applied sciences

Article The FEM Model of the Pump Made of Dielectric Electroactive Membrane

Jakub Kołota Institute of Automatic Control and Robotic, Faculty of Control, and Electrical Engineering, Poznan, University of Technology, 60-965 Pozna´n,Poland; [email protected]; Tel.: +48-6166-5-2504

 Received: 9 March 2020; Accepted: 25 March 2020; Published: 27 March 2020 

Abstract: Dielectric electroactive (DEAPs) undergo large deformations when subject to an electric field, which make them an attractive material for use in novel systems. This article presents the possibility of using DEAPs to model an innovative pumping actuator structure. The model was used to map important object parameters at individual operating points of the modeled pump. The experimental work involved designing the membrane and testing its changes in elasticity under the influence of varying and supplies. In the further part of the work, a finite element model (FEM) of a pumping device was implemented. In the new construction of the pump, pressure is generated by membrane deformation. This is due to electrostatic compressive between two electrodes applied to the polymer surface and forces generated by permanent magnets. The results are presented graphically, confirming the compliance of the model with the measurements.

Keywords: DEAP; FEM; dielectric electroactive polymers; finite element method

1. Introduction Smart materials are ones that can change their properties in a controlled manner in response to environmental stimuli. Such materials can be used both as and [1,2]. Electroactive materials like DEAPs (dielectric electroactive polymers) are included in the group of smart materials and are increasingly used in many automation and robotics systems [3]. This work presents the concept of a device using an electroactive polymer that changes its mechanical properties under the influence of voltage excitation. Finite element model (FEM) modeling of the behavior of DEAPs is a useful tool to understand such systems better and help in the optimal design of prototypes. The modelling and simulating of DEAP actuators is a cost-effective way of providing a better understanding of the devices built with their usage. DEAPs offer excellent performance, are lightweight, flexible and inexpensive. Therefore, , provide many potential applications as micro-actuators. A technique to accurately model this material—taking into account its nonlinearities as well as its large deformations—is being developed in this study. This work presents a model of the innovative construction of the pump using DEAP membrane deformation. There are many works that describe different modeling methods and analytical models [4]. One of the works presents a technique for accurate modeling of electroactive polymer taking into account nonlinearities and large deformations [5]. The authors simulated an algorithm in the ANSYS environment that considers the electromechanical coupling of the EAP (electroactive polymers) model. Simulation of charge distribution and electrostatic force is the subject of another work which uses COMSOL software [6]. Another work focuses on studying the kinematics of forming and loading, along with thermodynamics [7]. While much theoretical and computational modeling effort has gone into describing the ideal, time-independent behavior of these materials, viscoelasticity is a crucial component of the observed mechanical response, and hence has a significant effect on electromechanical

Appl. Sci. 2020, 10, 2283; doi:10.3390/app10072283 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 2283 2 of 11 Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 of 13 crucialactuation. component In [8], the of authorsthe observed show thatmechanical viscoelasticity response, provides and hence stabilization has a significant that delays effect the onset on electromechanicalof instability under actuation. monotonic In [8], loading the authors and may show fully that suppress viscoelasticity instabilities provides under stabilization sufficiently that fast delayscyclic loading,the onset which of instability may be desirableunder monotonic for many loading applications. and may Although fully suppress all of these instabilities models provide under sufficientlyadvantageous fast insightscyclic loading, into the which behavior may of be the desirable elastomer for material,many applications. they lack the Although ability to all accurately of these modelspredict provide the complete advantageous DEAP actuator insights structure. into the They behavior focus onof severalthe elastomer aspects material, of modeling they but lack are the not abilityable to to reflect accurately the full predict complexity. the complete DEAP actuator structure. They focus on several aspects of modeling but are not able to reflect the full complexity. In this work, an FEM model was made that was verified faithfully with measurements. First, In this work, an FEM model was made that was verified faithfully with measurements. First, the the relationship between static deformation and applied force was presented. Next, three series of relationship between static deformation and applied force was presented. Next, three series of displacement measurements were compared for a given force during alternating voltage applied to the displacement measurements were compared for a given force during alternating voltage applied to membrane. The results prepared in this way allowed us to identify the values of the virtual Young’s the membrane. The results prepared in this way allowed us to identify the values of the virtual modulus corresponding to individual points of states by collecting a sufficient number of real object Young's modulus corresponding to individual points of states by collecting a sufficient number of measurements. Using the work state mapping method, it was possible to model devices with sufficient real object measurements. Using the work state mapping method, it was possible to model devices precision having the knowledge how the membrane will deform; it was not necessary to calculate the with sufficient precision having the knowledge how the membrane will deform; it was not necessary full complexity of the model each time. to calculate the full complexity of the model each time. 2. Principle of Operation of DEAP Actuator 2. Principle of Operation of DEAP Actuator A DEAP membrane is made of a silicone membrane that is pre-stretched during the production process.A DEAP Then, membrane the carbon is made grease of electrodes a silicone aremembrane printed. that In principle,is pre-stretched DEAPs during are similar the production to flexible process.capacitors. Then, As the can carbon be seen grease in the Figureelectrodes1, they are consist printed. of an In elastomeric principle, DEAPs film sandwiched are similar between to flexible two capacitors.electrodes. As Applying can be seen voltage in the to them Figure leads 1, they to compression consist of an of elastomeric the elastomer, film which sandwiched causes a changebetween in twocapacity electrodes. [9–13]. Applying voltage to them leads to compression of the elastomer, which causes a change in capacity [9–13].

FigureFigure 1. 1. PrinciplePrinciple of of operation operation of of dielectric dielectric electroactive electroactive polymer polymer actuator actuator

For thisFor circular this circular geometry, geometry, the deflection the deflection of the of DEA the DEA membrane membrane is described is described by the by radial the radial stretch stretch λr, λ λ λ circumferentialr, circumferential stretch stretch λc andc andthickness thickness stretch stretch λz [14].z [ 14]. + q =1 l2+d2 = = (1) = = l 0 z (1) λrλcλz = 1 λr= = λz = λc = const l0 l0 z0 = = sin () = (2) + l0 l0 ( ) d + z = z0 l = z0 q sin θ = q l2+d2 l2+d2 (2) I propose a dynamic model composed of a set0 of nonlinear, time-invariant0 differential equations describingI propose the adynamic dynamic modelrelationship composed between of a set the of nonlinear,input voltage time-invariant u and the diff erentialoutput equationsactuator displacementdescribing the d dynamic. I present relationship a model which between accurately the input describe utheand behavior the output of actuatorthe actuator, displacement both in transient and steady state, and for different loads. The vertical force equilibrium causing membrane deformation can be defined as: Appl. Sci. 2020, 10, 2283 3 of 11 d. I present a model which accurately describes the behavior of the actuator, both in transient and steady state, and for different mass loads. The vertical force equilibrium causing membrane deformation can be defined as: ..   md = mg + FL sin(θ) FM + F + F (3) − h f where d is a distance, θ is an angle, m is the mass of the initial load, g means standard gravity and FL defines the external load. The electromechanical coupling is produced by Maxwell Stress which compresses the membrane when a voltage u is applied to the DEAP. The actuation is in the thickness direction and due to constant volume produces the change of radial stretch yr. The value of FM is equal to: 2 FMsin(θ) = c1c2du (4) 2πrz ε ε c = 0 , c = 0 r (5) 1 1 2 l0 − zo where ε0 is vacuum and εr is the of the polymer material. The parameters z0, l0 and r describe the geometry of DEAP actuator and are defined in Table1. Force Fu considers dynamic viscoelastic process according to:   l2  0  F sin(θ) = c dσe  (6) h 1  2 + 2  l0 d

σe = kee + ke(λr 1) + σe (7) − − . ke ke e = e + (λr 1) (8) −ηe ηe −

Table 1. DEAP actuator model parameters.

Parameter Symbol Value Unit mass m 125 g standard gravity g 9.81 m/s2 12 vacuum permittivity ε0 8.85*10− F/m relative permittivity εr 8.73 - coefficient of k 3.11 MPa viscoelasticity e coefficient of ηe 13.4 MPa s viscoelasticity · damping coefficients b 1.45 Nms Ogden model coefficients β1 11.4 kPa Ogden model coefficients β2 50.3 kPa Ogden model coefficients β3 44.1 kPa Ogden model coefficients γ 118 kPa 1 − Ogden model coefficients γ 30.4 kPa 2 − Ogden model coefficients γ3 23.3 kPa

There are many hyperelastic strain energy density functions, such as Yeoh, Ogden, Arruda-Boyce, etc. Similar to [11], I use the Ogden model and define two parameters βi and γi to obtain to the expression for the hyperelastic stress σe:

X3  αi αi σe = β λ γ λ− (9) i r − i r i=1 where αi is 2, 4 and 6. The main difficulty encountered with the identification of the DEAP parameters in the dynamic case is to obtain the damping coefficients b and the coefficients of the viscoelastic ke and Appl. Sci. 2020, 10, 2283 4 of 11

ηe. In this case the optimization procedure is based on both a step response and a frequency response (the results of this work were published in the [15,16]). Viscous friction is defined as:

. . bdd2 F sin(θ) = bλrsin(θ) = (10) f 3 + 2 l0 l0d where b is damping coefficient.

3. Experiments

3.1. Static Characteristics In order to develop a faithful mathematical model, it was necessary to obtain real device parametersAppl. by Sci. measuring 2020, 10, x FOR PEER and REVIEW identification process. Figure2 presents the laboratory4 of set13 used to collect displacementIn order measurementsto develop a faithful depending mathematical on model, the static it was force necessary acting to onobtain the real DEAP device membrane. The laboratoryparameters set contained by measuring a high and identi voltagefication amplifier process.TREK Figure MODEL2 presents 10the/10B-HS, laboratory laser set used distance to Micro-Epsiloncollect optoNCDT displacement ILD1320–10 measurements with depending 1 µm accuracyon the static and force an acting Inteco on RT-DAC the DEAP/USB membrane. data acquisition card. TheThe actuator laboratory presented set contained in a our high research voltage amplifier was made TREK MODEL of 3M VHB10/10B-HS, tape laser which distance was sensor pre-stretched Micro-Epsilon optoNCDT ILD1320–10 with 1 μm accuracy and an Inteco RT-DAC/USB data µ during theacquisition production card. process The actuator (1 mm presented to 200 in ourm thickness).research was Themade dimensions of 3M VHB tape of thewhich actuator was pre- created by the authorstretched are presented during the in production detail in process Table2 (1. mm to 200 μm thickness). The dimensions of the actuator created by the author are presented in detail in Table 2.

Figure 2.FigureThe 2. laboratory The laboratory kit kit with with dielectricdielectric electro electro active active polymer polymer actuator actuator..

Table 2.TableDimensions 2. Dimensions of of dielectric dielectric electroactive electroactive polymer polymer actuator. actuator.

ParameterParameter Symbol Symbol Value Value Unit Unit Pre-stretch tape diameter - 94 mm Pre-stretchPost-stretch tape diameter tape diameter 2 - - 210 94 mm mm Post-stretchPre-stretch tape diametertape thickness 2 - - 2101 mm mm Pre-stretchPost-stretch tape thickness tape thickness - z0 200 1 μm mm Post-stretchInternal tape plate thickness radius z0 r 25200 mm µm InternalExternal plate radius plate outer diameter r - 210 25 mm mm ExternalExternal plate outer plate diameterinner diameter - - 180 210 mm mm ExternalElectrode plate inner width diameter - l0 65 180 mm mm The obtainedElectrode data were width used to identify the value l0 of Young's 65modulus in mmthe developed FEM model. Figure 3 shows the membrane’s FEM model loaded with force 1.42 [N]. Table 2 and Figure 4 provide a faithful comparison of simulations with measurements in the full load range of the actuator. The obtained data were used to identify the value of Young’s modulus in the developed FEM model. Figure3 shows the membrane’s FEM model loaded with force 1.42 [N]. Table3 and Figure4 provide a faithful comparison of simulations with measurements in the full load range of the actuator. Appl. Sci. 2020, 10, 2283 5 of 11 Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 13

FigureFigure 3. The 3. finiteThe finite element element model model (FEM) (FEM) simulation simulation of di dielectricelectric electro electro active active polymer polymer actuator actuator for for staticstatic force force 1,42 1,42 [N]. [N]. Table 3. The comparison of the displacements for different load force. Table 2. The comparison of the displacements for different load force. Force ForceDisplacement Measurement [mm] DisplacementDisplacement Model [mm] [N] Displacement Measurement [mm] [N] model [mm] 0.00 0.00 0 0,00 0,00 0 0.64 6.83 5.93 1.030,64 10.376,83 5,93 9.54 1.231,03 12.0810,37 9,54 11.39 1.421,23 13.4812,08 11,39 13.15 1.671,42 15.8013,48 13,15 15.46 1.981,67 18.4215,80 15,46 18.33 2.101,98 19.3418,42 18,33 19.44 2.222,10 20.5019,34 19,44 20.56 2.332,22 21.2920,50 20,56 21.57 2.452,33 22.2721,29 21,57 22.69 2.612,45 23.5522,27 22,69 24.17 2.772,61 24.9523,55 24,17 25.65 Appl. Sci. 2020, 10, x2.92 FOR PEER2,77 REVIEW 26.4824,95 25,65 27.04 6 of 13 2,92 26,48 27,04

measurement "model"

30.00

25.00

20.00

15.00

10.00 Distance [mm] 5.00

0.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 Force [N]

Figure 4. Comparison of membrane distances for different load forces. Figure 4. Comparison of membrane distances for different load forces.

3.1. The identification process The operation of the device in an actuator mode using the DEAP membrane requires a supply. To collect reference data, a number of experiments on the deformation of the actuator membrane were carried out under the influence of various supply voltages. The relatively narrow range of membrane supply was sampled at nine points from 5.0 [kV] to 7,0 [kV] with a 0.25 [kV] step. This approach allowed the determination of the virtual value of Young's modulus compensating for the nonlinear structure of the DEAP model. To verify the correctness of the actuator modelling methodology, measurements were made for three different forces acting on the actuator. Tables 3–5 show the results of measurements and simulations carried out respectively for loads 1.03 [N], 1.23 [N] and 1.42 [N]. Figures 5–7 demonstrate the obtained convergence of the model with experimental measurements for the mentioned values of forces.

Table 3. The comparison of the displacements for different voltage supply and load force 1.03[N].

Voltage Displacement virtual Young Displacement Measurement supply model modulus [mm] [kV] [mm] [MPa] 5,00 12,02 12,08 7100 5,25 12,39 12,36 6950 5,50 12,80 12,76 6700 5,75 13,23 13,25 6450 6,00 13,76 13,71 6250 6,25 14,33 14,28 6000 6,50 15,00 15,03 5700 6,75 15,73 15,73 5450 7,00 16,56 16,56 5150

Appl. Sci. 2020, 10, 2283 6 of 11

3.2. The Identification Process The operation of the device in an actuator mode using the DEAP membrane requires a power supply. To collect reference data, a number of experiments on the deformation of the actuator membrane were carried out under the influence of various supply voltages. The relatively narrow range of membrane supply was sampled at nine points from 5.0 [kV] to 7.0 [kV] with a 0.25 [kV] step. This approach allowed the determination of the virtual value of Young’s modulus compensating for the nonlinear structure of the DEAP model. To verify the correctness of the actuator modelling methodology, measurements were made for three different forces acting on the actuator. Tables4–6 show the results of measurements and simulations carried out respectively for loads 1.03 [N], 1.23 [N] and 1.42 [N]. Figures5–7 demonstrate the obtained convergence of the model with experimental measurements for the mentioned values of forces.

Table 4. The comparison of the displacements for different voltage supply and load force 1.03 [N].

Voltage Supply Displacement Displacement Model Virtual Young Modulus [kV] Measurement [mm] [mm] [MPa] 5.00 12.02 12.08 7100 5.25 12.39 12.36 6950 5.50 12.80 12.76 6700 5.75 13.23 13.25 6450 6.00 13.76 13.71 6250 6.25 14.33 14.28 6000 6.50 15.00 15.03 5700 6.75 15.73 15.73 5450 7.00 16.56 16.56 5150

Table 5. The comparison of the displacements for different voltage supply and load force 1.23 [N].

Voltage Supply Displacement Displacement Model Virtual Young Modulus [kV] Measurement [mm] [mm] [MPa] 5.00 14.12 14.06 7300 5.25 14.51 14.43 7100 5.50 14.96 14.88 6900 5.75 15.51 15.52 6625 6.00 16.09 16.06 6400 6.25 16.78 16.83 6075 6.50 17.51 17.50 5850 6.75 18.39 18.39 5575 7.00 19.33 19.26 5300

Table 6. The comparison of the displacements for different voltage supply and load force 1.42 [N].

Voltage Supply Displacement Displacement Model Virtual Young Modulus [kV] Measurement [mm] [mm] [MPa] 5.00 16.47 16.47 7200 5.25 16.90 16.87 6975 5.50 17.43 17.34 6775 5.75 18.00 18.02 6600 6.00 18.76 18.8 6300 6.25 19.60 19.58 6050 6.50 20.40 20.35 5800 6.75 21.30 21.40 5550 7.00 22.40 22.24 5300 Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 13

measurement "model"

17

16

15

14

13 Distance [mm] 12

11 Appl. Sci. 2020, 10, 22834.5 5 5.5 6 6.5 7 7.5 7 of 11 Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 13 Voltage [kV]

measurement "model" Figure 5. Comparison of membrane distances for static load force 1.03 [N]. 17 Table 4. The comparison of the displacements for different voltage supply and load force 1.23[N]. 16 Voltage Displacement virtual Young 15 Displacement Measurement supply model modulus [mm] [kV] 14 [mm] [MPa] 5,00 14,12 14,06 7300 5,25 13 14,51 14,43 7100 Distance [mm] 5,50 14,96 14,88 6900 12 5,75 15,51 15,52 6625 6,00 11 16,09 16,06 6400 6,25 4.5 516,78 5.5 6 16,83 6.5 76075 7.5 6,50 17,51 Voltage [kV] 17,50 5850 6,75 18,39 18,39 5575

7,00 19,33 19,26 5300 FigureFigure 5. Comparison5. Comparison of of membrane membrane distances for for static static load load force force 1.03 1.03 [N]. [N].

Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 13 Table 4. The comparison of the displacementsmeasurement for different "model"voltage supply and load force 1.23[N]. VoltageTable 5. The comparison of the displacements for differentDisplacement voltage supply and loadvirtual force 1.42[N]. Young 20 Displacement Measurement supply model modulus Voltage [mm] Displacement virtual Young [kV] 19 Displacement measurement [mm] [MPa] supply model modulus 5,00 18 [mm]14,12 14,06 7300 [kV] [mm] [MPa] 5,25 14,51 14,43 7100 5,00 17 16,47 16,47 7200 5,50 14,96 14,88 6900 5,25 16,90 16,87 6975 5,75 16 15,51 15,52 6625 5,50 17,43 17,34 6775

6,00Distance [mm] 16,09 16,06 6400 5,75 15 18,00 18,02 6600 6,25 16,78 16,83 6075 6,00 18,76 18,8 6300 6,50 14 17,51 17,50 5850 6,25 19,60 19,58 6050 6,75 13 18,39 18,39 5575 6,50 20,40 20,35 5800 7,00 4.555.566.577.519,33 19,26 5300 6,75 21,30 21,40 5550 7,00 22,40 Voltage [kV] 22,24 5300

The results of these experimentsmeasurement show that the parameters"model" obtained from the voltage test Figure 6. Comparison of membrane distances for static load force 1.23 [N]. sufficiently accuratelyFigure describe 6. Comparison the behavior of membrane of the distances membrane for staticand canload be force used 1.23 in [N]. the model. 20 19 measurement "model"

1823 1722 1621 20 Distance [mm] 15 19 14 18

Distance [mm] 13 17 4.555.566.577.5 16 Voltage [kV] 15 4.5 5 5.5 6 6.5 7 7.5 Figure 6. Comparison of membraneVoltage distances [kV] for static load force 1.23 [N]. ` Figure 7. Comparison of membrane distances for static load force 1.42 [N]. Figure 7. Comparison of membrane distances for static load force 1.42 [N].

4. Pump FEM model with DEAP membrane Based on the obtained values of the virtual Young's module, a pump model was developed. In order to increase the force and thus the scope of the pump operation, two permanent magnets were implemented (see Figure 8). The model used neodymium magnets with a diameter of 17 [mm] and a height of 2 [mm]. The mass of each magnets was 3.4 [g] and the lifting capacity was about 2 [kg] for the induction of remanence 1.22–1.25 [T]. The force between two identical cylindrical magnets was approximately calculated according to Gilbert’s model [17,18]. This force in the start position without the pump supply was 0.8 [N], which caused a static deformation of membranes equal to 8 [mm]. Appl. Sci. 2020, 10, 2283 8 of 11

The results of these experiments show that the parameters obtained from the voltage test sufficiently accurately describe the behavior of the membrane and can be used in the model.

4. Pump FEM Model with DEAP Membrane Based on the obtained values of the virtual Young’s module, a pump model was developed. In order to increase the force and thus the scope of the pump operation, two permanent magnets were implemented (see Figure8). The model used neodymium magnets with a diameter of 17 [mm] and a height of 2 [mm]. The mass of each magnets was 3.4 [g] and the lifting capacity was about 2 [kg] for the induction of remanence 1.22–1.25 [T]. The force between two identical cylindrical magnets was approximately calculated according to Gilbert’s model [17,18]. This force in the start position without the pump supplyAppl. Sci. 2020 was, 10, 0.8 x FOR [N], PEER which REVIEW caused a static deformation of membranes equal9 of to 13 8 [mm].

Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 13

Figure 8. FigureA pump 8. A pump concept concept built built using using aa DEAPDEAP membrane membrane and permanent and permanent magnets. magnets.

A pump chamber was proposed (see Figure 9), which is a segment of a sphere with a radius of 9 [cm]. A pump chamberFigure was 8. proposed A pump concept (see built Figure using a 9DEAP), which membrane is aand segment permanent ofmagnets. a sphere with a radius of 9 [cm]. A pump chamber was proposed (see Figure 9), which is a segment of a sphere with a radius of 9 [cm].

Figure 9. Cont. Appl. Sci. 2020, 10, 2283 9 of 11

Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 13 Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 13

Figure 9. AFigure pump 9. A conceptpump concept built built using using aa DEAP membrane membrane and permanent and permanent magnets. magnets. The boundaryThe boundary conditionsFigure 9. Aconditions pump applied concept applied built in in ANSYS using ANSYS a DEAP software software membrane closely closely and match permanent matchthe boundary magnets. the boundaryconditions conditions during the experiments. In the experiments the frame of the DEAP was clamped rigidly and the force during the experiments.wasThe applied boundary at the In conditions central the experiments part applied of the actuator. in theANSYS frame The software actuator of the wasclosely DEAP designed match was to the clampedmove boundary out of rigidly plane conditions by and the force was appliedduringadding at thethe a experiments.fixed central boundary part In co ofthenstraint theexperiments actuator. to the perimeter the Theframe of actuator ofthe the materi DEAPal. was Thewas designedscheme clamped of therigidly to applied move and forces the out force of plane by adding a fixedwasand applied boundary the mechanical at the constraintcentral boundary part conditions toof the the actuator. perimeter applie dThe in ANSYS ofactuator the are material. was as shown designed Thein the to scheme Figuremove 10.out of of theplane applied by forces adding a fixed boundary constraint to the perimeter of the material. The scheme of the applied forces and the mechanical boundary conditions applied in ANSYS are as shown in the Figure 10. and the mechanical boundary conditions applied in ANSYS are as shown in the Figure 10.

Figure 10. Mechanical Boundary conditions for experiments and ANSYS Environment.

Since, the outer plastic frame was rigid as compared to the elastomer, the outer end of the Figureelastomer 10.Figure couldMechanical 10. be Mechanical considered Boundary Boundary fixed to conditions a conditions rigid frame. fo forr Similarly,experiments experiments the and circular andANSYS ANSYSpart Environment. of the Environment. frame at the center of the EAP was rigid and hence no radial displacement of the polymer occurs at the inner end. Since, theTheSince, outer force the plasticwas outer applied frameplastic at wasframe the rigid innerwas asrigidboundary compared as compared in the to the verticallyto elastomer,the elastomer, downward the the outer direction.outer end end ofForof the the elastomer could be consideredelastomerelectromechanical could fixed be consideredtosimulations, a rigid fixed frame.the area to a Similarly,surroundingrigid frame. thethe Similarly, EAP circular was the specified partcircular of as part the air. offrameThe the directional frame at the at the center of the centerdistribution of the EAP of forces was rigidattracting and hencmagnetse no is radial shown displacement in the Figure of11. the polymer occurs at the inner end. EAP was rigid and hence no radial displacement of the polymer occurs at the inner end. The force was The force was applied at the inner boundary in the vertically downward direction. For applied atelectromechanical the inner boundary simulations, in the verticallythe area surrounding downward the direction.EAP was specified For electromechanical as air. The directional simulations, the area surroundingdistribution of theforces EAP attracting was magnets specified is shown as air. in the The Figure directional 11. distribution of forces attracting magnets is shownAppl. inSci. 2020 the, 10 Figure, x FOR PEER 11 REVIEW. 11 of 13

Figure 11. AFigure grid 11. view A grid in view the in FEMthe FEM model model and and distribution distribution of force vectors. of force vectors.

In the suction cycle, the inlet slot (valve) opens and the outlet slot (valve) closes. In order to pump out the medium, the valves change their state and a voltage of 7 [kV] is applied to the membranes. The change in volume of one pump chamber ranges from 135.64 cm3 to 379.81 cm3 and its graph over time is shown in Figure 12. In the suction cycle, the inlet valve opens, and the outlet valve closes. In order to pump out the liquid, the valves change their state and the voltage on the membrane electrodes is switched on. Appl. Sci. 2020, 10, 2283 10 of 11

In the suction cycle, the inlet slot (valve) opens and the outlet slot (valve) closes. In order to pump out the medium, the valves change their state and a voltage of 7 [kV] is applied to the membranes. The change in volume of one pump chamber ranges from 135.64 cm3 to 379.81 cm3 and its graph over time is shown in Figure 12. In the suction cycle, the inlet valve opens, and the outlet valve closes. In order to pump out the liquid, the valves change their state and the voltage on the membrane electrodesAppl. Sci. 2020 is, 10 switched, x FOR PEER on. REVIEW 12 of 13

Figure 12. Supply voltage waveform and volume change of one pump chamber. Figure 12. Supply voltage waveform and volume change of one pump chamber. 5. Conclusions 4. Conclusions DEAPs are a subclass of electroactive polymers in which actuation is produced by an elastic deformationDEAPs resultingare a subclass from theof electroactive compressive electrostaticpolymers in forces. which Their actuation most importantis produced advantages by an elastic like energydeformation efficiency, resulting lightweight from andthe compressive scalability are electr key featuresostatic forforces. actuators Their in most applications important such advantages as pumps. Dielectriclike energy electroactive efficiency, lightweight polymer technology and scalability is able are to fulfillkey features requirements for actuators as well in as applications commonly usedsuch technologyas pumps. e.g.,Dielectric solenoids, electroactive but its limitations polymer concerntechnology relatively is able low to force fulfill [19 ,requirements20]. One way toas increasewell as forcecommonly generated used by technology these devices e.g., can solenoids, be the application but its limitations of neodymium concern permanent relatively magnets. low force The concept[19,20]. forOne pump way presentedto increase in thisforce paper generated consists by of athese stack devices of DEAP can membranes be the application combined with of aneodymium permanent magnets.permanent These magnets. two components The concept are for combined pump presente in a noveld pumpin this construction, paper consists which of a allows stack aof compact DEAP designmembranes by integrating combined thewith biasing a permanent mechanism magnets. with theThese DEAP two membranes.components Subsequently,are combined in the a singlenovel componentspump construction, are manufactured, which allows tested, a compact and theirdesign force-displacement by integrating the characteristic biasing mechanism is documented. with the IdentifyingDEAP membranes. the parameters Subsequently, experimentally the single allowed compon toents develop are manufactured, a faithful FEM tested, model and of their the pumpforce- devicedisplacement and make characteristic its simulations is documented. for different voltage.Identifying FEM the modeling parameters of the experimentally DEAP membrane allowed behavior to isdevelop a useful a toolfaithful to understand FEM model such of systemsthe pump better device and and helps make to create its simulations the optimal for design different of prototypes. voltage. TheseFEM modeling modeling of eff theorts DEAP must accountmembrane for thebehavior electromechanical is a useful tool coupling to understand in order tosuch accurately systems predict better theirand helps response to create to multiple the optimal loading design conditions of protot expectedypes. during These real modeling operating efforts conditions. must account Because for of the complexityelectromechanical of the nonlinearcoupling processes,in order theto finiteaccurate elemently predict analysis their is reasonableresponse toto bemultiple used and loading study theconditions behaviour expected of dielectric during elastomer real operating pump. Theconditions. experiments Because under of staticthe complexity and transient of considerationsthe nonlinear haveprocesses, been performedthe finite element to validate analysis the design is reasonable and implementation. to be used and It is study shown, the that behaviour the model of isdielectric in close agreementelastomer pump. with the The measured experiments responses under ofstatic the and real tr DEAPansient actuator. considerations The collected have been data performed allow for the to analysisvalidate ofthe various design otherand implementation. device designs using It is DEAPshown, membrane that the model in the is future. in close agreement with the measured responses of the real DEAP actuator. The collected data allow for the analysis of various Funding:other deviceThis designs research using was fundedDEAP bymembrane NATIONAL in the SCIENCE future. CENTRE (Poland), project SONATA13 grant number No. 2017/26/D/ST7/00092. The APC was funded by NATIONAL SCIENCE CENTRE (Poland). Funding: This research was funded by NATIONAL SCIENCE CENTRE (Poland), project SONATA13 grant Conflicts of Interest: The author declare no conflict of interest. number No. 2017/26/D/ST7/00092. The APC was funded by NATIONAL SCIENCE CENTRE (Poland).

Conflicts of Interest: The author declare no conflict of interest.

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