Anal. Chem. 1990, 62,1514-1519
Network Analysis Method Applied to Liquid-Phase Acoustic Wave Sensors Arlin L. Kipling' a n d Michael Thompson*
Department of Chemistry, University of Toronto, 80 St. George Street, Toronto, Ontario, Canada M5S IAl
The network analysls method was used to completely characterlre 5- and 9-MHz bulk acoustic wave quartz crystal sensors wtth alumhum and gold electrodes in the Uquld phase for ail conditions of the crystals. The behavlor of the sensors is evaluated In terms of the serles resonant frequency, frequency of maxhnum phase, minimum Impedance, motional reslstance, maxlmum phase, and slope of phase. The dlfference In resonant frequency In liquid relatlve to alr for crystals with gold and aiumlnum electrodes strongly lmplles that the electrode-llquld Interface plays a signlflcant role In acoustic wave propagatlon Into the liquld. Previous theories associated with liquid-phase sensor behavlor based on the Sauerbrey expression fail to explaln experlmental measurements of serles resonant frequency as a functlon of viscosity.
INTRODUCTION The use of piezoelectric bulk acoustic wave (BAW) devices as microgravimetric sensors for the gas phase has its origins in the work of Sauerbrey ( 1 , 2 ) ,which demonstrated that a thin film applied to a BAW device could be treated in sensor operation as an equivalent mass change of the crystal itself. This overall concept has been exploited extensively in the fabrication of chemically selective sensors for the gas phase, where a binding agent is incorporated into a film deposited on BAW (3) or surface acoustic wave (SAW) ( 4 ) structures. It was stated some time ago that operation of BAW sensors in liquids would not be possible due to viscous damping effects (5). Despite this prediction there has appeared in recent times a flurry of publications describing liquid-phase experiments, particularly those involving immunochemical sensing (6-9). Previous studies of BAW devices in liquids have almost exclusively used the oscillator method. The oscillator most often used is essentially two TTL inverters connected in series to give a noninverting amplifier (zero phase shift between input and output voltages) with the quartz sensor connected from the output to input of the amplifier to produce positive feedback (10-12). Oscillators with single transistors have also been used (13). Only one electrical quantity is measured by this method, the series resonant frequency of the quartz sensor. This frequency by itself constitutes an incomplete characterization of the sensor. Furthermore, it has been reported that this method gives conflicting results in some cases. Yao and Mo (14) found that the resonant frequency of a quartz crystal in an electrolyte could be made to increase, remain constant, or decrease as the conductivity of the electrolyte increases, by simply changing the value of a capacitor in series with the quartz crystal. However, there is a much more serious limitation associated with this method. The oscillator will not function in certain situations, notably when the crystal is in a liquid with a viscosity greater than some particular value. It is important to understand that this is not a practical limitation due to, for example, insufficient power produced Present address: De artment of Physics, Concordia University, 1455 de Maisonneuve Bgd., Montreal, Quebec, Canada H3G 1M8. 0003-2700/90/0362-1514$02.50/0
by the amplifier of the oscillator. Rather, this is a fundamental limitation due to the fact that the resonant frequency ceases to exist, as will be discussed later. Another quantity in addition to the resonant frequency can be measured by using a marginal oscillator. In this configuration the oscillator applies additional feedback internally, which changes the gain of the oscillator amplifier in response to a change of energy dissipated in the quartz crystal, such that the amplitude of the output voltage of the amplifier remains constant. The feedback voltage is a measure of the amount of energy dissipation. Thompson et al. (6, 15) used this approach but did not measure energy dissipation. An impedance analyzer has recently been used to measure the series resonant frequency and the resistance of the equivalent circuit of the quartz crystal (16, 17). Essentially, this instrument measures the voltage applied across the crystal and the current flowing through it and then computes the ratio of voltage and current, which by definition is the impedance. The series resonant frequency measured by this method is the same as that measured by the oscillator method. The resistance is related to the electrical energy dissipated in the quartz crystal. However, these quantities alone only partially characterize the sensor. The network analyzer method has been used in this work to completely characterize the BAW quartz device. For the case of a two-terminal device such as this type of sensor, the voltage incident on and reflected from the load (the sensor) is measured by the network analyzer as a function of frequency. From these basic measurements, all electrical quantities that characterize the load can be found, including impedance. EXPERIMENTAL SECTION Apparatus and Materials. The instrument used to characterize the BAW quartz crystal in liquid was the HP 4195A network/spectrum analyzer (Hewlett-Packard, Palo Alto, CA). The operational frequency range is from 10 Hz to 500 MHz with a resolution of 1 mHz. The HP 41951A impedance test kit and HP 16092A spring clip fixture were used to make impedance measurements directly. The frequency range of impedance measurements with the 41951A is 100 kHz to 500 MHz. The values of the four parameters of the equivalent circuit of the quartz crystal are calculated internally by the 419514 from the measured data. Four piezoelectric crystals were studied in detail. They are denoted 5AU1,5ALl, 9AU1, and 9AL2 where 5AU1 and 5AL1 are 5-MHz crystals with gold and aluminum electrodes, respectively, and similarly 9AU1 and 9AL2 are 9-MHz crystals with gold and aluminum electrodes. The crystals were supplied by Leigh Instruments, Toronto, and International Crystal Manufacturing, OK. The liquids employed were water-glycerol mixtures ranging from pure water with a kinematic viscosity of 1.00 mmz/s to pure glycerol with a kinematic viscosity of 1400 mm2/s. The mixtures used were percentages by weight of glycerol in water of 10% to 90% in increments of 10%. The values of kinematic viscosity for the liquids were taken from ref 18. Procedure. The magnitude and phase angle of impedance as a function of frequency were measured for various quartz crystals without any coatings on the electrodes. Each crystal was first measured in air and then with the liquids in contact with one 1990 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 62, NO. 14, JULY 15, 1990
defined by w = 2af, and f is the frequency (in hertz, Hz). The impedance of the equivalent circuit, 2, is the impedance of the series combination of R,, L,, and C,, denoted by Z ,, in parallel with the impedance of Co denoted by 2,:
2 = ZmZo/(Zm
The impedances 2, and Zo,in terms of X,, the reactance of L , and C, in series, and X o , the reactance of Cotare
2, = R Flgure 1. Display of a typical measurement of magnitude of impedance, IZI, and phase angle of impedance, 8, as a function of frequency. The scale for 8 is linear; the scale for IZI is logarithmic.
2,= j X o
, Equation 1 can be written in the standard form
Figure 2. Equivalent circuit of the quartz crystal giving the parameters and impedances of the circuit elements.
electrode (and air in contact with the other) starting with pure water and ending with pure glycerol. Measurements were first made with a frequency span that extended over the entire resonance region of the crystal. Then the frequency span was reduced, and the center frequency changed to find more accurate values of the maximum, minimum, and zero of impedance magnitude and impedance phase angle and their corresponding frequencies. The experimental method consists of making many measurements of the two curves, magnitude, and phase angle of impedance as a function of frequency. A representative printout of the two curves is given in Figure 1, where the frequency span covers the complete resonance region. On the vertical axis, the impedance magnitude, 121,is plotted on a logarithmic scale, which is not shown, and the phase angle, 0, is plotted on a linear scale. The frequency is plotted along the horizontal axis on a linear scale. These quantities are defined and discussed later. Contact angles of the electrode surfaces of various crystals were also measured.
RESULTS AND DISCUSSION Equivalent Circuit. The detailed theory of the piezoelectric crystal has been derived by Cady (19). A simplified version of the theory of Cady is given by Bottom (20). The starting point of the theory is the equation of motion of each point of the quartz plate. The final result of the analysis is an expression for the current density flowing through the crystal in terms of the voltage across the crystal. It has the form, i = (complex quantity)u, where i is the current density, u is the voltage, and the complex quantity (complex meaning that the quantity has both a real and imaginary part) is a function of several properties of the quartz and a function of the frequency. This relationship between i and u has the form of Ohm's law, and so the complex quantity can be regarded as an admittance, or its reciprocal as an impedance. The complex quantity can be interpreted as a circuit that contains four circuit elements, but only after considerable mathematical manipulation and some approximations. This circuit is an electrical model of the quartz crystal, called the equivalent circuit of the crystal, which responds to an applied voltage or current in the same way as the crystal itself. The equivalent circuit of the quartz crystal is given in Figure 2. The four circuit elements with parameters R, (in ohms, O), 15, (in henrys, H), C, (in farads, F), and Co (in farads) correspond to one mode of vibration of the quartz crystal. The impedance (in ohms) of each circuit element is also given, where w is the angular frequency (in radians/second, rad/s)
by multiplying the numerator and denominator by the complex conjugate of the denominator. Both R and X are complicated expressions that are functions of the four parameters R,, L,, C,, and Co and the angular frequency, w . The magnitude of 2, 121(in ohms), is
1 2 1 = (R2 + X2)1/2
The phase angle of 2, 8 (in radians), is
8 = tan-'
The phase angle, 0, is usually called simply the phase. Mathematically, a complex number is represented by a point in a plane, called the complex plane. The quantities R and X in eq 4 are the rectangular coordinates, and 121and 8 in eq 5 and 6 are the polar coordinates of the point, 2, in the complex plane. Physically, impedance, 2, is the voltage across the crystal, u, divided by the current flowing through the crystal, i, that is, Z = u/i. Both u and i vary sinusoidally with time at the same frequency. The magnitude of impedance, 121, is the magnitude (maximum value) of u divided by the magnitude of i. The phase angle of impedance, 8, is proportional to the difference in the time between when the voltage across the crystal is a maximum and the current flowing through the crystal is a maximum. If 6 is positive, u is a maximum a t an earlier time than when i is a maximum, and one says that the voltage leads the current. If 8 is negative, the voltage maximum occurs a t a later time than the current maximum; the voltage lags the current. For example, if B = -a14 rad, then u is a maximum ( a / 4 ) / 2 ~= period of oscillation after i is a maximum, which for a 9-MHz crystal is ' 1 8 X 0.111 = 0.014 ps. For an inductor, 8 = +7r/2 rad, and for a capacitor, 8 = -a12 rad. From (6) the reactance, X,has the same sign as 8, and therefore the reactive part of Z in eq 4 is inductive when 0 is positive and capacitative when 9 is negative. The theoretical calculations of the characteristic values of 121and 8 shown in Figure 1 from eqs 5 and 6 are tedious. T o alleviate some of the mathematical difficulty, X, and X , + X o in eq 1 (after substituting eq 2 into (1))can be replaced by expressions linear in w that are good approximations for the case of the quartz crystal. The reasoning follows. The reactance, X,, from eq 3 is
x, = (W2L,Cm
The quantity l/L,C, is the square of the series resonant angular frequency (defined later) when R, = 0, wso: w s 2 = l/LmC,. Substituting and rearranging give
- ~ s O ) ( w+ Wso)/Wso2wCm
For a 9-MHz crystal, fso = w s o / 2 a is nominally 9 MHz and f
ANALYTICAL CHEMISTRY, VOL. 62, NO. 14, JULY 15, 1990
= w / 2 r is the value of the frequency in the resonant region in Figure 1 which has a span of the order of lo4 Hz. So both fsp and f have order of magnitude lo7Hz, and their maximum difference has order of magnitude lo4 Hz. Therefore, to very good approximation w wso = 2w, and the final approximate expression for X, is
fsa - is,
x, = 2L,(w
5AU1 - 0
x, + xo = 2L,(w
where wpo is the parallel resonant angular frequency (defined later) when R, = 0:
Circuit Elements. The circuit elements depend on properties of the quartz. With the notation of Bottom (20), the expressions for the circuit element parameters are the following:
Kinematic Viscosity, mm2/s Figure 3. Frequency difference, f, (seriesresonant frequency in air) minus f , (series resonant frequency in liquid), versus kinematic viscosity of the liqukl for two 5-MHz and two 9-MHz quartz crystals. Both scales are logarithmic. 100000
R, = e3r/8Ac2 L , = e3p/8At2 C, = 8Ac2/r2ec Co = ktoA/e
The two geometrical properties, thickness, e, and area, A , of the quartz plate appear in the equations for all four parameters. The piezoelectric stress constant, t, is responsible for the existence of the three circuit elements with parameters R,, L,, and C,. If t = 0, then
R, = wL, = l/wC, =
leaving only the impedance of Co in the equivalent circuit in Figure 2. The circuit elements with parameters R,, L,, and C, are called the motional circuit elements (subscript m for motional) because they are associated with the large amplitude of vibrational motion caused by the piezoelectric effect in the resonant region shown in Figure 1. The dissipation coefficient, r, density, p , and elastic constant, c, are properties of the quartz plate that appear in the expressions for R,, L,, and C,, respectively. The dielectric constant of quartz, k, and permittivity of free space, to, both appear only in the expression for Co. Note that Co does not depend on the piezoelectric effect because t is not present in the equation for Co. The physical significance of the circuit elements follows from the expressions for the parameters, eq 9. The motional resistance, R,, represents the dissipation of electrical energy in the quartz crystal. There are both internal and external losses of electrical energy; electrical energy is converted into thermal energy in the quartz, and electrical energy flows out of the quartz into the fluid (gas or liquid) in contact with the surface of the quartz crystal in the form of acoustic waves. The latter loss of energy depends on the bulk properties of the fluid and the nature of the interaction between the fluid and the surface of the coating on the quartz crystal. The expression for R, must be generalized to account for the external dissipation of energy. The motional inductance, L,, represents the mass of the quartz plate, including the coating on its surface and the effect of the fluid in contact with the coating. The expression for L , must also be generalized to account for the external influences on the crystal. The motional capacitance, C,, represents the elastic properties of the quartz. The electrostatic
Klnernatlc Viscosity, rnm2/s Figure 4. Frequency difference, f d a (frequency of maximum phase in air) minus fmB(frequency of maximum phase in liquid), versus kinematic viscosity of the liquid for two 5- and two 9-MHz quartz crystals. Both scales are logarithmic. ohms
I 1 8 1 1 1 8
! . , I # ,
Kinematic Viscosity, rnrn2/s Figure 5. Minimum value of magnitude of impedance, IZI,,,,,,, versus kinematic viscosity of the liquid for two 5-MHz and two 9-MHz quartz
crystals. Both scales are logarithmic.
capacitance, Co, is the capacitance of the two parallel-plate metal electrodes on the quartz surfaces. Characteristic Quantities. Six of the characteristic quantities shown in Figure 1 have been extracted from the measurements and are plotted versus kinematic viscosity in Figures 3-8. (1) Series resonant frequency: There are two frequencies a t which the phase is zero: f, is the low frequency of zero phase, called the series resonant frequency, and fp is the high frequency of zero phase, called the parallel resonant frequency. f, is plotted in Figure 3. fp is not plotted. The theoretical expressions for f, and f,, are obtained by solving 0 = 0, which is a quadratic function of frequency. 0 is given by eq 6. The series resonant frequency is the quantity measured by the oscillator method.
ANALYTICAL CHEMISTRY, VOL. 62, NO. 14, JULY 15, 1990 10000
, , , , , , ,,,
, , , , , ,, ,,
, , , , , , , ,, 1000
Kinematic Viscosity, mm2/s Flgure 6. Motional resistance of the equivalent circuit, R , versus kinematic viscosity of the liquid for two 5-MHz and two 9-MHz quartz crystals. Both scales are logarithmic. eo
Bmax, d e g r e e s
Kinematic Viscosity, mm2/s Flgure 7. Maximum phase angle of impedance, 8, versus kinematic viscosity of the liquid for two 5-MHz and two 9-MHz quartz crystals. The scale for kinematic viscosity is logarithmic; the scale for phase angle is linear.
Kinematic Vlscoslty, mm2/s Flgure 8. Slope of phase angle of impedance, deldf, at the series resonant frequency, f,, versus kinematic viscosity of the liquid for two 5-MHz and two 9-MHz quartz crystals. Both scales are logarithmic.
(2) Frequency of maximum phase: The series resonant frequency, as well as the parallel resonant frequency, ceases to exist if the value of maximum phase is less than zero, since in this case the phase curve in Figure 1 does not cross the 8 = 0 axis. However, fms,the frequency at which the phase is a maximum, exists always. fmeis plotted in Figure 4. The theoretical expression for f,# is found by solving d8/df = 0 for the frequency, where 8 is given by eq 6. (3) Minimum impedance: The magnitude of impedance has one minimum and one maximum. The minimum magnitude of impedance, IZlmin,is plotted in Figure 5 . The maximum magnitude of impedance, IZI-, and the frequencies at which the minimum and maximum occur, fminand f,=, respectively, are not plotted. However, f- has been measured and is discussed later. T o find the theoretical expression for IZlmi,,, solve dlZl/df = 0 for fminand substitute fmin for f in the
expression for 121, where 121is given by eq 5. (4) Motional resistance: The motional resistance of the equivalent circuit, R,, is computed by the network analyzer and is plotted in Figure 6. The theoretical expression for R , is given in eq 9. (5) Maximum phase: The maximum phase of impedance, Omart is plotted in Figure 7 . (The frequency of maximum phase is plotted in Figure 4.) To find the theoretical Omax, solve d6/df = 0 for fd and substitute fm8for f i n the expression for 6, where 8 is given by eq 6. (6) Slope of phase: The slope of the phase as a function of frequency, [email protected]
/df,at the series resonant frequency, f,, is plotted in Figure 8. This slope has no meaning if the maximum phase is less than zero since f, does not exist. A corresponding quantity that always exists is the maximum slope; this has not been plotted. The theoretical expression for (d8/dflf#is obtained by substituting f, for f i n dO/df, where 8 is given by eq 6. Series Resonant Frequency. The difference between f, in air, f,, and f, in liquid is plotted in Figure 3 as a function of kinematic viscosity on logarithmic scales. This quantity will be discussed in detail since it has been measured by several workers using the oscillator method. An explanation of the results in Figure 3 based on the Sauerbrey equation is not possible. A theoretical expression for Af = f, - f,, has been derived by Bruckenstein and Shay (12) by substituting into the Sauerbrey equation a term for the effective amount of mass added to one or both electrodes of the quartz crystal due to the contact of the electrodes with the liquid. With their notation, the equation is
Af = -2.26 X 10-6f/2dJ/2
for liquid in contact with one electrode of the quartz crystal where Af is the change of frequency in hertz, f is the frequency of oscillation in hertz (f, in this paper), dL is the density of the liquid in g/cm3, and v is the kinematic viscosity in cmz/s. Kanazawa and Gordon (21) also derived an equation for Af from a model that couples the stress wave in the quartz to a damped stress wave in the liquid and obtained the same functional dependency for Af as in eq 10. The variation of v in Figure 3 is from 1.00 to about 20 mm2/s (1 mm2/s = cm2/s) for all but one of the curves. Over the same range, the density of the liquid (glycerol-water mixture) varies from 1.00 to about 1.18 g/cm3, from ref 18. f decreases by less than 3 kHz for the 5-MHz crystals and less than 7 kHz for the 9-MHz crystals, a change of frequency of less than 0.1%. So in eq 10, dL is constant to a good approximation and f is constant to a very good approximation. Thus, eq 10 has the form lAfl = cull2
where C = 2.26 X 104f/2dL, which is constant to a good approximation for crystals with the same nominal frequency. Equation 11is transformed into an equation of a straight line by taking the logarithm of both sides: log lAfl = log C + 0.5 log v (12) where the new variables are log IAfl and log v with intercept, log C, and slope, 0.5. Equation 12 predicts that in Figure 3 there will be two straight lines, one for the 5-MHz crystals and one for the 9-MHz crystals, and both with a slope of 0.5. Equation 11 predicts that the ratio of IAfl for the 9- and 5-MHz crystals at each viscosity and corresponding density is the ratio of C for each crystal: 93/2/53f2= 2.4. None of the predictions that follow from eq 10 agree with the results in Figure 3. The curves in Figure 3 are not linear over the complete range of viscosity. Furthermore, the curves for the two 5-MHz crystals are not coincident, but rather the crystal with the gold
ANALYTICAL CHEMISTRY, VOL. 62, NO. 14, JULY 15, 1990
electrode has a greater lAfl then the one with the aluminum electrode and similarly for the two 9-MHz crystals. The slope of the curves is less than 0.5 as predicted by eq 10. In the low-viscosity region, the slope is about 0.30 for both the 5and 9-MHz crystals with A1 electrodes and about 0.27 for the crystals with Au electrodes. Finally, the ratio of IAfl for the 5- and 9-MHz A1 crystals a t u = 1.00 mm2/s (pure water) is 3.1 and for the 5- and 9-MHz Au crystals, 3.4, values that are significantly higher than the predicted value of 2.4. It is clear that IAfl depends on the surface properties of the electrodes and therefore on the interaction of the surface with the liquid, since the curves for the two crystals at each frequency are separated. If there was no interaction between the surface and the liquid, the vibrating molecules on the surface would slip freely past the molecules of the liquid in contact with the surface and f , would be unaffected by the liquid, that is, f , would have the same value as in air and therefore IAfl would be zero. A t the other extreme, if there was a very strong force of attraction between species at the electrode surface and adjacent molecular layer in the liquid there would be no slippage between them (the so-called "no-slip" boundary condition often used to solve hydrodynamic problems). The maximum effective mass would be added to the crystal, resulting in the maximum reduction of series resonant frequency, f,, and therefore maximum IAfl. Measurements were made of the contact angles of the surface of each of the four crystals and the results are as follows, to the nearest 5': 5AU1,35'; 5AL1, 55'; 9AU1, 45'; 9AL2, 55'. This suggests that the Au surface is more hydrophilic than the A1 surface and so the interaction between the Au surface and liquid is greater, and this is consistent with the larger value of IAfl for Au than A1 in Figure 3. f , is found theoretically from eq 6 by setting 0 = 0. The two roots are f , and f p , the zero-phase frequencies shown in Figure 1. The equation for f, is f
The first term is the Sauerbrey expression since ( C / P ) ' / ~ is the acoustic wave velocity and eq 13 reduces to this term when r = 0, that is, when there is no dissipation of energy in the quartz crystal. The effective thickness of the quartz plate increases as the viscosity increases according to the theory of Bruckenstein and Shay. But r also increases with viscosity because the flow of acoustic energy into the liquid increases with viscosity. So, as viscosity increases, the first term decreases and the second term increases. Equation 13 accounts for the shape of the curves in Figure 3; the first term dominates at low viscosities and f , decreases as viscosity increases. But at high viscosities the second term more than compensates for the first term and f , begins to increase. However, eq 13 does not take into account the properties of the surface of the electrodes and so does not predict that the two curves for each frequency are separated. Frequency at Maximum Phase. There are no experimental points in Figure 3 for viscosity greater than about 50 mm2/s, although measurements were taken up to 1400 mm2/s (pure glycerol). The reason is that f , does not exist above about 50 mm2/s because the maximum phase is negative. With this in mind, a frequency that characterizes the quartz crystal and exists for all conditions must be used instead of f,. With reference to the phase curve in Figure 1, the most obvious choice of such a frequency is the frequency of maximum phase, ims. The difference between f m s in air, fmsa, and f m s in liquid is plotted in Figure 4 as a function of viscosity on logarithmic scales. The slope at low viscosity is about 0.25 and increases to about 0.47 at high viscosity.
Minimum Impedance and Motional Resistance. The minimum magnitude of impedance, IZI-, is plotted in Figure 5 as a function of viscosity on logarithmic scales. The motional resistance, R,, is plotted versus viscosity in Figure 6 on exactly the same scales as used for IZlmin. At low viscosity, lZlminand R, are about the same in magnitude and in slope, the slope being about 0.40. Their values are higher for Au then Al electrodes for both the 5- and 9-MHz crystals, which means that more energy is dissipated by the crystals with Au electrodes. At high viscosity the slope of 121- decreases to about 0.18. The values of IZlmhare higher for the 5-MHz than the 9-MHz crystals. In contrast, the slope of R, remains constant at high viscosity, and all crystals have the same value of R, within experimental error. At the highest viscosity, 1400 mm2/s, R, is greater than lZlminby approximately a factor of 3 for the 5-MHz crystals and by a factor of 4 for the 9-MHz crystals. The frequency of IZlmin, fmin, has not been plotted in this paper but has been measured. f- is always less than the series resonant frequency, f,. When the crystal electrodes are in air, the difference between the two frequencies is close to 10 Hz. But in the liquid phase, fmin is significantly less than f,, the difference being a sensitive function of kinematic viscosity, U. For the 5-MHz crystal with the Au electrode, 5AU1, f , fmin is 260 Hz for u = 1.00 mm2/s (pure water in contact with one electrode) and 2700 Hz for v = 19.3 mmz/s (70% glycerol in water, one electrode). For the 9-MHz Au crystal, 9AU1, f, - fmin is 720 Hz at 1.00 mmz/s and 8200 Hz at 19.3 mm2/s. The results are similar for the crystals with Al electrodes. For both the 5- and 9-MHz crystals, f, - f- varies approximately as YO.*, but of course only for values of u below the value at which f , no longer exists. The theoretical expression for this frequency difference from eq 5 and 6 in terms of the equivalent circuit parameters is where fSo2 = l/(LmCm),the series resonant frequency when R, = 0. Maximum Phase. The maximum phase, Om=, is plotted on a linear scale versus the viscosity on a logarithmic scale in Figure 7. The most striking feature of these results is that the maximum phase is negative for the 5-MHz crystals a t viscosities above about 50 mmz/s and for the 9-MHz crystals above about 20 mm2/s. As mentioned previously, above the values of viscosity a t which 8-, becomes negative, the oscillator method cannot be used since in principle the crystal will not oscillate (when the oscillator amplifier has zero phase shift, which is always the case). The maximum phase is greater for the 5-MHz than the 9-MHz crystals and depends on the electrode material and therefore the surface properties of the electrodes. To a reasonable approximation, the curves in Figure 7 are linear, which implies that the functional relationship between the quantities is u =
where u is the kinematic viscosity and (Y is the slope, which is negative. From Figure 7, (Y is approximately -0.023 deg-' = -1.3 rad-l and, of course, the constant is the value of u when om,, = 0. Slope of Phase. The quantity (dB/dflr, is the slope of B versus f a t the series resonant frequency, f,. It is plotted as a function of viscosity in Figure 8 on logarithmic scales. At low viscosity the slopes of (dB/fl, versus viscosity range from -0.4 to -0.5. The values of (dO/dflfsfor the 5-MHz crystals are about a factor of 2 greater than for the 9-MHz crystals. This quantity also depends on the properties of the.electrodes as does f,, - f , in Figure 3, but the dependence is
Anal. Chem. 1990, 62, 1519-1522
different. (dO/df)f, is greater for A1 than for Au electrodes at each frequency; the opposite is true for f, - f,.
CONCLUSIONS The network analyzer method completely characterizes a quartz crystal sensor for all conditions of the crystal. This method replaces the oscillator method, which is unsatisfactory for two reasons. First, only one quantity that characterizes the crystal is measured, and so the electrical characterization is incomplete. More importantly, however, the oscillator method does not function in all circumstances. There is no oscillation when the maximum phase of the quartz crystal is less than zero. This is the case, for example, when viscosity of the liquid in contact with the crystal exceeds a certain value. Modified Sauerbrey expressions fail to explain the experimental measurements of series resonant frequency, f,, as a function of viscosity. The basic reason is that these equations do not take into account the dissipation of electrical energy in the quartz crystal. In the liquid phase, the energy dissipation is due principally to the flow of acoustic waves into the liquid, which depends on the properties of the liquidcrystal interface, as well as the bulk properties of the liquid. Registry No. AI, 7429-90-5; Au, 7440-57-5;quartz, 1480860-7. LITERATURE CITED (1) Sauerbrey, G. 2.Phys. 1959, 155, 206-212. (2) Sauerbrey, G. Z. Phys. 1984, 778, 457-471. (3) McCallum, J. J. Analyst 1989, 7 74, 1173-1 189.
(4) Fox, C. G.;Alder, J. F. Analyst 1989, 774, 997-1003. (5) Gullbault. G. G. I n Lu, C.. Czanderna, A. W., Eds. Applications offiezoelechic Quartz Crystal Microbalances: Vol. 7 of Methods and phenomena, Their Applications to Science and Technology, Elsevier: New York, 1984; p 251. (6) Thompson, M.; Dhaliwal, G. K.: Arthur, C. L.; Calabrese, G. S. I€€€ Trans. Ultrason. Ferroeiec. Freq. Con*. 1987, UFFC-34, 127-135. (7) Muramatsu, H.; Dicks, J. M.: Tamiya, E.; Karube, I.Anal. Chem. 1987, 59, 2760-2763. (8) Ebersole, R. C.; Ward, M. D. J. Am. Chem. Soc. 1988, 770, 8623-8628. (9) Davis, K.A.; Leary, T. R. Anal. Chem. 1989, 67, 1227-1230. (10) Nomura, T.; Nagamune, T. Anal. Chim. Acta 1983, 755, 231-234. (11) Nomura, T.; Watanabe, M.; West, T. S. Anal. Chim. Acta 1985, 175, 107-1 16. (12) Bruckenstein, S.: Shay. M. Elecfrochim. Acta 1985, 30,1295-1300. (13) Nomura, T.; Okuhara, M. Anal. Chim. Acta 1982, 142, 281-284. (14) Yao, S.-2.: Mo, 2.-H. Anal. Chim. Acta 1987, 793, 97-105. (15) Thompson, M.; Arthur, C. L.; Dhaliwal, G. K. Anal. Chem. 1988. 58, 1206- 1209. (16) Muramatsu, H.; Tamiya, E.; Karube, I.Anal. Chem. 1988, 6 0 , 2 142-2 146. (17) Muramatsu, H.; Tamlya. E.: Suzuki, M.; Karube, I. Anal. Chim. Acta 1988. 275, 91-98. (18) Weast, R. C., Ed. CRC Handbook of Chemistry and Physics, 61th ed.; CRC Press: B o a Raton, FL, 1980-81. (19) Cady. W. G. Piezoeiectrlcity; Dover: New York, 1964. (20) Bottom, V. E. Infroduction to Quartz Crystal Unir &sign; Van Nostrand Reinhold: New York, 1962. (21) Kanazawa, K. K.; Gordon 11, J. G. Anal. Chlm. Acta 1985, 775, 99-105.
RECEIVED for review January 25,1990. Accepted March 29, 1990. Support for this work from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
High-Performance Liquid Chromatography/Chemical Ionization Mass Spectrometric Analysis of Pyrolysates of Amylose and Cellulose Peter W. Arisz,* James A. Lomax, and Jaap J. Boon Unit for Mass Spectrometry of Macromolecular Systems, FOM-Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands I n-source pyrolysis chemlcal lonlzatlon mass spectrometry (PyCIMS) of polysaccharldes shows a number of Ion series that correspond to anhydroollgosaccharldes and ollgosaccharides wkh attached sugar rlng-cleavage fragments. To conflrm that these observed Ion serles are produced upon pyrolysis and not by cluster formatlon or fragmentation In the Ion source, off-line Curle polnt pyrolysates of amylose and cellulose were prepared, and thelr condensates were per-0benzoylated. The desorptlon chemkal lonlzatlon (DCI) mass spectra of the pyrolysates show lon serles that correspond to those In the PyCI mass spectra, taklng Into account mass changes upon derlvatlzatlon. Chromatographlc separatlon (high-performance llquld chromatography (HPLC)) of these derlvatlzed pyrolysates lndlcates a range of lsomerlc products correspondlng to each of the major Ions found In the DCI mass spectra. The fact that the retentlon tlmes of the main peaks In the dlmer and the trlmer region of the amylose and cellulose pyrolysates In the chromatograms are qulte dmerent conflrms that the conformation of the glycosldlc bonds Is preserved upon pyrolysis. There are strong indications that pyrolysates of polysaccharides contain anhydrooligosaccharides. The presence of anhydrocellobiose (cellobiosan) in pyrolysates of cellulose
has been reported by Radlein et al. (1): between 6% and 15% cellobiosan (dimer) was found in the syrup, and the presence of higher oligosaccharides was indicated. Studies on larger polysaccharide fragments have been carried out by using in-source pyrolysis mass spectrometry (PyMS), usually with positive ion chemical ionization with ammonia (2-5). In-source PyMS is a rapid method, and almost no sample pretreatment is required. The most characteristic ion series found correspond to anhydrooligosaccharides, but other series, corresponding to (anhydro)oligosaccharides with attached ring-cleavage fragments, are also present ( 2 , 4 ) . The spectra contain information at the submonomer, monomer, and oligomer levels and are of analytical interest because of the structural insights they could provide. However, it is still possible that the observed ion series are recombination products of the anhydro monomer, which is the most abundant pyrolysis product (6). Such synthetic oligomers would contain a mixture of linkages (7),but these would not be distinguished by conventional MS, which is not sensitive to isomeric differences. To obtain information about the nature of the monomers and their mode of linkage, chromatographic methods such as gas chromatography (GC), supercritical fluid chromatography (SFC),or high-performance liquid chromatography (HPLC) must be used in combination with MS.
0003-2700/90/0362-1519$02.50/0 0 1990 American Chemical Society