Validation of a model for an ionic electro-active polymer in the static case
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Validation of a model for an ionic electro-active
polymer in the static case
M. Tixier1 & J. Pouget2
1 Laboratoire de Mathématiques de Versailles (LMV), UMR 8100, Université de
Versailles Saint Quentin, 45, avenue des Etats-Unis, F-78035 Versailles, France
arXiv:2101.10146v1 [physics.class-ph] 25 Jan 2021
2 Sorbonne Université, CNRS, Institut Jean le Rond d’Alembert, UMR 7190,
F-75005 Paris, France
E-mail: mireille.tixier@uvsq.fr, pouget@lmm.jussieu.fr
April 2020
Abstract. IPMCs consist of a Nafion® ionic polymer film coated on both sides
with a thin layer of metallic electrodes. The polymer completely dissociates when
it is saturated with water, releasing small cations while anions remain bound
to the polymer chains. When this strip is subject to an orthogonal electric
field, the cations migrate towards the negative electrode, carrying water away
by osmosis. This leads to the bending of the strip. We have previously published
a modelling of this system based on the thermodynamics of irreversible processes.
In this paper, we use this model to simulate numerically the bending of a strip.
Since the amplitude of the deflection is large, we use a beam model in large
displacements. In addition, the material permittivity may increase with ion
concentration. We therefore test three permittivity models. We plot the profiles of
the cations concentration, pressure, electric potential and induction, and we study
the influence of the strip geometry on the tip displacement and on the blocking
force. The results we obtain are in good agreement with the experimental data
published in the literature. The variation of these quantities with the imposed
electric potential allow us to discriminate between the three models.
Keywords : Electro-active polymers, Multiphysics coupling, Polymer mechanics,
Nafion, EAP modelling, Ionic polymer, EAP beam
PACS numbers : PACS 47.10.ab, PACS 83.60.Np, PACS 82.47.Nj, PACS 77.65.-j
published in Smart Materials and Structures 29(085019), 2020
https://doi.org/10.1088/1361-665X/ab8fcaValidation of a model for an ionic electro-active polymer in the static case 2
1. Introduction conductive polymers are such media. One of the
advantages of the these polymers is that they can be
The present work proposes a study of a thin strip activated by very low difference of electric potential of
of electro-active polymer (EAP). More precisely, the about 1 − 5 Volts. However, they can only be operated
study focuses on Ionic Polymer-Metal Composite within electrolyte medium.
(IPMC) belonging to the ionic class. We investigate Most efforts have been devoted on electronic con-
responses of the EAP subject to an applied difference ducting polymers (E.C.P.) belonging to the ionic class
of electric potential on the metallic electrodes and in order to improve strain, output forces and response
an applied punctual force at the tip of the strip. times. These polymers are used for multilayer poly-
In previous works we have constructed step by mer composite combined with a polymer which can
step a micro-mechanical model to establish the be considered as ion reservoir to improve ion transfer.
conservation laws for electro-active polymers (Tixier Trilayer actuators made of solid polymer electrolyte
& Pouget 2014). Following this work, the constitutive film sandwich between two electronic conducting poly-
equations were deduced from the hypothesis of local mers have been reported in Festin et al. (2014) and
thermodynamics equilibrium and the Gibbs relation Nguyen et al. (2018). One of the advantage of these
using the thermodynamics of irreversible processes electro-active polymers is that they can operate in air
(Tixier & Pouget 2016). The present study follows on and they are good candidates for biomimetic devices.
from the spirit of the previous works and we want to Efforts have been conducted to increase the integrat-
characterize the behavior of a thin blade, especially the ing biocompatible conducting polymers into contin-
chemical, electrical and mechanical parameters under uum micro-robots. Developments of such biocompat-
electro-mechanical loading. ible conducting polymers with the aim at designing
The behavior of electro-chemical-mechanical interac- accurate position control for the trajectory of the tele-
tions of EAP is of great interest for research and engi- scopic soft robots have been studied by Chikhaoui et
neering advanced technology. Simply, we can say that al. (2018).
an EAP is a polymer exhibiting a mechanical response,
such a stretching, contracting or bending for example, Modeling EAP is not an easy task, especially the model
when subject to an electric field. Conversely the EAP must include electro-mechanical and chemical-electric
can produce electric energy in response to a mechani- couplings of ion transport, electric field action and
cal load (Shahinpoor et al. 1998, Shahinpoor & Kim elastic deformation. Different kinds of approach have
2001, Bar-Cohen 2005, Pugal et al. 2010, Park et al. been proposed in the literature according to the un-
2010). This particular property is highly attractive for derlying physics of chemical activation (Shahinpoor &
applications. For instance, we can quote biomimetic Kim 2001, Brunetto et al. 2008, Deole et al. 2008,
devices (robotics, bio-inspired underwater robots such Bahramzadeh & Shahinpoor 2014). An instructive and
as fishes (Chen 2017), haptic actuators (artificial skin, comprehensive review paper devoted to IPMC has been
tactile displays or artificial muscle (Deole et al. 2008, proposed by Jo et al. (2013). The authors present the
Matysek et al. 2009, Bar-Cohen 2005) and this ma- chemico-physical mechanisms involved in IPMCs. As a
terial is an excellent candidate for energy harvesting complement to the previous references, a set of interest-
(Farinholt et al. 2009, Tiwari et al. 2008, Aureli et ing studies has been devoted to the modeling of EAP.
al. 2010, Jean-Mistral et al. 2010, Cha et al. 2013). Among them, Wallmersperger et al. (2009) develope a
In addition promising applications to micro-mechanical thermodynamically based mechanical model involving
systems (MEMS) at the sub-micron scale are also ex- the chemico-electric transport phenomenon. Numeri-
pected in medical engineering for accurate medical con- cal simulations are proposed for a strip of EAP made
trol or investigation, for instance Fang et al. (2007) and of Nafion® 117 Li+ . The authors deduce the profiles,
Chikhaoui et al. (2018). within the strip thickness, of the electric charge density,
electric potential, electric field and strain. Nardinoc-
Electro-active polymers can be categorized into chi et al. (2011) deduce a model based on the 3-D
two main groups depending on their activation theory of linear elasticity. The thermodynamics allows
mechanisms. The first one is electronic electro-active them to introduce chemo-electro-mechanical coupling
polymers and dielectrics which are subject to Coulomb and they deduce the constitutive equations of the ma-
force ; their volume change is due to the application terial, especially a Nernst-Planck like equation is de-
of an electric field. For instance dielectric elastomers duced. Their study continues with numerical illustra-
belongs to this category as well as piezoelectric and tions for a thin strip of EAP. Moreover, time evolution
electrostrictive polymers. The second group of EAP is at low frequency is proposed as well. It is notewor-
the ionic electro-active polymers. The latter are driven thy that these papers are partly, in their spirit, rather
by the displacement of ions inside the material. The close to the present model. Nevertheless, their results
polyelectrolyte gels, ionic polymer-metal composites,Validation of a model for an ionic electro-active polymer in the static case 3
depend strongly on adjustable parameters to fit the the dielectric permittivity.
experimental results. In their model, the Fourier law,
Darcy law and the generalized rheological constitutive 2. Modelling of the polymer
equation are not presented. Moreover, using their ad-
justable phenomenological parameters the authors de- 2.1. Description of the material and hypothesis
duce a dielectric permittivity greater than ours.
A continuum approach for multiphasic materials We study an IPMC (ionic polymer-metal composite):
has been investigated by Bluhm et al. (2016). The au- the system consists in an ionic electro-active polymer
thors write down the conservation laws and the entropy (EAP) blade of which the two faces are covered with
inequality based on the theory of porous media for an thin metal layers acting as electrodes. The model
arbitrary number of individual constituents. In spite of we developed applies for example to Nafion® , an
the rather thorough investigation, the authors do not ionic polymer well documented in the literature that
compare their results with those proposed in the liter- we will use for the validation of our constitutive
ature. However, their constitutive laws are related to equations. When saturated with water, this polymer
the material constituents and not to the macroscopic dissociates quasi completely, releasing small cations in
medium, and their coefficient are not numerically eval- water whereas anions remain bound to the polymer
uated. backbone (Chabe 2008). When a potential difference
is applied between the two electrodes, the cations
The present model is mainly based on an averag- migrate towards the negative electrode, carrying the
ing method of the microscopic description of phases water away by osmosis. As a result, the blade contracts
in order to deduce the macroscopic behavior of the on the side of the positive electrode and swells on the
material. Thanks to this approach, the constitutive opposite side, causing its bending (Figure 1).
coefficients for the whole material are computed with
the help of the microscopic physico-chemical properties
of the constituents (volume fractions, mass densities,
-
chemical potentials, electric charges, etc.) and physi-
+
cal meanings are identified.
An important polymer property that we would like to
address is the dependency of the dielectric permittiv-
ity with the cation concentration in the polymer. As + + - _
-++- -+ ++ -
_ _ _
+ _
+ + _
_ + + +
matter of fact, the dielectric constant of the polymer is - - _
+
_
+
_
not absolutely homogeneous within the strip thickness. +
_
_
+_
__ +
+
_ +
_
Accordingly, it seems that the ionic transport when a +
difference of electric potential is applied to the poly-
mer strongly modifies the dielectric constant along the
thickness direction of the blade. This question will be
examined and discussed in details in the forthcoming
sections.
The paper is organized as follows: the next section is -
devoted to the description of the IPMC and chemico- +
physical process of activation is briefly described. + -+ +
This section reports also the way of modeling the -++-+-+ -++-
EAP, especially, the conservation laws and constitutive - -- - -
equations are summarized. Section 3 focuses on the
application of the model to a slender beam made of
thin layer of EAP. A part of the section highlights Figure 1. Deformable porous medium: (a) Undeformed strip
the influence of the cation concentration within the (b) Strip bending under an applied electric field
thickness on the dielectric permittivity. Three kinds
of dielectric laws will be considered. Numerical To model the electro-active polymer, we used a
simulations are presented in Section 4. The profiles of ”continuous medium” approach. Negatively charged
the relevant variables and the scaling laws are reported. polymer chains are assimilated to a deformable,
Comparisons to the experimental data available in the homogeneous and isotropic porous medium in which
literature are discussed according to the chosen law of flows an ionic solution (water and cations). The
system is therefore composed of two phases and threeValidation of a model for an ionic electro-active polymer in the static case 4
components which move relative to each other: the the Maxwell equations (5) for the complete material
cations, the solvent (water) and the porous solid. (Tixier & Pouget 2014)
The solid and liquid phases (water + cations) are
∂ρ
→
−
separated by an interface without thickness. The + div ρ V = 0 , (1)
three components are respectively identified by the ∂t
→
−
subscripts 1, 2 and 3; 4 denotes the solution (1 + 2) DV →
−
and i the interfaces. Gravity and magnetic induction ρ = div σ + ρZ E , (2)
Dt e
are supposed to be negligible; the only external force
exerted is therefore the electric action. The different
X
D U
phases are supposed to be incompressible and the ρ = σk : grad V~k
Dt ρ 3,4 f
solution diluted. It is further recognized that solid
deformations are small. X
+ ~ I− ρk Zk V~k · E
~ − div Q
~ , (3)
k=3,4,i
2.2. Basis of the model
We used a coarse-grained model developed for two- →
− ∂ρZ
div I + = 0, (4)
component mixtures (Ishii & Hibiki 2006). We define ∂t
two scales. The conservation equations are first written →
− →
− →
− →
− →
−
rot E = 0 , div D = ρZ , D = εE , (5)
at the microscopic level for each phase and for the
interfaces. At this scale (typically about 100 A◦ ), where ρk denotes the densities relative to the volume
→
−
the elementary volume contains one phase only but is of the whole material, V the velocity, σ the stress
→
−
large enough so that the medium can be considered tensor, Z the electrical charge per unit of mass, E
e
→
−
as continuous. The macroscopic equations of the the electric field, U the internal energy density, I
→
− →
−
material are deduced by averaging the microscopic the current density vector, Q the heat flux, D the
ones using a presence function for each phase and electrical induction and ε the dielectric permittivity.
interface. For each phase k and for the interfaces, Subscripts refer to a phase, interface or a constituent
a Heaviside-like function of presence is defined. The and quantities without subscript are relative to the
macroscale quantities are obtained by averaging the whole material.
corresponding microscale quantities weighted by the We verify that the stress tensor of the whole
functions of presence. This volume average is material is symmetrical. The second member of
assumed to be equivalent to a statistical average equation (3) highlights the source terms of the internal
(ergodic hypothesis). At the macroscopic scale, the energy (viscous dissipation and Joule heating) and its
representative elementary volume (R.E.V.) must be →
−
flux Q . Equations (4) and (5) show that an EAP
large enough so that these averages are relevant, but behaves like an isotropic homogeneous linear dielectric.
small enough so that the average quantities could be In the last Maxwell equation, the permittivity of the
considered as local. According to Gierke et al. (1981) whole material is obtained by a local mixing law, and
and Chabe (2008), its characteristic length is about is therefore likely to vary over space and time.
1 µm.
To write the balance equations, it is necessary 2.4. Constitutive equations
to calculate the variations of the extensive quantities
for a closed system in the thermodynamic sense. At We make the hypothesis of local thermodynamic
the microscopic scale, we use the particle derivative equilibrium. The Gibbs relation of the whole material
or derivative following the motion of a constituent is deduced from the Gibbs relations introduced by de
or an interface. At the macroscopic scale, the Groot et Mazur (1962) for a deformable solid and for
three constituents velocities are different; we introduce a two-constituent fluid
a ”material derivative” or derivative following the d S d U d 1
D T = +p
movement of the three constituants Dt , which is a dt ρ dt ρ dt ρ
weighted average of the particle derivatives related to X
d ρk 1
→
−
each constituent (Coussy 1995, Biot 1977). − µk − p1 + σ e : grad V ,(6)
dt ρ ρ e f
k=1,2,3
2.3. Conservation laws where T is the absolute temperature, S the entropy
density, p the pressure, µk the mass chemical
We thus obtain balance equations of mass (1),
potentials, 1 the identity tensor and σ e the equilibrium
momentum (2), total, kinetic, potential and internal e d
stress tensor. dt denotes the derivative following the
f
energy densities (3), entropy, electric charge (4) and →
−
barycentric velocity V . At equilibrium, it is assumedValidation of a model for an ionic electro-active polymer in the static case 5
that the material satisfies Hooke’s law and that the + (static
electric+field +
+ +case). The other end A is either free
+ + +
liquid phase is newtonian and stokesian. _ _ to
or subject a shear force
_ _ _Consider
preventing its displacement
+
By combining the balance equations of internal (blocking force). _ _ an EAP +strip of length L,
_ _ + +
energy and entropy with the Gibbs relation, we of width 2l and of thickness 2e; we_ define
_ +
a coordinate E
_ the+length,
deduced the entropy production of the system. The system Oxyz such that the Ox axis is along _ _
thermodynamics of linear irreversible processes makes _
the Oy axis along its width and the Oz axis parallel
then possible to identify the generalized forces and to the imposed electric field (see figure 2). For all
fluxes and to deduce the constitutive equations of numerical applications, we choose the Nafion® Li+ ,
the electro-active polymer. According to the Curie an EAP well documented; in the nominal case, the
symmetry principle, a coupling between a force and dimensions of the strip are L = 2 cm, l = 2.5 mm
a flux of different tensorial ranks is impossible because and e = 100 µm and it is subject to an electric
of the isotropy of the medium. potential difference ϕ0 = 1 V . Considering these values
We thus obtained a Kelvin-Voigt type stress-strain and the exerted forces, this is a two-dimensional (x, z)
relation (7) and generalized Fourier’s, Darcy’s (8) and problem. The strip being thin and given the high
Fick’s laws (9). Given the orders of magnitude of the values of the deflection, we used a beam model in large
different physico-chemical parameters of the polymer displacements.
(in particular, we admit that the solution is diluted),
these equations can be written in the isothermal case
on a first approximation (Tixier & Pouget 2016) Z −p
→ −−→
p MAp
σ = λ tr 1 + 2G + λv tr˙ 1 + 2µv ˙ , (7)
e e e e e e e A X
−
→ − → K h
→
−
i
grad p − CF − ρ02 Z3 E ,
V4 − V3 ' − (8) 0 −
→
η 2 φ4 Fp
L
−
→ − →
V1 = V2 (9)
Figure 2. Forces exerted on the beam
D Z 1 M1 C →
− Cv 1 M v
1 2
− grad C − E+ 1− grad p ,
C RT RT M 2 v1 The force − system applied to the beam can be
Z →
ez →
−
where denotes the strain tensor, λ the first Lamé modelled by a distributed lineic force pp , a bending
−−→
constant, MAp applied
e G the shear modulus, λ and µ the
v v moment
O to the end A and in some cases
viscoelastic coefficients; η2 is the dynamic viscosity a shear force − →θ n
F p in A. The strip being very slender,
of the solvent, φk the volume fractions, K the we venture the hypothesis →
−
ex that the distributed force is
intrinsic permeability of the solid, C the cations molar independent of the x tcoordinate along the beam and is
concentration, F = 96487 C mol−1 the Faraday’s orthogonal to it. The internal electrostatic forces of the
constant, ρ02 the mass density of the solvent, D the mass stripθcancel R
each other. The electric force produced by
diffusion coefficient of the cations in the liquid phase, the electrodes also vanishes due to the electroneutrality
Mk the molar masses, vk the partial molar volumes and condition. We deduce that the distributed force is zero
R = 8.31J K −1 the gaz constant. everywhere.
The generalized Darcy’s law models the motion
of the solution compared with the solid phase. This
movement is caused by the pressure forces, and also + + + ++
+ + +
by the electric field, which reflects the electro-osmosis __ _ _
_ _ +
_ _ +
phenomenon. The third equation expresses the motion _ _ + + E
−
→ _ ++
of cations by convection (V2 ), by mass diffusion, under _
_ _
the actions of the electric field and the pressure field; _
it can be identified with a Nernst-Planck equation
(Lakshminarayanaiah 1969). An estimation of these Figure 3. EAP bending strip
different terms in the case of Nafion® Li+ shows that
the pressure one is negligible. When a potential difference ϕ0 is applied between
the two faces, the cations and the solvent move towards
3. Application of the model to a static the negative electrode, causing a volume variation and
cantilevered beam the bending of the strip (figure 3). The bending
moment
z is therefore exerted along the Oy axis and
3.1. Beam model on large displacements results from thepp pressure forcesMpA p = − σxx
3
Z l Z e Zx e
To validate this model, we apply it to an EAP strip p A
clamped at its end O under the action of a permanent MA = − σxx z dz dy =p 6l p z dz . (10)
O
−l −e F −e
L
z ez
O nValidation of a model for an ionic electro-active polymer in the static case 6
We made the usual hypotheses of a beam Fp MAp
θ= s (2L − s) − s. (16)
model: we assumed that the straight sections of the 2EI p EI p
strip remain flat and normal to the neutral fibers The deflection w is obtained by integrating the relation
after deformation (Bernoulli hypothesis) and that the dz
ds = sin θ.
stress and strain distributions are independent of the If F p = 0, the beam is circle shaped (Rp is
Z
application points of the−
→pexternal forces (Barré Saint
−−→ constant)
Venant hypothesis). We p define a local coordinate MAp h p i
Mp
→
− p M
system where t and → −
n are the tangent and normal w = EI
Mp
cos EIAp L − 1 , θ = − EIAp L . (17)
A
vectors; s and s denote the curvilinear abscissas along X A
It should be noted that this deflection calculation
the
0 beam at the rest and deformed configurations −→ becomes incorrect if the angle of rotation exceeds 90◦ .
Fp
respectively, n the coordinate in the normal direction
With the hypothesis of small displacements, we would
L of a cross-section (figure 4).
and θ the angle of rotation
have obtained
Let choose the point O as the origin of the curvilinear
abscissas. No normal effort is applied, so we assumed Mp
ws = − Ap L2 . (18)
that there is no beam elongation (ds = ds). 2EI
If a blocking force is exerted, the deflection is zero
Z −
→ which provides on small displacements
ez
3MAp
Fsp = . (19)
O θ 2L
n
On large displacements, F p verify
MAp
Z L
−
→
p
ex F
w= sin x(x − 2L) + x dx = 0 . (20)
t 0 2EI p EI p
Let us compare this result with the calculation
θ R on small displacements. The above condition can be
written
x2
RL h i
2
w = 0 = 0 sin x1∗2 (x + x0 ) − x∗02 dx
2
x
= x∗ cos x∗02 [S (x1 /x∗ ) − S (x0 /x∗ )] (21)
2
x
Figure 4. Beam on large displacements: coordinate system −x∗ sin x∗02 [C (x1 /x∗ ) − C (x0 /x∗ )] ,
The bending moment in the current section M p where
and the radius of curvature Rp are p p
q
2EI p MA MA
x∗ = Fp , x0 = Fp − L, x1 = Fp , (22)
1 dθ
M p = F p (s − L) + MAp , = . (11)
Rp ds and where S and C denote the Fresnel functions
Let →−
u the displacement vector; its gradient with Rx +∞
n x4n+3
S(x) = 0 sin t2 dt
P
respect to the reference configuration (beam at rest) = (−1) (2n+1)! (4n+3) ,
n=0
is Rx +∞ (23)
2 n x4n+1
P
1 − Rnp cos
θ−1 − sin θ
C(x) = 0
cos s dt = (−1) (2n)! (4n+1) .
→
−
Grad u = . (12) n=0
1 − Rnp sin θ
cos θ − 1 ®
In the case of Nafion Li+ , E = 1, 3 108 P a (Bauer
One deduces the strain tensor et al. 2005, Barclay Satterfield & Benziger 2009,
1h T i
Grad →− Grad →
− Silberstein & Boyce 2010). Using the blocking force
= u +1 u +1 −1 . (13)
e 2 e e e values provided by the literature in the nominal case
The beam is thin, so |n|Validation of a model for an ionic electro-active polymer in the static case 7
3.2. Additional hypothesis and static equations for a
Table 1. Dielectric permittivity values
bending strip
ε0 (F m−1 ) εmoy (F m−1 )
Let us evaluate the variations of the volume fraction of
constant 5 10−7 5 10−7
the solution φ4 . Consider a small volume dV located linear 0 10−4
at a distance z from the beam axis. According to affine 5 10−7 10−4
Bernoulli’s hypothesis, this volume takes the value
|Rp |+z
|Rp | dV when the beam bends with a radius of
curvature Rp . The volume of the solid phase does The permittivity strongly depends on the conduc-
not change, and the variation of liquid phase volume tivity, hence of the electric charge; it increases with the
fraction is about water uptake (Deng & Mauritz 1992, Nemat-Nasser
φ3 z Mp 2|w| 2002), therefore with the cations concentration. We
dφ4 = p = (1 − φ4 ) p z ' (1 − φ4 ) 2 z . (25)
|R | EI L assume that it satisfies a mixing law
φ4 ' 0.38 (Cappadonia et al. 1994, Chabé 2008, εmoy − ε0
Nemat-Nasser & Li 2000). In the nominal case, w ∼ ε = ε0 + αC , with α= , (29)
Cmoy
1 mm (Nemat-Nasser 2002, Newbury & Leo 2002,
where εmoy denotes the average permittivity of the
Newbury 2002). φ4 therefore varies less than 0.1%
material. We have considered three models of
over the thickness of the beam. As a consequence, we
permittivity: constant, linear and affine. We choose
assume that the volume fraction φ4 is constant.
our permittivity values in order that deflections and
Considering the dimensions of the strip, we
blocking forces are in agreement with the literature
assume that the problem is two-dimensional in the
data, namely 0.5 < w < 1.5 mm (Nemat-Nasser 2002,
Oxz plane. On a first approximation, we suppose that
Newbury 2002) and 0.6 < F p < 1.3 mN (Newbury
the electric field and induction are parallel to the Oz
→
− →
− 2002, Newbury & Leo 2002, Newbury & Leo 2003) in
axis: E ' Ez → −
ez and D ' Dz → −
ez . We further admit
the nominal case (table 1).
that C, Ez , Dz , p, ρZ and the electrical potential ϕ
The Nafion relative permittivity found in the
only depend on the variable z. Finally, we neglect the
literature are very scattered, but compatible with
pressure term of the equation (9), an assumption that
those we have chosen. The average permittivity
we will verify later. The equation system of our model
was measured by Deng & Mauritz (1992) for
then becomes
hydrated perfluorosulfonate ionomer membranes of the
Ez = − dϕdz ,
dDz
dz = ρZ , Nafion® family with different water contents. They
Dz = εEz , ρZ = φ4 F (C − Cmoy ) , (26) obtained permittivity values between 10−7 F m−1 and
dp 0
dC FC 10−6 F m−1 by electrical impedance measurements.
dz = CF − ρ2 Z3 Ez , dz = RT Ez ,
Nemat-Nasser (2002) deduced the permittivity of
where capacity measurements by assimilating the IPMC strip
(1 − φ4 ) ρ03 Z3 to a capacitor; for Nafion® 117 Li+ , he got ε '
Cmoy = − . (27)
φ4 F 2.7 10−3 F m−1 . Farinholt & Leo (2014) deduced
Cmoy denotes the cations average concentration. The a close value from their measurements but did not
anions being attached to the polymer chains, they specify their method. Wang et al. (2014) measured
® +
are uniformly distributed within the material; their the permittivity of Nafion 117 N a by time domain
concentration is therefore constant and equal to Cmoy dielectric spectroscopy. For samples obtained with
considering the electroneutrality condition. From different manufacturing methods and a water content
−5
Nemat-Nasser & Li (2000), the mass density of dry about 22%, the permittivity ranges from 5 10 to
−3
0 3 −3
Nafion is ρ = 2 10 kg m , and its equivalent weight, 5 10 .
3
that is the weight of polymer per mole of sulfonate
groups, is Meq = 1.1 kg eq −1 (Chabé 2008, Colette 3.3. Resolution with different permittivity models
2008), which provides Z3 = − MFeq = −9 10−4 C kg −1 .
Let us introduce dimensionless variables
We deduce Cmoy = 3080 mol m−3 . We also choose an
E = Eϕz0e , ϕ = ϕϕ0 , D = eφ4 FDCz moy ,
absolute temperature T = 300 K.
C ρZ p
The boundary conditions and the electroneutrality C= Cmoy , ρZ = φ4 F Cmoy , p= F ϕ0 Cmoy , (30)
condition are z= z εϕ0
Re e , ε= e2 φ4 F Cmoy .
ϕ(−e) = ϕ0 , ϕ(e) = 0 , −e
ρZ dz = 0 . (28)
The system of equations and the boundary conditions
According to (26), this last condition is equivalent to
Dz (e) = Dz (−e).Validation of a model for an ionic electro-active polymer in the static case 8
become
Table 2. Center and boundary values of the different quantities
E = − dϕ
dz , D = εE ,
−1 center 1
dD
dz = ρZ , ε = A0 C + A1 ,
C B1 exp (−A2 ) 1 B1
dp dC (31)
dz = C + A3 E , = A2 CE , ρZ B1 exp (−A2 ) − 1 0 B1 − 1
dz ln B1
ϕ 1 0
ρZ = C − 1 , D(1) = D(−1) , A2
D D(1) 0 D(1)
ϕ(−1) = 1 , ϕ(1) = 0 , ε A1 A0 + A1 A0 B 1 + A1
B1 exp(−A2 )
p − B3 − A3 1
− A3Aln B1 B1
with A 2 A 2 2 A2
ϕ0 (εmoy −ε0 ) ϕ0 ε0
q
1
A0 = e2 φ4 F Cmoy , A1 = e2 φ4 F Cmoy , where D(1) = A2
[A0 + 2A1 (A2 − 1 − ln B1 )]
(32)
F ϕ0 ρ02 Z3
A2 = RT ∼ 38.7 , A3 = − Cmoy F ∼ 0.303 ,
in the nominal case. We deduce the following relations The bending moment is given by
R1
C = B1 exp (−A2 ϕ) , (33) MAp = A5 M = A5 −1 ( AC2 − A3 ϕ)z dz ,
2
R1
D
1 A0 2
B2 with M = −1 p z dz ,
= C + (A1 − A0 )C + A1 ϕ + , (34) (41)
2 A2 2 2 A5 = 6le2 F ϕ0 Cmoy ,
R1 A0
C −1
Cz dz = 2D(1) − A 2 B1 − A1 .
p= − A3 ϕ + B 3 , (35)
A2 R1
Iϕ = −1 ϕ z dz must be numerically evaluated. We
where B1 , B2 and B3 are three constants. The deduce w, ws , F p , Fsp and θ using (17) to (20).
polymer strip behaves like a conductive material. The The resolution of the equation system is tricky
electric field, displacement and charge are then zero from a numerical point of view because of the steepness
throughout the strip except near the sides. We can of the functions near the boundaries. We used different
deduce methods according to the permittivity model to obtain
1 the best precision.
B2 = (A0 − 2A1 − 2A1 ln B1 ) . (36) In the case of a constant permittivity (A0 = 0),
A2
we use the variable y = ln C which verifies
The positive constant B1 satisfies the electroneu-
trality condition (31) d2 y A2 y
dz 2 A1 (e − 1)q
= ,
2A1 A2 dy
' dzdy
= 2AA1 (A2 − lnA2 − 1) , (42)
2
A0 (1 + e−A2 )B12 + 2(A1 − A0 )B1 − = 0 .(37) dz
1 R −1
1 − e−A2 1 R1
Iϕ = −1 ϕ z dz = − A12 −1 yz dz .
When ϕ0 & 1 V , e−A2Validation of a model for an ionic electro-active polymer in the static case 9
In the case of an affine permittivity, the following the constant case and 0.03 µm in the affine one. The
equation can be numerically integrated over the entire linear model predicts a permittivity tending towards
interval 0 and is therefore incorrect in this range. Near the
negative electrode, there is an accumulation of cations
d2 D A2 1 + dD dz
= D. (45) over a characteristic length depending on the chosen
dz 2
A0 1 + dD + A1 permittivity model: close to 0.1 µm for a constant
dz
permittivity and 1 µm in linear and affine cases.
The concentration on the negative electrode is twenty
4. Simulation results times higher with the constant permittivity model
than in the two other cases. Nemat-Nasser (2002),
4.1. Hypothesis validation
Wallmersperger et al. (2009) and Nardinocchi et al.
The dimensionless equation (9) provides (2011) obtained a similar profile, although less steep.
The electric potential profiles look similar to
dC
= A2 C E 1 + A4 C + A3 , (46) those obtained by Wallmersperger et al. (2009) and
dz Nardinocchi et al. (2011), although they are less
where (Tixier & Pouget 2016) steep in the vicinity of the electrodes. The linear
2
Cmoy M1 M 2 and affine models give almost identical results for the
M 1 v2
A4 = Cmoy v1 1 − ' 02 ' 1.2 10−3 , other profiles. The constant model distinguishes by
M 2 v1 ρ2
its steepness near the boundaries, with a characteristic
−1 −1
(M1 = 6.9 g mol and M 2 = 18 g mol length close to 0.02 µm near the negative electrode,
for Nafion® Li+ ). A4 C + A3 corresponds to the twenty times smaller than the other models; near the
pressure term; its maximum value (about 0.04) is positive electrode, its characteristic length is 0.1 µm,
reached near the negative electrode in the case of a five times larger than the other models for electric
constant permittivity, and it is of the order of A4Validation of a model for an ionic electro-active polymer in the static case 10
40
35
30 1 5
25
20 0.5 2.5
15
10 0 0
-1 -0.995 -0.99 0.99 0.995 1
5
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Figure 5. Variation of the dimensionless cation concentration in the thickness of the strip; the distribution close to the boundaries
are detailled in insets. The constant permittivity model is in blue, the linear one in red and the affine one in green.
Dimensionless electric potential
1
0.9
0.8
1 0.1
0.7
0.6
0.5 0.5 0.05
0.4
0.3 0 0
0.2 -1 -0.995 -0.99 0.99 0.995 1
0.1
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Figure 6. Variation of the dimensionless electric potential in the thickness of the strip; the distribution close to the boundaries are
detailled in insets (same colours as in figure 5).
Dimensionless electric displacement
0.002
0.0016
0.002 0.002
0.0012
0.001 0.001
0.0008
0.0004 0 0
-1 -0.995 -0.99 0.99 0.995 1
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Figure 7. Variation of the dimensionless electric displacement in the thickness of the strip; the distribution close to the boundaries
are detailled in insets (same colours as in figure 5).
and therefore depends only on ϕ0 and the material. the center of the strip and is very steep near the
The pressure profile is almost constant throughout boundaries. It can be modelled by a constant betweenValidation of a model for an ionic electro-active polymer in the static case 11
1
0.3
0.8
0
0.6
0.1
-0.2
0.4
-0.4 -0.1
0.2 -1 -0.9975 -0.995 0.995 0.9975 1
0
-0.2
-0.4
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Figure 8. Variation of the dimensionless pressure in the thickness of the strip; the distribution close to the boundaries are detailled
in insets (same colours as in figure 5).
two values −z1 and z2 with z1 , z2 . 0.99. Assuming
Table 3. Expected scaling laws
for example 0 < z1 < z2
R −z Rz L l e ϕ0 ϕ0 ϕ0
M = −1 1 p z dz + −z1 1 p z dz ε constant linear affine
Rz R1 (50)
+ z12 p z dz + z2 p z dz . MAp
- l e ϕ0
3/2
- f (ϕ0 )
3/2
Given the low values of δ1 = 1 − z1 and δ2 = 1 − z2 , θ L - e−2 ϕ0 - f (ϕ0 )
3/2
we can expand in Taylor series up to the second order Fp L−1 l e ϕ0 - f (ϕ0 )
3/2
the first and last integrals ws L2 - e−2 ϕ0 - f (ϕ0 )
R −z1
−1
p z dz = −p(−1)δ1 + o δ12 ,
R z1
−z1
p z dz = 0 , When the imposed electric potential is very small,
R z2 (51) B1 tends to 1 for all permittivity models, C tends to
p z dz = p(0) z2 2 − z1 2 ,
z1 2 Cmoy and ϕ to ϕ0 over the entire thickness of the strip.
R1
z2
p z dz = p(1)δ2 + o δ22 . We check that all the mechanical quantities become
Hence, in all cases and at the first order in δ null, which is in agreement with the experimental
results.
B1 1
M = A3 δ 1 + δ2 + (1 − A3 ln B1 ) (δ1 − δ2 ) . (52)
A2 A2
4.4. Influence of the strip geometry
δ1 and δ2 can be roughly evaluated using the following
formulas (thanks to eq. (31)) Our numerical simulations ascertain the results of the
previous section with an excellent correlation: for the
dp p (−z1 ) − p(−1) dp p(1) − p (z2 )
∼ , ∼ .(53) three permittivity models, the bending moment varies
dz −1 δ1 dz 1 δ2 linearly with the width and thickness of the strip and
M is therefore independent of L and l and approxi- is independent of its length.
mately inversely proportional to e. Its dependence on The blocking force is proportional to the width
ϕ0 is more complex and linked with the chosen permit- and inversely proportional to the length, which is in
tivity model. Given the values of a0 , a1 , a2 , A3 and good agreement with the results of Newbury & Leo
B1 in the nominal case, we obtain with a precision of (2003). It is also proportional to the thickness.
about 20% The predictions of the previous section are also
p a1 √ϕ0 well fitted by the deflection in large displacements; it
Constant case M ' a3 e ,
q2 is independent of the width and approximately pro-
Linear case M ' aa03 eϕ1 0 , portional to L2 : according to the chosen permittivity
2
M ' aa03 feϕ (54) model, the power law that best approximates our sim-
q (ϕ0 )
Affine case 0
, ulations has an exponent between 1.90 and 1.96, and
2
a1 2
a a ϕ
2 3 0
2
+2
a1
a ϕ
a0 2 0 +1 the variations of the ratio w/L2 are less than 15% in
with f (ϕ0 ) = a0 q a . all cases (figure 9). This is consistent with the results
1+2 a1 a2 ϕ0
0
of Shahinpoor (1999). It varies almost like e−2 : we
We deduce the scaling laws presented in table 3.Validation of a model for an ionic electro-active polymer in the static case 12
(b) (c)
70 1.2 100
60
Angle of rotation (deg)
1
80
Blocking force (mN)
Deflection (mm)
50
0.8
60
40
0.6
30
40
0.4
20
20
0.2
10
0 0 0
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
Length (mm) Length (mm) Length (mm)
Figure 9. Influence of the length: (a) on the deflection; (b) on the blocking force; (c) on the angle of rotation. The constant
permittivity model is in blue disks, the linear one in red diamonds and the affine one in green triangles. Fitting by power laws (solid
curves) or linear law (dashed curves).
(a) (b) (c)
12 1
300
10
Angle of rotation (deg)
Blocking force (mN)
0.8 250
Deflection (mm)
8
200
0.6
6
150
0.4
4 100
0.2
2 50
0 0 0
0 50 100 150 200 0 50 100 150 200 0 50 100 150 200
Thickness (µm) Thickness (µm) Thickness (µm)
Figure 10. Influence of the thickness: (a) on the deflection; (b) on the blocking force; (c) on the angle of rotation (same colours,
marks and curves as in figure 9).
find an exponent between −1.90 and −1.96 according
to the chosen permittivity model and the variations of 40
the product we2 are less than 14% (figure 10). This
Bending moment (µNm)
result is corroborated by the measurements of He et 30
al. (2011) as well as by the simulations of Vokoun et
al. (2015). 20
We also observe that the charge of the negative
electrode F C(e) = F B1 Cmoy is independent of the
10
thickness e with the three permittivity models, which
agrees with the results obtained by Lin et al. (2012)
for a close material. 0
0 1 2 3 4
Imposed electric potential (V)
4.5. Influence of the imposed electric potential
Unlike scaling laws, the relation between the different Figure 11. Influence of the electric potential on the bending
mechanical quantities and ϕ0 depends on the chosen moment: fitting by power law (solid curve), affine laws (dashed
curves) and with equations (54) (dotted curves); same colours
permittivity model. According to equations (47), the and marks as in figure 9.
angle of rotation, the blocking force, the deflection in
small displacements and the bending moment vary over
the imposed potential in the same way. First, we check tally an approximately linear relation between the de-
that the bending moment tends to 0 when ϕ0 tends flection and the potential for actuators, and Shahin-
to 0 in all three cases using Taylor expansions. For poor et al. (1998) and Mojarrad & Shahinpoor (1997)
imposed potentials close to 1 V , we have seen that did the same for the sensor effect. Our simulations
MAp is proportional to ϕ0 in the case of a constant
3/2
show a quite good correlation with a linear law if
permittivity, is almost constant if the permittivity is the permittivity is constant (the variation of the ra-
linear and is a complex function f (ϕ0 ) in the case of tio w/ϕ0 is about 30%); on the contrary, the results
an affine permittivity (table 3, figure 11). obtained with the other two models of permittivity do
Bakhtiarpour et al. (2016) observed experimen- not agree with these experimental results. More pre-Validation of a model for an ionic electro-active polymer in the static case 13
(a) (b) (c)
14 3 100
12
Angle of rotation (deg)
2.5
80
Blocking force (mN)
Deflection (mm)
10
2
60
8
1.5
6
40
1
4
20
0.5
2
0 0 0
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
Imposed electric potential (V) Imposed electric potential (V) Imposed electric potential (V)
Figure 12. Influence of the electric potential: (a) on the deflection; (b) on the blocking force; (c) on the angle of rotation (same
colours marks and curves as in figure 9).
cisely, in the constant case, the curve that best fits our The hydrated IPMC can be considered as a five-
numerical results is a power law of exponent 1.16 (fig- layer capacitor: the electrodes of thicknesses e1 and e5 ,
ure 12); this curve can hardly be distinguished from an the central area of thickness e3 very large compared to
experimental linear curve (Tixier & Pouget 2018). The the previous thicknesses and of zero electric charge,
blocking force and the angle of rotation follow approx- and two very thin zones of the polymer of respective
imately the same trend (good correlation with a linear thicknesses e2 and e4 in the vicinity of the positive and
law and with a power law of exponent 1.26). This re- negative electrodes. Each element has a permittivity
sult is in good agreement with Hasani et al. (2019) εi and a capacity Ci = εi eLi , and the five elements
experimental data, which can be well fitted by a power are in series. The permittivity of a material being
law of exponent 1.5. In the linear case, the moment closely related to its conductivity, so here to the cation
is independent of the imposed potential for ϕ0 & 1, concentration, ε1 = ε5 >> ε2 >> ε3 & ε4 . In addition
as well as the deflection, the blocking force and the e3 is much greater than e1 , e2 , e4 and e5 . We deduce
angle of rotation, which does not correspond to the e
experimental observations. In the affine case, the cor- ε = e1 +e5 e2 e3 e4 ' ε 3 ,
ε1 + ε2 + ε3 + ε4
relations with a linear law and with a power law are
wrong for all the quantities. The variation of the dif- where e denotes the total thickness. The overall
ferent quantities with the imposed electric potential is permittivity of the strip subject to an electric field is
thus discriminating for the permittivity model: only a therefore very close to the permittivity of the central
constant permittivity gives results compatible with the part, which is that of the hydrated polymer without
experimental studies. electric field. Our average permittivity values range
We can evaluate the average permittivity of the from 5 10−7 F m−1 to 10−4 F m−1 depending on
strip by analogy with a capacitor of thickness e and the permittivity model and are thus compatible with
surface Ll subject to a potential difference ϕ0 . We the permittivity values measured by Deng & Mauritz
saw that the cations accumulate near the negative (1992) and Wang et al. (2014), which range from
electrode over a thickness of e0Validation of a model for an ionic electro-active polymer in the static case 14
electrical impedance measurements, but significantly L: length of the strip;
lower than those deduced from capacity measurements. Mk : molar mass of component k;
The resolution of the equations of our model enabled
Meq : equivalent weight (weight of polymer per mole
us to plot the cation concentration, pressure, electric
of sulfonate groups);
potential and displacement profiles over the thickness
of the strip. These quantities are almost constant in ~ p (M~ p ): bending moment;
M A
the central part, but vary drastically in the vicinity of ~n (n): normal vector (coordinate) to the beam;
the electrodes, which is characteristic of a conductive
p: pressure;
material. The scaling laws obtained for the deflection →
−p
and the blocking force are in good agreement with p : distributed electric force
the experimental data published in the literature: in ~ heat flux;
Q:
particular, the deflection varies as the square of the R = 8, 314 J K −1 : gaz constant;
strip length and is inversely proportional to the square
Rp : radius of curvature of the beam;
of its thickness; the blocking force is proportional to the
width and the thickness and it is inversely proportional s (s): curvilinear abscissa along the beam at rest
to the length. The variation of the mechanical (deformed)
quantities with the imposed electric potential depends S: entropy density;
on the chosen permittivity model; only the constant T : absolute temperature;
permittivity model provides results compatible with
~u: displacement vector;
the experimental data and will therefore be retained
for further works. We now plan to apply our model U : internal energy density;
to other materials close to the Nafion® and to study vk : partial molar volume of component k (relative to
other configurations such as a strip clamped at its two the liquid phase);
ends. V (V~k ): velocity;
~
w (ws ): deflection of the beam on large (small)
Notations displacements;
k = 1, 2, 3, 4, i subscripts respectively represent Z (Zk ): total electric charge per unit of mass;
cations, solvent, solid, solution (water and cations) ε (ε0 , εmoy ): permittivity (average permittivity);
and interface; quantities without subscript refer to the : strain tensor;
whole material. Superscript 0 denotes a local quantity;
η2 : dynamic viscosity of water;
e
the lack of superscript indicates average quantity at
the macroscopic scale. Superscript T indicates the θ: angle of rotation of a beam cross section;
transpose of a second-rank tensor. Overlined letters λv , µv : viscoelastic coefficients;
denote dimensionless quantities. µk : mass chemical potential;
Ai , Bi : dimensionless constants; ρ (ρk ): mass density relative to the volume of the
C, Cmoy : cations molar concentrations (relative to whole material;
the liquid phase); ρ0k : mass density relative to the volume of the phase;
D: mass diffusion coefficient of the cations in the σ (σk ), σ e : stress tensor, equilibrium stress tensor;
liquid phase;
e f f
φk : volume fraction of phase k;
~
D: electric displacement field; ϕ (ϕ0 ): electric potential (imposed electric potential);
e: half-thickness of the strip;
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