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Fast Fourier transform admittance analysis method applied to thickness-shear-mode acoustic wave sensors
ANALYTIC4 CHIMICA ACTA ELSEVIER
Analytica Chimica Acta 353 (1997) 29-35
Fast Fourier transform admittance analysis method applied to thickness-shear-mode acoustic wave sensors Huwei Tan’, Jinhua Chen, Ronghui Wang, Xiaoli Su, Lihua Nie, Shouzhuo New Material Research Institute.
Chemistry and Chemical Engineering
Yao*
College. Hunan University, Changsha 410082, China
Received 4 June 1996; received in revised form 12 May 1997; accepted 14 May 1997
Abstract A fast Fourier transform admittance analysis (FFT-AA) method, applied to thickness-shear-mode (TSM) acoustic wave sensors, is first proposed and the corresponding theory is presented. Based on the FFT-AA, an oscillatory waveform with the feature of a damped free oscillation is obtained experimentally. It can illustrate dynamically and directly the vibratory characteristics of TSM sensors, which can be considered as an important advantage over an admittance spectrum obtained by admittance analysis. Compared with the motional resistance in equivalent circuit, the decay time, obtained by LevenbergMarquardt algorithm through the fit of the oscillatory equation to oscillatory waveform, is more suitable for characterization of the electrical energy dissipated in the quartz crystal. The oscillatory features of TSM sensor are investigated by F;1;TAA in different medium. The decay times are 0.1807, 0.0761 and 0.2300 ps with the media being air, water and dry coating film respectively. Finally, the proposed method has been successfully applied to monitor a curing process of coating polymer. It is therefore shown to be a useful technique to study oscillatory behavior of TSM acoustic wave sensors. rQ 1997 Elsevier
Science B.V. Keywords: Fast Fourier transform; Admittance
analysis;
Thickness-shear-mode
1. Introduction
*Corresponding author. ‘Present address: Department of Mathematics, Mechanics, Changsha Railway University.
Physics
and
0003-2670/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SOOO3-2670(97)00373-5
wave sensors; Coaling;
Curing
Kanazawa and Gordon [3] which predict the change of frequency of a sensor immersed in a liquid medium. The frequency shift arises from coupling the oscillation of the crystal, involving a standing shear wave, with damped propagation shear wave in the liquid. A simple relationship was derived which express the change in resonant frequency of a piezoelectric crystal, due to the total contact of one f&e of the crystal with liquid, in terms of.parameters that are characteristic of the crystal and the liquid phase. According to Yao and Zhou [4], the frequency response of a TSM sensor in the liquid phase depends on the dielectric constant, conductance, viscosity and
and
The use of thickness-shear-mode (TSM) acoustic wave sensors as chemical sensors has its origins in the work of Sauerbrey [I] who derived the linear relationship between frequency shift and mass change from the shear-mode vibration and thickness of the quartz crystal in the gas phase. In 1985, two simple physical models were developed by Bruckenstein and Shay [2]
acoustic
30
H. Tan tv al./Analytica Chimica Acta 353 (lYY7) 29-35
density effects of the liquid. Based on their experimental measurements, an empirical relationship was derived. In studies of TSM acoustic wave sensors, the oscillator method has almost exclusively been used. Only one electrical quantity (the series resonant frequency of the acoustic wave sensor) is measured by this method, and this frequency constitutes an incomplete characterization of the sensor. Furthermore, it has been reported by Yao and MO [5] that this method gives conflicting results in some cases. Thus, there are some limitations associated with this method. In recent years, the admittance analysis (AA) has been used to measure the series resonant frequency and the resistance of the equivalent circuit of the quartz sensor [6-91. This is commonly accomplished with impedance analyzers and can provide more informations on the characteristics of TSM acoustic wave sensors than the oscillator method does. Rut the resistance, an electrical parameter considered to be related to the electrical energy dissipated in the quartz crystal, could not describe directly the damped vibratory characteristics of the shear waves of TSM sensors. These properties, however, are of importance in characterization of TSM acoustic wave sensor. Fourier transform, as an eminent physical relationship between frequency spectrum and waveform, is widely used in spectrophotometers with the advantages of high resolution, high signal-to-noise ratio and wide spectral range [lO-121. With the development of cornputative technique, the fast Fourier transform (FFT) numbering method is easy to implement. In this paper, a FFI-admittance analysis (FFT-AA) and a corresponding theory are proposed. The oscillatory waveforms describing the vibratory characteristics of TSM sensors in different medium are measured by FIT-AA. They are fitted to the oscillatory equation by using Levenberg-Marquardt algorithm, and decay time is estimated. Finally, the proposed method is applied to monitor a practical interface process.
CO
‘1
_I-i Gl
WC,
WC0
R,
KU r4T
u
d
6
Fig. 1. Equivalent circuit of the TSM acoushc wave sensor with (a) parameters and (b) impedance of the circuit elements.
sinusoidal voltage at various frequencies across the terminals of the crystal. The voltage and current flowing through the crystal are measured. The ratio of voltage and current, defined as the impedance spectrum, is calculated. The impedance spectrum, Z(w), of the TSM sensor is complex: Z(w) = R(w) +jX(w), where R(w) is the real part of Z(w), the resistive part, and X(w) is the imaginary part of Z(w), the reactance. The quantity, w, is the angular frequency (in rad sP’) and is defined by w = 27-j, where f is the frequency in Hz. The impedance Z(w) is the function of w. Cady [ 131 and Bottom [ 141 have derived the equivalent circuit of the TSM sensor (Fig. l), which illustrates the main features of the behavior of the TSM sensor in both gas and liquids of low viscosity. For the equivalent circuit, the expression of the admittance spectrum (Fig. 2), Y(w), is simpler than the expression for Z(w). By definition, Y(w) is the reciprocal of Z(w), therefore if Z(w) is complex, so is Y(w): Y(w) = G[w) +$3(w)
(11
where the conductance, G(w) = R,/R,’ + (wL.,,, ( ~/wC,))~ is the real part of Y(w) and the susceptance, B(w) = -(w&,, - (l/~c,))/(R,~ + (WJ& - (~/wC,))~) + wCO, is the imaginary part of Y(L3).
2.2. Fourier transform
In the passive nected externally
_L iWL.. -j
Lm
2. Theory 2.1. Admittance
4
r-L
ofadmittance spectrum
spectrum method the quartz crystal is conto an instrument which applies a
Considering a delta function as a unit electrical impulse (in V) acted on the TSM sensor, the delta function at t=0 is identified by the symbol d(f) and has
H. Tan et al. /Analytica 12 t
31
Chimica Acta 353 (1997) 29-35
I
respectively.
10 8
4(t) = &
8 4
J jw -m
O” Y(w) expCiwt) dw
(3)
and B s
00
0’
i(t) = &
-2 4
Y(w) exp(jwt)dw
(4)
-cc
-8 -8 -10 -12 -2
0
2
4
8
8
10 12
14 18 18 20
G @w Fig. 2. A typical admittance spectrum of TSM sensor in air.
the following s(t) =
J
properties
O { greater than any assumed value
forallt # 0 for t = 0 (2)
Of G(t)dt = 1
where i(t) is electrical current waveform in the equivalent circuit at time t. Both q(t) and i(t) describe the features of a damped free oscillation in the equivalent circuit of TSM sensor, and depend on the viscous damping friction produced by TSM sensor and the medium with which TSM sensor contacts. Since the exact formulas of Eqs. (3) and (4) are not known analytically by the Fourier integral, a numbering method of m algorithm is employed. Thus, oscillatory waveform, q(t) or i(t) can be obtained experimentally by coupling FFI algorithm with impedance analyzer. Fig. 3 shows a typical oscillatory waveform of a TSM sensor in air. Its projection to the real axis and time axis plane (the dotted line) represents the change of the amplitude of oscillatory waveform over time. Therefore, the oscillatory waveform may illustrate the oscillatory behavior of TSM sensor.
I
O-
2.3. The damped free-oscillatory
The response function or oscillatory waveform of the TSM sensor to this unit electrical impulse can be obtained by using Fourier transform of Eqs. (1) and
When a linear system of one degree of freedom, e.g. a TSM acoustic wave sensor, is excited by an electric
equation
(2).
44 = &
m
Y(w) exptjwt) dw
J -cc
jw S(T)exp(-jw7)dT
1;
where the oscillatory waveform, q(t), is electric charge at time t, Y(w) is an admittance spectrum defined in Eq. (l), j is imaginary unity, and l/6 sy” C?(T)exp( -jwT)dT , the Fourier transform of delta function equals l/fi. Therefore, the oscillatory waveform of charge and current with time t in the equivalent circuit of TSM sensor are expressed
Fig. 3. Oscillatory waveform of the TSM sensor in air.
32
H. Tan et al. /Analytica
Chimica Acta 353 (1997) 29-35
impulse at a given instant, its oscillation will be free and damped by the viscous damping friction. Generally, its oscillatory equation in equivalent circuit will be of the form: 4(t) =
QO ed-kt)
exp0’(wt
+ 4))
(3
where Q. is the initial charge and 4 is the initial angle, which is determined from initial condition of q(0) and dq(O)/dt, and k is the decay constant which depends on physical characteristics of the medium, its reciprocal is the characteristic decay time X. The oscillatory equation (Eq. (5)), which is identical with Eq. (3), is known analytically, so it can be used to express the characteristics of the oscillatory behavior of TSM sensor and to estimate the parameters, e.g. Qo, k, X and 4 from oscillatory waveform data. 2.4. Levenberg-Marquardt
algorithm
A nonlinear least squares technique, the Levenberg-Marquardt method [ 15,161, is used for the three parameters estimation (Qo, k and 4) of oscillatory equation (Eq. (5)). The program searches for the best set of parameters which allow the minimization of: Aq(Qo>
k, 4) =
~(44 - q@, Qo, 4 4))* t
where q(t) represents the measured charges (oscillatory waveform) to be fitted at time c, Term q(t,Qo,k,4) is the expectation charge at time t based on Eq. (5). In order to assess the adequacy of the computed results, the F-statistical test is used as criteria, according to the following formula: R2 =
Qolk
C,Mh
2X4(t)
4) - 4(t))*
(7)
- iin)*
where q(t) = (l/M) C,q(t), M is the number of measured charges. When R-+1, then residual sum of squares Q-0. it shows that the nearer R tends to 1, the better is the goodness of fit. Hence R is the goodness-of-fit.
3. Experimental 3.1. Apparatus
and materials
The instrument used to characterize the TSM sensor in different media was a 4192A-LF impedance ana-
lyzer, which was connected through an HP-IB interface (Hewlett-Packard) to an IBM compatible 486DX88 computer. The quartz crystals (JA-5 model; Peking Factory No. 707) used were AT-cut crystals (12.5 mm diameter) having silver electrodes (6 mm diameter) on each side. One of the electrodes of the crystal was used as the working electrode. The specification of FOl-1 phenolic varnish was of the standard (Chinese Standard ZBG51018-87). The solvents used as diluters were: 200’ gasoline (Chinese Standard GB444-64) and dimethylbenzene. 3.2. Program and procedures All programs described in this paper were written in MATLAB (version 4.0) with toolbox (version 1.0). The admittance measurements of the Ag/QC oscillator systems were performed by using a set of abovementioned instruments. The admittance data were transferred to the computer, on which FFT and other calculation were carried out, and were graphically presented in the form of oscillatory waveform q(t) vs. time and phase angle vs. time. Then oscillatory waveform data were fitted to oscillatory equation (Eq. (5)) and decay constant k, characteristic decay time X and goodness-of-fit R were obtained.
4. Results and discussion 4.1. FFT-AA and AA In the equivalent circuit of TSM acoustic wave sensor, the oscillatory waveform and the admittance spectrum are Fourier transform of each other (Eqs. (3) and (4)). An oscillatory waveform obtained by FFTAA and an admittance spectrum obtained by AA when the quartz crystal is in air, are shown in Figs. 2 and 3, respectively. As can be seen, a damped free oscillation of quartz crystal excited by an electric impulse, is illustrated dynamically by the oscillatory waveform, while only the admittance feature of TSM sensor is shown by the admittance spectrum. Moreover, the periodic change of phase angle of q(t) can be evidently seen in Fig. 3, and the decay constant (k) and the characteristic decay time (X) can be obtained by using Levenberg-Marquardt algorithm from the oscillatory waveform. Therefore, FFT-AA method, as an admit-
H. Tan et al. /Analytica
Fig. 4. Oscillatory waveform of the TSM sensor when one side of crystal was in contact with a drop of water.
tance analysis method, provides dynamically and directly the inherent oscillatory laws of TSM acoustic wave sensor. 4.2. Oscillatory waveform of TSM sensor in different medium Different oscillatory waveforms, obtained by FFTAA when the TSM sensor contacts with air, a drop of water and dry film, are shown in Figs. 3-5, respectively. Apparently, the decay of damped oscillation, when one side of quartz crystal contacts with a drop of water, is heaviest (Fig. 4). The periodic change of phase angle of q(t) disappears completely at ca. 0.3 ps, which is attributed to the strong effects of viscosity and surface tension in a drop of water. By using Levenberg-Marquardt algorithm, the oscillatory waveforms in different medium obtained by FFT-AA are fitted to damped free-oscillatory equation (Eq. (5)). The decay constant k, the characteristic decay time (X) and goodness of fit (R) are listed in
Table 1 Estimated
parameters
Medium Air Water drop Phenolic varnish film
33
Chimica Acta 353 (1997) 29-35
Fig. 5. Oscillatory waveform of the TSM sensor when one side of crystal was in contact with drying polymer coating.
Table 1. It is apparent that the fit is successful and the best goodness-of-fit (R) is 0.99804. In addition, the motional resistance R, obtained by AA are also listed in Table 1. Both X and R, are related to the electrical energy dissipated in the quartz crystal. The characteristic decay time (X) is the time when the damped free oscillation of acoustic wave of TSM sensors disappears, due to the electrical energy dissipated by the viscous damping friction both in medium and quartz crystal. The motional resistance (R,) is a mechanical parameter and has been used to described the same characteristic of quartz crystal. But the former has more manifest physical meaning. Thus, the characteristic decay time (X) may be more suitable to characterization of the damped inherent law of quartz crystal than the motional resistance R,. In this work, The decay times estimated were 0.1807,0.0761 and 0.2300 ps with the media being air, water and dry coating film respectively, while R, estimated were 50.00, 180.9 and 43.75 R with the three same media, respectively.
of TSM sensor in different media by FFT-AA and AA Goodness R 0.99152 1.01350 0.99804
of fit
Decay constant k (s-l)
Decay time of the shear wave
Motional resistance
W)
Rm W)
5.5345x 106 1.3147x107 4.3481 x lo6
0.1807 0.0761 0.2300
50.00 180.9 43.75
34
H.
Tan er al./AnuZytica Chimica Acta 353 (i997J 29-35
Table 2 Estimated parameters of TSM sensor in curing process of polymer coating by FFT-AA and AA 7ime (min)
-
0 I .50 3.00 4.50 6.00 7.50 9.00 10.50 12.00 15.00 18.00 21.01 24.01 27.01 30.01 33.01 36.01 41.01 46.01 51.01 56.01 61.01 66.02 71.02 76.02 81.02 86.02 91.02 96.02 101.02
Goodness- Decay of-fit constant
Decay time
Motional resistance
R
k (s-l)
ols)
&, (0)
2.3179x 10’ 9.0508x 106 6.6946x lo6 6.1495~10~ 5.8144x 10’ 5.6955 x 106 4.9842x IO6 5.1889x 106 5.2429 x lo6 4.9970x lo6 4.9015x 106 4.8039x lo6 4.6600x lo6 4.5928% lo6 4.5070x 106 4.4200x ld 4.4158~10~ 4.3778x ld 4.3480x ld 4.3545 x 106 4.3369x 10” 4.3465 x lo6 4.3369x 106 4.3426x lo6 4.3275x IO6 4.3465x lo6 4.3489x lo6 4.3518x lo6 4.3551 x 106 4.3481 x lo6
0.0431 0.1105 0.1494 0.1626 0.1720 0.1756 0.2006 0.1927 0.1907 0.2001 0.2040 0.2082 0.2146 0.2177 0.2219 0.2262 0.2265 0.2284 0.2300 0.2297 0.2306 0.2301 0.2306 0.2303 0.2310 0.2301 0.2299 0.2298 0.2296 0.2300
260.85 115.97 72.53 66.76 59.22 58.15 51.10 54.44 54.61 50.75 49.74 48.93 47.41 46.60 45.81 44.75 44.62 44.49
0.98885 0.99301 0.99699 0.99704 0.99680 0.99780 0.99789 0.99437 0.99619 0.99773 0.99786 0.99787 0.99780 0.99786 0.99765 0.9979 1 0.99793 0.99738 0.99780 0.99753 0.99776 0.99770 0.99761 0.99786 0.99789 0.99801 0.99774 0.99791 0.99797 0.99804
-
22
18
5I4 10
3mt
44.16 44.10
43.98 43.98 43.98 43.82 43.80 43.78 43.95 43.98 43.72 43.75
‘O” t
-a0
0
m
40
80
80
loo
120
Th10 (mln)
4.3. Application in monitoring of the curing process of the coating The typical behavior of the curing of a coating (coating thickness is 32.446 nm) has been measured by FFT-AA and AA, respectively. The changes of the decay constants (k), the characteristic decay time (A) and motional resistance [I?,) are listed in Table 2. A typical. curve expressing the change of these three parameters with curing time is shown in Fig. 6. The tendencies of X and R, are the same, At first they increase quickly and then approach to a stationary value, which coincides with the curing
Fig. 6. A typical curing process of a 32.446 nm coating layer witi (a) the decay constant k (solid line) and the decay time (dotted line) and (b) motional resistance R, vs. curing time pIot.
behavior of the coating. Thus the characteristic decay time X can also be used to monitor the surface process.
5. Conclusion The FFT-AA is a very useful method for complete characterization of TSM sensor. An oscillatory wave-
H. Tan et al. /Analytica
form, experimentally measured by FFI-AA, provides dynamically and directly the inherent oscillatory behaviors of the TSM acoustic wave sensor. The characteristic decay time (X) can be used to specify quantitatively the oscillatory energy of the sensor dissipated by damping friction. In addition, the oscillatory waveform is similar to the propagation of the transverse shear wave from the TSM sensor into the media. Therefore, its potential application may be associated with the figure of the propagation of the thickness shear acoustic wave of the crystal in media. The proposed method may be widely used.
Acknowledgements Financial support from the National Natural Science Foundation and Education Commission Foundation of China is gratefully acknowledged. In addition, we would like to thank Dr. Jianhua Hu, Mr. Yinchun Luo and Mr. Changwen Liu for their help in preparing this paper.
Chimica Acta 353 (1997) 29-35
35
References 111G.Z. Sauerbrey, Phys. 115 (1959) 206. I21 S. Bruckenstein, M. Shay, Eletrochim. Acta 169 (1985) 257. [31 K.K. Kanazawa, J.G. Gordon, Anal. Chim. Acta 175 (1985) 99. [41 S.Z. Yao, T.A. Zhou, Anal. Chim. Acta 212 (1988) 61. PI S.Z. Yao, Z.H. MO, Anal. Chim. Acta 193 (1987) 97. WI H. Muramatsu, E. Tamiya, I. Karube, Anal. Chem. 60 (1988) 2142. t71 T.A. Zhou, L.H. Nie, S.Z. Yao, J. Electroanal. Chem. 293 (1990) 1. Chem. 316 181 Z.H. MO, L.H. Nie, S.Z. Yao, J. Electroanal. (1991) 79. A.L. Kipling, W.C. Duncan-Hewitt, L.V. 191 M. Thompson, Rajalovic, B.A. Cavic-Vlasak, Analyst 116 (1991) 881. WI P Luc, Spectrochim. Acta Part A 51 (1995) 1171. r111 J.M. Pland, A. Perrin, C. Camy-Peyret, Spectrochim. Acta A 51 (1995) 1217. WI D.K. Graff, J. Chem. Bduc. 72 (1995) 304. N31 W.G. Cady, Piezoelectricity, Dover, New York, 1964. t141 V.E. Bottom, Introduction to Quartz Crystal Unit Design, Van Nostrand Reinhold, New York, 1982. [W D.J. Marquardt, Sot. Ind. Appl. Math. 11 (1963) 431. WI H.W. Tan, L. Deng, L.H. Nie, S.Z. Yao, Analyst 122 (1997) 179.
Analytica Chimica Acta 353 (1997) 29-35
Fast Fourier transform admittance analysis method applied to thickness-shear-mode acoustic wave sensors Huwei Tan’, Jinhua Chen, Ronghui Wang, Xiaoli Su, Lihua Nie, Shouzhuo New Material Research Institute.
Chemistry and Chemical Engineering
Yao*
College. Hunan University, Changsha 410082, China
Received 4 June 1996; received in revised form 12 May 1997; accepted 14 May 1997
Abstract A fast Fourier transform admittance analysis (FFT-AA) method, applied to thickness-shear-mode (TSM) acoustic wave sensors, is first proposed and the corresponding theory is presented. Based on the FFT-AA, an oscillatory waveform with the feature of a damped free oscillation is obtained experimentally. It can illustrate dynamically and directly the vibratory characteristics of TSM sensors, which can be considered as an important advantage over an admittance spectrum obtained by admittance analysis. Compared with the motional resistance in equivalent circuit, the decay time, obtained by LevenbergMarquardt algorithm through the fit of the oscillatory equation to oscillatory waveform, is more suitable for characterization of the electrical energy dissipated in the quartz crystal. The oscillatory features of TSM sensor are investigated by F;1;TAA in different medium. The decay times are 0.1807, 0.0761 and 0.2300 ps with the media being air, water and dry coating film respectively. Finally, the proposed method has been successfully applied to monitor a curing process of coating polymer. It is therefore shown to be a useful technique to study oscillatory behavior of TSM acoustic wave sensors. rQ 1997 Elsevier
Science B.V. Keywords: Fast Fourier transform; Admittance
analysis;
Thickness-shear-mode
1. Introduction
*Corresponding author. ‘Present address: Department of Mathematics, Mechanics, Changsha Railway University.
Physics
and
0003-2670/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SOOO3-2670(97)00373-5
wave sensors; Coaling;
Curing
Kanazawa and Gordon [3] which predict the change of frequency of a sensor immersed in a liquid medium. The frequency shift arises from coupling the oscillation of the crystal, involving a standing shear wave, with damped propagation shear wave in the liquid. A simple relationship was derived which express the change in resonant frequency of a piezoelectric crystal, due to the total contact of one f&e of the crystal with liquid, in terms of.parameters that are characteristic of the crystal and the liquid phase. According to Yao and Zhou [4], the frequency response of a TSM sensor in the liquid phase depends on the dielectric constant, conductance, viscosity and
and
The use of thickness-shear-mode (TSM) acoustic wave sensors as chemical sensors has its origins in the work of Sauerbrey [I] who derived the linear relationship between frequency shift and mass change from the shear-mode vibration and thickness of the quartz crystal in the gas phase. In 1985, two simple physical models were developed by Bruckenstein and Shay [2]
acoustic
30
H. Tan tv al./Analytica Chimica Acta 353 (lYY7) 29-35
density effects of the liquid. Based on their experimental measurements, an empirical relationship was derived. In studies of TSM acoustic wave sensors, the oscillator method has almost exclusively been used. Only one electrical quantity (the series resonant frequency of the acoustic wave sensor) is measured by this method, and this frequency constitutes an incomplete characterization of the sensor. Furthermore, it has been reported by Yao and MO [5] that this method gives conflicting results in some cases. Thus, there are some limitations associated with this method. In recent years, the admittance analysis (AA) has been used to measure the series resonant frequency and the resistance of the equivalent circuit of the quartz sensor [6-91. This is commonly accomplished with impedance analyzers and can provide more informations on the characteristics of TSM acoustic wave sensors than the oscillator method does. Rut the resistance, an electrical parameter considered to be related to the electrical energy dissipated in the quartz crystal, could not describe directly the damped vibratory characteristics of the shear waves of TSM sensors. These properties, however, are of importance in characterization of TSM acoustic wave sensor. Fourier transform, as an eminent physical relationship between frequency spectrum and waveform, is widely used in spectrophotometers with the advantages of high resolution, high signal-to-noise ratio and wide spectral range [lO-121. With the development of cornputative technique, the fast Fourier transform (FFT) numbering method is easy to implement. In this paper, a FFI-admittance analysis (FFT-AA) and a corresponding theory are proposed. The oscillatory waveforms describing the vibratory characteristics of TSM sensors in different medium are measured by FIT-AA. They are fitted to the oscillatory equation by using Levenberg-Marquardt algorithm, and decay time is estimated. Finally, the proposed method is applied to monitor a practical interface process.
CO
‘1
_I-i Gl
WC,
WC0
R,
KU r4T
u
d
6
Fig. 1. Equivalent circuit of the TSM acoushc wave sensor with (a) parameters and (b) impedance of the circuit elements.
sinusoidal voltage at various frequencies across the terminals of the crystal. The voltage and current flowing through the crystal are measured. The ratio of voltage and current, defined as the impedance spectrum, is calculated. The impedance spectrum, Z(w), of the TSM sensor is complex: Z(w) = R(w) +jX(w), where R(w) is the real part of Z(w), the resistive part, and X(w) is the imaginary part of Z(w), the reactance. The quantity, w, is the angular frequency (in rad sP’) and is defined by w = 27-j, where f is the frequency in Hz. The impedance Z(w) is the function of w. Cady [ 131 and Bottom [ 141 have derived the equivalent circuit of the TSM sensor (Fig. l), which illustrates the main features of the behavior of the TSM sensor in both gas and liquids of low viscosity. For the equivalent circuit, the expression of the admittance spectrum (Fig. 2), Y(w), is simpler than the expression for Z(w). By definition, Y(w) is the reciprocal of Z(w), therefore if Z(w) is complex, so is Y(w): Y(w) = G[w) +$3(w)
(11
where the conductance, G(w) = R,/R,’ + (wL.,,, ( ~/wC,))~ is the real part of Y(w) and the susceptance, B(w) = -(w&,, - (l/~c,))/(R,~ + (WJ& - (~/wC,))~) + wCO, is the imaginary part of Y(L3).
2.2. Fourier transform
In the passive nected externally
_L iWL.. -j
Lm
2. Theory 2.1. Admittance
4
r-L
ofadmittance spectrum
spectrum method the quartz crystal is conto an instrument which applies a
Considering a delta function as a unit electrical impulse (in V) acted on the TSM sensor, the delta function at t=0 is identified by the symbol d(f) and has
H. Tan et al. /Analytica 12 t
31
Chimica Acta 353 (1997) 29-35
I
respectively.
10 8
4(t) = &
8 4
J jw -m
O” Y(w) expCiwt) dw
(3)
and B s
00
0’
i(t) = &
-2 4
Y(w) exp(jwt)dw
(4)
-cc
-8 -8 -10 -12 -2
0
2
4
8
8
10 12
14 18 18 20
G @w Fig. 2. A typical admittance spectrum of TSM sensor in air.
the following s(t) =
J
properties
O { greater than any assumed value
forallt # 0 for t = 0 (2)
Of G(t)dt = 1
where i(t) is electrical current waveform in the equivalent circuit at time t. Both q(t) and i(t) describe the features of a damped free oscillation in the equivalent circuit of TSM sensor, and depend on the viscous damping friction produced by TSM sensor and the medium with which TSM sensor contacts. Since the exact formulas of Eqs. (3) and (4) are not known analytically by the Fourier integral, a numbering method of m algorithm is employed. Thus, oscillatory waveform, q(t) or i(t) can be obtained experimentally by coupling FFI algorithm with impedance analyzer. Fig. 3 shows a typical oscillatory waveform of a TSM sensor in air. Its projection to the real axis and time axis plane (the dotted line) represents the change of the amplitude of oscillatory waveform over time. Therefore, the oscillatory waveform may illustrate the oscillatory behavior of TSM sensor.
I
O-
2.3. The damped free-oscillatory
The response function or oscillatory waveform of the TSM sensor to this unit electrical impulse can be obtained by using Fourier transform of Eqs. (1) and
When a linear system of one degree of freedom, e.g. a TSM acoustic wave sensor, is excited by an electric
equation
(2).
44 = &
m
Y(w) exptjwt) dw
J -cc
jw S(T)exp(-jw7)dT
1;
where the oscillatory waveform, q(t), is electric charge at time t, Y(w) is an admittance spectrum defined in Eq. (l), j is imaginary unity, and l/6 sy” C?(T)exp( -jwT)dT , the Fourier transform of delta function equals l/fi. Therefore, the oscillatory waveform of charge and current with time t in the equivalent circuit of TSM sensor are expressed
Fig. 3. Oscillatory waveform of the TSM sensor in air.
32
H. Tan et al. /Analytica
Chimica Acta 353 (1997) 29-35
impulse at a given instant, its oscillation will be free and damped by the viscous damping friction. Generally, its oscillatory equation in equivalent circuit will be of the form: 4(t) =
QO ed-kt)
exp0’(wt
+ 4))
(3
where Q. is the initial charge and 4 is the initial angle, which is determined from initial condition of q(0) and dq(O)/dt, and k is the decay constant which depends on physical characteristics of the medium, its reciprocal is the characteristic decay time X. The oscillatory equation (Eq. (5)), which is identical with Eq. (3), is known analytically, so it can be used to express the characteristics of the oscillatory behavior of TSM sensor and to estimate the parameters, e.g. Qo, k, X and 4 from oscillatory waveform data. 2.4. Levenberg-Marquardt
algorithm
A nonlinear least squares technique, the Levenberg-Marquardt method [ 15,161, is used for the three parameters estimation (Qo, k and 4) of oscillatory equation (Eq. (5)). The program searches for the best set of parameters which allow the minimization of: Aq(Qo>
k, 4) =
~(44 - q@, Qo, 4 4))* t
where q(t) represents the measured charges (oscillatory waveform) to be fitted at time c, Term q(t,Qo,k,4) is the expectation charge at time t based on Eq. (5). In order to assess the adequacy of the computed results, the F-statistical test is used as criteria, according to the following formula: R2 =
Qolk
C,Mh
2X4(t)
4) - 4(t))*
(7)
- iin)*
where q(t) = (l/M) C,q(t), M is the number of measured charges. When R-+1, then residual sum of squares Q-0. it shows that the nearer R tends to 1, the better is the goodness of fit. Hence R is the goodness-of-fit.
3. Experimental 3.1. Apparatus
and materials
The instrument used to characterize the TSM sensor in different media was a 4192A-LF impedance ana-
lyzer, which was connected through an HP-IB interface (Hewlett-Packard) to an IBM compatible 486DX88 computer. The quartz crystals (JA-5 model; Peking Factory No. 707) used were AT-cut crystals (12.5 mm diameter) having silver electrodes (6 mm diameter) on each side. One of the electrodes of the crystal was used as the working electrode. The specification of FOl-1 phenolic varnish was of the standard (Chinese Standard ZBG51018-87). The solvents used as diluters were: 200’ gasoline (Chinese Standard GB444-64) and dimethylbenzene. 3.2. Program and procedures All programs described in this paper were written in MATLAB (version 4.0) with toolbox (version 1.0). The admittance measurements of the Ag/QC oscillator systems were performed by using a set of abovementioned instruments. The admittance data were transferred to the computer, on which FFT and other calculation were carried out, and were graphically presented in the form of oscillatory waveform q(t) vs. time and phase angle vs. time. Then oscillatory waveform data were fitted to oscillatory equation (Eq. (5)) and decay constant k, characteristic decay time X and goodness-of-fit R were obtained.
4. Results and discussion 4.1. FFT-AA and AA In the equivalent circuit of TSM acoustic wave sensor, the oscillatory waveform and the admittance spectrum are Fourier transform of each other (Eqs. (3) and (4)). An oscillatory waveform obtained by FFTAA and an admittance spectrum obtained by AA when the quartz crystal is in air, are shown in Figs. 2 and 3, respectively. As can be seen, a damped free oscillation of quartz crystal excited by an electric impulse, is illustrated dynamically by the oscillatory waveform, while only the admittance feature of TSM sensor is shown by the admittance spectrum. Moreover, the periodic change of phase angle of q(t) can be evidently seen in Fig. 3, and the decay constant (k) and the characteristic decay time (X) can be obtained by using Levenberg-Marquardt algorithm from the oscillatory waveform. Therefore, FFT-AA method, as an admit-
H. Tan et al. /Analytica
Fig. 4. Oscillatory waveform of the TSM sensor when one side of crystal was in contact with a drop of water.
tance analysis method, provides dynamically and directly the inherent oscillatory laws of TSM acoustic wave sensor. 4.2. Oscillatory waveform of TSM sensor in different medium Different oscillatory waveforms, obtained by FFTAA when the TSM sensor contacts with air, a drop of water and dry film, are shown in Figs. 3-5, respectively. Apparently, the decay of damped oscillation, when one side of quartz crystal contacts with a drop of water, is heaviest (Fig. 4). The periodic change of phase angle of q(t) disappears completely at ca. 0.3 ps, which is attributed to the strong effects of viscosity and surface tension in a drop of water. By using Levenberg-Marquardt algorithm, the oscillatory waveforms in different medium obtained by FFT-AA are fitted to damped free-oscillatory equation (Eq. (5)). The decay constant k, the characteristic decay time (X) and goodness of fit (R) are listed in
Table 1 Estimated
parameters
Medium Air Water drop Phenolic varnish film
33
Chimica Acta 353 (1997) 29-35
Fig. 5. Oscillatory waveform of the TSM sensor when one side of crystal was in contact with drying polymer coating.
Table 1. It is apparent that the fit is successful and the best goodness-of-fit (R) is 0.99804. In addition, the motional resistance R, obtained by AA are also listed in Table 1. Both X and R, are related to the electrical energy dissipated in the quartz crystal. The characteristic decay time (X) is the time when the damped free oscillation of acoustic wave of TSM sensors disappears, due to the electrical energy dissipated by the viscous damping friction both in medium and quartz crystal. The motional resistance (R,) is a mechanical parameter and has been used to described the same characteristic of quartz crystal. But the former has more manifest physical meaning. Thus, the characteristic decay time (X) may be more suitable to characterization of the damped inherent law of quartz crystal than the motional resistance R,. In this work, The decay times estimated were 0.1807,0.0761 and 0.2300 ps with the media being air, water and dry coating film respectively, while R, estimated were 50.00, 180.9 and 43.75 R with the three same media, respectively.
of TSM sensor in different media by FFT-AA and AA Goodness R 0.99152 1.01350 0.99804
of fit
Decay constant k (s-l)
Decay time of the shear wave
Motional resistance
W)
Rm W)
5.5345x 106 1.3147x107 4.3481 x lo6
0.1807 0.0761 0.2300
50.00 180.9 43.75
34
H.
Tan er al./AnuZytica Chimica Acta 353 (i997J 29-35
Table 2 Estimated parameters of TSM sensor in curing process of polymer coating by FFT-AA and AA 7ime (min)
-
0 I .50 3.00 4.50 6.00 7.50 9.00 10.50 12.00 15.00 18.00 21.01 24.01 27.01 30.01 33.01 36.01 41.01 46.01 51.01 56.01 61.01 66.02 71.02 76.02 81.02 86.02 91.02 96.02 101.02
Goodness- Decay of-fit constant
Decay time
Motional resistance
R
k (s-l)
ols)
&, (0)
2.3179x 10’ 9.0508x 106 6.6946x lo6 6.1495~10~ 5.8144x 10’ 5.6955 x 106 4.9842x IO6 5.1889x 106 5.2429 x lo6 4.9970x lo6 4.9015x 106 4.8039x lo6 4.6600x lo6 4.5928% lo6 4.5070x 106 4.4200x ld 4.4158~10~ 4.3778x ld 4.3480x ld 4.3545 x 106 4.3369x 10” 4.3465 x lo6 4.3369x 106 4.3426x lo6 4.3275x IO6 4.3465x lo6 4.3489x lo6 4.3518x lo6 4.3551 x 106 4.3481 x lo6
0.0431 0.1105 0.1494 0.1626 0.1720 0.1756 0.2006 0.1927 0.1907 0.2001 0.2040 0.2082 0.2146 0.2177 0.2219 0.2262 0.2265 0.2284 0.2300 0.2297 0.2306 0.2301 0.2306 0.2303 0.2310 0.2301 0.2299 0.2298 0.2296 0.2300
260.85 115.97 72.53 66.76 59.22 58.15 51.10 54.44 54.61 50.75 49.74 48.93 47.41 46.60 45.81 44.75 44.62 44.49
0.98885 0.99301 0.99699 0.99704 0.99680 0.99780 0.99789 0.99437 0.99619 0.99773 0.99786 0.99787 0.99780 0.99786 0.99765 0.9979 1 0.99793 0.99738 0.99780 0.99753 0.99776 0.99770 0.99761 0.99786 0.99789 0.99801 0.99774 0.99791 0.99797 0.99804
-
22
18
5I4 10
3mt
44.16 44.10
43.98 43.98 43.98 43.82 43.80 43.78 43.95 43.98 43.72 43.75
‘O” t
-a0
0
m
40
80
80
loo
120
Th10 (mln)
4.3. Application in monitoring of the curing process of the coating The typical behavior of the curing of a coating (coating thickness is 32.446 nm) has been measured by FFT-AA and AA, respectively. The changes of the decay constants (k), the characteristic decay time (A) and motional resistance [I?,) are listed in Table 2. A typical. curve expressing the change of these three parameters with curing time is shown in Fig. 6. The tendencies of X and R, are the same, At first they increase quickly and then approach to a stationary value, which coincides with the curing
Fig. 6. A typical curing process of a 32.446 nm coating layer witi (a) the decay constant k (solid line) and the decay time (dotted line) and (b) motional resistance R, vs. curing time pIot.
behavior of the coating. Thus the characteristic decay time X can also be used to monitor the surface process.
5. Conclusion The FFT-AA is a very useful method for complete characterization of TSM sensor. An oscillatory wave-
H. Tan et al. /Analytica
form, experimentally measured by FFI-AA, provides dynamically and directly the inherent oscillatory behaviors of the TSM acoustic wave sensor. The characteristic decay time (X) can be used to specify quantitatively the oscillatory energy of the sensor dissipated by damping friction. In addition, the oscillatory waveform is similar to the propagation of the transverse shear wave from the TSM sensor into the media. Therefore, its potential application may be associated with the figure of the propagation of the thickness shear acoustic wave of the crystal in media. The proposed method may be widely used.
Acknowledgements Financial support from the National Natural Science Foundation and Education Commission Foundation of China is gratefully acknowledged. In addition, we would like to thank Dr. Jianhua Hu, Mr. Yinchun Luo and Mr. Changwen Liu for their help in preparing this paper.
Chimica Acta 353 (1997) 29-35
35
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