COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Finite Element Analysis of Electrochemical-Poroelastic Behaviors of
Conducting Polymer (PPy) Films
W. S. Jung*, Y. Toi
Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo, 153-8505 Japan
Email: [email protected]
Abstract Computational modeling has been formulated for the electrochemical-poroelastic behaviors of conducting
polymers. The three-dimensional continuum modeling based on poroelastic theory for the passive poroelastic
behavior of Polypyrrole (PPy) films given by Della Santa et al. has been extended to the analysis of active
electrochemical-poroelastic behavior. The system of governing equations has been established by combining the
above modeling with the one-dimensional ionic transportation equations given by Tadokoro et al. The established
equations have been discretized by the finite element method. The validity of the proposed computational modeling
has been illustrated by comparing the calculated solutions for the passive and active behaviors of polypyrrole films
with the theoretical and experimental results.
Key words: Polypyrrole, electrochemical-poroelastic behaviors, finite element method, conducting polymer
INTRODUCTION
In many fields of engineering application, as artificial muscles, minimally invasive microsurgery, microhydraulics and
micromachines demanding a high level of miniaturization and simple structure, the development of new
microactuators based on smart materials represents a primary task [1-5]. The ideal actuator must feature properties,
such as linearity, Durability (long lifetime, high stability), high power-to-weight ratio, a large degree of compliance
and the possibility to be direct-driven. Among smart materials, Conducting polymers (CPs) show, in addition to
electrical and optical properties, interesting mechanical behavior in response to various external stimuli. [4-5]:
actuators based on the conducting polymers show significant dimensional changes upon doping of CPs (from 1% to
10% linear) and the high stresses developed (from 1 to 500Mpa) can provide a corresponding increase in work
capacity per cycle [2].
Among various CPs, Polypyrrole has been more extensively investigated in order to realize an actuator. The analytical
solution for the passive mechanical behavior of PPy film saturated by fluid has been already discussed by Della Santa
et al [1]. Tadokoro et al. supposed actuator model on the basis of physicochemical hypotheses that explain the
movement of ion in the actuator membrane [6]. The finite element formulation for the electrochemical–mechanical
behavior of IPMCs (ionic conducting polymer-metal composites) caused by eigenstrains is given by Toi et al [7-8].
In this paper the finite element analysis for the passive mechanical behavior of PPy film is conducted and also extend
to include electrochemical coupling.
FUNDAMENTAL EQUATIONS
In order to describe the mechanical behavior of the PPy film we utilize the poroelastic theory of Biot [9], successfully
modified and applied to the passive model of fluid-saturated PPy film and extend here to the active model by
considering doping and de-doping. The following assumptions are used: small deformations (as in the classical theory
of elasticity); a porous elastic and isotropic solid matrix; globally incompressible biphasic aggregate [1].
1. Poroelastic theory The stress tensor in a porous material is the sum of the fluid and solid tensor. The stress
tensors of the solid phase, σ ijs , and fluid phase, σ ijf ,are defined as follows: if we consider a cube of unit size of the
⎯ 581 ⎯
bulk porous material,
σ ijs represents the tension applied to the solid portion of the cube faces while σ ijf represents the
pressure applied to the fluid portion of the cube faces. The previous definition implies that
σ ijf = − βPδ ij , where β is
δ ij is the delta of Kronecker. In the hypothesis of a conservative system, the constitutive
the porosity of the material and
equations of the porous material are:
σ ijt = σ ijs + σ ijf
σ ijs =
(1a)
νE
E s
ε ij +
e s δ ij
(1 + ν )(1 − 2ν )
1 +ν
(1b)
σ ijf = − β Pδ ij
(1c)
The equilibrium equation of the total stress field also holds.
∂σ ijt
=0
∂x j
(2)
Finite element formulation based on the principle of the virtual work for Eq.(2) leads to the following equation:
∫ δ ({ε } + {Δε } )({σ }+ {Δσ })dV = 0
T
T
t
t
(3)
Ve
where
{ε }: total strain tensor,
{σ }: total stress tensor,
t
If external force is zero, the new form of finite element formulation is obtained by substituting Eq. (1a) into the Eq. (3)
∫ δ {Δε } {Δσ }dV = −∫ δ {Δε } {Δσ }dV
T
T
s
Ve
f
(4)
Ve
The incremental nodal displacement vector {Δu} and the variation of total pressure ΔP may also be written in a
form consisting of shape function, [N u ] and [N ] which are a bilinear function of 8-node rectangular hexahedronal
element.
{Δu} = [Nu ]{ΔuN }
(5a)
ΔP = [N ]{ΔPN }
(5b)
Substituting Eq. (5a) and Eq.(5b) into Eq. (1b) and Eq (1c), we have
{Δ σ } = [D ]{Δ ε } = [D ][B ]{Δ u }
(6a)
{Δσ } = −β [H ] ΔP
(6b)
s
s
s
N
T
f
Finally, the stiffness equations for porous material are obtained using Eq. (6a) and Eq. (6b)
[K ]{ΔuN } = β [B* ]{ΔPN }
(7)
where
[K ] = ∫V [B]T [D s ][B ]dV
(8a)
[B ] = ∫ [B] [H ][N ]dV
(8b)
e
T
*
Ve
2. Poisson equation The Onsager-like laws for the coupled fluxes of charges (ions) J and mass V f are [10]
J = K11∇ϕ + K12∇P
V
f
(9a)
= K 21∇ϕ + K 22 ∇P
(9b)
⎯ 582 ⎯
where K ij are phenomenological coefficients relating the two fluxes with the two driving forces: the electrical
potential gradient ∇ϕ and the pressure gradient ∇P . Eq. (10) is obtained by applying divergence operator to Eq. (9a)
and Eq. (9b) and eliminating ∇ 2ϕ .
⎛
K K
∂e s K 21
=
∇J + ⎜⎜ K 22 − 21 12
∂t
K 11
K 11
⎝
⎞ 2
⎟⎟∇ P
⎠
(10)
The following equations also hold.
div V f =
∂e f
∂t
(11)
∂e s
β ∂e f
=−
∂t
1 − β ∂t
(12)
(1 − β ) ∇ 2 P
∂e f
=−
f
∂t
(13)
The new form of Poisson equation for electrochemical model can be derived by substituting Eq. (11) and Eq. (12) into
Eq. (10) and in contrast to that of passive model.
∂e s
−β
β
= C1
∇J + C 2 ∇ 2 P
(1 − β )
∂t
f
(14)
The volume change and charge ratio are assumed to be bilinear function of the element coordinates in each element
as given by the following equations:
∂e s
= [N ] e& Ns
∂t
{ }
∇J = −
(15a)
∂c
= −[N ]{c& N }
∂t
(15b)
The following equation is obtained by the Galerkin finite element formulations for Eq.(14)
T ⎛ ∂e
∫V [N ] ⎜⎜
s
⎝ ∂t
e
+ C1
β
(1 − β )
β
⎞
∇ 2 P ⎟⎟dV = 0
f
⎠
∇J − C2
(16)
Substituting Eq.(15a) and Eq.(15b) into Eq.(16), we have
[S ]{e& Ns } = C1
β
(1 − β )
[S ]{c& N } − C 2
β
f
[A ]{PN }
(17)
where
[s] = ∫V [N]T [N]{e&Ns }dV
(18a)
e
T
T
T
⎞
⎛
[A] = ∫V ⎜⎜ ∂[N] ∂[N] + ∂[N] ∂[N] + ∂[N] ∂[N] ⎟⎟dV
e
⎝ ∂x
∂x
∂y
∂y
∂z
(18b)
∂z ⎠
3. Motion law The following equation can be derived by means of relation Eq.(11), Eq. (12) and Eq. (13)
(1 − v )E ∇ 2 e s = β ∇ 2 P
(1 + v )(1 − 2v )
(19)
Substituting Eq. (14) into Eq. (19), we have the motion law of the system.
L∇ 2 e s =
1 ∂e s C1 β
+
∇J
C2 ∂t C2 (1 − β )
(20a)
where
⎯ 583 ⎯
L=
(1 −ν )E
f (1 + ν )(1 − 2ν )
(20b)
The following equation is obtained by the Galerkin finite element formulations for Eq. (20a),
⎛
∫ [N ] ⎜⎜⎝ L∇ e
T
2 s
Ve
⎞
1 ∂e s C1 β
−
∇J ⎟dV = 0
C2 ∂t C2 (1 − β ) ⎟⎠
−
(21)
The previous equation, Eq. (21), changes to
T
s
∫V [N] [N]{e&N }dV = C1
e
⎛ ∂[N]T ∂[N] ∂[N]T ∂[N] ∂[N]T ∂[N] ⎞ s
β
T
⎟{eN }dV
⎜
&
{
}
[
N
]
[
N
]
c
C
J
+
+
−
2 ∫ ⎜
N
V
(1− β ) ∫
∂y ∂y
∂z ∂z ⎟⎠
⎝ ∂x ∂x
(22)
e
and also is rewritten by using Eq. (18a) and. Eq. (18b)
[S ]{e&Ns } = C1
β
(1 − β )
[S ]{c& N } − C 2 J [A]{e Ns }
(23)
4. One-dimensional ion transport equations When PPy film is under an electric field, the volume change is
accelerated by doping and de-doping. Ions are subjected to the electric forces to the cathode sides and viscous
resistance force and diffusion forces to the anode directions. Based on the balance of these forces, the total charge
Q ( x , t ) is expressed as follows [6]:
⎫
∂Q ( x, t )
∂ 2 Q ( x, t ) ∂Q ( x, t ) ⎧ e ⎡ t
= kT
−
× ⎨
η1
i (τ )dτ + Q ( x, t ) − Q( x,0)⎤ ⎬
2
∫
⎢
⎥
0
⎦⎭
∂t
∂x
∂x
⎩ε e S x ⎣
(24)
Notations for the material and physical constants included in Eq. (24) are given in Table 1. The left-hand side term, the
first term and the second term in the right-hand side correspond to the electric forces, the viscous resistance force and
the diffusion forces respectively.
The initial and boundary conditions are given in Eq.(25a) and Eq.(25b) [7-8]. i is the electric current and d is
thickness of PPy film. The initial charge density is assumed that co = c ( x,0 ) = 1792mol / m3 .
{Q( x,0)} = N a eS x co x
(25a)
{Q(0, t )} = 0,
(25b)
{Q(d , t )} = N a eS x co d
Table 1 Material parameters for active, electrochemical-poroelastic response analysis
Coefficient of viscosity for hydrated Na + moving through membrane
η1 = 1.18 × 10 -11 (N ⋅ s/m 2 )
Oltzman constant
k = 1.380 × 10 -23 (N ⋅ m/K)
Absolute temperature
Elemental charge
T = 293(K)
Dielectric constant of hydrated Nafion membrane
ε e = 2.8 ×10 -3 (C 2 /N ⋅ m 2 )
Time interval
Space interval
Cross section area
Δt = 5 × 10-6 s
h = 1.75 μm
Avogadro number
N a = 6.02 ×10-23 (/mol)
e = 1.6 ×10 -19 (C)
S x = 33.0 × 10-6 (m 2 )
The charge density c ( x , t ) is calculated from the total charge by the following equation:
x
Q ( x, t ) = N a eS x ∫ c(ξ , t )dξ
(26)
0
The finite element formulation is conducted for Eq. (24) and Eq.(26). The total charge and charge density is assumed
to be a bilinear function of element coordinates as given by the following equation:
⎯ 584 ⎯
{Q( x, t )} = [N ]{QN } = ⎡⎢1 − x
⎣
h
{c( x, t )} = [N ]{QN } = ⎡⎢1 − x
⎣
h
x ⎤ ⎧Qi ⎫
⎨ ⎬
h ⎥⎦ ⎩Q j ⎭
(27)
x ⎤ ⎧ ci ⎫
⎨ ⎬
h ⎥⎦ ⎩c j ⎭
(28)
where {Qi }and {ci } are the nodal total charge and charge density. The following equation is obtained by the finite
element formulation based on Galerkin method.
∂Q ( x, t )
∂ 2 Q ( x, t ) ∂Q ( x, t ) ⎧ e ⎡ t
T ⎛
⎤ ⎫ ⎞⎟dx = 0
⎜
[
N
]
η
kT
i
(
τ
)
d
τ
Q
(
x
,
t
)
Q
(
x
,
0
)
×
+
−
+
−
⎨
1
∫0
∫
⎜
⎢0
⎥⎦ ⎬ ⎟
∂x
∂t
∂x 2
⎩ε e S x ⎣
⎭⎠
⎝
h
(29)
Substituting Eq. (27) into Eq. (29), the following ordinary differential equation is obtained.
⎡ A11
⎢A
⎣ 21
A12 ⎤ &
B12 ⎤
⎡B
QN + ⎢ 11
⎥
⎥{QN } = 0
A22 ⎦
⎣ B21 B22 ⎦
{ }
(30)
where
⎡ A11
⎢A
⎣ 21
⎡h
A12 ⎤ ⎢ 3
=⎢
A22 ⎥⎦ ⎢ h
⎢⎣ 6
h⎤
6⎥
⎥
h⎥
3 ⎥⎦
⎡ B11
⎢B
⎣ 21
⎡ kT
e ⎡ t
kT
Δte ⎡ t
⎤
⎤
⎤
⎢ η h − 2η ε S ⎣⎢ ∫0 i (τ )dτ ⎦⎥ + B1 − η h + 2η ε S ⎣⎢ ∫0 i (τ )dτ ⎦⎥ + B2 ⎥
B12 ⎤ ⎢ 1
1 e x
1
1 e x
⎥
=
⎥
t
⎢
⎥
B22 ⎦
kT
e ⎡
kT
e ⎡ t
⎤
⎤
i (τ )dτ + B3
i (τ )dτ + B4 ⎥
−
+
⎢−
∫
∫
⎢
⎥
⎢
⎥
0
0
⎣
⎦
⎣
⎦
η1h 2η1ε e S x
⎢⎣ η1h 2η1ε e S x
⎦
(31a)
⎡1
(Qi ( x, t ) − Qi ( x,0)) + 1 (Q j ( x, t ) − Q j ( x,0))⎤⎥
⎢
η1ε e S x ⎣ 3
6
⎦
e ⎡1
(Qi ( x, t ) − Q j ( x,0)) + 1 (Q j ( x, t ) − Q j ( x,0))⎤⎥
B3 = −
⎢
η1ε e S x ⎣ 6
3
⎦
B1 = −
(31b)
e
(31c)
B2 = − B1 , B 4 = − B3
The following equation is obtained by the Galerkin finite element formulation for Eq. (26).
h
∫ [N ]
T
0
⎛ ∂Q ( x, t )
⎞
− N a eS x c( x, t ) ⎟dx = 0
⎜
⎝ ∂x
⎠
(32)
Substituting Eq. (27) and Eq. (28) into Eq. (32) leads to the following form of equation:
⎡ D11 D12 ⎤
⎡C11 C12 ⎤
⎥{QN }
⎢C C ⎥{cN } = ⎢ D
22 ⎦
⎣ 21 D22 ⎦
⎣ 21
(33)
where,
⎡ N a eS x h
⎡C11 C12 ⎤ ⎢ 3
⎢C
⎥=⎢
⎣ 21 C22 ⎦ ⎢ N a eS x h
⎣ 6
⎡ D11
⎢D
⎣ 21
⎡ 1
D12 ⎤ ⎢− 2
=
D22 ⎥⎦ ⎢⎢− 1
⎣ 2
N a eS x h ⎤
6 ⎥
N a eS x h ⎥
⎥
3 ⎦
(34a)
1⎤
2⎥
1⎥
⎥
2⎦
(34b)
⎯ 585 ⎯
5. Computational procedure of the electrochemical-mechanical analysis The solution of the Eqs. (30) and (33)
with the initial and boundary condition leads to the electrochemical response during doping-dedoping. The volume
change and the pressure are calculated by solving Eq. (23) and Eq. (17). Finally, the mechanical response is obtained
by solving Eq. (7)
NUMERICAL EXAMPLES
We give two typical examples: stress-relaxation and electrochemical-mechanical response
1. The case of stress-relaxation The mechanical model is applied here to a stress-relaxation. We suppose that the
PPy film is elongated by the initial strain (along the z direction) and calculate the pressure, solid stress and total stress
during stress-relaxation. The initial strain and material property of the PPy film are shown in Table 2. PPy film shown
in Fig. 1 is analyzed by the three-dimensional finite element method.
Fig. 2 shows the pressure distribution of PPy film during the stress-relaxation. The pressure decreases from the side of
PPy film by the suction of fluid. Then it asymptotically approaches the reference external bath pressure, equal to 0.
The numerical results are in good agreement with the one-dimensional theoretical solution as shown in Fig. 3. After the stretch,
the elastic stress occurs and increases exponentially in PPy film. At the same time, the negative pressure decreases by the
suction of fluid from the external bath.
Table 2 Material parameters for poroelastic response analysis
Initial strain
e0 = 0.00340909
Poisson’s ratio
v = 0.412
Frictional coefficient
Young’s modulus
f = 1.29 × 10 20 ( Nsm 4 )
E = 1290( MPa)
Porosity
β = 0.108
Thickness
Δt = 0.005 sec
l = 30mm
a = 35μm
Width
b = 1.1mm
Time increment
Length
Figure 1: Polypyrrole film used for passive, poroelastic response analysis
⎯ 586 ⎯
Figure 2: Pressure distribution in polypyrrole film
Figure 3: Total stress, stress on the solid and fluid pressure vs. time
2. The case of electrochemical-mechanical response When a PPy film is under an electric field, the volume change
of the PPy film is accelerated by doping and de-doping. In this paper, we suppose the exchanged charge, which is
directly correlated to the amount of Na + entering or coming out the polymer, results in a corresponding elongation
and contraction of the PPy film. Table 1 shows the material constants and calculation conditions to solve Eq. (30) and
Eq. (33).
At first, the electrochemical response of the PPy film has been analyzed under the current patterns shown in Fig. 4(a).
Charge density of the PPy film increases by the intake of the Na + cations shown as Fig. 4(b). Fig. 4(c) shows that the
volume change is accelerated during reduction phase: positive pressure means that the volume is over-swelled due to
the doping of Na + and gradually approaches the reference external bath pressure when currents stop.
Figs. 5(a-c) show the result during the oxidation phase. The release of Na + cations causes the negative pressure and decreases
the volume of the PPy film.
⎯ 587 ⎯
(a) current patterns
(b) time variations of charge density distribution (case 3)
(c) time histories of pressure
Figure 4: Electrochemical-mechanical response of polypyrrole film (cation entering)
(a) current patterns
(b) time variations of charge density distribution (case 3)
(c) time histories of pressure
Figure 5: Electrochemical-mechanical response of polypyrrole film (cation emitting)
⎯ 588 ⎯
In Fig. 6, the length change wave is plotted versus time during a SW stimulation with I=5mA, T=62.5s. Material
constants for calculation are given in Tables 1-2. The sample elongates during the reduction semi-cycle and it
contracts during the oxidation one. The goodness of fit between experimental and numerical result confirms the
validity of the present formulation.
Figure 6: Isotonic length change of the polypyrrole sample during switching stimulation
CONCLUSIONS
We have presented finite element formulations for the poroelastic and electrochemical-mechanical behaviors of PPy
film. The generalized Poisson equation is extended to include electrochemical coupling by the Onsager-laws and
one-dimensional ion transport equations given by Tadokoro. et al.. Some numerical studies show the validity of the
present formulation.
REFERENCES
1. Della Santa A et al. Passive mechanical properties of polypyrrole films: a continuum, poroelastic model.
Material Science and Engineering, 1997; C5: 101-109.
2. Della Santa A et al. Performance and work capacity of a polypyrrole conducting polymer linear actuator.
Synthetic Metals, 1997; 90: 93-100.
3. Della Santa A et al. Characterization and modeling of a conducting polymer muscle-like linear actuator. Smart
Materials and Structures, 1997; 6: 23-34.
4. Cortes MT, Moreno JC. Artificial muscles based on conducting polymers. E-Polymers, 2003; 41: 1-42.
5. Hara S et al. Artificial muscles based on polypyrrole actuators with large strain and stress induced electrically.
Polymer Journal, 2004; 36(2): 151-161.
6. Tadokoro S et al. An actuator model of ICPF for robotic applications on the basis of physicochemical
hypotheses. Proceedings of the 2000 IEEE International Conference on Robotics and Automation, 2000,
pp.1340-1346.
7. Toi Y, Kang SS. Finite element modeling of electrochemical-mechanical behaviors of ionic conducting
polymer-metal composites. Transactions of the Japan Society of Mechanical Engineers, Series A, 2004;
70(689): 9-16.
8. Toi Y, Kang SS. Finite element analysis of two-dimensional electrochemical-mechanical response of ionic
conducting polymer-metal composite beams. Computers and Structures, 2005; 83: 2573-2583.
9. Biot MA. Theory of elasticity and consolidation for a porous anisotropic solid. Journal of Applied Physics,
1954; 26: 182-185.
10. Katchalsky A, Curran PF. Non-equilibrium Thermodynamics in Biophysics. Harvard University Press, 1967.
⎯ 589 ⎯