Anal. Chem. 1994,66, 1955-1964
Quartz Crystal Resonators as Sensors in Liquids Using the Acoustoelectric Effect Zack A. Shana and Fablen Josse' Department of Electrical and Computer Engineering, Marquette University, Milwaukee, Wisconsin 53233 Piezoelectric quartz crystal resonators (QCRs) have been investigated as detectors in liquid environments. In all the applications, mass loading and viscous coupling are the main interaction mechanisms which result in changes in the QCR resonant frequency. However, other interaction mechanisms such as the acoustoelectric interaction due to fringing fields at electrode ends arise which contribute to the total change in frequency, in particular, the parallel resonant frequency. In the present work, it is shown that by modifying the geometry of the electrode at the QCR surface in contact with the solution, a transition region can be created in which the lateral decaying acoustic field is enhanced. The electric field can then interact with an adjacent conductive/didectric solutionwhich will result in relatively large changes in the parallel resonance conditions of the QCR. An equivalent circuit is proposed to analyze the loaded QCR with a modified electrode geometry. It is shown that this circuit is a general circuit which can be used to analyze all cases of a loaded QCR with one side in contact with a given viscous, conductive, or dielectric liquid. Especially, expressions are obtained for the parallel resonant frequency of the loaded QCR in terms of the solution dielectric constant and conductivity. It is shown, using 11-MHz devices on AT-cut quartz, that the modified QCRs can be used as effective and reliable detectors in conductive liquid environments to detect ionic solutes and their dielectric properties. Other applications are suggested. Acoustic wave devices are increasingly being studied both for the physical measurement of liquid properties and in biosensor applications. This is because a need has been recognized for rapid, sensitive, inexpensive high-performance microsensors capable of performing in liquid environments. Initial work relating to the use of acoustic wave devices in liquid-phase sensing applications utilized conventional bulk acoustic wave (BAW) piezoelectric crystal Commonly used in these applications is the AT-cut quartz crystal resonator (QCR) in which thickness-shear horizontal vibrations are excited to set up standing waves. The AT-cut QCR typically consists of a thin quartz disk sandwiched between two circular, concentric, metallic electrodes of equal diameters. The electrodes are used to excite the resonator whose frequency is determined by the resonance of the standing wave across the thickness of the plate. The device is then used as the frequency-determining element of an oscillator circuit. The acoustic wave is at an antinode at the surface of the quartz plate and thus can interact with an adjacent medium. (1) Schulz, W. W.; King, W. H. J. Chromatogr. Sci. 1973, 11, 343-348. (2) Konash, P. L.; Bastiaans, G. J. Anal. Chem. 1980, 52, 1929-1931. 0003-2700/94/03681955$04.5Q/Q 0 1994 American Chemical Society
Although the ability of the QCR to oscillate in a liquid environment is often impaired due to the dissipation of energy in the liquid (from the vibrating crystal), properly designed QCRs with appropriate liquid cell and volume can effectively oscillate with one face or both faces in contact with the solution. The piezoelectric QCR has been used in various applications such as mass detectors for the determination of ions in solution^,^.^ as bi~sensors,~ and as electrochemical microbalances or nanobalances.6~~In those devices, the change in oscillation frequency of the crystal is measured and related to the analyte property being determined. Mass loading is the main interaotion mechanism which results in the frequency change. In these applications, the surface of the device is coated with a suitable selective thin coating film that absorbs or binds the species of interest, resulting in an increased mass loading of the device. The film thickness and the extent of the penetration of the shear wave determine the frequency. However, because the added mass or change in mass is very small, the device perturbation is small and sometimes negligible. As a result, the effects of other liquid parameters on the QCR frequency of oscillation were also investigated. In addition to the mass, it was shown that the viscosity and density of the solutions can affect the oscillation frequency.8 Specifically, AT-cut quartz crystal resonators have been utilized to monitor the viscosity of fluids. Theoretically, a relationship was derived for the change in the oscillation frequency of the QCR in terms of the material parameters of the liquid (density andviscosity) and the quartzcrystal (density and elastic modulus).8 This relation was obtained from a simple physical model which coupled the shear wave in the quartz crystal to a damped shear wave in the viscous liquid. On the basis of this model, the resonance phenomenon was treated as arising from the matching of the appropriate boundary conditions on the shear waves. Subsequently, a more rigorous calculation was obtained by modifying the stiffness term in the resonance equation to include piezoelectric stiffening of the crystal9 which gave a result of the same form. In all these studies, the change in the device resonant frequency can be related to a change in the mechanical boundary conditions. However, some work has also appeared in the literature in which mass loading or viscous coupling is (3) Nomura, T.; Minemura, A. Nippon Kagaku Kafshi 1980, 1621-1625. (4) Nomura, T.; Wanatak, M.; West, T. S. A M / . Chim. Acra 1985, 175, 107-
116.
M.;Kipling, A. L.; Duncan-Hewitt, W.C.; Rajakovic, Lj. V.; Cavic-Vlasak, B. A. Analyst 1991, 116, 881. (6) Bruckenstein, S.; Shay, M. Electrochim. Acra 1985, 30, 1295-1300. (7) Shumacher, R.;Borges, G.; Kanazawa, K. K. Surf. Sci. 1985, 163, L621(5) Thompson,
L626.
(8) Kanazawa, K. K.; Gordon, J. G., 11. A m / . Chim. Acra 1985, 175,99-105. (9) Shana, 2.A.; Radtke, D. E.; Kelkar, U.R.; Josse, F.;Haworth, D. T. AMI.
Chim.Acra 1990, 231, 317-320.
Analytical Chemfstty, Vol. 66, No. 13, July 1, 1994
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not the dominant interaction mechanism. For example, a ductive properties of thin film coated on a QCR surface can coated BAW detector has been used successfully to monitor also play an important role in the change in the parallel phase transition in liquid crystals and lipid multilayer films.I0J resonant frequency. Furthermore, it is known that the series In these studies, the sensor response cannot be attributed to resonant frequency is affected only by the changes in the mass loading alone, but rather tovisaelastic effect and changes mechanical properties of the contacting liquid and the parallel in dielectric constant that accompany phase changes. Moreresonant frequency shift is a function of both the mechanical over, several workers showed that a liquid dielectric constant and electrical properties of the liquid.12 Therefore, by and conductivity can be measured by immersing the two sides monitoring the changes in both resonant frequencies (series of the QCR in the ~olution.l~-'~ Some experiments have also and parallel), these QCR sensors can potentially be used to suggested that the device could be used as a sensor for dilute isolate the effects of the acoustoelectric interactions. QCR ionicliquidswith only oneside in contact with t h e s o l u t i ~ n . ~ ~ J ~sensors based on the acoustoelectric effect could be developed The results clearly were not due to mechanical loading since at low cost and with relatively high sensitivity. the density and viscosity of thedilute ionic solution are basically In the present work, it is shown that the development of those of the pure solvent. However, no adequate explanation such sensors is possible. By modifying the geometry of the was provided for the observed responses of the QCR in which electrodeon thesurfacein contact with thesolution,a transition theconductiveliquid is in contact with thegrounded electrode. region is created in which the lateral decaying acoustic field Note that even in the absence of imposed electric fields, an is enhanced. This field can interact with an adjacent electric double layer may form due to the inherent surface conductive solution which will result in a relatively large change charge at the crystal/liquid interface at most pH ~a1ues.l~ in the parallel resonance frequency of the QCR. Experiments However, the change in the electric double layer with solution are conducted with such electrode geometrieson AT-cut quartz properties would be too small to explain the observed response. which show that, when appropriately designed in conjunction In the studies, the two electrodes are concentric and have the with the volume of the cell, the ability of the modified QCR same diameter. It has been pointed out that the absence of to oscillate in liquids can be maintained while creating an a surface potential on a grounded electrode will prevent a acoustoelectric interaction. Examples are given by using proper acoustoelectric interaction with the conductive liquid. aqueous and nonaqueous ionic solutions to demonstrate the Such interaction can only occur on a charged unplated surface in the dielectric constant. The results are effect of change or an electrified plated surface.18 In the case of a charged in terms of a proposed equivalent circuit for the discussed unplated surface, there exists a surface potential associated liquid loaded QCR. It is shown that the proposed circuit is with the acoustic vibration. This surface potential generates a general circuit which can be used to analyze all particular an external electric field that can extend into an adjacent in contact with a viscous cases of a loaded QCR with one side conductive or dielectric solution, interacting with ions and and/or conductive liquid. dipoles in the solution. In the studies,l5J6since the liquid was in contact with the grounded electrode, any measured frequency change can only be attributed to interaction due to EXPERIMENTAL SECTION fringing fields at electrode edges. It is known that for two Electrode Design. A number of studies have been circular, concentric electrodes of equal diameter, the acoustic r e ~ r t e d I ~ on - ~ the 3 effect of electrode thickness and lateral vibration amplitude is very small at electrodeedges and decays dimensions on the vibrational pattern associated with each rapidly in the unelectroded region.I9 It is this rapidly decaying resonance of the QCR. The configuration and thickness of field in the nonelectroded region which somewhat interacts the partial electrodes on AT-cut quartz disks have a significant with the liquid, resulting in some perturbation. Note that the effect on the shape and strength of the thickness-shear results were not reproducible and found to vary with the displacement and on the resonant frequencies. Crystal position of the O-ring, the QCR, and the liquid cell. A slight vibration at the resonance is confined to the electroded region. misalignment was found to produce different results which An accurate description of the mass sensitivity distribution of clearly indicates effects of fringing fields in the unelectroded QCR resonators in liquid media has also shown that the mass region. sensitivity, df/dm, is maximum at the center of the circular It has become apparent that if one can create the electrode and low at the electrode edges.23 Calculations have appropriate conditions for an acoustoelectric interaction, also shown that for QCRs with identical, partial, concentric, changes in dielectric constant and/or conductivity of liquids circular electrodes, the trapped mode amplitude at the edge can be effectively measured using the parallel resonant of the electrodes has decreased to about 20% of its maximum frequency. Moreover, the change in dielectric and/or conamplitude. The mode then rapidly decreases exponentially thereafter in the unelectroded region. The profile of the (10) Muramatsu, H.; Suda,M.; Ataka, T.; Seki, A.; Tamiya, E.; Karubc, I. Scns. Actuators 1990, A21-A23, 362. acoustic vibration in both the electroded region and the (1 1) Wohltjen, H.; Dessy, R. AMI. Chcm. 1979, SI, 1470. unelectroded region can be obtained by solving the wave (12) Zhou, T. A.; Nie, L. H.; Yao, S. 2.J. EIectroaMI. Chem. 1990, 293, 1-18. (13) Nomura, T.; Maruyama, Mr. AMI. Chim. Acta 1983, 147, 365. equation in each region, subject to the appropriate boundary (14) Yao, S. 2.; Zhou, T. A. Anal. Chim. Acta 1988, 212, 61. conditions at the crystal surface and boundary between (15) Shana, Z. A,; Radtke, D. E.; Kelkar, U. R.; Haworth, D. T.; Jossc, F. IEEE Ultras. Symp. Proc. 1989, 567-571. electroded and nonelectroded regions. From the constitutive (16) JOSSC, F.; Shana, Z. A,; Radtke, D. E.; Haworth, D. T. IEEE Trans. Ultras. Ferrocl. Frea. Control 1990. 37. 359-368.
(17) Bockris, J. OM.; Reddy, A. K. N. Modern Electrochemistry; Plenum Prcss: New York, 1979; Vol. 2. (18) Yang, M.; Thompson, M. AMI. Chcm. 1993,6S, 3591-3597. (19) Martin, B. A.; Hager, H. E. J . Appl. Phys. 1989, 65, 2630-2635.
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(20) Byme, R. J.; Lloyd, P.; Spencer, W. J. J . Acoust. Soc. Am. 1%8,43,232-238. (21) Sckimoto, H. IEEE Tram. Sonics Ultras. 1984, 31, 664-669. (22) Tiersten, H. F. J . Acoust. Soc. Am. 1976,59,879-888. (23) Hillier, A. C.; Ward, M. D. AMI. Chem. 1992, 64, 2539-2554.
upper electrode (diameter di)
upper electrode (‘hag‘ elclectode)
~
‘lower
electrode
b) Cross-sectionalmew Mth sohm’on
Figure 1. (a) Schematic diagram showing the geomehy of both electrodes on A l e u t quartz did. Lower “hot” electrode diameter d, = 7 mm. Upper electrode diameter d, = 3, 4, 5, 6, OT 7 mm. (b) Cross-sectlonai view (through the diameter) of the QCR with liquid on the upper electroded surface.
equations in a piezoelectric the mechanical displacement and the electric potential in the quartz are coupled by the piezoelectric constant. This implies the existence of a surface potential whose lateral amplitude also decreases exponentially in the nonelectroded region. This surface potential in the nonelectroded region results in the so-called fringing fields at the electrode edges. From waveguide theory, reducing the upper electrode area, relative to the lower electrode area, will undoubtedly increase the amplitudeof the acoustic field within the transition area. This is due to the fact that a partially electroded region is created between the fully electroded region and the nonelectroded region, This partially electroded region represents a transition region in which the electrodes do not overlap. The amplitude of the acoustic vibration in the transition region is larger than it would be in the nonelectroded region. As a result, the amplitude of the surface potential in this transition region increases as does the corresponding electric field whose lines originate from the charged surface and ends on the grounded electrode. This electric field can then extend into an adjacent conductive medium which will result in a relatively large acoustoelectric interaction. Following the above reasoning, severalelectrode geometries and sizes were designed for 11-MHz AT-cut quartz disk using the partially concentric, circular, and ring configurations as shown in Figures 1 and 2. Figure l a shows the geometry which employs two partially concentric circular electrodes. The lower electrode which will face air has a diameter dz = 7 mm while the upper electrode which will face the liquid has a smaller diameter, dl. Several devices were fabricated with the upper electrode having a diameter dl = 3 , 4 , 5,6, and 7 mm. One such a device, for example, with dl = 4 mm and d2 = 7 mm,will be referred to as the “4-7” QCR. The second geometry, known here as the ”ring” configuration, is shown in Figure 2a with the upper electrode region defined by a radius, r, such that 1.5 mm G r 6 3.5 mm. Also shown in Figures 1b and 2b are the cross-sectional views through the ~~
(24) Tientcn,
1969.
lower electode
H.F. Linear Piezoelectric Hare Yibrarion; Plenum: New York,
b) Cross-sectionalview with solullon Figure 2. (a) SchemaUc diagram of the “ring“ electrode QCR. The m&ilkd of the is SUOh l.5 d 3*5 mm. (b) Cross-sectional view (through the diameter) of the “ring” electrode QcR with liquidOn the upper ‘Iectroded surface*
diameter of these QCRs with a conductive solution on the modified upper electroded surface. Apparatus. Several quartz crystal resonators used in this study were fabricated from 11-MHz AT-cut quartz crystals as described in Figures 1 and 2 by Marden Electronics (Burlington, WI). Silver electrodes about 1600 A thick deposited on an adhesion layer of chromium about 50 8, thick were used. A flow system was designed having a continuous sample delivery which consists of a miniature Plexiglas liquid flow cell, peristaltic pump (Cole-Parmer, Masterflex 755330), glass reservoir for the solution mixing, and a drain valve. The sample delivery system as well as the Plexiglas flow cell are detailed in ref 16. The cell was used with the previously described upper electrode surface of the AT-cut quartz crystal resonator facing the solution. The crystal was secured to the cell by a spring fixture using an O-ring for a liquid tight seal. The O-ring diameter is approximately 7 mm. The peristaltic pump driven with an electronic speed controller (Masterflex) was employed to ensure continuous sample delivery at a controlled rate of approximately 7 mL/min. A tuned-input, tuned-outputoscillator circuit was selectedI6in which the QCR is used in place of the input tank. In this type of configuration, the circuit was properly tuned so that the crystal can operate at its parallel resonant frequency. Loading of the QCRcaused by the introduction of a dilute ionic solution will be reflected at the output of the oscillator circuit. The properly tuned circuit was connected to a Hewlett-Packard (Model 5334B) frequency counter. Note that the upper electrode in contact with the solution is grounded. The output voltage measured on a Tektronixoscilloscope(Model 2213) as well as the parallel resonant frequency were monitored for about 20 min to ensure stability. Procedure. A measured amount of deionized water was first introduced into the flow system. A frequency drop of about 5000 Hz was observed. The parallel resonant frequency was then monitored for a short period of time to ensure stability. The experiments then consist of adding various amounts of concentrated conductive solutions into the sample tank to produce different dilutions. All solutions were prepared from reagent-gradechemicals. Test solutionswere made by dilution Analytical Chemisby, Voi. 00, No. 13, Ju& 1, 1994
1957
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CONDUCTIVITY (W’m-’) Flgure 3. Measured parallel resonant frequency shifts vs conductMty for aqueous dllute KCI solutions for the varlous devices of Flgures 1 and 2.
CONDUCIWITY (W’m’‘) Flguo 4. Measured parallel resonant freqquenoy shmS vs conductMty for the “4-7” QCR (d, = 4 mm and Cr, = 7 mm) for aqueous and nonaqueow dilute KCI sdutkns: 0, aqueow, eokrtkn; 0 , methanol SOiutiOn.
of stock solutions of potassium chloride or potassium nitrate. Successive aliquots of stock solution were added and mixed into the sample reservoir to produce different dilutions. Note that the pump is turned off and the solution is added and mixed in the tank before the pump is turned on again. This procedure has a negligible effect on the device frequency shift. The frequency drop due to the addition of each aliquot was then recorded after adequate time was allowed to ensure complete mixing and to achieve system stabilization. To ensure reproducibility of the QCR performance, the experiments were repeated after flushing the flow system and/or by using a new device (QCR). All experiments were conducted at a temperature of 24 OC. The various solutions’conductivities were also independently measured using a conductivity meter (Jenway 4010).
This problem can, however, be avoided by using a larger diameter O-ring, which will move the O-ring away from the edge of the electrodes. Figure 4 shows the results of the experiment using the “4-7” QCR with a nonaqueous KCl solution with methanol as the solvent. The lower dielectric constant solvent 6 = 32.9~~ is seen to affect the rate of change in the oscillation frequency as well as the total change in the frequency. It is seen that QCRs with those particular electrode geometries can be used as relatively high sensitivity detectors to effectively monitor dilute ionic solutions. For example, the highest measurable conductivity of aqueous solutions within the linear range of the frequency shift, using the 11-MHz QCRs, is approximately 0.16 i2-l m-I. The results clearly indicate the presence of an electrical interaction with the conductive liquid. They are not due todensity and/or viscosity change since the dilute KCl solutions in the range of the experiment have approximately the same density and viscosity as the pure solvent. For example, the density and viscosity of a 0.35wt 8KCI aqueous solution at 24 OC are about 1.002 g/cm3 and 0.999 cp, respectively, which are basically those of pure water. For such solution concentration, the series resonant frequency will decrease only by about 7 Hz. The responses in Figures 3 and 4 are analogous to those of the acoustic plate mode (APM) devices in contact with a dilute ionic s o l ~ t i o n . 2Such ~ ~ ~responses are characteristics of an acoustoelectric interaction between the solution and the fields associated with the APM. As shown in Figure 3, this interaction increases with the strength of the electric field. In the case of the ‘7-7” QCR with identical partial, circular,
RESULTS AND DISCUSSION The tested solutions include aqueous and nonaqueous potassium chloride solutions up to 0.35 wt 8. Methanol is used as the solvent for the nonaqueous solutions. The parallel resonant frequency shifts, A&, are plotted relative to the appropriate solvent. Figure 3 shows the measured frequency shifts as a function of the solution conductivity for several devices. These devices include QCRs as described in Figure la with the upper electrode diameter dl = 7, 6,5 , and 4 mm as well as QCR with upper “ring” electrode configuration. The “4-7” QCR (with dl = 4 mm) shows lower, but relatively high and comparable, sensitivity to the ”ring” QCR. This indicates a relatively strong interaction between the solution and the QCR. Note that the results were reproducible as indicated by indistinguishable measurements obtained by either using the same device after flushing the flow system or by using new but similar QCR. However, reproducibility could not be obtained with the “7-7” QCRs which, at times, show much lower sensitivity than that indicated in Figure 3. 1958 Analytlcel Chemistry, Vd. 66,No. 13, July 1, 1994
(25) JOSCC,F.; Haworth, D. T.; Kelkar, U. R.; Shana, 2.A. Electron. Lett. 1989, 25.
1446-1447.
(26) Josse,F.; Shana, 2.A.; Haworth, D. T.; Licw, S.; Grunzc, M. Sens. Actuators 1992, 9, 97-1 12. (27) Josse, F.;Shana, 2.A. IEEE Trans. Ultras. Ferroel. Frea. Control 1991.38. 297-304.
concentric electrodes of diameter dl = d2 = 7 mm, the results of the interaction are due to the fringing field at electrode ends. As indicated earlier, from the vibrational pattern of a resonator utilizing such electrode configuration, the trapped mode amplitude at the electrode edges is non-zero. The remaining vibration in the unelectroded region is basically a ‘nonguided” mode whose amplitude decreases rapidly from theelectrodeedges. By reducing theupper electrodediameter, a transition region is created between the trapped energy region and the region with evanescent wavesolution. In this transition zone in which the excitation electrodes do not overlap, the mode is a “weakly guided” mode. The acoustic vibration decays radially away from the electrode edges. However, the decay coefficient is small compared to that in the unelectroded region. In the transition region, the acoustic vibration is still relatively strong which results in a proportional surface potential and hence electric field. As the upper electrode radius decreases, the upper surface in contact with the liquid approaches the nonrealizable free surface condition in which the acoustoelectric interaction is strictly dictated by the piezoelectric electromechanical coupling coefficient.I6 In that case, maximum interaction would occur with the solution as suggested by the trend in the results of Figure 3. For QCRs, this maximum interaction case is nonrealizable since two electrodes are still needed to excite the acoustic vibration. Moreover, an electrode area covering at least 5% of the piezoelectric quartz plate area is required for an effective excitation20and oscillation of the crystal in the air. The actual minimum required diameter for the upper electrode will be dictated by the ability of the QCR to oscillate in a liquid environment. Such oscillation as indicated earlier is often impaired due to the dissipation of energy in the liquid from the vibrating crystal. A more detailed explanation of the acoustoelectric interaction which can be found e l ~ e w h e r e l will ~ . ~ be ~ summarized here. In the present case of a modified QCR, the acoustic vibration in the intermediate region generates a surface potential. The resulting electric field, whose lines originate from the surface in the intermediate region and end on the grounded electrode, extends into the conductive liquid resulting in acoustoionicinteraction. This interaction induces movement of ions and possible dipole reorientation. As a result, the ions in the conductive solution tend to redistribute to form a very thin double layer at the surface. As the ion concentration increases, the thin double layer tends to block the field from penetrating the bulk of the solution. This phenomenon is observed in the results which show that the rate of change of the frequency tends to zero as the ion concentrationincreases. This basically represents a screening of the electric field by the ions, which lowers the acoustoelectric interaction, as the solution concentration increases. Note, however, that the redistribution of ions to form a very thin double layer at the substrateliquid interface may also cause a local change in the density and viscosity of the solution near the surface. As a result, some change in the series resonant frequency may also occur due to the local change in the density andviscosity of the fluid near thesurface. The acoustoelectric induced change in the series resonant frequency is being further investigated.
t
I
f
a)
Figure 5. (a)Eleotrlcal equivalent circult of AT-cut quark crystal wlth two circular, cancentrlc electrodes of equal dlamtw. (b) Electrical equivalent circuit of AT-cut quartz crystal wlth electrode geometry as shown In Figures 1 or 2.
THEORY Equivalent C’trcuit. The results of Figures 3 and 4 can also be understood from an equivalent circuit of the loaded QCR. The circuit can be analyzed to obtain expressionsfor the change in the various critical frequencim in terms of the solution conductivity 6 and dielectric constant EL and the surface geometry. Equivalent circuits have been obtained to analyze QCRs with ofie or both electrodes immersed in a viscous l i q ~ i dor~ Wifh ~ . ~both ~ electrodes immersed in a conductive liquid.I2 However, to date, no circuit has been derived for the cases of a roaded QCR with electrode geometry as shown in Figures 1 and 2. When such a QCR with one face of the crystal is in contact with a liquid, it would be affected by the mechanical acoustic load as well as the electrical load of the liquid. It is convenient to use a lumped-parameter equivalent circuit to characterize a QCR for a narrow range of frequencies near resonance. In suc‘h a frequency range, the circuit parameters can be considered constant. Figure Ja shows the equivalent circuit of the unloaded QCR with equal partial concentric electrode oscillating in air. C, is the static capacitance arising from the two electrodes separated by the insulating quartz. C, is a function of the electrodes size, shape, and configuration. In general, when the QCR is encased, C, is slightly modified by a parasitic OY stray capacitance. Moreover, since quartz is piezoelectric, the electromechanicalcoupling gives rise to the motional arm branch (%,&, Cq)which is parallel with C,,. The motional inductance L, and capacitance C, are proportional to the mass and compliance of the vibrating crystal, respectively. The motional resistance 4 represents the mechanical dissipation of the QCR. For the case of a QCR with a smaller upper electrode or ‘ring” electrode as described in Figures 1 and 2, a more accurate representation of the equivalent circuit is shown in Figure 5b. In this case, the electrical arm of the circuit consists of the capacitance Coin parallel with the series combination of Cuand Ca. C, is the capacitancebetween the two electrodes (28) Muramatsu, H.;Tamiya,E.;K a ~ b c I., AMI. Chcm. 1988,60,2142-2146. (29) Martin,S.J.; Granstaff,V. E.; Fryc, G. C. AMI. Chcm. 1991,63,2272-2281.
A n e r L r c a I C t ? ~ Vd. , 66,No. 13, Ju& 1, 1994
VSSS
across the quartz plate in the intermediate region (rl Q r Q rz), and Carepresents the capacitance loading of the free space (or air) in the same intermediate region. In Figure 5b, the capacitance elements C,, C,,, and C, are defined as
I
'2zAo c, = d
cd1 c, = 7 &a
where e22 and eo are the dielectric constants of quartz and free space, respectively. d is the thickness of the quartz plate which is equal to a half wavelength (d = X/2). 1, represents the penetration length of the electric field from the transition region surface into free space to the grounded electrode. Ao Flguo 6. Electrical equivalent ckcult used to model the llqukcloaded is the surface area of the fully electroded region, and A1 is the QCR with electrode geometries as shown in Figures 1 and 2. The ionic conductive sdution is in contact with the modiRed electrodd upper area of the lower electrode which corresponds to the upper surface. This electrode is assumed grounded. surface area of the intermediate region in which the electrodes do not overlap. The exact values of these areas depend on the electrode configuration: (i) For the case of a concentric, circular electrode with equal diameter, radius r = rl = rz, A0 = ?rrz and A1 = 0. where A is the crystal/liquid interfacial area. The electrical loading of the liquid is represented by the (ii) For the QCR with electrodes configuration of Figure impedanceZ L = 1/ YLwhere the admittance YL= GL jwCL. 1, A0 = ? r q 2 and A1 = ?r(rzz - rlZ). The capacitance CLand resistance RL (or conductance GL) (iii) For the QCR with the upper ring electrode as in Figure are defined, respectively, as 2, A0 = ?r(rz2 - r02)and A1 = ?rr02(ro is the inner unelectroded circle radius). The series combination of C,, and Ca represents a model of a partially filled parallel plate capacitor, with a dielectric and (quartz, €22) placed on the lower plate leaving an air (e,) gap between it and the upper plate. The area of the plates is assumed to be AI. When one side of the QCR is in contact with a liquid as where 1~ is the penetration length of the electric field into the shown in Figure 1b or 2b, the equivalent circuit of the loaded liquid. Note that for pure dielectric solvents, u = 0 and RL QCR must be modified. The total mechanical properties of becomes an open circuit. In the case of dilute ionic solutions, , viscosity, r ] ~ relate , to the motionalthe liquid density, p ~and the liquid is treated as lossy dielectric material with a small arm parameters, while the electrical properties, dielectric conductivity u. Note that the dielectric constant of dilute constant EL, and conductivity u relate to the parallel capacitance solutions is approximately equal to that of the pure solvent. C,. Figure 6 shows the equivalent circuit of the loaded Analysis of the Critical Frequencies of the Loaded QCR modified QCR with one side in contact with a conductive Equivalent Circuit. By analyzing the equivalent circuit of the liquid. In Figure 6, the resistance, R,, and inductance, L.,,,, unloaded and loaded QCR,in the narrow range of frequencies are associated with the mechanical loading of the liquid and near resonance, various critical frequencies can be defined. related to the viscous losses and the fluid inertia, respectively. The series resonant frequency, b, is the critical frequency Because of the nature of the acoustic wave in the AT-cut which is most often used to characterize the mechanical loading quartz crystal, only the shear horizontal (SH) particle of a liquid.12.31 It also represents the frequency of maximum displacement is considered in the liquid. The hydrodynamic admittance. However, sincef, is invariant with changes in coupling between the QCR and the liquid is described in terms the electrical arm of the circuit, it is not a suitable indicator , the electrical loading of the liquid. All the other critical of an oscillating boundary layer of thickness6 = ( ~ ~ L / W P L ) ' / ~of where 6 is the decay length of the SH wave radiated into the frequencies, including the parallel resonant frequency,& are liquid and w is the angular frequency. The surface mechanical affected by changes in the electrical arm of the circuit and acoustic impedance of a semiinfinite layer of a Newtonian thus can be used to monitor electrical loading. The parallel liquid, represented by the entrained layer of effectivethickness frequency is given by the frequency of maximum impedance. In what follows, the admittance/impedance characteristics 6/2, can be expressed as30
+
(30) Mattin, S.J.; Frye, G. C.; Ricco, A. J.; Senturia, S.D. Anal. Chcm. 1993, 65. 2910-2922.
IS60
Analytical Chemktry, Vol. 66, No. 13, July 1, 1994
(31) ANSI/IEEE Std. 177-1966, Definitions and Methods of Measurement for Piezoelectric Vibrators.
of the circuit will be used to obtain expressions for the changes in both the series and parallel resonant frequencies,& andf,, as a function of the mechanical and electrical properties of a liquid. The admittance/impedance of the equivalent circuit of the liquid loaded QCR of Figure 6 can be characterized as follows. The admittance is expressed as
liquid loading. In that case, both the motional arm and the electrical arm are perturbed as indicated by eqs 6 and 5b. For example, in eq 5b, one can clearly observe the modification to Codue to the liquid loading. Following these techniques, eq 10 reduces to
Y=G+jB (4) where the conductance, G, and susceptance, B, are defined, respectively, by
where
Following the same perturbation approach, the nontrivial solution to eq 11 is
and
B=
-4
+wco+ R,’ + X12
wC,[GLZ + w2CL(CL + C,)] G:
+ w2(CL+ C,)’
(5b) or
In the above expressions, the quantities R1 and XIare defined by R , = R,
+ R,
(6a)
x, = w L , - - + w1 L ,
c q
(7)
1
(13)
co + cx
Solving eq 13 for the parallel resonant frequency wp (= 274,) yields
(6b) wc, The impedance associated with the same circuit is given by 1 Z == R +j X Y
+ WpZL, = 1 +-
u;Lq
UP=[--L [
+1 L, -+c, co+cx
(14)
For the case of QCR oscillating in air at the parallel resonant frequency, w,,,,, eq 13 reduces to
where the resistance, R, and reactance, X , are defined, respectively, as
R=- G G2
+
where Cxois obtained from eq 12 as (8d
CuCa cxo= cu + ca
and
X = - -B
+
G2 B2
The condition for the series resonance is that the reactance of the motional arm should vanish; that is
with Ca given by eq IC. Subtracting (13) from (15) and assuming that (w,,,, - wp)/wp0
