Suppleentary Inforation for Design of Bending Multi-Layer Electroactive Polyer Actuators Bavani Balakrisnan, Alek Nacev, and Elisabeth Sela University of Maryland, College Park, Maryland 074 1 Analytical Model This discussion is for readers without a echanical engineering background. We begin by introducing the trilayer and ulti-layer curvature odels, and then we present the -layer expressions for blocked force. 1.1 Tri-Layer Curvature Model A trilayer DEA configuration was illustrated in Figure 1 of the ain text. Bensliane et al. presented a trilayer odel [13] for the curvature under zero load as: (8) = Et 6Ett t Et 6t t Et Et 6t t Et6t 1t 1t 1 1 1 1 3 3 3 13 3 3 3 1 1 1 3 4 4 4 E1t1EtE3t3EE3tt3 8t7tt33t3 Et 1 1 4Ett13tt 1 te3t38t14tt 1 4t7tt 1 314tt33t3 The variables and effective oduli were defined in the ain text of the paper, and the ajor assuptions stated there. As it should, Equation (8) reduces to the well-known Tioshenko equation [54] as the thickness and/or odulus of the third layer go to zero. (9) Et 6Et 1 1 t1 t 4 4 Et 1 1Et EtEt 1 1 t13tt 1 t Equations (8) and (9) can be used to calculate curvatures using a hand calculator or software packages such as Excel, but with additional layers this becoes ipractical. To obtain Equation (4) in the ain text, note fro Equation (9) that goes as t / t = /t. 1. -Layer Curvature Model DeVoe et al. [14] showed that for an -layer bending actuator, the curvature under zero external load is given by: (10) 1 1 DA C -1 R DA B,
where A, B, C, and D are atrices. The atrix D is associated with the requireent that the oents at any cross-section of the actuator su to zero at equilibriu, i1 M i 0. It can be written as the product of two coponents. The first coponent is the inverse of the su of the products E i I i, which are the flexural rigidities of the layers. The second coponent is a vector giving the center positions of each layer, having thickness t i, relative to the botto of the ultilayer stack. For the botto-ost layer, i = 1, as in Figure 1 of the ain text. (11) 1 1 t1 t t D t 1... ti i1 EI i1 i i The atrix A contains the inverse products of the cross-sectional areas A i and the Young s oduli E i for each layer and for the layer iediately above it. These ters are related to the axial coponents of the strain. The cross sectional area A i is given by t i w i, where w i is the width of the layer (perpendicular to the plane of the page in Figure 1). The E i in Equation (1) are again the effective oduli: the Young s oduli divided by 1 - i. The last row of 1s in the atrix arises fro the requireent for axial force equilibriu, Fi 0. (1) 1 1 0... 0 AE 1 1 AE 1 1 0 0... AE AE 3 3 A............... 1 1 0... 0 A E A E 1 1 1 1 1 1 1 i1 The colun vector C contains the differences in the actuation strains i between each layer and the one above it, and originates fro equating the strains at the -1 interfaces. The last row is zero. 1 3 (13) C... 1 0 Finally, the colun vector B contains the su of the thicknesses of adjacent layers, t i-1 + t i, and it also originates fro equating the strains at the interfaces. The last row is again zero.
(14) t t B t 1 3 1 t t... t 0 This odel gives the sae results as the Bensliane et al. odel for three layers, and the sae results as the Tioshenko equation for two layers. 1.3 -Layer Blocked Force Model To deterine the blocked force, the coposite bea was first transfored into an equivalent single-aterial bea. The ratio of the Young s odulus of layer i relative to that of layer 1 is: (15) n i = E i /E 1. The layer widths w i were then expressed in ters of the odulus ratios: (16) w i = n i w 1. (Here w 1 is used for all i because the actuator has a constant width. If this is not the case, than all widths ust be adjusted by the odulus ratios n i.) This transfors the description of the bea to one with each layer having a varying width but the sae odulus as layer 1; this bea behaves equivalently to the original coposite bea. The distance d i, in the thickness direction, fro the botto of layer 1 to the iddle of each layer is given by: (17) i1 1 d t t. i i j j1 The neutral axis of a bea with layers can then be found by: (18) t neutral i i i i1. i1 w t d w t i i To calculate the blocked force, the parallel-axis theore was used to copute the oent of inertia I c for the equivalent bea [34]. This is possible because the neutral axis is parallel to all the center of ass axes of the individual layers. By this theore, I c is the su of the individual layer oents I i plus ters that depend on the square of the distance between each layer and the neutral axis. (19) I wt /1 i 3 i i (0) Ic [ Ii wt i i( di tneutral ) ] i1
The blocked force was described as the load applied to the end (tip) of the deflected bea necessary to push it back to the original undeflected position. Therefore, the deflection of a bea due to an applied point load was found starting with the differential equation for bea deflection [34], 3 d ( x) (1) EIc F 3 b, dx and using the following boundary conditions: d ( L) d (0) () 0, 0, (0) 0, dx dx where x is the position along the length of the bea, δ(x) is the deflection of the bea, and F b is the blocked force. Equations () state that the oent at the end of the bea (x = L) is zero and the slope and deflection at the fixed point (x = 0) is zero. The displaceent along the bea can then be written by integrating Equation (1) three ties and using the boundary conditions of Equation () to deterine the particular solution. 3 1 F blx Fb x (3) ( x). EI c 6 At the end of the bea, x = L, and (4) F L 3 b ( L). 3EIc By using the following curvature approxiation [34], for sall deflections, (5) L /, the blocked force can be written as: (6) F 3EI L. b c This is proportional to t because I c ~ t 3 (Equation (19)) and ~ 1/t [35]. To obtain Equation (5) in the ain text, note that by Equation (9) ~ /t and that by Equation (19) I ~ wt 3 and substitute into equation (6): (7) 3 Fb E wt / t L Ewt L 1.4 Calculation of Work Assuing a linear load curve: (8) F ( F) ( L) 1, Fb
where F is the load applied at the tip of the actuator, the work perfored by the actuator can be calculated. The work is defined as the product of the force and the tip displaceent: F (9) W(F ) F (F ) F (L) (L) Fb The force at which the work is axiu can be deterined fro the first derivative of the work: dw( F ) F (30) 0(L) (L), df Fb which is axiu at F = F b /: Fb Fb Fb (31) Wax 4. 4 To obtain Equation (6) in the ain text, cobine equations (19), (5), and (6): 3 ax b c c W F EI L L EI L E wt L E wt L t (3) Plot Liits The liits for the plots in the paper were chosen to be beyond the typical ranges of thicknesses and oduli for DEAs, and they are shown in SI Table 1 along with the resulting plot liits. SI Table 1. Thickness and odulus of electrode and elastoer. Thickness () () () Modulus (Pa) (Pa) Electrode t elec in 0.01 1E-08 E elec in 1 kpa 1E+03 t elec ax 10 1E-05 E elec ax 100 GPa 1E+11 Elastoer t elas in 1 1E-06 E elas in 10 kpa 1E+04 t elas ax 1000 1E-03 E elas ax 10 MPa 1E+07 X Liits (unitless) Y Liits (unitless) Plot X in = t elec in / t elas ax 1E-05 Y in = E elec in / E elas ax 1E-04 X ax = t elec ax / t elas in 1E+01 Y ax = E elec ax / E elas in 1E+07 3 Exaple: Obtaining Diensional Estiates for a Biorph DEA In this section, a 5-layer configuration is used to deonstrate how to convert the nondiensional data fro the paper into actual thicknesses and oduli for use in fabricates devices.
For a 5-layer DEA-type device, the axiu bending occurs when the relative electrode thicknesses ratios t 1, t 3, t 5 are < 10-1 and when the relative electrode oduli ratios E 1, E 3, E 5 are < 10 - to < 10 3, depending on the thickness ratio (SI Figure 1). Thus, if the elastoer (active) layer thicknesses (t, t 4 ) are 10, then for axiu bending the corresponding electrode thickness should be t 1, t 3, t 5 < 1. If the elastoer layers have oduli (E 1, E 3 ) of MPa, then the electrode oduli should be at least 100x saller at t 1, t 3, t 5 = 10-1, aking E 1, E 3, E 5 < 0.0 MPa. The electrodes can have greater oduli if they are thinner. SI Figure 1. Figure 8a in the paper for a 5-layer actuator coprising stacked DEAs with t = t 4 = 0.5, E = E 4 = 1, = %, and 4 = 0. This discussion assues that the perforance of the device can be roughly estiated fro its diensions and aterial paraeters. If the elastoer breakdown field E breakdown = 100 MV/ (1000 V for a 10 fil thickness) and if the relative dielectric constant 0 = 3, then the axiu actuation strain based on this siple odel is coputed as follows to be 6.6%: 1 0 r 0 r V breakdown 38. 8510 1000 breakdown 6 6 E E t * E 100% 100% 100% 6. 6% 10 10 10 The expected axiu curvature (at = 1 in Figure 8a) can be found using Equation (4): C N 15. 10 di = 0.015 * 6.6 * 1 / (10 ) = 9900-1. t t To convert the curvature to a displaceent, recall (Equation (5)) that for sall curvatures, the displaceent of the tip of the bea is given by: 0.5 L κ. If the actuator length is 100, then the expected deflection will be 49.5 :
0.5 * (100 ) * 9900-1 = 49.5 * 10-7. SI_Figure. Figure 8b in the paper for the sae 5-layer actuator. For this device, the blocked force increases with t 1, t 3, t 5 but is independent of E 1, E 3, E 5. Thus, to obtain the highest blocked force without coproising curvature, a axiu electrode thickness of t 1, t 3, t 5 = 1 should be used. At that thickness, the noralized blocked force is F b = 0.01 (Figure 8b). Using Equation (5), F di = [(C FN α E t ] / (L /w)] F b = [(0.11α E t ) / (L /w)] *0.01 and for a width of 1000 the expected blocked force is: F di = [(0.11 * 6.6 * MPa * (1 ) ] / (100 /1000 )] * 0.01 = 145 µn. At the point t 1, t 3, t 5 = 1 and E 1, E 3, E 5 = 0.0 MPa, W = 7x10-5. The axiu work is given by Equation (6): WN 0 1405 Wdi C EwLt W. EwLt W, where given by: W di = 0.1405 * 6.6 * MPa * 1000 * 100 * 10 * 7x10-5 = 8.5x10-10 J. 4 Optiization of a PZT Actuator Coplete data for a PZT bending actuator have been published by Ballas et al. [1], allowing us to run a diensional optiization.
SI Figure 3. Scheatic of the PZT actuator [1] optiized in this section. Paraeter values are given in SI Table. SI Table. Paraeters used in our odel to optiize the PZT actuator [1]. The actuator was 19. long and 8 thick. Layers Material Thickness Thickness Modulus Poisson s Strain in Device in odel* (GPa) Ratio** (%) (µ) (µ) 1,3,5,7,9,11 Ag/Pd not given 0.001-1 100** 0.4 0,4,6,8,10 PZT 48 1-100 70.7 0.4 0.06 1 glass 00 1-1000 88 0.3 0 13 Ni-steel 100 100 157 0.3 0 * Only one aterial was varied at a tie. ** Assued. A 13-layer odel was used to describe the ultilayer PZT uniorph [1], whose geoetry is shown in SI Figure 3 and whose paraeters are given in SI Table. The structure is essentially a bilayer, considering the PZT + electrodes as the active layer and the glass + Ni-steel as the passive layer. The PZT coefficient, d 31, was given in the paper as 350x10-1 /V, and using it we coputed the strain under 84 V (one of the voltages at which a force-deflection curve had been obtained): α = (Vd 31 /t)*100 = (84 V*350x10-1 V -1 / 48x10-6 )*100 = 0.061%. Using this strain, the device paraeters in SI Table (soe of which were unknown and had to be assued), and an electrode thickness of 0.01, the odel gave an expected curvature of 1.54-1 and a blocked force of 1.4 N. The tip deflection was then found fro the curvature: 0.5 L κ = 0.5*(19.x10-3 ) *1.54-1 = 84 µ, which closely atches the easured ~310. The odel predicted a blocked force of 1.4 N, copared to the easured value of 1.13 N, and work of 100 J.
Because the odulus is fixed for these aterials, only the layer thicknesses were varied. Varying the thickness of the electrode layers showed, as expected since they are within the actuating region of the bilayer, that they should be as thin as possible. Varying the PZT thickness while keeping the passive glass and Ni-steel layer thicknesses fixed and using 0.01 for the electrodes showed that for axiu bending ( = 1.6-1 ) the PZT layers should each be 35, that for axiu force they should be as thick as possible, and that for axiu work (170 J) they should be 164 (which ay not be feasible to fabricate). Of course, increasing the PZT thickness would require a concoitant increase in voltage to aintain the sae strain. The published device was therefore close to being optial for bending given those substrate thicknesses. Fro a practical point of view, it can be difficult to increase the PZT thickness, so we next kept the PZT fixed and varied the glass and Ni-steel thicknesses. As shown in SI Figure 4, the axiu curvature increased 50% (to =.4-1 ) by thinning the two passive layers to 50 each, while the axiu work increased just slightly (fro 100 to 107 J) if both passive layers were 190 thick. (Again, the axiu force resulted fro aking the passive layers as thick as possible.) SI Figure 4. Dependence of curvature and work on the thickness of the passive layers.