ELECTRICAL LOADING EFFECTS ON LEAKY LAMB WAVES FOR PIEZOELECTRIC PLATE BORDERED WITH A FLUID: ANALYSIS AND MEASUREMENTS

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1 12 th A-PCND 26 Asia-Paciic Conerence on ND, 5 th 1 th Nov 26, Acland, New Zealand ELECRICAL LOADING EFFECS ON LEAKY LAMB WAVES FOR PIEZOELECRIC PLAE BORDERED WIH A FLUID: ANALYSIS AND MEASUREMENS Yng-Chn Lee and Shi Hoa Ko Department o Mechanical Engineering, National Cheng Kng University ainan 71, aiwan, ROC Abstract Leay Lamb waves are important research topics in the area o NDE and acostic wave sensing technology. For waves propagating in a piezoelectric plate immersed in a lid (dielectric or condctive), both mechanical and electrical loading eects present and reslt in phase velocity variation and wave attenation. o qantitatively characterize these loading eects play an important role in varios applications o these plate waves. In this wor, we apply the partial wave theory or analyzing the lid s dielectric and condctive loading eects on an XZ-ct LiNbO plate. Both phase velocity variation and wave attenation as a nction o the lid s permittivity and condctivity are nmerically determined. Frthermore, a new dierential type measrement system is developed to accrately and sensitively measre the loading eects on the leay Lamb waves. Good agreements between experimental data and nmerical reslts are observed, which veriies the analytic model o the leay Lamb wave and can be sel or rther development o dielectric or condctivity sensors based on leay Lamb waves. 1. Introdction Lamb waves are gided elastic waves propagating in a plate placed in vacm or air, so that both plate sraces are traction-ree. It is also nown as the ree or pre mode o Lamb wave. I one or both o the plate sraces are in contact with a lid, part o the energy will leas into the adjacent lid. his attenated gide plat wave can be easily ond in many applications sch as biosensors or condctivity sensors [1~] and is called leay Lamb wave in order to distingish it rom the ree mode Lamb wave. In above applications, the wave gides are sally the piezoelectric plates sbjected to lid loadings. he gided plate waves lea energy into adjacent lids rom mechanical and electrical loadings throgh the mechanical and electrical copled bondary conditions respectively at the plate/lid interaces. he mechanical loading, also nown as mass loading, is rom the mass o the lid molecles loading on the plate srace. he electrical loadings are indced rom the electrical ields inside the piezoelectric plate copled into the electrical ields in the adjacent lids, which can be distingish in dielectric and condctivity loadings. In practice, all inds o these loadings will combine together and aect the wave propagation behaviors in two ways, phase velocity change and wave attenation. Since 198, Adler [4] and Nayeh [5] have been analyzed this problem with an approach nown as partial wave theory. However, only the lid mechanical loading eects are considered in Nayeh s research. Seqentially, Yang and Chimenti [6-8] analysis leay Lamb wave problems theoretically and experimentally. Both the mechanical and electrical (dielectric) loading eects are considers in their investigations. Frther, the same theory has been also adopted by Yang et al [9-11] or analyzing similar problems involving lid condctivity loading eects. In reviewing all previos wors, we ind that the lid electrical loading eects can strongly dominate the leay Lamb wave on piezoelectric plate in theoretical analysis [1, 11]. he conspicos case is the ndamental symmetry mode (S mode) Lamb wave propagating along 4 o azimthal angle o XZ-LiNbO waer [11]. Under certain conditions, the S mode Lamb wave velocity can be changed abot 9m/s [11] lid loading nder lid loadings. However, in previos calclations [6-11], the leay Lamb wave problems o piezoelectric plates are only partially solved since the wave attenation is missing. It is becase in previos wors [4-11], an incident acostic wave is always involved, which will vanish the wave attenations o leay Lamb wave and orce the wave nmber as a real nmber[12]. hereore, the exact soltions o partial wave theory are not completely given yet.

2 Besides, a variety o methods have been developed or leay Lamb wave measrement in previos wors sch as immersion-type dal-transdcer wave relection measrements [8, 9], plsed laser ltrasond techniqe with a 2D FF waveorm processing method [1, 11], and directly monted inter-digital transdcer (ID) on the piezoelectric plate or wave generating and receiving []. However, there are no convincing experimental measrements or the leay Lamb waves o piezoelectric plates to spport the theoretical reslts. his is becase the lid-loading eects can be very small and the experimental methods in previos wors may be inadeqate to detect them. hereore, a more precise and accrate measrement method is reqired to detect these small variations in the propagating characteristics o leay Lamb waves. Hence, in this paper, we will analysis the leay Lamb wave o piezoelectric plate sbjected to lid electrical loadings theoretically and experimentally. A 5µm thicness XZ-LiNbO waer will be applied as a sample plate and we will ocs on the S-mode leay Lamb wave propagates along 4 o azimthal angle. he sample plate will be immersed in the lid with dierent dielectric and condctivity loadings. In theoretical wors, we will re-visit the partial wave theory with the goal to establish an accrate way o calclating the characteristics o leay Lamb wave. Both the leay Lamb wave velocity and wave attenation inlenced by dielectric and condctivity lid loadings can be determined qantitatively. In experimental wors, a novel measrement system is developed to determine the lid loading eects with a dierential measrement concept. his new concept allows this system to measre the lid loading eects with high accracy. With this novel system, this paper tries to oer some convincing experimental measrements to spport the partial wave theory or not. 2. Analysis o analyze the leay Lamb waves, the partial wave theory is applied. he basic idea o partial wave theory is irst to reconstrct the ormla based on sperposition o homogeneos and inhomogeneos waves in the plate and the lid, then the bondary conditions at the plate/lid sraces are applied to solving the wave velocity and attenation o the leay Lame wave. his standard derivation can be ond in Res [5-11] as well as in a monograph by Nayeh [1], however, there is no incident wave involved in or approach Wave Fields in Piezoelectric Plate Consider an XZ-LiNbO plate having a thicness 2h and oriented along the Cartesian coordinate system as shown in Fig.1. In Fig.1, the (X, Y, Z) coordinates are the crystolograpical system and the wave propagation coordinates is (x 1 -x -x 2 ). he (x 1 - x ) plane is the sagittal plane o Lamb wave propagation and the coordinate x 1 indicates the wave propagating direction which is at an azimthal angle θ measred rom Y-axis direction.. x X x2 Figre 1: he coordinate systems o XZ-LiNbO plate. With the coordinate system shown in Fig.1, the constittive eqations or a piezoelectric material are, where ij = CijlSl eije (1) D = e S + ε E (2) ij ij i i ij is the stress tensor, θ Z Y x1 2h C ijl is the elastic constants, S l is the strain tensor, e ij is the piezoelectric constants, E is the electric ield vector, D is the electric displacement vector and ij i ε is the permittivity. For a dielectric material in qasi-static electrical ield assmption, the government eqation o the harmonic wave can be derived rom Eq.(1)and (2), C + e φ = ρ&&, () ijl lj, ij, j i il li, εiφ, i e =, (4) where is the displacement, φ is the electrical potential. hen assming the ormal soltions o the displacement and electrical potential, One can obtain the wave eqation o the φ and by solving the Christoel s eqation derived rom Eq.() and (4),

3 1 2 8 q q q q q iξ( x1+ α x) iωt. (5) q= 1 q (,,, φ) = ( U, V, W, Φ ) A e e where α and ( U, V, W, Φ ) (q=1~8) are the q q q q q eigenvales and eigenvectors o the Christoel s matrix. ξ is the complex wave nmber or the leay Lamb wave which can be expressed as, ω ξ = ( 1+ i γ) = ( 1+ i γ). (6) c where c and are the phase velocity and the real part o wave nmber along x 1 direction, respectively, and γ is a non-dimensional attenation coeicient. With Eq.(5), one can derive the stress and electrical displacement components as, q 2q q 4q q q= 1 i ξ( x 1+ α x ) i ω t q (,,, D ) = ( D, D, D, D ) A e e (7) 2.2. Mechanical Wave Fields in Inviscid Flid he wave eqations o an inviscid lid can be modeled as a ind o isotropic material rom the government eqations, ij = CijlSl. (8) Neglecting the shear modls, the wave eqations o the displacement and stress or an inviscid lid can be presented as, ( 1 2 ) 2,,, (1,, α, ρω / ξ) = i and l ( 1 2 ) iξ( x1+ α x) iωt F e e,( x >+ h) = α iρω ξ l 2,,, (1,,, / ) where (1) l iξ( x1+ α x) iωt l F e e, ( x < h). (9) ρ is the mass density o the lid and the sperscripts and l descript the pper and lower plate srace respectively. he α and α are the constant and have to satisy, α where =±. (11) ξ l 2, α ( ) 1 = ω is the wave nmber in the lid. c l 2.. Electrical Wave Fields in Inviscid Flid For a piezoelectric plate, the electrical potential in the plate can indce the electrical ield distrbance in the adjacent lid. his electrical ield o the lid E can be presented as, E = φ. (12), where φ is the electrical potential o the lid. Apparently the electrical potential φ is a harmonic wave having the same anglar reqency ω as the mechanical wave in the plate. hereore, the electrical potential and the electric displacement o the pper and lower srace lids can be represented as, (, 1 ) (1, ) x i x φ ξε and ξ ξ iωt D = Θ e e e (1) (, 1 ) l (1, ) x i x φ ξε l ξ ξ iωt D = Θ e e e (14) respectively. where D is the electric displacement in x direction and ε is the permittivity o the lid. I the lid is condctive, the permittivity can be replaced by a complex nmber ε, where * * ε ε = ε + i ( σ / ω ) (15) σ is the condctance o the lid Bondary conditions When the piezoelectric plate is immersed in a lid, there are three inds o bondary condition cases nder consideration, namely the dielectric, shorted and condctive loading srace bondary conditions [12, 14, 15]. In the dielectric case, the electrical condctivity o the adjacent lid is very small and can be neglected so that the eective lid permittivity in Eq.(15) is a real nmber. hereore, the only electrical loading rom the lid is it s permittivity. he opposite case o the dielectric type is the shorted case. Owing to the lid s high condctivity and its shielding eect, the electrical bondary conditions at the plate/lid interace are eectively shorted. Between these two extreme types is the condctive loading case. he condctivity is in a medim level so both the condctivity and dielectric loading can inlence the

4 plate. hereore, the eective lid permittivity in Eq.(15) is a complex nmber. With the mechanical loading in inviscid lid, the bondary conditions o dielectric, shorted and condctive loading cases are presented as ollows. First, or the dielectric inviscid lid case, the bondary conditions are =, =, 1 =, 2 =, φ = φ and D = D, which yields a linear algebra system in matrix orm 1 2 φ φ D D x =+ h = 1 2 φ φ D D x = h. (16) For the shorted case, the bondary conditions are =, =, 1 =, 2 = and φ = which yields a 1 1 linear algebra system, 1 2 φ. (17) x =+ h = 1 2 φ x = h For condctive loading case, the bondary conditions are =, =, 1 =, =, φ = φ and D = D, which yields a matrix orm, 1 2 φ φ D D x =+ h = 1 2 φ φ D D x = h. (18) For non-trivial soltions, the matrixes discssed in Eq.(16), (17) and (18) can be rther yield into the characteristic eqations o the leay Lamb wave dispersion relations. Given all material properties o the plate and lid and a propagation direction θ in Fig.1, one can irst choose a reqency ω and then search or the wave nmber ξ which will satisy the characteristic eqation. Owing that the wave nmber ξ in Eq.(6) is a complex nmber, both the real and image parts need to be determined simltaneosly. In this paper, a compter program in Matlab (he MathWors Inc., MA, USA) programming langage is developed to carry ot the 2D search o complex ξ or solving the characteristic eqation o leay Lamb wave.. Measrement Method For measring the lea Lamb wave eect, a dierential measrement system aiming at accrately determining the leay Lamb wave velocity change at chosen reqency is developed. he experiment Setp is shown in Fig.2 Figre 2: Schematic diagram o experiment setp. In Fig.2, a loading ixtre is designed and constrcted to mont both the XZ-LiNbO plate

5 nder measrement and the or laboratory-made line PZ transdcers. hese PZ transdcers are the ey elements, which consists o a long slender PZ bar with trapezoid cross-section. he length, height, widths in the bottom and top sraces are 12.7,.5, 1., and.2 mm, respectively. he PZ bar is pacaged with a copper bloc, tngsten/epoxy mixtre, and a metal case, as well as electrically connected and shielded. It can act as an acostic wave transmitter when applying electrical voltage to the PZ element, or an acostic wave receiver reversely. In this system, these or PZ transdcers have the same conigrations. wo o them (1 and 2) are or wave generation and the other two (R1 and R2) are or wave receiving. Fig. and Fig.4 show the explosive and composite diagram o one o these PZ transdcer. In order to generate Lamb waves at a chosen reqency, a tone-brst electrical system is constrcted. he original tone-brst signal is generated by a nction generator and ampliied by a RF ampliier. hen the tone-brst electrical signal is divided by a o power splitter into two signals which are in-phased and o eqal amplitde or exciting the transdcers 1 and 2 simltaneosly. Ater traveling or a certain distance, the plate waves are receiving by R1, R2 and ampliied by a preampliier. his signal is recorded by a digital oscilloscope. A personal compter controls and synchronizes the generation o the exciting tonebrst signal as well as the acqisition o the received signal waveorms rom the digital oscilloscope. his PC also connects to a linearstage or controlling the vertical position o the loading ixtre monted with sample plate and transdcers..1. Velocity Change Figre : Explosive diagram o line PZ transdcer. Figre 5: he conigrations o experimental measrement method (I). Figre 4: Composite diagram o line PZ transdcer. Dring the measrement, the 1 and 2 transdcers are carelly monted at the symmetrical positions on opposite sides o the XZ-LiNbO plate or generating the symmetrical modes o Lamb waves, thereore, the generated acostic waves can propagate in the plate along the direction perpendiclar to the long axes o 1 and 2 transdcers. he other two PZ transdcers, R1 and R2, are aligned parallel to those o 1 and transdcer so that the plate waves can be most eectively received. In the beginning o the measrement as shown in Fig.5, the sample plate with R1, R2 receivers are partially immersed in the lid. he distance between the transmitters and the receivers are L which is ixed dring the measrements. he distance between the receivers to the air/lid interace is x. Let τ 1 be the time-o-light o a plate wave traveling orm the transmitters to the receivers, one has, τ L x x o 1 = + (19) co c where o c and c are the wave velocities o the ree Lamb wave and the leay Lamb wave, respectively. o

6 x x LO 1 R1 2 R2 XZ-LiNbO Waer Line PZ ransdcer Lamb Wave Propagation Line PZ Receiver Figre 6: he conigrations o experimental measrement method (II). Sbseqently, moving the whole sample plate and transdcers assembly into the lid by a distance o x as shown in Fig.6, one can has the time-olight becomes, τ L x x x+ x o 2 = + (2) co c From Eq.(19) and Eq. (2), the dierence in timeo-light between Fig.5 and Fig.6 is, x x τ = τ2 τ1 = +. (21) c c Re-writing Eq.(2), one has, τ 1 c = + x co 1 o. (22) hereore, with given wave velocity co o ree Lamb wave, the leay Lamb wave velocity can be directly determined by measring the linear correlation between the change in time-o-light ( τ) and the change in distance ( x). his dierential measrement type is very sensitive to the wave velocity change and is less vlnerable to measrement errors and noises..2. Amplitde Attenation he measrement o wave attenation o leay Lamb wave shares the same principle as in the measrement o velocity change, and thereore ollows a similar measrement approach. he two basic assmptions needed to be made here are: (a) the wave attenation o ree Lamb wave is mch smaller than that o a leay Lamb wave and hence can be negligible; (b). the otpt signals o transdcer R1 are linearly proportional to the inpt acostic waves in the plate. Based on these two assmptions, the measrement o wave attenation actor can be easily obtained by, ( A2 A1 ) ln / 1 γ = x. (2) where A 1 and A2 can be considered as the wave amplitde measred in Fig.5 and Fig.6 respectively. γ is a non-dimensional attenation actor in Eq.(6). Again, Eq.(2) is a dierential type o measrement ocsing on the amplitde variation o received signal waveorms cased by dierent length o liqid-covered path. o be even more accrate in the measrement, one can tae a nmber o waveorms continosly when moving the sample plate into the lid over a wide range o x. 4. Condctivity Loading Experiment In analysis the leay Lamb wave sbjected to condctivity lid loadings, the water adding with dierent amont o salt is sed to apply the lid condctivity loadings. he condctivity o the salt soltion is continosly varying rom.1 S/ m to.5 S/ m and monitored by a condctivity sensing meter. Fig.7 (a) and (b) show the calclation reslt o the velocity changed and wave attenation o S mode leay Lamb waves sbjected to condctivity loadings. (a) (b) Attenation Factor, γ S/m.2 S/m.5 S/m 45.1 S/m.5 S/m Freqency,, MHz S/m.2 S/m.5 S/m.1 S/m.5 S/m Freqency,, MHz Figre 7: he loading eects o lid condctivity on the S mode o Leay Lamb

7 waves (a) the wave velocity and (b) the wave attenation actor In experiment, the condition settings are ollowing the theoretical analysis. Fig.8 and Fig.9 displays the measred velocities and attenation actor as nctions o water s condctivity at 2,, 4, and 5 MHz. (a) =2MHz Experimental 6 (b) =MHz 595 (c) =4MHz 6 (d) =5MHz 58 Figre 8: Condctive loading eects o the leay Lamb wave velocity, both theoretical and experimental data at (a) 2 MHz, (b) MHz, (c) 4 MHz, and (d) 5MHz.

8 (a) Attenation Factor, γ (b) Attenation Factor, γ (c) Attenation Factor, γ (d) Attenation Factor, γ =2MHz.5 =MHz =4MHz =5MHz Figre 9: Condctive loading eects o the leay Lamb wave attenation, both theoretical and experimental data at (a) 2 MHz, (b) MHz, (c) 4 MHz, and (d) 5MHz. in Fig.8. When the condctivity is varied rom.1 S/m to 1 S/m, the wave velocity continosly drops by abot 4 % (2MHz) to 6 % (5MHz). For the wave attenation, the vale o the attenation actor are irst increasing rapidly to a pea vale arond.12 S/m and then drop to a constant level. his phenomenon is de to electrically shielding eect. hogh there are some deviations in magnitde between the measred and theoretical reslt, this electrically shielding phenomenon still has been caght in or experiments. 5. Dielectric Loading Experiment In dielectric loading analysis, the ethanol/water soltions are sed to provide the varying dielectric loading eects. By varying the concentration o ethanol in water, the permittivity o the soltion can sbstantially be changed rom 8 (pre water) to 26 (pre ethanol) with varying other material properties (density, wave velocity) in an acceptable level. he theoretical calclations show that the lid s permittivity can have signiicant and dominate inlence on the wave velocities o leay Lamb waves [16]. hereore, the experiments abot dielectric loadings will ocs on the phase velocity change only and Fig.1 shows the experiment reslt. (a) (b) =2MHz Relative permittivity o Flid =MHz Relative permittivity o Flid For the wave velocities, good agreements between the experimental and theoretical data are observed

9 (c) =4MHz Experimental Relative permittivity o Flid distances between ree and leay waves. hereore, this system can avoid possible errors or ncertainties dring the measrements. Comparing with other experimental methods, this system shows its niqeness and strength in measring the characteristic o the leay Lamb wave. With this paper, the nmerical approaches and the measrement data cold be very sel in designing or analyzing new types o liqid acostic wave sensors based on leay Lamb waves. (d) =5MHz Relative permittivity o Flid Figre 1: Comparison between experimental and theoretical leay Lamb wave velocities at, (a) 2 MHz, (b) MHz, (c) 4 MHz, and (d) 5 MHz. he measred velocities o leay Lamb wave gradally decay when the relative permittivity o the lids is increasing rom 26 to 8. his experiment reslts qantitatively conorm the simlation reslts. hereore, the partial wave theory shows good agreements between experimental and theoretical data o dielectric loading eects. 6. Conclsion In this paper, the wave velocity changes and the wave attenation actors o leay Lamb waves indced by the lid dielectric and condctivity loading eects are sccesslly determined both theoretically and experimentally. Using a XZ- LiNbO piezoelectric plate as a sample plate, the partial wave theory with 2D search method provides a good prediction abot leay Lamb wave velocity changes in theoretical analysis. For the wave attenations, althogh the experimental reslts do not perectly match the theoretical predictions, bt the discrepancies are still in a small and acceptable ranges and the important eatres have been predicted sccesslly. Besides, the experiment system introdced in this paper has demonstrated its otstanding perormance in terms o good sensitivity and accracy. hat is becase the measrement method is designed and operated nder exactly the same mechanical and electrical conditions except or the relative traveling 7. Reerences [1] J. C. Andle, J.. Weaver, J. F. Vetelino, D. J. McAllister, Selective acostic plate mode DNA sensor, Sens. Actators B: Chem., vol. 24, n. 1-, pp , [2] J. C. Andle, J.. Weaver, D. J. McAllister, F. Josse, J. F. Vetelino, Improved acostic-plate-mode biosensor, Sens. Actators B: Chem., vol.1, n 1-, pp , 199. [] S. Liew., F. Josse, Haworth and Z. A. Shana, Applications o lithim niobate acostic plate mode as sensor or condctive liqids, IEEE Ultrason. Symp. Proc., pp , 199. [4] Nassar and E. L. Adler, Propagation and electromechanical copling to plate modes in piezoelectric composite membranes, IEEE Ultrason. Symp. Proc., pp.69-72, 198. [5] H. Nayeh and H.. Chien, he inlence o piezoelectricity on ree and relected waves rom lidloaded anisotropic plates, J. Acost. Soc. Am., vol. 91, n., pp , [6] C.-H. Yang and D. E. Chimenti, Acostic waves in a piezoelectric plate loaded by a dielectric lid, Appl. Phys. Lett. vol. 6, n.1, pp.128-1, 199. [7] C.-H. Yang and D. E. Chimenti, Gided plate waves in piezoelectric immersed in a dielectric lid. I. Analysis, J. Acost. Soc. Am., vol. 97, n. 4, pp , [8] C.-H. Yang and D. E. Chimenti, Gided plate waves in piezoelectric immersed in a dielectric lid. II. Experiments, J. Acost. Soc. Am., vol. 97, n. 4, pp , [9] C.-H. Yang and C. J. She, Gided waves propagating in a piezoelectric plated immersed in a condctive lid, ND&E International, vol. 4, n., pp , 21. [1] C.-H. Yang and M.-F. Hang, A stdy on the dispersion relations o Lamb waves propagating along LiNbO plate with a laser ltrasond techniqe, Jpn. J. Appl. Phys., vol. 41, n. 11A, pp , 22. [11] C. H. Yang and Y. A. Lai, Lamb waves propagating in an LiNbO plate nder the mechanical and dielectric loadings o a lid, Jpn. J. Appl. Phys., vol. 4, n. 4A, pp , 24. [12] Y. C. Lee and S. H. Ko, Leay Lamb wave o a piezoelectric plate sbjected to condctive lid loading: theoretical analysis and nmerical calclation, J. Appl. Phys. (Unpblished) [1] H. Nayeh, Wave Propagation in Layered Anisotropic Media: with Application to Composites, Netherlands, Amsterdam: Elsevier Science, 1995.

10 [14] Ald, Acostic Fields and Waves in Solids, 2nd Ed, Vol. I, Malabar, FL: Krieger, 199, Chap. 8. [15] J. J. Campbell and W. R. Jones, Propagation o srace waves at the bondary between a piezoelectric crystal and a lid medim, IEEE rans. Sonics and Ultrason., vol. SU-17, no. 2, pp.71-76, 197. [16] Y. C. Lee and S. H. Ko, Flid dielectric loading on leay Lamb wave o a piezoelectric plate Appl. Phys. Letter(Unpblished)