Supplementary Information
Multiphysics of Ionic Polymer Metal Composites Actuator
Zicai Zhu1,2, Kinji Asaka3, Longfei Chang1,2, Kentaro Takagi4 and Hualing Chen1,2*
1 State Key Laboratory of Mechanical Structure Strength and Vibration, Xi’an
Jiaotong University, Xi’an, 710049, PRC
2 School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, 710049, PRC
3 Health Research Institute, National Institute of Advanced Industrial Science and
Technology (AIST), Ikeda, Osaka 563-8577, Japan
4 Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
* To whom correspondence should be addressed. E-mail:
hlchen@mail.xjtu.edu.cn
A. Blocking effect of electrode
In Fig. 5, the moment balance equation is,
F1  tw  te   M1  2M 2  M 0 .
(A1)
Where te and tw are the thickness of the electrode and the total IPMC
respectively. As te<<tw , the moment M2 in electrodes can be ignored. Then we have,
F1tw  M1  M 0 .
(A2)
Using the linear strain  xx in Eq.Error! Reference source not found., M1 can be
figured out as,
M 1   ENW  xx hydy.
(A3)
Where ENW is the effective elastic modulus of the wet substrate membrane, h is
the width of the cantilever. And the strains are continuous at the contact surface
between the electrode and the substrate, so we get,

F1
M 1 tw

.
hte EPd
ENW I1 2
(A4)
Where EPd is the effective elastic modulus of the electrode, I1 is the area moment
of inertia of the substrate. The deformation under the bending moment M0 will be
much larger than that under the moment M1 because of the blocking force F1. The
electrode effect is equivalent to an external mechanical load, the blocking moment
F1tw.
M ext  F1tw 
6hte EPd
 xx ydy
tw 
(A5)
Then the corresponding elastic stress is,
 eB 
M ext
y
I1
(A6)
B. Electrostatic stress
According to the effective electrical field theory, the total electric field
influenced the local electrostatic stress in an ionic cluster is the sum of three electric
fields, Ee induced by electrons in the electrodes, Eout induced by the redistributed
ionic charges excluded the local cluster and Ein induced by the ionic charges in the
local cluster. The directions of Ee and Eout are along the thickness, so they cause the
cation migration and have little contribution to the local electrostatic stress. Then the
electrostatic stress in a cluster is mainly decided by the ionic charges in it.
Based on the RVE model, the total charge of cations in a cluster is,

4
1 
Q   cI F VS  VW   cI F  r 3 1 

3
 wV 
(A7)
If the cations distribute uniformly in the pore as in Fig. 6(a), the volume density
of the positive charges is,

Q
1 
 
 cI F 1 

VW
 wV 
(A8)
If the cations distribute uniformly on the liquid/solid surface as in Fig 6(b), the
surface density of the positive charges is,
 
Q
r
1 
 cI F 1 

2
4 r
3  wV 
(A9)
For the fixed anion charges, the total charge is,
Q   c  F VS  VW 0   c  F
4 r 3 
1 
1 

3 1  V   wV 
(A10)
So the surface density of the negative charges is,
 

Q
r
1 
 c F
1 

2
4 r
3 1  V   wV 
(A11)
For the electric charge system composed of positive charges in Eq. (A8) and
negative charges in Eq. (A11), the total electrical energy of the charge system is,
W

4 r 3    2 r 2
3
    r    2 

3 W  5
2

(A12)
2
 3  Q  2
Q  






Q Q 
4 W r  5
2 


1
Where W is the dielectric constant of water. Using the electric energy we can
calculate the electric force on the outer surface of the electric charge system, i.e. the
liquid/solid surface.
2
 2

Q  

1  3Q 
 W 
 

Fer   
Q Q 
 
2
2 
 r Q ,r 4W r  5


(A13)
And the electrical stress is,
2
2

c 

Fer
r0 2 F 2  wV  3 
1  3 2
c

pe1 

cI  cI


 1 

4 r 2
9W  wV 0   wV   5
1  V 2 1  V 2 


2
(A14)
For the electric system composed of positive charges in Eq.(A9) and negative
charges in Eq. (A11), the electrical stress is,
2
2
r 2 F 2  wV  3 
1  
c 
pe 2  0
1

c


 
  I

18W  wV 0   wV  
1  V 
2
(A15)
C. Model simplification
Dimensionless method is used to simplify coefficients for the comparison. A few
characterized parameters are presented for the dimensionless model as follows:
Length scale: ls 
Time scale: ts 
 RT
F 2c 
 RT
d II F 2 c 
.
.
Voltage scale: Vs 
RT
.
F
Pressure scale: Ps 
RT
.
VW
Concentration scale: cs  c  .
For the water based IPMC, the voltage scale is about 25.85mV, the pressure scale
is about 138.57MPa, while the others are dependent on the IPMC properties.
Furthermore three ratios are also presented,
Initial water number:  
cW 0
c
Ratio of diffusion coefficients  
dWW
d II
Ratio of hydraulic permeability coefficient  
KPs
dWW
The ratio of hydraulic permeability coefficient means the ratio of the convection
flux to the diffusion flux induced by the same pressure gradient. For Nafion based
IPMC, usually the concentration and mobility of water are larger than that of cation in
IPMC, then  and  are both larger than 1, and the convection flux is no less than
the pressure induced diffusion flux,  is no less than 1.
Using    *Vs ， cI  cI *c  ， cW  cW *c  ， p  p* Ps ， xi  xi*ls and t  t *t s (the
variable with a superscript * is the corresponding dimensionless form), then the
transport equation system including Eq. Error! Reference source not found.,
Error! Reference source not found., Error! Reference source not found. and
Error! Reference source not found. can be transformed to,
 2 *  1  cI * ;



VI * *
 cI *
*
*
*
*
*
*
*
*



c

zc



c

p


n

c

c

p


c

p






I
I
I
dI
W
W
I
 t *
V
W







 c *

 (A16)

VI * *
cI p    cW *p* 
 W*     cW *  cW *p*   ndW  cI *  zcI * * 
VW
 t





N
 p* 
pi*


i 1
For Nafion based IPMC, usually the concentration and mobility of water are
larger than that of cation in IPMC, then  and  are both larger than 1, and the
convection flux is no less than the pressure induced diffusion flux,  is no less than
1.The partial molar volume of cations are far less than that of water, so
VI
VW
1   ,
then the diffusion terms of cation induced by the pressure can be ignored. Compared
the convection flux with diffusion flux induced by the pressure in the water equation,
the former is larger than the later (   1 ) even if we use the smaller hydraulic
permeability measured by Evans. And in experiments we can not distinguish the
convection from the pressure diffusion flux, it means the hydraulic permeability is
obtained from the sum of the two fluxes. So we keep the convection term and ignore
the pressure diffusion term in the final form. Then we get a simplified equation as,
 cI
 
z Fc


   d II  cI  I I    ndI dWW cW  cI K p  .

RT

 t
 


 cW    d c  n d  c  z I FcI    c K p  .
dW II 
I
 W
 WW W

 t
RT





(A17)
However, the cluster size will change when the water content declines. Smaller
radius of the cluster can make the convection flux reduce greatly. Then the pressure
diffusion turns to be the dominant mechanism, so the following form is also right in
some occasions,
 cI
 
z Fc


   d II  cI  I I    ndI  dWW cW  cW K p   .

RT

 t
 


 cW    d c  c K p  n d  c  z I FcI    .
dW II 
I

 WW W W
 t
RT




(A18)
When   1 , equation (A18) turns to be the case dominated by the pure
diffusion mechanism. Although Eq. (A17) and Eq. (A18) are different on the pressure
term, numerical results of them are almost the same.
D. Cation and water distribution along the thickness varied with time
(1) Model B. The concentrations of the cation and water along the thickness
varied with time of Model B (ndw=4) are shown in Fig. A1 (Thickness axis 0: Cathode,
2×10-4m: Anode). In Fig. A1(a) cations accumulate near the cathode quickly after
applying the electrical stimulation. The highest steady concentration can achieve 40
times of the average. When shorting the electrodes, the cations at the cathode quickly
go back to the anode. In Fig. 8(b) water concentration also rises up quickly at the
cathode and falls at the anode. The concentration difference between the highest and
the lowest is no more than 15% of the average concentration. After the maximum
concentration point at the cathode, water concentration decreases with time slowly
due to the concentration diffusion. It will take tens of seconds for all the stored water
to go back to the anode and recover the initial state. When the electrodes are shorted,
the water molecules firstly accumulate near the anode, and then also diffuse back to
the initial state.
(a) Cation distribution
(b) Water distribution
Fig. A1 Model B: Concentration distribution along the thickness varied with time
(2) Model C. The dynamic process of the cation distribution along the thickness
is similar to the result in Fig. A1(a) of Model B. While that of the water distribution is
quite different from Fig. A1(b) and is shown in Fig. A2, the water is not highly
concentrated in the cathode any more, and the profile of the water concentration is
almost linear with the thickness.
Fig. A2 Model C: Water distribution along the thickness varied with time
(3) Model D.
For a specific case ndw=4, nhy=3, the dynamic distribution processes of the cation,
the free water and the total water are illustrated in Fig. A3. Compared Fig. A3 (a) with
Fig. A1 (a), the steady cation concentration at the cathode of the former is much lower,
only about 5 times of the average concentration in the bulk. In Fig. A3 (b) the free
water distribution along the thickness shows a contrary trend compared with the
distribution of the cation and the later total water. Much free water is driven to the
anode after applying the stimulation. In Fig. A3 (c), in the bulk the total water
concentration profile still displays a uniform gradient across the thickness and agrees
with the experimental results of the neutron imaging. While in the two electrode
boundaries, especially in the cathode, the total water concentration shows a sudden
change. It is caused by the distribution of the hydrated water concentration that is
much high in the cathode and low to zero in the anode.
(a) Cation distribution
(b) Free water distribution (revised thickness direction)
(c) Total water distribution
Fig.A3 Model D: Cation and water distribution along the thickness varied with time