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(17) Yokoyam, K.; Sode,K.; Tamlya. E.; Karube, I . Anal. Chem. Acfa 1989, 218, 137. (18) &egg, B.; Heller. A. J . phvs. Chem. 1991, 95, 5970. (19) &egg, B.; Heller, A. J . phvs. Chem. 1991, 95, 5976. (20) &egg, 8. A.: Heller, A. Anal. Chem. 1990, 62, 258. (21) Frederick, K. R.; Tung, J.; Emerick, R. S.; Masiarz, F. R.;Chamberlin, S. H.; Vasevada, A.; Rosenberg, S.; Chakraborty, S.; Schopter, L. M.; Massey, V. J . 8bl. Chem. 1990, 265, 3793. (22) Wightman, R. M.: May, L. J.; Baur, J.; Leszczyszyn, D.; Kristensen, E. ACS Symp. Ser. 1989, 403, 114. (23) Andrleux, C.; Saveant, J. J . Nectroanal. Chem. Interfacial Electroochem. 1980, 1 1 1 , 377. (24) a n g . L.: Reed, R. A.; Kim, M.-H.; Wooster, T. T.; Oliver, B. N.; Egek828, J.: Kennedy, R. T.;Jorgenson, J. W.; Parcher, J. F.; Murray, R. W. J . Am. Chem. Soc.1989, 1 1 1 , 1614. (25) Oliver, 8 . N.; Egekeze, J. 0.; Murray, R. W. J . Am. Chem. SOC. 1988, 110, 2321.
(26) Gang, L.; Longmire, M. L.; Reed, R. A.; Parcher. J. F.; Barbour. C. J.; Murray, R. W. Chem. Meter. 1989, 1 . 58. (27) Dayton, M. A.; Ewlng, A. 0.;Wlghhnan, R. M. Anal. Chem. 1980, 52. 2392. (28) Pishko. M. V.; Katakls, I.; Lindquist, S . I . ; Heller, A.; Degani, Y. Mol. Ctyst. Liq. Cryst. 1990, 190, 221. (29) Pishko, M. V.; Katakis, 1.; Lindquist. S . I . ; Ye, L.; &egg. B. A,; Heller. A. Angew. Chem., Int. Ed. Engl. 1990, 29(1), 82. (30) Gough. D. Horm. Metab. Res., Sup@. Ser. 1988, No. 20, 30. (31) Gough, D.; Leypokft, J.: Armour, J. Dlebetes Care 1982, 5 , 190.
RECEIVED for review February 27, 1991. Accepted July 25, 1991. This work is supported by the Office of Naval Research, the Robert A. Welch Foundation, the Texas Advanced Research Project, and the National Science Foundation.
Characterization of a Quartz Crystal Microbalance with Simultaneous Mass and Liquid Loading Stephen J. Martin,* Victoria Edwards Granstaff, and Gregory C. Frye
Sandia National Laboratories, Albuquerque, New Mexico 87185
WHh a contlnuum electromechanical model, the eiectrlcal admfflance has been derived for an AT-cut quartz crystal mlcrobalance (QCM) simultaneowly loaded by a surface m a s layer and a contactlng Newtonian liquid. This admittance expression Is used to derlve a lumped-element equivalent clrcull model that describes the near-resonance electrlcai characterlstlcs of the OCM under these loading conditions. The resulting model Is a modlfled Butterworth-Van Dyke equivalent ckcull, havlng clrcult elements that are explicltly related to phydcal properties of the quartz, perturbing mass layer, and contactlng liquid. The effects of mass and liquid loading on the fundamental and thlrd harmonic resonances are predlcted from the model and compared wlth experimental impedance analyzer measurements. Surface m a s accumulation causes a simple trandatlon In frequency of the resonance peak, whHe Increasing the denslty-viscoslty product of the contactlng solution causes both a translation and a damplng of the resonance peak. With the model, changes in surface mass can be dlfferentlated from changes In solution propertles.
INTRODUCTION The quartz crystal microbalance (QCM) is commonly configured with electrodes on both sides of a thin disk of AT-cut quartz. Due to the piezoelectric properties and crystalline orientation of the quartz, the application of a voltage between these electrodes results in a shear deformation of the crystal. The crystal can be electrically excited into resonance when the excitation frequency is such that the crystal thickness is an odd multiple of half the acoustic wavelength. At these frequencies, a standing shear wave is generated across the thickness of the plate (I), as depicted in Figure 1 for the fundamental and third harmonic resonances. QCMs were originally used in vacuo to measure deposition rates. As shown by Sauerbrey, changes in the resonant frequency are simply related to mass accumulated on the crystal (2). The QCM is typically instrumented as the frequency-
* To whom correspondence should be addressed. 0003-2700/91/0363-2272$02.50/0
control element of an oscillator circuit; a precise microbalance is realized by monitoring changes in oscillation frequency (3). More recently, QCMs have been shown to operate in contact with liquids (4), enabling their use as solution-phase microbalances. This microbalance capability has facilitated a number of solution measurements. Some examples include deposition monitoring (5),species detection (6),immunoassay (7),liquid chromatographic detection (81, corrosion monitoring (91,and electrochemical analysis (10, 11). Kanazawa and Gordon have shown that QCMs operating in solution are also sensitive to the viscosity and density of the contacting solution (12). Viscous coupling of the liquid medium to the oscillating device surface results in both a decrease in the resonant frequency of the QCM and damping of the resonance. Thus, since the resonant frequency is affected by both mass and liquid loading, measurement of the resonant frequency alone cannot distinguish changes in surface mass from changes in solution properties. When the electrical characteristics are measured over a range of frequencies (f, near resonance, however, we will show that the QCM can be sufficiently well characterized to differentiate between these loading mechanisms. Coupling between mechanical displacement and electrical potential in the piezoelectric quartz causes mechanical interactions between the QCM and contacting media to influence the electrical characteristics of the QCM, particularly near resonance, where the amplitude of crystal oscillation is greatest. The QCM electrical characteristics can be evaluated by using the electrical admittance, defined as the ratio of current flow to applied voltage (the reciprocal of impedance). This parameter contains information about the energy stored and the power dissipated in both the QCM and the perturbing media. It is convenient to use an equivalent circuit model to describe the electrical behavior of the QCM. With only a few lumped elements, this model simulates the electrical characteristics of the QCM over a range of frequencies near resonance. Ideally, the model should explicitly relate the circuit elements to physical properties of the QCM as well as the surface mass layer and contacting liquid. Fitting the circuit model to electrical measurements then allows extraction of surface mass and liquid properties. In addition, the behavior 0 1991 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 63. NO. 20, OCTOBER 15, 1991
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Liquid: I
I
I
I I
I
I
I I
I I I
I
I
I I
I
!
!
F@we 1. Shear dlsplacement profiles across the QCM thickness for the fundamental (N = 1) and the third harmonlc (N = 3) resonances.
P I I r‘“ I
t
R1
FI@m2. Buttemorth-Van Dyke equivalent ckcult for the unpertwbed quartz aystai “ b a l a n c e (OW): a static capacitance C, in parallel wtth a motional branch ( L l , C , , R l ) .
of the QCM in combination with oscillator circuitry can be more easily predicted with a model that describes the electrical characteristics of the loaded QCM. Another advantage of an equivalent circuit model is that standard circuit analysis software can be used to extract information from electrical measurements. The Butterworth-Van Dyke (BVD) equivalent circuit typically used to describe the unperturbed (without mass or liquid loading) QCM is shown in Figure 2. A “static” capacitance C, arises between the electrodes located on opposite sides of the insulating quartz. Since the quartz is also piezoelectric, electromechanical coupling gives rise to an additional ’motional” contribution (&, C1,Rl) in parallel with the static capacitance. The static capacitance dominates the admittance away from resonance, while the motional contribution dominates near resonance. While it has not been rigorously demonstrated that the BVD circuit model is an appropriate description for the loaded QCM, a number of workers have shown that impedance measurements made on liquid-loaded QCMs can be fit with this model (13-16). Using an analogy between a lumped-element mass-springdashpot oscillator and the LCR resonant circuit, Muramatsu et al showed that for a liquid-loaded QCM, the motional resistance R1is proportional to (pq)lI2, where p and q are the liquid density and viscosity, respectively (13). Beck et al. argued that a Newtonian liquid also causes a proportional change in the motional inductance Ll (14). To show rigorously that a BVD equivalent circuit appropriately describes the loaded QCM and to relate the equivalent circuit elements to physical properties of the QCM and contacting media, it is necessary to start with a continuum electromechanical model of the QCM and the contacting media. The
Air
Figure 3. Cross-sectional view of a QCM simultaneously loaded on one side by a surface mass layer and a contacting Newtonian liquid. Shear displacement, Re[u,(y,t*)],is shown at a moment t o when quartz displacement Is maximum. QCM surface displacement causes synchronous motion of the surface mass layer and entrainment of the contacting liquid.
electrical characteristics can then be approximated near resonance to obtain an equivalent circuit model. Reed et al. have described a continuum electromechanical model for the QCM in contact with a single viscoelastic medium of arbitrary thickness (13,but this has not been reduced to an equivalent circuit formulation. QCM sensing applications typically involve sorption or deposition of solid mass onto the device surface from a contacting liquid phase. In the present paper, we approximate this condition with a continuum model that describes the QCM simultaneously haded by a thin surface mass layer and a semiinfinite Newtonian liquid. The electromechanical model results in an analytical expression for the QCM admittance as a function of excitation frequency. By approximating the admittance near resonance, an equivalent circuit model is derived that explicitly relates the circuit elements to physical properties of the QCM and the perturbing mass and liquid. The reader interested only in the resulting model may wish to bypass the theory section, noting only the equivalent circuit of Figure 4, whose elements are given by eqs 25. In the experimental section, the admittance vs frequency predictions obtained from this equivalent circuit model are compared with measurements made on mass- and liquid-loaded QCMs.
THEORY Figure 3 depicts the cross-sectional geometry of the QCM loaded from above by a surface mass layer and a contacting liquid. Excitation electrodes, assumed to be infinitesimally thick, are also located at the upper and lower quartz surfaces. The mass layer is assumed to be very thin compared to the acoustic wavelength and rigidly attached to the QCM, ensuring synchronous motion with the oscillating surface. When liquid contacts this oscillating surface, a damped shear wave is radiated into the liquid, as shown in Figure 3. As long as the liquid thickness is large compared to the decay length of the radiated shear wave, the liquid may be considered semiinfiiite. QCM Admittance under Mass and Liquid Loading. The admittance of the QCM is obtained by solving a boundary-value problem that includes the mass layer and contacting liquid. Continuity of particle displacement and shear stress must be maintained at the boundaries between the QCM, mass layer, and contacting liquid. In addition, the electrical potential at the quartz surfaces must match the applied potential. A QCM having only one side contacted by liquid (and no mass layer) must satisfy five boundary conditions. The addition of a finite-thickness layer between the
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QCM and the liquid introduces an extra interface with two associated boundary conditions. When the film thickness hf is small compared to the acoustic wavelength in the film, however, the influence of the film can be included by considering the film's mass to be concentrated in an infinitesimally thin sheet at the quartz/liquid interface. This eliminates the increased complexity resulting from an additional interface; however, the acceleration of this mass layer leads to a discontinuity in shear stress at the liquid/solid interface that must be taken into account. When the QCM electrode diameter is large compared to the crystal thickness, h, variations in shear displacement in planes parallel to the surface may be neglected in comparison with variations across the thickness, allowing a one-dimensional model to be applied. The quartz shear displacement u, results from a superposition of two shear waves propagating in opposite directions along the y axis (17): u,(y,t) = (AdsY Be-j&)dUt (1)
+
where A and B are integration constants, w is the angular excitation frequency (w = 274, j3 is the wavenumber describing shear wave propagation in the quartz,t is time, and j = (-l)ll2, The electrical potential, 4, in the piezoelectric quartz is (17)
where C and D are integration constants and e26and c22 are the piezoelectric stress constant and permittivity of the quartz, respectively. The liquid velocity field, u,, can be determined by solving the Naviel-Stokes equation for onedimensional plane-parallel flow
(3) where p and 7 are the liquid density and shear viscosity, respectively, and ir, = au,/at. The solution to this equation for an oscillatory shear driving force at the solid/liquid boundary is (18) u,(y,t) = Eerb-h)ej"t (4) where E is an integration constant and y is a complex decay constant for the liquid velocity field. Substituting eq 4 into eq 3 yields two solutions for y. Only the root having a positive real part, y = ( w p / 2 ~ ) ' / ~ ( 1j )+, satisfies the requirement that u, 0 as y m, With this complex y, eq 4 represents a critically damped shear wave radiated into the liquid by the oscillating QCM surface (Figure 3). The decay length 6 = ( 2 7 / p ~ ) ' (12, / ~ 19) has a value of 0.25 pm in water at 20 "C when f = 5 MHz. There is evidence for moderate viscosity enhancements in the first few monolayers near a liquid/solid interface (20). Since the decay length is much greater than a monolayer thickness, however, it is appropriate to use bulk viscosity values for the liquid. The integration constants A through E are determined by applying the five boundary conditions that arise at the top and bottom surfaces of the QCM. The first boundary condition requires that displacement be continuous across the solid/liquid interface (the nonslip condition). Noting that u, = jwu, enables the liquid velocity (eq 4) to be converted to displacement. Equating liquid and solid displacements at the upper quartz surface yields
- -
A@
+ Be-iPh + j -E w
=0
(5)
Although Kipling and Thompson suggest that the nonslip boundary condition may be inappropriate with certain surface treatments (15),the nonslip assumption in the present model
leads to good agreement between calculated and measured admittances, provided the electroded quartz surface is sufficiently clean and smooth. The latter condition is necessary to ensure that plane-parallel laminar flow is generated in the liquid by the oscillating quartz surface. When the ratio of shear stiffness pf to density pf in the film is such that pf/pf >> (fhfI2,the shear stress T varies linearly across the film while the displacement is nearly constant. In this case, the onedimensional continuum equation of motion, aT,/ay = pp. (211,leads to the following boundary condition regarding the mass layer and the shear stresses acting upon it:
TL - TQ = Psox(h)
(6) where TL is the shear stress imposed in the x direction on the mass layer by the liquid, TQis the shear stress exerted on the quartz by the mass layer, and ps = p& The effect of a thin, rigid film between the QCM and liquid can thus be approximated by considering the film's areal mass density pa (mas/area) to be concentrated in an infinitesimallythin sheet at the quartz/liquid interface. This gives rise to a discontinuity in stress at the interface given by eq 6. From the piezoelectric constitutive equations (21),the shear stress TQexerted by the mass layer on the quartz is
where eqs 1 and 2 are used for u, and 4 and ces is a quartz elastic constant. The x-directed stress exerted on the mass layer by the liquid is
where eq 4 is used for u,. Using these identifications for TQ and TL in eq 6 yields the following relationship from the second boundary condition:
j@i-!ss(Ad~h - Be-jsh) + e& + ( ~ +y jwp,)E = 0 (9) where EM = (ces + eB2/c,) is the "piezoelectrically stiffened" quartz elastic constant. The third boundary condition requires zero stress at the quartz/air interface. Evaluating eq 7 at y = 0 rather than y = h yields the shear stress at this interface;equating this stress to zero yields jflCBS(A- B ) + ez6C = 0 (10) The fourth and fifth boundary conditions require that the electrical potential at the upper and lower quartz surfaces match the applied potential, taken to be -4,dwtand 4,dW', respectively. Matching eq 2 to these potentials yields
%(Adoh e22
+ Be-jbh) + Ch + D = -4,
e26
-(A e22
+ B ) + D = 4,
(11)
(12)
The boundary condition equations (eqs 5,9,10,11, and 12) constitute five linear equations in the five unknowns A through E. Solving these equations by a full Gauss-Jordan elimination is quite tedious. Fortunately, this is unnecessary since the following analysis indicates that the admittance depends only on C (17). From Gauss' law, the surface charge density induced on the lower Cy = 0) electrode is (13)
ANALYTICAL CHEMISTRY, VOL. 63, NO. 20, OCTOBER 15, lQQl 2275
...
Therefore, the current flow out of the lower electrode (and into the upper electrode) is I = q,A = joq,A, where A is the electrode area. The term &$lay in eq 13 is obtained from eq 2, noting that the lower boundary is stress-free, so that au,/dy = 0 (21,221. Then the admittance, defined as the ratio of terminal current I to applied potential V (V = 24) is related to C by Y = I / V = q,A/24 = - j w ~ ~ ~ A C / 2 4 , (14)
At resonance, $ is approximately Nn ( N = 1, 3, 5, ), SO that the trigonometric functions in eq 19 can be expanded about the poles as (23)
where eq 13 has been used to represent q, in terms of C. Solving the boundary condition equations for C and applying eq 14 yields the desired admittance for a QCM under simultaneous mass and liquid loading: Y(w)= -jwC,(sin $ A cos $)/ ((P/$)(l - cos $)(1- cos $ + A sin $) (sin $ + A cos $)[1 - (@/$I sin $1) (15)
Substituting eqs 20 into eq 19 and retaining only the terms up to first order in the small quantities A and - +2] yields
+
where C, = tn2A/his the static capacitance of the QCM, K2 is the electromechanical coupling constant for quartz, and $ = Oh is the frequency-dependent phase shift undergone by the shear wave in propagating across the quartz. For lossless quartz, €P= K,2 1 eas2/(t&cd and $ = $, = wh(pq/csa)’12,where pq is the quartz mass density. To account for quartz losses, we define a complex quartz elasticity (17): EM = E,& +it), where = wqq/cM and qq is the effective quartz viscosity. The effect of this dissipation is to make $ and K2 complex:
v
At parallel resonance, the condition most closely resembling the natural mechanical resonance of the QCM (I7),the phase shift $o across the QCM is Nn. Electrically excited resonance (i.e., maximum electrical admittance), however, occurs at the series resonance, at which $,2(w,) = ( N d 2- 8K: (23). Since $, is proportional to w, the frequency-dependent form of can be evaluated: +:(o) = ( ~ / w , ) ~ [ ( N-nSK:]. ) ~ Using eq 16a, the phase shift for a lossy QCM is then
+,
for w near w,, where w, = 2nf, and f, is the series resonant frequency of the unperturbed QCM. Substituting eq 22 into eq 21 and keep only terms up to first order in the small quantities 4, A,, Ai, K:, and [ ( N d 2 - G2] yields
2
It is interesting to note that the influence of the mass and liquid perturbations in eq 15 are contained in a single complex factor A: A = A, - jAi where
Noting that A, contains two terms allows this motional impedance to be written in the form
A, represents a change in QCM stored energy caused by the perturbation, i.e., due to the kinetic energy of the bound mass and/or entrained liquid layer. Ai represents power dissipation due to the radiation of a damped shear wave into the liquid by the oscillating QCM surface. Since moving mass does not cause power dissipation, p, does not appear in eq 17b. Equivalent Circuit for the Loaded QCM. An equivalent circuit model can be derived that closely approximates the admittance vs frequency behavior of eq 15 in the vicinity of resonance. The model is a modified form of the Butterworth-Van Dyke equivalent circuit: a static capacitance C,, dominating the admittance away from resonance, appears in parallel with a motional arm, dominating the admittance near resonance (see Figure 2). To derive the elements in the motional arm, we decompose the admittance into contributions from the two branches: 1 Y = joC, + (18)
zm
where 2, is the impedance of the motional arm of the equivalent circuit. Algebraic manipulation of eqs 15 and 18 gives
Comparison between eqs 23 and 24 (using eqs 17 for A, and Ai) enables identification of the motional impedance elements (C, is also included for completeness): c, = t22A
h
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ANALYTICAL CHEMISTRY, VOL. 63,NO. 20, OCTOBER 15, 1991
P 1 I,I
T"'
T
T
""z-
c1
appear from the parallel combination of C, and C, in Figure 4 that C, would be indistinguishable from C, However, since C, arises from fields across the quartz, which ab0 excite the mechanical response of the QCM, C, enters into the motional circuit elements as well (see eqs 25);in contrast, the parasitic capacitance C, arises from fields external to the QCM that do not influence the motional elements. Consequently, by measurement of the resonance and broad-band admittance characteristics, C, can be separated from C,. The total admittance, including the parasitic contribution, is 1 Y = j W ( C , C,) (26)
+
I
I
L
LLS
L3
MarLordlng
+
Flgun 4. Equivalent circuit for a QCM under mass and liquid loading including parasitic capacitance in the test fixture C,. Equatlons 25
+zm
Together, eqs 24-26 fully describe the QCM admittance in terms of the physical properties of the QCM, surface mass layer, and contacting liquid. These results were derived for maw and liquid contacting only a single side of the QCM. For two-sided contact, the elements L2, R2, and L3 are doubled. In addition, Tiean et al. (16)have pointed out that conduction current between electrodes must also be considered when both are immersed. Model Predictions. An initial check on the equivalent circuit model is the predicted change in resonant frequency with maas or liquid loading. Since the series resonant frequency f , = 1/(27r(LC1)112),where L = L1+ L2 + L3,and C1 is unchanged by mass or liquid loading, the fractional change in f , is
relate circuk elements to physical parameters.
Figure 4 shows the equivalent circuit that uses these elementa to approximate the admittance of a QCM perturbed by a thin mass layer and a contacting liquid. This model gives the QCM admittance over a range of frequencies near series resonance. For the unperturbed QCM, p8 = pq = 0; eqs 25 indicate that L2 = R2 = L3 = 0, and Figure 4 reduces to the usual BVD equivalent circuit for the unloaded QCM (23). Equations 25a-d agree with previous expressions for the unloaded QCM elements (23) with the exception of the ( w / w , ) ~ factor in R,; however, this factor is approximately unity near resonance. As surmised by Beck et al., liquid loading causes an increase in both the motional inductance (contributed by L,) as well as resistance (contributed by R2with R2 = wLZ) (14). In contrast, mass loading increases only the motional inductance, contributed by La. An interesting feature of the equivalent circuit model is that the elements that arise from mass and liquid loading are related to the unperturbed QCM parameters. Consequently, characterizing the unperturbed QCM fully determines the response to a given mass or liquid perturbation. Equations 25 indicate that R1 varies as 02, while Lz and R2 vary as w-lI2 and all2,respectively. Thus, the assumption that a BVD equivalent circuit, having frequency-independent elements, can be used to represent the loaded QCM is strictly correct only for the case of a lossless QCM with mass loading. However, the admittance behavior at resonance arises mainly from cancellation between the reactive elements (jwL and l/jwC terms), rather than changes in the circuit elements themselves. Consequently, frequency-independent elements, obtained by evaluating eqs 25 at w = w,, can be used in eq 24 without incurring significant error (less than 0.25% in admittance magnitude over a 1% bandwidth). To simplify the analysis, this approach has been used to generate admittance vs frequency curves in this paper. In fitting measured QCM admittances to the BVD equivalent circuit, it was found that an additional element was needed to account for the parasitic capacitance arising in the teat fixture. This parasitic capacitance, C,, depends on the geometry of the test fixture and the QCM electrode pattern. Values between 3 and 15 pF were found in this study. It might
When mass and liquid loading are small, Le., (L, + L3)
