Sensors and Actuators B 190 (2014) 954–967

Contents lists available at ScienceDirect

Sensors and Actuators B: Chemical
journal homepage: www.elsevier.com/locate/snb

IPMC bending predicted by the circuit and viscoelastic models
considering individual inﬂuence of Faradaic and non-Faradaic
currents on the bending
Kota Ikeda a , Minoru Sasaki b , Hirohisa Tamagawa b,∗
ab

Graduate School of Advanced Mathematical Sciences, Meiji University, 4-21-1, Nakano, Nakano-ku, Tokyo, 165-8525, Japan
Department of Human and Information Systems, Faculty of Engineering, Gifu University, 1-1 Yanagido, Gifu, Gifu, 501-1193, Japana r t i c l e

i n f oArticle history:
Received 25 July 2013
Received in revised form 3 September 2013
Accepted 4 September 2013
Available online xxx
Keywords:
IPMC
Circuit model
Viscoelasticity
Faradaic
Non-Faradaic

a b s t r a c t
Electrically induced bending behavior of a highly dehydrated Selemion CMV-based IPMC was analyzed
theoretically. The experimental observation suggested that even the highly dehydrated IPMC bending wasunder the inﬂuence of not only Faradaic current but also non-Faradaic current. Hence, it was essential
to know the Faradaic and non-Faradaic current inﬂuences on the IPMC bending separately for predicting
the IPMC bending behavior. A circuit model was proposed for deriving the time-dependent behavior of
charge, which was caused by both Faradaic and non-Faradaic currents, passing through the IPMC. In thecourse of derivation of the time dependence of charge, electrical properties of the IPMC such as resistanceand capacitance were obtained, and those physical quantities were reasonable from the stand pointof physical chemistry. The time dependent charge behaviors obtained by employing the circuit model
successfully resulted in the prediction of bending behavior of the IPMC. The electric properties obtainedby the analysis of circuit model was combined with a viscoelastic model we proposed for predicting
the bending behavior of IPMC, and the newly proposed viscoelastic model successfully reproduced theexperimentally observed behavior of Selemion CMV-based IPMC bending.
© 2013 Elsevier B.V. All rights reserved.

1. Introduction
IPMC (Ionic Polymer Metal Composite) has been widely investigated for the purpose of fabricating a practical bending mode
polymer-based actuator [1–6]. IPMC is electrically activated inhighly hydrated state. So, the IPMC can be activated even in an
aqueous solution unlike other type of actuators, and it is one of
IPMCs’ featuring facets. Despite intensive efforts made by a number of researchers toward the ultimate research goal of fabricatinga practical bending mode actuator of IPMC, the research progress isstagnant recently. There are some challenging tasks to achieve forfabricating a practically utilizable IPMC. The IPMC has to be able togenerate higher force and has to exhibit more precisely controllable
bending behavior.
The IPMC is usually electrically activated in highly hydrated
state. It has been ﬁrmly believed that the bending of IPMC is dueto the shift of mobile hydrated ions inside the IPMC under applied
voltage [7,8]. This means that a large quantity of water is needed
for inducing the bending of IPMC. Contrary to that bending mechanism ﬁrmly believed and accepted, however, it was found that one∗ Corresponding author. Tel.: +81 582932529; fax: +81 582932529.
E-mail address: tmgwhrhs@gifu-u.ac.jp (H. Tamagawa).
0925-4005/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.snb.2013.09.016

of IPMCs consisting of a Selemion CMV coated with silver exhibited
relatively large bending under applied voltage even in the highly
dehydrated state [9–12], (Selemion CMV is an ion exchange membrane containing SO3 H functional atomic groups, manufactured
by Asahi Glass Co., Ltd., Tokyo, Japan). A completely dehydrated
Selemion CMV-based IPMC cannot exhibit bending because of theloss of electrical conductivity, however, highly dehydrated (notcompletely dehydrated) Selemion CMV-based IPMC can exhibit
large bending with improved bending controllability [9]. Naﬁon
is one of ion exchange membranes, whose chemical structure isquite similar to Selemion CMV. It was found that Naﬁon-based
IPMC (Naﬁon coated with silver) exhibited bending under applied
voltage, too, even in the highly dehydrated state, though its bending curvature was quite small [13]. Furthermore, the Naﬁon-based
IPMC bending controllability was largely improved in the highly
dehydrated state compared with that in hydrated state. According
to the experimental facts so far described, the dehydration treatment to the Selemion CMV-based IPMC and Naﬁon-based IPMC
improved their bending controllability. Therefore, we have thoughtthat minimizing the occurrence of non-Faradaic current in the
IPMC body resulted in the improvement of IPMC bending controllability. However, our current study revealed that even the highly
dehydrated IPMC is under heavy inﬂuence by the non-Faradaic
current. Because of the importance of precise controllability forK. Ikeda et al. / Sensors and Actuators B 190 (2014) 954–967

a practical IPMC actuator, we attempted to elucidate the rolesof non-Faradaic current as well as Faradaic current for the IPMC
bending controllability. First, we proposed a circuit model of thehighly dehydrated IPMC and derived its electrical properties suchas its resistance, capacitance and etc. Then the bending behaviorof Selemion CMV-based IPMC was predicted. We moved onto thenext step: by employing those electrical properties derived in thecircuit model, we investigated the bending behavior of the same
Selemion CMV-based IPMC from the stand point of elasticity andsuccessfully predicted it’s bending behavior.

direction of the curvature and current are shown in Fig. 1(a). Whenthe top and bottom surfaces of CMV IPMC were short-circuited after
the 1.5 V voltage application process, the current ﬂowing throughthe 10 -resistor of r was measured as a function of time, where
the deﬁnition of the positive and negative of current is shown in
Fig. 1(b). At the same time, the vertical displacement of the CMVIPMC surface 10 mm away from its clamping point was measured
with a laser displacement sensor as a function of time. Fig. 2(a)shows the time dependence of current, when the voltage application and short-circuit tests were carried out, where the thickcurve represents the current induced under voltage application of
1.5 V and the ﬁne curve represents the current ﬂowing through theresistor when short-circuited.
Absolute humidity (AH) is quite substantial factor for determining the bending behavior of IPMC in response to external
electric stimulation. All the experimental measurements shownin this paper were all carried out at the low absolute humidity
of AH ∼9 gm−3 . Unexpectedly, Fig. 2(a) suggests that even highly
dehydrated CMV IPMC exhibited the discharge of non-negligible
quantity of charge from t = 20 s. The charge was stored in the CMVIPMC in the middle of 1.5 V voltage application, and it was relatively large quantity of charge compared with the charge imposedon the CMV IPMC from t = 0 s to 20 s. Fig. 2(b) shows the relationshipbetween the total charge induced from t = 0 s and the CMV IPMC
bending curvature. Thick curve represents the CMV IPMC bending curvature vs. total charge induced under voltage application of
1.5 V. Fine curve represents the CMV IPMC bending curvature vs.
the sum of the charge induced from t = 0 s to 20 s under 1.5 V andthe charge ﬂowing thorough the resistor r from t = 20 s onward by
the discharging process. What is quite intriguing about Fig. 2(b)is that the decay of the bending curvature was induced when discharged. Furthermore, we noticed that the bending curvature in the
decaying process did not trace back its trajectory from t = 0 s to 20 s.
We carried out another experiment. A voltage increasing at therate of 10 mV s−1 from 0 V to 1.5 V was applied on a CMV IPMC
and its current and curvature were measured. Fig. 3 shows voltage
dependence of current ﬂowing through the unit surface area of the
CMV IPMC and the voltage dependence of CMV IPMC curvature. It
is noticeable that the current and the curvature started increasing

2. Circuit model
2.1. Experiment

OFF

IPMC

A
ammeter

ON

A

+

OFF

r
A

positiveandnegativecurrent flowing through r

10 mm
+

positive and negative curvature

laser

positive and negative current

positive and negative curvature

To begin with, we would like to explain the essential conditions
for building a circuit model of the Selemion CMV-based IPMC, then
we introduce a circuit model. Hence, above all, we experimentallystudied the bending behavior of Selemion CMV-based IPMC.
A Selemion CMV-based IPMC was fabricated simply by coatinga Selemion CMV surface with silver by the silver mirror reaction[14]. The resultant Selemion CMV-based IPMC was dehydrated bystoring it in a desiccator with desiccant, resulting in a highly dehydrated (not-completely dehydrated) Selemion CMV-based IPMC. It
was cut into a strip of 20 mm × 2 mm ×120 ␮m (hereafter the strip
is called CMV IPMC). One end of the CMV IPMC was horizontallyclamped with a pair of electrodes as illustrated at upper part in
Fig. 1(a). By using this experimental setup, we carried out the following bending testing of CMV IPMC (see the CMV IPMC in the bent
state as illustrated at the lower part in Fig. 1(a)): 1.5 V constantvoltage was applied on the CMV IPMC from time t = 0 s to 20 s, andsubsequently its top and bottom surfaces were short-circuited witha resistor as illustrated in Fig. 1(b). During the 1.5 V voltage application process for the initial 20 s, vertical displacement of the CMVIPMC surface 10 mm away from its clamping point was measured
with a laser displacement sensor as a function of time, as illustrated in Fig. 1(a). The measured vertical displacement was later
converted into curvature. Simultaneously, the current was measured as a function of time. Deﬁnitions of the positive and negative

(a)

955

A

(b)

Fig. 1. (a) A CMV IPMC set to the bending testing apparatus at ON and OFF stages. (b) The CMV IPMC, when short-circuited with a 10  resistor.

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K. Ikeda et al. / Sensors and Actuators B 190 (2014) 954–967

1.5 V impose

short-circuited

0.009
curvature / mm-1

current / mAmm-2

0.06
0.03
0.00
-0.03

0.006
0.003

1.5 V impose

short-circuit

0.000

-0.06
0

30

60

0.00 0.05 0.10 0.15 0.20 0.25

90

total charge / mCmm-2

time / s(a)(b)

Fig. 2. (a) Time dependence of current, where the thick curve represents the current induced under voltage application of 1.5 V and the ﬁne curve represents the current
ﬂowing through the resistor when short-circuited. (b) CMV IPMC bending curvature vs. total charge imposed Thick curve represents the CMV IPMC bending curvature vs.
total charge induced under voltage application of 1.5 V. Fine curve represents the CMV IPMC bending curvature vs. the sum of the charge ﬂowing through the CMV IPMC
from t = 0 s to 20 s and another charge ﬂowing thorough the 10 -resistor r from t = 20 s onward.

around at 0.7 V as indicated by a dashed arrow in Fig. 3. As alreadyreported for the past several years, redox reaction of Ag ↔ Ag2 O
starts around above 0.7 V, that is, 0.7 V is threshold voltage for the
induction of redox reaction [11]. Effective bending of CMV IPMC andeffective current induction were always simultaneously observed
and they were inevitably accompanied by the redox reaction ofAg ↔ Ag2 O. Ag on the anode was oxidized and Ag2 O on the cathodewas reduced. Polarity change of voltage caused opposite reaction,
hence, it is reversible redox reaction. CMV IPMC bending, current
induction and redox reaction are undoubtedly intimately correlated one another.
Next, we investigated the surface color change behavior of CMV
IPMC in response to the polarity change of applied voltage. Fig. 4
shows the relationship among CMV IPMC bending direction, voltage polarity and redox reaction (surface color change of CMV IPMC).
Initially as shown in Fig. 4(a), both top and bottom surfaces of CMV
IPMC were bright because of white Ag color. Once the 1.5 V voltage was imposed, bottom surface (anode surface) darkened due tothe creation of Ag2 O as shown in Fig. 4(b). Then the discharging byconnecting the top and the bottom surfaces with a resistor caused
backward bending of CMV IPMC and reduction of Ag2 O into Ag on
the bottom surface, and the oxidation of Ag on the top surface into
Ag2 O was also induced, as shown in Fig. 4(c). Of course these tworeactions were induced simultaneously.
2.2. Building a circuit model

0.009

0.009

0.006

0.006

0.003

0.003

0.000
0.0

0.5

1.0

current/ mAmm -2

curvature / mm-1

What is so far described brought us some essential requirements
for building an appropriate circuit model explaining CMV IPMC
bending behavior. Those essential requirements are given below:

0.000
1.5

voltage/ V
Fig. 3. CMV IPMC bending curvature vs. voltage imposed, and current ﬂowing
through the CMV IPMC vs. voltage imposed.

(i) As shown in Fig. 2(a), the CMV IPMC can store a certain quantityof charge.
(ii) As shown in Figs. 2(b) and 3, increase and decrease of charge
or current has decisive role for the increase and decrease of
bending curvature of CMV IPMC, respectively.
(iii) Effective redox reaction of Ag ↔ Ag2 O is induced around at or
above 0.7 V applied voltage as shown in Fig. 3.
(iv) Below at 0.7 V, the redox reaction is not induced effectively,
but a small current can ﬂow through CMV IPMC. That current
is ineffective for the induction of CMV IPMC bending as shownin Fig. 3.
(v) Even discharging process can cause the redox reaction ofAg ↔ Ag2 O as shown in Fig. 4.
As the circuit model has to satisfy the above ﬁve requirements,
we propose a circuit model illustrated in Fig. 5(a) with the comments (a)–(d) below that explain further our proposal.
(a) Selemion CMV part of the CMV IPMC is approximated by acapacitor C and a resistor R. Since C can store a certain quantity of charge, the requirement (i) is satisﬁed, where we will
denote the capacitance of C and resistance of R by C and R,
respectively. Silver layers are approximated by r , where its
resistance is denoted by r , too. Although there are two silver
layers on the surfaces of CMV IPMC, both surface resistances
are combined into one, r . We hypothesized that charge storage in the capacitor C induces the CMV IPMC bending. It is thebending induced by non-Faradaic current, and also we hypothesized that the occurrence of current Ir (see Fig. 5(a)) ﬂowing
through the resistor r inevitably causes the redox reaction,
resulting in the induction of bending. It is the bending induced
by Faradaic current. It was hypothesized that the downwardbending of CMV IPMC was induced, when Q > 0 or Ir > 0, whereQ is the charge stored in C (see Fig. 5(a)).
(b) Since it has been reported occasionally that the curvature ofhighly dehydrated silver-plated Selemion CMV IPMC (virtually the same as the CMV IPMC) is proportional to the total
charge imposed on it, we further hypothesized that the curvature proportional to Q and another curvature proportional to
the total charge passing through r could be induced. So, whatis described about C and r here satisﬁes the requirement (ii).
(c) Another capacitor Cs and resistor rs were introduced into thecircuit model relating to the (iii) and (iv), where the capacitanceof Cs and resistance of rs are denoted by Cs and rs , respectively. We hypothesized that the current ﬂowing though Cs
and rs cannot cause any bending of CMV IPMC. Namely, thisK. Ikeda et al. / Sensors and Actuators B 190 (2014) 954–967

957

Fig. 4. The relationship between CMV IPMC bending direction, voltage polarity and redox reaction (surface color change of CMV IPMC).

capacitance and resistor is a model of the path for the experimentally observed current causing ineffective bending of CMV
IPMC below 0.7 V shown in Fig. 3. Since effective bending of
CMV IPMC was not induced up to 0.7 V, most of the current
must ﬂow though this path and the total charge induced up
to 0.7 V was quite small compared with that at or above 0.7 V,
we hypothesized that both Cs and rs were quite small. We theauthors would like to urge the reader to ﬁrmly bear in mind
that Cs and rs are just a path for a current causing no bending
in this circuit model, and only the charge stored in C and thecharge ﬂowing through r can cause the CMV IPMC bending.
(d) Now imagine that the voltage of 1.5 V was applied on CMV IPMC
and subsequently discharging was carried out. When in the discharging process, the direction of current ﬂowing though r isreversed as illustrated in Fig. 5(b). Therefore, the redox reaction during discharging process predicted by this circuit modelsatisﬁes the requirement (v).

and the contraction of the anode side, resulting in the downwardbending, as illustrated in Fig. 6(b-1). The state shown in Fig. 6(b1) is interpreted as the storage of charge Q in C from the standpoint of the circuit model as illustrated in Fig. 6(b-2). The current
charging C is non-Faradaic current. The experimental observationof CMV IPMC bending by non-Faradaic current is well explained by
the circuit model. As described earlier, it was hypothesized that the
CMV IPMC bending is caused by Faradaic current as well, inducing
the redox reaction of Ag ↔ Ag2 O. In fact, we observed the redox
reaction of Ag ↔ Ag2 O and the bending simultaneously, during theapplication of 1.5 V on the CMV IPMC. As easily understood, thecharge storage (or release) process inevitably causes the current
passing through r in the circuit model shown in Fig. 6(b-2). The
direction of the current Ir passing through r is shown in Fig. 6(b-2)
and it is known that CMV IPMC exhibits bending in the anode direction, though the mechanism of bending induced by Faradaic current
has not been elucidated yet. Hence the circuit model is so far quiteconsistent with the experimental evidences. We further assessedthe validity of our idea that Faradaic and non-Faradaic current haveindividual inﬂuence of inducing IPMC bending by studying anothertype of IPMC as described below.
Selemion AMV is another type of ion exchange membrane manufactured by ASAHI GLASS Co., Ltd., Tokyo, Japan. Since it is an anion
exchange membrane, it contains immobile cations and mobile
anions in hydrated state (even in the highly dehydrated state unlessin the completely dehydrated state). As easily understood, the electrical structure of Selemion AMV is just the opposite to that of theSelemion CMV. Coating the surface of the Selemion AMV resulted ina Selemion AMV-based IPMC (hereafter called AMV IPMC). Bending

connected to the power supply terminal

Now, we would like to prove the appropriateness of our circuit
model by comparing the model with the experimental evidences.
The CMV IPMC is featured especially by the functional atomic
groups of SO3 H contained in it. It comes to contain immobile
anions of SO3− and mobile cations of H+ in the hydrated state(even in the highly dehydrated state unless in the completely dehydrated state) as illustrated in Fig. 6(a). Once the constant voltage
1.5 V is imposed on it, it exhibits downward bending (in the anodedirection). This bending process is explained from the stand pointof physical chemistry. All cations are shifted toward cathode side of
CMV IPMC. It causes the swelling of cathode side of the CMV IPMC,

ON

-Q ℓ
Cℓ

Cs

V

OFF

Cℓ
+Qℓ

R

R

r

+Qℓ
Ir

Ir
rs

rs

rℓ

under1.5 V
(a)

-Q ℓ

Cs

Ir

rℓ

dischargingby theuseof resistorr(b)Fig. 5. (a) Circuit model (b) under 1.5 V and in the discharging process.

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K. Ikeda et al. / Sensors and Actuators B 190 (2014) 954–967

Fig. 6. (a) Selemion CMV in the hydrated state (b-1) bending under applied voltage and shift of mobile cations toward cathode and (b-2) its circuit model.

test of the AMV IPM was carried out by employing the same setup as
illustrated in Fig. 1(a). 1.5 V constant voltage was imposed for 180 s
on the AMV IPMC horizontally clamped with a pair of electrodes
and followed by discharging process by connecting its top and bottom surfaces with a resistor (short-circuit). Fig. 7 shows the timedependence of AMV IPMC bending curvature. The AMV IPMC initially exhibited upward bending (negative curvature). But aroundat time t = 30 s it changed the bending direction downward. This
unusual behavior is explained as follows: Once the voltage of 1.5 V
was imposed at t = 0 s, immediately mobile anions tended to shiftdownward as illustrated in Fig. 8(a), causing the upward bendingof the AMV IPMC. While in progress of the upward bending, redox
reaction of Ag ↔ Ag2 O is in progress, too, and gradually the inﬂuence of the redox reaction on the induction of AMV IPMC bendingcomes to be dominant. Since the silver plated IPMC has tendency tobend in the anode direction by Ag ↔ Ag2 O redox reaction, the AMV
IPMC comes to exhibit downward bending around after t = 30 s.
Long after t = 30 s but before starting discharging process at t = 180 s,

already the redox reaction was dominant for the induction of AMV
IPMC bending. Although mobile anions were mostly located at theanode side of the AMV IPMC right before t = 180 s, and since redox
reaction was dominant for the induction of bending as illustrated inFig. 8(b), the curvature of AMV IPMC is positive. Once the discharging process started at t = 180 s, all anions were liberated from anodeside and tended to homogeneously disperse in the AMV IPMC bodyas illustrated in Fig. 8(c), causing further downward bending as
indicated by a solid arrow in Fig. 7. This discharging process causes
the redox reaction, too, according to the circuit model (see Fig. 5(b)).
During the redox reaction after t = 180 s, Ag on the top surface wasoxidized into Ag2 O, while the Ag2 O on the bottom surface was
reduced into Ag. Therefore, the redox reaction caused in the discharging process has inﬂuence of inducing upward bending. Hence,
bending relaxation started around at t = 200 s as shown in Fig. 7.
Concerning what has been described so far, the circuit modelwith considering Faradaic and non-Faradaic current appears to well
explain the bending behavior of IPMCs.
2.3. Parameter estimate

1.5 V impose

short-circuitedcurvature / mm-1

0.030
0.020

We hypothesized that the curvature is induced by the charge,
Qr , of Faradaic current passing through the resistor r of silver layer
and by the another charge, Q , of non-Faradaic current stored in
the capacitor C of Selemion CMV phase. Qr and Q are supposed toinduce the curvature BF and BnF , and BF and BnF are assumed to begiven by Eqs. (2.1) and (2.2), respectively. The total curvature, B, isthe sum of BF and BnF as represented by Eq. (2.3):
BF = kF Qr

0.010

(2.1)

Coefﬁcient kF is determined experimentally.

0.000

BnF = knF Ql(2.2)

Coefﬁcient knF is determined experimentally.

-0.010
0

100

200

300

400

time / sFig. 7. Time dependence of AMV IPMC curvature in the middle of 1.5 V voltage
impose followed by discharging.

B = BF + BnF(2.3)

Fig. 9(a) shows the time dependence of CMV IPMC curvature
under rectangular oscillating voltage with the amplitude = 1.5 V and
its frequency = 1/60 Hz. This experimental result is rearranged into
curvature vs. charge as shown in Fig. 9(b-1 and 2). The charge was

K. Ikeda et al. / Sensors and Actuators B 190 (2014) 954–967

959

Fig. 8. Explanation of AMV IPMC bending.

0.005
0.000
-0.005
0

100

200

300

time / s

(a)

0.005
0.000
-0.005-0.010-0.4-0.010(b-1)

curvature / mm-1

curvature / mm-1

curvature / mm-1

0.010

0.010

0.010

-0.2

0.0

0.2

charge/ mCmm-2

0.005
0.000
-0.005-0.010
-0.4

0.4

-0.2

0.0

0.2

0.4

charge/ mCmm-2(b-2)

Fig. 9. (a) Time dependence of CMV IPMC curvature under rectangular oscillating voltage (amplitude 1.5 V, frequency 1/60 Hz). (b) Curvature vs. Charge about CMV IPMC(b-1) from t = 0 s through 300 s and (b-2) from t = 0 s through 75 s, where the dashed arrows represent the progression of time.

measured by the ammeter shown in Fig. 1. This means that the
charge shown in Fig. 9(b-1 and 2) is the total charge passing throughthe whole circuit normalized by the unit surface area of CMV IPMC.
We would like to make comment on the Fig. 9(b-1 and 2), where
the dashed arrows indicate the progression of time. The diagram
forms parallelogram shape trajectory. It is a quite typical behavior
between the curvature and charge in case where the CMV IPMC is
activated under rectangular oscillating voltage. In order to know
the curvature of BF and BnF , it is necessary to know the Qr and Q .
In order to obtain them, it is necessary to estimate the parameters( s ,   , Qs , Q , Cs , C , rs , r , R) given in Fig. 10, where Qs represents the charge stored in Cs , and  s and   are characteristic time
and their exact deﬁnitions are given in Eq. (2.1), where I(t) represents the entire current generated by the power supply (see Fig. 10).
Derivation procedure of Eq. (2.1) is given in Appendix A.
I(t) =

 t

Qs
exp −
s
s

+

 t

RCl − l
Ql exp −l RCl
l
CrR

+

Ql
RCl

R
l
where  s = rs Cs , Qs = Cs V, l = r l+R
and Ql = r +R
Cl V .
l

l

(2.4)

V

-Qs
Cs
+Qs

-Qℓ
Cℓ
+Qℓ I

ℓ

R
IR

rs
rℓ
I

Is

Ir

Fig. 10. Circuit model with parameters.

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K. Ikeda et al. / Sensors and Actuators B 190 (2014) 954–967

0.010
curvature / mm-1

current / mAmm-2

0.08
0.04
0.00
-0.04

0.005

t = 15 s

0.000
direction of time

-0.005t = 45 s

-0.08
0

100

200

-0.010-0.4

300

time / sFig. 11 shows the time dependence of current, I, when the experiment for Fig. 9 was carried out. Eq. (2.4) is valid for t = 0 s throught = 15 s. By curve ﬁtting technique to the current data from t = 0 s
through 15 s in Fig. 11, all the parameters were determined. But we
did not use these parameters for the analysis of CMV IPMC bending behavior. Because empirically we found that the parameters
determined by the use of current data for t = 0 s through 15 s neverbrought us so appropriate outcomes. Although we have not ﬁguredout the exact reason of inappropriateness about the use of parameters obtained by the use of current data for the initial 15 s, we are
now close to ﬁnding out the reason. It is to be shown in the futurereport from us. Those parameters were good only for explaining
the CMV IPMC bending behavior at early stage. Therefore, thoseparameters were used only for obtaining Qs T and Q T introduced
below:



where V = 1.5 V and T = 15 s.

QlT ≡ Ql (t = T ) = Ql 1 − exp T 

(2.5)

s T 

(2.6)

lwhere V = 1.5 V and T = 15 s.
At time t = 15 s, the voltage polarity was reversed and anotherequations are derived for the circuit, and the t = 15 is redeﬁned ast’ = 0 s. Therefore, Eqs. (2.5) and (2.6) can be respectively regarded
as Qs (t’ = 0) and Q (t’ = 0).
Under deﬁning t = 15 s as t’ = 0 s, all the parameters are redeﬁned by placing a prime on the shoulder of individual parameters.
Then we derived the entire current generated by the power supply,
I’(t’), given in Eq. (2.7). Derivation procedure of Eq. (2.7) is given in
Appendix B.
I  (t  ) = Is (t  ) + Il (t  ) + IR (t  ) =+(QlT − Ql )(l − R Cl )R Cl lQs − QsTt
exp − 
s
s



exp

−

t
l

0.0

0.2

0.4

charge/ mCmm-2

Fig. 11. Time dependence of current induced in the CMV IPMC under rectangularoscillating voltage (amplitude 1.5 V, frequency 1/60 Hz).

QST ≡ QS (t = T ) = QS 1 − exp

-0.2



+

Ql



R Cl

(2.7)

By the curve ﬁtting technique to the current data from t = 15 s
through 45 s in Fig. 11, all the parameters were ﬁnally determined.
They are summarized in Table 1. The parameters in Table 1 are to
be used for the analysis of CMV IPMC bending behavior.
2.4. Prediction of bending behavior
Now we move on to the process of deriving the kF and theknF . We focus on the charge and curvature behavior from t = 15 s
through 45 s shown in Figs. 9 and 11 and those experimental data
were arranged into curvature vs. charge as shown in Fig. 12. Bear

Fig. 12. Curvature vs. Charge about CMV IPMC from t = 15 s through t = 45 s.

in mind that since the capacitance of Cs is negligibly small compared to that of C , the charge shown in Fig. 12 virtually consistsof Faradaic current passing through r and non-Faradaic current
stored in the capacitor C . Right before t = 45 s in Fig. 12, the curvature change with time was primarily caused by the Faradaic
current passing through the silver layer of r , since the capacitances Cs and C were full of charge. Therefore, the curvature atand right before t = 45 s is represented simply by Eq. (2.1), and kF
is given as the gradient of tangential line of curve right beforet = 45 s in Fig. 12. Dashed line in Fig. 12 has the slope kF , where
the line passes through the data point at t = 15 s. This line reaches
the point indicated by a circle at t = 45 s in Fig. 12. The curvature atthe point indicated by the circle is totally different from the experimentally observed curvature at t = 45 s. This curvature difference
must be caused by the charge stored in the capacitor C from t = 15
through 45 s. Since it is possible to compute the charge quantitychange and the curvature difference between t = 15 s and 45 s, knF
is easily obtained based on Eq. (2.2). kF is 0.0159 mC−1 and knF is
0.0071 mC−1 . By the use of these kF and knF and the parameters
shown in Table 1, we computed the bending curvature behaviorof CMV IPMC under the rectangular oscillating voltage (amplitude
1.5 V, frequency 1/60 Hz). Fig. 13 shows the computed result of therelationship between the CMV IPMC bending curvature and charge
from t = 0 s through 75 s. This diagram is in good agreement withthe experimental data in Fig. 9(b-2).
Researchers have already reported various circuit models of
IPMCs for characterizing the electrical and electromechanical properties of IPMCs [15–18]. The circuit model proposed in this paper is
one of such models. However, our circuit model is distinct from others, that is, our circuit model takes into consideration the types ofcurrents, Faradic and non-Faradaic currents. Unlike the other models, the inﬂuence of Faradic and non-Faradaic currents on the CMVIPMC bending is analyzed individually.
Some may argue that there is nothing meaningful about theresults given in Fig. 13, since all the parameters were determinedso that the computed result shown in Fig. 13 reproduces theexperimental result shown in Fig. 9(b-1) and (b-2). In order todefy such an argument, we performed further analysis of CMV
IPMC bending behavior. Using some concept employed in thecircuit model, we did an analysis of CMV IPMC bending behavior based on the viscoelasticity. Namely, the bending behaviorof CMV IPMC is explained from the standpoint of physics andthe circuit model we proposed. It is described in the next section.
3. Viscoelastic analysis
The meaning of our circuit model should be clearly shown here.
For the purpose of it, we will attempt to explain the free bending

K. Ikeda et al. / Sensors and Actuators B 190 (2014) 954–967

961

Table 1
Parameters for the circuit model.
 s ’/s  ’/s

Qs ’/mC mm−2Q ’/mC mm−2

Cs ’/mF mm−2C ’/mF mm−2

rs ’/k  mm2r ’/k  mm2

R’/k  mm2

0.18

6.7

0.002470.202

0.00165

0.168

109.2

49.8

200.2

0.005
0.000
-0.005-0.010-0.4(a)0.0025curvature / mm-1

curvature / mm-1

0.010

-0.2

0.0

0.2

-0.0025-0.1

0.4

charge/ mCmm-2

0.0000

(b)

0.0

0.1

charge/ mCmm-2

Fig. 13. Curvature vs. Charge predicted by the circuit model about CMV IPMC (b) shows the part of (a) magniﬁed by a factor of around 2.

behavior of a CMV IPMC employing a certain idea from the circuit
model. More concretely, we will employ the viscoelastic theory
associated with the circuit model proposed in the previous section.
Our viscoelastic model expressed by Eq. (3.1) (to be explained indetail in the Section 3.1) has the same form as the one in ref. [19].
In ref. [19], the authors studied the bending behavior of a Naﬁon
IPMC one end of which was horizontally clamped with a pair ofelectrode and the other end of it 15 mm away from the clamping
point was placed on the stage of balance as illustrated in Fig. 14.
A voltage applied on the Naﬁon IPMC induced the bending in theanode direction, and the authors made measurements of the timecourse of Naﬁon IPMC shape evolution and simultaneously of timecourse of blocking force it generated recorded by the precisionbalance (hereafter the bending test condition illustrated in Fig. 14
is called “constraint bending” in contrast to the free bending). The
experimental data of the blocking force and the shape evolutionof the Naﬁon IPMC led to the computational derivation of thetime-dependent Young’s modulus of the Naﬁon IPMC.
For explaining the free bending behavior of a CMV IPMC by using
viscoelastic theory, we needed to obtain the time-dependent dataof Young’s modulus of CMV IPMC, too. However, the bending behavior of CMV IPMC in our question is free bending, and it is different

from the condition of constraint bending. Hence, in this work, we
further improved the existing viscoelastic model introduced in ref.
[19] and obtained the time-dependent Young’s modulus of CMV
IPMC in the process of free bending. Then we attempted to explain
the free bending behavior of CMV IPMC.
3.1. Viscoelastic model of CMV IPMC
The free bending here means the bending of a CMV IPMC
under a rectangular oscillating voltage (amplitude = 1.5 V, frequency = 1/60 Hz), which is clamped with a pair of electrodes justlike Fig. 1. The polarity of the applied voltage was reversed at t = 15 s
and 45 s. When the CMV IPMC exhibits bending under the applied
voltage, the force f indicated by arrows in Fig. 15, must be exerted
on the CMV IPMC from its inside. More concretely, f represents
the force per unit width (1 mm) of the CMV IPMC. The bending
behavior of the CMV IPMC can be expected to obey the general lawof viscoelastic materials [20]. We put forth a theoretical model bymodifying that of ref. [19] and applying it to our cases.
Suppose that the applied voltage is low and the CMV IMPC does
not exhibit large deformation in the free bending, which implies
that the displacement of x-direction of CMV IPMC is almost 0. Let y(t,Fig. 14. Experimental setup introduced in ref. [19]. Using this setup, blocking force, F(t), and shape of Naﬁon IPMC (more concretely, the coordinates of representative points
of Naﬁon IPMC (xi (t), Yi (t)) were measured as a function of time).

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K. Ikeda et al. / Sensors and Actuators B 190 (2014) 954–967

time-dependence of the Young’s modulus, Ecb (t), of CMV IPMC in
the constraint bending state. By the use of Ecb (t), we derived Efb (t)of the CMV IPMC in the process of free bending. The derivation
process of Efb (t) and f (t) as well as Ecb (t) is described in the next
section.
3.2. Derivation of Efb (t) and f (t)Fig. 15. Fine arrows suggest the induction of force generated from the CMV IPMC
inside, which causes the downward bending of IPMC. Coordinate (x, y) of a representative point on a CMV IPMC in the free bending state is shown.

Employing the experimental and theoretical techniques
described in ref. [19], we ﬁrst obtain the time-dependent Ecb (t) ofa CMV IPMC in the constraint bending state. It is detailed below.
One end of the CMV IPMC was horizontally clamped with a pairof electrodes. The other end of the CMV IPMC was placed on thestage of balance as illustrated in Fig. 14. Once constant 1.5 V was
imposed on the CMV IPMC, it tended to bend downward. Hence, thetip force was exerted on the stage of precision balance, and the precision balance recorded the blocking force F as a function of time.
At the same time the CMV IPMC deforms, and vertical displacement of ten representative points on the surface of CMV IPMC in theconstraint bending state, denoted by Yi (t) at xi (i = 1–10), were measured with a 2-D laser displacement sensor as a function of time,
where all xi are constant, independent of time. According to theformula (12) given in ref. [19], Ecb (t) can be theoretically calculated
by Eq. (3.5):
2F̂(p)l,3p(p)Iˆm

Êcb (p) =x) be the vertical position of the CMV IPMC beam at given positionx at time t as illustrated in Fig. 15. Then a viscoelastic model isrepresented by

 t

Efb (t − )Im ẏ (, x)d + f (t) = 0,y (t, 0) = 0,y (t, l) = 0,

(3.2)

where we denote the length of the CMV IPMC by l. Before moving on to the next step of solving Eq. (3.1), we would like to urge
the readers to pay careful attention to Eq. (3.1). The f was earlier
deﬁned as the force per unit width (1 mm) of the CMV IPMC, andthe width of actual CMV IPMC was 2 mm. Hence, its dimension is[N/mm]. Eq. (3.1) was derived for the CMV IPMC whose width was
the same as the unit width of 1 mm. Namely, we considered a beammodel of CMV IPMC, 20 mm length × 1 mm width × 120 ␮m thickness, for deriving Eq. (3.1). Therefore f (t) in Eq. (3.1) represents not
f (t) [N/mm] but f (t) [N] given by f (t) [N/mm] × 1 [mm] = f (t) [N].
By solving Eq. (3.1) under the conditions Eq. (3.2) with respect
to x, the displacement y(t, x) can be represented byy(t, x) =u(t) 2 1 3(lx − x ),
2Im
3

 ∞

ĥ(p) =

where Efb (t) is the time-dependent Young’s modulus of the CMVIPMC in the free bending and Im is the moment of inertia of thecross section of the beam. The dot and the prime represent thedifferentiation with respect to t and x, respectively. The boundary
conditions are given byy(t, 0) = 0,

where (t) denotes the curvature of CMV IPMC at the left end of the
CMV IPMC, and the wedge “ˆ’’ represents the Laplace transformation,
generally deﬁned by

(3.1)

0

(3.3)

h(t)e−pt dt

0

for a function h(t). Actually, Y(t, x) and (t) have to satisfy
Y (t, x) =(t) 2x (l − x)
4l

(3.6)

for any 0 < x < l.
From these formulas, it is clear that making measurements of thetime course of blocking force and the shape evolution are necessaryfor estimating the Young’s modulus. We attempted to estimate theLaplace transformation of F(t) and (t) as described next: Fig. 16(a)shows the time dependence of the blocking force, F(t), (normalized
by the width of CMV IPMC). The experimental data shows approximately linear relationship. Using the data of CMV IPMC shapeevolution – the coordinate sets of representative points on the CMVIPMC (xi , Yi (t)) (see Fig. 15) –, (t) was calculated by the least squares
method shown in ref. [19], and it is shown in Fig. 16(b). Here weassume that the F(t) and the (t) are given by a simple model as
below, where 0 , 1 , F0 and F1 were obtained by applying the leastsquares method the experimental data shown in Fig. 16(a) and (b),respectively.
(t) = 0 + 1 t

where u(t) is a solution of(3.7)

o = 0.00268 1 = 0.00011

 t

Efb (t − )u̇()d = f (t).

(3.5)

(3.4)

F(t) = F0 + F1 t.

0

For solving Eq. (3.4) with respect to u(t), it is necessary to obtain
Efb (t) and f (t). For deriving Efb (t), we investigated the constraint
bending of CMV IPMC ﬁrst. The coordinates (xi , Yi (t)) representing the shape evolution of CMV IPMC in the constraint bending
state and the blocking force F(t) the CMV IPMC generated (seeFig. 14) were obtained by carrying out the same experiment
explained in ref. [19]. It enables us to theoretically estimate the(3.8)

Fo = 0.0856 F1 = 0.00499
The Laplace transformations of F(t) and (t) are explicitly givenbyF̂(p) =

F0
F1
,
+
p
p2

(p)
ˆ
=

0
1
.
+
p
p2

(3.9)

By substituting these functions into Eq. (3.5) and applying
the inverse Laplace transformation, the Young’s modulus Ecb (t) is

963

0.02

0.6
curvature / mm-1

blocking force / mNmm-1

K. Ikeda et al. / Sensors and Actuators B 190 (2014) 954–967

0.4
0.2
0.0

0.01

0.00
20

0

40

(a)

60

80

0

100

20

(b)

time / s

40

60

80

100

time / sFig. 16. (a) Time course of blocking force (normalized by the width of CMV IPMC) the CMV IPMC generated under 1.5 V constant applied voltage open circle: experimental
linear line: ﬁtting linear line. (b) Time course of the CMV IPMC bending curvature under 1.5 V constant applied voltage × mark: experimental linear line: ﬁtting linear line.

explicitly represented by:
Ecb (t) =

F
2l

1

3Im

1

+

F

0

0

−


F
1

1

e


− 1 t0

= 3.15 − 0.937e−0.0409t .
(3.10)

4.0
3.0
2.0
1.0
0.0
0.0

E (q1 , q2 ) = e0 + e1 (1 − e−˛q1 ) + e1 (1 − e−˛q2 ).

0.2

0.4

0.6

0.8

1.0

total charge / mCmm-2
Fig. 17. Young’s modulus (Ecb ) vs. Total charge imposed on the CMV IPMC.

(3.11)

Eq. (3.11) is a function of charge, q1 and q2 , instead of a functionof time. The q1 represents the charge quantity passing through thebottom surface, involved in the redox reaction Ag ↔ Ag2 O, while
q2 represents the charge quantity passing through the top surface,
involved in the redox reaction Ag ↔ Ag2 O. The e0 represents theYoung’s modulus of intact CMV IPMC. The second term and third
term of Eq. (3.11) represent the variation of Young’s modulus of
CMV IPMC caused by the redox reaction of silver layers of bottom
and top surfaces, respectively. The ˛ is the decay rate associatedwith increase/decrease of Ag2 O quantity on the individual surfacesof a CMV IPMC. Now, we will estimate the parameters in Eq. (3.11).
As described earlier, Ecb (t) is expressed by Eq. (3.10), when theconstant voltage is imposed on the CMV IPMC. In this experimentillustrated in Fig. 14, a constant voltage of 1.5 V continued being
imposed on the CMV IPMC, and no polarity change was performed.
Hence, Ag2 O kept on emerging on the bottom surface by the oxidation of Ag, while the Ag on top layer exhibited no response to theapplied voltage. What we would like to mention here is that if someblocking force / mNmm-1

Young’s modulus / GPa

Eq. (3.10) implies that the Young’s modulus Ecb (t) increases
monotonically and approaches 3.15 GPa as time grows, whichmeans that the CMV IPMC becomes harder and harder. This result
is different from that of the Naﬁon IPMC in ref. [19], where it wasreported that the Young’s modulus estimated for the Naﬁon IPMC
ﬁrst increases with time till it reaches the maximal peak at a certain time, and eventually decreases as time passes. Hence, Eq. (3.10)
indicates the particular feature of the CMV IPMC. However, wecannot use Eq. (3.10) as is. The behavior of CMV IPMC in the freebending state under oscillating voltage is in our question. Since
Eq. (3.10) represents Ecb (t) in the constraint bending state underconstant voltage, it is not applicable for explaining the behaviorof the free bending CMV IPMC. It is necessary to ﬁnd an alternative method of estimating Efb (t). For the purpose of ﬁnding out amethod of estimating Efb (t), we obtained the relationship Ecb vs.
Q ﬁrst, where Q represents the total charge imposed on the CMVIPMC per its unit surface area in the constraint bending state. Ecb
vs. Q was obtained by the following procedure: In the experimentfor obtaining Fig. 16, the current I ﬂowing through the CMV IPMC
was measured as a function of time. We numerically calculated thetotal charge Q(t) imposed on the CMV IPMC by Q(ti ) = I(tj ) (tj – tj-1 )(j = 1, 2, 3,. . ., N, and we denote t0 = 0 s and tN = 100 s) for obtaining
Q vs. t. Since we could obtain Q vs. t here, and already we obtained
Ecb vs. t by Eq. (3.10), Ecb was associated with Q so as to resulting in
Ecb vs. Q shown in Fig. 17.
Since it has been repeatedly reported that the redox reaction
Ag ↔ Ag2 O was induced on the surface of Selemion CMV-based
IPMC, which was virtually the same as CMV IPMC, in the processof its bending [10–12], and it was strongly speculated that the
IPMC Young’s modulus had intimate correlation to the creation andvanishing of Ag and Ag2 O [21]. Previously it was experimentally
conﬁrmed that the quantity of Ag2 O formed on the IPMC surface
was almost proportional to the total charge imposed on the IPMC
[22]. Hence, it is speculated that Efb is charge-dependent. Judging
from this empirical speculation and the shape of curve representing Ecb vs. Q shown in Fig. 17, we put forth Eq. (3.11) as a general
expression of Young’s modulus of CMV IPMC, which means Eq.
(3.11) represents the CMV IPMC Young’s modulus regardless of theexperimental condition the CMV IPMC is in. Appropriately choosing parameters in Eq. (3.11) results in a curve resembling to thediagram Fig. 17.

0.6
0.4
0.2
0.0
0

0.2

0.4

0.6

0.8

1

total charge / mCmm-2
Fig. 18. Blocking force CMV IPMC generated vs. total charge imposed on the CMV
IPMC.

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K. Ikeda et al. / Sensors and Actuators B 190 (2014) 954–967

1.0
surface darkness

voltage / V

2.0
1.0
0.0
-1.0
-2.0

0.0

-2 0 2 4 6 8 10 12 14 16 18
(a)

0.5

time / s-2 0 2 4 6 8 10 12 14 16 18(b)time / sFig. 19. (a) Wave form of voltage imposed on the CMV IPMC (b) Surface darkness index of CMV IPMC vs. time.

f = K(q1 − q2 ),(3.12)

where q1 and q2 are the same charges as in Eq. (3.11).
Now we estimated K in Eq. (3.12). As described earlier, the blocking force in Fig. 16(a) was measured along with the current I as afunction of time. Those experimental data – time course of blocking
force (same as f) shown in Fig. 16(a) and time course of current I– were rearranged into the blocking force vs. total charge imposedon the CMV IPMC as in Fig. 18. As is the case with the procedureof determining the parameters of Eq. (3.11), q2 is zero. Linear line
shown in Fig. 18 is a ﬁtting linear line of blocking force vs. total
charge imposed on the CMV IPMC. The slope of this linear ﬁtting
line is K and is 0.639.
3.3. Prediction of bending behavior based on the viscoelasticmodel
In the previous section, we derived formulas of Efb and f bothas a function of charge. Then we attempt to solve Eq. (3.4) withrespect to u as analytically as possible. In order to verify the validity of our viscoelastic model, we check up if u of our viscoelastic
model can reproduce the oscillatory bending behavior, for example,
Fig. 9(b-1).
Firstly, we need explicit forms of the charge behavior. Since
the current was measured experimentally and the charge can beeasily obtained by integrating the current with respect to time,

we propose a model of the current behavior. Based on the circuit
model introduced in the section 2, it was assumed that the current
takes the forms represented by
I1 (t) = I10 + I11 e−ˇ11 t + I12 e−ˇ12 t
I2 (t) = I20 + I21 e−ˇ21 (t−15)

I3 (t) = I30 + I31 e−ˇ31 (t−45)

0 s ≤ t ≤ 15 s,+ I22 e−ˇ22 (t−15)

15 s ≤ t ≤ 45 s,+ I32 e−ˇ32 (t−45)

45 s ≤ t ≤ 75 s.

(3.13)

i = 1,2,3; j = 0,1,2; k = 1,2 for Ii , Iij and ˇik .
By the procedure described in Appendix C, all the parameters
are given by Eq. (3.14).
I10 = 0.00435,

I11 = 0.0103,

I20 = −0.00427,
I30 = 0.00443,

I21 = −0.042,
I31 = 0.0173,

I12 = 0.0104,

ˇ11 = 11.5,

I22 = −0.00264,
I32 = 0.031,

ˇ12 = 1.88,

ˇ21 = 1.45,

ˇ31 = 12.7,

ˇ22 = 0.167,

ˇ32 = 0.418.

(3.14)

The current is positive from t = 0 s to 15 s and from 45 s to 75 s,
while it is negative from t = 15 s to 45 s.
Electrical properties of CMV IPMC are quite sensitive to the environmental humidity. Although we carried out all the experiments
by maintaining the environmental absolute humidity at 9 gm−3 ,it was practically impossible to achieve the perfect 9 gm−3 absolute humidity environment. Unfortunately, slight deviation of say
0.5 gm−3 from the absolute humidity from of 9 gm−3 can cause largechange of some properties of CMV IPMC such as current and bending curvature. However, basically the bending curvature vs. charge
relationship of CMV IPMC exhibits similar behavior to the diagram
shown in Fig. 9(b-1), even though the slight deviation of absolute
humidity happens, as long as the environmental absolute humidity
is low enough (empirically less than 10 gm−3 ). More concretely, thefollowing characteristics (I)–(III) are observed in the low absolute
humidity environment, even though the absolute humidity slightly

Young’s modulus / GPa

Ag2 O existed on the top surface, undoubtedly it would be reduced
into Ag, but actually not. By the deﬁnition of q1 and q2 described earlier, these charges represent only the charge involved in the redox
reaction of Ag ↔ Ag2 O. Hence q2 was identical to 0. Thus the parameters in Eq. (3.11) are estimated as e0 = 2.18, e1 = 0.999 and ˛ = 3.59.
Hence, we have obtained the Young’s modulus for the oscillatory
free bending IPMC as a function of charge. Namely, Efb = E(q1 , q2 ).
One might argue that this viscoelastic model treats whole current sothat it is involved in the redox reaction, although not all the charge
ﬂowing through the CMV IPMC is involved in the redox reaction ofAg ↔ Ag2 O according to the circuit model. Concerning the circuit
model, in fact, the current ﬂowing though rs and Cs (see Fig. 5(a))
is not related to the redox reaction of Ag ↔ Ag2 O. Since the current
ﬂowing through rs and Cs is virtually negligible, the viscoelastic
model is consistent with the circuit model.
Next we would like to derive the concrete form of f in the freebending state. As described above, it is also difﬁcult to measure
it experimentally. According to ref. [21], it was observed that theblocking force by a Selemion CMV-based IPMC (same as a CMV
IPMC) was roughly proportional to the charge imposed on the CMV
IPMC. Furthermore, based on the basic concept of strength of materials, f is the same as the blocking force. Hence, f could be assumed
to be proportional to the charge imposed on the CMV IPMC as given
by Eq. (3.12)

4.0
3.0
2.0
1.0
0.0
0

25

50

75

time / sFig. 20. Time dependence of Young’s modulus (Efb ) of CMV IPMC.

K. Ikeda et al. / Sensors and Actuators B 190 (2014) 954–967

0.001
0.000
-0.001-0.002
-0.2

(a)

0.002
curvature / mm-1

curvature / mm-1

0.002

965

-0.1

0.0

0.1

0.000
-0.001-0.002-0.2

0.2

charge/ mCmm-2

0.001

(b)

-0.1

0.0

0.1

0.2

charge/ mCmm-2

Fig. 21. Curvature vs. Charge of CMV IPMC from t = 0 s through 75 s (a) Theoretical prediction by viscoelastic model, where the arrows suggest the time progression (b)Experimental result.

deviates from the desired. In fact, the diagram in Fig. 9(b-1) bears
all the three characteristics below, (I)–(III).
(I) Diagram of bending curvature vs. charge takes a very long parallelogram form (it roughly looks a single straight line).
(II) If the diagram is approximated by a single straight line as
described in (I) above, its slope usually takes a particular value
not far different from the order of 0.01 mC−1 .
(III) Time direction of diagram of bending curvature vs. charge isclockwise (though it often turns counter-clockwise in highabsolute humidity environment).
From Eqs. (3.13) and (3.14), Eq. (3.15) was obtained (seeAppendix C).
Q1 (t) = 0.00643 + 0.00435t − 0.0009e−11.5t − 0.00553e−1.88t ,Q2 (t) = 0.0267 − 0.00427(t − 15) + 0.0291e−1.45(t−15) + 0.0158e−0.167(t−15) ,Q3 (t) = −0.0259 + 0.00443(t − 45) − 0.00137e−12.7(t−45) − 0.074e−0.418(t−45) .

(3.15)
Although the explicit forms of the charge Eq. (3.15) can be completely determined by experimental data, exactly speaking, it is
impossible to estimate q1 and q2 in Eqs. (3.11) and (3.12) only by
the experimental data, since not all the experimentally measured
charge was involved in the redox reaction Ag ↔ Ag2 O. Namely,
according to the circuit model, a certain quantity of charge ﬂowed
through rs and Cs , which was not involved in the redox reaction atall. But as described earlier repeatedly, the quantity of such charge
was usually negligibly small. Therefore, we hypothesized that thewhole charge was involved in the redox reaction unless speciﬁcallycommented. Hence, the quantity of Ag oxidized at anode surface
was the same as that of Ag created by the reduction of Ag2 O. This
speculation is supported by the experimental observation of CMV
IPMC surface color below.
Rectangular oscillating voltage shown in Fig. 19(a) was imposedon the CMV IPMC. Darkness of the CMV IPMC bottom surface wasmonitored, during the voltage application, by the use of high speed
microscope VW-9000 (Keyence, Osaka, Japan). When the voltage
was positive, the bottom surface was anode, and when the voltage
was negative, the bottom surface was cathode. Fig. 19 (b) shows thetime dependence of darkness index of CMV IPMC bottom surface,where the darkness index was normalized by the maximal darkness
index. Dark surface suggests the creation of Ag2 O by the oxidationof Ag, because Ag2 O is black. Bright surface suggests the creation of
Ag by the reduction of Ag2 O, because Ag is white. Maximal darkness
index was quantitatively repeatable as clearly indicated by solidovals in Fig. 19(b), while minimal darkness index was quantitatively
repeatable, too, as indicated by dashed ovals. This experimental
observation suggests that the whole Ag2 O created by the Ag

oxidation was reduced into Ag again and whole Ag created by
the Ag2 O reduction was oxidized into Ag2 O. This result is further
interpreted as impose of the a certain quantity of charge, say ␦[Coulomb] which is involved in the redox reaction, on one surfaceof CMV IPMC indicates that quantity of charge −␦ [Coulomb] is
involved in the redox reaction on the opposite surface of the CMV
IPMC. Considering the discussion so far, we derived the explicit
forms of q1 and q2 at arbitral time, where the detail is described inAppendix D.
Substituting the functions q1 and q2 (see Appendix D) deﬁned
above into the Eq. (3.11), we obtained the time-dependent Young’s
modulus Efb (t), which is shown in Fig. 20. Taking a close look at thisdiagram reveals that the Young’s modulus exhibits a bit wavy form
as expected but approximately constant. In order to simplify theprocess of solving Eq. (3.4) analytically, we consider Efb (t) as constant and takes as the value of the Young’s modulus of CMV IPMC,
which was calculated as E0 = 2.54 GPa. Then Eq. (3.4) is rewritten
into
E0 u(t) = f (t),(3.16)

where the force f(t) was obtained by the direct substitutions of thefunctions q1 and q2 into Eq. (3.12).
As a result, we could theoretically predict the relationshipbetween the free bending CMV IPMC curvature and charge imposedon the CMV IPMC by the viscoelastic model. The theoretical prediction is shown in Fig. 21(a) along with typical experimental result
shown in Fig. 21(b) obtained by the use of experimental setup for
viscoelasic model study. Though the theoretical prediction a bit
underestimated the curvature, it still can be said that the curvaturevs. charge was successfully predicted quantitatively. Furthermore,theoretical result agrees with the characteristics (I)–(III) introduced
earlier. We have not found yet the reason of underestimate ofcurvature by the theoretical prediction, but we strongly suspectthat it is due to the experimental error in making measurement ofblocking force described in the Section 3.2. The blocking force wasquite small and its measurement was so critical. Even slight environmental air ﬂow causes undesired noise to the precision balance.
4. Conclusions
The bending behavior of CMV IPMC (Selemion CMV-based
IPMC) was quantitatively well reproduced employing circuit
model. Especially we would like to emphasize that the Faradaic
and non-faradaic current were found to have distinguishedinﬂuence on the bending induction. Employing a concept of thecircuit model, viscoelastic analysis on the bending behavior of
CMV IPMC was carried out. The computational result obtained by
the viscoelastic model successfully predicted the experimental
bending behavior of CMV IPMC. Hence, the appropriateness of thecircuit model was validated by a concept of physics, viscoelasticity.

966

K. Ikeda et al. / Sensors and Actuators B 190 (2014) 954–967

It gives the circuit model a certain level of usefulness for predicting
the bending behavior of IPMC.
To our best knowledge, no one has succeeded in quantitatively
predicting the far future bending behavior of IPMC. Therefore, prediction of far future bending behavior of CMV IPMC is our nextchallenging task.

These equations were solved for Qs ’(t’) and Q ’(t’), where
V’ = −1.5 V, Qs ’(t’ = 0 s) = Qs T , and Q ’(t’ = 0 s) = Q T . Following equations were derived: t  Qs (t  ) = Qs 1 − exp −s t 

+ QST exp −

(B.3)

s

where QS = CS V  (V’ = −1.5 V), t’ = t–T and s = rs Cs

Acknowledgment
We would like to express our gratitude to Mr. Yoshihisa
Kinoshita (KEYENCE) for his technical assistance to conduct thisresearch. This research was conducted under the ﬁnancial supportof the Ministry of Education, Science, Sports and Culture, Grant-inAid for Scientiﬁc Research (C), 24560291,2012.
Appendix A.
We derived the circuit equations about the circuit shown in
Fig. 10 as below:
I = Is + Il + IR

(A.1)

Ir = Il + IR

(A.2)

V=

Qs
+ Is rs
Cs

(A.3)





+ QlT exp

−

t
l



(B.4)
C  r  Rlwhere Ql = r  R+R Cl V  (V’ = −1.5 V), t’ = t–T and l = r l +Rll

Next we derived an equation of I’(t’) just like Eq. (2.4) as follows:







I  s (t  ) =

d  
Q s
t
Q s (t ) =  exp − 
s
s
dt 

I  l (t  ) =

d  
Q
t
Q l (t ) =  l exp − 

l
l
dt

I  R (t  ) = t 
Q T − q l
Q lq=  l + l   exp − 


RCl
RCl
RCl
l





QsT
t
exp − 
s
s

−

−

QlT


l



 t 

exp −

l

(B.5)

(B.6)

(B.7)

Solving the above equations resulted in Eq. (2.7).

We estimated all the parameters in Eq. (3.13) by the following
procedure: Owing to the Eq. (3.13), the charge formulas are given
as simple expressions as below:
I11
I12(1 − e−ˇ11 t ) +(1 − e−ˇ12 t )ˇ11ˇ12

0 s ≤ t ≤ 15 s,

Q2 (t) = Q1 (15) + I20 (t − 15) +I21
I22(1 − e−ˇ21 (t−15) ) +(1 − e−ˇ22 (t−15) )ˇ21
ˇ22

15 s ≤ t ≤ 45 s,

Q3 (t) = Q2 (45) + I30 (t − 45) +I31
I32(1 − e−ˇ31 (t−45) ) +(1 − e−ˇ32 (t−45) )ˇ31
ˇ32

45 s ≤ t ≤ 75 s.

Ql+ Ir rl
Cl

(A.5)

dQl
+ Il
dt

(A.6)

V = IR R + Ir rl

(A.7)

Ql
+ IR R
Cl(A.8)

Solving the above equations resulted in Eq. (2.1) where Qs andQ were both zero at t = 0 s.
Appendix B.
We derived the circuit equations after t’ = 0 about the circuit
shown in Fig. 10 as below:

V =

t
l

(A.4)

Q1 (t) = I10 t +

V =

−

Appendix C.

dQs
+ Is
dt

V=

Ql (t  ) = Ql 1 − expQ sdQ  s 
+
rs
Cs
dt 

(B.1)

Q l
dQ  l Q+
r l +  l r  l


Cl
RCl
dt

(B.2)

(C.1)

All the parameters in Eq. (C.1) are employed in Eq. (3.13).
Since the charge changes continuously, Q2 (15) = Q1 (15) andQ3 (45) = Q2 (45), respectively. We estimated all the parametersin Eq. (C.1) by the use of the current experimentally obtainedby applying the rectangular oscillating voltage (amplitude = 1.5 V,
frequency = 1/60 Hz) on a CMV IPMC in the free bending state.
Employing the Newton method, the parameters in Eq. (C.1) (or in(3.13)) are estimated as given by Eq. (3.14).
Here, we have to make a comment that we did not use the experimental data of current shown in Fig. 11, which was used in thecircuit model study, for obtaining the results given by Eq. (3.14).
Instead of it, we chose to use the current data obtained by using theexperimental setup for viscoelastic model study (data not shown inthis paper). It is due to the following reason: Since our experimental
setup employed for viscoelastic model study was not exactly sameas that employed for circuit model study, we thought that it was
better to use the current data obtained by employing the setup forthe viscoelastic model study, as long as we performed viscoelastic
model study. But the relationship curvature vs. charge was basically the same as in Fig. 9(b-1) regardless of the experimental setup
employed.
Appendix D.
Explicit form of q1 and q2 at arbitral time were derived by thefollowing discussion:

K. Ikeda et al. / Sensors and Actuators B 190 (2014) 954–967

It is clear that q1 and q2 for 0 s ≤ t ≤ 15 s are given by the following
equations:q1 = Q1 (t)(D.1)

q2 = 0(D.2)

After the polarity of the applied voltage is reversed, the following two equations are valid until q1 becomes 0. Hence, the following
two equations are derived:
q1 = Q2 (t)(D.3)

q2 = Q1 (15) − Q2 (t)(D.4)

Since Q2 (t) is equal to 0 at t = 22.3 s from the Eq. (C.1), q1 = 0 and
q2 = Q1 (15) at t = 22.3 s. Since reduction of Ag2 O on a surface stops
at t = 22.3 s, all energy is consumed only for oxidization of Ag on
the surface of the other side, which implies that q2 = Q1 (15) – Q2 (t)from t = 22.3 s to 45 s.
q1 = 0(D.5)

q2 = Q1 (15) − Q2 (t)(D.6)

By the same argument, following four equations were derived:
q1 = Q3 (t) − Q2 (45)(D.7)

q2 = Q1 (15) − Q3 (t)(D.8)

where the above two equations are valid for 45 s ≤ t ≤ 67 s.
q1 = Q3 (t) − Q2 (45)(D.9)

q2 = 0 at t = 67 s and onward(D.10)

where the above two equations are valid for 67 s ≤ t ≤ 75 s.

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Biographies
Kota Ikeda obtained PhD from Tohoku University, Japan in 2008. He is a speciallyappointed lecturer in Meiji University, Japan, and currently involved in the researchon pattern formation problems and reaction diffusion equations.
Minoru Sasaki obtained PhD from Tohoku University, Japan in 1985. He is a professor in Gifu University, Japan, and currently involved in the research on robotics andcontrol theory.
Hirohisa Tamagawa obtained PhD from Tokyo Institute of Technology, Japan in
1997. He is an assistant professor in Gifu University in Japan and currently involvedin the research on the biomaterials, especially electroactive polymers.