Langmuir 1992,8, 420-423
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Effect of Particle Size Distribution and Aggregation on Electroacoustic Measurements of Potential Michael James, Robert J. Hunter,* and Richard W. O’Brien School of Chemistry, University of Sydney, N.S.W ,2006, Australia Received March 18, 1991. In Final Form: October 3, 1991 The electroacoustic procedure allows one to estimate the high-frequency mobility of colloidal particles in moderately concentrated suspensions. From the mobility it is possible to calculate the electrokinetic (0 potential, provided one can estimate the appropriate particle size for the kinetic units in the system. We show that the mass average particle size, determined from light scattering, is the measure that best estimates the size of the kinetic units for the electroacoustic method, even when the system is undergoing coagulation. A comparison is also made between the mobility formulas of OBrien and of Babchin. The former is found to give much better agreement with {potential measurements obtained on dilute suspensions of the same system. A more recent formulation by Babchin is in better agreement but neglects the permittivity of the particles.
Introduction Although electroacoustic measurements provide a convenient method for determining the zeta potential (0 in systems of moderate concentration,’ the calculation requires a knowledge of the particle size. In this paper we calculate { potentials from electroacoustic titration data on polydisperse suspensions using both O’Brien’s theory2 and that of Babchin.’ O’Brien et al.3 obtained good agreement between the c potentials calculated from electroacoustic and dc electrophoresis measurements for a well-dispersed suspension of smooth, spherical, monosized cobalt(I1) phosphate particles. In the same study, two polydisperse titania suspensions were investigated. Good correlation was observed for the disperse region below the point of zero charge (pzc) by introducing a multiplicative correction factor. The correspondence between electroacoustics and electrophoresis broke down, however, once the suspensions were taken through their isoelectric point (iep). This result was attributed to the occurrence of aggregation near the iep, a process which was not taken into account in the particle size estimate. In the electrophoresis method, the particle concentrations are significantly smaller and what little aggregation occurs has a negligible effect on the measured mobility. In the ensuing Faraday discussions, it was suggested,” on the basis of the study to be described here, that the mass average particle size was the parameter necessary to produce the required multiplicative correction factor. Scales and Jones5confirmed agreement with the O’Brien equations for well-dispersed suspensions of a monosized silica sol. They also attempted to take account of particle size distribution for a more polydisperse Ti02 system and obtained a reasonable level of agreement, though there are some unnecessary approximations used in their analysis (e.g., the particle permittivity is assumed to be small and that is not so for TiO2). They claimed that the size required to give perfect agreement was “somewhat less than the mass average size ...measured on the Coulter LS130”, but did not calculate the expected values of { corresponding (1) Babchin, A.J.;Chow, R. S.; Sawataky,R. P. Adu. Colloid Interface Sci. 1989, 30, 111. (2) O’Brien, R. W. J . Fluid Mech. 1988,190, 71-86. Midmore, B. R.; Lamb, A.; Hunter, R. J. Faraday (3) O’Brien, R. W.; Discuss. Chem. SOC.1990,90, 301-312. (4) Hunter, R.J. Faraday Discuss. Chem. SOC.1990, 90, 359. (5) Scales, P.; Jones, E. Langmuir 1992, 8, oo00.
0743-7463/92/2408-0420$03.00/0
to the measured size from which one could determine the sensitivity to the radius value in the region in question. In this study we begin with a polydisperse material and seek to correlate the static and dynamic t potential using the particle size distribution as determined on a dilute, highly disperse sample. That gives the mass average radius, a, which proves to be the appropriate parameter, but only if one can ensure complete dispersion. We have taken the process one step further by taking account of the effective (or apparent) mass average particle size (a,ff) in systems where the degree of aggregation may vary during the course of the experiment. The values of aeff are measured a t each pH of the suspension using photon correlation spectroscopy (PCS).
Theory The electroacoustic measurements reported here use the electrosonic analysis (ESA) method6 and were Carrie& out on the ESA-8000 (Matec Applied Sciences, Hopkinton, MA). In this device, a 1-MHz alternating electric field is applied to the colloid, causing an oscillatory particle motion which generates sound waves. The theory for the ESA effect was developed by O’Brien2 who showed how the sound wave signal could be used to estimate the dynamic (frequency dependent) mobility. For thin doublelayer systems (Ka >> 1)this quantity can be related, for dilute suspensions, to the static (dc) mobility and, hence, the {potential of the particles.2 The assumption that the double layer is thin compared to the particle radius is usually valid for colloids of size greater than about 0.1 wm in nondilute electrolyte solution. The first step in analyzing the data is to determine the dynamic mobility from the measured ESA signal. That requires3 a knowledge of the product 4 Ap ( = A m ) where 4 is the particle volume fraction and Ap is the difference in density between solid and liquid. Note that if this quantity is determined by measuring the mass fraction of solid in the suspension, then any uncertainties caused by the porosity of the aggregate structure are relatively unimportant. There remains, however, some uncertainty when one wishes to calculate the { potential from the measured dynamic mobility because that involves a knowledge of Ap and not Am. (It can be shown (from eqs 1and 2 below) that overestimating Ap by 20% leads to a corresponding overestimate of {, of about 10%.) (6) Oja, T.;Petersen, G.; Cannon, D., U S . Patent 4,497,207, 1985.
0 1992 American Chemical Society
Langmuir, Vol. 8, No. 2, 1992 421
Electroacoustic Measurements of {Potential If surface conduction is unimportant (as is usually the case12for modest values of 0 ,and the particle permittivity is much smaller than that of the solvent, the form factor, f, in O’Brien’s formula3 can be set equal to 0.5. In that case, the dynamic mobility Pd is given by
where t and r] are the dielectric permittivity and viscosity of the fluid. G ( a ) is the frequency and size-dependent inertial contribution to the dynamic mobility, given by
The pH probe, a combination glass electrode, was calibrated against standard buffer solutions of pH 4.00, 7.00, and 10.00, The sign of the particle charge and the instrument calibration factor were determined using a suspension of colloidal silica (Du Pont Ludox TM, pp = 2.37 g ~ m -a,~ ,= 15 nm, and volume fraction 4 = 10% of known apparent {potential (-28 mV at pH These were the values suggestedby Matec 9.75 and 25 0.2 “0). Applied Sciences. The density, ppof Ludox is only 2.20 according to Alexander and Iler: but that is immaterial for the present purposes. The value we use in calibration must be the same as that used by Matec in order to preserve the validity of their calibration procedure, as should be clear from the following description. For dilute suspensions, the ESA signal S is given by2
*
= c4 Ap Md
(3) where C is an instrument constant, and C$is the particle volume Here a = wa2/v, where w is the angular frequency of the fraction. The quantity C was originallydetermined in the Matec applied field (2aMHz in this case), a is the particle radius, laboratories9with a monodisperse latex suspension of diameter p is the density, and Y is the kinematic viscosity of the 350 nm. The {potential of the particles was determined by electrophoresis, and the expected dynamic mobility was then fluid. Accurate estimates of G(a) are essential in the decalculated using O’Brien’s theory. The latex sample was not termination of l potential values from measurements of sufficiently stable to be a standard so they chose to use a 10% the dynamic mobility. volume fraction silica suspension (Ludox TM) as the standard. The resulting values of l can then be compared with They therefore compared the signal from the silica sample with values calculated from microelectrophoresis measurethat from the latex to arrive at an apparent dynamic mobility ments. for the silica. It is not the true value because the theory for such high particle concentrations is not yet established, but that is Experimental Section unimportant. The silica is used only as a transfer standard to calibrate the sensitivity of the probe. A further problem was the Colloid Systems. (a) Alumina Powders. Three alumina temperature sensitivity of the calibration. To overcome this to powders were used to prepare suspensions for investigation. some extent, they converted the estimated apparent dynamic UCAR (agglomeratefree) B-AFand C-AF aluminapowders were mobility into an apparent { potential, using the Smoluchowski supplied by Union Carbide, with particle densities (pp = p + Ap) equation, that is, the -28-mV value quoted above. It bears little of 3.5 and 3.7 g ~ m -respectively. ~, The third alumina powder resemblance to the true {potential but will adjust the calibration (DegussaC) was supplied by DegussaAustralia, and had a particle for (small) changes in viscosity if the temperature departs from density of 2.9 g ~ m - ~ . 25 “C. (b) Silicon Nitride Powder. The SNE-10 silicon nitride In most of our work we used a 5.8% sample of Ludox for powder was supplied by Ube Industries, with pp = 3.4 g ~ m - ~ . calibration. The ESA signal for silica is not a linear function of Suspensions (250 mL) were prepared without further purifi4 for values above about 5%. As noted above, the initial Matec M KCl solution at a cation, by dispersion of the powder in calibration used a 10% sample and our measurements of the particle volume fraction of 0.5 % . dependence of the ESA signal on 4 suggest that we should adjust Initial estimates of particle radii for the aluminas were made our expected {for a 5% sample to -31 mV. That correction has using a submicron particle analyzer (Malvern 4700C) on dilute been incorporated into the results shown below. The ESA suspensions prepared by dispersion of a small quantity (ca. 1.5 measurements were made at resonance; maximum signals were x particles by volume) of powder in M potassium obtained for this suspension at frequencies between 910 and 920 chloride solution. UCAR B-AF and C-AF alumina samples kHz. showed mass average particle radii (a,) of 220 and 310 nm, The pH of the suspension in the SSP-1titration assembly was respectively,while the Degussa C alumina gave a, = 250 nm. An varied by dropwise addition of either 1.0 M HCl or 1.0 M NaOH. initial estimate for the SNE-10 silicon nitride sample was a, = After each addition of acid or base, the system was allowed to 275 nm, taken from a study by Malghan and Lum.’ come to equilibrium before its electroacousticcharacteristicswere Transmission electron micrographs of UCAR B-AF and Derecorded. A sample of approximately 25 mL was removed and gussa C both showed structures composed of crystals of approxcentrifuged (12 000 rpm for 10 min) and the supernatant used imately 20-nmdiameter. These crystals were assumedto be fused to dilute the particle concentration and so record the dc elecinto much larger structures, since we have no evidence of the trophoretic mobility (ME). This sample was reconstituted and separation of individual particles, even after some ball-milling returned to the SSP-1titration assembly prior to the next addition and protracted stirring, shaking, and ultrasonication. Microof reagent. The effectiue mass average radius was measured on graphs of the UCAR C-AF and SNE-10 samples revealed a sample obtained by taking a few drops of the suspension at aggregated structures of larger particles ranging between 400 nm each pH and rapidly diluting them with the clear supernatant. and 1 pm in diameter. We assume that the aggregate can be Five measurements of the dynamic mobility (&J) were made at treated as a single entity when assessing its inertia, so that the each point; typically the scatter in { values was about 0.1 mV. total mass of each aggregate is what determines its apparent The electroacoustic signal of the background electrolyte (the dynamic mobility. “electrolyte effect”)was measured with the software parameters Electroacoustic Measurements. Measurements were made set equal to those of the colloid, giving an “apparent {potential” on the ESA-8000,using the SP-80 dip probe and SSP-1titration of the electrolyte. This term was at most a few millivolts, and cell assembly. The suspensions were kept under constant appeared to be independent of pH, and so could be neglected agitation with an impeller and a magnetic stir bar. compared to the particle signal. Microelectrophoresis. Static electrophoretic mobility meaMeasurementsweremade in the single-pointmode. The Matec surements were made using a particle microelectrophoresis software for calculating {by eq 1requires the particle density, radius, and volume fraction for each suspension. The values apparatus (Rank Brothers Mark 11)with a flat, rectangular,closed used for particle radii were those given by the above initial quartz cell and very dilute samples (ca. lo+% particlesby volume) prepared as indicated above. All measurements of the dc elecestimates, except where otherwise stated. trophoretic mobility were made at the stationary levels of the (7) Malghan, S. G.; Lum, L. Technical Report, Ceramics Division, National Institute of Standards and Technology, Gaithersburg, MD, 1989.
(8) Alexander, G. B.; Iler, R. K. J. Phys. Chem. 1953,57, 932. (9)Cannon, D. W. Private communication, 1991.
James et al.
422 Langmuir, Vol. 8, No. 2, 1992 30.0
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Figure 1. {potentialasafunctionofpHforUCAFtB-AFalumina in 10-3 M KCl solution: X, microelectrophoresis results; 0,electroacoustic results using the initial estimate of mass average particle size (a, = 220 nm);+, electroacoustic result on a welldispersed sample taken from pH 4 up to pH 8;,. corrected electroacoustic resulta usinglight scattering estimatesfor the effective mass average radius at each pH. cell, at an applied potential of 50 V; the estimated precision of the {potential measurementswas 3 mV. The pH of these dilute samples was measured using an Activon Model 209 pH/millivolt meter with an Ionode combined electrode. Particle Size. Initial calculations of the electroacoustic { potential were based on the particle size estimated from light scattering studies of the highly dispersed, dilute suspensions. For the alumina, the radius, a,, was 250-310 nm, and for silicon nitride, a, was 550 nm. In almost all cases those values turned out to be underestimates of the inertia effect. Only when special care was taken to ensure complete dispersion were we able to obtain satisfactory agreement with this =high dilution” radius value. For most systems, it was necessary to measure the apparent mass average particle size distributionat different pH values by PCS. The effective mass average particle radius (a,~) was calculated from the resulting distribution. It is the value of a, obtained at each pH as a result of incomplete dispersion. The measurementa were carried out on samples formed by rapidly diluting 6 drops of the suspension from the SSP-1 titration cell assembly with approximately 10 mL of the centrifuged supernatant so that flocfloc collisions were prevented,and the degree of aggregation could be assumed fixed. These values are assumed to be reasonable estimates of the size of the effective kinetic unit in these more concentrated systems.
Results Figure 1 shows the comparison of potential values obtained by static (electrophoresis) and dynamic (electroacoustic) methods with the first of the alumina samples. In this case the initial pH (ca. 6.5) was lowered to improve the degree of dispersion and the sample was titrated from low to high pH. The sample was obviously not disperse, and using the initial ’high dilution” particle size (open circles) gives very poor agreement with the static mobility (shown as X). If, however, we first adjust the pH to 4, then sonicate for 15 min to obtain good dispersion, and then move up in pH, the agreement (pointa shown as +) is very good on the acid side of the iep. Alternatively, we can allow the system to undergo some degree of aggregation, and provided we use the effective particle size we again get very good agreement (filled circles). Essentially the same degree of agreement was obtained with the other two alumina samples. Table I shows values for the measured effective mass average radii (a& and the corresponding inertial contributions Geff(a) as a function of pH for the alumina and
-*O -100 -120
t1
I
I I
1
I.
Figure 2. As for Figure 1 but with silicon nitride SNE-10 in 10-3 M KCl. Pointa shown as are the {potentials calculated from Babchin’s equation (from ref 1). Table I. Effective Mass Average Radii and Corresponding G(4 suspension pHrange aeff,w Gda) UCAR B-AFalumina 5.00-6.50 0.895 0.4784
UCAR C-AFalumina
Degussa C alumina
SNE-10silicon nitride
7.12 8.75 9.63-10.30 4.13 5.95-6.62 7.75-8.70 9.00 9.40-10.02 10.25 3.22-3.97 4.95-6.60 7.40 8.75 9.57-9.87 10.35 3.03-3.92 5.55 6.03 6.51 7.15-7.72 8.50-9.82
0.927
0.4646
iep
1.048 0.850 0.702 1.293
0.4180 0.4849 0.5650 0.3339
iep
1.203 1.116 0.609 0.528 0.813
0.3570 0.3823 0.7533 0.7408 0.5643
iep 1.038 0.842 0.598
0.4670 0.5499 0.6521
iep 1.199 1.266 1.425 1.128
0.3747 0.3667 0.3199 0.3958
silicon nitride systems studied. It should be noted that values of aeffvary by a t least 20% and sometimes by more than 100%over the pH range. In the pH region close to the iep, rapid particle aggregation was observed by PCS, and so corresponding effective mass average particle radii are not given. The unusual order of the first two aeffvalues for the UCAR C-AF sample may be an indication that the steady-state aggregate structure had not been established prior to measurement. Figure 2 shows a plot of f potential versus pH for the silicon nitride suspension. Mean electroacoustic t potentials are shown, calculated using inertial contributions (G’(a)) based on the initial estimates of mass average particle radii (am),and corrected for the electrolyte effect. Also displayed are t potential values corrected by multiplication by the ratio of Geff(a) to G’(a). For each of the alumina and silicon nitride suspensions, excellent agreement is found between the tpotential data obtained from microelectrophoresis and from corrected measurements of the ESA effect.
Langmuir, Vol. 8, No. 2,1992 423
Electroacoustic Measurements of { Potential Also shown in Figure 2 are {potentials calculated using Babchin’s formula for the dynamic mobility’ (asgiven in ref 10). To calculate {potentials using Babchin’s formula for pd, we first recalculated the C value from eq 3, which was found to be 1.35 times the value obtained using O’Brien’s formula. The { potentials were then calculated using Babchin’s relation for p,j and eq 3 for s.
Discussion The comparison made in Figures 1 and 2 is based on the assumption that the surface conductivity is negligible and the particle permittivity is much less than that of the solvent. It was suggested by O’Brien et aL3 that, for a polydisperse system in a pH range where the sample should remain disperse, a multiplicative factor would be required to give correspondencebetween microelectrophoresisand electroacoustic estimates of the {potential. Figures 1and 2 lend confirmation to the suggestion4 that, for welldispersed systems, this multiplicative factor can be quantitatively estimated from O’Brien’s theory, given the mass average particle size. Furthermore, in aggregated systems, provided that the aggregates are not changing size too rapidly, it seems that the effective mass average particle size, as estimated by PCS, is the appropriate quantity required to estimate the 1: potential from the electroacoustic signal. It should be pointed out that, if aggregation occurs, there is some additional uncertainty about the appropriate value of Ap to use in estimating G because it will depend on the degree of compaction of the aggregates. The compaction will also affect the distribution of the electric field around the particle; the looser the floc, the more the field lines will be able to penetrate, thereby reducing the tangential electric field a t the outer surface of the floc. Since this field drives the electrophoretic motion, the result will be a reduced mobility. On the other hand, the decrease in density associated with the open structure will tend to increase the mobility, since the inertial effect is smaller. Judging from our experimental results, it seems that the two effects tend to cancel one another. It must also be noted that there are real problems in ensuring that the electrophoretic measurement is, in every case, a correct representation of the { potential of the particle surface. The experimental technique requires a high degree of skill, and even given that, the opportunities for contamination abound. One also cannot rule out the possibility of a change in surface properties as the solid concentration is reduced. Figure 2 shows that using Babchin’s original formula, the mass average size gives a very poor estimate of {. A more recent formulation by Babchinll gives results iden(10) James, R. 0.; Texter, J.; Scales, P. Langmuir 1991,7,1993-1997. (11) Babchin, A.; Sawatzky, R. P.; Chow, R. S.; Isaace, E. E.; Huang, H. Znternatioml Symposium on Surface Charge Character, Fine Particle
Society, San Diego Meeting.
tical to those of the O’Brien relation? but it does not take account of the particle permittivity. It would be expected to be in error for suspensions like Ti02 where tp can no longer be assumed to be small. Although the effective mass average size can be used as the input to the Matec software to obtain a good estimate of {, the more rigorous method of accounting for particle size variation, according to OBrien’s theory, is as follows. It is first necessary to replace G(a) in eq 1 by the more general form
G’ = C G ( a )p ( a ) da (4) wherep(a) da is the fraction (by mass) of particles having radius between a - da/2 and a da/2. That integration can be carried out numerically, using a program which can handle complex functions along with numerical values of p ( a ) da. Alternatively, one can break the integral into its real and imaginary parts as follows. Using A = a(3 + 2 A p / p ) / ( 9 [ 1 + (2a)1/2+ a]),we can
+
write
G = (1+ -A) M
+
i ( l + -)A M
I +
iy
(5)
where
+
+
M = 11 + -AI2 [ ( l -)AI2 (6) For a suspension in water a t 25 “C, and a frequency of 1 MHz, a = 7.04~~ where a is in microns. It is then necessary to find the integrals J m X p ( a )da = I , JomYp(a)da = Z2 (7) and then G’ = (112+ 1 ~ ~ ) ’(Note / ~ . that the functions a, A, p ( a ) da, and G’ are all dimensionless.)
Conclusions The ESA effect has been shown to be a useful means of determining electrokinetic properties such as electrophoretic mobility and { potential. In order to make accurate measurements of { potential using the present Matec instrument, however, it is necessary to have accurate knowledge of the effective mass average particle size. In systems where coagulation may cause significant changes to the effective particle size, the appropriate radius for the O’Brien formula can be estimated from light scattering data. It seems that estimates of the mass average particle size taken by dispersing the sample a t low particle density can be quite misleading unless care is taken to ensure that the sample being measured is still well dispersed. In most cases it is preferable to use the effective value, estimated under conditions that preserve the degree of aggregation present in the measuring system; this can give excellent correlations with the static electrophoresis results.
Acknowledgment. We thank the AustralianResearch Council for supporting this project and Matec Applied Sciences for the loan of the ESA-8000 system. (12)An exception is the polymer latex which appears to exhibit anomalous surface conductance.3 This has discouraged us from using latex samples to validate the theory.
