Anal. Chem. 1996, 68, 2219-2227
QCM Operation in Liquids: An Explanation of Measured Variations in Frequency and Q Factor with Liquid Conductivity Michael Rodahl*
Department of Applied Physics, Chalmers University of Technology & Go¨ teborg University, S-412 96 Gothenburg, Sweden Fredrik Ho 1o 1k
Department of Biochemistry and Biophysics, Chalmers University of Technology & Go¨ teborg University, S-413 90 Gothenburg, Sweden Bengt Kasemo
Department of Applied Physics, Chalmers University of Technology & Go¨ teborg University, S-412 96 Gothenburg, Sweden
Recently, several reports have shown that when one side of a quartz crystal microbalance (QCM) is exposed to a liquid, the parallel (but not the series) resonant frequency is influenced by the conductivity and dielectric constant of the liquid. The effect is still controversial and constitutes a serious complication in many applications of the QCM in liquid environments. One suggestion has been that acoustically induced surface charges couple to charged species in the conducting liquid. To explore this effect, we have measured the parallel and the series mode resonance frequencies, and the corresponding Q factors, for a QCM with one side facing a liquid. These four quantities have all been measured versus liquid conductivity, using a recently developed experimental setup. It allows the simultaneous measurement of the resonant frequency and the Q factor of an oscillating quartz crystal, intermittently disconnected from the driving circuit. Based on these results, a simple model together with an equivalent circuit for a quartz crystal exposed to a liquid is presented. The analysis shows that it is not necessary to infer the existence of surface charges (or other microscopic phenomena such as electrical double layers) to account for the influence of the liquid’s electrical properties on the resonant frequency. Our results show that the contacting conductive liquid, in effect, enlarges the electrode area on the liquid side and thereby changes the parallel resonant frequency. By proper design of the QCM measurement, perturbing effects due to the liquid’s electrical properties can be circumvented. The quartz crystal microbalance (QCM) has for a long time been used in vacuum and gaseous environments as an ultrasensitive weighing device,1 especially for film thickness monitoring. It consists of a thin disk of single-crystal quartz, with one metal electrode deposited on each side. When the crystal is connected to an oscillator, the crystal can be excited to oscillate at the (1) Czanderna, A. W.; Lu, C. In Applications of piezoelectric quartz crystal microbalances; Lu, C., Czanderna, A. W., Eds.; Elsevier: Amsterdam, 1984; Vol. 7, pp 1-18. S0003-2700(95)01203-0 CCC: $12.00
© 1996 American Chemical Society
combined system’s resonant frequency, f. The most common type of crystal is the so-called AT-cut crystal, where the crystal oscillates in a thickness shear mode. The principle of operation as a microbalance is that mass added to or removed from the crystal’s electrode(s) induces a frequency shift, ∆f, related to the mass change, ∆m.2 The high inherent sensitivity (,1 ng/cm2, i.e., submonolayer amounts of adsorbed species) derives from the high stability of the oscillator and the high resolution with which even very small frequency changes can be measured. Since Nomura and Okuhara3 showed that a crystal completely immersed in a liquid can also be excited to stable oscillation, the QCM has more recently become a tool in electrochemistry, the so-called electrochemical quartz crystal microbalance (EQCM).4,5 Several investigations have also shown that the QCM may be used as a sensor in biomedical sciences.6,7 However, these and other applications in the liquid phase are hampered by unsatisfactorily explained side effects, casting doubts on how to interpret the measured frequency changes. Several recent reports show that when one side of a QCM is exposed to a liquid, the parallel (but not the series) resonant frequency is influenced by the conductivity of the liquid.8-11 In a recent EQCM review article, Kanazawa and Melroy5 point out that “[this conductivity effect] would be a very serious additional contribution to the frequency shift; it would have to be understood in order to preserve the quantitative interpretation of frequency shift data”. The results presented here clarify this effect, and, equally importantly, they show that it can be fairly easily eliminated or accounted for in liquid phase measurements. (2) Lu, C. Applications of piezoelectric quartz crystal microbalances; Lu, C., Czanderna, A. W., Eds.; Elsevier: Amsterdam, 1984; Vol. 7, pp 19-61. (3) Nomura, T.; Okuhara, M. Anal. Chim. Acta 1982, 142, 281-284. (4) Buttry, D. A. Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker, Inc.: New York, 1991; Vol. 17, pp 1-86. (5) Kanazawa, K. K.; Melroy, O. R. IBM J. Res. Dev. 1993, 37, 157-171. (6) Ebersole, R. C.; Ward, M. D. J. Am. Chem. Soc. 1988, 110, 8623-8628. (7) Okahata, Y.; Ebato, H. Anal. Chem. 1991, 63, 203-207. (8) Josse, F.; Shana. Z. A.; Zong, H. Proc. IEEE Ultrason. Symp. 1993, 1, 425430. (9) Shana, Z. A.; Josse, F. Anal. Chem. 1994, 66, 1955-1964. (10) Dunham, G. C.; Benson, N. H.; Petelenz, D.; Janata, J. Anal. Chem. 1995, 67, 267-272. (11) Yang, M.; Thompson, M. Anal. Chem. 1993, 65, 3591-3597.
Analytical Chemistry, Vol. 68, No. 13, July 1, 1996 2219
Figure 1. Schematic illustration of an electrode layout used by Shana and Josse9 that showed an increased change in fp with increasing liquid conductivity, compared to the normal electrode layout, where the size of the electrode facing the liquid is equal to or larger than the air-side electrode size.
Variations in the resonant frequency with changes in liquid conductivity similar to those noted as for the QCM have earlier been observed with shear horizontal acoustic plate mode (SHAPM) devices with one side facing the liquid.12 A SH-APM device consists of a thin piezoelectric plate (e.g., quartz) onto which two interdigital transducers have been deposited on the side of the plate not facing the liquid. One of the transducers excites the acoustic wave, which is then detected by the other transducer. The strain that accompanies the propagation of an acoustic wave in a medium induces a separation of charges, provided that the strain is in a direction in which the medium is piezoelectrically active.13 Consequently, an uncompensated surface charge may propagate with the acoustic wave. If the surface is in contact with a liquid, these surface charges can interact with ions and dipoles in the liquid. This so-called acoustoelectric interaction affects the velocity and the attenuation of the propagating acoustic wave if the response times of the ions and the dipoles are short enough so that they reorganize themselves due to the varying surface charges.12 By coating the surface facing the liquid with a thin conductive layer, like gold, the surface charges are screened out and the acoustoelectric effect is avoided. In contrast to SH-APM devices, the quartz crystal in a QCM setup is, in almost all applications, coated with electrodes which then should screen any surface charges. However, if the electrode facing the liquid is smaller than or equal in size to the second electrode (not facing the liquid), the crystal oscillation will extend outside the electrode area on the liquid side.14 Josse et al. have suggested that an electrical field resulting from the lateral decaying acoustic field in the nonelectroded region interacts with the adjacent conductive solution.8,9 They showed that the conductivity-induced change in the parallel resonant frequency was greatly enhanced when the upper electrode (facing the liquid) only partially overlapped with the lower electrode, as illustrated in Figure 1.8,9 In the present work, we show that it is not necessary to infer the existence of oscillating surface charges in order to understand the change in fp with liquid conductivity. In effect, an increase of the liquid’s conductivity increases the effective electrode area facing the liquid. This, in turn, raises the shunt capacitance of the QCM and thereby changes the parallel resonant frequency. (12) Niemczyk, T. M.; Martin, S. J.; Frye, G. C.; Ricco, A. J. J. Appl. Phys. 1988, 64, 5002-5008. (13) Salt, D. Hy-Q handbook of Quartz Crystal Devices; Van Nostrand Reinhold Co. Ltd: Padstow, Cornwall, UK, 1987. (14) Sauerbrey, G. Arch. Elektrisch. U ¨ bertragung 1964, 18, 617-624.
2220 Analytical Chemistry, Vol. 68, No. 13, July 1, 1996
Figure 2. (a) Equivalent circuit of a quartz crystal resonator near resonance. When the crystal is freely oscillating in a gas or in vacuum, Z0 can be replaced by a capacitance, C0. Ztot represents the total impedance of the crystal. (b) When an oscillating crystal is shortcircuited, the electrical energy that is stored in the motional arm will be dissipated in R1 via the current im. Since no current will flow through Z0, the equivalent circuit which determines the resonant condition (i.e., the resonant frequency and dissipation factor) for the crystal in shortcircuit condition (series mode) is Zs, i.e., R1, C1, and L1 in series. (c) Under open-circuit condition (parallel mode), the current im will go through Z0, and hence the equivalent circuit will be equal to Zp, i.e., R1, C1, L1, and Z0 in series.
We will demonstrate that the resistive losses occurring in the liquid at intermediate liquid conductivities cause the parallel, but not the series, Q factor to decrease. EQUIVALENT CIRCUITS The electrical equivalent circuit of a quartz crystal resonator near resonance is depicted in Figure 2a.15 The circuit elements can be related to the physical properties of quartz, the perturbing mass layer, and the contacting liquid.15 The motional capacitance, C1, can be seen as a representation of the compliance of the quartz; the motional inductance, L1, represents the total oscillating mass (quartz plate + electrodes + overlayer); and the motional resistance, R1, characterizes the sum of the losses due to the motion of the quartz, including internal friction in the quartz, losses in the liquid, mounting losses, etc. In most cases, for a crystal operated in a nonconducting fluid, the parallel impedance, Z0, consists of a capacitance, C0eff, that comes from the overlap of the electrodes, C0, and various stray capacitances, e.g., between the QCM holder and the QCM electrodes. However, as shown in this work, Z0 cannot generally be represented by a single capacitor when the crystal is exposed to a conductive liquid. Precise analysis of the equivalent circuit reveals several characteristic frequencies.13 In this work, we will mainly consider the two that correspond to the natural modes of the resonator under short-circuit and open-circuit conditions, which we will term the series resonant frequency, fs, and the parallel resonant frequency, fp, respectively. When the crystal is short-circuited, (15) Martin, S. J.; Granstaff, V. E.; Frye, G. C. Anal. Chem. 1991, 63, 22722281.
the resonant condition will be determined solely by the motional arm impedance (Figure 2b), while under open-circuit condition it will be determined by the impedance of the motional arm in series with Z0 (Figure 2c). Series and parallel resonance, as defined above, occur at the frequencies where Zs and Zp, respectively, are real (purely resistive). [The series and the parallel resonant frequencies are usually defined respectively as the lower and the higher frequencies where the resonator’s impedance, Ztot, is purely resistive. In this definition, the series resonant condition depends (to some degree) on the parallel impedance. It is therefore not used in the present study. In most practical cases, there is no major difference between the two definitions.] The series resonant frequency is given by
1
fs )
2πxL1C1
(1)
and, in the case when Z0 can be equated by a single capacitance, C0eff, the parallel resonant frequency is given by
x
fp ) 1/2π
C0effC1 L1 eff C0 + C1
(2)
The dissipation factor, D, is proportional to the energy dissipated in the oscillatory system. The dissipation factor is the inverse of the more familiar Q factor and is defined by
D)
1 Edissipated ) Q 2πEstored
R1 2πfsL1
R1 + Re(Z0) 2πfpL1
(6)
where Fq and νq are the specific density and the shear wave propagation velocity in quartz, respectively, tq is the thickness of the quartz plate, Ff and tf are the density and thickness of the added film, respectively, and κ is the so-called sensitivity factor of the QCM to a change in the areal mass density. When the film is deposited the change in L1 is
∆L1 ) -
2∆fsL1 2L1 ) Ft fs Fqtq f f
(7)
With Fq ) 2648 kg/m3, νq ) 3340 m/s, and fs ) 6 MHz, we have κ ) 81 Hz µg-1 cm2. The proportionality between the change in resonant frequency and the evenly deposited mass has been the basis for using the piezoelectric quartz resonator as a microbalance.2,14 In eqs 6 and 7, it is assumed that the added mass is much smaller than the mass of the quartz disk, i.e., ∆fs/fs , 1 (∆L1/L1 , 1). It is also assumed that the mass is rigidly attached to the electrodes, with no slip or deformation due to the oscillatory motion, and that it does not change the oscillatory coupling to the surrounding medium. In this case, the change in dissipation factor due to the added mass is negligible. The changes in fs and Ds when the QCM is taken from air (vacuum) and exposed on one side to a fluid with viscosity ηf and density Ff are17,18
∆fs ) -
xfs
xFfηf
(8)
xFfηf
(9)
2xπtqFq
and
∆Ds )
1
xπfstqFq
(4) respectively. The corresponding changes in L1 and R1 are
∆L1 )
and similarly, we have the dissipation factor of the parallel resonance,
Dp )
2fs2 fs Fftf ) F t ) -κFftf Fqνq Fqtq f f
(3)
where Edissipated is the energy dissipated during one period of oscillation and Estored is the energy stored in the oscillating system.16 If the crystal is oscillating in its series mode, then the corresponding dissipation factor, Ds, is given by16
Ds )
∆fs ) -
(5)
(16) Smith, K. L. Electron. Wireless World 1986, July, 51-53. (17) Stockbridge, C. D. Vacuum Microbalance Techniques; Plenum Press: New York, 1966; Vol. 5, pp 147-178.
xFfηf
tqFqxπfs
(10)
and
∆R1 ) (provided that Z0 only consists of resistors and capacitors). The change in fs due to a mass load, ∆fs, can, for a film covering one of the electrodes evenly [The term “evenly” refers to the fact that the sensitivity of the quartz crystal microbalance varies over the electrode area14 (see below). The quoted mass sensitivity below is an average sensitivity for a uniform distribution of the mass over the electrode. The deposited mass is therefore conveniently expressed as mass per unit area.] be expressed as
L1
≈
2L1xπfs R1∆L1 Fη + tqFq x f f L1 2L1xπfs Fη tqFq x f f
(11)
Equations 8 and 10 assume small liquid loading (∆L1/L1 , 1). To a good approximation, the induced changes in fp and Dp due to the application of a film or liquid are equal to the changes in fs and Ds, respectively, if the film or the liquid does not induce a change in Z0. However, as shown below, the application of a (18) Rodahl, M.; Kasemo, B. Sens. Actuators B, in press.
Analytical Chemistry, Vol. 68, No. 13, July 1, 1996
2221
Figure 3. Experimental setup used to measure the parallel resonant frequency and the parallel dissipation factor. See text for details.
film or a liquid on one side of the QCM can under certain conditions change Z0 significantly. The changes in fp and Dp are then due to a combination of the changes in L1, R1, and Z0, and their magnitudes can in such cases be quite different from the corresponding series mode values. EXPERIMENTAL SECTION Recently, we developed an experimental system to measure the series and parallel mode resonant frequencies and the corresponding absolute dissipation factors of a QCM operating in vacuum, gas, or liquid.19,20 The method is based on the principle that when the driving power to a quartz oscillator is switched off at t ) 0, the voltage over the crystal, U, decays as an exponentially damped sinusoidal:
U(t) ) U0e-t/τ sin(2πft + φ), t g 0
(12)
where τ is the decay time constant and φ is the phase. The dissipation factor is inversely proportional to τ:
D ) 1/πfτ
(13)
A block diagram of the technical solution of the simultaneous fp and Dp measurements is shown in Figure 3 and is discussed in detail in ref 19. Extension to fs and Ds measurements is described briefly below and in more detail in ref 20. In brief, the measurements were done by periodically connecting and disconnecting the quartz crystal to/from the driving circuit via a computercontrolled relay. Disconnection causes the oscillating crystal to switch to its parallel resonant frequency and its amplitude of oscillation to die out exponentially in time with a time constant τp ∝ 1/Dp (eqs 5 and 13). The decay of the QCM oscillation is recorded on a digitizing oscilloscope using a high-impedance probe. The decay curve is subsequently transferred to a computer, and a numerical fit of eq 12 is performed. From the fit, both the resonant frequency and the dissipation factor are obtained (19) Rodahl, M.; Ho¨o ¨k, F.; Krozer, A.; Brzezinski, P.; Kasemo, B. Rev. Sci. Instrum. 1995, 66, 3924-3930. (20) Rodahl, M.; Kasemo, B. Rev. Sci. Instrum, in press.
2222 Analytical Chemistry, Vol. 68, No. 13, July 1, 1996
Figure 4. Experimental setup used to measure the series resonant frequency and the series dissipation factor. See text for details.
simultaneously. The sampling rate (including curve fitting) of fp and Dp is usually around 1 Hz, i.e., approximately one measurement per second. Hence, both the parallel resonant frequency and dissipation factor are measured simultaneously. The series mode setup used to simultaneously measure fs and Ds, shown in Figure 4, is similar to the parallel mode setup,20 but for the series mode, the crystal oscillation is sensed by measuring the current that flows through the crystal rather than the voltage over the crystal as in the parallel setup. For this measurement, the crystal is short-circuited by a 15 cm copper lead that ensures that the crystal oscillation is controlled by the motional arm elements and not by the parallel arm parameters. For both measurements, the stability with one crystal face immersed in water is better than 3 Hz in f and better than 2 × 10-6 in D for a 6 MHz, AT-cut sensor crystal. In air, the stabilities in f and D are better by a factor of more than 10. In the present measurements, we used mainly 6 MHz, polished AT-cut quartz crystals from Maxtek Inc. (Torrance, CA). The electrodes were made of a 15 nm thick evaporated chromium adhesion film with a 160 nm thick gold film evaporated on top. The electrode configuration is shown in Figure 5. The upper (liquid side) electrode is larger than the lower (air exposed) electrode. The electrode facing the liquid is grounded, and the other electrode is connected to the signal generator. The tab of the liquid-side electrode goes around the edge of the crystal to the air side so that both electrodes can be contacted from the lower (air) side. The crystal is placed in a home-built liquid cell as indicated in Figure 5. The liquid cell was made from Teflon and has a total volume of ∼2 mL. The design of the liquid cell is similar to the Maxtek Inc. liquid QCM cell. RESULTS In this section, we will first present the changes in f and D (series and parallel mode) caused by exposing one side of the QCM crystal to pure (nonconducting) water. Then we show how f and D vary as the conductivity of the water is increased by the addition of salt. Finally, we describe some important, complementary results, obtained by depositing liquid drops (conducting and nonconducting) at various positions on the QCM. The experimental results are analyzed and explained in the next section.
Figure 5. Electrode configuration of the Maxtek crystals. The liquidside electrode is going around the edge of the crystal to the air side, where both electrodes are contacted with spring loaded contacts.
Figure 6. Changes in fs, fp, Ds, and Dp as the conductivity of the liquid was increased by the addition of LiCl. The solid lines are the calculated changes in fs and Ds, eqs 8 and 9, respectively due to the change in liquid viscosity and density.
When one side of the 6 MHz QCM crystal was exposed to pure water, the changes, compared to operation in air, of the series and parallel resonant frequencies were -954 ( 3 Hz (close to the theoretical value of -948 Hz predicted by eq 8) and -2167 Hz, respectively. Note the large difference between ∆fp and ∆fs. The corresponding changes in the series and parallel dissipation factors were 34.2 × 10-5 ( 0.2 × 10-5 and 34.4 × 10-5 ( 0.2 × 10-5, respectively, which both are close to 31.8 × 10-5, the value predicted by eq 9. The changes in fs, fp, Ds, and Dp as the conductivity of the liquid is increased by the addition of LiCl are shown in Figure 6. The ∆f and ∆D values for pure water were set to zero. Over the whole conductivity range, ∆fs decreases slightly by ∼0.1 kHz and ∆Ds increases by about 3 × 10-5. In contrast, ∆fp decreases monotonically by as much as 1 kHz, i.e., about 10 times more than ∆fs.
∆Dp first increases by about 15 × 10-5 and then decreases again as the conductivity of the liquid increases. At the highest conductivity studied here, ∆Ds and ∆Dp are almost equal, in spite of the fact that ∆Dp was about 40 times larger than ∆Ds at an intermediate conductivity. These measurements of ∆fp, ∆fs, ∆Dp, and ∆Ds are central results of this study and will be analyzed in detail in the next section. The conductivities of the LiCl solutions were obtained by interpolation of tabulated values.21 The measured ∆f and ∆D values showed no hysteresis on going up and down in salt concentration. Use of other salt solutions (potassium chloride and potassium phosphate at pH 7) to change the liquid conductivity gave the same results, within experimental accuracy, as in Figure 6. Also shown in Figure 6 (solid lines) are the calculated variations in fs and Ds respectively due to the changes in viscosity and density of the liquid according to eqs 8 and 9. The viscosity and density values were taken from literature.21 The calculated values reproduce ∆fs and ∆Ds quite well, while both ∆fp and ∆Dp vary much more with the variation in the liquid’s conductivity, i.e., with the LiCl concentration. As a complement to the measurements of Figure 6, the responses in f and D to localized water droplets were studied as reported in ref 22. The f and D responses to droplets (2.8 mm in diameter), deposited at various positions on the QCM sensor, peaked in the center of the crystal and diminished to virtually zero at the edge of the upper (larger) liquid-side electrode. The behavior is very similar to how ∆f varies with position for solid deposits, first measured by Sauerbrey.14 The measured responses were well described by Gaussian curves. Droplets on the bare quartz outside the electrode area cause no significant changes in f or D. Since the f and D sensitivities to Newtonian liquids (like water) are proportional to the amplitude of oscillation, we conclude that there is no significant motion of the bare quartz outside the upper electrode. These results, for individual water droplets, were also found to be independent of the conductivity of the water drops and of the mode of crystal oscillation (series or parallel). We thus conclude that a QCM operated in air does not sense any difference between the deposition of nonconducting or conducting droplets. This result is at first sight quite surprising since a large difference was observed between fp and fs and between Dp and Ds when one side of the QCM was immersed into liquids of varying conductivity. In addition, similar measurements with droplets were made as follows. When an ∼10 µL drop (∼4 mm in diameter) of salt solution (conductivity ≈ 0.05 S/cm) was placed on the edge of the upper electrode at position 1, as depicted in Figure 7, the changes in both fs and fp were about -8 Hz, and the corresponding changes in Ds and Dp were ∼2 × 10-7 (Table 1). When a second drop (drop 2) was placed over the tab for the lower electrode, as illustrated in Figure 7, the resonant frequencies and dissipation factors (series and parallel) did not change significantly. Drops 1 and 2 were then linked together by gently poking the second drop with the tip of a pipet, causing the two drops to merge into a larger drop. The accompanying changes in fs and Ds were again negligible. However, the change in the parallel resonant frequency and the parallel dissipation factor were quite dramatic. (21) Weast, R. C., Ed. CRC Handbook of Chemistry and Physics; CRC Press Inc.: Boca Raton, FL, 1981-82. (22) Rodahl, M.; Kasemo, B. Submitted to Sens. Actuators B.
Analytical Chemistry, Vol. 68, No. 13, July 1, 1996
2223
Figure 7. Placement of saltwater drops on the top (liquid) side of the crystal. The numbers in the drops indicate the order in which they were placed on the crystal. See text for details. Table 1. Changes in fs, fp, Ds, and Dp Due to the Deposition of Saltwater Drops with a Conductivity of 0.05 S/m, as Shown in Figure 7a drop
∆fs (Hz)
∆fp (Hz)
∆Ds (10-7)
∆Dp (10-7)
1 -7 ( 4 (-8 ( 4) -8 ( 4 (-9 ( 4) 2 ( 4 (2 ( 4) 2 ( 3 (4 ( 4) 2 -1 ( 2 (0 ( 2) -3 ( 3 (-2 ( 3) 1 ( 3 (2 ( 3) 1 ( 3 (1 ( 3) 1 + 2 0 ( 1 (0 ( 1) -26 ( 2 (-10 ( 1) 0 ( 1 (0 ( 1) 8 ( 2 (0 ( 2) a The corresponding values for pure water drops are given in parentheses.
They changed by -26 Hz and +8 × 10-7, respectively, i.e., about 4 times more than when drop 1 was deposited. This effect was only seen when drop 1 was in contact with the grounded (top) electrode and drop 2 was placed above the tab of the lower electrode, as illustrated in Figure 7. When the above experiment was repeated with nonconducting (i.e., pure) water instead of conducting salt water, ∆fs and ∆Ds were, within measurement errors, the same as above. ∆fp and ∆Dp were also the same as with conducting drops in all but one case, namely when drops 1 and 2 were linked together (Table 1). In the latter case, fp decreased only 10 ( 1 Hz, and Dp did not change at all (within measurement errors). As discussed below, this effect on fp and Dp is a consequence of the liquid’s influence on the fringing field present on the liquid side between the upper and the lower electrodes. DISCUSSION For the analysis of the above results, it is important to consider the changes in stray capacitive and resistive pathways caused by the liquid via its dielectric and conducting properties, respectively. In air, the origin of the shunt capacitance, C0eff, is mainly (but not entirely) due to the capacitor formed by the two electrodes and the quartz. As schematically illustrated in Figure 8a, there is a small “spillover” of the electrical field, a so-called fringing field, that extends outside the electrode overlap when the crystal oscillates in the parallel mode (cf. Figure 2c). When the crystal oscillates in the series mode (as shown in Figure 2b), the electrodes are short-circuited, and hence there is no fringing field. (In our measurement setup, the crystal is short-circuited by a 15 cm long copper wire which has a small self-inductance of ∼90 nH and a negligible resistance. This small impedance of the 2224 Analytical Chemistry, Vol. 68, No. 13, July 1, 1996
Figure 8. (a) Schematic (not to scale) illustration (of a part) of the QCM seen from the side. The dotted lines are a qualitative representation of the electric field lines between the two electrodes when the crystal is driven in the parallel mode. (b) QCM exposed on one side to a nonconducting liquid. (c) QCM exposed on one side to a perfectly conducting liquid. (d) Electrode geometry used in Shana and Josse’s work9 (left) and in this work (right) are shown for comparison.
copper wire is neglected in the subsequent analysis, and the electrodes will be considered to be perfectly short-circuited in the series mode measurements.) This fringing field causes a stray capacitance that contributes to the shunt capacitance of the QCM. The size of this stray capacitance depends on the geometry and the dielectric constant of the media which the field lines pass through: the higher the dielectric constant, the larger the capacitance. Air and quartz have relative dielectric constants of 1 and 4.55, respectively. When the crystal is oscillating in air, the stray capacitance will be relatively small since the fringing field goes through a medium with low dielectric constant. When the crystal is submerged into a liquid with higher dielectric constant than air, like pure water with a relative dielectric constant of 81 at 18 °C,21 the stray capacitance is increased. When the liquid is nonconducting, the fringing field lines will go through the liquid, as schematically shown in Figure 8b. However, for a perfectly conducting liquid, the fringing field will be almost exclusively confined to the volume between the liquid and (the tab of) the lower electrode, as shown in Figure 8c. The stray capacitance will in this case be even larger than when the liquid is nonconducting, because the effective area of the liquid-side electrode is increased. When the liquid has an intermediate conductivity, the field lines will not be totally excluded (screened) from the liquid, which means that any free charges (ions) present in the liquid will move
Figure 10. Predicted (solid lines) and experimental (filled markers) ∆fpe and ∆Dpe versus liquid conductivity.
Figure 9. (a) Simple model to explain the conductivity effect. The resistance, Rf, is inversely proportional to the conductivity of the fluid on the upper side of the QCM. The capacitance, Cf, is proportional to the dielectric constant of the fluid. Cq is the capacitance formed between the lower electrode and the fluid. (b) Simple equivalent circuit of a quartz crystal resonator with one side contacting a liquid. Z0 is the elements within the dashed line.
along these field lines, i.e., a current, il, will flow in the liquid. This current will cause resistive energy dissipation proportional to the product of il2 and the resistivity of the liquid. There will be no such energy dissipation when the liquid is nonconductive (since there is no current), nor when the liquid is perfectly conducting (since there will be no resistivity). The energy losses will therefore have a maximum at intermediate liquid conductivity, explaining the maximum in Dp in Figure 6. An important observation from the drop experiments is that only when one of the drops was in contact with the upper electrode and the other drop was placed directly above the tab of the lower electrode was a shift in the parallel mode parameters observed. This suggests that the fringing fields are mainly confined to the space between the upper electrode and the tab of the lower electrode, as indicated in Figure 8. A Simple Equivalent Circuit. The qualitative picture above is all that is needed to (qualitatively) understand the influences of the liquid’s electrical properties on ∆f and ∆D, and why these influences are different in the series and parallel modes. To make a quantitative analysis requires a precise estimate of the stray capacitance and the resistive losses in the liquid due to the fringing fields. This is a rather complicated task but can be solved using, e.g., finite element methods. However, the main features of the system can be modeled using the simple equivalent circuit shown in Figure 9. This equivalent circuit has before been suggested by Josse et al.8,9 The stray capacitance is modeled by two independent capacitors, Cf and Cq, in series (Figure 9). They correspond to the part of the fringing field that goes through the fluid (air or liquid) on the upper side of the QCM and the quartz, respectively (Figures 8 and 9). Cf and Cq are therefore proportional to the dielectric constants of the fluid and the quartz, respectively. The resistive losses in the fluid are modeled by a resistor, Rf, which is inversely proportional to the fluid conductivity and connected in parallel with Cf. (All three parameters are, of course, influenced by the exact geometry of the electrodes and the fluid.) In Appendix A, it is shown how Cf, Rf, and Cq can be estimated.
Performance of the Simple Model. The measured changes in fp and Dp are due both to the changes in the fluid’s electrical properties (conductivity and dielectric constant) and the changes in its mechanical influences on the crystal oscillation (adsorption, compressional waves,23,24 viscosity, and density). It is possible to show, for our case, that small changes in the fluids mechanical properties induce virtually identical changes in fp and fs and Dp and Ds, respectively. Thus, in order to extract the liquid’s electrical effect on the parallel mode parameters, it is useful to define ∆fpe ≡ ∆fp - ∆fs and ∆Dpe ≡ ∆Dp - ∆Ds. Figure 10 shows the predicted (full lines) and measured (filled symbols) changes in ∆fpe and ∆Dpe versus conductivity. The fit is surprisingly good, considering the simplicity of the model. Note also that there is only one fitting parameter, namely a geometry factor, teq, that describes how the geometry of the electrode layout relates Rf and Cf respectively with the liquid’s dielectric constant and conductivity, as described in Appendix A. Role of Acoustically Induced Surface Charges. It has been suggested that an electrical field resulting from the lateral decaying acoustic field in the nonelectroded region interacts with the adjacent conductive solution.9 In the present work, we see no need for such an explanation since all the observed phenomena can be explained by the fringing fields outside the electrodes (Figure 8). This does not exlude acoustically induced surface charges, but if they exist, they do not contribute significantly to the observed conductivity effects for the following reason. As we have shown earlier,22 the motion in the quartz crystal is virtually identical during series and parallel operations. This means that any acoustically induced surface charges would be the same in the series and parallel modes. Further, any charges in the nonelectroded region caused by the acoustic motion in the quartz crystal will create an electric field that will not be affected by a short-circuiting of the electrodes. This implies that the series and the parallel mode resonant frequencies and dissipation factors should be affected in a similar manner by acoustoelectric interactions (acoustically induced surface charges). However, as clearly shown in Figure 6, the parallel and series modes are influenced very differently by the conductivity of the liquid (as has previously been noted for the parallel and series resonant frequencies8,9). It should be noted that it follows that the equivalent circuit shown in Figure 9 is not applicable to acoustoelectric interactions. The results from the drop experiments provide further support for the present model and against the acoustoelectric effect (23) Martin, B. A.; Hager, H. E. J. Appl. Phys. 1989, 65, 2627-2629. (24) Lin, Z.; Ward, M. D. Anal. Chem. 1995, 67, 685-693.
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(acoustically induced surface charges) as an explanation for the variation in parallel resonant frequency with liquid conductivity. The oscillatory motion in the quartz decreases in a Gaussian fashion away from the center of the electrodes,22 implying that the relative strength of an acoustoelectric interaction should decrease as the distance from the electrode center increases. However, this was not the case in the drop experiments. Only when (i) drop 1 was in contact with the grounded (top) electrode, (ii) drop 2 was placed directly above the tabs of the lower electrode, and (iii) the two drops were connected (Figure 7) did fp and Dp change significantly. No effect was observed if drop 2 was placed somewhere else on the bare quartz surface. Note that in no position on the bare quartz surface does a single liquid drop cause a significant response in ∆fp, ∆fs, ∆Dp, or ∆Ds, and that this statement is valid for both pure water and concentrated salt solution drops. (As shown in Appendix B, the effective extension of the electrode area by increasing the conductivity does not mean that the quartz oscillates with a significant amplitude in the extended electrode areas due to the energy trapping of the acoustic wave by the metal electrodes.) We therefore conclude that acoustoelectric interactions cannot explain the observed shifts in parallel mode resonant frequency and dissipation factor versus liquid conductivity. FINAL REMARKS In this work, we have shown that the fringing fields and thereby also the QCM electrode tabs, which are often neglected, may play an important role for both the resonant frequency and the dissipation factor responses of a QCM operated in a liquid. It is therefore important to consider all possible stray capacitance and resistance pathways when constructing a liquid QCM setup. In most applications where the QCM is operated in a liquid, the influences of the liquid’s conductivity and dielectric properties on the resonant frequency and dissipation factor are unwanted contributions that can interfere with the measurements. However, these effects can easily be avoided either by measuring fs and Ds (as defined in this work) or by letting the upper electrode cover the entire liquid side of the crystal. The latter eliminates/ minimizes the effective extension of the upper electrode and the dissipative losses due to an electrical current flowing in the liquid. The model presented here for how the dielectric constant and conductivity can change fp and Dp is not restricted to liquid overlayers. If, for example, the QCM is covered with a polymer film that changes its electrical properties upon sorption of some species, similar changes in fp and Dp are predicted to occur as in the liquid case. Also, as shown in Appendix C, the resonant frequency and dissipation factor (attenuation) of an SH-APM device can be affected by changes in stray capacitance and resistance due to the presence of a liquid in the same way as the QCM. As far as we have found, the dependence of the magnitude of the dissipation factor on the crystal mode has been neglected in the literature on QCM measurements. This study has shown not only the value and importance of measuring the dissipation factor (together with f) but also that additional information can be obtained by measuring it in both the series and the parallel modes. ACKNOWLEDGMENT The authors thank Ulrika Engstro¨m, Eric Steffen, and Andreas Kasemo for assistance with the measurements. We also thank 2226
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Craig Keller for fruitful discussions and express our gratitude for Vasile Mecea’s, Lennart Lindberg’s, and Guy Portnoff’s (Quartz Pro AB, Stockholm, Sweden) suggestions on the manuscript. This work was financially supported by the Swedish Research Council for Engineering Sciences (No. 92-951) and the Swedish Biomaterials Consortium (NUTEK and NFR, No. 90-02859). APPENDIX A Let Su be the quartz surface on the liquid (upper) side defined by the air (lower) electrode. The part of the air electrode covered by the liquid electrode is not included in Su. Further, let Cq be the capacitance formed between Su and the air electrode. If we make the approximation that Su is an equipotential surface, then Cq can be estimated using the equation for a parallel-plate capacitor:
Su Cq ≈ 0q tq
(14)
where q is the relative dielectric constant of AT-cut quartz. This approximation is reasonable since the major part of Su is a narrow band (∼2 mm wide) that lies close (0.28 mm) to a true equipotential surface (the lower electrode), while the mean distance to the upper electrode is ∼4.5 mm. We estimate that Su ≈ 58 mm2 in our setup, which gives Cq ) 8.4 pF using q ) 4.55.13 The equivalent capacitance, Cf, representing the influence of the fluid’s dielectric properties, is formed between Su and the upper electrode and is proportional to the fluid’s dielectric constant, i.e.,
Cf ) 0fΓ
(15)
where f is the relative dielectric constant for the fluid and Γ is a factor that depends on the geometry, on the electrical properties of the materials of the liquid cell, and on the layout of the QCM electrodes. Cf, or rather Γ, can be estimated by the measured values of fs and fp in air and pure water (Rf ) ∞) in the following manner. We define the effective shunt capacitance, C0eff, as C0 in parallel with Cf and Cq (cf. Figure 9a with Rf ) ∞), i.e.,
C0eff ) C0 +
CqCf Cq + Cf
(16)
From eqs 1 and 2, we have that
C0eff )
fs2C1 fp2 - fs2
(17)
The motional capacitance (in fF) can be approximated by13
C1 ≈ 0.105d2f
(18)
where d is the diameter of the smaller (in our case the lower) electrode and f the resonant frequency. We estimated C1 ) 31 fF, using d ) 7 mm. The motional inductance L1 can then be calculated from eq 1.
Table 2. Equivalent Circuit Parameters for Our Crystal Oscillating in Air and Watera parameter
air
water
fs (MHz) fp (MHz) C0 (pF) C1 (fF) L1 (mH) R1 (Ω) Cq (pF) Cf (pF)
5.975 424 5.981 376 15.5 31.0 22.9 11.6 8.4 0.1
5.974 470 5.979 209 15.5 31.0 22.9 308 8.4 7.7
a
Rf depends on the liquid conductivity and can be obtained from eq 21.
When the crystal is taken from air to a nonconducting fluid (other than air), Cf is increased by a factor equal to the relative dielectric constant of the fluid (the relative dielectric constant of air is taken as unity). This changes C0eff, and Cf can then be estimated as
Cf|air )
(C0eff|fluid,σf)0 - C0eff|air)Cq Cq(f - 1) - (C0eff|fluid,σf)0 - C0eff|air)f
(19)
In our case, we obtained, Cf ) 0.095f pF (i.e., Cf ) 7.7 pF in pure water with f ) 81). The equivalent resistor Rf is, like Cf, formed between Su and the upper electrode. In our setup, Rf is confined to the same geometry as Cf; we can hence use Cf to estimate Rf. By comparing eqs 14 and 15, we can define an equivalent fluid thickness, teq, as
teq ≡ Su/Γ
(20)
This means that Cf is equivalent to a parallel-plate capacitor with area Su and thickness teq filled with a dielectric with relative dielectric constant f. For a parallel-plate resistor with area Su and thickness teq filled with a medium of conductance σf, the resistance is equal to teq/σfSu. We therefore have
Rf )
teq 0f ) σfSu σfCf
(21)
The best fit (shown in Figure 10) was obtained for teq ) 5.5 mm, which is a quite reasonable value for our geometry (cf. Figure 5). An increase in teq increases the magnitude of the ∆Dpe peak and increases ∆fpe at zero conductivity but does not change the transition region, i.e., the conductivity at which ∆Dpe peaks or ∆fpe drops. This transition region is determined by the relationship between Rf and Cf (eq 21) and by the resonant frequency. The transition region can be shown to occur at
(
σfmax D ) 0f 1 +
C0Cq (C0 + Cq)Cf
)
(22)
Increasing the fundamental resonant frequency moves the transition region toward higher conductivity since Cq increases as the thickness of the quartz plate decreases. The equivalent circuit parameters obtained from the fit are given in Table 2. APPENDIX B The resonant frequency at which the crystal oscillates is basically determined by two factors, (i) the thickness of the quartz
Figure 11. Schematic illustration of the equivalent circuit model for the interaction of the SH-APM device with a liquid that is not due to acoustoelectric interactions. For simplicity, only two fingers of the interdigital transducers are shown. See text for details.
plate and (ii) the mass loading. The deposition of one of the metal electrodes (∼160 nm of Au on top of 15 nm of Cr) causes a decrease in the resonant frequency by more than 25 kHz for a 6 MHz crystal. This implies that the part of the crystal that is not coated by gold and instead is excited due to the presence of the conductive liquid (acting as an extended electrode) and has a resonant frequency more than 25 kHz higher than the driving frequency. Using a Q factor of 2800 (typical for a 6 MHz crystal oscillating with one side exposed to water), it is possible from the theory of the simple harmonic oscillator to calculate that the amplitude of oscillation in the uncoated area is reduced by a factor of more than 20. In addition, the vibration amplitude at the edges of the crystal is further damped because of the proximity to the O-ring mount. This, in turn, implies that the mass sensitivity in this extended electrode area is much smaller than that where the gold electrodes overlap. APPENDIX C If the interdigital transducers of a SH-APM device are located directly underneath the liquid, as indicated in reports on acoustoelectric effects measured with SH-APM devices,12 it is possible that the resonant frequency and the damping can change due to the conductivity and dielectric constant of the liquid for reasons other than acoustoelectric interactions. A model very similar to the one proposed for the conductivity effects for a QCM, oscillating with one side facing a liquid (Figure 9), is schematically illustrated in Figure 11. The capacitances across the plate (Cp) are formed between the uncoated side of the plate and the metal transducers on the other side of the plate. The fluid components, Cf and Rf, are formed between the surfaces on the uncoated plate directly above the interdigital transducers, as depicted in Figure 11. The magnitudes of the fluid elements depend, in the same way as described above, on liquid conductivity and dielectric constant. The relative importance of the mechanism proposed here (based on the influence of the liquid’s conductivity and dielectric constant on the equivalent circuit) and the acoustoelectric mechanism suggested by Niemczyk et al.12 for SH-APM devices remains to be tested. Received for review December 12, 1995. Accepted March 26, 1996.X AC951203M X
Abstract published in Advance ACS Abstracts, May 1, 1996.
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