

ORIGINAL ARTICLE 

Year : 2022  Volume
: 6
 Issue : 3  Page : 429437 

A novel experimental static deflection equation for specific cantilever beam made of ionic polymer–metal composite
Amin Nasrollah^{1}, Poopak Farnia^{2}, Saba Hamidgorgani^{3}, Jalaledin Ghanavi^{2}
^{1} Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran ^{2} Mycobacteriology Research Center, National Research Institute of Tuberculosis and Lung Disease, Shahid Beheshti University of Medical Sciences, Tehran, Iran ^{3} Department of Energy Engineering and Economics, Science and Research Branch of Islamic Azad University, Tehran, Iran
Date of Submission  27May2022 
Date of Decision  09Aug2022 
Date of Acceptance  29Aug2022 
Date of Web Publication  17Sep2022 
Correspondence Address: Jalaledin Ghanavi Mycobacteriology Research Centre (MRC), National Research Institute of Tuberculosis and Lung Disease (NRITLD), Shahid Beheshti University of Medical Sciences, Tehran Iran Poopak Farnia Mycobacteriology Research Centre (MRC), National Research Institute of Tuberculosis and Lung Disease (NRITLD), Shahid Beheshti University of Medical Sciences, Tehran Iran
Source of Support: None, Conflict of Interest: None
DOI: 10.4103/bbrj.bbrj_180_22
Background: Nowadays, ionic polymer–metal composites are widely used in various industries. They are in the group of electroactive polymers and smart materials with electromechanical properties. By applying a small amount of voltage, the nonlinear stress inside them will happen and their deformation can be seen. The energy transformation from electrical to mechanical is observable during the process of giving voltage to a specimen. The aim of this study is to investigate a novel experimental static deflection equation for specific cantilever beam made of ionic polymer–metal composite. Methods: In this paper, an ionicpolymermetal composite is provided; the main core is based on an electroactive Fluoropolymer named Nafion, and the coated electrodes are made of Platinum. The length of the specimen is 27.131 mm and its width is 5.728mm. Voltage from 1.5 to 4.3V was applied to the specimen used in this study; the ydirectional displacement of the IPMC at each step is measured and recorded; then, a finite element analysis was performed. Curve fitting of the data for the experimental analysis was also done. Moreover, the governing relations of IPMC according to the Nernst–Planck equation were investigated in this study. Results: The results have been validated in two forms of finite element method and experimental analysis. The results of finite element analysis showed that the ion flux in the polymer is calculated by the equation: [INSIDE:1]. In other words, this equation, which is called Nernst–Planck, is the basic equation of this type of material. This equation is the main governing equation to describe the transfer phenomena of IPMC materials. Furthermore, in order to calculate the deflection of IPMC membrane, 19 equations designed in this study were used. In the next step, the results of the experimental analysis showed that, based on the field emission scanning electron microscope images, the Nafion surface is completely sandblasted and its area is completely uniform. The right image taken by DinoLite shows the thesis effect on the electrode. Furthermore, the results showed that IPMC has highquality coated electrodes. Conclusions: It is shown that a nonlinear equation governs the behavior of IPMCs' deflection versus voltage.
Keywords: Experimental analysis, finite element method, ionic polymer–metal composites, microelectromechanical systems, smart material
How to cite this article: Nasrollah A, Farnia P, Hamidgorgani S, Ghanavi J. A novel experimental static deflection equation for specific cantilever beam made of ionic polymer–metal composite. Biomed Biotechnol Res J 2022;6:42937 
How to cite this URL: Nasrollah A, Farnia P, Hamidgorgani S, Ghanavi J. A novel experimental static deflection equation for specific cantilever beam made of ionic polymer–metal composite. Biomed Biotechnol Res J [serial online] 2022 [cited 2023 Jan 31];6:42937. Available from: https://www.bmbtrj.org/text.asp?2022/6/3/429/356152 
Introduction   
The growing trend of these materials has led to the production of a type of smart material that is used in various industries such as electromechanics, defense, and aerospace industries.^{[1]} Mechanical flexibility, lightweight, and smooth processing are a number of the inherent properties of all polymers;^{[2]} further to these properties, a few polymers show off massive property changes in reaction to electric stimulation which provides great benefits to their potential applications.^{[3]} One cause for interest in IPMC is that they are known as electroactive polymers (EAPs), which are taken into consideration to be options for conventional actuators and sensor substances.^{[3]} Although there are various ways to produce energy such as using Stirling engines^{[4],[5]} or simply fossil fuels like oil, using IPMC as an energy harvester can be a huge help to future progresses.^{[6],[7]} EAPs are substances that reply mechanically to electrical stimulation with considerable shape and size alterations.^{[8]} They may be categorized into main classes based on the activation mechanism; for instance, if they are electronic or ionic EAPs.^{[9]} Coulomb forces drive the electronic EAPs, which include electrostrictive, electrostatic, piezoelectric, and ferroelectric.^{[10],[11]} Ionic polymer gels, ionic polymer–metal composites, conductive polymers, and carbon nanotubes are several examples of ionic EAPs. Ionicbased polymers require a totally small activation voltage.^{[12]} The utility of active substances as sensors and actuators in smart systems has currently extended considerably intending to enhance the intelligence, reliability, and overall performance of the systems.^{[13]} The most common polymers consist of perfluorinated alkane containing short side chains terminated with an ionic group such as SO3, and there are three main polymeric ion exchange membranes consisting of Nafion, Flemion, and Aciplex.^{[14]} The membrane is composed of noble metals such as silver, gold, platinum, and palladium in some chemical processes.^{[15]} Nafion is also used as a conductor for proton exchange in fuel cells and water filtration and allows cations to move, coated with gold.^{[16]}
Nowadays, one of these smart materials, which are used as actuators and sensors^{[17]} based on EAPs,^{[18]} has received a lot of attention due to their good capabilities and flexibility;^{[19]} the field of ionic polymers has seen rapid growth in academic^{[20]} attention as well as industrial^{[21]} and medical applications in recent decades.^{[22]} The use of ionic polymers as artificial muscles,^{[23]} actuators,^{[24]} and sensors is one promising area.^{[25]}
The shrinking of mechatronic systems makes robots increasingly important in micro and nano sizes, which will play a major role in future developments.^{[26]} Sensors and actuators are made of EAPs due to their special properties.^{[27]} The characteristics of these materials include their response to low voltages which means <5V,^{[28]} high sensitivities to external stimulus, and high power;^{[29]} however, IPMCs also have negative properties such as short service life and short durability and lifetime, as well as a rapid reduction in efficiency due to ionic and electrolytic properties.^{[30]}
Piezoelectric materials are known as one of the most important smart materials in the world with electromechanical properties and are among the most commonly used materials.^{[31]} IPMCs withstand significant bending deformations at low driving voltages.^{[32]} When IPMCs operate above 1.23V in the air, waterbased IPMCs lose a considerable amount of their solvent content.^{[33]} They have been modeled as both capacitive and resistive element actuators that operate like biological muscle and offer an appealing way of actuation for biomechanic and biomimetic applications.^{[34],[35],[36],[37]} In this paper, a novel method is used to investigate the attitude of the ionic polymer–metal composites with both finite element method (FEM) and experimental tests.
Methods   
In this study, referring to the electroactivity of the actuator made from IPMC due to the presence of the Nafion polymer, the displacement of the membrane under the application of voltage was investigated in both theoretical and practical ways. The structure of these smart materials is described in “[Figure 1].”  Figure 1: Structure of ionicpolymermetal composite before and after applying voltage
Click here to view 
According to [Figure 1], red and blue circles are free cations and water molecules, respectively. Electromechanical stress and, as a result, a deflection occur through the ionic polymer metal composites by applying a DC voltage; this voltage must be applied to the electrode domain which is specified with yellow color; the voltage will be transferred to the EAPs which in this case is Nafion. The study protocol was approved by the ethical committees of Shahid Beheshti University of Medical Science (IR.SBMU.NRITLD.REC.1401028).
Results   
The results have been validated in two forms of FEM and experimental analysis, and they are discussed in the following.
Finite element analysis
The FEM and the governing equations of the materials with electromechanical properties have been used for numerical analysis of the abovementioned material. The ion flux in the polymer is calculated by Equation 1; in the other words, this equation, which is named Nernst–Planck, is the fundamental equation of this type of material. This equation is the main governing equation for describing IPMC material transfer phenomena. In order to calculate the deflection of the IPMC membrane, equations 1–19 have been used as below:^{[38]}
Eq. 1
In “Equation 1,” C is the cationic concentration, is cation flux, the diffusion coefficient is shown by D, μ is the mobility that can be obtained from Equation 2, the symbol of charge number is Z, F is the Faraday constant, ∇V and is the molar volume; ∇V is the electric potential gradient which is explained by the Poisson equation; ∇ϕ has a reverse direction in the case of mechanicalelectrical transitions because the ionic current is not controlled by the applied voltage; it is controlled by a pressure gradient under a bending pressure ∇P, which is the solvent pressure. Equation 2 explains the mobility of cations; in this equation, R is the gas constant and T is the absolute temperature.
Some of the values mentioned are given in [Table 1].^{[34]}  Table 1: Properties of the ionic polymer–metal composite and other parameters
Click here to view 
Eq. 2
Eq. 3
In Equation 3, ρ is the charge density and ϵ is the effective dielectric permittivity. ϵ can be written as ϵ = ϵ_{0}ϵ_{r}; ϵ_{0} is the dielectric permittivity in vacuum and is equal to 8.85 × 10 − 12F. m^{−1}. Furthermore, C_{a} is known as anion concentration. C is the cation concentration calculated by Nernst–Planck equation which was mentioned in Equation 1. Then, the charge density can be calculated from Equation 4.^{[39]}
Eq. 4
Due to Equation 5, the displacement vector is represented by u, in which the anion concentration is related to the local volumetric strain. A positive value of volumetric strain means an increase in volume strain and a negative value means a decrease in volume.
Eq. 5
Volume changes in the matrix of polymer affect the concentration of local anions because the anions are part of the backbone of the polymer. Hence, the concentration of the anion can be calculated from Equation 6; in this respect, C_{0} is the primary focus of cations or anions. It should be noted that for most practical calculations, C_{a} = C_{0} can be approximated.
Eq. 6
According to the momentum conservation, the relation between solvent pressure 'nabla P' and polymer pressure 'nabla p' are explains in Equation 7.
Eq. 7
In Equation 8, E is Young's modulus of the material, and ν is Poisson's ratio.
Eq. 8
Knowing these constants, Navier's equation for displacements can be written as Equation 9; in that, F is the force per unit volume:
Eq. 9
In Equation 10, Newton's second law is used to describe timedependent deformation. In this equation, rho is the density of the material, while C is the Navier's constant.
Eq. 10
In order to investigate the effect of the electrodes in the model, the ionic current in the polymer is coupled with the electric current in the electrodes. Unlike many physicsbased models, electrodes are not considered ideal, but they are limited electrical conductors. In electrodes, the Ohm's law differential for current density is expressed as in Equation 11.
Eq. 11
In Equation 11, σ is the electrical conductivity of the electrodes, and V and j are the electric potential and current density in the electrodes, respectively. It should be noted that the electrical potential ϕ inside the polymer and the electrical potential V at the electrodes are two different variables.
The ionic current at the electrode boundaries is calculated by applying Gauss's law directly. As it is observable in [Figure 2], in order to use finite element method, whole membrane is divided to 2 main domains; electrode domain ψ_{a} and polymer domain Ω. ∂Γ as a contour can be considered in a way that covers the entire domain Ω and the boundary between Ω and ψa and a small part of ψb. In the contour ∂Γ, the integral form of Gauss's law can be written as Equation 12.
Eq. 12
where n is a normal unit vector in ∂Γ. Given that in Ω, the net charge density is zero and using the Poisson equation in Equation 3, the expression can be simplified as Equation 13.
Eq. 13
There is no charge density inside the electrode, except on its surface. Load density is indicated by ρb. Since the potential gradient in the x direction is very small and the values of the linear integrals on the segments are small, the equation can be simplified with opposite values as Equation 14.
Since the potential gradient in the Y direction at the electrode is expected to be negligible, the second component of Equation 14 on the left is omitted. The integral on the right is equal to the load accumulated in length and is denoted by q' in Equation 15.
Eq. 15
If d is assumed to be the width of the IPMC in the zdirection, the total current is shown in Equation 16.
Eq. 16
To calculate j, which is the current density, a large number of lines Γ_{1} . . . Γ_{n} are considered. While n → ∞, the length of the contour parts is specified by dΓ_{i}, Γ_{ai}, and Γ_{b}^{i} for the vertical, top, and bottom parts, respectively. Given that line integrals at the internal boundaries cancel each other out, the lefthand side of the equation contains the following expressions as shown in Equation 17.
Eq. 17
As mentioned earlier, the voltage gradient in the sections Γ and Γ_{n+1} is negligible and there is no potential gradient in the y direction in the lower sections of ∂Γ_{bi}; the current contour Γ_{ai}_{ }can be expressed in Equation 18.
Eq. 18
Therefore, the density of the local ion current in the ydirection on the electrode boundary is shown in Equation 19.
Eq. 19
The electrode and polymer properties are illustrated in [Table 2], respectively. Like all computeraided engineering methods, this analysis follows the FEM. First of all, the twodimensional model was designed with the characteristics mentioned in [Table 2]. Next step in finite element method is applying material properties; these properties are added from [Table 1] and 2.^{[34]} In the third step, the model has meshed. Totally 48 elements were created.
Then, the boundary conditions which were the whole physics, fixed edges, and applying voltage, exactly like the abovementioned equations, were applied. By using its result, which was the force per volume, in the deflection and stress equations, the von Mises stress and the deflection of the designed model had been calculated.
Creating mesh is vital in finite element method. after doing a mesh convergence, it is understood that 480 elements are the highest number that can lead the simulation to precise results ^{[40],[41]} After solving the model using FEM, the deflection of the membrane and its stress were calculated using the Timoshenko beam theory. This analysis was time dependent, and the voltage increased from 0 to 4.3V. [Figure 3] illustrates the deformation of membrane and its stress while applying 3.3V.
Due to [Figure 3], the stress is highlighted by colors; red and blue are the least and highest amounts of stress in N/m^{2}. The displacement can be observed from the axis of the plot.
[Figure 4] shows the graph of deflection which is relative to the voltage from 0 to 4.5V while x = 27.131 mm. It is obvious that from 0 to 3.27V, the graph follows an increasing path; its peak occurred with 3.2V which is 25.1 mm. After it, with rising the amount of applied voltage, the displacement will decrease.  Figure 4: The deflection of the membrane while x = 27.131 mm by changing voltage
Click here to view 
Then, by comparing numerical values and curve fitting the experimental data, the governing equation was obtained. According to Nernst–Planck equation, which is mentioned in Equation 1, the nonlinear behavior of the ionic polymer–metal composite is caused by the amount of cation concentration through the prepared EAPs; because of the deflection equation of distributed load on a cantilever beam following Timoshenko beam theory, the curve fitted graph must be a sinbased equation; the fitted curve is observable in Equation 20 that illustrates the deflection attitude of the IPMC membrane while the xdirectional location is equal to 27.131 mm.
This graph shows that the maximum displacement appears while the applied voltage is 3.3V.
Eq. 20
y (v)= (183.7sin (0.6024 v + 0.9567) + 178sin (0.624 v + 4.165)
+1.191sin (2.585 v + 1.547) + 0.1674sin (4.039 v + 2.321)
+0.01537sin (7.782 v2.582) + 0.1609sin (5.389 v + 1.814))
From Equation 20, it is obvious that at the beginning of the test, while the applied voltage is infinity, the cation concentration is extremely high.
Experimental analysis
In this step, the deflection of the manufactured actuator from IPMC material is investigated. The specimen was manufactured using Nafion and Platinum with complex chemical processes; then, the membrane was checked using field emission scanning electron microscope (FESEM) and DinoLite AM4113ZT which are observable in [Figure 5]a and [Figure 5]b, respectively.
In the FESEM image, it is shown that the surface of the Nafion is perfectly sandblasted, and its area is pretty uniform; the right image taken by the DinoLite shows the dissertation effect on the electrode; this effect must have happened to the coated metal, and it occurs correctly; it means that the IPMC has highquality coated electrodes.
[Figure 6] shows the IPMC crosssection. This picture was taken with the purpose of IPMC measurement; with the use of [Figure 6], the thickness of the polymer core of material made of Nafion and electrodes made of platinum were considered; as measures are shown in the picture, the total thickness of our specimen is 134.813 μm and 72.588 μm and 24.189 μm are the thicknesses of the polymer and electrodes, respectively. These dimensions were used for finite element analysis and for experimental tests. An electronic microscope was used to calculate and capture the thickness of the Nafion and platinum layers.
First of all, a constant voltage was given to the sample, and after spending a minute, its maximum ydirectional deflection was calculated; the deflection was observed using a calibrated microscope name DinoLite AM4113ZT, and then by using the image processing method, the deflection was calculated. While there no voltage is applied, the image processing is used to measure the dimensions of the specimen; the length of the beam was 27.131mm.
In the next step, the voltage was increased to 1.5V; this boosting of the voltage was because of the initial working voltage of the power supply; the diagram of the power control system is shown in [Figure 7]. Then, the voltage increased from 1.5 to 4.3V.
Due to [Figure 7], the AC will be converted to DC with an adjustable power supply, MW LRS 350–12; the voltage will smoothly increase and drive the IPMC membrane; this is for preventing the sample from damaging by an electric shock; the deflection will be measured with the abovementioned microscope and logged in a computer for creating its plot.
Each time that a voltage in the range of 1.5V to 4.3V is applied, the IPMC sample undergoes an amount of deflection; after the maximum deflection rate, the displacement resulting from the voltage rise at the end of the beam was recorded. This operation was recorded for each step; after recording each voltage and deflection, the amount of displacement was calculated and recorded using DinoCapture software, after calibrating it. After recording the numbers related to voltage and displacement, the graph of these changes was drawn using MATLAB. The membrane is on a wooden clamp; this clamp made a fixed constrain for the cantilever beam. Clamps which are made of copper and wood are also observable in [Figure 8].
The clamp is made of wood, copper, and tin; the copper plates are glued to the wooden clamp and then they are soldered to the lowpower wires.
At last, all of the measured displacements after applying voltage from 0 to 4.5V are gathered in a plot shown in [Figure 9]. This plot illustrates voltage versus displacement of the specimen; totally, the displacement increases from 0 to 3.3V; however, after increasing the voltage from 3.3V to 4.3V, the displacement followed a stable path. This means that in this range of voltage, the IPMC has no efficient deflection.  Figure 9: Deflection of the IPMC while applying voltage and xdirectional location is 27.131 mm
Click here to view 
Discussion   
The IPMC consists of three main layers; including electrodes and a polymer layer that usually consists of a Nafion or a Flemion.^{[42]} Nafion is composed of various elements such as carbon, sulfur, fluorine, oxygen, and some surface impurities, and the chemical formula of this polymer is C_{7}HF_{1}3O_{5}SC_{2}F_{4}.^{[43]} A few number of reasons that affect the IPMC actuation are the hydration concentration and the environmental conditions such as temperature and humidity.^{[44]} In the 1960s, polymer electrolyte membranes and their special place in fuel cells were first introduced by Grubb and Niedrachat after combining water and lithium hydride to release hydrogen.^{[40]} Nowadays, most ionic polymer metal composites are manufactured by coating platinum on the surface of the membrane. Due to the high research cost,^{[41]} the study and preparation of lowcost^{[41]} electrode materials have become an important research direction in the field of IPMC.^{[45]} For describing the attitude of actuating and the whole behavior of the IPMCs, a huge number of models have been presented until now.^{[46]}
Our results indicate that from derivation of Equation 20 up to about 1V, the speed will be decreased. Then, when the voltage from 1V reaches about 2.75V, the speed will be raised again; however, the velocity will not be as high as it was at the beginning. All of these behaviors are because of the nonlinear attitude of IPMCs. The negative velocity of the IPMC means that the deflection direction will be reversed. In Equation 20, v is the applied voltage, and Y_{x} is the deflection of the beam while the xdirectional location is 27.131 mm. Pugal et al. used FEM to analyze the IPMC's behavior,^{[38]} In this study, in order to simplify the entire process, an equation has been found.^{[18]} Min Yu measured the maximum deflection of the IPMC membranes with different platinumbased electrode thicknesses; they only presented a plot of the best magnitude of the specimen under diverse constant voltages.^{[47]} Then, Kyoung Kwan Ahn designed and manufactured micropumps with changing their diaphragm shape; he concluded that squareshaped proposed diaphragm is the best choice with maximum deflection.^{[39]} Then, Kyoung Kwan Ahn designed and manufactured micropumps with changing their diaphragm shape; he concluded that squareshaped proposed diaphragm is the best choice with maximum deflection.^{[48]} Furthermore, Kwang J. Kim used platinum coating for a bioinspired multidegree of freedom actuator based on a novel cylindrical ionic polymer–metal composite material; in his article, bending deflection and angle in the diagonal direction with diverse positive actuation signals was investigated.^{[49]} Tetsuya Horiuchi researched voltagecontrolled IPMC actuators for accommodating intraocular lens systems; the material used in his research was made of gold.^{[50]} WanHasbullah MohdIsa analyzed IPMC membrane under mechanical excitation with a shaker; he measured the displacement and generated voltage.^{[51]} Xuan Hui, used an optimized electrode plating method to create silverbased ionic polymer metal composites. He presented his results in a plot which indicates the performance of his membrane.^{[52]} Dilip Kumar Biswal used orthogonal array method for bending response optimization of an ionic polymer–metalcomposite actuator under various voltages; the material used as electrodes was made of silver.^{[53]} Ankur Gupta, in the same year, numerically studied the method for controlling position of an underwater biomimetic IPMC membrane.^{[54]} Min Yu measured axial displacements of the helical IPMC actuator under different signals.^{[55]} Then, Burawudi Kennt K used a comparative method in order to predict IPMCbased soft actuators – with platinum electrodes. The maximum displacement in 2, 3, and 4 volts altered from 0 mm to 20 mm.^{[56]} Jose Emilio Traver tried to present an equation to predict and control the IPMC membrane; unfortunately, his method was not accurate.^{[43]}
Conclusions   
After comparing experimental data with the results of FEM, it is understood that the equation which was curve fitted from experimental data has a 19.072% error; it should be mentioned that this amount is extremely negligible. Another method is also used for measuring the error of this equation; the total areas of the graph of [Figure 9] limited by the x and y axes from 0 to 4.3V for that FEM and experimental test have been calculated; Equation 20 is formed based on the given voltage, and as the result, the amount of displacement can be calculated in the mentioned lengths and static situation.
Limitation of the study
The error measured by this method was 12.86%. The error can be caused by the matter of time or in the other words, dehydration phenomenon of the polymer, negligible problems while manufacturing it, or the noises caused by the used power supply. Either of the abovementioned problems can cause an error in the curvefitted equation while comparing it with FEM.
Financial support and sponsorship
Nil.
Conflicts of interest
The authors declare that none of the authors have any competing interests.
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[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9]
[Table 1], [Table 2]
