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Evaluation of electromechanical coupling factor for a piezoelectric quartz crystal in liquid phase
Analytica Chimica Acta 419 (2000) 251–254
Evaluation of electromechanical coupling factor for a piezoelectric quartz crystal in liquid phase Anhong Zhou, Qingji Xie, Yu Yuan, Youyu Zhang, Shouzhuo Yao∗ Chemical Research Institute, Hunan Normal University, Changsha 410081, PR China Received 30 July 1999; received in revised form 7 March 2000; accepted 30 March 2000
Abstract Based on the Butterworth–Van Dyke (BVD) equivalent circuit model for a quartz crystal, the general equations upon the series resonant-frequency fs , and parallel resonant-frequency fp , were obtained by solving the quadratic equation of frequency at susceptance B=0. Consequently, the electromechanical coupling factor (K) in liquid was first evaluated by using fs and fp . In sucrose solutions, it was seen that K2 decreases as (ρη)1/2 increases, where ρ and η were the density and viscosity of the solution, respectively. It was also interesting to find that K2 linearly correlates with (ρη)1/2 of the media. The regression equation was K2 =−1.216×10−6 (ρη)1/2 +5.204×10−6 , correlation coefficient r2 =0.9954. Regardless of unknown reason for this linear relationship, it was evident that the values of K2 showed apparent difference when a quartz crystal worked in different media. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Electromechanical coupling factor; Piezoelectric quartz crystal
1. Introduction Piezoelectric quartz crystal impedance analysis technique has been widely applied to interfacial science for its ability to offer multiple chemical information [1,2]. It is well-known that the QCM oscillator functions as an electromechanical transducer, and the value of electromechanical coupling factor K, is obviously very important to characterize resonance of a QCM, however, little attention has been paid to evaluation of this parameter in liquid. Resonance of an unperturbed QCM can be represented by the Butterworth–Van Dyke (BVD) equivalent circuit model [3]. The electrical behavior of a ∗ Corresponding author. Fax: +86-731-8865515. E-mail address: [email protected] (S. Yao).
QCM represented by BVD model in a Newtonian liquid has been examined by impedance analysis method [4]. The admittance Y, for the BVD model can be written as a complex of conductance (G) and susceptance (B) ! U R1 + j ωC0 − 2 (1) Y = G + jB = 2 R1 + U 2 R1 + U 2 where angular√ frequency ω=2πf, U = ωL1 − (1/ωC1 ), j = −1. As for the four equivalent circuit parameters, C0 is the static capacitance, R1 , L1 , C1 are the motional resistance, motional inductance, and motional capacitance, respectively. The series resonant-frequency fs , and parallel resonant-frequency fp , can be obtained by solving the quadratic equation of frequency at B=0.
0003-2670/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 3 - 2 6 7 0 ( 0 0 ) 0 0 9 1 8 - 1
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A. Zhou et al. / Analytica Chimica Acta 419 (2000) 251–254
ωs2 = (2πfs )2
Z=
L1 − C0 R12 + 2L1 C0 /C1 −(C02 R14 + L21 =
−2L1 C0 R12 − 4L1 C02 R12 /C1 )1/2 2C0 L21
(2)
ωp2 = (2πfp )2 L1 − C0 R12 + 2L1 C0 /C1 +(C02 R14 + L21 =
−2L1 C0 R12 − 4L1 C02 R12 /C1 )1/2 2C0 L21
(3)
It should be noted that the forms of these two equations depend on the electrical equivalent circuit model of a quartz crystal. For a QCM in air, the value of R1 is very small and can be neglected (for a 9 MHz crystal used in the present work, R1 ≈5 in air, R1 ≈240 in pure water). Therefore, Eqs. (2) and (3) would be respectively reduced to the equations of fs and fp reported for a QCM resonator in vacuum [3]: 1/2 1 1 (4) fs = 2π L1 C1 1/2 1 1 1 + (5) fp = 2π L1 C1 L1 C0 Additionally, the impedance analyzer can offer resonant frequency f0 , at which G reaches its maximum value [4–6] 1/2 1 1 (6) f0 = 2π L1 C1 It is clear that f0 is virtually equal to fs (as shown by Eq. (4)) for a lossless crystal. However, for a crystal in liquid, these two frequency values show apparent difference since fs would be obtained from Eq. (2) in this case. On the other hand, for a lossless quartz resonating 2 /(c ε ), where K in ideal mode, K 2 = K02 ≡ e26 0 66 22 is the electromechanical coupling factor of a lossless quartz crystal (K0 =7.899×10−3 ), c66 the ‘piezoelectrically stiffened’ quartz elastic constant, e26 and ε22 are the piezoelectric stress constant and permitivity of the quartz, respectively. For a piezoelectric quartz crystal operated as a thickness-shear mode resonator, the impedance Z is given by [7,8]
1 j2π C0
π fs 2 fp 2 K tan 1− π fs 2 fp
(7)
For a crystal resonating at f = f0 (i.e. Z = 0), we have π fs π fp − fs 2 tan (8) K = 2 fp 2 fp For a liquid-loaded QCM, K2 would not be a constant as those in ideal mode, because the values of fs and fp obtained from Eqs. (2) and (3) could be varied as the changes of solution physico-chemical properties (e.g. density–viscosity). In this communication, the expressions of fs and fp that are suitable both in liquid and gaseous phases have been proposed, and consequently the values of K in different media are evaluated.
2. Experimental An AT-cut 9 MHz quartz crystal with diameter 12.5 mm was commercially obtained from National 707 Factory (Beijing, China). The electrode contacting with the solution was coated with evaporated gold by using an Eiko IB-3 ion coater (Japan). Conductance G and susceptance B in the admittance spectra of the QCM resonance were measured synchronously by a HP 4395A impedance/network/ spectrum analyzer (Hewlett-Packard), which is controlled by a program written in VB 5.0. Each group of G and B data was fitted to the BVD model to obtain the equivalent circuit parameters, based on the Gauss–Newton nonlinear least square fitting algorithm. The values of density (ρ) and viscosity (η) for sucrose solutions were obtained from the standard table [9]. Sucrose solid was analytical grade. All the experiments were conducted in 20±0.1◦ C. Doubly distilled water was used throughout experiments.
3. Results and discussion Table 1 shows the comparison of the values of series resonant frequency obtained from Eq. (2) and those synchronously recorded by a HP 4395A impedance analyzer. For a QCM in air or in liquid, the values of fs calculated Eq. (2) agreed well with those found by
A. Zhou et al. / Analytica Chimica Acta 419 (2000) 251–254 Table 1 Comparison of the recorded and calculated values of the series resonant frequency in different media (ρη)1/2 (kg m−2 s−1/2 )
fs,HP a (Hz)
fs,calc b (Hz)
4.47×10−3c 0.973 1.06 1.15 1.25 1.43 1.76
8983190.7 8980794.2 8980684.4 8980594.6 8980496.2 8980406.3 8980320.6
8983190.7 8980794.8 8980685.9 8980596.2 8980498.3 8980407.4 8980318.7
a The values of the series resonant frequency recorded by HP 4395A impedance analyzer. b The values of the series resonant frequency calculated from Eq. (2). c This value is referred to (ρη)1/2 of air.
the impedance analyzer, with deviations generally smaller than 2 Hz. For instance, for a 9 MHz crystal in pure water, fs calculated from Eq. (2) is 8980794.8 Hz, compared with that provided by a HP 4395A impedance analyzer, 8980794.2 Hz. Fig. 1 shows the shifts of f0 , fs and fp in sucrose aqueous solutions with different (ρη)1/2 . It is clear that 1f0 holds a good linearity with (ρη)1/2 of the liquid. The regression equation is 1f0 =−1899(ρη)1/2 +1792, correlation coefficient r2 =0.9989. Moreover, Fig. 1 shows that 1fs has smaller frequency change than 1fp at a certain (ρη)1/2 . As expected by Eqs. (2) and (3), fs and fp would not exist when (ρη)1/2 higher than
253
some critical value. It is found that fs and fp vanish when (ρη)1/2 >1.76 kg m−2 s−1/2 under our experimental conditions, because of the negative value of the square root items in Eqs. (2) and (3) when over a certain (ρη)1/2 range. In fact, it was also previously observed by other authors that the phase angle of Y, would not be 0 at higher (ρη)1/2 [10] (Fig. 8 in this ref.). Fig. 2 illustrates the variation of K2 values with the changes of (ρη)1/2 of the sucrose solutions. It is seen that K2 decreases as (ρη)1/2 increases. For a quartz crystal in air ((ρη)1/2 =4.472×10−3 kg m−2 s−1/2 ), K2 is calculated to be 5.18×10−5 , in contrast with that obtained for a crystal in pure water, 4.00×10−5 . It is also interesting to find that K2 shows a good linearity with (ρη)1/2 . The regression equation is K2 =−1.216×10−6 (ρη)1/2 +5.204×10−6 , correlation coefficient r2 =0.9977. However, the results also show that K2 vanishes when (ρη)1/2 >1.76 kg m−2 s−1/2 . The reason for this phenomenon is allowed for the disappearance of fs and fp at that critical value of (ρη)1/2 . The disappearance of K2 at higher (ρη)1/2 implies that (i) the four-parameter BVD model should be modified in order to more effectively characterize quartz crystal in liquid (in other words, the expressions upon fs and fp as depicted in Eqs. (2) and (3) would be modified as the equivalent circuit model changes) and (ii) the equation used to obtain K2 (as described in Eq. (8)) should be developed in the case of a crystal in liquid.
Fig. 1. The shifts of frequencies f0 , fs and fp for a 9 MHz quartz crystal in sucrose aqueous solutions. The frequency shifts are all relative to respective frequency in pure water.
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A. Zhou et al. / Analytica Chimica Acta 419 (2000) 251–254
Fig. 2. Dependence of the electromechanical factor (K) and the (ρη)1/2 of different media.
In conclusion, the generally applicable equations upon fs and fp are obtained. It is shown that 1f0 linearly correlates with (ρη)1/2 of the liquid. Although the reason for the linear relationship between K2 and (ρη)1/2 is unknown presently, it is evident that the value of K2 in gaseous phase differs considerably from those obtained in liquid phase. Additionally, we note that the electromechanical coupling factor K is defined as the ratio of the output electrical energy to the total input mechanical energy for the direct piezoelectric effect, and vice versa for the converse piezoelectric effect. Thus, from the angle of energy transformation, the studies upon the electrical behavior of piezoelectric quartz crystal in different media would be more general and meaningful.
Acknowledgements This work was supported by the National Natural Science Foundation of China and the Science and
Technology Foundation of Hunan Province for the Youth. References [1] D.A. Buttry, M.D. Ward, Chem. Rev. 92 (1992) 1355. [2] S.Z. Yao, Piezoelectric Chemistry & Biosensors, Hunan Normal University Press, Changsha, 1997 (in Chinese). [3] V. Bottom, Introduction to Crystal Unit Design, Van Nostrand Reinhold, New York, 1982. [4] H. Muramatstu, E. Tamiya, I. Karube, Anal. Chem. 60 (1988) 2142. [5] Z.X. Lin, C.M. Yip, I.S. Joseph, M.D. Ward, Anal. Chem. 65 (1993) 1546. [6] M.A.M. Noël, P.A. Topart, Anal. Chem. 66 (1994) 484. [7] T. Yamada, N. Niizeki, Proc. IEEE 58 (1970) 941. [8] Y. Li, Z.K. Qin, Z.G. Zhou, Measurement of Piezoelectric and Ferroelectric Materials, Scientific Press, Beijing, 1984, p. 213 (in Chinese). [9] R.C. Weast (Ed.), CRC Handbook of Chemistry and Physics, 66th Edition, CRC Press, Boca Raton, FL, 1985–1986, p. D262. [10] S.J. Martin, V.E. Granstaff, G.C. Frye, Anal. Chem. 63 (1991) 2272.
Evaluation of electromechanical coupling factor for a piezoelectric quartz crystal in liquid phase Anhong Zhou, Qingji Xie, Yu Yuan, Youyu Zhang, Shouzhuo Yao∗ Chemical Research Institute, Hunan Normal University, Changsha 410081, PR China Received 30 July 1999; received in revised form 7 March 2000; accepted 30 March 2000
Abstract Based on the Butterworth–Van Dyke (BVD) equivalent circuit model for a quartz crystal, the general equations upon the series resonant-frequency fs , and parallel resonant-frequency fp , were obtained by solving the quadratic equation of frequency at susceptance B=0. Consequently, the electromechanical coupling factor (K) in liquid was first evaluated by using fs and fp . In sucrose solutions, it was seen that K2 decreases as (ρη)1/2 increases, where ρ and η were the density and viscosity of the solution, respectively. It was also interesting to find that K2 linearly correlates with (ρη)1/2 of the media. The regression equation was K2 =−1.216×10−6 (ρη)1/2 +5.204×10−6 , correlation coefficient r2 =0.9954. Regardless of unknown reason for this linear relationship, it was evident that the values of K2 showed apparent difference when a quartz crystal worked in different media. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Electromechanical coupling factor; Piezoelectric quartz crystal
1. Introduction Piezoelectric quartz crystal impedance analysis technique has been widely applied to interfacial science for its ability to offer multiple chemical information [1,2]. It is well-known that the QCM oscillator functions as an electromechanical transducer, and the value of electromechanical coupling factor K, is obviously very important to characterize resonance of a QCM, however, little attention has been paid to evaluation of this parameter in liquid. Resonance of an unperturbed QCM can be represented by the Butterworth–Van Dyke (BVD) equivalent circuit model [3]. The electrical behavior of a ∗ Corresponding author. Fax: +86-731-8865515. E-mail address: [email protected] (S. Yao).
QCM represented by BVD model in a Newtonian liquid has been examined by impedance analysis method [4]. The admittance Y, for the BVD model can be written as a complex of conductance (G) and susceptance (B) ! U R1 + j ωC0 − 2 (1) Y = G + jB = 2 R1 + U 2 R1 + U 2 where angular√ frequency ω=2πf, U = ωL1 − (1/ωC1 ), j = −1. As for the four equivalent circuit parameters, C0 is the static capacitance, R1 , L1 , C1 are the motional resistance, motional inductance, and motional capacitance, respectively. The series resonant-frequency fs , and parallel resonant-frequency fp , can be obtained by solving the quadratic equation of frequency at B=0.
0003-2670/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 3 - 2 6 7 0 ( 0 0 ) 0 0 9 1 8 - 1
252
A. Zhou et al. / Analytica Chimica Acta 419 (2000) 251–254
ωs2 = (2πfs )2
Z=
L1 − C0 R12 + 2L1 C0 /C1 −(C02 R14 + L21 =
−2L1 C0 R12 − 4L1 C02 R12 /C1 )1/2 2C0 L21
(2)
ωp2 = (2πfp )2 L1 − C0 R12 + 2L1 C0 /C1 +(C02 R14 + L21 =
−2L1 C0 R12 − 4L1 C02 R12 /C1 )1/2 2C0 L21
(3)
It should be noted that the forms of these two equations depend on the electrical equivalent circuit model of a quartz crystal. For a QCM in air, the value of R1 is very small and can be neglected (for a 9 MHz crystal used in the present work, R1 ≈5 in air, R1 ≈240 in pure water). Therefore, Eqs. (2) and (3) would be respectively reduced to the equations of fs and fp reported for a QCM resonator in vacuum [3]: 1/2 1 1 (4) fs = 2π L1 C1 1/2 1 1 1 + (5) fp = 2π L1 C1 L1 C0 Additionally, the impedance analyzer can offer resonant frequency f0 , at which G reaches its maximum value [4–6] 1/2 1 1 (6) f0 = 2π L1 C1 It is clear that f0 is virtually equal to fs (as shown by Eq. (4)) for a lossless crystal. However, for a crystal in liquid, these two frequency values show apparent difference since fs would be obtained from Eq. (2) in this case. On the other hand, for a lossless quartz resonating 2 /(c ε ), where K in ideal mode, K 2 = K02 ≡ e26 0 66 22 is the electromechanical coupling factor of a lossless quartz crystal (K0 =7.899×10−3 ), c66 the ‘piezoelectrically stiffened’ quartz elastic constant, e26 and ε22 are the piezoelectric stress constant and permitivity of the quartz, respectively. For a piezoelectric quartz crystal operated as a thickness-shear mode resonator, the impedance Z is given by [7,8]
1 j2π C0
π fs 2 fp 2 K tan 1− π fs 2 fp
(7)
For a crystal resonating at f = f0 (i.e. Z = 0), we have π fs π fp − fs 2 tan (8) K = 2 fp 2 fp For a liquid-loaded QCM, K2 would not be a constant as those in ideal mode, because the values of fs and fp obtained from Eqs. (2) and (3) could be varied as the changes of solution physico-chemical properties (e.g. density–viscosity). In this communication, the expressions of fs and fp that are suitable both in liquid and gaseous phases have been proposed, and consequently the values of K in different media are evaluated.
2. Experimental An AT-cut 9 MHz quartz crystal with diameter 12.5 mm was commercially obtained from National 707 Factory (Beijing, China). The electrode contacting with the solution was coated with evaporated gold by using an Eiko IB-3 ion coater (Japan). Conductance G and susceptance B in the admittance spectra of the QCM resonance were measured synchronously by a HP 4395A impedance/network/ spectrum analyzer (Hewlett-Packard), which is controlled by a program written in VB 5.0. Each group of G and B data was fitted to the BVD model to obtain the equivalent circuit parameters, based on the Gauss–Newton nonlinear least square fitting algorithm. The values of density (ρ) and viscosity (η) for sucrose solutions were obtained from the standard table [9]. Sucrose solid was analytical grade. All the experiments were conducted in 20±0.1◦ C. Doubly distilled water was used throughout experiments.
3. Results and discussion Table 1 shows the comparison of the values of series resonant frequency obtained from Eq. (2) and those synchronously recorded by a HP 4395A impedance analyzer. For a QCM in air or in liquid, the values of fs calculated Eq. (2) agreed well with those found by
A. Zhou et al. / Analytica Chimica Acta 419 (2000) 251–254 Table 1 Comparison of the recorded and calculated values of the series resonant frequency in different media (ρη)1/2 (kg m−2 s−1/2 )
fs,HP a (Hz)
fs,calc b (Hz)
4.47×10−3c 0.973 1.06 1.15 1.25 1.43 1.76
8983190.7 8980794.2 8980684.4 8980594.6 8980496.2 8980406.3 8980320.6
8983190.7 8980794.8 8980685.9 8980596.2 8980498.3 8980407.4 8980318.7
a The values of the series resonant frequency recorded by HP 4395A impedance analyzer. b The values of the series resonant frequency calculated from Eq. (2). c This value is referred to (ρη)1/2 of air.
the impedance analyzer, with deviations generally smaller than 2 Hz. For instance, for a 9 MHz crystal in pure water, fs calculated from Eq. (2) is 8980794.8 Hz, compared with that provided by a HP 4395A impedance analyzer, 8980794.2 Hz. Fig. 1 shows the shifts of f0 , fs and fp in sucrose aqueous solutions with different (ρη)1/2 . It is clear that 1f0 holds a good linearity with (ρη)1/2 of the liquid. The regression equation is 1f0 =−1899(ρη)1/2 +1792, correlation coefficient r2 =0.9989. Moreover, Fig. 1 shows that 1fs has smaller frequency change than 1fp at a certain (ρη)1/2 . As expected by Eqs. (2) and (3), fs and fp would not exist when (ρη)1/2 higher than
253
some critical value. It is found that fs and fp vanish when (ρη)1/2 >1.76 kg m−2 s−1/2 under our experimental conditions, because of the negative value of the square root items in Eqs. (2) and (3) when over a certain (ρη)1/2 range. In fact, it was also previously observed by other authors that the phase angle of Y, would not be 0 at higher (ρη)1/2 [10] (Fig. 8 in this ref.). Fig. 2 illustrates the variation of K2 values with the changes of (ρη)1/2 of the sucrose solutions. It is seen that K2 decreases as (ρη)1/2 increases. For a quartz crystal in air ((ρη)1/2 =4.472×10−3 kg m−2 s−1/2 ), K2 is calculated to be 5.18×10−5 , in contrast with that obtained for a crystal in pure water, 4.00×10−5 . It is also interesting to find that K2 shows a good linearity with (ρη)1/2 . The regression equation is K2 =−1.216×10−6 (ρη)1/2 +5.204×10−6 , correlation coefficient r2 =0.9977. However, the results also show that K2 vanishes when (ρη)1/2 >1.76 kg m−2 s−1/2 . The reason for this phenomenon is allowed for the disappearance of fs and fp at that critical value of (ρη)1/2 . The disappearance of K2 at higher (ρη)1/2 implies that (i) the four-parameter BVD model should be modified in order to more effectively characterize quartz crystal in liquid (in other words, the expressions upon fs and fp as depicted in Eqs. (2) and (3) would be modified as the equivalent circuit model changes) and (ii) the equation used to obtain K2 (as described in Eq. (8)) should be developed in the case of a crystal in liquid.
Fig. 1. The shifts of frequencies f0 , fs and fp for a 9 MHz quartz crystal in sucrose aqueous solutions. The frequency shifts are all relative to respective frequency in pure water.
254
A. Zhou et al. / Analytica Chimica Acta 419 (2000) 251–254
Fig. 2. Dependence of the electromechanical factor (K) and the (ρη)1/2 of different media.
In conclusion, the generally applicable equations upon fs and fp are obtained. It is shown that 1f0 linearly correlates with (ρη)1/2 of the liquid. Although the reason for the linear relationship between K2 and (ρη)1/2 is unknown presently, it is evident that the value of K2 in gaseous phase differs considerably from those obtained in liquid phase. Additionally, we note that the electromechanical coupling factor K is defined as the ratio of the output electrical energy to the total input mechanical energy for the direct piezoelectric effect, and vice versa for the converse piezoelectric effect. Thus, from the angle of energy transformation, the studies upon the electrical behavior of piezoelectric quartz crystal in different media would be more general and meaningful.
Acknowledgements This work was supported by the National Natural Science Foundation of China and the Science and
Technology Foundation of Hunan Province for the Youth. References [1] D.A. Buttry, M.D. Ward, Chem. Rev. 92 (1992) 1355. [2] S.Z. Yao, Piezoelectric Chemistry & Biosensors, Hunan Normal University Press, Changsha, 1997 (in Chinese). [3] V. Bottom, Introduction to Crystal Unit Design, Van Nostrand Reinhold, New York, 1982. [4] H. Muramatstu, E. Tamiya, I. Karube, Anal. Chem. 60 (1988) 2142. [5] Z.X. Lin, C.M. Yip, I.S. Joseph, M.D. Ward, Anal. Chem. 65 (1993) 1546. [6] M.A.M. Noël, P.A. Topart, Anal. Chem. 66 (1994) 484. [7] T. Yamada, N. Niizeki, Proc. IEEE 58 (1970) 941. [8] Y. Li, Z.K. Qin, Z.G. Zhou, Measurement of Piezoelectric and Ferroelectric Materials, Scientific Press, Beijing, 1984, p. 213 (in Chinese). [9] R.C. Weast (Ed.), CRC Handbook of Chemistry and Physics, 66th Edition, CRC Press, Boca Raton, FL, 1985–1986, p. D262. [10] S.J. Martin, V.E. Granstaff, G.C. Frye, Anal. Chem. 63 (1991) 2272.