Fully Coupled Simulation Model for Transient Response of
Mechanical Sensors using Electro-active Polymers
Seongwon YOO 1 and Yutaka TOI 1
11

Institute of Industrial Science, University of Tokyo, Japan

Introduction

The conducting polymer(CP) showed a multitude ofpotentials as a novel material and exhibited prospective
characteristics: low voltage, high sensitivity, miniaturization,lightweight, easy access, flexible, silent and chemically stableproperties. The CP has attracted attention as a candidate ofactuators or sensors [2-6].
The aim of the present study is to develop a computationalsystem for design and control of the mechanical CP sensors. TheCP sensors, responding to mechanical stimulation, showtransient behaviors with the relaxation of reaction forces. Whena bending deformation occurs as shown in Fig.1, ions andsolvent are pushed out from compressive sides and enter into thetensile sides. The transport reduces reaction force and causes
instant electric potential by charge density gradient, and then theelectric potential decreases [4].
Even though the CP sensors, conversely to the well-known
CP actuators, generate an electrical voltage when a displacementis enforced, the output voltage from the input displacement isvery much smaller than the input voltage to the same
displacement [5,6]. Hence, the computational model of the CPsensors has not been well established even though many
researchers have conducted the analysis of the CP actuators.
For the estimation of the sensor output, the simplified model[8] was developed from the observation that output voltage is
proportional to mechanical solid stress. However, the simplified
model cannot express the transient behavior of the CP sensor.
Furthermore, it cannot explain the aforementioned difference ofinput-output relation between the actuator and sensor. In orderto overcome this problem, the present study focused on theinteractions between solid, fluid and ion transport in the polymer
matrix. First, the mechanical behavior of the sensor is assumed
as the bending of layered Timoshenko beam model. Second, thefluid pressure in the polymer matrix is obtained by Biot poroelastic model with electric potential and solid strain. Third, ionFig. 1 Mechanical sensor using conducting polymer

concentration and electric potential are calculated by PoissonNernst-Planck equations with the fluid pressure. Therefore, thetransient response with the relaxation is numerically simulated.
Lastly, the numerical results are compared with experimentalresults.
2Formulation

2.1

Layered Timoshenko Beam Model
The CP sensor consists of different layers having different
properties, and its mechanical deformation can be assumed as
the bending of a beam in solid mechanics[3,6]. Therefore,layered Timoshenko beam theory [7] is employed as shown inFig.2.
2.2

Biot Poro-elastic Model
The CP has two phases, a polymer matrix and electrolyte.
The polymer matrix is solid, and the electrolyte is fluid in thepolymer matrix. When the CP sensor is deformed, the inducedpressure difference of the matrix causes fluid flow through thepores of the matrix. For the analysis of the porous CP saturated
with fluid, Biot poro-elastic model [1] is employed. In case ofisotropy, Biot linear stress-strain relations for Timoshenko beamwith no leakage of fluid on boundary, is obtained as follows.

 x  E x  bp  Eud  bp  E   z
  bpxdx 

(2.2.1) w u  dw   G x z  dx
(2.2.2)

 zx  G zx  G

where,  x is nominal stress,  x is nominal strain,  zx is transverse
shear stress,  zx is transverse shear strain; p is pore pressure, uis axial displacement, w is lateral displacement,  is rotation of
the normal section, z is distance from neutral axis, E is elasticmodulus, G is shear modulus, b is Biot coefficient, and  isshear correction factor.
Next, the fluid pressure can be obtained by Darcy law inwhich Biot linear stress-strain relations are embedded so that theflux of the fluid in the porous media can be obtained. Adding
electric force into the Biot poro-elastic model, leads to thefollowing differential equation of the pore pressure in the CPs.
h  2 p3( u   ) 2V 
3 
 2  FC 2   x  p   z
B z  2GB(1   )(1   u ) t (2.2.3)

where,  h is hydraulic permeability,  is dynamic viscosity, is porosity, F is Faraday constant, C is ion concentration, V iselectric potential, B is Skemptom coefficient,  is drained Poisson’s ratio, and  u is undrained Poisson’s ratio.
2.3

Fig. 2 Layered Timoshenko beam model|

Poisson-Nernst-Planck (PNP) Model
CP molecules and large dopants are immobile, so the movement of mobile ions causes voltage as sensor output [3,5].
Therefore, PNP equations [3] are employed to the analysis of

the ion transport in electric field. Adding the pore pressure intothe PNP equations, ion concentration distribution and electric
potential are obtained with the below equations.
2h 2 p
C  h D   2 C
 F  V nC  2 Ct  z 2RT z 2z(2.3.1)

 2V FCkez 2(2.3.2)

where, n is species charge of mobile ion, R is the gas constant,
T is absolute temperature, k e is electric permittivity, and D isdiffusivity coefficient.
Using the layered Timoshenko beam, Biot poro-elastic andPNP models, the interactions between fluid pressure, bending
stress, and electric field at each time step are fully coupled.
3Fig. 5 Mobile ion concentration over thickness at beam root
during loading(left) and relaxation(right)

Numerical Results

A typical conducting polymer sensor in Figs.1 and 2 wasselected in this simulation. The physical parameters of theaforementioned models for the CP are determined by experiments in macro-scale. The reservoir for electrolyte is TBA.PF6of 0.05M in solvent propylene carbonate. The ion movement ismuch more dominant factor than oxidation-reduction reaction,
because of the dependence of the polarity of voltage on the typeof counter-ions [5]. Furthermore, the ion flux applies only tomobile ions and excludes the polymer molecules and large
dopants [3]. A monotonic case of a prescribed displacement
enforced at the beam was simulated and its results are shown in
Fig.3~6.

Fig. 6 Electric potential output with respect to time(left) andelectric potential over thickness at root(right)
4

Conclusion

In the present study, the interactions of solid stress, fluidpressure, ion transport and electric potential were fully coupled
and the transient response with the relaxation of the reaction
force was analyzed with the time-marching. As a result, themechanism of the CP sensor was understood through thenumerical analysis. Mechanical factors, especially porosity,
have a large effect on the responses. Above all, the noninvertible problem between actuator and sensor models, inwhich the current black box model cannot be used for both ofsensor and actuator, was solved through the present model using
the same theories in the actuation models.
References

Fig. 3 Time history of tip displacement and reaction force[1]

[2]

[3]

[4]

Fig. 4 Pressure distribution over thickness at beam tip
during loading(left) and relaxation(right)
As shown in Fig.3, when the prescribed displacement isenforced, the reaction force elastically responses and thengradually reduced with time. As illustrated in Fig.4, the fluidpressure was obtained with the interactions between fluidpressure, bending stress, and electric field. These results revealthat the relaxation of the reaction force is due to the change ofthe fluid pressure which means the fluid flow in the pores of thepolymer matrix. The mobile ion concentration in Fig.5 is alsochanged with the fluid pressure. Finally, electric potential areobtained with the mobile ion concentration as shown in Fig.6.
As a result, mechanical deformation results in differences ofionic charge and voltage as sensor output. Furthermore, thetransient behavior and relaxation were numerically expressed.

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