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ScienceDirect Procedia IUTAM 12 (2015) 134 – 144

IUTAM Symposium on Mechanics of Soft Active Materials

Modelling of viscoelastic dielectric elastomers with deformation dependent electric properties Anna Aska , Andreas Menzela,b,∗, Matti Ristinmaaa b Institute

a Division of Solid Mechanics, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden of Mechanics, Department of Mechanical Engineering, TU Dortmund, Leonhard-Euler-Str. 5, D-44227 Dortmund, Germany

Abstract This work deals with electro-viscoelastic modelling and simulation of dielectric elastomer actuators (DEA), including the case of deformation dependent electromechanical coupling. A large deformation modelling framework is adopted, and specific thermodynamically consistent material models are established. The general framework is applied to VHB49 polyacrylic polymers which are commonly used in DEA applications. The effects of viscosity and deformation dependent electric permittivity are studied with regards to the stability behaviour and also in view of predicting experimentally observed electromechanical behaviour using numerical simulations. © byby Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license c 2014 2014The TheAuthors. Authors.Published Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of Konstantin Volokh and Mahmood Jabareen. Peer-review under responsibility of Konstantin Volokh and Mahmood Jabareen.

Keywords: Electromechanics; Coupled problems; Finite strain; Viscoelastic materials; Electrostriction

1. Introduction Dielectric elastomer actuators (DEA) are electromechanical transducers which can transform electric energy into mechanical energy by responding to electric loading with deformation. The typical setup for a DEA is a parallel-plate capacitor, with a slab of dielectric elastomer sandwiched between two compliant electrodes, such as a spray-coated layer of carbon grease on each side. A voltage difference is applied over the electrodes, resulting in a build-up of surface charge. The deformation response of the DEA actuator is generally attributed to electrostatic forces and to corresponding stresses of electric origin due to the interaction between these charges. The electromechanical behaviour of common DEA materials is complex and, in practical applications, phenomena such as electromechanical instabilities, dielectric breakdown, current leakage and viscoelastic effects have been observed. Electromechanical instability can occur when the mechanical stresses in the material no longer balance the electrostatic forces, resulting in pull-in behaviour and, in general, a subsequent electric collapse of the material, which looses its insulating properties 1,2,3,4,5 . The deformation response due to the electric loading of dielectric elastomer actuators is directly coupled to the material’s electric permittivity. It is possible for the permittivity to depend on the state of deformation of the material body, in which case the material is sometimes said to be electrostrictive. The ∗

Corresponding author. Tel.: +49-231-755-5744/7978 ; fax: +49-231-755-2688. E-mail address: [email protected]

2210-9838 © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Peer-review under responsibility of Konstantin Volokh and Mahmood Jabareen. doi:10.1016/j.piutam.2014.12.015

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Anna Ask et al. / Procedia IUTAM 12 (2015) 134 – 144

presence of electrostriction in a dielectric elastomer actuator affects the stability behaviour as well as the general deformation response due to electrostatic loading. As the specimens are generally subjected to considerable prestretches before electric loading is applied, the associated decline in the permittivity reduces the performance of the actuator in terms of deformation. A very common material for DEA applications is VHB49, manufactured by 3M, and specifically the VHB4910 product family. These elastomers exhibit rubber-like elastic behaviour and pronounced viscosity 6,7,1,8 and have been observed to suffer from unstable electromechanical behaviour 1 . The effect of electrostriction in VHB49 is debated. Some studies suggest that the permittivity of VHB49 remains constant or varies very little even at large deformations 9,10 whereas other studies report a pronounced deformation dependence of the dielectric properties, with a significant reduction in relative dielectric permittivity at large deformations 11,12,7 . The reported values of the electric permittivity for VHB49 vary between the different studies, and the measured values seem to tend towards larger deformation dependency for e.g. carbon grease electrodes than for gold electrodes 13 . It has been suggested that the pronounced deformation dependence observed using certain types of electrodes is an artefact from the measurements and not representative of any actual behaviour of the polymer 9,14 . In terms of electromechanical modelling, the general framework is well established, although electromagnetic body forces and couples acting on polarisable bodies, and correspondingly stress tensors of electromagnetic origin, are derived in somewhat different manners in the literature, and are given different physical interpretations 15,16,17,18,19 . From a modelling point of view, however, there is no real inconsistency between different formulations as long as care is taken that corresponding formats of the electromagnetic body force and stress tensor are used together. In this work, the approach of, for instance, Toupin and later Maugin and Eringen is adopted 15,20,21,22 . Electroactive behaviour in the quasi-electrostatic regime is well suited for finite element analysis, as the electric field in this case can be derived from a scalar potential. This allows for an extension of existing finite element formulations by simply including the electric potential as an additional degree of freedom, resulting in a fully coupled finite element framework 23,24,25,26,27 . Different formats for the deformation dependence of the relative permittivity have been suggested in the literature for small 28,29,18 and finite 30,12,31,14 deformations respectively. The influence of using a deformation dependent permittivity on the overall mechanical response, and particularly on the stability behaviour of dielectric elastomer actuators, has also been studied 30,14,31,32 . The aim of this study is to investigate the behaviour of dielectric elastomer actuators undergoing finite deformations in response to electric loading. Specifically, the influence of including viscoelastic behaviour and electrostriction in the material model is studied. 2. Governing equations for the electromechanically coupled problem In view of modelling electromechanically coupled behaviour, the governing equations of continuum mechanics need to be extended so as to include the governing equations for the electromagnetic fields. In this study, the problem is restricted to quasi-electrostatics and the corresponding field equations. The usual kinematics apply, where the motion between the undeformed configuration B0 at time t0 and the deformed configuration Bt at a later time t is described by the (sufficiently smooth) mapping ϕ, transforming position vectors X of material points in B0 to position vectors x of material points in Bt as x = ϕ(X, t). The deformation gradient tensor is then given by F = ∇X ϕ. An isochoricvolumetric split F = J −1/3 F, with J = det(F), gives the isochoric, volume-preserving part of the deformation gradient as F. Viscoelasticity in the finite deformation setting is modelled by introducing (possibly several parallel) multiplicative splits of the deformation gradient into elastic and viscous components 33,34 . Since incompressible materials are considered here, the multiplicative splits are performed on the isochoric part of the deformation gradient. The split corresponding to viscosity element α is given by F = Feα · Fvα .

(1)

In the following, it is assumed that det Feα = det Fvα = 1. Deformation measures are introduced in the usual manner, with the right and left Cauchy-Green type deformation tensors represented by C = Ft · F and b = F · Ft t t 2 respectively, with their isochoric counterparts given by C = F · F = J −2/3 C = 3A=1 λA N A ⊗ N A and b = F · F =

136


2 J −2/3 b = 3A=1 λA nA ⊗nA . Deformation measures related to the viscous deformation are introduced as Cvα = Ftvα ·Fvα t t and beα = Feα · Feα = F · C−1 vα · F . The relevant electric quantities in Bt are the electric field e, the electric polarisation π and the electric displacement d which is defined to be (2) d = ε0 e + π , where ε0 is the electric permittivity of vacuum. In the case of electrostatics, the electric field is curl-free and can therefore be derived from a scalar electric potential φ such that e = − ∇x φ. The corresponding referential electric field in B0 is found through the transformation E = e · F, which is equivalent to E = − ∇X φ. The dielectric field d in Bt is governed by Gauss’s law which, in local form and in the absence of free charges, is given by ∇x · d = 0 .

(3)

Its referential counterpart is found by a Piola transformation, i.e. D = d · cof(F) where cof(F) = ∂ F J = J F−t . At surfaces of discontinuity, including the boundary ∂Bt , the electric field and the electric displacement must fulfil the jump conditions [[d]] · n = 0 and [[e]] × n = 0 , (4) where brackets [[•]] indicate a jump and where n is the outward unit normal to the surface of discontinuity. In the presence of electromagnetic fields, a dielectric body experiences forces and torques of electromagnetic origin, which means that the balance laws need to be modified accordingly. In the literature, the specific format of the electromagnetic force and corresponding tensor varies to some extent 17,18,19 . The approach followed here is to introduce the local electric body force for the case of no free charges as 15,20,21 f e = ∇x e · π .

(5)

The force (5) is often referred to as the Kelvin force density. It can be related to a stress tensor t e , denoted the electromagnetic stress tensor, through the relation f e = ∇x · t e , given for the choice (5), together with (2) and (3), by t e = e ⊗ d − 12 ε0 [ e · e ] I ,

(6)

where I is the second order identity tensor. This enables the introduction of a total stress tensor σ = t + te ,

(7)

which can be shown to be symmetric, whereas t represents the generally unsymmetric Cauchy stress tensor 22 . The quasi-static mechanical balance of linear momentum can then be written in terms of the total stress as ∇x · σ + ρt f = 0 ,

(8)

where f is the body force of mechanical origin and ρt is the spatial mass density. Specifically, the total stresses take the representation (9) σ = t + e ⊗ π + ε0 e ⊗ e − 12 [ e · e ] I , where the last term in (9), related to ε0 , is the Maxwell stress tensor due to the electric field in free space 16,20 . The continuity condition for the total stress tensor on ∂Bt is given by [[σ]] · n = 0 .

(10)

The governing equations need to be closed by specifying suitable constitutive equations that relate the electric and mechanical quantities to each other. Thermodynamically consistent constitutive relations for the total Piola-type stresses and the material electric displacements are obtained as 21,22,25 σ=

2 ∂Ω F· · Ft , J ∂C

d=−

1 ∂Ω · Ft , J ∂E

where Ω is an augmented free energy 22 such that Ω = ρ0 Ψ − 12 ε0 J C−1 : [ E ⊗ E ] with ρ0 = J ρt .

(11)

137


3. Constitutive model It is assumed in this work that the free energy can be split additively into contributions related to volumetric (Ψvol ), long-term elastic (Ψ∞ ), viscous (Ψα ) and electromechanical (Ψmel + Ψel ) behaviour respectively. 3.1. Viscoelastic behaviour The volumetric part of the free energy is chosen as Ωvol (J) = ρ0 Ψvol (J) =

1 2

K [ J − 1 ]2 ,

(12)

where K is the bulk modulus. In order to capture the rubber-like elastic behaviour of VHB49, the hyperelastic model introduced by Ogden 35 is adopted μ p αp αp αp Ω∞ C = ρ0 Ψ∞ C = λ1 + λ2 + λ3 − 3 , (13) αp p where λA , A = 1, 2, 3 are the principal stretches of the isochoric elastic strain. The parameters μ p are related to the shear modulus μ while α p are dimensionless parameters to be determined for the specific material. The behaviour of the elastic elements in the viscosity model is assumed to be governed by an energy format of neo-Hooke type cf. 36,25 μα Ωα C, Cvα = ρ0 Ψα C, Cvα = C : C−1 (14) vα − 3 , 2 where μα is the shear modulus of viscosity element α. The contribution from each viscosity element to the total stress is then given by ∂Ωα 2 σα = F · · Ft = μα J −5/3 beα − 13 [ beα : I ] I . (15) J ∂C A thermodynamically consistent evolution law for the viscous deformation measures is given by 36 t C˙ vα = Γ˙ α Cvα · Mdev vα ,

Mvα = ρ0 C ·

∂Ψα ∂C

= ρ0

∂Ψα ∂C−1 vα

· C−1 vα =

μα C · C−1 vα , 2

(16)

where Γ˙ α is a constant parameter and where superscript dev denotes the deviatoric part of the respective tensor with respect to the identity. 3.2. Electromechanical behaviour In many works dealing with VHB49 dielectric elastomer actuators, the material is assumed to be a linear dielectric with constant relative electric permittivity εr , obeying the constitutive relation d = ε0 εr e .

(17)

Ωmel (C, E) = ρ0 Ψmel (C, E) − 12 ε0 J C−1 : [ E ⊗ E ] = − 21 ε0 εr J C−1 : [ E ⊗ E ]

(18)

This format is obtained by choosing

and Ωel (E) = ρ0 Ψel (E) = 0. The stress contribution related to the electromechanical behaviour is then σmel =

∂Ωmel 2 1 F· · Ft = ε0 εr [ e ⊗ e − [ e · e ] I ] . J ∂C 2

(19)

For electrostrictive materials, the electric parameters (in this case the permittivity) depend on the deformation. A first approach is to assume a linear dependency on the principal in-plane stretches 30,12,14 εr = εr 1 + a I 0 , (20)

138


where I 0 = λ1 +λ2 +λ3 −3, a is a dimensionless parameter which must be calibrated to the specific material, and where I 0 ≥ 0 as λ1 λ2 λ3 = 1. By using this expression rather than a constant value for εr in (18), the electric displacement is given by d = ε0 εr 1 + a I 0 e . (21) Adopting the perturbation method outlined in 37 , to account for the case of coinciding principal stretches, the related stresses can be expressed as 3 3 λA − 13 λB nA ⊗ nA . σmel = ε0 εr 1 + a I 0 e ⊗ e − 12 [ e · e ] I − ε0 εr a [ e · e ] A=1

(22)

B=1

It may be noted that, in the undeformed state, λ1 = λ2 = λ3 = 1, which means that the equations for the linear dielectric, (17) and (19), are recovered. A second alternative for the deformation dependence of the permittivity with a logarithmic ansats 31 is also considered ⎤ ⎡ ⎥⎥ ⎢⎢ a (23) εr = εr ⎢⎢⎣ 1 + ⎥⎥⎦ , ln I 0 /Jm lim

where Jm is a material parameter, determining the stretchability limit of the chains and given by Jm = I 1 − 3, where I 1 = C : I. For this choice of energy function, the stress can be expressed as σmel = ε0 εr 1 +

a

ln( I 0 /Jm )

e⊗e−

3 3 1 a 1 [ e · e ] I + ε0 εr [ e · e ] λ − λB nA ⊗ nA . (24) A 2 3 B=1 I 0 ln(I 0 /Jm ) 2 A=1

4. Homogeneous deformation In the case of incompressible homogeneous deformation, the general system of equations can be reduced to scalar expressions. This is useful for the calibration of material parameters and for the study of the stability behaviour. 4.1. ODE formulation For the parallel-plate capacitor setup, unconstrained motion in the planar direction, corresponding to uniaxial compression, is considered. Let e1 denote the unit vector in the direction of loading (not to be confused with the spatial electric field e). For an incompressible material, the deformation gradient is then given by F = λ e1 ⊗ e1 + λ−1/2 [ I − e1 ⊗ e1 ] = F ,

(25)

and the (incompressible) viscous strains by Cvα = λ2vα e1 ⊗ e1 + λ−1 vα [ I − e1 ⊗ e1 ] .

(26)

The referential and spatial electric fields are given by E = E e1 and e = E λ−1 e1 respectively. For this particular case, incompressibility is enforced directly through a pressure term p I so that the stresses read σα + σmel = p I + σmech + σmel . (27) σ = p I + σ∞ + α

The only non-zero stress contribution is the component in the e1 -direction, i.e. σ = e1 · σ · e1 , and it is a combination of a mechanical term due to the viscoelastic behaviour and a coupled part due to the electromechanical coupling σ = σmech + σmel .

(28)

139

2.5

5

2

4.5

Relative permittivity

True stress [MPa]


1.5 1

3.5 3

0.5 0 0

4

200

(a) (b)

400 600 Time [s]

800

2.5 0

1000

2

4 2 λ +1/λ2−3 r

6

8

r

Fig. 1. Comparison of simulated and experimental 6 relaxation curves for the calibrated viscoelastic parameters in (a) and simulated and experimental 11 values for linear (solid line) and logarithmic (dashed line) electrostriction respectively in (b). Circles represent experimental values. Table 1. Material parameters calibrated to experimental data 6 .

Elastic parameters μ1 = 2.5799 kPa α1 = 2.8436 Electrostriction εr = 4.7

μ2 = 0.1813 MPa α2 = 0.1171 Linear a = −0.0608

Viscosity parameters μv1 = 9.4072 kPa Γ˙ 1 = 1.0392 10−7 Logarithmic a = 1

μv2 = 39.28 kPa Γ˙ 2 = 2.6487 10−6 Jm

= 167

Taking the time derivative of the stress will result in a system of ordinary differential equations (ODEs) which, for given load conditions and boundary conditions together with initial conditions, can be solved using an appropriate ODE solver. The system is given by σ ˙ = f λ˙ +

Nv

gα λ˙ vα + h E˙ ,

f = fmech + fmel ,

(29)

α=1

where f , gα and h denote the partial derivatives of the stress σ with respect to λ, λvα and E, respectively. 4.2. Material parameters Experimental data from a study of VHB49 found in the literature 11,6 is used for the calibration of the material parameters. The resulting fit for the viscoelastic material parameters is shown in figure 1(a). For the electrostriction, one fit is done for the linear model and one for the logarithmic model respectively for the same data. The resulting fits are shown in figure 1(b). All the calibrated material parameters are given in table 1. 4.3. Stability behaviour To find the point where the mechanical stress no longer counteracts the stress due to electric loading, the ODE system (29) is rearranged as Nv ˙ − gα λ˙ vα /h , (30) E˙ = − f λ/h α=1


1.3

25

1.25 r

30

In−plane stretch λ

Electric field [MV/m]

140

20 15 10 5 0 0

1.2 1.15 1.1 1.05

2

(a) (b)

4 6 In−plane stretch λr

8

1 0

10

10 20 Electric field [MV/m]

30

30

35

25

30 Electric field [MV/m]

Electric field [MV/m]

Fig. 2. Study of stability behaviour for the non-electrostrictive case. Electric field versus in-plane stretch for constant compressive stretch rate loading in (a) and in-plane stretch versus electric field for electric loading at constant rate in (b). Solid lines represent purely elastic behaviour. For the viscoelastic response in (a), the load rates are 1/1000 s−1 (dashed line), 1/100 s−1 (dashed-dotted line) and 1/10 s−1 (dotted line) and in (b), dashed lines represent E˙ = 1 MV/m s−1 , dashed-dotted lines E˙ = 10 MV/m s−1 and dotted lines E˙ = 100 MV/m s−1 .

20 15 10 5 0 1

(a) (b)

25 20 15 10 5

2

3 4 In−plane stretch λr

5

6

0

2

4 6 8 In−plane stretch λr

10

Fig. 3. Comparison of behaviour with constant (dashed lines) or deformation dependent (solid lines) permittivity for linear electrostriction (a) and logarithmic electrostriction (b). Lines without markers represent purely elastic behaviour, circles and squares represent viscoelastic behaviour and a constant stretch load rate of 1/100 s−1 and 1/10 s−1 , respectively.

˙ under the assumption that the specimen is free to contract and solved to find the value of E, for a given stretch rate λ, so that the total stress remains zero and, consequently, its rate vanishes. In the very first step, if the start value of E is zero, the coefficient h is also zero. This can be circumvented by setting h to a low but non-zero value in the first step. Figure 2(a) shows the result of these simulations, where a constant permittivity of εr = 4.7 was used, i.e. no electrostriction was assumed. In both figures, solid lines represent purely elastic behaviour, and for the viscoelasline) and 1/10 s−1 (dotted line). tic response, the load rates are 1/1000 s−1 (dashed line), 1/100 s−1 (dashed-dotted √ The electric field is plotted as a function of the in-plane stretch λr = 1/ λ. The material response is stiffer when viscoelastic behaviour is active, and consequently the point of instability (which is where increasing electric loading would result in pull-in or snap-through) is reached for a higher level of the electric field. This is illustrated in figure 2(b), where the stretch as a function of the electric field is plotted up to the the point of instability. The influence of electrostriction is shown in figure 3(a) and (b), for linear electrostriction and logarithmic electrostriction respectively. Dashed lines show the behaviour without electrostriction to allow for direct comparison.


(a) (b)

Fig. 4. Initial geometry of the circular actuator for the simulations with prestretch λ pr = 3 (a) and the actuator with applied voltage in the middle (b). Only one eighth of the actuator is modelled as symmetry boundary conditions apply.

5. Finite element analysis of prestretched actuators In the study by Wissler and Mazza 6 , tests were carried out on circular actuators. The elastomers were prestretched radially to values 3, 4 and 5 of the radial stretch λ pr respectively and then fixed to a circular frame with a radius of 75 mm. The specimens were then allowed to fully relax before electric loading was applied. Each specimen was spraycoated with compliant electrodes forming a circle of radius 7.5 mm in the centre of the specimen. Voltages of 2 kV, 2.5 kV, 3 kV and 3.5 kV were applied and held constant for 900 s. In this study, the circular actuator experiments are simulated by using the finite element method (the theory and implementation of fully coupled electromechanics in a finite element framework are well documented elsewhere 23,38,25,39 and will not be reviewed here) and the constitutive models discussed in previous sections together with the calibrated material parameters. The initial geometry used for simulations of the experiments with prestrain λ pr = 3 is shown in figure 4(a), and the actuated state in figure 4(b). Symmetry conditions apply radially and in the thickness direction, so only one eighth of the actuator is modelled in the analysis. The initial radius of the actuator is 25 mm, 18.75 mm or 15 mm for prestretch λ pr of 3, 4 or 5 respectively, and the initial thickness is 1 mm (which means that the thickness in figure 4(a) is 0.5 mm). In the simulations, the value of the bulk modulus K is calculated from the shear modulus and the Poisson’s ratio, where the latter is chosen to be ν = 0.495 to (almost) account for the incompressibility of the elastomer material. The overall shear modulus for the Ogden model is calculated from the formula 2 μ = p μ p α p . This gives the bulk modulus as K = 1.4236 MPa. The response of the actuator using a constant permittivity is shown in figure 5. For εr = 4.7, the simulated response, shown in 5(a), is much larger than the experimental response for a prestretch of λ pr = 3, and the simulations do not converge for the highest voltage. Although not shown, the simulated response is even more inaccurate for larger prestretches. A second set of simulations using a lower value of the permittivity is shown in figure 5(b). While the simulations now correspond better with the experimental values, the same tendency is present, i.e., for higher prestretches and voltages the simulations predict a response which is too large. For higher prestretches, the simulations failed to converge for the higher voltages. It was not possible to find a value of εr for which all experimental curves could be captured using a constant permittivity. Simulations were also performed allowing for electrostriction. The simulated response with linear electrostriction is shown in figure 6. While the response at a prestretch λ pr = 3 is captured nicely, for λ pr = 4 the simulated response is much too low, indicating that the modelled decline in permittivity with deformation is too pronounced. For the logarithmic model of the electrostriction, the response looks better as can be seen in figure 7. By using this model and the calibrated parameters, all three sets of experimental curves can be captured.

141


35

35

30

30 Nominal strain [%]

Nominal strain [%]

142

25 20 15 10 5 0 0

(a) (b)

25 20 15 10 5

500

Time [s]

0 0

1000

500

Time [s]

1000

20

80

15

60

10 5 0 0

(a) (b)

Nominal strain [%]

Nominal strain [%]

Fig. 5. Simulated response for the circular actuator with constant permittivity and prestrain λ pr = 3. In (a) εr = 4.7 is used while in (b) εr = 3.4 is used. Experiments 6 are shown as markers only and simulations as solid lines with markers. Triangles refer to 2 kV loading, circles refer to 2.5 kV loading, squares refer to 3 kV loading and stars refer to 3.5 kV loading.

500

Time [s]

1000

40 20 0 0

500

Time [s]

1000

Fig. 6. Response with linear electrostriction included for the actuator prestretched to λ pr = 3 (a) and λ pr = 4 (b). Experiments 6 are shown as markers only and simulations as solid lines with markers. Triangles refer to 2 kV loading, circles refer to 2.5 kV loading, squares refer to 3 kV loading and stars refer to 3.5 kV loading.

143


80

25 Nominal strain [%]

Nominal strain [%]

20 15 10 5 0 0 (a) (b)

500

60

40

20

0 0

1000

Time [s]

500

Time [s]

1000

Nominal strain [%]

40 30 20 10 0 0 (c)

500

Time [s]

1000

Fig. 7. Response with logarithmic electrostriction, a = 1, included for the actuator prestretched to λ pr = 3 (a), λ pr = 4 (b) and λ pr = 5 (c). Experiments 6 are shown as markers only and simulations as solid lines with markers. Triangles refer to 2 kV loading, circles refer to 2.5 kV loading, squares refer to 3 kV loading and stars refer to 3.5 kV loading.

Acknowledgement Financial support for this work was provided by the Swedish Research Council (Vetenskapsrådet) under grant 2011-5428, which is gratefully acknowledged.

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