Anal. Chem. 1998, 70, 237-247
Analysis of the Radial Dependence of Mass Sensitivity for Modified-Electrode Quartz Crystal Resonators Fabien Josse* and Youbok Lee
Microsensor Research Laboratory and Department of Electrical and Computer Engineering, P.O. Box 1881, Marquette University, Milwaukee, Wisconsin 53201-1881 Stephen J. Martin and Richard W. Cernosek
Microsensors Research & Development Department, Sandia National Laboratories, P.O. Box 5800, MS 1425, Albuquerque, New Mexico 87185-1425
The radial dependence of mass sensitivity of the sensing surface is analytically calculated for two examples of “modified-electrode” quartz crystal resonators (QCR). The term “modified-electrode” QCR is used here with respect to the conventional QCR which has two identical circular and concentric electrodes. For these QCRs, the sensing surface is divided into a fully electroded, a partially electroded, and an unelectroded region, and the efficiency of each region is evaluated in terms of the electrode mass loading factor. Such QCRs are typically investigated for sensor applications in which the electrical properties of the liquid load or the coating deposited on the sensing surface (electroded and partially electroded regions) are being measured in addition to mass loading. While modified-electrode QCRs can be viewed as a simple capacitance sensor in those applications, the use of a piezoelectric crystal resonator in the narrow range of frequencies near resonance and antiresonance allows for a direct measurement of the capacitance through the antiresonant frequency, provided that the device damping (motional resistance) is not too high or that the resonance quality factor, Q, is high enough for a stable vibration under the load. It is shown that, for some values of the electrode mass loading factor, the off-electrode efficiency (partially electroded and unelectroded region efficiency) can still have a significant contribution to the overall surface area mass sensitivity. Knowledge of the efficiencies is needed to determine the loading area required for stable QCR sensor operation. This is because additional dissipation of energy into the load can occur, especially for cases where the sample load extends to the unelectroded surface, which has a nonnegligible particle displacement amplitude. It is also shown that, for some applications involving a liquid load and for some values of the electrode thickness, the shear particle displacement profile is such that compressional wave generation can contribute significantly to device damping, thus making the device unstable. Experimental measurements of the S0003-2700(97)00603-3 CCC: $15.00 Published on Web 01/15/1998
© 1998 American Chemical Society
mass sensitivity profile on the surface are also performed for those QCRs and compared to theory. The quartz crystal resonator (QCR) has been extensively used as a sensitive device for the measurement of the density and thickness of thin films. Over the last two decades, chemical sensors utilizing QCRs have been also investigated for the detection of chemical compounds in the gas phase.1-5 In these applications, a planar QCR with two identical circular concentric electrodes is used, with the electrodes uniformly coated with a chemically sensitive film. The change in the device operating characteristics is attributed to the change of mass in the coating during the detection process. The measured frequency shift is proportional to the change in mass. While these devices suffer, at times, from reproducibility problems, they do operate adequately in the gas environment, since the reduction in the quality factor, Q, of the resonance under the gaseous load is not substantial enough to result in a frequency instability of the resonator. More recently, new applications of the QCR as a detection device in liquid environments have emerged in which one surface of the device is exposed to the liquid.6-16 QCRs with various (1) Sauerbrey, G. Z. Phys. 1959, 155, 206-222. (2) Go ¨pel, W., Hesse, J., Zemel, J. N., Eds. Sensors: A Comprehensive Survey; VCH Verlag: Weinheim, Germany, 1991; Vol. 2. (3) King, W. H., Jr. Anal. Chem. 1964, 36, 1735. (4) Ho, M. H. In Applications of Piezoelectric Quartz Microbalances; Lu, C., Czanderna, W., Eds.; Elsevier: New York, 1984; pp 351-388. (5) Guibault, G. G.; Jordan, J. M. CRC Crit. Rev. Anal. Chem. 1988, 19, 1. (6) Kanazawa, K. K.; Gordon, J. G., II. Anal. Chem. 1985, 57, 1770-1771. (7) Martin, B. A.; Hager, H. E. J. Appl. Phys. 1989, 65 (7), 2627-2629. (8) Roederer, J. E.; Bastiaans, G. J. Anal. Chem. 1983, 55, 2333-2336. (9) Muramatsu, H.; Dicks, J. M.; Tamiya, E.; Karube, I. Anal. Chem. 1987, 59, 2760-2763. (10) Josse, F.; Shana, Z. A.; Radtke, D. E.; Haworth, D. T. IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 1990, 37 (5), 359-368. (11) Josse, F.; Shana, Z. A.; Zong, H. Proc. IEEE Ultrason. Symp. 1993, 1, 425430. (12) Shana, Z. A.; Josse, F. Anal. Chem. 1994, 66, 1955-1964. (13) Yang, M.; Thompson, M. Anal. Chem. 1993, 65, 3591-3597. (14) Martin, S. J.; Frye, G. C.; Wessendorf, K. O. Sens. Actuators A 1994, 44, 209-218.
Analytical Chemistry, Vol. 70, No. 2, January 15, 1998 237
electrode geometries on the sensing surface have been shown11,12 to have good sensitivity to both the mechanical and the electrical properties of the load. In particular, the sensitivity to the electrical properties of the load has been shown to be a function of the strength of the electric fringing fields on the sensing surface.17 This sensitivity, caused by changes in the electrostatic capacitance, viewed as an external electric load by the vibrating crystal, often depends on and requires the use of various electrode geometries, sizes, and configurations on the sensing surface. This, in effect, makes the device equivalent to a simple capacitance sensor. However, the use of a piezoelectric crystal resonator within the narrow range of frequencies near resonance and antiresonance allows the change in the capacitance to be measured through the device antiresonant frequency with greater accuracy than other methods. Applications of the QCR as a detector in liquid environment suffer from various problems, which include the generation of and interference from longitudinal waves and a drastic drop in the quality factor, Q. Longitudinal waves in QCR applications in liquids are generated primarily due to the finite dimension of the crystal plate and electrodes. They result from contributions from nonuniform displacement profiles along the shear direction. These waves can form a standing wave when reflected off the liquid/cell interface. Measurements of the longitudinal standing wave frequency and amplitude have shown the influences of the crystal contour, liquid properties, liquid/cell interface, and radial dependence of the shear wave amplitude.18,19 The results have shown that, by careful design of the liquid cell, with the upper surface at an appropriate distance different from the crystal surface, compressional standing wave patterns are not formed, and constructive interference can be avoided.18,19 The drastic drop in the quality factor, Q, of the crystal, a more serious problem, is due to energy absorption by the loading medium. The energy absorption by the adjacent medium is even more severe for modified-electrode QCRs, where the displacement amplitude outside the electrode can be nonnegligible. As a result, the ability of the crystal plate to maintain a stable vibration under the liquid load is often impaired due to the electrode geometry. This is because the electrode geometry dictates the energy trapping process, and hence the energy distribution profile on the sensing surface. The mass sensitivity profile of a QCR with two identical circular concentric electrodes has been well analyzed.16,20-23 While there can be a non-negligible acoustic vibration amplitude just off the electroded area, most applications involve the load covering only the electrode area. However, this is not the case for applications involving modified-electrode QCRs, where the amplitude vibration profile can vary depending on the electrode geometry and thickness. Thus, an analysis of the sensitivity profile on the sensing surface will help in the prediction and analysis of (15) Martin, S. J.; Granstaff, V. E.; Frye, G. C. Anal. Chem. 1991, 63, 22722281. (16) Hillier, A. C.; Ward, M. D. Anal. Chem. 1992, 64, 2539-2554. (17) Lee, Y.; Everhart, D.; Josse, F. Proc. IEEE Freq. Contr. Symp. 1996, 577585. (18) Lin, Z.; Ward, M. D. Anal. Chem. 1995, 67, 685-693. (19) Schneider, T. W.; Martin, S. J. Anal. Chem. 1995, 67, 3324-3335. (20) Sekimoto, H. IEEE Trans. Son. Ultrason. 1984, 31, 664-669. (21) Mindlin, R. D. Appl. Math. 1961, 19, 51-61. (22) Tiersten, H. F.; Smythe, R. C. J. Acoust. Soc. Am. 1979, 65, 1455-1460. (23) Martin, B. A.; Hager, H. E. J. Appl. Phys. 1989, 65, 2630-2635.
238 Analytical Chemistry, Vol. 70, No. 2, January 15, 1998
Figure 1. Top half view and cross-sectional (through the center) view of two types of QCRs with modified electrode configurations. (a) “n-m”-type electrode, where n and m represent the upper and lower electrode diameter, respectively. (b) Ring electrode, where the outer diameter of the upper ring electrode is equal to the lower electrode diameter.
device sensors utilizing such electrode configurations in which the load may extend to the unelectroded surface, as well as the prediction of the compressional wave generation. The generation of such waves arises from a gradient in the in-plane surface shear displacement. In the present study, the mass sensitivity profile on the sensing surface is analytically calculated for two examples of modifiedelectrode QCRs used in liquid phase sensing applications. For those QCRs, the sensing surface is divided into a fully electroded, a partially electroded, and an unelectroded region, and the efficiency of each region is evaluated in terms of the electrode mass loading factor. It is shown that, for some values of the electrode mass loading factor, the off-electrode efficiency (partially plus nonelectroded regions efficiency) can still be significant and thus must be accounted for in the design of sensor surfaces. In liquid-phase sensing applications, knowledge of the compressional wave generation is needed to determine the resulting device damping, critical for stable QCR operation. Using the generated surface displacement profiles for those QCRs, the effect of the compressional wave generation is analyzed. Experimental measurements of the mass sensitivity profile on the sensing surface were also performed for the QCRs and compared to theory. THEORY The QCRs of interest are made of thin circular AT-cut quartz plates with a metallic electrode on each side of the plate. Figure 1 shows two examples of electrode configurations, referred to here as the “modified-electrode” QCR configurations. The term “modified-electrode” QCR is used with respect to the conventional QCR, which has two identical circular and concentric electrodes. The quartz plate and electrode in Figure 1 are assumed to have
thickness 2h and 2h′, respectively. The sensing surface is assumed to be the surface with the modified or smaller electrode. This is because the electric fringing fields are relatively large on that surface, and some of the applications may involve measuring the sensitivity to the electrical loading and specifically the capacitive loading. It is known that the electrical loading sensitivity of a QCR device is due to the interaction of surface electric fields and electrical properties of the loading medium. The electric field contributing to this sensing mechanism of the device is due mainly to the electric fringing fields caused by the electrode configuration and potential difference between the electrodes. The electrode geometries of Figure 1 have relatively large electric fringing fields on the sensing surface, compared to the device with two identical concentric electrodes. Unfortunately, these electrode geometries which result in strong electric fringing fields on the sensing surface will not necessarily yield a stable vibration. This is because the stability of QCRs is due to energy trapping caused by the finite dimension of the metal electrodes and to the drop in the quality factor, Q, under the load. The effect of an electrode on the piezoelectric crystal plate is to reduce the resonance frequency of the bare crystal plate. Because of the different electrode boundary conditions along the radial direction, the various regions of the QCR have different resonance conditions and frequencies. In modified-electrode QCRs, a total of three different resonant frequencies coexist. Due to the different dimensions of the upper sensing electrode, three regions can be defined along the radial direction. For the geometry of Figure 1a, they are the fully electroded region (r e a), the partially electroded region (a < r e b), and the unelectroded region (r > b). For the geometry of Figure 1b, the electroded and partially electroded regions are defined by (a e r e b) and (r < a), respectively. The geometry in Figure 1a is referred to as the “nm”-type electrode QCR with n e m. The “n” and “m” stand for the diameter (in millimeters) of the upper and lower electrode, respectively. The geometry in Figure 1b is referred to as a ring electrode QCR. Because of the metallic electrode effect and the different mechanical boundary conditions, it is expected that each region will have a different resonance condition and, hence resonance frequency. However, because the electrodes are, in general, very thin, the resonance frequencies of the three regions are very closely spaced. For the AT-cut quartz, only the shear horizontally polarized particle displacement u1 (in the X1 direction) is coupled to the electric potential. The change in the resonance frequency due to the mechanical loading is referred to as the mechanical sensitivity or mass sensitivity. This sensitivity is proportional both to the added mass and the square of the particle displacement amplitude at the point at which it is added. In a QCR device with finite surface dimension, the frequency change, ∆f, due to the addition of mass on the surface of the QCR is given by1
∆f0 ) -Sf(r,θ) ∆m(r,θ)
(1)
where ∆m(r,θ) is the added mass, and Sf(r,θ) is the mass sensitivity function; r and θ are the polar coordinates of the point at which the mass is added. The mass sensitivity function is given by
Sf(r,θ) )
|u ˜ 1(r,θ)|2
∫∫ ∞
0
Cf
2π
r|u ˜ 1(r,θ)|2 dθ dr
0
(2)
where u ˜ 1(r,θ) is the particle displacement amplitude function on the surface. Cf is Sauerbrey’s sensitivity constant, with units of (Hz)(m2)/kg. The unit of the sensitivity function, Sf(r,θ), is found as Hz/kg. In a QCR device operating at the fundamental frequency, the particle displacement amplitude is invariant with the angular direction, θ. Equation 2 is thus reduced to
Sf(r) )
|u ˜ 1(r)|2
∫ r|u˜ (r)|
2π
∞
1
0
2
Cf dr
(3)
The denominator in eq 3 is the intensity of the total available particle displacement amplitude in a given device, and it becomes a constant after integration. The sensitivity function, also called the differential mass sensitivity profile, can then only be calculated after evaluation of u ˜1(r,θ). In what follows, the vibration characteristics of the modified-electrode QCRs shown in Figure 1 are analyzed. Analysis of Mode Amplitude. The plate surface plane is defined by (X1,X3). A time-varying electric potential applied across the electrodes generates the acoustic wave within the crystal medium. In AT-cut quartz, only the u1 component of the particle displacement is coupled to the electric potential. u1 describes the X1-polarized particle displacement (wave propagation is assumed in the X2 direction). Assuming that the excited acoustic wave is a time-varying field of the form
u1(x1,x2,x3,t) ) u ˜ 1(x1,x3) sin(k2x2)ejωt
(4)
where k2 is the shear horizontal acoustic wavenumber in the X2 direction. The equation governing the shear horizontal acoustic wave can be reduced to24
( )
( )
[
( )]
C11 ∂2u ˜1 C55 ∂2u ˜1 C h 66 + + k2 - k22 u ˜ )0 C66 ∂x 2 C66 ∂x 2 C66 1 1 3
(5)
where k ) ω/v is the wavenumber of the driving frequency; C h 66 ) C66 + e262/22 is the acoustically stiffened elastic constant; v is the acoustic wave velocity in the crystal, given by (C66/Fq)1/2; is the mass density of the crystal; Cij, ejk, and jj represent the elastic stiffness constant, piezoelectric constant, and dielectric constant, respectively; and ω is the excitation frequency. The wave propagation constant, k2, can be determined from the resonance condition of the device, i.e., the boundary conditions. In a bare crystal plate, k2 ) n1π/2h, where n1 is an integer number and 2h is the thickness of the crystal plate. The resonance condition is found when the amplitude of the shear stress field or particle displacement becomes maximum.24 The condition is obtained by solving the wave equations, subject to the boundary conditions. For the fully electroded region, this condition is given by24 (24) Tiersten, H. F. Linear Piezoelectric Plate Vibration; Plenum Press: New York, 1969.
Analytical Chemistry, Vol. 70, No. 2, January 15, 1998
239
k2h (k262
)
2
2
+ (R/2)h k2 )
sin(k2h) cos(k2h)
) tan(k2h)
where k262 ) e262/C h 6622 is the electromechanical coupling constant; R ) 2h′F′/hFq and is referred to as the relative frequency lowering factor, also known as the electrode mass loading factor; 2h′ and F′ are the electrode thickness and mass density, respectively. Solving for k2 allows the fundamental resonance frequency, f0E, for the electroded region to be obtained as
f0E )
( )
ˆ 66E 1 C 4h Fq
f0
( )
ˆ 66P 1 C ) 4h Fq
f0U )
P
u ˜ 1(r,θ) ) u ˜ r(r)u ˜ θ(θ)
(7)
(13)
where u ˜ r(r) is the solution term represented by the r-dependent terms, and u ˜θ(θ) is the solution term represented by the θ-dependent term. Substituting eq 13 into eq 12 results in two equations given by
∂u ˜r ∂2u ˜r + [(rkr)2 - n2]u r2 2 + r ˜r ) 0 ∂r ∂r
(14)
(8) and
1/2
∂2u ˜θ
(9)
∂θ2
where Cˆ 66P ) C h 66(1 - R - (8k262/π2)). Comparing eqs 7-9, it is seen that the spectrum of the resonance frequencies satisfies E
Equation 12 is a scalar Helmholtz wave equation representing the particle displacement amplitude distribution in a cylindrical coordinate system. The first two terms of the equation show no θ variation but have variations along the r direction. The third term in the equation is independent of r but has a variation with θ. Equation 12 can be solved by the method of separation of variables. The solution can be written as follows
1/2
( )
h 66 1 C 4h Fq
(12)
1/2
with Cˆ 66E ) C h 66(1 - 2R - (8k262/π2)), referred to as the piezoelectrically stiffened effective elastic constant, which decreases as the thickness of the electrode increases. Using procedures similar to those in the fully electroded region, the fundamental resonant frequencies in both the partially electroded and unelectroded regions are found to be, respectively, P
∂2u ˜ 1 ∂u ˜1 ˜ 1 ∂ 2u + 2 + (rk′)2u ˜1 ) 0 r2 2 + ∂r ∂r ∂θ
(6)
+ n2u ˜θ ) 0
(15)
where
kr2 ) k′2 ) k2 - kc2
(16)
U
f0 < f0 < f0
The particle displacement amplitude profile in the QCR is found by solving eq 5 for the QCR examples of Figure 1. Equation 5 is given in a Cartesian coordinate system. This equation is best solved in polar (r,θ) coordinates to coincide with cylindrical boundaries of a typical QCR. Equation 5 can be rearranged as
()
()
2 2 ˜1 ˜1 1 ∂u 1 ∂u + + k′2u ˜1 ) 0 2 2 2 2 R ∂x1 β ∂x3
[
( )]
k′2 ) k2 - k22
) k2 - kc2
u ˜ (r,θ) )
h 66/C66) kc2 ) k22(C
Transformation into the cylindrical coordinate system gives eq 10 as Analytical Chemistry, Vol. 70, No. 2, January 15, 1998
∑ [AJn(krr) + BNn(krr)][C cos nθ + D sin nθ]
n)0 n)∞
∑ [AIn(krr) + BKn(krr)][C cos nθ + D sin nθ]
n)0
(17)
(11)
with
240
{
n)∞
(10)
where R2 ) (C66/C11) and β2 ) (C66/C55), and
C h 66 C66
and n is the harmonic constant with n ) 0, 1, 2, 3, 4, .... Note that when k < kc in eq 16, kr becomes imaginary, which results in an evanescent wave. kc is referred to as a cutoff wavenumber. Equation 14 is a Bessel’s differential equation, and the solution is given by Bessel functions. Equation 15 is a harmonic equation, and the solution is given by harmonic functions. General solutions of eqs 14 and 15 are of the form25
where A, B, C, and D are unknown amplitude constants to be determined by the boundary conditions. Jn(krr), which has a finite limit as krr approaches zero, is called the Bessel function of the first kind with order n. Nn(krr), which has no finite limit (i.e., is unbounded) as krr approaches zero, is called the Bessel function of the second kind with order n (also known as a Neumann function), In(krr) and Kn(krr) are called the modified Bessel functions (25) McLachlan, N. W. Bessel Functions for Engineers; Clarendon Press: London, UK, 1955.
of the first kind and the second kind with order n, respectively. In(krr) has a finite limit as krr approaches zero, and Kn(krr) has no finite limit as krr approaches zero. For a QCR operating at a fundamental mode, for which n ) 0, the particle displacement amplitude is invariant of angular direction but varies only in the radial direction. Equation 17 can then be rewritten as
u ˜ 1(r) )
{
[AJ0(krr) + BN0(krr)] for (krr)2 > 0 [AI0(krr) + BK0(krr)] for (krr)2 < 0
kr2 )
{
{
( )( ) ( )( ) ( )( ) 1
4h2
f662 1
π2
4h2 f662 π2 1 4h2 f662
[f 2 - fce2] in fully electroded region [f 2 - fcp2] in partially electroded region [f 2 - fcu2] in unelectroded region (19)
where
fce )
(4h1 )x CFˆ
) cutoff frequency in fully
q
( )x F
1 fcp ) 4h
C ˆ P66
electroded region ) cutoff frequency in partially
q
electroded region C h 66 ) cutoff frequency in Fq unelectroded region C66 ) resonance frequency in Fq quartz crystal plate
( )x
1 fcu ) 4h f66 )
66
fcu > f > fcp > fce (a) or fcu > fcp > f > fce (b)
(21)
(18)
where J0 and N0 are the Bessell functions of the first and second kind, respectively, with order 0, and I0 and K0 are the modified Bessel functions of the first and second kind with order 0, respectively. A and B are unknown amplitude constants. Equation 18 represents possible solutions for the particle displacement amplitude distribution in the modified-electrode QCR. The exact solution type can be chosen on the basis of the conditions of the argument of the Bessel functions.25 The various wave propagation constants for the thickness direction (X2) for each region have been found based on the resonance conditions. These propagation constants obtained in a Cartesian coordinate system correspond to the wave propagation constants for the X2 direction in a cylindrical coordinate system. Therefore, by substituting eqs 7-9 into eq 16, the radial component of the wave propagation constant defined in eq 18 can be rewritten in the various regions in terms of the operating frequency, f, and cutoff frequency, fc, as
π2
smaller than (or equal to) that of the lower electrode. Assuming that the excited acoustic wave is mostly confined within the electroded region, an appropriate or stable condition of the operating frequency, f, is
(20)
(4h1 )x
Substituting kr from eq 19 into eq 18, the variations of the particle displacement amplitude can be found. (a) “n-m” Electrode QCR (Figure 1a). In this configuration, the upper electrode diameter on the sensing surface is always
depending on the electrode mass loading factor R. The choice of this operating frequency condition determines the sign of kr in each region. For the frequency conditions 21a and 21b, the solution of the particle displacement in each region can be selected, respectively, as
{
0erea aereb ber
