- Email: [email protected]
Acoustoelectric effect in piezocomposite sensors
Materials Science and Engineering C 21 (2002) 177 – 181 www.elsevier.com/locate/msec
Acoustoelectric effect in piezocomposite sensors F. Teston a,*, C. Chenu a, N. Felix b, M. Lethiecq a a
LUSSI/GIP Ultrasons, Rue de la Chocolaterie, BP 3410, 41034 Blois Cedex, France b VERMON S.A., Rue du Ge´ne´ral Renault, 37000 Tours, France
Abstract The acoustoelectric effect, given by conductive liquid loading, on the first symmetric mode of Lamb wave in piezoelectric composites was theoretically investigated. This S0 mode shows a low phase velocity and a high electromechanical coupling factor. Results obtained by first order perturbation theory are compared with those of a model based on the effective permittivity function of a multi-layer. The observed differences are being due to the high value of the eT dielectric constant and to the high coupling coefficient in the piezocomposites. The permittivity model predicts a nonzero attenuation, contrary to the first order perturbation calculations. It is shown that in the case of piezocomposites, only the permittivity model takes into account the strong interaction between the acoustic wave and the liquid. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Acoustic sensors; Acoustoelectric effect; Effective permittivity; Conductivity of liquids
1. Introduction In the past years, considerable attention has focused on the use of acoustic devices for gas or liquid sensor applications for chemistry or biology. In general, the response of these sensors is linked to mass loading, viscoelastic stiffening and for electrical property variations. The majority of these acoustic sensors have employed bulk shear wave (Quartz Crystal Microbalance) or surface acoustic waves (SAW) [1]. However, other acoustic modes, which propagate in plate thickness, have been studied in order to achieve higher sensitivities or to isolate the interdigited electrodes from the liquid. All these Acoustic Plate Mode devices consist of a delay line made of two interdigital transducers. Among these modes, the Lamb waves are particularly interesting. Today, only the first lowest antisymmetric mode A0 has been used, due to a lower phase velocity than the sound velocity in liquids, which minimizes the radiation loss. Moreover, very thin plates with high amplitude displacement can then be used, which leads to a high coupling between the plate and an adjacent layer. Typically, such devices are fabricated on a substrate such as lithium niobate, zinc oxide or aluminum *
Corresponding author. Tel.: +33-2-54-55-84-33; fax: +33-2-54-5584-45. E-mail address: [email protected] (F. Teston).
nitride. Recent works have proposed ceramics substrates which can bring an improvement on the piezoelectric coupling factor [2 –4]. In the previous paper, authors have shown the characteristics of Lamb wave propagation in 1 – 3 connectivity piezoelectric composites [5]. These materials have been widely used in medical ultrasonic probes because of their high conversion efficiency for what and relative low acoustic impedance. Many methods have been reported to calculate the effective properties of these composites considered as a homogeneous material. In these materials, the first symmetric Lamb wave S0 shows a low phase velocity (Vp < 1800 m s 1) and a very high electromechanical coupling factor (K > 50%), which are expected to produce significant improvements in Lamb wave sensors performance. During the last decade, several studies have been devoted to sensors using the interaction between an acoustic wave propagating in a piezoelectric plate and the electrical properties of a liquid [6– 9] or conductive layer in contact with the substrate [10] (i.e. acoustoelectric interactions). The objective of this paper is to investigate the theoretical characteristics and the interest of S0 Lamb wave sensors on piezocomposite substrates in which the acoustoelectric interaction is used to characterize the relative permittivity and the conductivity of liquids.
0928-4931/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 8 - 4 9 3 1 ( 0 2 ) 0 0 0 9 9 - 1
178
F. Teston et al. / Materials Science and Engineering C 21 (2002) 177–181
2. Theoretical results 2.1. Dispersion relation The propagation characteristics of a Lamb wave propagating along a solid/liquid interface can be analyzed numerically by extending the effective permittivity function. The geometry of the problem under consideration assumes that the acoustic waves propagate along the x1 direction in a piezoelectric plate of thickness h and decays in the opposite x3 direction. The plate is bordered by a liquid at x3 = h, and the interface is electrically free. The mechanical boundary conditions are assumed to be unperturbed; only the electrical boundary conditions are perturbed at the upper surface. In solving the problem, the liquid can thus be treated as a lossy dielectric materials characterized by a complex permittivity e = eV j(r/x), with eV the dielectric constant and r the conductivity. For pure water, eV= 79.5e0 and r = 0. Mass loading and viscosity effects being neglected, only conductivity and dielectric permittivity are considered as variables in our calculations. Thus, using the effective permittivity concept, the velocity and the attenuation of the Lamb modes were calculated. Fig. 1 shows the phase velocity and normalized attenuation of the S0 mode, for a plate loaded by liquid, over the normalized thickness. The influence of the liquid conductivity on the propagation characteristics was evaluated for two values of the conductivity: r = 0 mS cm 1 (pure water) and 10 1 mS cm 1 (arbitrary conductivity). Fig. 2 shows the longitudinal (u3) and transverse (u1) particle displacements of the S0 mode versus the normalized thickness for two cases: plate with or without liquid loading. From these figures, it can be seen that for low values of normalized thickness, the difference between the velocities is not significant, the phase velocities are higher than the sound velocity in water and the attenuation is moderate. For example, the attenuation is 3 10 2 dB/k for h/k = 0.1. This is due to a dominant component of the wave in the propagation direction (u1) with high amplitude. As the normalized thickness increases, u 1 decreases and u 3 increases which is confirmed by increasing attenuation. The energy of Lamb waves can only be coupled into the liquid through the u3 component of the displacement at the plate surface, which explains why more energy is radiated into the liquid when the normalized thickness increases. A logarithm dependence is observed for the attenuation versus normalized thickness. A maximum attenuation is observed when the phase velocity is near 1500 m s 1, and its value is 3.89 dB/k (Fig. 1). When normalized thickness is just over 1.2, the phase velocity of the mode with pure water loading is lower than that of the liquid, which causes a sharp decrease of the attenuation. It is interesting to note also in Fig. 2 that the amplitude displacement, as well as u1 that u3, is not perturbed by liquid loading. So a Rayleigh wave could propagate in piezocomposite thicker plate bordered by water without consequent attenuation.
Fig. 1. Phase velocity and normalized attenuation of the S0 Lamb wave propagating in a piezocomposite plate (Vf = 56%) in contact with pure water (solid line) and a liquid with a conductivity of 10 1 mS cm 1 (dashed line).
In Fig. 1, it can be seen that the variations of the liquid conductivity affect the propagation characteristics of the wave. The situation is characterized by significant increase of attenuation without significant change of velocity. For h/k < 1.2, attenuation has the same behavior for the two liquid but with a higher value in the case of a conductive liquid. When h/k>1.2, attenuation is still significant, and cannot be explained by radiation into the liquid but solely to the acoustoelectric effect since propagation velocity is lower than sound velocity in the liquid. Surface waves then propagates at the liquid/surface interface, which leads to higher sensitivity with regard to dielectric perturbation. 2.2. Acoustoelectric interaction In this paper, two approaches have been considered to predict the sensor response to acoustoelectric perturbation, such as a dielectric property changes in a liquid: a complete model based on the effective permittivity function, previously described, which includes all the boundary conditions
F. Teston et al. / Materials Science and Engineering C 21 (2002) 177–181
179
electromechanical coupling factor of the considered acoustic mode, K2, effective dielectric constant of the piezoelectric substrate, es, liquid conductivity, r, and liquid permittivity, el. It is important to note that the perturbation theory involves the dielectric tensor eijT (not eijS) of the substrate. This is due to the assumption that the intensive variable stresses Tij remain constant during the perturbation. Contrary to the permittivity model, the constitutive equations for the acoustic and electromagnetic fields are as follows: Dij ¼ ejkl Ul; jk eSjk /; jk
ð3Þ
E Tij ¼ Cijkl Ul; jk þ ekij /; jk
ð4Þ
where Ui is the particle displacement and / is the electric potential in the piezoelectric plate. In order to calculate the electromechanical coupling factor K2 for the acoustic mode, a definition of this coefficient widely used for the SAW mode, is given by equation: K2 ¼ 2
Fig. 2. Normalized displacements of the S0 Lamb wave propagating in a piezocomposite versus normalized thickness in plate without liquid loading (solid line) and in the case of plate bordered with pure water (dashed line).
at the two interfaces without restrictions, and a simplified model where only the material properties of the perturbing layer are taken into account. The latter is based on a first order perturbation method and therefore has a limited range of validity (i.e. only for small perturbations). In the case of an acoustoelectric perturbation, this problem has been treated previously [7,8] and the relations are as follow: DV 1 es r2 ¼ K2 V 2 es þ el r2 þ x2 ðes þ el Þ2
ð1Þ
and 1 es xrðes þ el Þ c ¼ K2 2 2 es þ el r þ x2 ðes þ el Þ2
ð2Þ
Eqs. (1) and (2) give the relative phase velocity variation and the normalized attenuation by wavelength in terms of
DV Vm Vf ¼ m f f V Vf
ð5Þ
Here, DV corresponds to the phase velocity variation between two different electrical conditions at the plate surfaces. In the case of a finite thickness plate, two electrical boundary conditions are considered: both surfaces metalf lized (V m m ) and both surfaces free (V f ) [11– 13]. Fig. 3 shows the relative frequency variations and the normalized attenuation variations versus the liquid conductivity obtained from the rigorous theoretical analysis based on the effective permittivity function and the approximate perturbation model. These variations are calculated considering that the reference state is the plate bordered by pure water. For the calculations, a value of 0.3 for the normalized thickness has been used, from which a value of 0.36% for the electromechanical coupling K has been calculated with Eq. (5). For the value of the plate permittivity es in Eqs. (1) and (2), the following definition has been used: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2 eT11 e T T es ¼ e33 T 23 e33 eT33
ð6Þ
For example, the homogenized values in the composite T T material are e11 = 8.85e0 and e33 = 2778e0, which lead to a eT value of 156e0. This value also corresponds to the effective permittivity at infinite frequency. A significant deviation is observed for the relative frequency variations as well as for the normalized attenuation variations, between the results obtained by the permittivity model and the perturbation method. As demonstrated, the relative frequency variations increase when the liquid conductivity increases. With the permittivity model, the maximum occurs for a liquid conductivity of approximately 102 mS cm 1 and the relative frequency variation is then
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F. Teston et al. / Materials Science and Engineering C 21 (2002) 177–181
However, for materials with a high dielectric constant, the perturbation approach cannot be used and only models taking into account the strong interaction at the plate/liquid interface can be used. It is important to note that the normalized attenuation variations calculated with the permittivity model does not tend towards zero for high values of conductivity, but to a finite value of 0.4 dB/k. Finally, in order to adjust all curves for the two theoretical approach, a fitting procedure on the (eT, K) couple is attempted. It is observed that no couple of values allows to fit both relative frequency and normalized attenuation variations. The best fitting is obtained for the values eT = 1750e0 and K2 = 37.7% for relative frequency variations, whereas eT = 1550e0 and K2 = 38.8% give the best fit for normalized attenuation variations. A comparison for the last case is shown in Fig. 4, on which a good agreement is observed for low conductivities only. The values obtained by Eq. (5) and by fitting are very slightly different, the discrepancy may be
Fig. 3. Relative frequency shift and normalized attenuation versus conductivity calculated by the permittivity model (solid line) and by the perturbation method (dashed line).
6.3 104 ppm. The perturbation method yields lower values, respectively, 4 mS cm 1 and 2.7 104 ppm. The normalized attenuation variations have Gaussian-like behaviors for the two models. However, the peaks of the normalized attenuation curves are not localized at the same values. They occur at 0.3 mS cm 1 (with a peak value of 0.8 dB/k) for the perturbation approach and 2 mS cm 1 (with a peak value of 1.9 dB/k) for the permittivity model. As shown previously by Zaitsev et al. [9], in order to make correct use of the perturbation approach, it is necessary that the substrate and liquid have similar permittivity tensors and that the substrate has a low electromechanical coupling factor. At the present time in the literature, the perturbation approach is only used to confirm experimental results for materials such as ST-cut quartz (SH-APM) [14,6], ZX-LiNbO3 (APM waves) [8,10] and 36YX LiTaO3 (SH-SAW) [7], which all have a relatively low dielectric constant and a weak electromechanical coupling factor. In these cases, a good agreement is observed.
Fig. 4. Relative frequency shift and normalized attenuation versus conductivity calculated by the permittivity model (solid line) and by the perturbation method (circles).
F. Teston et al. / Materials Science and Engineering C 21 (2002) 177–181
due to the limited range of validity of Eq. (5) (i.e. for weak coupling only). The values of the dielectric constant determined by fitting fall in the range of 1550– 1750e0, which is much higher than that calculated by Eq. (6). The difference may be due to the fact that the components of dielectric permittivity at constant stress, eTij, were not measured but were calculated from homogenized constants (i.e. elastic E stiffness, Cijkl , and dielectric permittivity at constant strain, S eij ). Only part of the homogenized constants have been verified experimentally to this date.
3. Conclusion We have demonstrated the interest of using of first symmetric Lamb wave in piezocomposite plates as liquid sensors. Although the phase velocity of the S0 mode is slightly higher than that of sound in water, the acoustic wave can propagate without significant attenuation in the plate bordered by liquids. This is due to the fact that the main component of the particle displacement is in the propagation direction. The acoustoelectric interaction has been investigated in these piezocomposite materials for the S0 mode. The normalized attenuation and the relative frequency variations have been calculated for a plate bordered by a liquid with different conductivity values using two models: an approach based on the first order perturbation theory and a model based on the effective permittivity function. We have identified some limitations in the use of the perturbation method. First, the knowledge of the exact values of the dielectric constant eT and the electromechanical coupling factor K in the piezocomposite are problems that are still not solved; secondly, the perturbation method is limited to low acoustoelectric interactions.
Acknowledgements The authors thank Vermon for the manufacturing of the composite materials. This research was supported by the
181
Action Concerte´e Incitative en Biotechnologie of the French Ministery of Research.
References [1] D.S. Ballantine, et al., in: R. Stern, M. Levy (Eds.), Acoustic Wave Sensors: Theory, Design, and Physico-Chemical Applications, Academic press, San Diego, 1997, pp. 70 – 99. [2] G. Sugizaki, et al., Propagation characteristics of Lamb waves in PbZrO3 based ceramic substrate, Japanese Journal of Applied Physics 31 (1992) 3048 – 3050. [3] N. Kaewkamnerd, et al., Material characterization of (Bi1/2Na1/2)0.94 Ba0.06TiO3 ceramic as lamb wave device substrate, Sensors and Materials 9 (1) (1997) 47 – 55. [4] P. Luginbuhl, et al., Microfabricated lamb wave device based on PZT sol gel thin film for mechanical transport of solid particles and liquids, Journal of Microelectromechanical systems 6 (4) (1997) 337 – 345. [5] F. Teston, et al., Propagation of lamb waves in 1 – 3 piezocomposites and their application in liquid sensors, Ferroelectrics 224 (1999) 13 – 20. [6] S.J. Martin, et al., Characterization of SH acoustic plate mode liquid sensors, Sensors and Actuators 20 (1989) 253 – 268. [7] J. Kondoh, S. Shiokawa, Shear surface acoustic wave liquid sensor based on acoustoelectric interaction, Electronics and Communications in Japan 78 (1) (1995) 338 – 347. [8] F. Josse, Z.A. Shana, Electrical surface perturbation of a piezoelectric acoustic plate mode by conductive liquid loading, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 39 (4) (1992) 512 – 518. [9] B.D. Zaitsev, et al., Shear horizontal Acoustic waves in piezoelectric plates bordered with conductive liquid, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 48 (2) (2001) 627 – 631. [10] B.D. Zaitsev, et al., Influence of conducting layer and conducting electrode on acoustic waves propagating in potassium niobate plates, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 48 (2) (2001) 624 – 626. [11] K.A. Ingebrigsten, Surface waves in piezoelectrics, Journal of Applied Physics 40 (7) (1969) 2681 – 2686. [12] S.G. Joshi, Y. Jin, Electromechanical coupling coefficients of ultrasonic lamb waves, Journal of the Acoustical Society of America 94 (1) (1993) 261 – 267. [13] F. Teston, et al., Characteristics of lamb waves in piezoelectric ceramics, IEEE International Ultrasonics Symposium, 1997, Toronto. [14] T.M. Niemczyk, et al., Acoustoeletric interaction of plates modes with solutions, Journal of Applied Physics 64 (10) (1988) 5002 – 5008.
Acoustoelectric effect in piezocomposite sensors F. Teston a,*, C. Chenu a, N. Felix b, M. Lethiecq a a
LUSSI/GIP Ultrasons, Rue de la Chocolaterie, BP 3410, 41034 Blois Cedex, France b VERMON S.A., Rue du Ge´ne´ral Renault, 37000 Tours, France
Abstract The acoustoelectric effect, given by conductive liquid loading, on the first symmetric mode of Lamb wave in piezoelectric composites was theoretically investigated. This S0 mode shows a low phase velocity and a high electromechanical coupling factor. Results obtained by first order perturbation theory are compared with those of a model based on the effective permittivity function of a multi-layer. The observed differences are being due to the high value of the eT dielectric constant and to the high coupling coefficient in the piezocomposites. The permittivity model predicts a nonzero attenuation, contrary to the first order perturbation calculations. It is shown that in the case of piezocomposites, only the permittivity model takes into account the strong interaction between the acoustic wave and the liquid. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Acoustic sensors; Acoustoelectric effect; Effective permittivity; Conductivity of liquids
1. Introduction In the past years, considerable attention has focused on the use of acoustic devices for gas or liquid sensor applications for chemistry or biology. In general, the response of these sensors is linked to mass loading, viscoelastic stiffening and for electrical property variations. The majority of these acoustic sensors have employed bulk shear wave (Quartz Crystal Microbalance) or surface acoustic waves (SAW) [1]. However, other acoustic modes, which propagate in plate thickness, have been studied in order to achieve higher sensitivities or to isolate the interdigited electrodes from the liquid. All these Acoustic Plate Mode devices consist of a delay line made of two interdigital transducers. Among these modes, the Lamb waves are particularly interesting. Today, only the first lowest antisymmetric mode A0 has been used, due to a lower phase velocity than the sound velocity in liquids, which minimizes the radiation loss. Moreover, very thin plates with high amplitude displacement can then be used, which leads to a high coupling between the plate and an adjacent layer. Typically, such devices are fabricated on a substrate such as lithium niobate, zinc oxide or aluminum *
Corresponding author. Tel.: +33-2-54-55-84-33; fax: +33-2-54-5584-45. E-mail address: [email protected] (F. Teston).
nitride. Recent works have proposed ceramics substrates which can bring an improvement on the piezoelectric coupling factor [2 –4]. In the previous paper, authors have shown the characteristics of Lamb wave propagation in 1 – 3 connectivity piezoelectric composites [5]. These materials have been widely used in medical ultrasonic probes because of their high conversion efficiency for what and relative low acoustic impedance. Many methods have been reported to calculate the effective properties of these composites considered as a homogeneous material. In these materials, the first symmetric Lamb wave S0 shows a low phase velocity (Vp < 1800 m s 1) and a very high electromechanical coupling factor (K > 50%), which are expected to produce significant improvements in Lamb wave sensors performance. During the last decade, several studies have been devoted to sensors using the interaction between an acoustic wave propagating in a piezoelectric plate and the electrical properties of a liquid [6– 9] or conductive layer in contact with the substrate [10] (i.e. acoustoelectric interactions). The objective of this paper is to investigate the theoretical characteristics and the interest of S0 Lamb wave sensors on piezocomposite substrates in which the acoustoelectric interaction is used to characterize the relative permittivity and the conductivity of liquids.
0928-4931/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 8 - 4 9 3 1 ( 0 2 ) 0 0 0 9 9 - 1
178
F. Teston et al. / Materials Science and Engineering C 21 (2002) 177–181
2. Theoretical results 2.1. Dispersion relation The propagation characteristics of a Lamb wave propagating along a solid/liquid interface can be analyzed numerically by extending the effective permittivity function. The geometry of the problem under consideration assumes that the acoustic waves propagate along the x1 direction in a piezoelectric plate of thickness h and decays in the opposite x3 direction. The plate is bordered by a liquid at x3 = h, and the interface is electrically free. The mechanical boundary conditions are assumed to be unperturbed; only the electrical boundary conditions are perturbed at the upper surface. In solving the problem, the liquid can thus be treated as a lossy dielectric materials characterized by a complex permittivity e = eV j(r/x), with eV the dielectric constant and r the conductivity. For pure water, eV= 79.5e0 and r = 0. Mass loading and viscosity effects being neglected, only conductivity and dielectric permittivity are considered as variables in our calculations. Thus, using the effective permittivity concept, the velocity and the attenuation of the Lamb modes were calculated. Fig. 1 shows the phase velocity and normalized attenuation of the S0 mode, for a plate loaded by liquid, over the normalized thickness. The influence of the liquid conductivity on the propagation characteristics was evaluated for two values of the conductivity: r = 0 mS cm 1 (pure water) and 10 1 mS cm 1 (arbitrary conductivity). Fig. 2 shows the longitudinal (u3) and transverse (u1) particle displacements of the S0 mode versus the normalized thickness for two cases: plate with or without liquid loading. From these figures, it can be seen that for low values of normalized thickness, the difference between the velocities is not significant, the phase velocities are higher than the sound velocity in water and the attenuation is moderate. For example, the attenuation is 3 10 2 dB/k for h/k = 0.1. This is due to a dominant component of the wave in the propagation direction (u1) with high amplitude. As the normalized thickness increases, u 1 decreases and u 3 increases which is confirmed by increasing attenuation. The energy of Lamb waves can only be coupled into the liquid through the u3 component of the displacement at the plate surface, which explains why more energy is radiated into the liquid when the normalized thickness increases. A logarithm dependence is observed for the attenuation versus normalized thickness. A maximum attenuation is observed when the phase velocity is near 1500 m s 1, and its value is 3.89 dB/k (Fig. 1). When normalized thickness is just over 1.2, the phase velocity of the mode with pure water loading is lower than that of the liquid, which causes a sharp decrease of the attenuation. It is interesting to note also in Fig. 2 that the amplitude displacement, as well as u1 that u3, is not perturbed by liquid loading. So a Rayleigh wave could propagate in piezocomposite thicker plate bordered by water without consequent attenuation.
Fig. 1. Phase velocity and normalized attenuation of the S0 Lamb wave propagating in a piezocomposite plate (Vf = 56%) in contact with pure water (solid line) and a liquid with a conductivity of 10 1 mS cm 1 (dashed line).
In Fig. 1, it can be seen that the variations of the liquid conductivity affect the propagation characteristics of the wave. The situation is characterized by significant increase of attenuation without significant change of velocity. For h/k < 1.2, attenuation has the same behavior for the two liquid but with a higher value in the case of a conductive liquid. When h/k>1.2, attenuation is still significant, and cannot be explained by radiation into the liquid but solely to the acoustoelectric effect since propagation velocity is lower than sound velocity in the liquid. Surface waves then propagates at the liquid/surface interface, which leads to higher sensitivity with regard to dielectric perturbation. 2.2. Acoustoelectric interaction In this paper, two approaches have been considered to predict the sensor response to acoustoelectric perturbation, such as a dielectric property changes in a liquid: a complete model based on the effective permittivity function, previously described, which includes all the boundary conditions
F. Teston et al. / Materials Science and Engineering C 21 (2002) 177–181
179
electromechanical coupling factor of the considered acoustic mode, K2, effective dielectric constant of the piezoelectric substrate, es, liquid conductivity, r, and liquid permittivity, el. It is important to note that the perturbation theory involves the dielectric tensor eijT (not eijS) of the substrate. This is due to the assumption that the intensive variable stresses Tij remain constant during the perturbation. Contrary to the permittivity model, the constitutive equations for the acoustic and electromagnetic fields are as follows: Dij ¼ ejkl Ul; jk eSjk /; jk
ð3Þ
E Tij ¼ Cijkl Ul; jk þ ekij /; jk
ð4Þ
where Ui is the particle displacement and / is the electric potential in the piezoelectric plate. In order to calculate the electromechanical coupling factor K2 for the acoustic mode, a definition of this coefficient widely used for the SAW mode, is given by equation: K2 ¼ 2
Fig. 2. Normalized displacements of the S0 Lamb wave propagating in a piezocomposite versus normalized thickness in plate without liquid loading (solid line) and in the case of plate bordered with pure water (dashed line).
at the two interfaces without restrictions, and a simplified model where only the material properties of the perturbing layer are taken into account. The latter is based on a first order perturbation method and therefore has a limited range of validity (i.e. only for small perturbations). In the case of an acoustoelectric perturbation, this problem has been treated previously [7,8] and the relations are as follow: DV 1 es r2 ¼ K2 V 2 es þ el r2 þ x2 ðes þ el Þ2
ð1Þ
and 1 es xrðes þ el Þ c ¼ K2 2 2 es þ el r þ x2 ðes þ el Þ2
ð2Þ
Eqs. (1) and (2) give the relative phase velocity variation and the normalized attenuation by wavelength in terms of
DV Vm Vf ¼ m f f V Vf
ð5Þ
Here, DV corresponds to the phase velocity variation between two different electrical conditions at the plate surfaces. In the case of a finite thickness plate, two electrical boundary conditions are considered: both surfaces metalf lized (V m m ) and both surfaces free (V f ) [11– 13]. Fig. 3 shows the relative frequency variations and the normalized attenuation variations versus the liquid conductivity obtained from the rigorous theoretical analysis based on the effective permittivity function and the approximate perturbation model. These variations are calculated considering that the reference state is the plate bordered by pure water. For the calculations, a value of 0.3 for the normalized thickness has been used, from which a value of 0.36% for the electromechanical coupling K has been calculated with Eq. (5). For the value of the plate permittivity es in Eqs. (1) and (2), the following definition has been used: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2 eT11 e T T es ¼ e33 T 23 e33 eT33
ð6Þ
For example, the homogenized values in the composite T T material are e11 = 8.85e0 and e33 = 2778e0, which lead to a eT value of 156e0. This value also corresponds to the effective permittivity at infinite frequency. A significant deviation is observed for the relative frequency variations as well as for the normalized attenuation variations, between the results obtained by the permittivity model and the perturbation method. As demonstrated, the relative frequency variations increase when the liquid conductivity increases. With the permittivity model, the maximum occurs for a liquid conductivity of approximately 102 mS cm 1 and the relative frequency variation is then
180
F. Teston et al. / Materials Science and Engineering C 21 (2002) 177–181
However, for materials with a high dielectric constant, the perturbation approach cannot be used and only models taking into account the strong interaction at the plate/liquid interface can be used. It is important to note that the normalized attenuation variations calculated with the permittivity model does not tend towards zero for high values of conductivity, but to a finite value of 0.4 dB/k. Finally, in order to adjust all curves for the two theoretical approach, a fitting procedure on the (eT, K) couple is attempted. It is observed that no couple of values allows to fit both relative frequency and normalized attenuation variations. The best fitting is obtained for the values eT = 1750e0 and K2 = 37.7% for relative frequency variations, whereas eT = 1550e0 and K2 = 38.8% give the best fit for normalized attenuation variations. A comparison for the last case is shown in Fig. 4, on which a good agreement is observed for low conductivities only. The values obtained by Eq. (5) and by fitting are very slightly different, the discrepancy may be
Fig. 3. Relative frequency shift and normalized attenuation versus conductivity calculated by the permittivity model (solid line) and by the perturbation method (dashed line).
6.3 104 ppm. The perturbation method yields lower values, respectively, 4 mS cm 1 and 2.7 104 ppm. The normalized attenuation variations have Gaussian-like behaviors for the two models. However, the peaks of the normalized attenuation curves are not localized at the same values. They occur at 0.3 mS cm 1 (with a peak value of 0.8 dB/k) for the perturbation approach and 2 mS cm 1 (with a peak value of 1.9 dB/k) for the permittivity model. As shown previously by Zaitsev et al. [9], in order to make correct use of the perturbation approach, it is necessary that the substrate and liquid have similar permittivity tensors and that the substrate has a low electromechanical coupling factor. At the present time in the literature, the perturbation approach is only used to confirm experimental results for materials such as ST-cut quartz (SH-APM) [14,6], ZX-LiNbO3 (APM waves) [8,10] and 36YX LiTaO3 (SH-SAW) [7], which all have a relatively low dielectric constant and a weak electromechanical coupling factor. In these cases, a good agreement is observed.
Fig. 4. Relative frequency shift and normalized attenuation versus conductivity calculated by the permittivity model (solid line) and by the perturbation method (circles).
F. Teston et al. / Materials Science and Engineering C 21 (2002) 177–181
due to the limited range of validity of Eq. (5) (i.e. for weak coupling only). The values of the dielectric constant determined by fitting fall in the range of 1550– 1750e0, which is much higher than that calculated by Eq. (6). The difference may be due to the fact that the components of dielectric permittivity at constant stress, eTij, were not measured but were calculated from homogenized constants (i.e. elastic E stiffness, Cijkl , and dielectric permittivity at constant strain, S eij ). Only part of the homogenized constants have been verified experimentally to this date.
3. Conclusion We have demonstrated the interest of using of first symmetric Lamb wave in piezocomposite plates as liquid sensors. Although the phase velocity of the S0 mode is slightly higher than that of sound in water, the acoustic wave can propagate without significant attenuation in the plate bordered by liquids. This is due to the fact that the main component of the particle displacement is in the propagation direction. The acoustoelectric interaction has been investigated in these piezocomposite materials for the S0 mode. The normalized attenuation and the relative frequency variations have been calculated for a plate bordered by a liquid with different conductivity values using two models: an approach based on the first order perturbation theory and a model based on the effective permittivity function. We have identified some limitations in the use of the perturbation method. First, the knowledge of the exact values of the dielectric constant eT and the electromechanical coupling factor K in the piezocomposite are problems that are still not solved; secondly, the perturbation method is limited to low acoustoelectric interactions.
Acknowledgements The authors thank Vermon for the manufacturing of the composite materials. This research was supported by the
181
Action Concerte´e Incitative en Biotechnologie of the French Ministery of Research.
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