Conductance fluctuations in the laminar flow of a colloid

Conductance Fluctuations in the Laminar Flow of a Colloid SERGEY M. BEZRUKOV, GILYARIY M. DRABKIN, AND A U G U S T I N I. SIBILEV Leningrad Nuclear Physics Institute, Gatchina Leningrad District, Leningrad, 188350 USSR Received July 12, 1985; accepted November 5, 1985 The laminar flow of the electrolyte containing suspended colloidal particles through a single capillary gives rise to conductance fluctuations whose spectrum is proportional to - I n f a t low frequencies, and t o f -2 at high frequencies. With a constant volume fraction of colloidal particles, intensity of the fluctuations grows as the third power of their radius. Two factors that limit applicability of this method for determination o f the average size of particles are the Johnson noise and the equilibrium fluctuations in the ionic concentration of the electrolyte. © 1986AcademicPress,Inc. INTRODUCTION

The conductance of an electrolyte containing nonconductive colloidal particles must show fluctuations that are caused by equilibrium deviations of the particle concentration from an average value. The amplitude and time characteristics of these fluctuations are related to the size and dynamics of the colloidal particles. Unfortunately, up to this day both the conductance and its fluctuations can be measured in experiments with electrolytes only in an area having linear dimensions of about 10 # m (1) or larger. This means that the characteristic time of the diffusion relaxation of the particle concentration in the area is sufficiently large. For example, for 0.1-~zm-diameter particles in an aqueous medium under normal conditions this time is a few tens of seconds. From this it follows that the fluctuations in question are most intensive at very low frequencies f < 10-2 Hz, which makes it very difficult, if not impossible, to measure them. Another cause of difficulties in measurements and interpretation of the results is that relatively high electric fields which accompany measurements of conductance fluctuations give rise to an intensive electrophoretic motion of colloidal particles.

These difficulties can be partly overcome if the electrolyte volume is "scanned" with the help of a capillary in which a flow of the electrolyte is maintained. In this case the characteristic time of relaxation of the colloidal particle concentration is determined by the macroscopic flow velocity, so that the measurements can be done with the use of relatively large capillaries. Limitations on their size will be discussed in the concluding section of this paper. FLUCTUATION SPECTRUM

We restricted ourselves to the case of fairly low flow velocities characteristic of the laminar flow of a colloid. The laminar flow in a long cylindrical channel is described by the velocity profile V(r)

= (AP/4~L)(R

2 -

r2),

[1]

where AP is the hydrostatic pressure difference across a channel of a length L and a radius R, is viscosity of the electrolyte, and r is the distance from the axis of the channel. In a fairly long channel (L >> R) the effects due to constriction contacts between the capillary and the reservoir can be neglected. This makes possible an assumption that each spherical nonconducting particle of a diameter d passing

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JournalofColloidandInterfaceScience,Vol. 113,No. 1, September1986

195

CONDUCTANCE FLUCTUATIONS IN FLOW

through the capillary increases its resistance by pd3/47rR 4 provided that the particle is relatively small (d < R) (2). The resulting change in the conductance may be written as

h = 7rd3/4pL 2,

[2]

where p is the resistivity of the electrolyte. This change in the conductance lasts for the duration of the particle's passage through the capillary: r = L/V(r), [31

The parameter rmi n is easily found from [ 1] as "/'min

[9]

= 471L2/APR 2.

The function S*(f'gmin) that determines the frequency dependence of spectral distribution of the conductance fluctuation intensity for the laminar electrolyte flow with suspended nonconducting colloidal particles cannot be expressed through elementary functions. Computer-generated results of the integration for 1-min 10-2 s are shown on the double logarithmic chart in Fig. 1 (solid curve). For comparison purposes the same figure gives a Lorentzian spectrum, i.e., a curve of the form A/{ 1 + (27rfr) 2} with the time r equal to Tmin (dashed line). The coefficient A that determines the position of the curve relative to the axis of ordinates is selected to have equal intensifies of fluctuations described by two spectra. In processes accompanied by a variation in the size of particles, it is often the volume fraction of particles F, rather than their concentration n, that is constant, ----

where V(r) is the velocity of the electrolyte layer in which the particle is captured. The random and interindependent relative position of particles in space makes it possible to consider a Poisson wave of such events with an average number of events per second v that has the spectrum (3)

S ( f r) = 2 u { g ( f r)} 2,

[4]

where g ( f r) is the Fourier transform of elementary change in conductance, i.e., a squarewave pulse with height h and duration r, g ( f , ~-)

sin 7rrf

h~- - -

=

~f

[51

(the term corresponding to the average value is omitted). It is self-evident that the spectrum of conductance fluctuations in the entire capillary SG(f) includes components of [4] with times from a certain Tmin (for particles at the capillary axis) to infinitely large times for particles at the capillary walls

S a ( f ) = 27rn

S { f r(r)} V(r)r dr,

[6]

where n is the concentration of nonconducting particles and v = 1. Integrating with the velocity profile [1 ] taken into account we obtain

F = 7rd3n/6.

[10]

Equation [7] for such processes can be written, for convenience purposes, in the form

S*(ftmin ) I

i

i

1 10-1 x

10-2

""

10-3

"\\, k

r/Tr3d6R2"rmin

SG(f)-

2-~

.

S (frmin) ,

[71

where S*(fTmin) =

~2 @rain °°

sin2x/2 dx/x 3.

[8]

I

I

1

10

I

100

1000 f, Hz

FIG. 1. Function S*(frmin) compared to the Lorentzian curve. Analysis of Eq. [8] at low and high frequencies demonstrates that this function is proportional to - l n f a t f - + 0 and decreases a s f -2 atf---~ oo. Journal of Colloid andlnterface Science, Vol. 113, No. 1, September 1986

196

B E Z R U K O V , D R A B K I N , A N D SIBILEV

bution, larger particles provide a major con[111 tribution to the spectrum even at very low relative concentration. The admixture of parfrom which it follows that with a permanent ticles with diameters 10 times that of the prinvolume fraction of particles the fluctuation cipal particles even at a relative concentration intensity increases in proportion to the third as low as 10-6 makes a decisive contribution degree of their diameter. Therefore measure- to the conductance fluctuations. This effect of ments of conductance fluctuations in the larger-particle admixtures may lead to grave presence of colloidal particles can be sensitive errors in determination of the average size of tool for determining the size of such particles, particles in a sample; however, this feature of as well as an indicator of coagulation, coales- the method may, in our opinion, be useful in cence, and similar processes. its own way. S~f) -

37r2R2d3rminF Z3p2 S*(f-rmin)

Influence of Particle Size Variance

Limit of Resolution

Equation [7] is valid for particles with a zero variance of the size, i.e., for those of strictly the same dimensions. Actually, particles always show a certain finite spread of their size about its average value. Sometimes the density of the probability distribution of particles by the size can be approximated by a Gaussian (the case of a unimodal monovariant distribution), ~(d) = ~

1

e -((a-a°)/¢~/2,

[ 12]

where do is the average size of the particles and a is the variance. Now, because Eq. [7] contains the sixth, and not the first, degree of the diameter of the particle, the diameter determined by the conductance fluctuation method with the help of Eq. [7] is not the average value but the sixth root of the sixth moment of the distribution in size. The required correction is found by appropriate formulas for relationships between moments of the normal distribution. Provided a ~ do it can be demonstrated that the particle diameter found by Eq. [7] is larger than the real average do by Ad for which the following equation is valid: Ad/do ~- 2.5(a/do) 2.

[13]

For example, for particles showing a 10% spread about the average size the correction is estimated to be 2.5%. The result is affected much more substantially by even small admixtures of larger particles. In the case of the bimodal size distriJournal of Colloid and Interface Science, Vol. 113, No. 1, September 1986

What is the minimum size of particles capable of generating a distinct fluctuation signal? To answer this question, we must consider the principal sources of fluctuations in the system under discussion. The easiest method for measuring conductance fluctuations is application of a constant direct current to the sample under study. Here the conductance noise appears as a voltage noise with a spectrum Sv(f) = ff2SG(f)G-2,

[141

where f is the average voltage drop across the sample and G is its conductance. Taking into account Eq. [ 1 1] we can write the above equation as Sv(f) -

3d3 rminFff 2 LR 2 S*(frmin).

[15]

One of the sources of the noise that covers the effect of colloidal particles is related to the thermal motion of the charge carders. It is well known that voltage fluctuations across a conductor in a thermodynamic equilibrium are described by the Nyquist theorem relating the electric noise of the conductor to its impedance properties (4). The spectrum density of voltage fluctuations of an ionic conductor in the frequency range where the capacitance shunting effect is negligible can be expressed by Sve(f) = 4kTG-1,

[ 16]

CONDUCTANCE FLUCTUATIONS IN FLOW

where k is Boltzmann's constant and T is the absolute temperature. Let us introduce the signal-to-noise ratio ~ as the ratio of the spectral density of the signal [15] to that of the Johnson noise [16]. The ~ is thus frequency dependent, and we shall find it for the frequency that corresponds to the characteristic time 7"min, i.e., f = (27rrmin) -1. By using the numeric value of the function S*(frmin) for this frequency (see Fig. 1), we derive 7rd3 rminFff 2 ~e ~-

4kTpL 2

[17]

Another as unavoidable source of fluctuations in the system under discussion is equilibrium fluctuations of the electrolyte concentration. According to (1), with this source taken into account the root-mean-square level of conductance fluctuations in a capillary is expressed as

ni(°P~ ]N?IG 2, ( AG)2 = -~ kOndp,rl

[18]

where ni is the ion concentration in the electrolyte and Ni is the total number of ions in the capillary. Speaking in general, the spectrum of the fluctuations described by Eq. [18] must be different from that obtained with the help of Eq. [8] for colloidal particles passing through the channel. The difference stems basically from the fact that in the case of particles we considered "unrelaxing" perturbations of the capillary conductance assuming that a colloidal particle generates a time-constant perturbation as long as it passes through the capillary. The authors of (5) believe the fluctuations described by Eq. [18] in the laminar flow of an electrolyte through a capillary to have a Lorentzian spectrum. The idea is difficult to accept because fluctuations of various scales that form the level [18] comprise fairly large-scale ones whose relaxation may be neglected for times of an order of the characteristic time 7"min , For purposes of this section no substantial error will be introduced by assuming the spectrum of conductance fluctuations due to swings of the electrolyte concentration

197

to be similar to the spectrum [8]. What is more, for electrolytes of moderate concentrations the term in brackets in Eq. [18] is close to one and thus can be omitted. These assumptions permit the expression 8Tminl~2 S * ( f r m i n ) . Svi(f)-

7rR2Zl, l~i

[19]

Hence, the expression for the signal-to-noise ratio taking into account the interference in question is ~i = (3/8)Trd3Fni. [20] Let us assume that reliability of measurements is assured by ~ > 1. Then expressions [17] and [20] make it possible to estimate the minimum average size of particles at unimodal monovariant distribution [12], at which the particles contribute measurably to voltage fluctuations in a capillary. Analysis shows that under typical experimental conditions, an example of which is given below, the most rigid constraints are imposed by the latter of the interferences considered, viz., the equilibrium fluctuations of the electrolyte concentration. Here the m i n i m u m diameter of particles can be written as dmin = ((3/8)TrFni) -1/3.

[21]

As an example let us take the values of the volume fraction of particles F = 10-3 and the ion concentration of the electrolyte ni = 1.2 X 1027 m -3 (that corresponds to t M of 1-1 electrolyte); the resulting drain is 10 nm. To conclude, it should be noted that estimate [21 ] is valid only when our assumptions are complied with, the most important assumptions being an interindependent spatial arrangement of particles and smallness of particles compared to the capillary. MATERIALS AND METHODS

Fluctuations in the conductance of a capillary were measured by the constant direct current method (5). A two-compartment cell with Ag/AgC1 electrodes, as shown in Fig. 2, was reliably screened against electromagnetic and vibration interference. The compartments Journalof ColloidandInterfaceScience,Vol. 113.No. 1, September1986

198

BEZRUKOV, DRABKIN, AND SIBILEV

1

FIG. 2. The cell for measuring conductance fluctuations. 1, Pair of Ag/AgC1 electrodes; 2, cell compartments inside bent quartz tubes (fused silicon dioxide); 3, quartz plate with a single cylindrical capillary in the center; 4, sliding guides for alignment of quartz tubes.

of the cell were partitioned by a quartz glass diaphragm (silicon dioxide) with a single capillary with a diameter 2R = 26 #m and a length L = 408 #m. The measurements were made by the two-electrode technique, the same electrodes being used both for maintaining a constant current through the compartment-capillary-compartment system and for conducting the fluctuation signal generated in the system. Such a cell differs favorably from fourelectrode arrangements in which each function is performed by a separate pair of electrodes by simplicity of its design. Errors in the two-electrode measurements stem from the fact that the capillary voltage drop, of importance for the experiment, is somewhat augmented by the voltage drop at the electrodeelectrolyte sections. The same applies to the voltage fluctuations. The magnitude of these errors can be measured in a separate experiment by placing the electrode pair in the cell without the diaphragm. When 0.1 M aqueous KC1 solution was used as the electrolyte, the voltage drop between the electrodes with a surface area of about 2 cm 2, with 2.5 × 10-5 A current passing (the magnitude of current in capillary experiments), was 1.5 X 10-2 V. The voltage fluctuation level under these conditions did not exceed 10-16 V2/Hz. The electronics for voltage fluctuation measurements was similar to that used in optical-mixing spectroscopy (6). The signal picked at the electrodes was amplified by a Journal of Colloid and Interface Science, Vol. 113,No. 1, September1986

low-noise preamplifier with the following input characteristics: voltage noise and current noise at 30 Hz, 1.5 × 10-16 V2/Hz and 5 X 10-3° A2/Hz, impedance, 5 X 1011 ohms, shunt capacitance, 5 X 10-12 F. The amplified signal was analyzed with the help of a laboratory-constructed multichannel analog spectrum analyzer of the type described in (7). It had 13 parallel channels, each channel comprising an active bandpass filter, a quadric detector, and an integrator. The central frequencies of filter bands (f~) were chosen to obey relations lg(f~+l/fn) = 0.25 thus splitting the frequency range 3 Hz-3 kHz equidistantly in logarithmic scale. The bandwidth for each filter wasfn/3, the averaging time was i00 s. Spherical particles of polystyrene latex of a 0.49-~m diameter were used as nonconductive colloidal particles; decimolar aqueous KC1 solution was invariably used as the electrolyte. The cell compartments were vacuum-filled with the filtration-cleaned solution, the CSSRmade Thinpore filters (of Millipore type) being employed. The resistance of the electrolyte in the capillary was (6.43 + 0.05) X 105 ohm. RESULTS AND DISCUSSION

Figure 3 shows arbitrarily selected fragments of the signals subjected to the spectral analysis. Before the signals were fed to the oscillograph the spectrum of the signals was limited to the 1.5- to 6000-Hz bandwidth. The vertical scale is shown with allowance for the gain of the amplifiers. Figure 4 shows results of the spectral analysis of fluctuations for the electrolyte without any specially added admixtures. It can be seen that with no current applied, voltage fluctuations in the capillary correspond to the equilibrium value calculated from the capillary conductance by the Nyquist formula. The electrolyte flow with a capillary-axis velocity corresponding to rmin= 2.5 X 10-3 S does not result in effects that are resolvable against the equilibrium noise background of the conductor. This observation is in a good agreement with the results of (8, 9) that deal

199

CONDUCTANCE FLUCTUATIONS IN FLOW

FIG. 3. Oscillograph records of voltage fluctuations at the electrodes of the cell filled with admixture-free electrolyte, 1, 2; and electrolyte with 3.2 × 10-6 volume fractions of latex particles, 3, 4. A 1.6 X 10-3 Pa hydrostatic pressure difference was applied to the capillary. 1, 3, The external current source is disconnected; 2, 4, 2.8 X 10-5 A current flows from the external source.

with studies of excess noise generated during Poiseuille flow of certain polymer solutions. The authors of these papers come to the conclusion that the excess noise induced by flow occurs only in non-Newtonian solutions. As 2.8 X 10 -5 A current is applied an excess voltage noise takes rise, which is related to fluctuations in the ion concentration of the

electrolyte. As seen from Fig. 4, the spectrum of this noise is close to the value calculated by Eq. [ 19] (the solid curve is plotted as the sum of [ 19] and [ 16]) somewhat exceeding it. The cause of this difference is not yet quite clear. Adding of millionth volume fractions of polystyrene latex to the electrolyte increases the power of the excess noise in the low-frequency region by several orders of magnitude. The spectral distribution curves shown in Fig. 5 relate to basically the same electrolyte and latex sample (F = 6.1 × 10 -6) which in the case of curves 2 and 3 was passed through 2.5and 1.5-#m filters, respectively. Changes in the latex concentration were determined from absorption of the samples in the 280-nm bandwidth before and after filtration. The volume fraction was found to be 5.4 × 10 -6 for curve 2 and 3.2 × 10 - 6 for curve 3. The results shown in Fig. 5 make it clear that the fluctuation power in different samples differs in excess of an order of magnitude while the latex concentration was lowered by filtration by less than a factor of 2. This is evidence

S(f) V2/Hz i.~

1,

,

,

I0-I0~~ ~ ~;~

s(f) 10-12 V2/Hz i

J

i

10-13

10-13

10-14 3'o

300 '

f, Hz

3000

]FIG. 4. Spectral density of voltage fluctuations at the electrodes of the cell filled with admixture-free electrolyte. The experiment conditions under which spectra 1 and 2 were obtained were identical to those mentioned in the legend to Fig. 3 for the oscillograph records 1 and 2, respectively.

~k~ ,

,

30

300

~"~ 3000 f, Hz

FIG. 5. Spectra of the excess voltage noise induced by latex particles. All the three spectra relate to basically identical electrolyte samples with 6.1 × 10-6 volume fractions of latex, which in the case of curves 2 and 3 were passed through filters with different pores (see the text). Other conditions were as indicated in the legend to the oscillograph record 4 (Fig. 3). Journal of Colloid and Interface Science, V o l, 113, No. 1, Se~ember 1986

200

BEZRUKOV, DRABKIN, AND SIBILEV

of an agglomeration sensitivity of these measurements. A spectacular proof of this is provided by curves 1 and 2; here the latex concentrations differ by less than 13% resulting, however, in a sixfold difference in the fluctuation power. The solid curve in Fig. 5 was plotted according to Eq. [15], the diameter of the nonconducting particles being taken to be 0.66 #m. This value exceeds the true dimension of latex particles used by 34%, which is evidence of a certain agglomeration inside the system. One of the possible reasons may be shear induced coagulation (10) associated with flow through the filter or the capillary itself. All the three experimental curves have a characteristic break at the frequency f = ~--' as predicted by the calculation. The results presented in Fig. 6 illustrate the dependence of the excess fluctuation spectrum on the hydrostatic pressure drop across the capillary. The cell was filled with the electrolyte and latex mixture that was passed through a 1.5-~tm filter; the pressure drops for curves 1 and 2 differed by a factor of 3.2. The behavior of the fluctuation spectrum is coincident with that predicted by Eq. [ 15] in that the spectrum corresponding to the lower pressure drop shifts s(f) V2/Hz

10-11 ~

,0-'3

\ I

3

I

30

I

300 Hz

FXG.6. Spectra of the latex-inducedexcess noise at a hydrostaticpressure drop AP of 1.6 × 103 Pa, 1; and 0.5 × 103 Pa, 2. Constantdirect current flowingthrough the capillarywas 2.8 × 10-5A. The electrolyteand latexsample was passed through a 1.5-urn filter,the volume fraction of latex being equal to 3.2 X 10-6. Journal of Colloid and Interface Science, Vol. 113, No. 1, September 1986

left and upward, the value of the shift being in a good agreement with the change in the pressure drop. The effect of the change in the conductance of a capillary as a particle passes through it has been long used in conductometric counters. Production instruments of this type appeared almost three decades ago, soon after Coulter patented his conductometric counting method in 1953 (11). Such counters operate on the principle of recording and processing pulse changes in the conductance of a microscopic aperture in the wall of the tube immersed in an electrolyte. The counting accuracy is governed greatly by the ability of the instrument to resolve two consecutive pulses; otherwise stated, a normal functioning of the instrument can be assured under conditions when the probability of two particles occurring in the capillary simultaneously is sufficiently low. In addition, the size of the particles must be in a certain accord with the dimensions of the capillary. Recently it was demonstrated (12, 13) that the sensitivity of the method can be substantially increased by using a new type of capillary (etched particle track pore in polycarbonate) and employing the phenomenon of electrophoretic mobility of microparticles. The diameter of the capillary used, however, must be only a few times the size of the particle. The authors of (13) also describe experiments on correlation analysis of glass capillaries conductance. They suggest that as the particles pass electrophoretically through a microhole, the autocorrelation function of the conductance fluctuations must be measured, which makes it unnecessary to resolve individual pulses generated by each particle; the number of pulses can be found from the amplitude of the autocorrelation function, and the capillary travel time can be determined from the point of intersection between the function and the abscissa axis. In conclusion they demonstrate the limiting effect of ion concentration fluctuations. In this aspect the work (13) anticipates our results in many ways. The basic difference and advantage of the

CONDUCTANCE FLUCTUATIONS IN FLOW size-measurement techniques in which the flow of particles through a capillary is characterized by the autocorrelation function or the spectrum of conductance fluctuations is that the same capillary can be used for measuring particles whose dimensions differ by many orders of magnitude and for observing, among other things, processes related to considerable changes in these dimensions. As can be seen from the signal-to-noise ratio expressions [ 17] and [20], the capillary radius has no effect whatsoever on the resolution limit attainable by this technique. Yet restraints on the dimensions of the capillary do exist, although they are related to the Joulean heat released in the capillary-filling electrolyte, rather than to the capillary-to-particle size ratio. As follows from [ 17] measurements cannot be made with no matter how small value of ff since the colloidal particle signal must exceed the equilibrium noise of the capillary. The heat released in the capillary is removed basically by the electrolyte flow and by the transfer across the walls of the capillary. The cooling efficiency of the former mechanism, with the linear velocity of the electrolyte maintained, is shown by estimates to be independent of the capillary radius, while in the latter case the cooling efficiency rapidly grows as the radius decreases. Let us consider a specific case of experimental conditions under which the results shown in Fig. 4 were obtained. With an average voltage drop across the capillary ff = 17.8 V, the electric noise due to fluctuations in the ion concentration of the electrolyte exceeds the Johnson noise at the frequency f = (271"Train)-1 by an order of magnitude, and the disturbance caused by the Johnson noise can be neglected. This produces conditions under which is attained the resolution limit described by Eq. [21 ]. It follows from the calculations that under these conditions the electrolyte in the capillary, if cooled only by the flow, would be warmed up by about 3°C, while with the heat removed across the walls it would be heated by a few tenths of a degree, i.e., the latter mechanism is the principal tool for re-

201

moving the Joulean heat. The calculations are confirmed by the results of the measurements that showed the capillary conductance to increase at *Y= 17.8 V, compared to the value at ff = 0.3 V, by about 1%, which at the temperature coefficient of conductance 2 × 10-2 corresponds to a heating of 0.5°C. Thus, the choice of dimensions of the capillary is govel-ned basically by the temperature sensitivity of the system under study. In conclusion let us consider reasons of disagreement between experimental results obtained for the low-frequency region ( f < 10 Hz) and theoretical calculations of the frequency dependence described by Eq. [8]. There appear to be two likely reasons for the discrepancy. First, in derivation of Eq. [8] we neglected the diffusion movement of particles during their flow through the capillary. The effect of the movement is that the distance r between a particle and the axis of the capillary varies in time. With the velocity profile [1] and the time distribution [3] taken into account, it can be shown that the diffusion movement does not call for substantial corrections in the spectrum in the case of particles flowing close to the axis of the capillary but is of great importance for the particles near its walls where the r coordinate velocity gradient is at a maximum. Therefore the time of travel in a wall layer is governed not by the velocity of the layer, as was assumed in deriving Eq. [8], but by the time of diffusion into higher-velocity layers. Second, because the center of a particle cannot obviously approach to the capillary wall closer than half its diameter d/2, the particle velocity cannot be below a certain value easily found from Eq. [1 ]. The two reasons suggest that the spectrum [5] must be integrated with respect to z not to infinite values but to a certain effective Zma~ whose magnitude is determined by the dimensions of the capillary, the diffusion coefficient, and the size of particles, as well as by the velocity of the electrolyte flow. The restraint on the upper limit of integration with Journal of Colloid and Interface Science, Vol. I 13, No. 1, September 1986

202

BEZRUKOV, DRABKIN, AND SIBILEV

r e s p e c t to r l e a d s to a f l a t t e n i n g o f t h e r a t e d s p e c t r u m at l o w f r e q u e n c i e s a n d little affects its h i g h - f r e q u e n c y b e h a v i o r , as r e v e a l e d b y the e x p e r i m e n t . A t the s a m e t i m e , h o w e v e r , e x p r e s s i o n [8], w h i c h is v a l i d for sufficiently large capillaries a n d describes satisfactorily the results o f o u r e x p e r i m e n t s , b e c o m e s e x t r e m e l y elaborate. ACKNOWLEDGMENT Our thanks are due to Dr. O. S. Chechik of the AllUnion Synthetic Rubber Research Institute, Leningrad, for providing specimens of latex used in this work and for discussing methods of stabilization of colloidal systems in concentrated electrolytes. REFERENCES 1. van den Berg, R. J., de Vos, A., and de Goede, J., Phys. Lett. 84A, 433 (1981).

Journalof ColloidandInterfaceScience.Vol.113,No. 1, September1986

2. Gregg, E. C., and Steidley, K. D., Biophys. J. 5, 393 (1965). 3. Verveen, A. A., and DeFelice, L. J., Prog. Biophys. MoL BioL 28, 189 (1974). 4. Nyquist, H., Phys. Rev. 32, 110 (1928). 5. Feher, G., and Weissman, M. B., Proc. NatL Acad. Sci. USA 70, 870 (1973). 6. Berne, B. J., and Pecora, R., "Dynamic Light Scattering," p. 7. Wiley, New York, 1976. 7. Bendat, J. S., and Piersol, A. G., "Measurement and Analysis of Random Data," p. 258. Wiley, New York, 1967. 8. Hedman, K., Klason, C., and Kubat, J., J. Appl. Phys. 50, 8102 (1979). 9. Hedman, K., Klason, C., Poupetova, D., Rheol. Acta 22, 449 (1983). 10. Green, B. W., and Sheetz, D. P., J. Colloid Interface Sci. 32, 96 (1970). 11. Coulter, W. H., U.S. Patent No. 2,656,508 (Oct. 20, 1953). 12. DeBlois, R. W., and Bean, C. P., Rev. Sci. Instrum. 41, 909 (1970). 13. Bean, C. P., and Golibersuch, D. C., in "Electrical Phenomena at the Biological Membrane Level" (E. Roux, Ed.), p. 311. Elsevier, Amsterdam, 1977.