Langmuir 1994,10, 931-935
931
The Electroacoustic Reciprocal Relation R. W. O’Brien, P. Garside, and R. J. Hunter’ School of Chemistry, University of Sydney, Sydney, NSW 2006, Australia Received August 13, 1993. I n Final Form: December 8 , 1 9 9 9 In this paper we present experimental verification of the reciprocal relation between the ultrasonic vibration potential ( U W )and the electrokineticsonic amplitude (ESA)effect for colloidaland electrolyte systems. Measurements of the ionic vibration potential at the lower end of the range of concentration8 (0.001 M) require correction for the high internal impedance of the solution, while measurements of the ESA effect require correctionat the high concentration end because the field appliedtothe cell is significantly reduced when the cell resistance is low. We also show how the ESA measurementson ionic solutions can be used to calibrate the Matec ESA-8000 device. The resulting calibration constant is in good agreement with that obtained by James et al. (1991)from measurements of a monodisperse suspension of spherical particles.
Introduction When a high-frequency sound wave is passed through an ionic solution, there is set up a macroscopic alternating voltage difference called the ionic vibration potential (IVP), which can be measured using high-impedance electrodes immersed in the solution. The theory for the IVP was set up by Debye’ and ita implications developed by Bugosh et alS2The first measurements on electrolytes were done by Yeager et aL3 but quantitative verification of the theory was not obtained until the work of Zana and Yeager.4 The IVP arises from differences in the inertia of the hydrated ions and differences in the magnitude of the frictional forces they experience. The result is that, as the sound wave passes through the solution, different ions move with different amplitudes and so a sinusoidally varying charge distribution is set up in the electrolyte, and this generates the IVP. A much larger potential is generated when a sound wave is passed through a colloidal suspension because the particles have a much larger inertia than the surrounding double layer, and so a much greater relative charge displacement is obtained. This colloid vibration potential (CVP)was examined theoretically by Hermans6 and by Enderby and Booth and Enderby? The early experimental work by Rutgerss and its extension by Yeager et al. have been reviewed by Zana and ye age^^ The IVP and CVP effects are referred to jointly as ultrasonic vibration potentials (UVP). More recently, it has been shown by Oja et al.1° that the obverse effect also occurs: when an alternating electric field is applied to a colloidal suspension, a sound wave is generated, This electrokinetic sonic amplitude (ESA) effect, as they called it, is the basis of an important new method for investigating the charge and size of colloidal Abstractpublishedin Advance ACS Abstracts, February 1,1994. (1) Debye, P. J. Chem. Phys. 1933,1, 13. (2) Bugoeh, J.; Yeager, E.; Hovorka, F. J. Chem. Phys. 1947,15,592. (3) Yeager, E.; Bugoeh, J.;Hovorka, F.; McCarthy, J. J. Chem. Phys. 1949, 17, 411. (4) Zana, R.; Yeager, E. J. Phys. Chem. 1966, 70,954; 1967, 71, 521. (5) Hermans, J. Philos. Mag. 1938, 26,426; 26, 679. (6) Enderby, J. R o c . Phys. SOC.1961,207A, 329. (7) Booth, F.; Enderby, J. R o c . Phys. SOC.l962,208A, 321. (8) Rutgers, A. Physica 1938,5, 46. (9) Zana, R.; Yeager, E. In Modern Aspects of Electrochemistry; Bockris, J. OM., Conway, E. E., White, R.E., Ede.; Plenum Press: New York, 1982; Vol. 14, pp 1-60, (10) Oja, T.; Peterson, G. L.;Cannon, D. W. United States Patent 1985, #4 497 207. @
particles.11 O’Brien12 has shown that the ESA effect is determined by the dynamic mobility Pd of the colloidal particles. For dilute suspensions (up to about 5 %I volume fraction), this relation takes the form ESA = 81+(AP/P) Pd
(1)
where ESA is measured in pascals per unit field strength (V m-l), is the volume fraction of the particles, and Ap is the difference between the particle density and that of the solvent, p. O’Brien has also shown that, for systems of arbitrary concentration, the ESA and UVP are linked by the reciprocal relation
+
ESA = Q2
K* UVP
(2)
where UVP is in volts per unit pressure gradient (Pa m-9 and K* is the complex conductivity. Q1 and 42 are apparatus constants which do not depend in any way on the properties of the system in the cell. Note that measurementa of the ESA effect can be used to obtain the dynamic mobility, Pd, directly, whereas the determination of Pd from the UVP measurement requires a knowledge of the complex conductivity. In an electrolyte solution the term +(Ap/p)pd in eq 1 is replaced by
J
(3) P
where nj is the number of ions per unit volume of type j of valency zj and frictional coefficient f j . AMj is the mass of ion j corrected for bouyancy.
Experimental Section Apparatus. Measurementswere performed on the ESA-8000 apparatus (Matec Applied Sciences, Hopkinton, MA) with the SP-80 dip-type probe and an external frequency generator (Wandeland Golbrmann,PSM-6)in place of the usual frequency synthesizer supplied. In the standard ESA-8000the frequency of the applied field is automatically adjusted by the Matec software to achieve a resonance in the cell with an accuracy of about 10 kHz. By using the external frequency generator, we were able to fix the resonance frequencymanuallywith a precision of 1kHz, and this gave some improvement in the precision of the final results. (11) O’Brien, R. W.; Midmore, B. R.; Lamb,A.; Hunter, R. J. Faraday Discllss. Chem. SOC.(London) 1990, 90,301. (12) O’Brien, R. W. J. Fluid Mech. 1988, 190, 71; 1990, 212, El.
0143-1463/94/2410-0931$04.50/0 0 1994 American Chemical Society
932 Langmuir, Vol. 10, No. 3, 1994 Probe
O'Brien et al.
Preampllfler 1
Figure 1. Equivalent circuit of the SP-80 probe and the input to the preamplifier. C is the capacitance of the probe itself. Materiale. All water used was doubly distilled, the second time from alkaline permanganate, to remove possible surface active contaminants. The final conductivity was of order 1 &I cm. All alkali halide solutions were prepared from Merck @ro analysi) reagents. The poly(acrylic acid) (PAA, average molar mass 90 OOO) was obtained from Aldrich as a 25 wt '36 solution in water and used without further purification. A 30mMsolution was neutralized with Li, Na, and K hydroxides from Merck @ro analysi). The Rb and Cs hydroxideswere obtained from Aldrich as 50 wt '36 solutions in water and made up at a concentration of about 0.7M. (Note:all molarities for the polymer refer to the monomer concentration.) Impedance Corrections. ESA Correction at High Electrolyte Concentration. The applied voltagepulse which generates the ESA signal is reduced in amplitude at high electrolyte concentrations since the gated amplifier is unable to supply sufficient current to maintain the field through such a low impedance. Direct measurements of the field strength, using a high impedanceprobe tip, showedthat, whereasthe field strength at lowconcentrations (around1 mM) was 600 V/cm,at the highest concentrationmeasured (0.1 M) the field had dropped to 370 Vlcm. All ESA amplitudes have thereforebeen corrected by the measured fador to bring them to a common field strength of 500 V/cm; this correction was necessary for solutions with a conductivity greater than 0.3 Slcm. UVP Correction at Low Concentration. The problem here is that the measurementof the UVP shoulddrawa negligiblecurrent across the electrolyte solution, but this is not so at the lowest electrolyte concentrations. The total current in the electrolyte is given by K*
(4)
where V , is the voltage across the cell due to the sound wave and Pis the preesure difference between the two electrodes. A is the cross sectional area of the cell and L is the distance between the electrodes. This current formula can be derived by using the formulas 5.2 and 5.4 in O'Brien's 1990 paper, with the g[(Ap)/ p]Md in (5.4) replaced by the expression 3 for the electrolyte case of interest here. This expression has the same form as that for the current produced by a voltage source V, in series with an impedance Z,, where 2, = L/K*A is the cell impedance and
is the true IW voltage. Figure 1 shows the equivalent circuit of the probe and preamplifier used to sense the UVP voltage. V. is the voltage b are the electrical measured by the preamplifier. Zc and z impedances of the electrolyte solution and the preamplifier, respectively. The current, I,flows through the input resistance zb and the capacitor, C, representa the stray capacitance of the lead connecting the probe and preamplifier. If 2. is the total impedance of that combination then VJZ. = I. This can be written, using eq 4
T Figure 2. Equivalent circuit for the input lead. When 2. >> Z,, it is apparent that V, = V,, sothe voltagemeasured by the probe is the true IVP. When this condition no longer holds, it follows from eq 6 that the IVP is given by v,=IVP= V,(l+?)
(7)
To apply the correction, we must determine the complex impedances 2. and 2, Impedance Measurement. The magnitudeand phase angle of the various impedanceswere determined usingan impedance/ gain-phase analyzer (Hewlett-Packard 4194A). The connection to the instrument was via a 0.5-m coaxial cable and allowance must be made for the impedance of this link. This is done using the circuit representation shown in Figure 2, where 2, is the measured impedancewhen the end of the cable is an open-circuit, and 2, is the measuredimpedanceunder short-circuit conditions. It can readily be shown that the true load impedanceZTis related to the measured load impedance Zmby Z,=--
1
z=
z-z 1
1
(8)
The impedancesreportedhere have been corrected for thiseffect. To apply the correction (7),we require the impedance of the ESA probe and preamplifier Z., given by 1 = iwC + 1 -
2,
zb
The preamplifier impedance z b can be measured directly, but the determination of C involves measuring the probe impedance in air and water, as follows. The impedance of the probe at the connectionwhere the ESA signal is applied is given by 1
z = iwc
1 +
ZC
(10)
where 1/27, is the admittance of the cell. This quantity is proportionalto the complex conductivity (K* = KO+ iwe) of the medium in contact with the electrodes. Using the measured impedances in air, ZA, and doubly distilled water, ZW,we can determine the probe capacitance, C, from the followingformulas
where the subscriptsA and W refer to air and water, respectively, t is the dielectric permittivity of the medium, and a is a cell constant for the probe. Note that the in-phase, conductance, contributionto the admittance can be neglected at the frequency of the measurement (even for the water). Solving for a and C we get
and
Langmuir, Vol. 10, No.3, 1994 933
The Electroacoustic Reciprocal Relation
---1
1
A'
/
,lcScl
iO'*Z* iw*zw C= 1 1 ---
RbCl
KCI
'W
aCI
The estimated magnitudes of a and C were 0.20m and 170 pF, respectively. These values were checked by comparing the measured impedance for a range of electrolyte solutions ZTwith the expected value, which from eq 10 is given by 2, = (i0C + CuK*)-'
= pW + ~ + A P + ~ A P -
Br
(13)
Once the probe capacitance, C,and preamplifier impedance, zb, are known, the true ion vibration potential can be calculated usingeqs 7and 9. The correctionbecomesimportant for solutions with a specific conductivity below 0.013 S/m. Instrument Calibration, The ESA-8000system is normally calibrated against a standard suspension of silica (Ludox TM and Du Pont) of about 10% volume fraction. The magnitude of the signal is given by eq 1 and the calibration constant Q1 is obtained from the known value of ( $ A p / p ) p d for the silica once the value Of I.ld is known. Thevalue originallysuggestedby Matec was Pd = -0.028c/q,corresponding to a Smoluchowski zeta potential of -28 mV. (The figure is given in this form to take account of modest departures of the measuringtemperature from the notional value of 26 O C . It is not meant to imply that the system obeysthe Smoluchowski equation.) More recent studies by James et ~ 1 . suggest ~ 3 a figure of -42 mV. The measured ESA signal for the 10.9% Ludox ( A p = 1.2)was -17.232 X 10-8 Pa m V-1, which correspondsto Q1= 4403Pa s m-1for our measurement system, if ~.ca= -0.042dq. We have sought to use the data obtained on the electrolyte solutions to check this calibration constant, since this would render it independent of the colloid theory and would pave the way for a calibration standard which could be readily made up and would be less liable to aging effects. The method depends on identifyingthe relation betweenthe parameters which appear in eq 1 and the corresponding ionic properties. The density of the electrolyte, pet is related to the density of the water, pa, by P*
CI
(14)
where pt = mtlvl and 4t = ntu*/ww. For a dilute electrolyte solution, eq 1 takes the form (15)
Solving eqs 14 and 15 simultaneously then gives
The ionic mobilities, densities, and partial molar volumes are all known sothat Q1can be calculatedfromthe measuredESA signal. There is, however, another correction to be taken into account. According to eq 16 the ESA should be proportional to the electrolyteconcentration, but when the ESA amplitudeis plotted as a function of electrolyte concentration for the various alkali metal halides, the signal magnitudes for the lighter metals (Li+, Na+,and K+)do not pase through the origin at zeroconcentration, a~ shown in Figure 3. The heavier metals (Rb+ and Cs+)give much larger signals which do pass through the origin. We have examined this zero offset effect in some detail and have found that it can be varied by applying a dc bias to the electrodes. This has led us to conclude that it is due to the development of another ESA signal which comes from the electricaldoublelayer on the measuringelectrode. Thia electrode ESA effectmust be subtracted fromthe signal,as shown in Figure (13)Jamell, R. 0.;Texter, J.; Scales, P.J. Longmuir 1991, 7, 1993.
electrolyte concentration (mol/L)
Figure 3. Removingthe electrode ESA effectfromthe measured ESA values for the alkali metal chlorides. (Regressionlines are based on measurements up to 0.1 M in all cases.)
> ;. . . ' I #10-3
0
I
I
2
3
4
1
1
5
1
1
1
1
10-2
. . .
I
I
I
2
3
4
1
1
5
1
1
1
1
,
10-1
electrolyte concentration (mol/L)
Figure 4. Values of ( 4 A p ) ~ +as a function of concentration for potassium chloride and bromide using QI= 4332 Pa s m-l. 3,before it can be used for calibration. The signals from the Rb+ and Cs+ salts are evidently much less affected.
Results Calibration Constant. The constant Q1 was adjusted so that (#+Ap+) for the KC1 was the same as for the KBr, as should be the case for a dilute electrolyte. This gave Q1 = 4332 Pa s m-l. When the appropriate corrections (impedance and zero effect) have been applied to the ESA signals for the potassium salts, this value of Q1 gives a very consistent representation for the two salts KC1 and KBr, over the entire concentration range studied as is shown in Figure 4. The constant is in reasonable agreement with the value of 4403 obtained by James et aL13 The Reciprocal Relation. ( a ) For the Colloid. We have measured both the UVP (i.e. CVP) and the ESA signal for Ludox as a function of volume fraction from 1 to 10%. The result is shown in Figure 5. The figure of 6.5 that we have chosen for QZis a compromise that gives the best correlation of ESA and UVP for the Ludox and the electrolytes. Using a figure of 6.0 gives much better agreement, showing that the validity of the reciprocal relation is not in doubt. The value of QZshould, however, be independent of the system under study, but the origin of this discrepancy remains unresolved. ( b )For Ionic Solutione. For the electrolyte solutions, the ESA signal for the lighter metal salts (Li-K) must be corrected for the electrode ESA effect and when that is done the reciprocal relation is again satisfied with a value of QZ= 6.5, as is shown in Figure 6. (c) For Poly(acry1ic acid). The amplitude of the ESA signal obtained for unneutralized PAA is very small, comparable to that obtained for LiCl as shown in Figure 3. This is to be expected from eq 1 since the density
Langmuir, Vol. 10, No. 3, 1994 2o
r
16
-
12
-
8 0
O'Brien et al.
ESA ( x 1 0 - 3 Pa m / V )
UVP ( x 1 0-3 v s/m) m UVP xIK*Ix 6 5 n - UVP xIK*Ix 6 0
0
A
4 -
0' 0
'
"
I
2
"
"
4 6 8 volume fraction ( X ) Ludox TM
"
10
A
10-31"
"
10-3
12
2
" " 1
3
3
4
5 6
ESA Corrected UVPxlK-1x6 5 Uncorrected UVPxlK'lx6 5 '
10-2
2
*
3
1
4
~
5 6
~
10-1
PAA concentration (mal/L)
Figure 5. Reciprocal relation for Ludox TM as a function of volume fraction. Using Q2 = 6.6 does not give as good a fit to the ESA data as the value 6.0. The higher figure gives rather better agreement between UVP and ESA for the ionic and polyelectrolyte signals (Figures 6, 7,9, and 10).
Figure 7. Reciprocal relation for unneutralized PAA. The correction referred to is for the high impedance of the solutions at the lower acid concentrations.
3 L h
2 -
E
3
100:
>
0.4
n
0
4 -
3 -
v
2 -
10-1
a
%
:
4 -
s-
0.2
' I
nn
2 -
0
1
20
40
60
00
100
Z neutralization tlo-3
2
3
4
5 6
10-2
2
3
4
5 6
10-1
elec traiyte Concentration (moi/L)
Figure 6. Reciprocalrelation for the alkali-metalchlorides.IVP measurements at the lowest electrolyte concentrationhave been corrected for the high impedance of the solutions. Table 1. Electrical Conductivities of PAA Solutions concentration(mM) K" (S/m X lo41 F*l(S/m X 104) 100 30 10 3 1
409.9 176.6 80.6 32.2 13.2
412.6 182.4 92.7 66.1 47.8
difference Ap is small. (From the data of Zanaand Yeager14 we calculate that Ap = 0.26 kg m-s for 10 mM PAA, compared to about 1.29 kg m4 for 10 mM CsC1.) Such small signals must be corrected for the electrode ESA effect. The ESA signal increases steadily by a factor of about 10 over the 100-fold concentration range. The conductivities of the PAA solution at various concentrations are given in Table 1. The impedance of the solution at the lowest concentration is sufficiently high to make it necessary to correct the UVP signal for the impedance effect as describedabove. Note that a 100-foldincrease in the concentration produces less than a 10-foldincrease in the magnitude of the complex conductivity. The UVP signal in this case goes through a maximum in the region 10-30 mM as the increase in signal due to the larger number of charges present is balanced to some extent by increasing ionic interaction. Figure 7 shows that when the conductivity effect is included, the reciprocal relation is shown to be valid and again the value of Q 2 is 6.5. ( d ) For Neutralized PAA. The measured UVP signals for PAA (30 mM) as a function of added base, for the various alkali metal hydroxides, are shown in Figure 8. (14) Zana, R.;Yeager, E.J. Phys. Chem. 1967, 71, 3602.
Figure 8. UVP signalfor 30 mM PAA neutralizedwithdifferent alkali-metal hydroxides. (Error bars shown on alternate data points.) 0.3
2E
0.2
a0
n
0 8
v
0.1
W
0.0
0
20
40
60
80
100
Z neutralization
Figure 9. Reciprocal relation for PAA at various degrees of neutralization. They are qualitatively very similar to the results obtained by Zana and Yeager (1967) but they differ quantitatively, perhaps because of our inability to correct them for the electrode electroacoustic effect referred to earlier but also because of differences in the frequencyof our measurement and that of Zana and Yeager. The ESA signal (Figure 9) is rather more straightforward, since it is not confused by the simultaneous variation in the conductivity as the base is added. Once again Figure 9 shows that the reciprocal relation is obeyed over the whole range of the titration processfor each of the solutions and the value of Q 2 remains at 6.5. ( e ) For a Heteropolyion Solution. The success of the reciprocal relation and the magnitude of the signals obtained for suitable ionic solutions suggested that it should be possible to find inorganic salts which had electroacoustic signals comparable in magnitude to those generated by colloidal systems. Such a salt could act as
'
,
Langmuir, Vol. 10, No.3, 1994 936
The Electroacoustic Reciprocal Relation h
$
i‘
*
3 2 .
2
b
smaller ions is due to the difference in their dbnsities. Initially these bigger metal ions make the signal less negative whereas the smaller ions have too small a contribution to produce a significant effect. Ultimately the same process dominatesin all solutions: the increasing number of negative sites on the PAA chain produces an increasingly negative ESA signal. The negative signal is also increased at high degrees of neutralization by the fact that the counterions become bound to the chain, increasing its density.14
10’
r,,
.
I
I
I
, . . .
, I
. .
lo-3
.
I
I
.
.
.
.
8
.
I
10-2
eiectrolyte concentration (mol/L)
Figure 10. Reciprocal relation for solutions of potassium dodecatungstosilicate.
a reliable calibration standard for the ESA measurements and this would obviate some of the problems which occur with the present colloidal standards. (They can become contaminated with frequent use, the zeta potential then changes, and they may become partially aggregated.) Equation 1 suggests that one should look for a salt with a large density difference between cation and anion. The heteropolyacids and their salts are a class of compounds which have high solubility in water and a high anion density. Some of them, like dodecatungstosilicicacid (H4[ S i W l ~ O & z H 2 0 and ) , its salts, are easily prepared and also very stable. Figure 10shows the reciprocal relation for the potassium salt (K4) and it is clear that, once again, the value Q2 = 6.5 gives a good description of the data over the entire range. It should be noted that these solutions do not strictly obey eq 1, in that there is a zero effect here as there was for the lighter alkali metals. Nevertheless, the signal at a concentration of 10 mM is about half as large as that for the usual Ludox standard ( 5 % volume fraction) and that should be quite adequate as a calibrating standard.
The behavior of the UVP signal (Figure 8) is more complicated since we have also to take account of the simultaneous changes in the conductivity. The early increase in the UVP for the lighter ions is partly due to a lowering of the conductivity as the protons are replaced by metal ions of lower mobility. For the heavier ions that effect is more than offset by the effect of the added ions themselves, which is in the opposite direction. As more salt is added, the contribution of the middle term in eq 17 becomes less significant because growing counterion binding increases the size of the last term at the expense of the middle term. The general validity of the reciprocal relation allows us to choose in any given case whether to use the ESA or the UVP measurement to determine the electrokinetic properties of a system. In general the ESA method is more convenient because one does not need a knowledge of the complex conductivity, but there are some situations, especially where the conductivity is very low, where the UVP becomes much larger in magnitude and is therefore easier to measure. This will be particularly so in nonaqueous media. The problem then will be to ensure that the impedance effect is properly taken into account. The plot in Figure 10 suggests that a 10 mM solution of potassium dodecatungstosilicatecould serve as a stable and readily prepared standard for ESA measurements.
Discussion It is not clear why the zero effect shown in Figure 3 should be confined to the lighter metals of the series. It may be simply that the signals for Rb+ and Cs+ are so large that the effect is swamped, but we would have expected to see it if it were there. An alternative possibility is that the diffuse double layer is much less developed on the electrode in the presence of these two large ions. Both of them tend to be poorly hydrated and on a metal surface they may well be strongly specifically adsorbed, because of the large image force they generate in the metal surface. Since the zero effect is being attributed to an ESA effect generated by the diffuse layer on the electrode, a collapse of that layer would drastically reduce the magnitude of the effect. The very good correlation shown in Figure 4 needs little comment, and the consistent picture shown in Figures 5-8 also needs no further discussion. The value of Q 2 in our apparatus appears to be 6.5. The ESA signals shown in Figure 9 can be understood in terms of the equation
when one recognizes that the signal is negative. The very different behavior of Rb+ and Cs+ compared to the three
Conclusions We have obtained experimental confirmation for O’Brien’s reciprocal relation (eq 21, linking the ultrasonic vibration potential and the electrokineticsonic amplitude for colloidal dispersions, ionic solutions, and solutions of a polyelectrolyte. Measurements of the ESA effect, especially when the signal is low, are confused by the presence of a small signal which appears to emanate from the electrodes themselves and this must be subtracted from the measured signal to obtain the true ESA effect. By use of the standard Matec apparatus, the measured signal is also diminished in concentrated salt solutions, because the apparatus is unable to provide the necessary current to sustain a constant field strength when the electrical conductivity rises above about 0.3 S m-l. Corrections are also required for the measured UVP signal when the conductivity is too small, because under those conditions the input impedance of the measuring preamplifier is too smalland so the voltage is not measured under open-circuit conditions. We have also presented a method for obtaining the instrument calibration constant Q1 from measurements on electrolyte solutions. This procedure has the advantage that electrolyte solutions are readily available, and they give a stable signal over a long period of time.
