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IOP PUBLISHING SMART MATERIALS AND STRUCTURES
Smart Mater. Struct. 16 (2007) 14771488 doi:10.1088/0964-1726/16/4/063
A dynamic model for ionic polymermetalcomposite sensorsZheng Chen, Xiaobo Tan1, Alexander Will and Christopher Ziel
Smart Microsystems Laboratory, Department of Electrical and Computer Engineering,Michigan State University, East Lansing, MI 48824, USA
E-mail: firstname.lastname@example.org and email@example.com
Received 31 January 2007, in final form 8 June 2007Published 17 July 2007Online at stacks.iop.org/SMS/16/1477
AbstractA dynamic, physics-based model is presented for ionic polymermetalcomposite (IPMC) sensors. The model is an infinite-dimensional transferfunction relating the short-circuit sensing current to the applied deformation.It is obtained by deriving the exact solution to the governing partialdifferential equation (PDE) for the sensing dynamics, where the effect ofdistributed surface resistance is incorporated. The PDE is solved in theLaplace domain, subject to the condition that the charge density at theboundary is proportional to the applied stress. The physical model isexpressed in terms of fundamental material parameters and sensordimensions and is thus scalable. It can be easily reduced to low-order modelsfor real-time conditioning of sensor signals in targeted applications of IPMCsensors. Experimental results are provided to validate the proposed model.
(Some figures in this article are in colour only in the electronic version)
0 chargestress coupling constant (J C1) V volumetric change (m3 mol1)e dielectric permittivity (F m1) electric potential (V) charge density (C m3) induced stress (Pa)C+ cation concentration (mol m3)C anion concentration (mol m3)D electric displacement (C m2)d ionic diffusivity (m2 s1)E electric field (V m1)F Faraday constant (C mol1)F(t) external force (N)h distance from neutral axis to surface (m)I moment of inertia (m4)i short-circuit current (A)ip sensing current density (A m1)is surface current (A)J ion flux vector (A m2)L free length of IPMC beam (m)
1 Author to whom any correspondence should be addressed.
M bending moment (N m)p fluid pressure (Pa)Q sensing charge (C)R gas constant (J mol1 K1)r0 surface resistance per length and width ()v free solvent velocity field (m s1)W width of IPMC beam (m)w tip deflection (m)Y Youngs modulus (Pa)
Ionic polymermetal composites (IPMCs) form an importantcategory of electroactive polymers and have built-in actuationand sensing capabilities [1, 2]. An IPMC sample typicallyconsists of a thin ion-exchange membrane (e.g. Nafion),chemically plated on both surfaces with a noble metal aselectrodes . Transport of ions and solvent molecules withinan IPMC under an applied voltage leads to bending motionsof the IPMC, and hence the actuation effect. IPMC actuatorshave various promising applications in biomedical devices,bio/micromanipulation and biomimetic robotics . Onthe other hand, a mechanical stimulus on an IPMC causesredistribution of charges and produces a detectable electrical
0964-1726/07/041477+12$30.00 2007 IOP Publishing Ltd Printed in the UK 1477
Z Chen et al
signal, suggesting the use of IPMCs as mechanical sensorsfor force, pressure, displacement or velocity measurementin medical applications, structural health monitoring androbotics .
Comparing with the extensive work on modeling of IPMCactuators (see, e.g., ), research on IPMC sensingmodels has been relatively limited. De Gennes et al proposeda static model based on linear irreversible thermodynamics tocapture both actuation and sensing mechanisms of IPMCs .Using an analogy to piezoelectric materials, Newbury and Leopresented a geometrically scalable grey-box model for IPMCactuators and sensors [17, 26]. The latter model was furtherelaborated and verified by Bonomo et al . Farinholt andLeo derived the charge sensing response for a cantileveredIPMC beam under a step change in tip displacement .The derivation was based on a linear, one-dimensional partialdifferential equation governing the internal charge dynamics,which was first developed by Nemat-Nasser and Li forstudying the actuation response of IPMCs . A keyassumption in  is that the initial charge density at theboundary is proportional to the applied step deformation.Farinholt and Leo then obtained an analytical but approximatesolution by assuming that the solution is separable as a productof temporal and spatial components.
The contribution of the current paper is an explicit,dynamic, physics-based model for IPMC sensors. Thestarting point of the model development is the same governingPDE as in [18, 28] that describes the charge redistributiondynamics under electrostatic interactions, ionic diffusion andionic migration along the thickness direction. But thiswork extends previous studies significantly in three aspects.First, it incorporates the effect of the distributed surfaceresistance, which is known to influence the actuation andsensing dynamics [29, 30]. The consideration leads toadditional dynamics on IPMC surfaces along the lengthdirection. Second, an exact, analytical solution to the PDEis obtained by converting the original time-domain equationto the Laplace-domain version. Third, instead of limiting tostep deformation only, an arbitrary mechanical deformationstimulus is allowed, which is of interest for real applications.For the boundary condition, an assumption analogous to thatin  is made: the charge distribution at the boundary isproportional to the externally applied stress. The derived short-circuit sensing current is related to the mechanical deformationthrough an infinite-dimensional transfer function involvinghyperbolic and square-root terms. The transfer function modelis expressed in terms of fundamental physical parameters andsensor dimensions, and is thus geometrically scalable. It canbe further reduced to low-order models in the form of rationaltransfer functions, which are again scalable.
Experiments have been conducted to validate the proposeddynamic model for an IPMC sensor in a cantileverconfiguration. Good agreement, both in magnitude and inphase, has been achieved between the measured sensingresponse and the model prediction for periodic mechanicalstimulation from 1 to 20 Hz. The results show that consideringthe surface resistance leads to more accurate predictions.The geometric scalability of the sensor model has alsobeen confirmed without re-tuning of the identified physicalparameters. Experiments for an IPMC sensor under a damped
oscillatory deformation and a step deformation have furtherverified the reduced, low-order model. The influence of theIPMC hydration level on the sensing performance has alsobeen studied.
The salient feature of the proposed model is itscombination of the usually incompatible advantages ofphysics-based models and simple models. As a physicalmodel, its effectiveness in predicting sensing responsesin experiments provides valuable insight into the sensingmechanism of IPMCs. On the other hand, the model, as atransfer function, is suitable for analysis and design using therich set of system-theoretic tools. In particular, the reducedlow-order models can be conveniently used to constructcompensation circuits or algorithms for real-time conditioningof IPMC sensor signals.
The remainder of the paper is organized as follows. Thegoverning PDE is reviewed in section 2. In section 3 theexact solution and thus the model are derived, with andwithout considering the surface resistance. Model reductionis discussed in section 4. Experimental validation resultsare presented in section 5. Finally, concluding remarks areprovided in section 6.
2. The governing partial differential equation
The governing PDE for charge distribution in an IPMC wasfirst presented in  and then used by Farinholt and Leo for investigating the sensing response. Let D, E, and denote the electric displacement, the electric field, the electricpotential and the charge density, respectively. The followingequations hold:
E = De
= , (1)
D = = F(C+ C), (2)where e is the effective dielectric constant of the polymer,F is Faradays constant, and C+ and C are the cation andanion concentrations, respectively. The continuity expressionthat relates the ion flux vector J to C+ is given by
J = C+
Since the thickness of an IPMC is much smaller than its lengthor width, one can assume that, inside the polymer, D, E and Jare all restricted to the thickness direction (x direction). Thisenables one to drop the boldface notation for these variables.The ion flux consists of diffusion, migration and convectionterms:
J = d(
C+ + C+ F
RT + C
+ C+v, (4)
where d is the ionic diffusivity, R is the gas constant, T isthe absolute temperature, p is the fluid pressure, v the freesolvent velocity field and V is the volumetric change, whichrepresents how much the polymer volume swells after takingwater. From (2), C+ can be written as
C+ = 1F
+ C, (5)
A dynamic model for ionic polymermetal composite sensors
Figure 1. Geometric definitions of an IPMC beam in the cantilevered configuration.
where C is homogeneous in space and time-invariant sinceanions are fixed to the polymer backbone. Taking the gradientwith respect to x on both sides of (5) and using (1) and (2), onegets
C+ = eF
2 E . (6)Darcys Law  is used to relate the fluid velocity v to thepressure gradient p ,
v = k (C FE p), (7)where k denotes the hydraulic permeability coefficient.Neglecting the convection term , i.e. assuming v =