Anal. Chem. 1988, 60, 2142-2146
2142
Computation of Equivalent Circuit Parameters of Quartz Crystals in Contact with Liquids and Study of Liquid Properties Hiroshi Muramatsu Research and Development Department, Seiko Instruments, Inc., Takatsuka-shinden Matsudo-shi, Chiba 271,Japan
Eiichi Tamiya and Isao Karube* Research Laboratory of Resources Utilization, Tokyo Institute of Technology, Nagatsuta-cho, Midori-ku, Yokohama 227, Japan
Electrical equivalent circult parameters of quartz crystals In contact with a llquld were computed by using an impedance analyzer and a microprocessor. The equation for deficit resistance R which Is one of the parameters, was derlved as R , = (2?rFgq)1“A/k2,where F, Is the resonant frequency, A is the electrode area, k is an electromechanical coupling factor, and p and q are the density and viscoolty of the liquid, respectively. Water-ethanoi, ethanoi-methanoi, and waterglycerol mixtures were used for the experlments. R showed excellent lineartty to (pq)”* wlthln a wlde range of (pq)”’ values. A resonant frequency shift ( A F ) baslcaiiy showed linearity to (pq)”’, but an evident shlH from the linear line was obtained when the crystal quartz electrode was In contact with high-vlscoslty ilqulds and when both sides of the electrode were In contact with water. The shift of AF In water was attributed to the dlelectrlc polarization of water molecules and a slmuitaneous change of the physical property. The computatlon of R , is useful for measuring liquld viscosity, and the contrast of R , and AF offers the possibillty of providing new informatlon on llquid properties.
,,
,
It has been reported that it is possible for AT cut quartz crystals to oscillate in contact with liquids (1-3). Several applications have been published, including determination of ions ( 4 ) , immunoassays ( 5 , 6 ) ,liquid chromatography (l), and an electrochemical microbalance (7-9). The frequency shift of AT cut quartz crystals in contact with liquids has been studied (10-12)and also the frequency shift (AF) associated with mass loading or deposition onto the quartz surface (13-1r). Kanazawa et al. derived the following equation for A F from equations of the shear stress relationship in the quartz and for the liquid (10)
detailed discussion was presented. Simpson also studied the determination of solid/liquid interface structures by using a quartz crystal (12). Fluid properties in megahertz vibrations have been studied by using torsionally vibrating crystals. Mason et al. have estimated shear elasticity and viscosity of oil and polymer liquids on the basis of the equivalent electrical circuit for both the liquid and crystal (18,19). This method has been applied to measure other liquids (20-22). In this paper, electrical equivalent circuit parameters are computed, and an equation for the properties of R1 (a deficit resistance) and liquid properties is derived. In particular, AF (a resonant frequency shift) and R1 are discussed in relation to the viscosity and elasticity of the liquid.
THEORY An electrical equivalent circuit of an AT cut quartz crystal is shown in Figure la. An equation for electrical oscillation of the series of L,, C,, and R1 is shown as
d29 dq 1 R1-d t -q C, = (4) dt2 where q is the charge and V is the applied voltage. For the quartz crystal, it is well-known that the electrical model can be converted to a mechanical model by use of the electromechanical coupling factor k as an electromechanical analogy. The equation for the mechanical model shown in Figure 1b is given as d2x dx 1 mr-x = F (F = k V ) (5) dt2 dt Cm where x is displacement, m is mass, r is the coefficient of friction, and Cm is compliance. From a comparison of eq 4 and 5, the following relationship is obtained:
+
L,-
+
AAGC = -k,A(p~7)O.~ f ( A e )
(2)
AF = - k 2 A ( p ~ ) 0 .+5 g ( A e )
(3)
where AAGC is the automatic gain control change, Ae is the dielectricity change, and k,, k2, f , and g are proportional constants. The dielectric effect was suggested here, but no
v
+
+
= - F s 3 ’ z ( ~ ~ ~ ~ / ~ ~ g ~ )(1) 1’2 where F, is a resonant frequency of the quartz crystal, AF is the resonant frequency shift, I.L is the shear module of the quartz crystal, q L is the viscosity of the liquid, pQ is the density of the quartz, and pL is the density of the liquid. A similar study has been reported by Hager (11).He described both the resonant frequency shift and automatic gain control change of the AT cut quartz crystal in contact with a liquid. By starting from the equation for the fluid velocity of the surface of the resonator and considering viscous energy losses, he obtained the following equations:
+
L1 = m / k 2
(6)
C1 = Cmk2
R1 = r / k 2 Equations 6-8 show the correspondence of inductance-mass, capacitance-compliance, and resistance-friction factors. When F = F , cos ( u t ) , eq 5 may be solved as follows: x =
Fm
+
[ ( l / C m - mu2) cos (ut)
+
[ ( l / C m - mu2)2
ur sin ( u t ) ](9) A t the resonant state u: = l/(Cmm),and eq 9 is arranged as Fm
x = - sin (ut) ur
0003-2700/88/0360-2142$01.50/00 1988 American Chemical Society
(10)
ANALYTICAL CHEMISTRY, VOL. 60, NO. 19, OCTOBER 1, 1988
9
2143
+-gg F
r
B
A
b
Flgure 2. Schematic diagram of two types of cells: (A) only one side of electrode in contact with the liquid: (B) both sides in contact.
Liquid .z
Equations 1and 21 show basically that both AF and R1have ~ . the resonant frequency is linear relations to ( ~ q ) l / But expressed as
Q u a r t z piate
C Flgure 1. (a) Electrical equivalent circuit of AT cut quartz crystal. (b) Mechanical model for (a). (c) Coordinate model of oscillating quartz piate. where u, is the velocity. The mechanical model shown in Figure l b has been adapted to a practical model of the quartz plate, which is oscillating horizontally and has one of its sides in contact with the liquid, where the movement of the plate is defined only in the x direction (Figure IC). x and u, in eq 10 and 11 mean displacement and velocity of the quartz surface for Figure IC. In Figure IC,the displacement and velocity for the vertical coordinate location are shown as u(z,t) and u(z,t), respectively. The starting equations for eq 1, which express the shear stress relation for the quartz and liquid, are given according to Figure ICas follows (10):
F ( z , t ) / A= p d u ( z , t ) / d z
(12)
F(z,t)/A= q L d ~ ( ~ , t ) / d ~
(13)
and
( d F ( ~ , t ) / ddz ~ )= ~ Q dz A ~(z,t)/dt
(14)
( d F ( ~ , t ) / ddz ~ )= pLA dz ~ ( z , t ) / d t
(15)
where p is the shear module of the quartz crystal, qL is the viscosity of the liquid, pQ is the density of the quartz, pL is the density of the liquid, dz is the thickness, and A is the area. From eq 13 and eq 15, the following equation is derived: d2u(z,t)/dz2 = PL/?L[dU(Z,t)/dtI
(16) Equation 16 is equal to the equation dealt with in ref 23. The solution for u(0,t) is given under the boundary condition of u, = u ( z , t ) at z = 0 and u(z,t) = 0 at z = as follows:
-
u ( z , t ) = uoe-(wp/2~)’’2z cos ( a t - ( 0 p / 2 q ) ~ / ~ z (17) )
where uo is defined as u, = uo cos ( w t ) . The force of friction on the surface of the quartz plate is given from eq 13 as
F(O,t) -A
du(0,t)
-qLdz
(18)
The following equation for the force of friction was also derived from eq 17 and 18 as (23)
According to eq 10 and 11,Ll and C1are changed by surface mass load and the elasticity of the fluid, respectively. This ~ ’ also ~ the surface means that hF relates not only to ( ~ 7 ) but mass load and elasticity of the fluid.
EXPERIMENTAL SECTION Apparatus and Materials. Two types of cells were used, one with both sides of the quartz crystal electrode in contact with the liquid and the other with only one side of the electrode in contact with the liquid (Figure 2). A T cut 9-MHz quartz crystals, an impedance analyzer (Yokogawa Hewlett-Packard, Model 4192A), and a personal computer (Nippon Electric Co., Model 9801E) were used for admittance measuring. All the data shown in this paper was measured at a voltage of 0.01 V, but measurements using 0.1 and 1 V were also investigated. Distilled/deionized water, which has a conductivity below 1 pQ cm-2, and chemicals of analytical grade were used for the experiments. Computation of Electrical Equivalent Circuit Parameters. Admittance for the circuit of Figure l a is expressed as Y=
1
R1 + j w L l
+ l/jwC1 + jwCo
It is possible to obtain conductance G and susceptance B by using an impedance analyzer at scanned frequency. Equation 23 is separated to form
R1
G= R12
‘ B - wc, =
+ (wL1- 1/wC1)2
(24)
-(wL1 - l/wC1) R12
+ (wL1 - 1/wC1)2
(25)
Equations 24 and 25 are rewritten to give (G -
& )+
( B - WC,,)~=
(&!
(26)
The frequency giving the maximum value of G is the resonant frequency w, and the value of 1/G is equal to R1. Frequencies w1 and w2 are obtained from the maximum and minimum point of B , respectively, and the following relations are derived: Aw = R1/L1 (Aw =
~
1
4. -
In our system, frequencies for measuring admittance were set up by computer in a range of w1 - Aw to w2 + Aw to cover the area of resonant frequency. About 1000 points were measured in the range. The center point and radius for the circle drawn by eq 26 were computed by using the method of least squares for the circle. The value of B at the center point indicates the shift due to C,. The value of susceptance at the center point was subtracted from all the data of susceptance, and again computation was performed. This process was repeated until the revised value became small enough. Co was calculated from the total revised value of susceptance. w,, wl, and w2 were computed from the value of the center point, and the series of data on G or B, which corresponds to each frequency, by using a method of approxi-
2144
ANALYTICAL CHEMISTRY, VOL. 60, NO. 19, OCTOBER 1, 1988
Table I. Computed Data of Equivalent Circuit Parameters of Quartz Crystal in Contact with a Water-Ethanol Mixture, Obtained by Using a Two-sided Cell (A) and a One-sided Cell (B) at 30 "C
cell type
ethanol, wt %
A
F., Hz
AF,Hz
8 691 441 8 687 870 8 686 647 8 865 980 8 865 786 8 685 765 8 868 510 8 687 325 8719011 8 717 307 8 716 800 8 716 478 8 716 408 8 716 420 8716773 8717 168
100 76 54 44 35 17 0
B 100 76 54 44 35 17 0
3571 4794 5461 5656 5676 4931 4116 1704 2211 2533 2603 2591 2238 1843
c,, 10-14~
Ll, H
RIP fi
Co, 10-12F
2.318 2.486 2.287 2.579 2.589 1.954 2.367 2.596 2.197 2.317 2.316 2.322 2.321 2.324 2.314 2.309
0.014 47 0.014 68 0.014 68 0.013 02 0.012 97 0.017 18 0.014 18 0.012 93 0.015 17 0.014 38 0.014 39 0.014 36 0.014 36 0.014 34 0.014 41 0.014 44
11.18 569.9 759.9 844.1 871.7 865.8 746.7 589.8 28.12 321.7 422.1 497.1 491.1 487.4 416.6 331.3
13.86 12.92 13.68 14.50 14.93 15.39 16.11 16.75 3.555 6.111 6.164 6.195 6.203 6.247 6.263 6.261
h
G
600
v
E 300
-
0
0 0
.5
1.0
1.5
.5
0
1.5
1.0
Figure 3. Correlation of @q)'l2 and R for water-ethanol mixture with one-sided cell (0) and two-sided cell (0) at 30 O C . Numbers inside the figure show ethanol composition (wt %).
Figure 4. Correlation of @q)'/*and AF for water-ethanol mixture with a one-sided cell (0) and two-sided cell (0)at 30 O C . Numbers inside the figure show ethanol composition (wt %).
mation to a polynomial equation,
One possibility is that an electrical double layer behaves as an adsorption layer, causing a surface mass change. But the electrochemical behavior of quartz crystals has been studied previously, and the frequency shift for Pt electrode oxidation using 5-MHz quartz crystal is only 20 Hz (9). In addition, 9-MHz oscillation does not allow enough time to form a static electrochemical double layer, and the effect of this double layer was supposed to be small (25). The value of AF and R1did not change when measuring voltages of 1, 0.1, and 0.01 V were used. This result also denies the possibility of the electrochemical effect. The effect of conductivity for the two-sided cell can be considered as an addition of parallel resistance (R,) to the electrical equivalent circuit of the quartz crystal in Figure lb. The admittance is expressed again as
RESULTS AND DISCUSSION Table I shows the computed data of the resonant frequency (F,) and resonant frequency shift (AF),which are subtracted from the resonant frequency value in air. L1, C1,R1, and Co were measured in an ethanol-water mixture with the two types of quartz crystal cells at 30 "C. The error of measurement for L1 and C1is larger than the shift value expected from the frequency shift. Thus, the discussion is limited to AF and R1. Figure 3 shows the correlation of ( ~ q ) 'and / ~ R1for one-sided and two-sided cells. The values of p and q are calculated from data in which viscosity was measured with Ostwald's viscom/ ~ eter, and the data was shown to four digits (24). ( p ~ ) l and R1 have good linearity with both types of quartz crystal. The value of R1 for the one-sided cell is about half that of the two-sided cell. This is explained by the surface area A in eq 21, where twice the area gives twice the friction. Figure 4 shows the correlation of ( p q ) l I 2 and AF for onesided and two-sided cells. AF is almost linear with ( p ~ ) ' / ~ , but a remarkable shift was obtained with the two-sided cell at the points with a higher ratio of water. The linearity of the one-sided cell shows the value of ( p ~ ) ' is / ~proper. The unusual phenomenon of the resonant frequency shift for water interfaced with quartz crystal has been reported (5,6). The frequency shift is not induced by a viscosity change of the quartz surface, because R1shows linearity independent of fluid composition (Figure 3). The cause of the resonant frequency shift is supposed to be mass load on the surface or elasticity change of the liquid.
Y=
1
R1 + j w L l
+ l/jwC1
+jwco
+
1
-
RO
Equation 28 is taken from eq 27 instead of eq 26 as
Equation 28 shows that Ro can be estimated from the shift of the center point of the G-B circle on the G axis, and Ro does not affect the resonant frequency. The data shown in Table I include no effect of Ro, because these data were obtained by using the method of least squares for the circle, as shown in the Experimental Section.
ANALYTICAL CHEMISTRY, VOL. 60,NO. 19, OCTOBER 1, 1988
1
-
7000
2145
1
E000 .. N
0 O
0
I 5000 '.
o
u 0
LL
a
..
4000
Ethanol
t
:: f 3000
:
;
1
:
;
600
800
0 0
20
40
EO
80
0
100
Ethanol composition (wfolo)
Flgure 5. Correlation of the computer values of Coand the waterand two-sided cell (0)at 30 "C. ethanol ratio with a one-sided cell (0)
200
400
1000
Ri (Q) Figure 7. Correlation of R , and AF for a water-ethanol mixture with a two-sided cell at 25 "C.
I5000
EO00
/o
1
c
N
-
4000
h
s-
LL
a
10000
CI:
I, 5000
2ooo
0
0
0
300
EO0
0
PO0
IO
20
30
40
&q-($'~Cr;i3'2Cpl") Figure 8. Correlation of R , and AF for a water-ethanol mixture (0) and methanol-ethanol mixture (0) with a two-sided cell at 30 "C. Numbers inside the figure show ethanol composition (wt %).
The major property difference between water and ethanol is the dielectric constant. Water molecules have a large dipole moment and readily form hydrogen bonds. Figure 5 shows the computed value of COagainst the ethanol-water ratio for one-sided and two-sided cells. Although the C, value with the one-sided cell is constant, the Co value for the two-sided cell reflects the fluid dielectricity and shows a higher value at a higher ratio of water. Properly, Co itself cannot affect the value of hF but with a high dielectric constant can induce physical property changes in the fluid. Dielectric polarization of water molecules is faster than 9-MHz oscillation (26) and attended orientation of molecules (27). It is supposed that the polarization of water molecules results in a hydrogen-bonded system, which induces a change in elasticity. This view means that a Newtonian liquid, which has a large dipole moment like water, shows an elastic property under these defined conditions, i.e., dipping two sides of the electrode and oscillating at 9 MHz. Of course, there may still be a possibility that this is caused by an unknown electrical effect. Figure 6 shows the correlation of R1,which means ( p ~ ) l / ~ , and AF for the water-ethanol mixture and the methanolethanol mixture with the two-sided cell. The data obtained for the methanol-ethanol mixture shows more linearity than that for the water-ethanol mixture, as expected, but the value of 100% methanol also deviated from the linear. We believe this also to be a result of polarization and the dielectric constant of methanol. Figure 7 shows the correlation of R1and A F for a waterethanol mixture at 25 "C with the two-sided cell. The value of aF for water shows larger shifts than that a t 30 "C. This
Flgure 8. Correlation of
@r))'/* and R for a water-glycerol mixture in contact with 6MHz (e),9-MHz (0),and 12-MHz (0)quartz crystals with a one-sided cell at 25 O C .
h
N 60000
I
v
LL
0
40000
20000
0 0
IO
20
30
40
Figure 9. Correlation of (pq)"' and AF for a water-glycerol mixture in Contact with 6-MHz (0),9-MHz (U),and 12-MHz ( 0 )quartz crystals with a one-sided cell at 25 "C.
is because Brownian movement is less active at 25 "C, and therefore, the water molecules are more closely associated. Results for a water-glycerol mixture with various resonant frequency quartz crystals and single-sided cells give additional information. Figures 8 and 9 show the relation of (p7)lI2 with R1and (pq)'l2 with AF, respectively. The value of R1exhibits good linearity with (p~)'/~. But AF at high ( p ~ ) lvalues /~ shows a shift from the linear line. This result is similar to that for the case of large mass load (13). These shifts from eq 1 are caused by approximation in the derivation process of eq 1.
2146
ANALYTICAL CHEMISTRY, VOL. 60, NO. 19, OCTOBER 1,
BOO00
$
1988
As described in this paper, R1 shows good linearity to (p#12 in all cases and is practical for measuring viscosity of liquids. AF has, basically, a linear correlation to (pq)li2,but liquid elasticity and mass load onto the surface change the hF value too. A contrast of R1 and AF provides the possibility for obtaining new information on liquid properties.
60000
ACKNOWLEDGMENT
,
t
20000
-7 I
We thank Mark E. A. Downs for useful discussion. Registry No. Quartz, 14808-60-7;water, 7732-18-5; ethanol, 64-17-5; methanol, 67-56-1; glycerol, 56-81-5.
LITERATURE CITED 0
2
i
,
4
3
+( olo
=3/2
Figure 10. Correlation of F3’2and AF for 0% (0). 60% (O), 80% ( o ) and , 100% (A)glycerol solution. 15500
I DODO
G v
cc
5coo
,
II 0
/a A ’
200
400
600
EO0
Figure 11. Correlation of R , and (F)”2Afor 0 % (0),60% (O), 80%
( o )and ,
100% (A)glycerol solution.
The linear relation of AF and P12shows the proper relationship of eq 1 (Figure 10). Similarly, R1 should relate to (F)’12Alinearly, but this correlation is not linear when the values of the electrode area are used for A (Figure 11). Vibrations of AT cut quartz crystal have been studied in Perhaps relation to crystal shapes and electrode area (28,29). the reason is that the actual moving area of the quartz crystal is not equal to the electrode area. Another possible reason is an effect of frequency dependence on the biscosity of the sample liquid.
(1) Konash, P. L.; Bastiaans. G. J. Anal. Chem. 1980, 52, 1929-1931. (2) Nomura, T.; Minemura, A. Nippon Kagaku Kaishi 1980, 1980, 1821- 1624. (3) Nomura, T.; Okuhara, M. Anal. Chlm. Acta 1982, 142,281-284. (4) Nomura, T.; IIjima, M. Anal. Chim. Acta 1981, 737, 97-102. (5) Thompson, M.; Arthur, C. L.; Dhaliwal. G. K. Anal. Chem. 1986, 58, 1206-1209. (6) Muramatsu, H.; Dicks, J. M.; Tamiya, E.; Karube, I.Anal. Chem. 1987, 59,2760-2783. (7) Bruckenstein. S.;Shay, M. Nectrochlm. Acta 1985, 3 0 , 1295-1300. (8) Bruckenstein, S.;Shay, M. J . Electroanal. Chem. Interfacial Electrochem. 1985, 188,131-136. (9) Shumacher, R.; Borges, G.; Kanazawa, K. K. Surf. Sci. 1985, 163, L82 1 -L626. (IO) Kanazawa, K. K.; Gordon, J. G., 11. Anal. Chim. Acta 1985, 175, 99-105. (11) Hager, H. E. Chem. Eng. Commun. 1986, 4 3 , 25-38. (12) Simpson, R. Presented at the American Institute of Chemical Engineers Summer National Meeting, 1985. (13) Lu, C. S.;Lewis, 0. J. Appl. Phys. 1972, 4 3 , 4385-4390. (14) Pulker, H. K.; Benes, E.; Hammer, D.; Sollner, E. Thin Solid films 1976, 32, 27-33. (15) Glassford, A. P. M. J. Vac. Sci. Techno/. 1978, 75,1836-1843. (16) Mecea, V.; Burcur, V. Thin Solidfilms 1979, 6 0 , 73-84. (17) Crane, R . A.; Fisher, G. J. Phys. D : Appi. Phys. 1979, 12, 2019-2026. (18) Mason, W. P.; Hill, M. Trans. ASME 1947, 6 9 , 359-370. (19) Mason, W. P.; Baker, W. 0.; McSkimin, H. J.; Heiss, J. H. phvs. Rev. 1949, 75,938-946. (20) Barlow, A. J.; Lamb, J. Proc. R . SOC.London, A 1959, 253,No. 52, 52-69. (21) Wada, Y.; Sasabe, H.; Tomono, M. Biqoo/ymers 1987, 5, 887-897. (22) Moose, R. S.;McSkimin, H. J.; Gieniewski, C.; Andreatch, P., Jr. J. Chem. Phys. 1969, 50,5088-5100. (23) Landau, L. D.: Lifshtz, E. M. fluid Mechanics; Pergamon: Oxford, England, 1959;pp 88-98. (24) Kagaku-Binran Kisohen; Morino, Y., Ed.; Japanese Chemical Society: Maruzen, Tokyo, 1975;pp 583-584. (25) Randeles, J. E. B.; Somerton, K. W. Trans. faraday SOC. 1952, 48, 937-950. (26) Schwan, H. P. Ann. N . Y . Acad. Sci. 1977, 703, 198-213. (27) Kruus, P. Liquid and Solutions; Dekker: New York, 1977; pp 298-297. (28) Byrne, R. J.; Lloyd, P.; Spencer, W. J. J. Acoust. SOC. A m . 1967, 4 3 , 232-238. (29) Perman, G. T. J. ACOUS~. SOC. Am. 1969, 4 5 , 928-934.
RECEIVED for review October 20,1987. Accepted May 23,1988.
