlo

Pull-in and wrinkling instabilities of electroactive dielectric actuators

We propose a model to analyze the insurgence of pull-in and wrinkling failures in electroactive thin films. We take in consideration both cases of voltage and charge control, the role of prestretch and the size of activated regions, which are all crucial factors in technological applications of EAPs. Based on simple geometrical and material assumptions we deduce an explicit analytical description of these phenomena, allowing a clear physical interpretation of different failure mechanisms such as the occurrence of pull-in and wrinkling. Despite our simple assumptions, the comparison with experiments shows a good qualitative and, interestingly, quantitative agreement. In particular our model shows, in accordance with experiments, the existence of different optimal prestretch values, depending on the choice of the actuating parameter of the EAP.

Introduction

Electro-active polymers (EAPs) are very promising materials for several technological applications in the robotic, medical, energy harvesters, and biological fields [START_REF] Kornbluh | Ultrahigh strain response of field-actuated elastimeric polymers[END_REF][START_REF] Koh | Maximal energy that can be converted by a dielectric elastomer generator[END_REF][START_REF] Carpi | Dielectric elastomers as electromechanical transducers[END_REF]. These materials are characterized by important qualities such as lightweight, small size, low-cost, flexibility, fast response. The rapid technological advances in material science have recently improved EAPs performances in terms of their traditional limits, such as small actuation forces, low robustness and the requirement of very high electric fields. A prototypical device for electro-actuation based on EAPs is constituted by a thin layer of polymeric material sandwiched between two compliant electrodes. Application of a voltage to the two parallel electrodes generates an electric force (measured through the so called Maxwell stress [START_REF] Bustamante | Nonlinear Electroelastostatics: a variational framework[END_REF]) which induces a compression of the layer; since the polymer is typically nearly incompressible, the resulting transversal extension is used as a mean of actuation.

A typical drawback in technological applications based on the described electroactivated films is the insurgence of different types of instability, leading to failure (e.g. [START_REF] Moscardo | On designing dielectric elastomers[END_REF] and [START_REF] Plante | Large-scale Failure modes of dielectric elastomer actuators[END_REF]). One of the well known phenomena occurring in these devices is the so called pull-in instability. In order to describe this type of instability, one may consider a toy model composed by two rigid conducting plates connected by an insulated linear spring, with a rest distance equal to d. If the plates are subjected to a voltage V , the Coulomb forces between the plates will tend to attract them so that the spring will result compressed. It is easy to check that there exists a voltage threshold V * such that for larger voltages no more equilibrium is possible between the elastic and electric forces and the top plate slams onto the bottom plate. The resulting thumb rule used in EAPs and similar MEMS devices is that the critical distance associated with V * equals d/3.

Clearly, the quantitative determination of the pull-in voltage is not a simple task for real deformable devices: the failure mechanism of electro-activated thin films depends not only on electro-elastic interactions, but also on the actuator shape and on purely electrical effects. A further purely mechanical failure mechanism in EAPs is represented by the insurgence of compressive stresses inside the polymer film. In typical actuation systems (e.g. [START_REF] Kornbluh | Ultrahigh strain response of field-actuated elastimeric polymers[END_REF], [START_REF] Plante | Large-scale Failure modes of dielectric elastomer actuators[END_REF]) the polymeric film is constrained inside a rigid frame which serves to assembly the actuator, to provide electrical insulation, to transmit the actuation force, and to provide a pre-stretching of the polymer. Under the action of electric forces the constrained polymer film may undergo in-plane compressions, typically shed in evidence by the insurgence of a buckling type instability known as wrinkling in membrane theory ( [START_REF] Pipkin | The relaxed energy density for isotropic elastic thin layers[END_REF][START_REF] Steigmann | Tension-field theory[END_REF]). Another crucial aspect for technological applications is given by the observation that the polymer film is tipically activated only in an internal region, due again to both assembly and insulation motivations.

In the recent past several theoretical and numerical analysis have been proposed in order to elucidate the causes of the various failure mechanisms of EAPs actuation devices. The importance of prestraining has been clearly evidenced in [START_REF] Kornbluh | Ultrahigh strain response of field-actuated elastimeric polymers[END_REF] and [START_REF] Perline | High speed electrically actuated elastomers with strain greater that 100%[END_REF], where experimental observations have shown the existence of optimal prestraining values. An interesting analysis of the benefits of prestraining has been carried on in [START_REF] Kofod | The static actuation of dielectric elastomer actuators: how does pre-stretch improve actuation[END_REF] for an unconstrained layer activated on the whole surface; in the same work, the necessity of a correct description of the strongly non linear material behavior of the polymer has also been evidenced. A similar problem is considered in [START_REF] Patrick | Characterization of dielectric elastomer actuators based on a hyperelastic film model[END_REF] where the authors have discussed the relevance of an energetic analysis in the description of the pull-in instability. A numerical analysis for a circular planar actuator has been recently presented in [START_REF] Plante | Large-scale Failure modes of dielectric elastomer actuators[END_REF], where the main role of prestretch and stretch rate up to failure are numerically analyzed. An energetic analysis for a homogeneously deformed dielectric elastomers with prestress and pull-in is studied in [START_REF] Zhao | Method to analyze electromechanical stability of dielectric elastomers[END_REF] and [START_REF] Díaz-Callja | Effect of an electric field on the bifurcation of a biaxially stretched incompressible slab rubber[END_REF]. Finally, a theoretical analysis of the insurgence of deformation localization in a variational framework was recently proposed in [START_REF] Puglisi | Damage localization and stability in electroactive polymers[END_REF] and [START_REF] Zhao | Electromechanical hysteresis and coexistent states in dielectric elastomers[END_REF], where the role of damage and dissipation were considered.

In this paper we propose a simple prototypical model for a slender prestrained actuator with an inner region activated by assigned voltage or charge (see the scheme in Fig. 1). Based on simplifying geometrical and material hypotheses we elucidate the combined roles of prestrain, material properties and dimensions with respect to the wrinkling and pull-in failure modes. Due to the film thinness we neglect bending resistance and the formation of a boundary layer between activated and non activated regions, so that we assume a sharp interface between such regions [START_REF] Coleman | Necking and Drawing in Polymeric Fibers Under Tension[END_REF][START_REF] Carr | Structured phase transition on a finite interval[END_REF] and we ignore fringing effects [START_REF] Parker | Electric field outside a parallel plate capacitor[END_REF]. Our geometrical assumptions are well suited for linear actuators typically adopted in artificial muscle technologies [START_REF] Perline | High speed electrically actuated elastomers with strain greater that 100%[END_REF]. Of course our approach can be extended to more complex geometrical schemes and constitutive assumptions by using standards numerical methods. Moreover we do not consider dielectric strength thus neglecting the possibility of dielectric breakage [START_REF] Koh | Maximal energy that can be converted by a dielectric elastomer generator[END_REF]. The paper is organized as follows. In section 2 we recall the basic electromechanical equilibrium equations. In section 3 we introduce our prototypical example for a thin electroactivated system. In section 4 we obtain analytical solutions for both boundary value problems with assigned charge and voltage, as a function of given prestretch. Then, in section 5, we discuss the results of our model. In particular, we deduce analytically the existence of an optimal pre-strain (experimentally detected in [START_REF] Perline | High speed electrically actuated elastomers with strain greater that 100%[END_REF]) which depends on the material modulus of the polymeric layer and on the size of the activated region. Shortly, the model describes how prestretch has a positive effect on the wrinkling instability and a negative effect on the pull-in instability, leading to the existence of an optimal prestretch. The corresponding value sof the activation parameters are then deduced. In section 6 we show that the model well describes the experimental behavior. Finally, in the Appendix we deduce a necessary stability condition for the electroactive device here considered.

Basic Equations

In this section we collect the main equations that are at the base of our theoretical approach. We refer the reader to the recent paper [START_REF] Bustamante | Nonlinear Electroelastostatics: a variational framework[END_REF] and to the references therein for details. Let X ∈ B 0 and x ∈ B be the typical point in the reference B 0 and in the current configuration B of a continuous body, with x = f (X ), where f is the deformation field. Moreover, we denote by φ, D and E the potential, the displacement vector and the electric field in the current configuration, respectively. For a linear, homogeneous and isotropic dielectric we have

D = εE = -εgradφ, (2.1)
where ε = ε 0 ε d with ε 0 the permittivity of free space and ε d the dielectric constant of the material. The equilibrium equations for the electromechanical problem are

divD = ρ, divT = b in B D • n = q in ∂B q , Tn = ŝ in ∂B s . (2.2)
Here ρ and q (assigned on ∂B q ) are the bulk and surface charge densities and b and ŝ (assigned on ∂B s ) are the bulk and surface forces densities in the current configuration. T is the (symmetric) total stress tensor, that can be decomposed as the sum of the elastic (ela) term T ela and of the electric (elc) component T elc (i.e. the Maxwell stress)

T = T ela + T elc , (2.3)
with

T elc = ε(E ⊗ E -1 2 (E • E )I), T elc ij = ε( 1 2 E i E j -(E • E )δ ij ), (2.4)
where i, j range from 1 to 3 (here and in the following we will use, when necessary, both component and absolute notations). The corresponding nominal (Piola-Kirchhoff) stress can be obtained as

S = S ela + S elc = JT ela F -T + JT elc F -T , (2.5)
where F := ∇f and J := det F.

Setting of the problem

We consider a dielectric elastomer actuator which occupies, in its natural configuration, a right prismatic region [0, ] × [0, L] × [0, H]. We use the length of the elastomer to adimensionalize the thickness H = H/ and the width L = L/ and thus we consider the prismatic region [0, 1] × [0, L] × [0, H] (see Fig. 1, where we also indicate the chosen coordinate system). In particular, we assume that the dielectric is thin, in the sense that H L 1. For thin films the membrane approximation of the nonlinear elasticity may be adopted. The standard membrane assumption (e.g. [START_REF] Haughton | On the incremental equations in non-linear elasticity-Membrane theory[END_REF]) states that the bending stiffness is zero and any in-plane compressive stress leads immediately to the membrane buckling, with the appearance of a wrinkled regions. The presence of wrinkles is frequently encountered in experiments on thin dielectric elastomers and it represents one of the main observed failure mechanism of such devices.

We assume that voltage or charge can be controlled on a region of the thin film, more precisely on opposite strips of length χ ≤ 1 belonging to the upper and lower faces of the actuator. This region is placed centrally in direction of e 1 length and it covers the full length L in direction e 2 . The prismatic region of the actuator with upper and lower faces coinciding with the electrically controlled strips will be shortly denoted by active region, whereas the remaining part is simply called non active region. In accordance with the technological applications described in the introduction, we consider the possibility of a prestretch of the actuator along the e 1 direction (see again Fig. 1).

We describe the layer deformation by two homogeneous deformations with eigenvectors coinciding with the coordinate axes. More specifically, we assume in the non active region a deformation f such that non active region:

x 1 = λ 1 X 1 x 2 = λ 2 X 2 x 3 = λ 3 X 3 , F := ∇f =    λ 1 0 0 0 λ 2 0 0 0 λ 3    . (3.1)
In the activated region we denote by X = ( X1 , X2 , X3 ) and x = (x 1 , x2 , x3 ) the generic point in the reference and current configuration, respectively, and by f the deformation. For this region we assume active region:

x1 = λ1 X1 x2 = λ2 X2 x3 = λ3 X3 , F := ∇ f =    λ1 0 0 0 λ2 0 0 0 λ3    . (3.2)
The assumption of piecewise homogeneous deformation can be justified by the following hypotheses: piecewise homogeneous loading, thinness hypothesis, and constitutive assumption of homogeneous, isotropic, material behavior of the polymer. This type of deformations (which essentially neglects the presence of a geometricallycompatibile boundary layer at the interface between active and non active regions) is classical within the framework of phase-transitions [START_REF] Fu | Characterization and stability of two-phase piecewise-homogeneous deformations[END_REF] and of thin bodies subjected to homogeneous boundary loading (see, e.g., [START_REF] Coleman | Necking and Drawing in Polymeric Fibers Under Tension[END_REF][START_REF] Carr | Structured phase transition on a finite interval[END_REF]). The balance equation at the interface may be formulated in the weak form S e 1 = 0 , e 1 being the normal to the interface in the reference configuration.

As well as the geometrical boundary layer we here neglect edge effects of the electric field, known in literature as fringing fields. Across the interface between the active and non active regions any discontinuity of the electric field E must fulfill the jump condition (see e.g. [START_REF] Bustamante | Nonlinear Electroelastostatics: a variational framework[END_REF]) e 1 × E = 0 . The assumption of negligible fringing fields leads to an apparent violation of the jump condition, since it results that E = Ee 3 in the active region whereas E = 0 in the non active region, so that the component of E which is parallel to the interface is not continuous through the interface; this apparent violation is actually resolved by considering that in reality the electric field undergoes an abrupt, but continuous, variation from E to 0 in a region of negligible width. Also the assumption regarding electrical edge effects are acceptable, granted the thickness of the EAP is much smaller than its other dimensions (see e.g. [START_REF] Parker | Electric field outside a parallel plate capacitor[END_REF]).

Regarding the polymer material behavior we assume that this is incompressible, so that its deformations respect the isochoric constraint

λ1 λ2 λ3 = 1, λ 1 λ 2 λ 3 = 1. (3.3)
Moreover, in a typical application scheme, the actuator is pre-stretched in the direction X 1 , say of an amount equal to ν. By imposing that the total length of the actuator in the current configuration equals ν, we get

ν = χ λ1 + (1 -χ)λ 1 . (3.4)
To obtain analytic results we assume that the incompressible polymer is characterized by a neo-Hookean constitutive response

T ela = -πI + µB, S ela = T ela F -T , T ela ij = -πδ ij + µλ 2 i , S ela ij = -πλ -1 i δ ij + µλ i , (3.5)
where π is a Lagrange multiplier due to the incompressibility constraint, µ is the shear modulus and B := FF T is the left Cauchy-Green deformation tensor. By introducing the adimensionalized principal stresses and pressure

s i := S i µ , t i := T i µ , π := π µ ,
the non zero stress components in the non active region can be calculated by substitution of the deformation (3.1) in the constitutive response, which gives non active region:

s i = -πλ -1 i + λ i , t i = -π + λ 2 i (i = 1, 2 , 3). (3.6)
As anticipated above, we assume that the electric field in the active region coincides with E = (0, 0, E). By using (2.3) and (2.4) the principal stresses in the active region are given by active region:

ti = -π + λ2 i + ε E 2 µ δ i3 -1 2 , ŝi = -π λ-1 i + λi + ε E 2 µ δ i3 -1 2 λ-1 i .
(3.7)

The nature of the electric stresses, as well as their geometrical coupling with the deformation, changes according to the fact that voltage or electric charge are externally controlled by the device which drives the actuator. In the first case, the applied voltage V determines the electric field:

voltage control: D = (0, 0, εV H λ3 ), E = (0, 0, V H λ3 ). (3.8)
In the second case, the total electric charge Q is controlled on the electrodes, so that charge control :

D = (0,0, Q λ1 λ2 A ), E = (0,0, Q ε λ1 λ2 A ). (3.9)
Here A = χ L is the reference area where the charge is applied. In the following the two driving mechanisms will be considered separately.

Equilibrium for the voltage and charge controls

We first consider the following pre-stretch deformation f p in the e 1 direction

x p 1 = νX 1 x p 2 = ν 2 X 2 x p 3 = ν 3 X 3 , F p := ∇f p =    ν 0 0 0 ν 2 0 0 0 ν 3    , (4.1)
where ν = (ν 2 ν 3 ) -1 is the assigned longitudinal prestretch. (Here and in the following, for simplicity of notation, we sometime drop the index 1 when there is no ambiguity.)

The principal Cauchy stresses in the prestretched configuration are given by

t p i = -π p + ν 2 i ,
so that after imposing the conditions of vanishing stresses on the faces orthogonal to the axis X 2 and

X 3 t p 2 = t p 3 = 0, (4.2)
we obtain π p = ν -1 and ν 2 = ν 3 = ν -1/2 . Thus, the only non-zero stress component in the prestretched configuration amounts to

t p 1 = ν 2 -ν -1 , s p 1 = ν -ν -2 . (4.3)
Now we consider the electric activation of the EAP. Regarding the nonactive region the boundary conditions t 2 = t 3 = 0 give (λ 1 , λ 2 , λ 3 ) = (λ, λ -1/2 , λ -1/2 ) so that the stress in this region as in Eq.ns (4.3) is given by

t i = (λ 2 -λ -1 )δ i1 , s i = (λ -λ -2 )δ i1 . (4.4)
Concerning the electrically active region, after substitution of Eq.ns (3.8) in Eq.(3.7), we get

voltage control: ti = -π + λ2 i + 1 2µ εV 2 λ2 3 H 2 (2δ i3 -1), (4.5)
and, by using (3.3), (3.7) and (3.9) we get charge control: ti = -π

+ λ2 i + 1 2µ Q 2 λ2 3 εA 2 (2δ i3 -1). (4.6)
In order that the EAP maintains its total length equal to the given prestretch also after the application of charge or voltage (see Eq.(3.4)), we impose

ν = (1 -χ)λ + χ λ. (4.7)
This equation gives the following relation

λ = ν -χ λ 1 -χ , (4.8)
so that the condition λ > 0 implies that λ < ν/χ (4.9)

Using (4.4) and (4.8) we then obtain

s = ν -χ λ 1 -χ - ν -χ λ 1 -χ -2 . ( 4.10)
Let us now take in consideration the case when voltage is applied. In this case it is convenient to introduce the parameter κ V defined by

κ V = 1 2µ εV 2 H 2 ,
which represents the (adimensionalized) electric free energy density under the assigned potential. By imposing the boundary conditions t2 = t3 = 0, after simple calculations we obtain the equilibrium stress and strains voltage control:

                   ŝ = λ - 1 λ2 1 -2κ V λ2 , λ2 2 = 1 λ 1 -2κ V λ2 , λ2 3 = 1 -2κ V λ2 λ . (4.11)
When the total charge is assigned, after imposing the boundary conditions t2 = t3 = 0 and by introducing the (adimensionalized) electric energy density κ Q

κ Q = 1 2µ Q 2 εA 2 ,
we obtain charge control:

                 ŝ = λ - √ 1 + 2κ Q λ2 , λ 2 2 = √ 1 + 2κ Q λ , λ 2 3 = 1 λ√ 1 + 2κ Q . (4.12)
Finally, in order to deduce the equilibrium configurations we must impose the mechanical balance at the interface between the active and nonactive regions, which reads s = ŝ. (4.13) This equation let us determine λ and, hence, the equilibrium configurations of the system.

Wrinkling and pull-in instabilities

The two failure mechanisms of wrinkling and pull-in will be considered separately for the cases of voltage and charge control.

Voltage Control

When voltage is assigned (see Fig. 2) the equilibrium configurations, according with (4.13), (4.10), (4.11) correspond (if these exist) to the zeros of the function

g( λ) ≡ λ - 1 λ2 1 -2κ V λ2 ŝ -    ν-χ λ 1-χ 3 -1 ν-χ λ 1-χ 2    s ,
measuring the stress discontinuity at the interface. In order to obtain physically reasonable deformations (see (4.9) and (4.11) 1 ) we are interested to the solutions of g( λ) = 0 in the range (0, λ * ) where λ * is the min{1/ √ 2κ V , ν/χ}. We observe that the stress s in (4.10) is a (ν-dependent) monotonic decreasing function of λ, whereas an analysis of (4.11) shows that ŝ is a (κ V -dependent) concave function of λ, defined for λ ∈ (0, 1/ √ 2κ V ), with ŝ → -∞ at the boundaries of the definition domain. An inspection of the stress plots reported in Fig. 2 a shows that for given ν and χ the condition g( λ) = 0 may have two roots λ-≤ λ+ (equal or distinct) or no roots in the range of interest. If two roots exist it is possible to show that only one of them is stable in a sense which is discussed in detail in the Appendix; by means of this condition, it easy to check that the stable solution corresponds to the root λ-.

For given χ, ν we define κ comp V the value of the loading parameter in correspondence of which the stress evaluated at λis such that s = ŝ = 0 (see Fig. 2 b ): for values of κ V > κ comp V the equilibrium stress is negative and, due to the lack of bending resistance, the film exhibits wrinkling, determining the device failure. Always keeping ν, χ fixed, a second eventuality corresponds to values of the load parameter for which the two solutions coalesce (see again Fig. 2 b ), so that for values of κ V larger than this critical value κ pull V no more roots of s = ŝ exist: this instance represents the pull-in instability. For κ V > κ pull V the elastic response of the film is no more capable of balancing the compressive electric forces and the facing electrodes smash onto each other. Clearly, in correspondence of κ V = κ pull V it results g = 0 and dg/d λ = 0. The situation is depicted in Fig. 3, where both values of κ comp V and of κ pull V are plotted with respect to the prestretch ν. Under the particularly simple assumptions of our model, the expression of κ comp V can be deduced analytically by imposing s 1 = ŝ1 = 0, which gives

κ comp V (ν) = 1 2 1 λ2 (ν) - 1 λ8 (ν) , with λ(ν) = ν + χ -1 χ .
The function κ comp V (ν) is plotted with a bold line in Fig. 3; it attains a maximum in correspondence of a prestretch

ν comp opt = 1 + (2 1/3 -1)χ.
(5.1)

The maximum value of κ V which can be applied to the film without inducing compression then amounts to

κ comp V,max = 1 2 (2 -2/3 -2 -8/3 ), (5.2) attained when λ = λcomp opt = 2 1/3 .
The analytical expression of κ pull V is not easy to get, but the numerical solution of the conditions g = 0, dg/d λ = 0 gives κ pull V for each value of ν, which is represented by a light line in Fig. 3. It is possible to show that, for fixed χ, the curve κ comp V (ν) is always under the curve κ pull V (ν) but in a common point where these curves are tangent, corresponding to the simultaneous attainment of both wrinkling and pull-in failures. This point can be analytically determined by imposing that s = ŝ = 0 and dg/d λ = 0, which after simple algebraic manipulations yields the value of the prestretch

ν pull opt = χ( λpull opt (χ) -1) + 1, (5.3)
where

λpull opt (χ) := 4 -χ 1 -χ 1/6
.

(5.4)

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Stretch λ (χ = 0. 5) 1 ŝ (sol i d ) an d s (d ash ed ) ν k V a)
0.7 0.8 0.9

ν k pull V (ν) k comp V (ν) b) 1.2 1.25 1.3 1.35 1.4
k pull V = 0.2239 1 k comp V = 0.2362 c) Figure 2.
Stress strain curves in the activated and non activated region in the case of assigned voltage: continuous lines represent the stress ŝ1 ( λ, κ V ) in (4.11), of the activated region, whereas dashed lines represent the stress s 1 ( λ, ν) in (4.10) in the non active region. In b) we show the attainment, for a fixed ν of both the pull-in and wrinkling instability. In c) we show the stress curves s 1 ( λ, ν) corresponding to the maximum attained value of κ V It is easy to check that for all χ it results ν comp opt ≤ ν pull opt . It is interesting to observe that not all the values of the functions κ pull V (ν) and κ comp V (ν) are actually attainable; regarding κ pull V (ν), the analysis of the functions s( λ) and ŝ( λ) shows that for ν > ν pull opt the pull-in failure is attained at positive stress, whereas for ν < ν pull opt it is attained at negative stress, which is not possible in our model (dashed light line in Fig. 3). On the other side it is possible to check that the values of κ comp V corresponding to ν > ν pull opt are relative to the unstable solutions λ+ and hence these are not attainable (dashed bold line in Fig. 3). As a result, the film failure is due to wrinkling for ν < ν pull opt and it is due to pull-in for ν > ν pull opt . As it is clear from the plot, the highest value of the load parameter which can be applied without generating failure of both types coincides with the maximum value of κ comp V given in Eq.5.2, which is attained in correspondence of ν = ν comp opt ; in this sense ν comp opt is optimal if one is interested at the maximization of the stored electric energy.

A remarkable fact is that if one is interested at other useful activation parameters, such as the maximum actuating force or the maximum applicable electric field, then ν comp opt is no more optimal. This can be seen by evaluating the values of the stress gap with respect to the prestretched configuration, given by ∆S := µ[ŝ(ν)ŝ( λ)] and of the electric field E in correspondence of the initiation of both wrinkling and pull-in failures; as it is evident from the plots, the maximum values of both parameters are now attained for ν = ν pull opt , that is for λ = λpull opt (see Eqn.s(5.4)-(5.3)). We remind that this value of ν corresponds to the simultaneous attainment of the wrinkling and pull-in failures. After a short calculation, we deduce that the optimal value of ∆S

∆S pull opt = µ(ν pull opt -(ν pull opt ) -2 ), (5.5)
whereas the optimal value of the electric field is given by

E pull opt = 3µ ε (1 -χ) -1 6 (4 -χ) -1 3 . (5.6)
It should be underlined, in conclusion, that all the activation parameters ∆S, κ V and E depend on the elastic modulus µ, and that ∆S and E grow with the fraction χ of activated film. The resulting dependence of ν pull and ν comp with respect to the fraction of activated material is summarized in Fig. 5.

Charge Control

In the case of charge control, it is interesting to observe that the pull-in failure is not possible (this is in accordance with previous analyses [START_REF] Seeger | Charge control of parallel-plate, electrostatic actuators and the tip-in instability[END_REF]). This is due to the fact that for any ν and χ the stress s is a decreasing function of λ, whereas the stress ŝ is an increasing function of λ. On the contrary the wrinkling failure is possible and it is attained as soon as s

(ν, χ) = ŝ(κ Q ) = 0, that is λ = λcomp opt (ν) = 1 + ν -1 χ
which corresponds to a maximum load parameter

κ comp Q,max = 1 2 ( λcomp opt ) 6 -1 .
Since the load parameter is an increasing function of ν, it is clear that in the case of charge control it is not possible to find an optimal value of the prestretch.

Comparison with experiments

In this section we briefly compare the theoretical results of our model with the experimental behavior of electroactive polymer actuators [START_REF] Kornbluh | Ultrahigh strain response of field-actuated elastimeric polymers[END_REF].

In Fig. 7 a we reproduce the experimental behavior of silicone rubber films (Nusil CF19-2186) as reported in [START_REF] Kornbluh | Ultrahigh strain response of field-actuated elastimeric polymers[END_REF], where circles represent the values of the electric field at failure for different assigned prestretches. In Fig. 7 b we reproduce the results of our model, where we have considered an elastic modulus µ = 1MP a, as suggested by [START_REF] Kornbluh | Ultrahigh strain response of field-actuated elastimeric polymers[END_REF] for this material. It should be remarked that the considered experiments refer to circular actuators, whereas we here have focused attention on linear actuators. Nevertheless, the comparison between experiments and theoretical results are in nice quantitative agreement. Quite interestingly, a careful analysis of the experimental Fig. 7 a in light of the theoretical Fig. 7 b shows the existence of different behaviors, separated by a threshold prestretch which approximately corresponds to ν pull opt . For values of ν lower than ≈ 1.4 both the experimental and the theoretical results show an initial growth of the maximum applicable electric field with growing prestretch. According to our model this may be due to the initially positive effect of prestretch in avoiding the occurrence of wrinkling. Such positive effect, both in experiments and in our model, ceases to exist when the prestretch overcomes the critical value of about 1.4, since here the thickness reduction anticipates the occurrence of pull-in failure. A finer description of the film behavior for larger values of the prestretch certainly requires a better description of the exact film geometry and of the material response. To the price of renouncing to closed form solutions (which are generally possible under very simple assumptions such those embraced in this article), the methods we have discussed can be numerically applied to more complex geometries and material behaviors suitable for polymers undergoing large deformations, such as Mooney-Rivlin, Gent, Arruda-Boyce and others; some of these applications will be considered in a forthcoming paper. which is the necessary stability condition adopted in the paper.

Figure 1 .

 1 Figure 1. Scheme of the model and boundary conditions. The gray region represents the activated region.

Figure 3 .

 3 Figure 3. Critical values of the activation energy k V . Bold line represents the function k comp V (ν) corresponding to wrinkling instability, whereas light line represent the function k pull V (ν) corresponding to pull-in instability. Dashed lines correspond to unstable configurations.

1 E pull unstabl e 1 E pull stabl e 1 E c om p stabl e 1 EFigure 4 .Figure 5 .

 111145 Figure 4. Critical values of the activation stress ∆S a) and of the electric field E. Bold lines represent the values corresponding to wrinkling instability, whereas light lines represent the values corresponding to pull-in instability. Dashed lines correspond to unstable equilibrium states.

Figure 6 .

 6 Figure 6. Stress strain curves in the activated and non activated region in the case of assigned charge: continuous lines represent the stress ŝ1 ( λ, κ Q ) in (4.12), of the activated region, whereas dashed lines represent the stress s 1 ( λ, ν) in (4.10) in the non active region.

Figure 7 .

 7 Figure 7. Failure value of the electric field as a function of the prestretch: a) experiment (deduced from [13]), b) theoretical.

χ λ 1 ŝ 1 sχ λ+ζ 1 ŝ

 111 the undeformed configuration; since in this case the only non zero stress component iss we have E ( λ) HL = (x) dx + (1 -χ) λ (x) dx .This function must be minimized under the compatibility constraintν = χ λ + (1 -χ)λ, that gives λ = λ( λ) = ν -χ λ 1 -χ .Let then λ define a given configuration of the system, and let λ + ζ be a perturbed configuration that is obtained by shifting the interface between the active and non active regions. The perturbed energy isE ( λ + ζ) HL = (x) dx + (1 -χ) λ-ζ χ 1-χ 1 s(x) dx.Thus, a stable configuration λ must verifyE ( λ + ζ) ≥ E ( λ) for all ζ.By considering the following expansionE ( λ + ζ) = E ( λ) + ζχ ŝ( λ) -s(λ)the first variation gives the Euler Lagrange equilibrium condition ŝ( λ) = s(λ), whereas the positivity of the second variation gives dŝ(

APPENDIX

In this Appendix a necessary stability condition is obtained for the equilibrium configurations of a linear actuator undergoing piecewise homogeneous deformations. We observe that the homogeneity of the deformations in both the active and non active regions, together with the constraint of assigned prestretch ν, essentially reduces the present equilibrium problem to a one-dimensional problem. Under conservative hypotheses, the total energy E stored in the device at given λ can be obtained by integration of the internal working density along an arbitrary deformation path from


M Morsli

Linda Cattin

J C Bernède

P Kumar

S Chand

JV characteristics of dark and illuminated classical and inverted organic solar cells based on the CuPc/C60 heterojunction

Keywords: numbers: 68.35.bm, 73.40.Cg, 81.05.Fb, 88.40.H

The comparison of the performances of classical and inverted organic solar celles based on the junction copper phthalocyanine/fullerene (CuPc/C60) shows that the former devices give the best efficiency. The transport properties of charge carriers in the organic material and the interface properties have been investigated using a mathematical simulation taking into account the effect of bulk and interface properties. The good agreement between experimental and calculated values can be achieved using different parameter values following the type of solar cells. In classical solar cells, the current is space charge limited, while there is no barrier at the contacts electrode/organic material. In the case of inverted solar cells it is necessary to introduce a barrier contact at these interfaces to achieve a good fit between experimental and theoretical values. Therefore the smaller efficiency of inverted solar cells is due to barrier contact at the interface and smaller electrode work function difference.

Introduction

During the last fifteen years the interest to organic solar cells has been continually growing [1]. A limiting factor affecting power conversion efficiency is the exciton diffusion length that is small prior to recombination. Typical exciton diffusion length is in the range of 10 nm. The film thickness should be higher than 100 nm in order to absorb most of the light. This difficulty has been overcome by using the concept of bulk heterojunction. This bulk Confidential: not for distribution. Submitted to IOP Publishing for peer review 2 July 2010 heterojunction is obtained by blending an electron donor (D) and an electron acceptor (A).

Blend cells exhibit a large interface area and most excitons reach the D/A interface. Although efficiency higher than 6% has been attained with P3HT:PCBM bulk heterojunctions the morphology of the blend is difficult to control. Therefore, a possible way to control and stabilizes the structure is to deposit the organic materials onto a well structured anode such as ZnO nanorods [2]. Another way consists in the growth of tandem solar cells [3,4].

The fundamental processes involved in efficient organic solar cells are (a) absorption of light, (b) creation and separation of the charge carriers at the donor/acceptor interfaces and (c) transport of these charges through the bulk of the device from the creation site to the appropriate collecting electrode. Therefore, first, the absorption spectrum of at least one of the organic materials should correspond, as near as possible, to the solar spectrum.

After creation, the pair of carriers should be separated. The separation takes place at donor/acceptor interfaces. To achieve that goal, two configurations have been used, simple bilayer devices and bulk heterojunctions. An organic double layer solar cell contains an electron rich (electron donor, ED) and an electron deficient (electron acceptor EA) layer. In bulk heterojunction, both donor and acceptor are also present but the ED/EA interfaces are distributed in the whole bulk of the sample.

In the case of multi-layer thin film heterojunction the introduction of an electron blocking layer [5] between the cathode and the acceptor material allows achieving good efficiencies. In these devices, transparent conductive oxide, (TCO), coated substrates are usually used as transparent anode, while the cathode is an aluminium evaporated film.

Another possibility is the use of inverted solar cells. In that case the TCO is used as cathode, while a metallic electrode, with high work function, is used as anode.

The classical multi-heterojunction structures are usually based on ITO/ABL/CuPc/C 60 /CBL/Al devices where CuPc is the copper phthalocyanine, C 60 the fullerene, while ABL and CBL are the anode buffer layer and the cathode buffer layer respectively. The buffer layers (ABL and CBL) are necessary in view of the difficulties in organic optoelectronic devices of the charge carrier transport between the organic materials and the electrodes.

In the case of the anode/electron donor contact, a common solution is to introduce a thin ABL, which adjusts the electronic behaviour of the adjacent materials. We have shown that an ultra-thin metal film or a thin oxide film deposited onto the conductive substrate, whatever TCO can be used to improve the interface TCO/donor and therefore the devices performances [6][7][8]. Indeed, Au and/or MoO 3 allows achieving this goal [5], by simple vacuum deposition of an ultra-thin (0.5 nm) gold film or of a thin (3.5± 1 nm) MoO 3 film. Therefore, we have used Au as ABL buffer layer in classical solar cells, while MoO 3 of the MoO 3 /Ag/ MoO 3 can also be used as ABL in classical but also in inverted solar cells. As a matter of fact, we have shown that MoO 3 /Ag/ MoO 3 can be efficient transparent and conductive anode in organic solar cells [9].There is a threshold silver thickness value, 10 nm, where the structures commute from an insulating state to a conductive state. We attribute this commutation to the percolation of the silver conducting path. The transmittance of the films increases when the silver thickness increases from 8 to 10 nm, while further increase induces transmittance decrease. Such effect is attributed to surface plasmon resonance. The structure with the best factor of merit is obtained when the silver thickness is 10 nm. Therefore these structures can be used as anode in classical and in inverted organic solar cells.

In the present manuscript, both classical and inverted solar cells have been studied.

The cathode buffer layer (CBL) is a large band gap material, such as bathocuproine (BCP) [10] or aluminium tris(8-hydroxyquinoline) (Alq 3 ) [11,12], it is called exciton blocking layer (EBL). The EBL, not only blocks the excitons far from the cathode where they can be quenched, but, also, prevent from damage the electron acceptor film during cathode deposition. It should be transparent to the solar spectrum to act as a spacer between the photoactive region and the metallic cathode. It must, also, transport electrons to avoid high series resistance.

If the classical devices described above have given the best performances, development of inverted solar cells could allow more flexibility on designing tandem structures, which can be formed using a semitransparent cathode. The structure of the inverted solar cells studied in the present work is: ITO/CBL/C 60 /CuPc/MoO 3 /Ag/MoO 3 . The MoO 3 /Ag/MoO 3 is used as anode, while the first MoO 3 layer of this structure is also the ABL.

Up to now, the inverted organic solar cells studied are usually based on bulk heterojunctions [13,14], while no many works have been devoted to inverted multiheterojunction PV cells. The inverted bulk heterojunctions structure has been highly successful and represents the device case that can be manufactured [15,16]. The challenge to reversing the layer sequence of multi-heterojunction PV cells is achieving cathode and anode ohmic contacts.

In the present work we study the (J-V) characteristics of classical and inverted OPV cells based on the CuPc/C 60 couple.

Experimental

The glass substrates coated by an ITO thin film were provided by the SOLEMS. The other materials (Au, Ag, MoO 3 , CuPc, C60, Alq 3 ) were provided by Aldrich.

Classical organic solar cells

In the case of classical cells, an ultra thin gold film, (thickness t = 0.5 nm), used as ABL, was deposited onto some ITO anodes to increase the cells performances [6,7,17].

The organic solar cells structure is anode/organic electron donor/ organic electron acceptor/organic buffer layer/aluminium. The anode is ITO/Au (0.5 nm), MoO 3 /Ag/ MoO 3 and MoO 3 /Ag/ MoO 3 /Au. The electron donor is CuPc, the electron acceptor is C 60 , the buffer layer is Alq 3 [11,12]. The whole organic cells were prevented from air contamination by an amorphous selenium thin film [8,9,12]. The deposition conditions have been described earlier [17]. Finally, the structures used were: glass/anode/CuPc(35 nm)/C 60 (40 nm)/Alq 3 (9 nm)/Al(100 nm). Anode = ITO/Au (0.5 nm), MoO 3 /Ag/ MoO 3 or MoO 3 /Ag/ MoO 3 /Au.

Inverted organic solar cells

In the case of inverted cells, the thin film sequence is as follow: ITO/Alq 3 (9 nm)/C 60 (40 nm)/CuPc (35 nm)/ MoO 3 /Ag/ MoO 3 . The CBL is, here also the Alq 3 ,

The experimental conditions for thin films depositions are the same as those used for classical solar cells.

Electrical characterizations were performed with an automated I-V tester, in the dark and under sun global AM 1.5 simulated solar illumination. Performances of photovoltaic cells were measured using a calibrated solar simulator (Oriel 300W) at 100 mW/cm 2 light intensity adjusted with a PV reference cell (0.5 cm 2 CIGS solar cell, calibrated at NREL, USA).

Measurements were performed at an ambient atmosphere. All devices were illuminated through ITO electrodes.

Experimental results and discussion

Typical J-V characteristics, in the dark and under one sun global AM 1.5 simulated solar illumination, are presented in Figures 1, 2 and3. The open circuit voltage (Voc), the short circuit current density (Jsc), the fill factor (FF) and the power cells conversion efficiency ( ) are presented in table 1. As expected, the classical organic solar cell structures give better result, while inverted organic solar cells give promising but smaller efficiencies. It can be seen that the best results are obtained with ITO/Au. All the current densities are given for an area of 10 mm 2 . More precisely, it can be seen in table 1 that the short circuit currents density and the fill factors of inverted solar cells are smaller than those of classical solar cells. Also the open circuit voltage of the inverted cells is only half that of classical cells.

The slopes at the short circuit point and at the open circuit voltage are the inverse values of the shunt resistance (Rsh) and the series resistance (Rs) of the equivalent circuit scheme of a solar cell respectively [18]. It can be immediately seen from table 1 that the best cells correspond to the smaller Rs and higher Rsh.

In such devices, the process of carrier collection is one of the main factors which control the electrical characteristics and the efficiency of the devices. Therefore electrode modification can lead to good or bad performance solar cells.

In order to explain the different cell behaviours, the modelling of J-V characteristics could be helpful in understanding and optimisation of organic solar cells by providing a quantitative estimation for losses in the cells.

The current voltage (I-V) characteristics of metal/polymer/metal devices are controlled by two basic processes: injection of charge carriers from the electrodes into the organic material and vice versa and/or transport of charge in the bulk of the organic film. The current in the device is determined by the less effective mechanism, which limits charge carrier flow. It is crucial for the understanding and optimisation of organic devices to obtain an answer to the question of whether their characteristics are controlled by interface exchange or bulk transport. So far, different descriptions of the J-V behaviour of organic solar cells have been provided, using either Schottky barrier or space charge limited current (SCLC), but these models were exclusive. Recently, a model which includes both injection and bulk transport properties has been proposed [19]. According to this model the current can even be SCLC if the injection barrier is not too high. This model explains the effect of injection barrier height on SCLC. Most of the organic semiconductors are disordered materials and posses traps. In that case the transport equations and Poisson's equations will respectively be written as, ( ) ( )

x F x p q J µ = , (1)
and

( ) ( ) { } 0 εε x p x p q dx dF t + = (2)
where, q is elementary charge, µ is carrier mobility, p(x) is free carrier distribution, p t (x) is trapped carrier distribution, F(x) is electric field distribution and ε, ε 0 are respectively the dielectric constant and permittivity of free space. Usually in most of organic semiconductors the traps are distributed exponentially in energy space and for exponential distribution of traps,

( ) ( ) l v b t N x p H x p / 1 = (3)
where H b is total trap density, N v is effective density of states and l = T c /T, where T c is characteristic temperature of trap distribution and T is the absolute temperature. If p(x) is less than p t (x), p(x) can be ignored in Eq. ( 2) and Eqs. ( 1)-( 3) can be solved analytically to derive the J-V relation [20]. It is also shown that at high voltages p(x) can not be ignored compared to p t (x). In this case, from Eqs. ( 1) and ( 2) we get:

( ) ( ) + = d F F l v b d F N q J H F q J dF q dx 0 / 1 0 0 µ µ εε (4)
The analytical solution of Eq. ( 4) is very difficult therefore the solution is obtained by numerical calculations. The way to calculate the J-V characteristics in this case has been discussed by some of the authors of the present paper in ref 16.

We have compared our experimental data with the calculated characteristics using Eq. ( 4). The experimental data has been observed to be in good agreement with the calculated characteristics using Eq. ( 4). The experimental data was also compared with other models for example, Richardson-Schottky Thermionic Emission, Fowler-Nordheim tunneling, thermal assisted tunneling and mobility model [21,22]. The experimental data did not show any agreement with any of these models.

The J-V characteristics were calculated using Eq. ( 4) with the following values of parameters, ε = 3, d = 75 nm, N v = 10 19 cm -3 and T = 300 K. From the curves logJ-logV, we observe in figures 4, 5 and 6 that below a critical voltage the current density varies linearly on the voltage, which corresponds to an ohmic regime. For voltage above the critical voltage, the current density strongly increases which is characteristic for SCLC. The transition between the two different regimes is sharp for trap levels located at a single energy [23]. The gradual transition visible in the figures points to a distribution of trap-level energies.

It can be seen in figures 4, 5 and 6 that at low voltages curves show ohmic behaviour which can be attributed to background doping and thermally generated charge carriers. At higher voltages the experimental data (symbols) show good agreement with the numerically calculated curves (dashed curves) for SCLC with exponentially distributed traps in energy and space. The parameters values calculated are given in table 2. The main information given by this table 2 is that, to achieve a good agreement between experimental and calculated J-V characteristics the value of the barrier height should not be zero only in the case of inverted solar cells. About Tc, values of the same order of magnitude have already been observed in some other organic semiconductors as well [24,25].

As shown above, the low efficiency in inverted solar cells can be attributed to the low values of Voc, FF and Jsc. From the results summarized in table 2 it can be said that the low FF and Jsc are due to presence of a barrier at the contact ITO/electron acceptor, since it has already be shown that the interface CuPc/MAM and CuPc/Au/MAM are good quality contacts 6 .

As usual, the low Voc value corresponds to low shunt resistance. It may be attributed to the fact that in inverted solar cells the top anode MoO 3 /Ag/ MoO 3 may diffuse much more in CuPC than Al into C 60 in the classical case, since there is no protective interlayer in the former case while there is an Alq 3 intermediate layer in the former case. The diffusion may in turn result in conducting shorts in the active layer.

Moreover it is generally admitted that the Voc increases with the work function difference of the electrodes [26]. In the case of classical solar cells the work function difference, taking in account the buffer layers is Φ Au -Φ Al or Φ MoO3 -Φ Al that is to say it is around 0.9 Ev. For reversed solar cells the same estimation gives Φ ITO -F MoO3 that is to say around 0.6 Ev. Such electrode work function difference can explain the smaller value of the Voc of inverted solar cells.

Conclusion

In conclusion the efficiency difference between classical and inverted multilayers organic solar cells has been justified using a mathematical simulation on charge carrier transport in organic solar cells. The smaller efficiency of inverted solar cells can be attributed to the presence of a barrier at the interface cathode/electron acceptor and to its smaller electrode work function difference. Tables Table 1: Photovoltaic performance data and parasitic resistances under AM1.5 conditions of the devices of figures 1, 2 and 3.

Figures

Table 2: Parameters values calculated using Eq. ( 4).

  Figure 1 : Typical J-V characteristics of: glass/ITO(80nm)/Au(0.5nm)/CuPc(35nm)/C 60 (40nm)/Alq 3 (9 nm)/Al(100nm)structure, in the dark (full symbol) and under illumination of AM1.5 solar simulation (100 mW/cm 2 ) (open symbol).

Figure 2 :

 2 Figure 2 : Typical J-V characteristics, in the dark (full symbol) and under illumination of AM1.5 solar simulation (100 mW/cm 2 ) (open symbol), of: a-glass/MoO 3 /Ag/MoO 3 /Au(0.5nm)/CuPc(35nm)/C 60 (40nm)/Alq 3 (9 nm)/Al(100nm) structure, b-glass/MoO 3 /Ag/MoO 3 /CuPc(35nm)/C 60 (40nm)/Alq 3 (9 nm)/Al(100nm) structure.

Figure 3 :

 3 Figure 3 : Typical J-V characteristics of: glass/ ITO/Alq 3 /C 60 /CuPc/ MoO 3 /Ag/ MoO 3 . structure, in the dark (full symbol) and under illumination of AM1.5 solar simulation (100 mW/cm 2 ) (open symbol).

Figure 4 :

 4 Figure 4: Calculated (…..) and experimental ( ) J-V characteristics of glass/ITO(80nm)/Au(0.5nm)/CuPc(35nm)/C 60 (40nm)/Alq 3 (9 nm)/Al(100nm) organic solar cell.

Figure 5 :

 5 Figure 5: Calculated (…..) and experimental ( ) J-V characteristics of: a-glass/MoO 3 /Ag/MoO 3 /Au(0.5nm)/CuPc(35nm)/C 60 (40nm)/Alq 3 (9 nm)/Al(100nm). b-glass/MoO 3 /Ag/MoO 3 //CuPc(35nm)/C 60 (40nm)/Alq 3 (9 nm)/Al(100nm).

Figure 6 :

 6 Figure 6: Calculated (…..) and experimental ( ) J-V characteristics of: glass/ITO/Alq 3 /C 60 /CuPc/ MoO 3 /Ag/ MoO 3 .

  Figure 1

Figure 4

 4 Figure 4

Figure 6

 6 Figue 5

Table 1

 1

Table 2 .

 2 Alq 3 /C 60 /CuPc(35nm)/ MoO 3 /Ag/ MoO 3 .

	Sample	Jsc (mA/cm 2 )	(V) Voc	(%) FF	η (%) Rs	Rsh
	Classical solar cells
	ITO/Au/CuPc(35nm)/C 60 /Alq 3 /Al	5.84	0.49	53.9 1.53	174 8190
	ITO/Au/CuPc(40nm)/C 60 /Alq 3 /Al	5.08	0.48	52.9 1.30	216 9684
	MoO 3 /Ag/MoO 3 /CuPc(35nm)/C 60 /Alq 3 /Al	4.90	0.430 52	1.15	143 6315
	MoO 3 /Ag/MoO 3 /Au/CuPc(35nm)/C 60 /Alq 3 /Al 5.12	0.444 52	1.19	144 4050
	Inverted solar cells
	ITO/Surface area 0.15 cm 2	2.45	0.27	36.8 0.24	495 2535
	Surface area 0.10 cm 2 ITO/Alq 3 /C 60 /CuPc(35nm)/ MoO 3 /Ag/ MoO 3 .	2.91	0.24	35.3 0.25	460 2780
	Sample	µ (cm 2 V -1 s -1)	T c K	H b cm -3	φ eV
	Classical organic solar cells
	ITO/Au/CuPc(35nm)/C 60 /Alq 3 /Al	4.8×10 -4	2200	1.0×10 17 0
	3.5×10 -4 MoO 3 /Ag/MoO 3 /Au/CuPc(35nm)/C 60 /Alq 3 /Al 1.8×10 -3 ITO/Au/CuPc(40nm)/C 60 /Alq 3 /Al MoO 3 /Ag/MoO 3 /CuPc(35nm)/C 60 /Alq 3 /Al 2.5×10 -3	1950 2400 2550	1.2×10 17 0 1.0×10 17 0 9.0×10 16 0
	Inverted organic solar cells
	Surface area 0.15 cm 2 ITO/Alq 3 /C 60 /CuPc(35nm)/ MoO 3 /Ag/ MoO 3 .	4×10 -4	775	5×10 17	0.12