Physics-based modeling of mechano-electric transduction of tube-shaped ionic polymer-metal composite

Ionic polymer-metal composites (IPMCs), a type of electroactive polymer (EAP), have been a focus of study for actuation and sensory applications since the late 20th century.^1–3 Such studied applications have included implementing IPMCs for use in robotic actuators and sensors, energy harvesting devices, underwater robotics applications, and medical applications, such as active catheters.^4–11 IPMCs are sought after in many applications due to their ability to induce large deformation in case of an applied low-voltage input, referred to as electromechanical transduction, and produce an electric potential for an applied deformation, called mechanoelectric transduction.^3,12 This enables IPMC devices to be used in both actuation and sensory applications.^13–20 The ability to simulate IPMC actuation and sensor transduction phenomena through physics-based modeling is desirable to better develop IPMC fabrication specifications and enable advanced IPMC device design.¹² The focus of the current study is to fabricate and experimentally study tube-shaped IPMC sensors and simulate their transduction phenomena through physics-based modeling. This paper is organized as follows. A brief background into IPMC sensors will be provided. Then the physics-based mathematical model will be explained. Then fabrication and experimental setup will be presented. Finally, experimental and simulated results are presented followed by a summary of the research and findings.

The polymer membrane of an IPMC consists of anions fixed to the polymer backbone and freely positioned positive ions or cations. In a hydrated state, micro-channels in the polymer expand to allow free transport of cations and attracted water molecules through the membrane. When a voltage is applied to IPMC electrodes, the cations migrate away from the anode, dragging the attracted water molecules with them. This causes osmotic pressure change with localized swelling near the cathode interface and contraction near the anode, which results in an overall deformation of the IPMC. A basic schematic of an IPMC polymer structure and electromechanical transduction phenomenon is shown in Fig. 1.

IPMC actuators and sensors can be made in a variety of shapes, granted they have two or more separated electrodes to apply or measure electric potential. Much research has been done for rectangular IPMC actuators and sensors, which provide 1D transduction. Rectangular bar, cylindrical, or tube-shaped IPMCs are capable of producing 2D transduction and have been gaining considerable research attention.^7,21 IPMC geometry of interest for this study is tube-shaped, which is capable of multi-directional actuation and sensing. Additionally, tube-shaped IPMCs can allow transportation of fluids, tools, or other materials through the center of the actuator, which is desirable for medical applications, such as catheters. The focus of this research is to lay out the fundamental theory of transduction for tube-shaped IPMCs to facilitate development and fabrication of advanced functional devices of this kind.

The underlying cause of IPMC electromechanical and mechanoelectrical transduction is induced ion migration and resulting charge density in the vicinity of the electrodes. Ionic current in the polymer for both cases can be described by the Nernst-Planck equation.²² For a single species absent of chemical reaction, the following species flux terms are of significance to describe the transport of cations through the IPMC polymer membrane:

where j_diffusion is the cation diffusive flux, j_{migration in electric field} is the cation flux due to migration in the electric field, j_convection is the cation convective flux, C is cation concentration, D is the diffusion coefficient, F is Faraday's constant, z is the charge number, and ϕ is the electric potential in the polymer. The cation mobility, μ, can be expressed as D/RT, where R is the gas constant and T is the absolute temperature. The cation velocity, u, can be expressed in terms of the solvent pressure gradient as

where ΔV is the molar volume which quantifies the cation hydrophilicity and ∇P is the solvent pressure gradient. The change of cation concentration with time can be expressed by performing a species flux balance over a given volume; that is, the sum of the species flux gradients and time derivative of species concentration must equal to zero. Thus, the time-dependent transport of cations in the IPMC polymer membrane can be described by the Nernst-Planck equation as

An important difference between IPMC electromechanical and mechanoelectrical transduction is the magnitude, direction, and significance of individual terms in Eq. (5). Besides the concentration time derivative, the terms of Eq. (5) consist of three flux terms governed by the field gradients of electric potential, concentration, and solvent pressure. In case of actuation (i.e., electromechanical transduction), the electric potential gradient term is significantly more prevalent than the solvent pressure flux, that is, zF∇ϕ ≫ ΔV∇P, therefore the pressure flux term is often neglected in actuation model implementation. However, in case of sensing (i.e., mechanoelectric transduction), these terms are of similar significance to the diffusion flux term and neither should be neglected.²³ It is also interesting to note the direction of the electric potential gradient is opposite for sensing as compared to actuation because the ionic current is governed by an induced pressure gradient rather than an applied voltage.

The electric potential gradient term can be described by Poisson's equation

where ε is the absolute dielectric permittivity and ρ is charge density defined as

where C_a is local anion concentration.

While the cation concentration C is governed by the Nernst-Planck equation (5), the anion concentration is related to local volumetric strain. The volume changes in the polymer matrix affect the local anion concentration because anions are fixed to the polymer backbone. Hence, the anion concentration C_a is expressed as

where u₁, u₂, and u₃ are local displacements in the x, y, and z directions, respectively, and C₀ is the initial ion concentration.

Equations (5) and (6) comprise the Poisson-Nernst-Planck (PNP) model for IPMCs and describe the fundamental physics within the polymer membrane.

The linear elastic material model has been used to describe deformation of IPMCs. The constitutive relation of Hooke's Law was used to relate stress and strain in the polymer as

where ε_ij is the normal strain in the i-direction for i = j and a shear strain for i ≠ j. Stress terms σ_ij are defined similarly, where σ_ij is normal stress in i-direction for i = j and a shear stress for i ≠ j.

The constants μ and λ are Lame's constants, defined as

where E is Young's Modulus and ν is Poisson's ratio.

The system is in equilibrium if Navier's displacement equations are satisfied, given by the relation

where F is the body force vector per unit volume.

Newton's second Law is used to describe time-dependent deformation

where u is the local displacement vector and ρ_p is the polymer density.

It is beneficial to note that the mechanoelectrical IPMC modelling framework presented herein can be readily applied to IPMCs of arbitrary cross-sectional geometry; that is, it can be directly used to simulate sensing response for any shape IPMC by adjusting the model geometric domain and mesh. In the present study, a tube-shaped IPMC geometry has been selected.

Three-dimensional finite element simulations were performed using COMSOL Multiphysics 4.3b software to couple mechanical deformation with ionic concentration, diffusion, and migration physics. The mesh used for all simulations was extra fine in the radial direction with a set node distribution along the edges and sparsely swept along the longitudinal direction. The mesh and IPMC tip geometry are shown in Fig. 2.

The simulation utilizes the Solid Mechanics, Transport of Diluted Species, Electric Currents, and General Form PDE COMSOL physics modules to achieve the desired governing equations. A linear elastic material model was implemented for the solid mechanics physics, where the material property values used are shown in Table I. The transport of diluted species and general form PDE physics provide the PNP differential equations, which have been used to describe cation concentration and electric potential gradient during IPMC deformation. The model is split into two studies: (1) the deformation is calculated independently and (2) the PNP differential equation model is applied to find ionic concentration and electric potential throughout the IPMC. In the case of sensor simulation, study (1) is computed first followed by study (2). It is the opposite for actuator simulation.

The model input is a prescribed tip displacement of a rigid connecter on the IPMC tube's free-end with the opposite end held fixed, which is similar to the experimental setup. The solid mechanics physics describe stress, strain, and local displacement throughout the specimen for a prescribed tip displacement. The solid mechanics model is primarily coupled to the PNP model through the term, μCΔV∇P, from Eq. (5), where the solvent pressure is related to the polymer pressure by ∇P =−∇p, where p is the polymer pressure found from the solid mechanics model as the average normal stress at each location. Additionally, the PNP model is coupled to the solid mechanics model through anion migration, as indicated in Eq. (8). There are no empirical or experimentally determined coefficients apart from material properties for the given sensor model. The accuracy of the results will be discussed in Sec. II B 4.

Nafion pellets are heated above glass transition temperature and are extruded in the cross-sectional shape shown in Fig. 3. The polymer is tensioned after extrusion to facilitate polymer chain realignment and to achieve the desired mechanical properties. After the polymer is cooled to room temperature, it is cut into 20–25 mm segments for testing.

The extruded Nafion samples are activated by a hydrolysis reaction using a potassium hydroxide solution. After hydrolysis, the samples are cleaned and prepared for primary plating. The Nafion samples are plated using a Platinum impregnation-reduction reaction. Primary plating is repeated to increase the surface conductivity of the sample. When the surface conductivity reaches a desired amount, secondary plating is performed.

Secondary plating increases the surface conductivity of the samples by depositing platinum on the surface of the sample via a chemical deposition. After secondary plating, the surface conductivity of the samples is at an acceptable level for good IPMC performance. An ion exchange can now be conducted to achieve better transport of water molecules through the membrane. Hydrogen ions within the membrane are exchanged for lithium ions which give a better transport of water molecules through the membrane. After the ion exchange process, the samples are cleaned and stored in DI water.

The flanges are removed with precision razor blades to make separated electrodes. Separating the electrodes allow for multiple degree-of-freedom sensing motion when a voltage is applied. A finished IPMC sensor is shown in Fig. 4.

A collet is fabricated out of Nylon to clamp the IPMC tube at each electrode. The electrode contact patches of the collet are made out of copper foil. Individual wires are soldered to each contact patch and are glued to each of the collet fingers as seen in Fig. 5. The collet is hand tightened around each sample to securely fasten it. Previous clamping systems that have been used are restricted to a set diameter.²⁴ The current design allows for testing of variable diameter IPMCs.²⁵ 

Once the IPMC is mounted inside the collet, the collet is secured to a collet mount and can be lowered into the water to the desired level using an adjustable plastic slide. The other end of the IPMC is clasped between two pieces of Plexiglas for the sensor setup. This end is attached to a shaker to create the motion required to produce voltage changes in the IPMC, as shown in Fig. 6 (top). An illustration of the mechanoelectrical IPMC experimental setup is shown in Fig. 6 (bottom).

The signal data were recorded using LabVIEW with National Instruments USB-6008. The displacement was measured with a laser displacement sensor (optoNCDT-1401, Micro-Epsilon) and was recorded in LabVIEW.

Tube-shaped IPMCs were fabricated and tested as sensor devices with sinusoidal tip displacement inputs of half peak-to-peak amplitudes 0.5 mm and 1.5 mm at a frequency of 1 Hz. The induced voltage amplitude increase with increasing tip displacement is quite significant and is similar to that seen when using rectangular shaped IPMCs. The induced voltage amplitude, although low, is on similar scale to that seen with much larger rectangular IPMC samples.²⁶ Direct comparison of the performance is difficult however, due to the different sample dimensions and experimental conditions. These results suggest that tube shaped IPMCs are capable of multi degree of freedom displacement transduction; however, further research needs to be done to better understand the polymer-electrode interaction and improve the electrode plating process. Sample experimental traces can be seen in Fig. 7.

Finite element simulations were performed of mechanoelectric transduction for a tube-shaped IPMC with 1 mm outer diameter, 1/3 mm inner diameter, and 20 mm length. The electrodes were set to be 0.01 mm thick. Simulations were performed using COMSOL following the steps outlined in Sec. II. Modeling parameters are provided in Table II. Diffusion coefficient and molar volume change were adjusted to best fit the experimental results.

It should be noted that the electrode-polymer interface morphology consists of a diffusion layer (composed of dispersed metal particles that result in large electrode surface areas) with a corresponding capacitive effect. To adequately compensate for these effects, a large effective dielectric permittivity value has been used in the governing model physics.^17,23

Comparison of experimental and simulation results for sinusoidal tip displacement inputs of 0.5 mm and 1.5 mm amplitude at 1 Hz are shown in Fig. 8. These are seen to be in good agreement in terms of both phase and amplitude, indicating the given IPMC sensor model and selected parameters match the experimental results for the fabricated tube-shaped IPMCs for cases of simple input motion.

Cation and anion concentration variation, electric potential, and charge density within the polymer were simulated for sinusoidal tip displacement inputs. Results taken from a time at max displacement during sinusoidal input are presented as surface plots showing these distributions at the fixed end of the IPMC. A line path was also taken across this surface in the bending direction to better display cation and anion concentration changes. These are shown in Figs. 9 and 10.

As we would expect, the cation concentration rise in the subsurface layer near the IPMC boundary is small in comparison to actuator simulations, which show increase in concentration as large as 1000 mol/m³ in this same small subsurface area. This is likely due the high electrode charge driving the strong migration of cations in the case of actuation. Whereas for sensing, the induced electric potential is only due to rearrangement of ions from deformation, which results in small values for both concentration change and electric potentials as opposed to the case of actuation.

Experimental results and simulations have been presented which provide a framework for further developing tube-shaped IPMC sensors and other 3D IPMC models. These results show that 3D transduction of the fabricated IPMC devices is feasible and the given model can provide accurate simulations for simple input motion. Because there are no empirical relationships utilized in coupling the solid mechanics physics to the PNP physics in the given model, this model can readily be applied to other 3D complex shaped IPMC sensor devices. To the best of the author's knowledge, this is the first work which presents a multi-physics mechanoelectrical transduction model for 3D shaped IPMCs.

The signal in case of complex motion sensor simulation consists of both a position and ionic velocity component of the deforming IPMC. As an example, a flower pedal shape has been used as the prescribed displacement, as seen in Fig. 11. This type of tip displacement motion starts from the center (zero location) and passes through the center halfway through and at the end of each cycle. Simulations were carried out for two cycles. The resulting electrode signal is shown in Fig. 12. If the signal was comprised only of the IPMC deformation at any instant of time, one would expect at times t = 2 and t = 4 s the signals of the electrodes would all pass through 0 mV; however, this is not the case, and hence there is a significant velocity component to the IPMC sensor signal.

Mathematical modelling, fabrication, and experimental performance studies of tube-shaped IPMC sensors have been presented. The results suggest that IPMC transducers of this design are functional and simulations using the provided model are in good agreement with experimental results for simple harmonic input motion. Further work needs to be done to decouple the position and velocity components of the sensor signal. Further work also includes revising the solid mechanics model to include nonlinear material effects and electrode deformation. Improvements to IPMC fabrication, electrode plating, and patterning are also being sought. Finally, design of sized-down tube shaped IPMCs is being studied to develop smaller actuator and sensor IPMC devices.

This work was in part supported by the Office of Naval Research (N00014-13-1-0274) and National Aeronautics and Space Administration (NNX13AN15A). Additionally, the authors would like to thank Qi Shen for helping fabricate IPMCs.