Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 4, pp. 1309-1318, Warsaw 2016 DOI: 10.15632/jtam-pl.54.4.1309 STUDY OF A MEMS HYBRID THERMO-PZT MICRO ACTUATOR Hasan Pourrostami, Navid Seyed Kazem Viliani Department of Mechanical Engineering, Islamic Azad University, Abhar Branch, I.R. Iran e-mail: a.pourrostami@gmail.com The current studypresents ananalyticalmodel for aMEMScantilever basedhybrid thermo- -piezoelectric micro actuator. The micro actuator structure consists of a unimorph type bending piezoelectric beam and a passive layer which uses the thermal bimorph principle. Primarily, under electrical and thermal loadings, the governing equations of motion are derived. After verification of the results, the deflection analysis of the micro actuator is performed.Moreover, the equivalent force at the actuator tip is obtained.Toprovide efficient optimization, the effects of the micro actuator geometrical and materials properties on the actuator tip deflection and force are investigated. Keywords: hybrid micro actuator, thermo-piezoelectric, deflection, force analysis 1. Introduction Micro actuators are broadly used in miniaturized systems or devices such as micro-valves and micro-switches. These actuators are designed differently from classical actuators and are ma- inly employed to produce mechanical motion in these devices. In order to fulfill the required performances, micro actuators need to be designed with high resolution, good accuracy, quick response time and high range of positioning capability. Among different types of actuation mechanisms which can be classified into thermal, elec- trostatic, magnetic, piezoelectric and other categories, thermal and piezoelectric ones are the most prized due to their remarkable characteristics and high compatibility withMEMSdevices. Some of these noticeable features in piezoelectric actuators are high positioning accuracy, ra- pid response time, more mechanical power. While the distinguishable feature of electrothermal actuators is a large output force with reasonable displacement. Both piezoelectric and electrothermal micro actuators are used in the variety of applications which include micro gripper (Nguyen and Ho, 2004; Capparelli et al., 2002), scanning probe arrays (Zheng and Lu, 2004), micro positioning applications (De Cicco and Morten, 2009; Jain et al., 2005) and control valves (Roberts and Li, 2003). It should be noted that the use of electrothermalmicro actuators aremore common.This is due to the fact that they can be easily fabricated and operated at small drive voltages. This group of actuators is ideally suited for applications that demand relatively high-output forces.However, they consumemuchpower and show slow response time (Alwan and Aluru, 2009). On the other hand, piezoelectric actuators have the advantage of high positioning accuracy, more sensitivity and rapid response, but when compared to electrothermal micro actuators, they suffer from the limited range of deformation (Ha and Kim, 2002). Combination of these two classes ofmicro actuators as a hybridmicro actuator removes their limitations and leads tomodified characteristics such as lowpower consumption, better response time andmore deflection. Rakotondrabe and Ivan (2010) attempted to integrate electrothermal and piezoelectric actuation mechanisms. They employed the thermal bimorph principle and a piezocantilevermicroactuatorwhichwasmadeupof apiezoelectric andapassive layer.APeltier 1310 H. Pourrostami, N.S.K. Viliani module (as an external source of heat generation) was used to provide temperature variation. They used the thermal network approach to model the micro actuator. The main difference between the current research and the Rakotondrabe research is the problem-solution approach. They used the thermal network approach to model the micro actuator. In this research, the elasticity approach is used to model the micro actuator. In the elasticity approach, there is better control on designing parameters such as geometrical andmaterial properties of themicro actuator. Although physical modeling and temperature characterization of multi/bi layered piezoelec- tric cantilevers have also been studied by some researchers (Tadmor and Kósa, 2003; Ballas, 2007; Rakotondrabe et al., 2008), there is still a gap in knowledge for optimized selection of the micro actuator material and geometry based on the design requirements. Recently, the deflection analysis of a U shape hybrid thermo-piezoelectric micro actuator, in which the piezoelectric actuation mechanism was based on d33 strain coefficient, has been performed (Pourrostami et al., 2012). In the research, the piezoelectric material properties were varied by temperature and the effects of piezoelectric material properties on the deflection of micro actuator were studied which provided efficient optimization for selecting the piezoelectric material. The main objective of this research is the study of the effects of geometrical and material properties of the micro actuator to provide efficient optimization on its performance. In this paper, a MEMS hybrid cantilevered thermo-piezoelectric micro actuator with d31 actuationmechanism isdeveloped.Themicro actuator geometry is basedon thepresentedmodel by Rakotondrabe and Ivan (2010). In the current study based on strength of the material, the actuator tip deflection and force are studied. Furthermore, to provide efficient optimization, the effects of the micro actuator geometrical andmaterials properties are investigated. 2. Actuator modeling The configuration of the MEMS hybrid cantilevered thermo-piezoelectric micro actuator is il- lustrated in Fig. 1. This micro actuator is made of a Lead-Zirconnate-Titanate Piezoceramic (PZT) cantilever beam and a passive elastic layer (Rakotondrabe and Ivan, 2010). By applying an electrical field, the piezoelectric layer expands (or contracts) and consequently the cantilever beam bends over. Moreover, due to the existence of a difference between thermal expansions of different layers when themicro actuator is subjected to a temperature variation, the secondary bending will be induced. Fig. 1. The structure of the hybrid thermo-piezoelectric micro actuator (Rakotondrabe and Ivan, 2010) Piezoelectric ceramics haveveryhigh electrical resistivity (more than109Ωm)whichprevents them from any heat generation by passing electric current through the material. Therefore, in order to have thermal actuation in piezoelectric beams, an additional external heat source is needed. Based on the small size of the micro actuator, usually a uniform temperature variation on the surfaces of micro beams is considered (Rakotondrabe and Ivan, 2010). In this model, Study of a MEMS hybrid thermo-PZT micro actuator 1311 the classical Euler-Bernoulli beam theory (EBT) is considered, and it is assumed that the shear and electrostriction effects are negligible. Also x-z plane stress and x-y plane strain are enforced (Devoe and Pisano, 1997). 2.1. Constitutive relations For the piezoelectric materials, the general constitutive relations are (Kargarnovin et al., 2007) ε=SEσ−dTE F=dσ−ǫTE (2.1) In which σ, ε, E and D represents the columns of stress components, strain components, elec- trical field components and electric displacement components, respectively. Also SE, d and ε are the compliancematrix at the constant electric field, the piezoelectric strain constantmatrix and the dielectric permittivity matrix. On the assumption of the x-z plane strain which is reasonable for a wide flat micro beam, the strain component in the y-direction is zero. Setting εy =0 in equation (2.1)1, σy yields to σy =− 1 s22 (s12σx+s23σz +d31Ez) (2.2) When the temperature is applied, by substituting equation (2.2) into equations (2.1), the con- stitutive equation reduces to εx = s11σx+s13σz +d31Ez +α∆T εx = s13σx+s33σz +d33Ez Dz = d31σx+d33σz + ǫ33Ez (2.3) It should be noticed that because of small height andwidth of themicro actuator in comparison with its length, the thermal expansions in the y and z directions are neglected.Also the reference temperature is room temperature. s11, s13, s33, d31, d33 and ǫ33 are reducedmaterial constants of the piezoelectric for the plain stress assumed s11 = s11− s212 s22 s13 = s13− s12s23 s22 s33 = s33− s223 s22 d31 = d31− s12 s22 d31 d33 = d33− s23 s22 d31 ǫ33 = ǫ33− d231 s22 (2.4) Rewriting the constitutive relations in terms of stress-strain components, we get [ σx σz ] = [ c11 c13 c13 c55 ]([ εx εz ] − [ d31Ez d33Ez ]) (2.5) where c11, c13 and c33 are reduced stiffness components of the piezoelectric beam. They are expressed by c11 = s33 s11s33−s 2 13 c13 =− s13 s11s33−s 2 13 c33 = s11 s11s33−s 2 13 (2.6) For a transversely isotropic piezoelectric beam with a small aspect ratio, the thickness in the z-direction is stress free. Therefore, it is plausible to set σz =0. By setting σz =0 in equation (2.5), εz yields εz =− 1 c33 [c13(εx−d31Ez)− c33d33Ez] (2.7) 1312 H. Pourrostami, N.S.K. Viliani By substituting equation (2.7) into equations (2.3)3 and (2.5), when the uniform temperature variation is applied, the constitutive equation reduces to σpx =Ep(εx−d31Ezαp∆T) Dz = d31Epεx+(ǫ33−d 2 31Ep)Ez (2.8) where αp and Ep are the thermal expansion coefficient and the effective elasticity modulus of the PZT layer. For the one dimensional piezoelectric beam under considered assumptions, the effective elasticity modulus is Ep = c11− c213 c33 (2.9) For the elastic beam under plane strain and stress assumptions by applying temperature, the constitutive relation in terms of stress-strain components is σex =Ee(εx−αe∆T) (2.10) whereαe is the thermal expansion coefficient of the elastic beamandEe is the effective elasticity modulus of the elastic beam for the plane strain. It can be obtained from Ee = E 1−ν2 (2.11) 3. Deflection analysis The deflection of the HTP micro actuator can be derived by applying static equilibrium and strain compatibility between the layers (Devoe andPisano 1997). Figure 2 shows a cross section of the micro actuator. Fig. 2. The cross section of the hybrid thermo-piezoelectric micro actuator (Devoe and Pisano 1997) In Fig. 2, R represents radius of the curvature, he and hp are heights of the elastic and piezoelectric beams.P1,P2,M1 andM2 are axial forces andmoments at the cross-section of two layers. Furthermore,Ee andEp are the effective elasticitymodulusof the elastic andpiezoelectric micro beams, respectively. The curvature 1/R of the curve y(x) is expressed as follows 1 R = d2y dx2 (3.1) Then y= x2 2R (3.2) Themaximum deflection is found at the tip of the micro actuator (x=L). Using the following equations, the requirements for force and moment equilibrium are satisfied (Devoe and Pisano 1997) P1+P2 =0 he 2 P1+ ( he+ hp 2 ) P2+ Eeh 3 e +Eph 3 p 12R =0 (3.3) Study of a MEMS hybrid thermo-PZT micro actuator 1313 The total strain in the elastic layer is due to axial force, bending moment and thermal effect, while the piezoelectric effect causes an additional strain component in thePZTbeam.To satisfy the strain compatibility between two layers, we can write εaxial +εbending +εthermal = εaxial +εbending +εthermal +εpiezo P1 Eehe + he 2R +αe∆T = P2 Ephp − hp 2R +αp∆T + d31 hp V (3.4) whereV ,∆T and d31 are the applied voltage, temperature variation andpiezoelectric transverse strain constant, respectively. To determine the actuator curvature, the system of equations in a matrix form is rewritten       1 1 0 he 2 he+ hp 2 Eeh 3 e +Eph 3 p 12 1 Eehe −1 Ephp he+hp 2           P1 P2 1 R     =     0 0 (αp−αe)∆T + d31 hp V     (3.5) By solving equation (3.5) for the curvature 1/R, the actuator tip deflection using equation (3.2) is δ= 3L2(he+hp)EeEphehp E2ph 4 p+EeheEphp(4h 2 p+6hehp+4h 2 e)+E 2 eh 4 e ( − d31 hp V +(αp−αe)∆T ) (3.6) whereL is the total length of the actuator. 4. Deflection analysis The equivalent generated force of the micro actuator can be obtained by the using actuator tip deflection and stiffness F =Kδ= 3EeqIeq L3 δ (4.1) whereK is the stiffness,Eeq and Ieq are the equivalent elasticitymodulus andmoment of inertia of the two layers, respectively. Themethod used to calculate the equivalent beam is based on the construction of an equiva- lent homogeneous section that is mechanically equivalent to the initial, heterogeneous, section (see Fig. 3). To keep flexural rigidity in the equivalent homogeneous section, the method of normalization ofwidths by the elasticitymodulus is used. In thismethod, the ratio of the elastic beamwidth be to the piezoelectric beamwidth bp should be equal to the ratio of their elasticity moduli be bp = Ee Ep (4.2) To calculate the equivalentmoment of inertia, it is essential to find the location of theneutral axis of the equivalent homogeneous section heq heq = ∑ Aihi ∑ Ai = hpEp ( he+ hp 2 ) + 1 2 h2eEe hpEp+heEe (4.3) The equivalent moment of inertia is then expressed as follows: Ieq = bp 12 E2ph 4 p+4hphe ( h2p+ 3 2 hphe+h 2 e ) EpEb+E 2 eh 4 e (Eehe+Ephp)Ep (4.4) 1314 H. Pourrostami, N.S.K. Viliani Fig. 3. (a) The cross section of the actuator with piezoelectric and elastic layers, (b) the equivalent homogenous beam section The equivalent generated force of themicro actuator can be obtained using equations (3.6), (4.1) and (4.4) F = 3 4 3EpEehehp(he+hp)b L(Eehe+Ephp) ( − d31 hp V +(αp−αe)∆T ) (4.5) 5. Results and discussion Based on the developed relations, the governing derived equations for the deflection and force of the actuator are solved numerically using the data listed in Tables 1 and 2 for geometrical and material properties, respectively. In the current study, PZT-5H and copper are used as active piezoelectric and passive elastic layers whose geometry and materials are exactly the same as the one which was presented in the experimental model by Rakotondrabe and Ivan (2010). In the first step, the validity of the obtained results is verified by comparing with the reported and experimental results (Devoe and Pisano 1997; Rakotondrabe and Ivan, 2010). Then, in order to have an optimized design, the effects of geometrical and material properties on the actuator performance are studied. Several researches have been done on the temperature dependence of the PZT material propertieswhich showthat forPZT-5H: i) piezoelectric strain constantsd31 andd33 increasewith temperature while d15 remains almost constant, ii) piezoelectric dielectric constants ǫ11 and ǫ33 increase with temperature, iii) for the mentioned temperature range, the elastic constants do not remarkably change with temperature (Hooker, 1998). In the current study, for verification of the results (based on the experimental results byRakotondrabe and Ivan (2010)) theworking temperature of the PZT-5H is considered 50◦C. The PZT-5H material properties at room and working temperature are listed in Table 2. Also it is assumed that the variation of material properties of the copper layer with temperature are negligible (Rakotondrabe and Ivan, 2010). Table 1.Geometrical data for the actuator (Rakotondrabe and Ivan, 2010) Length of the actuator, L [mm] 15 Width of the actuator, b [mm] 2 5.1. Verifications From Eq. (3.6), it is clear that the actuator tip deflection is a linear function of the applied voltage and temperature. Substituting∆T =0 in Eq. (3.6), the hybridmicro actuator converts to the unimorph piezoelectric micro actuator (Devoe and Pisano 1997). Furthermore, based on the experimental results, two linearmodels have been proposed for variation of deflection versus Study of a MEMS hybrid thermo-PZT micro actuator 1315 Table 2.Material properties (Rakotondrabe and Ivan, 2010; Hooker, 1998) Title Symbol Value PZT-5H Compliance matrix component [m2/N] s11 16.5 ·10 −12 Compliance matrix component [m2/N] s33 20.7 ·10 −12 Compliance matrix component [m2/N] s12 −4.78 ·10 −12 Compliance matrix component [m2/N] s13 −8.45 ·10 −12 Piezoelectric transverse strain constant [V/m] d31 −100 ·10 −12 Expansion coefficient [1/◦C] αp 3.6 ·10 −6 Elastic layer (copper) Axial elastic constant [Pa] E 125 ·109 Poisson’s ratio [–] ν 0.35 Expansion coefficient [1/◦C] αe 1.7 ·10 −6 voltage and temperature (Rakotondrabe and Ivan, 2010). In this Section, the analytical results are compared with those from the proposed linear models. Figure 4 ashows the variation of the actuator tip deflection versus the applied voltage when ∆T =0◦C, hp =0.2mm and he =0.1mm. The results follow a linear trend and the analytical result is in a good agreement with the experimental result. In the next step, the actuator tip deflection as a function of the applied temperature is plotted (see Fig. 4b) at V =0when hp =0.2mmand he =0.1mm, respectively. As it is seen in Fig. 4b, when a uniform temperature distribution is applied, the deflection variation is changed linearly. Fig. 4. The actuator tip deflection versus applied voltage (a) and uniform temperature variation (b) The present difference between the two obtained results is due to the pyroelectric effect. The pyroelectricity is generation of an electric charge on a crystal by a change in temperature. The change in temperature slightlymodifies positions of atomswithin the crystal structure such that the polarization of thematerial changes. This polarization change gives a rise to the voltage across the crystal. As a result, the generated voltage causes greater deflection. 1316 H. Pourrostami, N.S.K. Viliani 5.2. Geometry effects In this Section, the variations of heights of the piezoelectric and elastic layers on the actuator deflection and force are studied. Figure 5a illustrates the variation of the actuator tip deflection versus height of the elastic layer for two different heights of the PZT layer when ∆T = 25◦C, V =40V. Referred to this figure, smaller heights of PZT and elastic layers enable the actuator to deflect more efficiently. Figure 5b shows the variation of the actuator force versus height of the elastic layer for two different heights of thePZT layer when∆T =25◦C,V =40V.Contrarily, greater heights of the PZT and elastic layers generate large forces in the actuator. Therefore, optimization of heights of the PZT and elastic layers can lead to the maximum efficiency of the micro actuator. Fig. 5. (a) The actuator tip deflection versus height of the elastic layer for different heights of the PZT layer; (b) the actuator force versus height of the elastic layer for two different heights of the PZT layer Fig. 6. The actuator tip deflection versus elasticity modulus of the elastic layer 5.3. Material property effects In the present Section, the effects of the piezoelectric strain constant d31 and the elasti- city modulus of the elastic layer E on the actuator deflection are investigated. The variation of actuator deflection versus elasticity modulus of the elastic layer are plotted in Fig. 6 for Study of a MEMS hybrid thermo-PZT micro actuator 1317 ∆T = 25◦C, V = 40V, hp = 0.2mm and he = 0.1 mm, respectively. There is a nonlinear variation trendwhen the elasticity modulus of the elastic layer increases. It means that a stiffer elastic layer leads to a greater deflection and force in the actuator. Furthermore, when the pie- zoelectric strain constant d31 increases indirectly with the temperature, the actuator deflection and force rise. 6. Conclusion In the present study, static analysis is performed for the proposedMEMShybridmicro actuator which applies both the piezoelectric and thermal actuation. To sum up, the most important conclusions arising from this study are as follows: • By increasing the applied voltage and temperature gradient, the actuator force and tip deflection linearly increase. • The actuator tip deflection is decreased upon an increase in the heights of PZTand elastic layers. • An indirect increase in the piezoelectric transverse strain constant d31 by growth of the temperature give a rise to the actuator force and tip deflection. • Making use of a stiffer elastic layer rises the nonlinear increasing trend of the actuator force and tip deflection. • Increasing the temperature not only increases thermal expansions of the layers and, as a result, increases deflection, but also affects the piezoelectric strain constant, which enlarges deflection as well. References 1. 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ZhengL., LuM., 2004,A large-displacementCMOSmicromachined thermal actuatorwith comb electrodes for capacitive sensing, Sensors and Actuators A: Physical, 136, 2, 697-703 Manuscript received August 17, 2015; accepted for print March 7, 2016