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Analysis of compressional-wave influence on thickness-shear-mode resonators in liquids
A
ELSEVIER
PHYSICAL
Sensors and ActuatorsA 60 (1997) 40-48
Analysis of compressional-wave influence on thickness-shear-mode resonators in liquids Ralf Lucklum *, Stefan Schranz, Carsten Behling, Frank Eichelbaum, Peter Hauptmann Otto-von-Guericke-Universitiit Magdeburg. Institut fiir Prozeflmefltechnik und Elektronik, PF 4120. D-39016 Magdeburg, Germany
Abstract The operation of a thickness-shear-mode (TSM) resonator contacting a fiaite liquid layer has been analysed to investigate the effect of eompressional-wave generation. This effect is mainly related to the non-uniform shear velocity profile across the surface of a TSM device. Hydrophone measurements show two coils of !ongitudinal waves. Their influence on the TSM resonator response is studied with impedance analysis, varying the spacing between resonator and reflector as well as the reflecting conditions on the top side of the liquid layer. A characteristic response with a periodicity of A/2 is observed when the spacing of the liquid cavity or the liquid layer thickness is changed. It indicates standing longitudinal waves in the cavity. Their influence can be modelled with an additional complex impedance in the motional arm of the Butterworth-van-Dyke equivalent circuit representing an own (compressional) transmission line. Keywords: Compressionalwaves;Thickness-shear-moderesonators
i. Introduction The high sensitivity of acoustic-wave devices to mass changes at their surface enables them to be used as general gravimetric detectors, often called microbalances. An appropriate coating of such devices with chemically active fihns turns them into chemical sensors which are capable of measuring mass changes in the nanogram range, which is, in terms of concentration, equivalent to the ppm range, They were used first in gas analysis. These devices rely on simple physical absorption or more complex interfacial chemical events that result in mass changes, which alter the resonator frequency. In most applications the resonator works as the frequency-determining element of an electrical circuit. The resonance frequency is monitored. Typical frequency shifts due to analyte interaction are in the range of a few tens of hertz to a few kilohertz. Based on the Sauerbrey equation [ ! ], the resonance frequency shill is directly related to the analyte concentration. Quartz-crystal microbalances (QCMs) are also becoming increasingly appreciated for numerous fundamental investigations in the liquid phase. Despite the loss in quality factor of the resonator, which is the ratio of power stored in the resonator to the energy dissipated per cycle, these devices are successfully applied for the detection of chemically or bio-
logically significant analytes in liquids. Even though the validity requirements of Sauerbrey's relation are not fulfilled, many authors still interpret the frequency shift due to analyte sorption as the mass increase of the coating. As discussed in Refs. [2-.4], for example, the resonance frequency of the quartz crystal is also sensitive to changes in liquid density and viscosity and to changes in material parameters of the coating. In practice, the liquid conditions are quite stable under suitable experimental conditions. The material properties of the coating, however, may change significantly upon analyte sorption. This effect can lead to unexpected results. Recently published analyses of the behaviour of acoustic sensors [5,61 explain with theoretical models why under typical conditions for chemical sensing the validity of the often assumed proportionality between mass increase and frequency decrease is limited. The analyses are based on a one-dimensional model of wave propagation in a multilayered system (e.g., quartz-coating-liquid) [7,81. The propagation through a layer i can be described with the complex wave propagation constant, Z, and the layer thickness, hi, whereas the transmission depends on the acoustic impedance, Z~:
(1)
0 * Corresponding author. Tel.: +49 391 671 8310. Fax: +49 391 671
2609. E-mail:[email protected] 0924-4247/97/$17.00 © 1997ElsevierScienceS.A. All rightsreserved PIIS0924 ~247(96}01420-3
exp()'~hi)
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~, -1
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R. Lucklum et a~ I Sensors and Actuators A 60 (1997) 40-.48
~
4!
L qumz
liquid
B
~ I
liqu~!
(a) Fig. I. Butterwonh-van-Dykemodel i a) and the moregeneraltransmission-linemodel(b) in a quaxtz-tiquidcavit]/-reflectorarrangement.2 ~ represents the effectiveacousticimpedanceof the compressional-wavetransmissionlinewhile ~,,~r~ ~" is its electricalanalogue.Gzcan be neglectedbecausethe lower surfaceof the quartz is capsuled wtth 7=joo~lp/G and Z = ¢"~',, where oJ is the angular frequency. G is the complex shear modulus and p is the density of the layer material. Under tossy conditions G has a complex value G = G' +jG". Considering the layer i as a quadrupole with an input stress, ui, and a particle velocity, i~, at the top side and an output stress, u,_ ~, and particle velocity, i , 1, at the bottom of the layer, the transformation of the acoustic wave passing the layer can be calculated with matrix multiplication: [IJi(Z)~
U,(z)] =
Tt-lpi -
tT[Ui+,(z+hi)~ "'~i,~,(z+h,)]
(2)
In particular, the effective impedance Z,,,,--"u,(z)/i~(z) of a finite layer becomes a function of h, in the case of a stressfree surface:
Z~,,= Z ~ r tanh(y,h,)
(3)
Z ~ = p,c~is a material constant where c is the speed of sound. The procedure to calculate the impedance seen at the surface of the resonator is to stack the matrices,
~ . [.,+,(z+ Eh,)~ il (Z) ] - ' " ,M ...~'liti, +1(z + F.h,) ]
The index q marks quartz parameters, K is the electromechanical coupling factor, and a is the wave phase shift. Under certain conditions this rather complicated equation can be translbrrned into the modified Butterworth-van-Dyke equivalent circuit shown in Fig. I. ~ can be transformed into R~. and Lt. and reflects the semi-infinite liquid. In the following we analyse an effect that appears when a real device with relatively small lateral dimensions works in contact with a liquid: the generation ofcompressional waves by a shear-mode resonator [9-11 ]. Finite lateral dimensions of the resonator is a violation of the model assumption mentioned above. However, we will show that longitudinal waves generated by a laterally small TSM resonator influence its complex electrical impedance and therefore its resonance frequency. The value of the resonance frequency shift is that of a typical QCM experiment. The effect can be modelled within the transmission-line concept and results in an additional complex impedance, Z~o,~,. in the motional arm of the Butterworth-van- Dyke equivalent circuit. Z¢ompneeds to represent a complete own transmission line.
(.,(z)~_.
(4)
starting with the knowledge of the acoustic load at the top of the resonator, e.g., stress free Zt.~,= 0 for a gas or ZL~,=~= J ~ P t ~1 for a Newtonian liquid, which is characterized by its density, t~, and its viscosity, rh. The acoustic impedance can be transformed into the electrical impedance, Zel, using a suitable model for piezoelectricity. In particular, the impedance of a quartz disc which is in contact with a semi-infinite liquid is { 1
2 tan(~'~) + 1"~ ~ q ~LL
_~
2. Experimental A%cut quartz crystals (resonance frequency 5 MHz/10 MHz, electrode diameter 6 mm/ 12.5 ram, reflectively) were sealed on the bottom of a cylindrical cell. The cell was mounted on a goniometer table to align the quartz pianoparallel to the liquid surface and the reflector. Th.e vibrational direction of the resonator was defined as the x-axis. An aluminium disc, a Pyrex disc of 3 mm thickness and a 300 ixm silicon wafer were used as reflector materials. The reflector was mounted on a micrometer which provides movement of the reflector m the vertical direction, z, with an accuracy of I ~m. The whole equipment was mounted on a heavy vibration-free table. The measuring conditions were kept constant at 23°(2 and 25% relative humidity.
42
R. Lucklum et al. / Sensors and Actuator* A 60 (I 997) .I0--48
The 'active' resonance frequency was measured with an oscillator described in Ref. [ 12]. A Schlumberger impedance bridge with an impedance test kit was used to measure the real and imaginary parts of the impedance with 100 Hz/step centred about the resonance frequency. The spatial mass sensitivity was investigated with a mierodrop system which allows the positioning of single microlitre drops in a grid of 200 p.m x 200 p,m. A drop of constant volume was placed at a defined position at the quartz crystal surface and the initial frequency shift was measured. After evaporation of the liquid, the next drop was placed at the next point of the raster. The measurement of the longitudinal wave distribution was performed with a hydrophone needle with a diameter of ] mill. It was measured at a 1.6 mm distance from the quartz surface, substantially longer than the shear-wave decay length. During every measurement the oscillator was turned off to avoid electrical coupling. The x-v resolution of the stepper used was 200 ptm. The frequency measurements and the impedance analyses started with a rough adjustment of the measuring set-up. Then the well-cleaned measuring cell was filled carefully with distilled water containing 1 ppm surfaetant, The experiments were performed with a liquid columns of 8 and 15 mm height. For the final alignment of the sensor to the reflector, the eLand the ",/-axes of the goniometer table were turned and the reflector moved up and down along the z-axis within a 250 Ixm range. The frequency difference of the oscillator was used as a measure.
. -i
3. Results and discussion 3.1. Shear velocity distribution and longitudinal-wave distribution
If the shear displaccment is uniform across the surface of the TSM sensor, the in-plane oscillation of the resonator causes only a critically damped shear wave, which is radiated into the contacting fluid. This situation is given for a resonator with inlinite lateral dimensions as assumed in tbe one-dimensional model. The mechanical coupling perturbs the vibrating behaviour of the resonator as reflected in the acoustic load ZL,,,,,,,. Since the decay length 6 = (2rh/~p3 n/... for the shear wave is extremely short (e.g., 6=0.25 v,m for a 5 MHz resonator) the sensor should be insensitive to the spacing between the resonator and any reflecting boundary (apart from hydrostatic effects). By contrast, the real lateral dimensions of a quartz crystal are rather small, even if compared with the crystal thickness. This results in a non-uniform motion across the electrode surface. Various techniques have been employed to map the surface displacement across the crystal. We measured the mass sensitivity and assume F,'oportionality between mass sensitivity and shear velocity. The deposition of a small constant mass results in a frequency shift. The response depends on the position of this mass and is a measure of mass sensitivity. Fig. 2(a) depicts a normalized shear velocity distribution of the quartz disc where the shear velocity ofthe centre was set to one, Our result agrees roughly with earlier published investigations, e.g. by Benes et al. 1131, Martin and
'-
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I mm
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Fig. 2. Map of the normalized shear-velocity distribution (a) and the point-to-point velocity difference (b) with respect to the direction of vibration, x, calculated from mass sensitivityat different points of the sensor surface.
R. L u c k l u m et a t / S e n s o r s
ltlartz
from the centre of the quartz disc along the x-axis in both directions. The in-plane variation of the c(~mpressional-wave coils is not uniform; the highest gradient was found near the centre of the quartz disc 2 mm along the y-axis in both directions. From that one can expect a significant recoupling of the in-plane variation in the compressional component to the shear component with the same mechanism discussed for the generation of compressional waves as suggested by Schneider and Martin [ 11 ].
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a n d A c t u a t o r s A 6 0 ( i 9 9 7 ) 40--48
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3.2. Compressional-wave influence on resontmce f~'equency
clprttcll(If~Ofsh~ar c[l~placemeat
Io
5
0 % /illffl
5
Fig. 4 depicts the dependence of the serial resonance frequency on the height of the liquid column without a reflector. The frequency difference was related to the serial resonance frequency measured with a deadjusted reflector, which simulates a semi-infinite liquid (pseudo-semi-infinite). The sensor response is periodic. The periodicity is approximately 75 ~tm, nearly identical to the A/2 of a 10 MHz longitudinal wave in water as calculated from the veloci,y of sound. This observation is attributed to the presence o flongitudinal waves generated by the TSM resonator and the reflection of these waves oft the upper liqtaid interface. The same result can be obtained with an aluminium, silicon or Pyrex plate as reflector, only the maximum frequency difference varies slightly. The shape of the cur,,c and frequency differencedepend much more on the alignment accuracy. The better the alignment, the higher the difference frequency; the minima and maxima become more pronounced ann the transition between minimum and maximum gets very sharp. Fig. 5 shows the maximum difference frequency upon a (x-axis) and y (y-axis) deadjustment at an aluminium reflector distance of 15 mm. The maximum at 0 ° represents best alignment. The frequency response decreases rapidly within 0.3 ° angle of disalignment. The corresponding shift of the compressional wave spot is about 0. t6 mm, while the cavity spacing changes with 5.25 p.m ram- ' from the quartz disc
I0
Fig. 3, Compressional-wa,~e distributionmeasured with a hydrophone needle
in a distanceof 1.5 mmfromthe quartz-crystalsurlace.Thedeeperthe gray, the higherthe compressional-wa,,e amplitude. Hager [ 141, or Hillier and Ward [ 15]. However, we have to vote disturbances from acoustic effects which caused unexpected errors. These disturbances do not occur in the centre of the quartz disc. They arise 1-3 mm away frem the centre along the x-axis during evaporation of the microdrop and show a periodicity in time. We assume that compressionalwave resonance effects are responsible for this behaviour. From the Navier-Stokes equation, which describes fluid velocity components under sinusoidal, steady-state excitation, and the continuity equation we expect longitudinal wave generation at a position where the shear velocity, t,~, varies in the x-direction [10,161. Fig. 2(b) shows normalized point-to-point differences with respect to x of the distribution shown in Fig. 2(a). From that we expect two areas of compressional-wave generation symmetrical to the y-ax;s through the centre of the quartz electrode. The longitudinal-wave amplitude distribution, which was measured with the hydrophone, is shown in Fig. 3 and agrees with that assumption. The maximum intensity was found at a distance of 3 mm 150
100 N Z
50 '
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position 5 .
.
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.
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-200
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~ t ~
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-150
~
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thickness c h a n g e I I~n Fig. 4. Sensor response to changes of the height of a liquid coluran due to liquid evaporation, The approximate height was 8 mm. The periodicity of 75 ~.m could be observed during the complete measurement. The system was calibrated in a separate experiment.
44
R. Luck/urn et al. / Sensors and Actualors A 60 (1997) 40-48
2500
2000
~ ,1500 1000
500
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..... -5
, -3
, -1
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~ 1
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Fig. 5. Dependenceof the maximumresonancefrequencydifferenceon angulardisalignmentin boti~the x-di,ection(a) and y-direction(7). centre along the x- or y-axis. This means that while turning the x- or y-axis already within the central part of the compressional wave coil different interferometric conditions may arise. From these results it is evident that standing longitudinal waves with uniform phase conditions across the compressional-wave spot influence the shear resonance frequency significantly. Destructive and constructive interference of these waves seem to account for the minima and maxima as discussed by Lin and Ward [ 10].
3.3. lmpedance analyses Before running the impedance analysis, the reflector (aluminium in the demonstrated example) was perfectly aligned and adjusted with the oscillator at different positions equivalent to those marked with ! to 5 in Fig. 4. After that ~the quartz was disconnected from the oscillator and connecried with the test kit of the impedance analyser. Due to the extremely high sensitivity against any movement of the =J,perimental set-up, the influence of cable handling on the resonance frequency was controlled. Differences up to 100 Hz, especially at position i, have to be noted. Fig. 6 presents the plots of magnitude and phase of the electrical impedance. A significant deviation from the 'unperturbed' impedance curve of quartz immersed in water can be observed in the plots measured at the positions 1..-4while the curve related to position 5 represents a typical impedance plot of quartz which is in contact with water. The deviation can be characterized as a secondary maximum at position 1-3 (a) and position 1,2, and 4 (b) or as a second:.u'y minimum at position 4 (a) or position 3 (b) in the magnitude or phase plot, respectively. We attribute this deviation to the influence of the compressional wave and call it a secondary extreme. A closer analysis of the deviation shows an additional slight shoulder a few kilohertz after the secondary extreme. Both effects change to higher frequencies when the reflector distance decreases. Note that fineperiodicity in z is equivalent to 50 kHz under the experiraer~,tal conditions. After a further decrease of the
reflector distar..ce one can find a response nearly equivalent to that measured at por,ition 1. With that result the frequency response found with an oscillator is easy to understand. Because many oscillators work at zero degrees phase angle (serial or parallel resonance), it will be discussed in the phase plot. While decreasing the reflector distance, the secondary extreme moves along the 'unperturbed' phase curve causing a more or less monotonic shift of the whole phase curve, which finally results in a change in resonance frequency. As soon as this secondary extreme reaches zero degrees, the frequency change becomes more pronounced. At a certain position the phase plot crosses the zero degrees line three times in a small frequency range of about I to 2 kHz. This value is identical to the best oscillator measurements with aluminium or silicon as reflector material. This is the moment where the resonance frequency of an oscillator jumps from the maximum resonance frequency to the minimum resonance frequency (see Fig. 4). The 'width' of the secondary extreme related to the periodicity in frequency is equivalent to about 1.5 p.m cavity thickness change. It is obvious that additional serial elements L, C and R in the motional arm of the equivalent circuit as suggested in Ref. [ 10] are not sufficient to explain the deviations from the 'classical' impedance plot of a TSM resonator. They could be summarized to the well-known (unmodified) Butterworth-van-Dyke equivalent circuit, which gives a 'classical' impedance plot of a damped quartz crystal. The admittance locus plot, which is calculated from the measured impedance values at positions 2 ( solid line ) and 5 ( dashed line) in Fig. 7, gives a better idea how to model the effect. The second circle indicates a second resonance structure with different characteristic parameters.
3.4. Modelling As discussed in Section 1, a TSM resonator which is in contact with a semi.infinite liquid can be modelled with a single three-port u'ansmission line with a complex load at one port. The load is nearly constant within the measuring range.
45
R. Lucklum et al. / Sensors and Actuators A 60 (1997) 40--48 10000
(a)
~
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-80 : : : i : , i0030 f I k["z~ Fig. 6. Plotof the electricalimpedancemagnitude(a) and phaseangle(b) at differentreflectorpositions,whichare markedin Fig.4, Thecavityspacingwas about 15 ram.
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-0003 Fig, 7, Admittancelocuscurvecalculatedfromthe impedancemeasurements at the reflectorpositions2 (solid) and 5 (dashed). The circle measured at position 5 (see Fig. 7) reflects this situation. This model is equivalent to a single damped resonating structure. The compressional wave can affect only the acoustic behaviour of the quartz. It is, in terms of the Butterworthvan-Dyke model, the motional arm or, in terms of the transmission-line model, the port EF. The quartz motional
I
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elements of the Butterworth-van-Dyk¢ model are L, C and R. The viscous shear load is represented as the complex impedance g':~'~. R, L and C together with g ~ form a complex impedance with a characteristic frequency near the serial resona, tce frequency. The compressional-wave effect must be modelled with an additional complex impedance. This complex impedance. ~*,,~,~,,is attributed to the second circle found in the locus plot at position 2 (see Fig. 7). In contrast to ~ , ~ , , cannot be simplified to serial frequency-independent elements. To calculate the additional impedance we assume a compressional-wave transmission line which is formed by the liquid cavity. To model accurately the influence of energy storage and dissipation in the cavity on the resonator response, acoustic loss mechanisms must be accounted for. Loss mechanisms which are related to the wave propagation in the cavity are propagation loss, reflection loss and diffraction losses duc to spreading of the acoustic beam. While the shear wave experiences severe attenuation in the liqt~id,compressional wave propagation is almost lossless i a water. Reflection losses are modelled with the acoustic Io~d at the end of the transmission line. Due to the lack of inf:)nnation on the sound field, the diffraction losses must be eslimated.
46
R. Lucklum et aL / Ser,sors and Actuators A 60 (/997) 40-48
the surface impedance, Z2,~'~"~, calculated from equations I and 2 with a constant thickness hj and a varying co. Recall that the periodicity in frequency is about 50 kHz. The electrical parameters of the complex impedance • lectr ~,,mp, were• separated from the measurement at position 2 and a measurement with a pseudo-semi-inlinite liquid. To relate both impedances, it is necessary to include a coupling factor, which includes both the generated compressional-wave amplitude due to the shear-velocity distribution as well as the recoupling of the additional compressional acoustic load to a shear acoustic load. Progress is being made in establishing a quantitative relation between in-plane surface displacement and compressional wave generation. A few different concepts exist about the recoupling mechanism [9-I1]. Following Schneider and Martin [ 11 ], the shear acoustic impedance Z~n,:,~and the compressional acoustic impedance Z~,,,,o at the boundary between quartz and liquid have the same origin and can be added. The compressional impedance is weighted by a profile factor, P. The profile factor is a measure of the gradient in the surface displacement, normalized by the magnitude of the surface displacement. Fig. 8 depicts the electrical impedance of the compressional part (solid line) and the calculated effective acoustic
A more important 'loss' mechanism is related to the alignment of the measuring se~.-up. As outlined in the previous section, even a small error in angular alignment results in a significant error in the cavity spacing. Consequently, different interferometric patterns arise across the compressional-wave spot. each with its own acoustic load. This is not a real loss mechanism in terms of energy dissipation. However, the resutiing acoustic load from compressional waves is an average value, the thickness- or frequency-dependent compressional-wave effect being damped. The slight dent after the secondary circle indicates the necessity of another transmission line instead of a constant acoustic load at the upper port of the line. it is related to the reflector. Due to its finite thickness the characteristic impedance, Z,.h,, of the material has to be replaced by an effective acoustic impedance, Z~ff. Unfortunately Eq. (3) is not valid. Due to the assembly of reflector plate and micrometer, the upper surface of the plate is not stress free. Thus the overall acoustic compressional impedance seen at the quartz-liquid interface has to be calculated from Eq. (4). M= represents the liquid cavity, M2 the reflector. The acoustic load Z3 = u~/i~ has to be estimated. However, for simplicity the effective impedance of the reflector was assumed to be constant and
15
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~8
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10010
10015
10020
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Fig. 8. Plotof the realand imaginarycompressionalpansofthe acousticandelectricalimpedance.The acousticimpedancewas calculatedfromthe transmissionline modelwitha constantreflectorimpedance,whilethe electricalimpedancewas separatedfromimpedancemeasurements.
R. Luckluraetal./Sensors and Actuators A 60 f 1997) 40--48
impedance of the liquid cavily. The plots are quite similar+ only the disregard of the finite thickness of the reflector results in a significant deviation, While the calculated acoustic impedance is straight in that frequency range, the electrical impedance exhibits a secondary maximum which we attribute to the reflector line, Most influence on the impedance of the TSM resonator can be recognized when the compressional wave resonates in the liquid cavity, The standing wave has a node at the quartz-liquid boundary, represented by the maximum in the real part of the compressionai impedance and a vanishing imaginary part, This node perturbs the generation of compressional waves and therefore the shear movement. This effect is significant in a small region around the longitudinal-wave resonance, which depends on frequency, liquidlayer thickness and parallelism of the liquid boundaries. ......t is defined only by the compressional-wave propa~compr gation and interference and represents acoustic energy storage and loss in the liquid cavity. It has its own characteristic parameters which differ from that of the quartz crystal. Actually, this is valid for any composite resonator. In contrast to the typical situation with coated resonators, the compressional transmission line includes a number of nodes ( i.e., a ~,,w periodicity in frequency) and is tess damped (i.e., a small bandwidth). The so-called film resonance effect, which arises when a quartz crystal (or other acoustic transducer) is coated with a rubbery mater ial of sufficient thickness, has the same physical origin. The quality factor of the coating layer is much less than that of the compressional-wave cavity, It results in an acoustic impedance which is constant with respect to o in the measuring range.
4. Conclusions Longitudinal waves are generated by laterally small TSM resonators. They influence its complex electrical impedance and therefore its resonance frequency if they form standing waves in a parallel liquid cavity. This can be modelled with a compressional transmission line at the acoustic port of the TSM resonator. Resonance of the compressional wave results in a node at the quartz-liquid boundary. In contrast to other acoustic loads, the compressional acoustic load is sensitive to changes in frequency of 100-200 ppm. The value of this effect is that of a typical QCM experiment and h,~s theretbre to be considered during QCM experiment design. A simple non-parallelism between resonator and housing should be sufficient to avoid a compressional-wave influence on the sensor resp,nsc.
47
is gratefully acknowledged. Part of the work was supported by the German Academic Exchange Service.
References [ I] G. Saucrbrey,Verwendungyon Schwingquarzenz,ur W ~ n g d~ner Schichtcnund Mikrow~igung,Z Phys., /55 ( 19595 206-222. [2 ] S+J.Martin,V.E.Gransmffand G.C. Frye,Characterizationof a quax~z crystalmicrobalancewithsimallaneousmassand liquidloading,Aria/. Chem,. 63 ( 1991) 2272-2281. [3 ] S.J. Marlinand GC. Frye,Surfaceacousticwaveresponseto changes in viscoelastic film properties, Appl. Phys. Lett.. 57 (1990) 18671869. [41 C. Behlmg,R. Lucklttmand P. Hauptmann,in A.T. Augousti(ed.), Sept.~ors and their Applicatirars VII. Instituteof PhysicsPublishing. Brislol+UK, 1995. pp. 370-375. [51 S_J.Martin.G.C. Frye ~d S.D. Sentuna. Dynamicsand responseof polymer-coatedsurface acousticwave devices: effect of viscoelastic properties and filmresonance,Anal. Chem., 66 (1994) 2201-2219, [61 R. Lucklum+S. Rtisler.P. Hauptmann,J. Augeand J. Hanmann.Online dclectionof organicpollutantsin waterby selectivequartzcrystal microbalance+Proe. 6th int. Meeting CheraicalSe~vors, Gaithersburg, MD. 22-25 July. 1996, p. 204,On-linedetectionof organicpollulanls in water by thickness shear mode resonators. (extended version). Sensorx and Acnmwrs B. 35-36 (1996) 103-1 i I. [71 V.E, Granstaffand SJ. Mart,n,Characterizationof a thick~ss-shear modequartz resonatorwith multiplenvnpiezoelectriclayers,J. Appl~ Pity,v.. 75 ( 19945 1319-1329, [81 C. FiliS.tre,G. Bard;zcheand M. Valentin.Transmission-linemodelfor immersedqum'tz-crystalsensors,Sensors and Actuators A, 44 (1994) 137-144, [9 ] L_Tessier, F, Patat,N, Schmitt,G. Feuitlardand M. Thompson,Effect of the generation of comprcssional waves on the response of the thickness-shear mode acoustic wave sensor in liquids,Anal Chem.. 66 ( 19945 3569-3574_ [ I01 Z. Lin and M.D.Ward.The roleof longitudinalwavesin quartzcrystal microh',danceapplications in liquids. Anal. Chem., 67 (1995) 685693, [ II I T.W. Schneider and S.J. Martin, Influenceof compressionalwave generationon thickness-shearmoderesonatorresponseina fluid,Anal. Chem. 67 (1995) 3324-3335. [ 12] J. Auge,P. Hauptmann.J. H,'~rtmanr~,S. ROslerand R. Lucklum,New design for QCM sensors in liquids,Sensors and Actuators 1). 24-25 ( 1995543-48. [ 13] E. Bones,M. Schmidand g. Kravchenko,Vibrationmodesof rossloaded planoconvexquartz crystalresonators,J. Acoust Soc. Am.. 90 ( 1991) 700-706. [ 14] BA. Martin and H.E. Hager, Flow profile above a quartz crystals vibrating in liquid,J. Appl. Phys., 65 (1989) 2627-2629. [151A.C. Hillier and MD. Ward, Scanning electrochemical mass sensitivitymappingof the quartzcrystalmicrobalancein Iiquidmedia. Anal. Chem.. 64 ( 19925 2539-2554, [ 161 B.A. Martin and H.E Hager, Velocity profile on quartz crystals oscillatingin liquids.J. AppL Phys.. 65 (1989) 2630-2635.
Biographies Acknowledgements The authors wish to thank Peter Payne from UMIST (UK), where the hydrophone measurements were done. The work was supported by the German Research Foundation, which
R a l f Luckh+m has been employed at the Otto-von-Guericke-University as a senior lecturer at the Department of Electrical Engineering since 1986. In 1977 he received his Ph, D. degree, His thesis work was in the area of polymer physics, As a physicist, his special field is the theoretical
48
R. Lucklumet al. / Sensors and ActuatorsA 60 (1997) 40--48
description and testing of sensing effects of different types of acoustic-wave-based chemical sensors. His research interests also include resonant micromechanical sensors and the development of new sensor principles. Stefan Schranz studied physics at the Technical University Braunschweig and graduated in 1994 in material science. Since 1995 he has been working as a research associate on a three-year project to model quartz resonators with FEM His actual interests are concentrated on effects which are related to real dimensions of the resonator. Carsten Behling graduated at the Otto-von-Guericke-University in the Department of Physics in 1993, He is currently working on the last year of a four-year appointment at the Department of Electrical Engineering. He is now preparing his Ph.D. thesis. His research activities are concentrated on modelling thickness shear mode (TSM) sensors.
Frank Eichelbaum received his Ph,D. degree at the Ottovon-Guericke-University in 1987. His thesis work was on sensor electronics. Since 1986 he has been working at the Institute for Measurement Technology and Electronics as a research assistant. He is our specialist in design and simulation of analog and digital electronics for sensors. Peter Hauptmann is head of the group of sensors and microsystems at the Otto-von-Guericke-University as well as vice director of the ifak. During the time from 1968 to 1985 he worked at the Technical College Leuna-Merseburg as a lecturer and senior lecturer. In 1973 he received the Ph.D. degree from the Technical College Leuna-Merseburg for a thesis about polymer physics. He joined the Otto-von-Guericke-University in 1985, bringing with him great experience in sensor development and application. The aims of his research are more accurate, economical and reliable sensors for chemical and biological processes as well as sensors for other industries.
ELSEVIER
PHYSICAL
Sensors and ActuatorsA 60 (1997) 40-48
Analysis of compressional-wave influence on thickness-shear-mode resonators in liquids Ralf Lucklum *, Stefan Schranz, Carsten Behling, Frank Eichelbaum, Peter Hauptmann Otto-von-Guericke-Universitiit Magdeburg. Institut fiir Prozeflmefltechnik und Elektronik, PF 4120. D-39016 Magdeburg, Germany
Abstract The operation of a thickness-shear-mode (TSM) resonator contacting a fiaite liquid layer has been analysed to investigate the effect of eompressional-wave generation. This effect is mainly related to the non-uniform shear velocity profile across the surface of a TSM device. Hydrophone measurements show two coils of !ongitudinal waves. Their influence on the TSM resonator response is studied with impedance analysis, varying the spacing between resonator and reflector as well as the reflecting conditions on the top side of the liquid layer. A characteristic response with a periodicity of A/2 is observed when the spacing of the liquid cavity or the liquid layer thickness is changed. It indicates standing longitudinal waves in the cavity. Their influence can be modelled with an additional complex impedance in the motional arm of the Butterworth-van-Dyke equivalent circuit representing an own (compressional) transmission line. Keywords: Compressionalwaves;Thickness-shear-moderesonators
i. Introduction The high sensitivity of acoustic-wave devices to mass changes at their surface enables them to be used as general gravimetric detectors, often called microbalances. An appropriate coating of such devices with chemically active fihns turns them into chemical sensors which are capable of measuring mass changes in the nanogram range, which is, in terms of concentration, equivalent to the ppm range, They were used first in gas analysis. These devices rely on simple physical absorption or more complex interfacial chemical events that result in mass changes, which alter the resonator frequency. In most applications the resonator works as the frequency-determining element of an electrical circuit. The resonance frequency is monitored. Typical frequency shifts due to analyte interaction are in the range of a few tens of hertz to a few kilohertz. Based on the Sauerbrey equation [ ! ], the resonance frequency shill is directly related to the analyte concentration. Quartz-crystal microbalances (QCMs) are also becoming increasingly appreciated for numerous fundamental investigations in the liquid phase. Despite the loss in quality factor of the resonator, which is the ratio of power stored in the resonator to the energy dissipated per cycle, these devices are successfully applied for the detection of chemically or bio-
logically significant analytes in liquids. Even though the validity requirements of Sauerbrey's relation are not fulfilled, many authors still interpret the frequency shift due to analyte sorption as the mass increase of the coating. As discussed in Refs. [2-.4], for example, the resonance frequency of the quartz crystal is also sensitive to changes in liquid density and viscosity and to changes in material parameters of the coating. In practice, the liquid conditions are quite stable under suitable experimental conditions. The material properties of the coating, however, may change significantly upon analyte sorption. This effect can lead to unexpected results. Recently published analyses of the behaviour of acoustic sensors [5,61 explain with theoretical models why under typical conditions for chemical sensing the validity of the often assumed proportionality between mass increase and frequency decrease is limited. The analyses are based on a one-dimensional model of wave propagation in a multilayered system (e.g., quartz-coating-liquid) [7,81. The propagation through a layer i can be described with the complex wave propagation constant, Z, and the layer thickness, hi, whereas the transmission depends on the acoustic impedance, Z~:
(1)
0 * Corresponding author. Tel.: +49 391 671 8310. Fax: +49 391 671
2609. E-mail:[email protected] 0924-4247/97/$17.00 © 1997ElsevierScienceS.A. All rightsreserved PIIS0924 ~247(96}01420-3
exp()'~hi)
T~=
~, -1
(|)
R. Lucklum et a~ I Sensors and Actuators A 60 (1997) 40-.48
~
4!
L qumz
liquid
B
~ I
liqu~!
(a) Fig. I. Butterwonh-van-Dykemodel i a) and the moregeneraltransmission-linemodel(b) in a quaxtz-tiquidcavit]/-reflectorarrangement.2 ~ represents the effectiveacousticimpedanceof the compressional-wavetransmissionlinewhile ~,,~r~ ~" is its electricalanalogue.Gzcan be neglectedbecausethe lower surfaceof the quartz is capsuled wtth 7=joo~lp/G and Z = ¢"~',, where oJ is the angular frequency. G is the complex shear modulus and p is the density of the layer material. Under tossy conditions G has a complex value G = G' +jG". Considering the layer i as a quadrupole with an input stress, ui, and a particle velocity, i~, at the top side and an output stress, u,_ ~, and particle velocity, i , 1, at the bottom of the layer, the transformation of the acoustic wave passing the layer can be calculated with matrix multiplication: [IJi(Z)~
U,(z)] =
Tt-lpi -
tT[Ui+,(z+hi)~ "'~i,~,(z+h,)]
(2)
In particular, the effective impedance Z,,,,--"u,(z)/i~(z) of a finite layer becomes a function of h, in the case of a stressfree surface:
Z~,,= Z ~ r tanh(y,h,)
(3)
Z ~ = p,c~is a material constant where c is the speed of sound. The procedure to calculate the impedance seen at the surface of the resonator is to stack the matrices,
~ . [.,+,(z+ Eh,)~ il (Z) ] - ' " ,M ...~'liti, +1(z + F.h,) ]
The index q marks quartz parameters, K is the electromechanical coupling factor, and a is the wave phase shift. Under certain conditions this rather complicated equation can be translbrrned into the modified Butterworth-van-Dyke equivalent circuit shown in Fig. I. ~ can be transformed into R~. and Lt. and reflects the semi-infinite liquid. In the following we analyse an effect that appears when a real device with relatively small lateral dimensions works in contact with a liquid: the generation ofcompressional waves by a shear-mode resonator [9-11 ]. Finite lateral dimensions of the resonator is a violation of the model assumption mentioned above. However, we will show that longitudinal waves generated by a laterally small TSM resonator influence its complex electrical impedance and therefore its resonance frequency. The value of the resonance frequency shift is that of a typical QCM experiment. The effect can be modelled within the transmission-line concept and results in an additional complex impedance, Z~o,~,. in the motional arm of the Butterworth-van- Dyke equivalent circuit. Z¢ompneeds to represent a complete own transmission line.
(.,(z)~_.
(4)
starting with the knowledge of the acoustic load at the top of the resonator, e.g., stress free Zt.~,= 0 for a gas or ZL~,=~= J ~ P t ~1 for a Newtonian liquid, which is characterized by its density, t~, and its viscosity, rh. The acoustic impedance can be transformed into the electrical impedance, Zel, using a suitable model for piezoelectricity. In particular, the impedance of a quartz disc which is in contact with a semi-infinite liquid is { 1
2 tan(~'~) + 1"~ ~ q ~LL
_~
2. Experimental A%cut quartz crystals (resonance frequency 5 MHz/10 MHz, electrode diameter 6 mm/ 12.5 ram, reflectively) were sealed on the bottom of a cylindrical cell. The cell was mounted on a goniometer table to align the quartz pianoparallel to the liquid surface and the reflector. Th.e vibrational direction of the resonator was defined as the x-axis. An aluminium disc, a Pyrex disc of 3 mm thickness and a 300 ixm silicon wafer were used as reflector materials. The reflector was mounted on a micrometer which provides movement of the reflector m the vertical direction, z, with an accuracy of I ~m. The whole equipment was mounted on a heavy vibration-free table. The measuring conditions were kept constant at 23°(2 and 25% relative humidity.
42
R. Lucklum et al. / Sensors and Actuator* A 60 (I 997) .I0--48
The 'active' resonance frequency was measured with an oscillator described in Ref. [ 12]. A Schlumberger impedance bridge with an impedance test kit was used to measure the real and imaginary parts of the impedance with 100 Hz/step centred about the resonance frequency. The spatial mass sensitivity was investigated with a mierodrop system which allows the positioning of single microlitre drops in a grid of 200 p.m x 200 p,m. A drop of constant volume was placed at a defined position at the quartz crystal surface and the initial frequency shift was measured. After evaporation of the liquid, the next drop was placed at the next point of the raster. The measurement of the longitudinal wave distribution was performed with a hydrophone needle with a diameter of ] mill. It was measured at a 1.6 mm distance from the quartz surface, substantially longer than the shear-wave decay length. During every measurement the oscillator was turned off to avoid electrical coupling. The x-v resolution of the stepper used was 200 ptm. The frequency measurements and the impedance analyses started with a rough adjustment of the measuring set-up. Then the well-cleaned measuring cell was filled carefully with distilled water containing 1 ppm surfaetant, The experiments were performed with a liquid columns of 8 and 15 mm height. For the final alignment of the sensor to the reflector, the eLand the ",/-axes of the goniometer table were turned and the reflector moved up and down along the z-axis within a 250 Ixm range. The frequency difference of the oscillator was used as a measure.
. -i
3. Results and discussion 3.1. Shear velocity distribution and longitudinal-wave distribution
If the shear displaccment is uniform across the surface of the TSM sensor, the in-plane oscillation of the resonator causes only a critically damped shear wave, which is radiated into the contacting fluid. This situation is given for a resonator with inlinite lateral dimensions as assumed in tbe one-dimensional model. The mechanical coupling perturbs the vibrating behaviour of the resonator as reflected in the acoustic load ZL,,,,,,,. Since the decay length 6 = (2rh/~p3 n/... for the shear wave is extremely short (e.g., 6=0.25 v,m for a 5 MHz resonator) the sensor should be insensitive to the spacing between the resonator and any reflecting boundary (apart from hydrostatic effects). By contrast, the real lateral dimensions of a quartz crystal are rather small, even if compared with the crystal thickness. This results in a non-uniform motion across the electrode surface. Various techniques have been employed to map the surface displacement across the crystal. We measured the mass sensitivity and assume F,'oportionality between mass sensitivity and shear velocity. The deposition of a small constant mass results in a frequency shift. The response depends on the position of this mass and is a measure of mass sensitivity. Fig. 2(a) depicts a normalized shear velocity distribution of the quartz disc where the shear velocity ofthe centre was set to one, Our result agrees roughly with earlier published investigations, e.g. by Benes et al. 1131, Martin and
'-
I
o Bo 0 ¢.,", 04O O20 M
~3.f',£ -0.2C E
-0.4(
-0,80 tOO _
coordinates
(a)
I mrn
coordinates
I mm
(b)
Fig. 2. Map of the normalized shear-velocity distribution (a) and the point-to-point velocity difference (b) with respect to the direction of vibration, x, calculated from mass sensitivityat different points of the sensor surface.
R. L u c k l u m et a t / S e n s o r s
ltlartz
from the centre of the quartz disc along the x-axis in both directions. The in-plane variation of the c(~mpressional-wave coils is not uniform; the highest gradient was found near the centre of the quartz disc 2 mm along the y-axis in both directions. From that one can expect a significant recoupling of the in-plane variation in the compressional component to the shear component with the same mechanism discussed for the generation of compressional waves as suggested by Schneider and Martin [ 11 ].
T " ' ~
I~
I0
N~
•. I i_
5
""~ '"~ ~! ¢ :~V i
'¢ " " ;
.i '
;o
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•
,-
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s
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i
i
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i
43
a n d A c t u a t o r s A 6 0 ( i 9 9 7 ) 40--48
/
eI
3.2. Compressional-wave influence on resontmce f~'equency
clprttcll(If~Ofsh~ar c[l~placemeat
Io
5
0 % /illffl
5
Fig. 4 depicts the dependence of the serial resonance frequency on the height of the liquid column without a reflector. The frequency difference was related to the serial resonance frequency measured with a deadjusted reflector, which simulates a semi-infinite liquid (pseudo-semi-infinite). The sensor response is periodic. The periodicity is approximately 75 ~tm, nearly identical to the A/2 of a 10 MHz longitudinal wave in water as calculated from the veloci,y of sound. This observation is attributed to the presence o flongitudinal waves generated by the TSM resonator and the reflection of these waves oft the upper liqtaid interface. The same result can be obtained with an aluminium, silicon or Pyrex plate as reflector, only the maximum frequency difference varies slightly. The shape of the cur,,c and frequency differencedepend much more on the alignment accuracy. The better the alignment, the higher the difference frequency; the minima and maxima become more pronounced ann the transition between minimum and maximum gets very sharp. Fig. 5 shows the maximum difference frequency upon a (x-axis) and y (y-axis) deadjustment at an aluminium reflector distance of 15 mm. The maximum at 0 ° represents best alignment. The frequency response decreases rapidly within 0.3 ° angle of disalignment. The corresponding shift of the compressional wave spot is about 0. t6 mm, while the cavity spacing changes with 5.25 p.m ram- ' from the quartz disc
I0
Fig. 3, Compressional-wa,~e distributionmeasured with a hydrophone needle
in a distanceof 1.5 mmfromthe quartz-crystalsurlace.Thedeeperthe gray, the higherthe compressional-wa,,e amplitude. Hager [ 141, or Hillier and Ward [ 15]. However, we have to vote disturbances from acoustic effects which caused unexpected errors. These disturbances do not occur in the centre of the quartz disc. They arise 1-3 mm away frem the centre along the x-axis during evaporation of the microdrop and show a periodicity in time. We assume that compressionalwave resonance effects are responsible for this behaviour. From the Navier-Stokes equation, which describes fluid velocity components under sinusoidal, steady-state excitation, and the continuity equation we expect longitudinal wave generation at a position where the shear velocity, t,~, varies in the x-direction [10,161. Fig. 2(b) shows normalized point-to-point differences with respect to x of the distribution shown in Fig. 2(a). From that we expect two areas of compressional-wave generation symmetrical to the y-ax;s through the centre of the quartz electrode. The longitudinal-wave amplitude distribution, which was measured with the hydrophone, is shown in Fig. 3 and agrees with that assumption. The maximum intensity was found at a distance of 3 mm 150
100 N Z
50 '
~-
0
.
i°o°
P,.
I
position 5 .
.
. .
.
.
. .
~ -50
-200
posl~on I
~ t ~
.
-150
~
i
\j position 2 "--~b~
thickness c h a n g e I I~n Fig. 4. Sensor response to changes of the height of a liquid coluran due to liquid evaporation, The approximate height was 8 mm. The periodicity of 75 ~.m could be observed during the complete measurement. The system was calibrated in a separate experiment.
44
R. Luck/urn et al. / Sensors and Actualors A 60 (1997) 40-48
2500
2000
~ ,1500 1000
500
0
..... -5
, -3
, -1
"
~ 1
;
'
~ 3
5
aklgle of disalignment I degrees
Fig. 5. Dependenceof the maximumresonancefrequencydifferenceon angulardisalignmentin boti~the x-di,ection(a) and y-direction(7). centre along the x- or y-axis. This means that while turning the x- or y-axis already within the central part of the compressional wave coil different interferometric conditions may arise. From these results it is evident that standing longitudinal waves with uniform phase conditions across the compressional-wave spot influence the shear resonance frequency significantly. Destructive and constructive interference of these waves seem to account for the minima and maxima as discussed by Lin and Ward [ 10].
3.3. lmpedance analyses Before running the impedance analysis, the reflector (aluminium in the demonstrated example) was perfectly aligned and adjusted with the oscillator at different positions equivalent to those marked with ! to 5 in Fig. 4. After that ~the quartz was disconnected from the oscillator and connecried with the test kit of the impedance analyser. Due to the extremely high sensitivity against any movement of the =J,perimental set-up, the influence of cable handling on the resonance frequency was controlled. Differences up to 100 Hz, especially at position i, have to be noted. Fig. 6 presents the plots of magnitude and phase of the electrical impedance. A significant deviation from the 'unperturbed' impedance curve of quartz immersed in water can be observed in the plots measured at the positions 1..-4while the curve related to position 5 represents a typical impedance plot of quartz which is in contact with water. The deviation can be characterized as a secondary maximum at position 1-3 (a) and position 1,2, and 4 (b) or as a second:.u'y minimum at position 4 (a) or position 3 (b) in the magnitude or phase plot, respectively. We attribute this deviation to the influence of the compressional wave and call it a secondary extreme. A closer analysis of the deviation shows an additional slight shoulder a few kilohertz after the secondary extreme. Both effects change to higher frequencies when the reflector distance decreases. Note that fineperiodicity in z is equivalent to 50 kHz under the experiraer~,tal conditions. After a further decrease of the
reflector distar..ce one can find a response nearly equivalent to that measured at por,ition 1. With that result the frequency response found with an oscillator is easy to understand. Because many oscillators work at zero degrees phase angle (serial or parallel resonance), it will be discussed in the phase plot. While decreasing the reflector distance, the secondary extreme moves along the 'unperturbed' phase curve causing a more or less monotonic shift of the whole phase curve, which finally results in a change in resonance frequency. As soon as this secondary extreme reaches zero degrees, the frequency change becomes more pronounced. At a certain position the phase plot crosses the zero degrees line three times in a small frequency range of about I to 2 kHz. This value is identical to the best oscillator measurements with aluminium or silicon as reflector material. This is the moment where the resonance frequency of an oscillator jumps from the maximum resonance frequency to the minimum resonance frequency (see Fig. 4). The 'width' of the secondary extreme related to the periodicity in frequency is equivalent to about 1.5 p.m cavity thickness change. It is obvious that additional serial elements L, C and R in the motional arm of the equivalent circuit as suggested in Ref. [ 10] are not sufficient to explain the deviations from the 'classical' impedance plot of a TSM resonator. They could be summarized to the well-known (unmodified) Butterworth-van-Dyke equivalent circuit, which gives a 'classical' impedance plot of a damped quartz crystal. The admittance locus plot, which is calculated from the measured impedance values at positions 2 ( solid line ) and 5 ( dashed line) in Fig. 7, gives a better idea how to model the effect. The second circle indicates a second resonance structure with different characteristic parameters.
3.4. Modelling As discussed in Section 1, a TSM resonator which is in contact with a semi.infinite liquid can be modelled with a single three-port u'ansmission line with a complex load at one port. The load is nearly constant within the measuring range.
45
R. Lucklum et al. / Sensors and Actuators A 60 (1997) 40--48 10000
(a)
~
.
~
E
8,...,. 1000
posZdon 1
pos~
3 /
pOSition 5
'E posi~on 2
E
i
100
80
i
i
i
i 1 10010
i
t
i
t i 10020
"
~
t
~
f
i i 10030
Jr
I
I fl kHz
(b)
60 40
positio
~
~ 2o ~
o
~ -20
{ 40
/1
oo,,,oo,
L
-60
\ N
#/
°°"°"'
-80 : : : i : , i0030 f I k["z~ Fig. 6. Plotof the electricalimpedancemagnitude(a) and phaseangle(b) at differentreflectorpositions,whichare markedin Fig.4, Thecavityspacingwas about 15 ram.
-I 00
l
I
I
i
I
~
I
I
10010
000, 0.004
..........
""..
.,
0,0O3
"
'--
0.001
0
.( 08
-0.001 -0002
"" ',.,.. . . . . . . , . .
-0003 Fig, 7, Admittancelocuscurvecalculatedfromthe impedancemeasurements at the reflectorpositions2 (solid) and 5 (dashed). The circle measured at position 5 (see Fig. 7) reflects this situation. This model is equivalent to a single damped resonating structure. The compressional wave can affect only the acoustic behaviour of the quartz. It is, in terms of the Butterworthvan-Dyke model, the motional arm or, in terms of the transmission-line model, the port EF. The quartz motional
I
;
f
,~
;
~
10020
elements of the Butterworth-van-Dyk¢ model are L, C and R. The viscous shear load is represented as the complex impedance g':~'~. R, L and C together with g ~ form a complex impedance with a characteristic frequency near the serial resona, tce frequency. The compressional-wave effect must be modelled with an additional complex impedance. This complex impedance. ~*,,~,~,,is attributed to the second circle found in the locus plot at position 2 (see Fig. 7). In contrast to ~ , ~ , , cannot be simplified to serial frequency-independent elements. To calculate the additional impedance we assume a compressional-wave transmission line which is formed by the liquid cavity. To model accurately the influence of energy storage and dissipation in the cavity on the resonator response, acoustic loss mechanisms must be accounted for. Loss mechanisms which are related to the wave propagation in the cavity are propagation loss, reflection loss and diffraction losses duc to spreading of the acoustic beam. While the shear wave experiences severe attenuation in the liqt~id,compressional wave propagation is almost lossless i a water. Reflection losses are modelled with the acoustic Io~d at the end of the transmission line. Due to the lack of inf:)nnation on the sound field, the diffraction losses must be eslimated.
46
R. Lucklum et aL / Ser,sors and Actuators A 60 (/997) 40-48
the surface impedance, Z2,~'~"~, calculated from equations I and 2 with a constant thickness hj and a varying co. Recall that the periodicity in frequency is about 50 kHz. The electrical parameters of the complex impedance • lectr ~,,mp, were• separated from the measurement at position 2 and a measurement with a pseudo-semi-inlinite liquid. To relate both impedances, it is necessary to include a coupling factor, which includes both the generated compressional-wave amplitude due to the shear-velocity distribution as well as the recoupling of the additional compressional acoustic load to a shear acoustic load. Progress is being made in establishing a quantitative relation between in-plane surface displacement and compressional wave generation. A few different concepts exist about the recoupling mechanism [9-I1]. Following Schneider and Martin [ 11 ], the shear acoustic impedance Z~n,:,~and the compressional acoustic impedance Z~,,,,o at the boundary between quartz and liquid have the same origin and can be added. The compressional impedance is weighted by a profile factor, P. The profile factor is a measure of the gradient in the surface displacement, normalized by the magnitude of the surface displacement. Fig. 8 depicts the electrical impedance of the compressional part (solid line) and the calculated effective acoustic
A more important 'loss' mechanism is related to the alignment of the measuring se~.-up. As outlined in the previous section, even a small error in angular alignment results in a significant error in the cavity spacing. Consequently, different interferometric patterns arise across the compressional-wave spot. each with its own acoustic load. This is not a real loss mechanism in terms of energy dissipation. However, the resutiing acoustic load from compressional waves is an average value, the thickness- or frequency-dependent compressional-wave effect being damped. The slight dent after the secondary circle indicates the necessity of another transmission line instead of a constant acoustic load at the upper port of the line. it is related to the reflector. Due to its finite thickness the characteristic impedance, Z,.h,, of the material has to be replaced by an effective acoustic impedance, Z~ff. Unfortunately Eq. (3) is not valid. Due to the assembly of reflector plate and micrometer, the upper surface of the plate is not stress free. Thus the overall acoustic compressional impedance seen at the quartz-liquid interface has to be calculated from Eq. (4). M= represents the liquid cavity, M2 the reflector. The acoustic load Z3 = u~/i~ has to be estimated. However, for simplicity the effective impedance of the reflector was assumed to be constant and
15
r-o-ReIz0a000~
~'13
I--
Re(Zc eIectr) I
c '° i
~9 ~
7
E
5
u
3
I
,
i
g 1 -1
.10
10n05
10010
10015
10020
10025
f/kHz
8
7.02Irn(Zc ac0ust)] I.--Im(Zc elect0 ~.~
~8
E 4o
o -70 ,~
10005
10010
10015
10020
t0025
f I kHz
Fig. 8. Plotof the realand imaginarycompressionalpansofthe acousticandelectricalimpedance.The acousticimpedancewas calculatedfromthe transmissionline modelwitha constantreflectorimpedance,whilethe electricalimpedancewas separatedfromimpedancemeasurements.
R. Luckluraetal./Sensors and Actuators A 60 f 1997) 40--48
impedance of the liquid cavily. The plots are quite similar+ only the disregard of the finite thickness of the reflector results in a significant deviation, While the calculated acoustic impedance is straight in that frequency range, the electrical impedance exhibits a secondary maximum which we attribute to the reflector line, Most influence on the impedance of the TSM resonator can be recognized when the compressional wave resonates in the liquid cavity, The standing wave has a node at the quartz-liquid boundary, represented by the maximum in the real part of the compressionai impedance and a vanishing imaginary part, This node perturbs the generation of compressional waves and therefore the shear movement. This effect is significant in a small region around the longitudinal-wave resonance, which depends on frequency, liquidlayer thickness and parallelism of the liquid boundaries. ......t is defined only by the compressional-wave propa~compr gation and interference and represents acoustic energy storage and loss in the liquid cavity. It has its own characteristic parameters which differ from that of the quartz crystal. Actually, this is valid for any composite resonator. In contrast to the typical situation with coated resonators, the compressional transmission line includes a number of nodes ( i.e., a ~,,w periodicity in frequency) and is tess damped (i.e., a small bandwidth). The so-called film resonance effect, which arises when a quartz crystal (or other acoustic transducer) is coated with a rubbery mater ial of sufficient thickness, has the same physical origin. The quality factor of the coating layer is much less than that of the compressional-wave cavity, It results in an acoustic impedance which is constant with respect to o in the measuring range.
4. Conclusions Longitudinal waves are generated by laterally small TSM resonators. They influence its complex electrical impedance and therefore its resonance frequency if they form standing waves in a parallel liquid cavity. This can be modelled with a compressional transmission line at the acoustic port of the TSM resonator. Resonance of the compressional wave results in a node at the quartz-liquid boundary. In contrast to other acoustic loads, the compressional acoustic load is sensitive to changes in frequency of 100-200 ppm. The value of this effect is that of a typical QCM experiment and h,~s theretbre to be considered during QCM experiment design. A simple non-parallelism between resonator and housing should be sufficient to avoid a compressional-wave influence on the sensor resp,nsc.
47
is gratefully acknowledged. Part of the work was supported by the German Academic Exchange Service.
References [ I] G. Saucrbrey,Verwendungyon Schwingquarzenz,ur W ~ n g d~ner Schichtcnund Mikrow~igung,Z Phys., /55 ( 19595 206-222. [2 ] S+J.Martin,V.E.Gransmffand G.C. Frye,Characterizationof a quax~z crystalmicrobalancewithsimallaneousmassand liquidloading,Aria/. Chem,. 63 ( 1991) 2272-2281. [3 ] S.J. Marlinand GC. Frye,Surfaceacousticwaveresponseto changes in viscoelastic film properties, Appl. Phys. Lett.. 57 (1990) 18671869. [41 C. Behlmg,R. Lucklttmand P. Hauptmann,in A.T. Augousti(ed.), Sept.~ors and their Applicatirars VII. Instituteof PhysicsPublishing. Brislol+UK, 1995. pp. 370-375. [51 S_J.Martin.G.C. Frye ~d S.D. Sentuna. Dynamicsand responseof polymer-coatedsurface acousticwave devices: effect of viscoelastic properties and filmresonance,Anal. Chem., 66 (1994) 2201-2219, [61 R. Lucklum+S. Rtisler.P. Hauptmann,J. Augeand J. Hanmann.Online dclectionof organicpollutantsin waterby selectivequartzcrystal microbalance+Proe. 6th int. Meeting CheraicalSe~vors, Gaithersburg, MD. 22-25 July. 1996, p. 204,On-linedetectionof organicpollulanls in water by thickness shear mode resonators. (extended version). Sensorx and Acnmwrs B. 35-36 (1996) 103-1 i I. [71 V.E, Granstaffand SJ. Mart,n,Characterizationof a thick~ss-shear modequartz resonatorwith multiplenvnpiezoelectriclayers,J. Appl~ Pity,v.. 75 ( 19945 1319-1329, [81 C. FiliS.tre,G. Bard;zcheand M. Valentin.Transmission-linemodelfor immersedqum'tz-crystalsensors,Sensors and Actuators A, 44 (1994) 137-144, [9 ] L_Tessier, F, Patat,N, Schmitt,G. Feuitlardand M. Thompson,Effect of the generation of comprcssional waves on the response of the thickness-shear mode acoustic wave sensor in liquids,Anal Chem.. 66 ( 19945 3569-3574_ [ I01 Z. Lin and M.D.Ward.The roleof longitudinalwavesin quartzcrystal microh',danceapplications in liquids. Anal. Chem., 67 (1995) 685693, [ II I T.W. Schneider and S.J. Martin, Influenceof compressionalwave generationon thickness-shearmoderesonatorresponseina fluid,Anal. Chem. 67 (1995) 3324-3335. [ 12] J. Auge,P. Hauptmann.J. H,'~rtmanr~,S. ROslerand R. Lucklum,New design for QCM sensors in liquids,Sensors and Actuators 1). 24-25 ( 1995543-48. [ 13] E. Bones,M. Schmidand g. Kravchenko,Vibrationmodesof rossloaded planoconvexquartz crystalresonators,J. Acoust Soc. Am.. 90 ( 1991) 700-706. [ 14] BA. Martin and H.E. Hager, Flow profile above a quartz crystals vibrating in liquid,J. Appl. Phys., 65 (1989) 2627-2629. [151A.C. Hillier and MD. Ward, Scanning electrochemical mass sensitivitymappingof the quartzcrystalmicrobalancein Iiquidmedia. Anal. Chem.. 64 ( 19925 2539-2554, [ 161 B.A. Martin and H.E Hager, Velocity profile on quartz crystals oscillatingin liquids.J. AppL Phys.. 65 (1989) 2630-2635.
Biographies Acknowledgements The authors wish to thank Peter Payne from UMIST (UK), where the hydrophone measurements were done. The work was supported by the German Research Foundation, which
R a l f Luckh+m has been employed at the Otto-von-Guericke-University as a senior lecturer at the Department of Electrical Engineering since 1986. In 1977 he received his Ph, D. degree, His thesis work was in the area of polymer physics, As a physicist, his special field is the theoretical
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description and testing of sensing effects of different types of acoustic-wave-based chemical sensors. His research interests also include resonant micromechanical sensors and the development of new sensor principles. Stefan Schranz studied physics at the Technical University Braunschweig and graduated in 1994 in material science. Since 1995 he has been working as a research associate on a three-year project to model quartz resonators with FEM His actual interests are concentrated on effects which are related to real dimensions of the resonator. Carsten Behling graduated at the Otto-von-Guericke-University in the Department of Physics in 1993, He is currently working on the last year of a four-year appointment at the Department of Electrical Engineering. He is now preparing his Ph.D. thesis. His research activities are concentrated on modelling thickness shear mode (TSM) sensors.
Frank Eichelbaum received his Ph,D. degree at the Ottovon-Guericke-University in 1987. His thesis work was on sensor electronics. Since 1986 he has been working at the Institute for Measurement Technology and Electronics as a research assistant. He is our specialist in design and simulation of analog and digital electronics for sensors. Peter Hauptmann is head of the group of sensors and microsystems at the Otto-von-Guericke-University as well as vice director of the ifak. During the time from 1968 to 1985 he worked at the Technical College Leuna-Merseburg as a lecturer and senior lecturer. In 1973 he received the Ph.D. degree from the Technical College Leuna-Merseburg for a thesis about polymer physics. He joined the Otto-von-Guericke-University in 1985, bringing with him great experience in sensor development and application. The aims of his research are more accurate, economical and reliable sensors for chemical and biological processes as well as sensors for other industries.