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Current fluctuations from particles flowing through a pore
Current Fluctuations from Particles Flowing through a Pore NICOLE RAKOTOMALALA, ~ LARS INGE BERGE, JENS FEDER, AND TORSTEIN J•SSANG Department of Physics, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo 3, Norway Received December 17, 1990; accepted June 10, 1991 We have studied experimentally the electrical fluctuations which result when colloidal particles suspended in an electrolyte flow through a current carrying pore. A pressure difference across the pore sets u p a Poiseuille flow that transports particles through the pore at low Reynolds numbers. The shape of the power spectrum depends on the particle transit time distribution, and the amplitude is proportional to the particle concentration, the particle size, and the average transit time. We cover the size range from large particle sizes, where distinct electrical particle pulses can be observed, to small particles, where individual particles can no longer be characterized. The aim of this work is to explore the particle flow noise in the transition from macroscopic to microscopic particle sizes, allowing the determination of particle size and concentration, and with applications to flow properties on the pore level, relevant to, for instance, flow in natural porous media. © 1992AcademicPress,Inc. INTRODUCTION
The resistive pulse technique (Coulter principle) is widely used to count and size small particles suspended in a saline solution by letting them pass one at a time through a constricted current path (a pore). When a particle enters the pore, the momentary change in the resistance of the pore is proportional to the particle volume. As the particle concentration is increased, the particle pulses tend to overlap one another and there is a transition to noise. Even if individual particle pulses cannot be observed, information about particle size and concentration, and the dynamics of the flow, are contained in the fluctuating signal. In addition to the noise from particles that have been added to the conducting solution, there are other noise sources that limit the resolution. These are mainly the amplifier noise, the Johnson noise of the pore resistance, and the ion flow noise. The first two are approximately uniform across the span o f frequencies (white noise) and they give a uniform background in the spectral density 1 To w h o m correspondence should be addressed.
representation. The ion flow noise is similar to the flow noise of particles suspended in the electrolyte. The flow noise is confined to the low frequency part of the power spectrum, where the minimum transit time defines a characteristic upper frequency. A large literature exists on noise theory and measurements. This literature is mostly devoted to physical systems and devices with electrons as charge carriers, while the noise from ionic systems with ions as charge carriers has received less attention. A variety of interesting phenomena can be studied by noise analysis of ionic systems. A review of membrane noise is given by Verveen and DeFelice (1). Feher and Weissman (2) measured the noise from a capillary separating two aqueous solutions of beryllium sulfate and they were able to determine the chemical reaction kinetics. Later, Bean and Golibersuch (3) reported some experiments concerning well characterized macroscopic systems of single pores, macroscopic spheres, and simple ions. They obtained the number of ions in a macroscopic pore from noise measurements. Their work has later been extended for small particle sizes (4) and ions (5).
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Iournal~fColloMand lnteffhceScience.Vol. 148,No. 1, January 1992
0021-9797/92 $3.00 Copyright© 1992by AcademicPress,Inc. All rightsof reproductionin any formreserved.
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RAKOTOMALALA ET AL.
In this paper, we give a more comprehensive treatment of the particle flow dynamics and the effect on the power spectrum for a large range of particle sizes. We first briefly present our experimental system. Then, the various noise sources that contribute to the fluctuating signal are discussed. The power spectrum of the noise signal associated with the pressure driven flow of particles through the pore can be modeled using an expression for the particle transit time probability distribution. Our measurements are found to be in good agreement with the theoretical power spectrum, except for some low frequency noise, the origin of which has not yet been established. Also, the particle concentration and the particle size can be obtained from measurements of the fluctuations. MATERIALS AND METHODS
The central part is a Plexiglas cell with two chambers ( ~ 1 cm 3) connected by a pore (6, 7). The complete experimental setup is shown schematically in Fig. 1. The particles to be analyzed are suspended in an electrolytic solution which fills the cell. Particles flow through the pore by pressure drive. Each chamber contains a silver, silver-chloride electrode connected to a constant voltage source. When a particle enters the pore, the resistance between the electrodes increases and consequently the current decreases. The current fluctuations are analyzed using a Keithley Model 427 current amplifier with a voRage output, connected to a D A T A 6000 waveform analyzer (Data Precision Corp., USA), which is interfaced to a personal computer. The waveform analyzer digitizes a portion o f the input signal (a trace) to a chosen n u m b e r of points (~<16 384), each with a preset time duration (>~ 10 ~zs). The cell is placed on a vibration damped table in a brass box that provides shielding against electrical interference. The brass box can be temperature stabilized by circulating water from a temperature controller. The temperature of the cell is measured with a thermistor calibrated against a quartz thermometer. Journal of Colloid and lnteffh6~, Science, Vol. 148, No. 1, January 1992
FIG. 1. Schematic experimental arrangement. The experimental cell connected to a constant voltage source and a current amplifier is the heart of our system. A low-pressure regulator maintains a constant pressure in the res-
ervoir. The signal from the current amplifier is analyzed with a waveform analyzer. The pressure is measured with differential pressure sensors, and the temperature of the cell is measured with a thermistor. The instruments are interfaced to a personal computer. The cell is placed on a vibration damped table in a brass box that provides shielding against electrical interference (not shown). The electrolyte used has been 0.15 M NaC1 in distilled water. The solution was filtered through a 0.22-/zm Millipore filter before monodisperse polystyrene spheres were added (Dynospheres, D y n o Industries, Oslo, Norway). The standard deviation o f the diameter o f the spheres is about 2%. For some o f the measurements using large particles, sucrose was added to match the density o f the particles. Experiments have been carried out using two cylindrical glass capillaries with diameters 27 and 70 # m and resistive lengths 540 and 570 #m, respectively. These pores have been m a d e by drawing down lengths of capillary tubing, epoxying t h e m to thin sheets o f plastic, and then grinding until the desired length is obtained. To m a k e a correction for end effects, the resistive pore length should be used instead o f the geometrical pore length. The resistive pore length exceeds the geometrical pore length by approximately 0.8 times the pore
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F L O W I N G PARTICLES
diameter (6). Also, a 9-gm equivalent diameter and 100-gin long mica pore has been used. The pore cross section is diamond shaped, and the pore has been made by the etched particle track process (8) by Dr. R. Brandt. The basis for the resistive pulse technique is the increase in resistance which accompanies the presence of a particle inside the pore. For long pores, the electrical pulse from a spherical particle flowing through the pore will be square-like, with a well defined height and width. The relative increase in resistance due to the presence of a sphere inside a long pore is given by Maxwell's first order approximation (6, 9) with a particle size correction factor based on the calculations by Smythe (10). The response of the instrument for spherical particles is (6, 7)
I / D2L A V = G~d3S-~D) ~-1) -1.
[1]
Here A V is the measured voltage change at the output of the current amplifier due to the presence of a sphere with diameter d inside a pore with diameter D and resistive length L. Further, I is the current through the pore (in the absence of particles), G is the current amplifier gain (volts/ampere), and S(d/D) is a size correction factor based on the calculations by Smythe. The size correction factor has been given a simple empirical form, S(d/D) = ( 1 -0.8(d/D)3) -1 ( 11 ). For very small particles, S = 1, and the response may be approximated by AV = IG(d3/D2L); i.e., the voltage increment is proportional to the particle volume divided by the pore volume.
1..5 ,
0.0
i 0
i
i
i
i
i 1.00
i 200
i 300
i 400
500
time (arbitraryunits) P3G. 2. Signals obtained in a pore 70 # m in diameter and 570 ttm long, using solutions of 15 # m diameter spheres at two concentrations ( N = 0.3 and 0.08; upper two signals), a concentrated suspension of 3-ttm diameter spheres ( N = 400), and only electrolyte (the voltages have been scaled differently). The gain is 108 for all cases.
pore 70 ttm in diameter and 570 #m long. The two upper signals were obtained with two different concentrations of 15-~tm diameter spheres, the next signal is for a concentrated suspension of 3-#m diameter spheres, and the bottom signal is the noise obtained for the pure electrolyte. We show that the number density and the minimum transit time may be recovered by Fourier analysis of the noise signal. In the following we present a discussion of the different noise sources that contribute to the measured signal.
The Measured Signal Current fluctuations due to fluctuations in the number of particles in the pore volume are amplified, and the measured signal at the output of the Keithley current amplifier is a voltage,
NOISE SOURCES
Vout = - ( / i n X G),
The resistive pulse technique is usually applied to systems where individual pulses can be detected (6, 7). When the number of particles in the pore becomes sufficiently large, this is no longer possible. Also, when the particle size becomes very small, the response from a single particle is not distinguishable from the background noise. Figure 2 shows examples of these different situations using a
where/in is the current at the input after suppressing the average current through the pore and G (volts/ampere) is the gain of the current amplifier. The relation between the measured voltage fluctuations and the actual current fluctuations is
V 2 s = G212~,
[2]
[31
Journal of Colloid and Inter/hoe Science, Vol. 148,No. 1, January 1992
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where V rm~ 2 -- (V - 12)2 is the mean square deviation of the output voltage from its average and I~ms 2 = ( I - / ) 2 is the mean square deviation of the input current from its average. The bar denotes a time average.
Amplifier Noise The amplifier noise m a y be approximated by white plus 1 / f n o i s e (12). The noise level of the Keithley current amplifier depends on the bandwidth, which varies with the gain and rise time settings. Also, the noise decreases as the input shunt capacitance is reduced. The input shunt capacitance is equal to 60 pF, and a noticeable reduction is not easily achieved with the present setup. The root mean square (rms) noise voltage generated by the amplifier with an open circuit at r o o m temperature is approximately 1.05 m V with G = 108 and rise time setting equal to 0.1 ms (noise bandwidth equal to about 5 k H z ) . This value was obtained by averaging several hundred traces (4096 points per trace). The amplifier noise level is approximately the same as the Johnson noise from a 1-M~ resistor with the same noise bandwidth at r o o m temperature (see below).
Johnson Noise When the electrolytic cell is included in the circuit, the inherent Johnson noise (13) of the pore resistance is an additional noise source which adds to the amplifier noise. The Johnson noise of the pore resistance determines the ultimate limit in resolution. When we apply a constant voltage across a resistance R, the mean square current associated with the fluctuations in resistance is measured as a m e a n square voltage, given by
V2
rms,J
= G2 4kTAf R
'
[4]
where k is the Boltzmann constant, T is the absolute temperature, and A f is the noise bandwidth. The index Y denotes Johnson noise. For uneorrelated noise sources, the mean square noise voltages add and the a m Journal of Colloid and lnte(/hce Science, VoL 148, No. 1, January 1992
ET AL,
plifier noise should preferably be smaller than the Johnson noise of the pore resistance in order to achieve a high resolution. For a 1M r pore resistance, Gms,s = 0.9 m V for G = 108 and A f = 5 kHz at r o o m temperature. Note that the current fluctuations decrease with increasing resistance when the voltage across the pore is constant. F o r small pore resistances, it is more favorable to use a constant current which gives voltage fluctuations proportional to R.
Other Noise Contributions The energy of the fluid in the pore fluctuates around its average energy. This gives rise to voltage fluctuations that m a y be expressed as
(2)
V~
T2k ( I OR] 2 -
c
OT]
(GI)2'
[5]
where c is the heat capacity and R is the pore resistance. The resistance o f the electrolyte in the pore decreases approximately 2% per degree increase in temperature. For our experimental conditions, these energy fluctuations have been small. As an example, V~s = 0.06 m V according to Eq. [ 5 ] due to energy fluctuations in a pore 27 g m in diameter and 540 # m long. The effect o f heating is assumed to be small since there is a constant flow rate through the pore. For small systems, diffusion of ions or particles m a y constitute a significant noise source. The transit time associated with longitudinal diffusion in a pore of length L and diffusion coefficient D is given by (2) L2 rdi~--~ 12D"
[6]
Here L2/12 is the m e a n square distance from the ends of the pore. Often m o r e important, transverse diffusion of ions or small particles in capillary flow m a y modify the transit time probability distribution due to mixing of particles. This becomes i m p o r t a n t when the characteristic diffusion path o f a particle
FLOWING
( D r m i n ) 1/2 is comparable to the radius of the
pore (5). For the large particles (compared to the ions) and long capillaries we have used, the characteristic time associated with longitudinal particle diffusion out of the pore is long and the corresponding frequency small. Also, mixing o f particles due to transverse diffusion has been negligible. For the ions, the characteristic diffusion path might be of the same order of magnitude as the pore radius. With a diffusion coefficient D -~ 10 -5 cm2/s, we obtain (D'rmin) 1/2 = 2.2 # m for a m i n i m u m transit time equal to 5 ms. Thus, transverse diffusion may be an important effect. Longitudinal diffusion becomes negligible for long pores. Using a 100-#m long p o r e , "/'diff ~-~ 0.8 S. This is a very long time compared to typical transit times set by the pressure difference across the pore ( ~ 5 ms) and therefore not important in the frequency regime we are considering.
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PARTICLES
level in the absence of particles. Note that all voltages are amplified currents and that in our setup V0 is an arbitrary voltage which depends on the current suppression of the amplifier. If there is no correlation between the particles, the probability of having n particles in the pore is given by Poisson's distribution P ( n ) = Nne-U/n!,
[8]
where N is the average number of particles in the pore. Thus, the average voltage is = Vo + N A V ,
[9]
and the mean square deviation of the voltage from its average is V2ms = N ( A V ) 2.
[101
The noise from very small particles may be significant if their number is large. The above results should apply whether N is greater or less than one. Note that measurements of l~ Vo and V~mscan be divided to obtain V-N. The number N is a measure of the particle concentration and it does not depend on the velocity profile or the flow rate. Also, the above relations may be applied to the case where the ions themselves are considered as non-interacting particles. Without any ions in the pore, the current is zero and it will increase as the number of charge carriers increases. The mean square noise voltage is then given by -
THEORY
There is an excess noise to the Johnson noise that arises from fluctuations in the number o f ions in the pore. When particles have been added to the electrolyte, there is also an additional noise from these particles. For both cases, this excess noise is confined to frequencies less than the frequency corresponding to the m i n i m u m transit time through the pore, which makes it possible to separate the flow noise from the background noise for rather small noise levels. We now turn to discussing flow noise in more detail. Statistics o f Uncorrelated Particles
For a particle-containing pore, we assume that the voltage at the output of the current amplifier is given by (see Ref. (3)) V ( n ) = Vo + n A V ,
[7]
where n is the number of particles in the pore, A V is the voltage increment associated with each particle (Eq. [ 1] ), and V0 is the voltage
2 Vrms, i
= G 2 12 Nion s ,
[]1]
where the index i denotes ions, I is the average current, and Nions is the average number of ions inside the pore. This relation has been shown by Bean and Golibersuch (3) to give the correct number of ions in a macroscopic pore. The excess noise due to the fluctuations in the number of ions in the pore covers a frequency range set by the pressure drop across the pore. It comes in addition to the amplifier and Johnson noise and it is similar to the noise from particles which have been added to the electrolyte. Journal of Colloidand Inter'faceScience, VoL 148, No. 1, January 1992
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RAKOTOMALALA
ET
AL.
Autocorrelation Function
Power Spectrum
The autocorrelation function, C ( r ) , is defined by (14)
The power spectrum S ( f ) is related to the autocorrelation function through the WienerKhintchine relation (15),
C(r) = V(t). V(t + r),
[12]
where the bar indicates a time average as before. If V(t) is taken to be the deviation of the voltage from its average value, then for r equal to zero, the autocorrelation function is simply 2 V rm~ as defined in Eq. [ 10 ] for suspended particles or as defined in Eq. [ 11 ] for the ions. The autocorrelation function is nonzero over a time span where the events are causally related in the voltage signal, determined by the particle transit time through the pore. In plug flow, there is only one transit time and the autocorrelation function is especially simple, consisting of a straight line segment. In Poiseuille flow, the transit time varies continuously between a minimum transit time (rmin) for particles entering on the pore axis and a maximum transit time (rm,x) for particles flowing close to the pore wall (7). The interval of possible transit times depends on the particle size, smaller particles can come closer to the pore wall where the fluid velocity is less, which results in a longer m a x i m u m transit time. For square particle pulses with a spread in transit times, the symmetric autocorrelation function can be modeled by a sum of straight line segments ( r > 0):
l
r ~ r ....
[13]
For times longer than the m a x i m u m transit time (rmax), the autocorrelation function is equal to zero. It measures the fraction of particles remaining in the pore at time r. The sum includes all classes of transit times r~ >~ r. Here Pg denotes the probability of having transit times in a subinterval about r~. A model for the probabilities Pi will be discussed later. Journal o f Colloid and lnte('[ace Science, Vol. 148, No. 1, January 1992
S(f) =
f° C(r)cos(27rfr)dr,
[14]
oo
where f is the frequency. This gives for square particle pulses [ sin r~fr~ \2 7rfr,--- ]l .
S(f)= V~s~Piri!k
[151
The amplitude of the power spectrum is lim S ( f )
=
2
V r m s r- .
[16]
f~0
The average transit time 7 = Zi Pir~, which depends on the particle size, approaches its maximum value 2train for very small particles in pure Poiseuille flow. The integrated power spectrum gives the mean square deviation of the voltage from its average,
f]° s ( f ) d f = V~s. co
[17]
From the above considerations we can estimate the ratio of the particle flow noise to the background noise at zero frequency, A = V ~ , s ( 1 + a2) '
[18]
w h e r e f is the sampling frequency, oz = Vrms,J V~ms,J, and V ~ in the numerator may either be the ion noise (Eq. [11]) or added particle noise (Eq. [ 10] ). The index a denotes the amplifier noise. For the ion case, the above ratio may be expressed as
RI27r f A - 4kTNion~ Af ( 1 + oz2) "
[19]
TO obtain a large ratio A, the value of Vr.... should not be much larger than Vrm~,jand a sampling frequency f two times larger than the frequency bandwidth o f the signal 2xf is desirable. In the following, the normalized power spectrum Su(fu) = S ( f ) / V ~fi, where
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FLOWING PARTICLES
fN = frmi~, i.e., the amplitude of the power spectrum and the frequency corresponding to the minimum transit time are both normalized to one. Flow Characteristics Particles are driven through the pore by pressure drive which sets up a Poiseuille flow. The probability for having transit times in an interval about ~; is . I~'i+l / 2
Pi =
P('r)dz,
[20]
~ri--l]2
where the probability density may be expressed in the form (7)
P(~-) = c ~ × cl+-~c2x 3 1-~x
a
.
[211
Here C~ is a normalization constant and v f is the electrophoretic particle velocity inside the pore (for charged particles in an electric field). For a pore diameter D and a particle diameter d, Cl = 1 - ( 2 / 3 ) ( d / D ) 2 and c2 = 23.36(1 - C l ) . The m a x i m u m fluid velocity of the parabolic velocity profile is Vm = D2&P/16~L, where the pressure drop across the pore length L is &P, and n is the fluid viscosity. The mini m u m particle transit time is "/'min ~-~ " r 0 / ( C l + V d V m ) , where V~ is the algebraic sum o f v f and v~ (the electro-osmotic fluid velocity in a charged pore). The m i n i m u m transit time for a particle of negligible size in pure Poiseuille flow is r0 = L / V m . The relative radial position is x = 2 r / D , where r is the radial position. The velocity of neutrally buoyant spheres in Poiseuille flow is smaller than the fluid velocity at the center of the sphere (16). This effect diminishes with decreasing particle size and the correction factor (the last term in Eq. [ 21 ]) approaches one. For a particle diameter to pore diameter ratio equal to 0.26, the correction factor varies between 0.81 and 1.04, while for a 10 times smaller particle di-
ameter, the correction factor varies only between 1 and 1.01. If electrophoretic effects are negligible for small particle sizes, then the probability density simplifies to P ( r ) ~ ~-3. For large particles in long pores, the probabilities P~ are not directly deducible from the fluid velocity profile, due to radial migration (motion transverse to streamlines) ( 7, 17, 18). This is illustrated in Fig. 3, which shows a measured (filled circles) particle transit time probability distribution for 7.1-urn diameter spheres transported in Poiseuille flow through a pore 27 t~m in diameter and 540 #m long. The dotted curve is the theoretical probability distribution based on Eq. [ 21 ] and the solid curve is a calculated distribution which includes the effect of radial migration (see Ref. (7)). For large particle sizes, radial migration plays an important role, but the effect is strongly dependent on particle size and it diminishes rapidly when the particles become small. Figure 4 shows four theoretical power spectra illustrating the effect of particle size, radial migration, and electro-osmosis on the shape of the power spectrum. The frequency corre-
0.4
i
i
i
1.5
2.0
2.5
0.2
0.1
0.0 1.0
-"
3.0
FIG. 3. Measured (filled circles) and two theoretical particle transit time distributions for 7.1-1~m diameter spheres flowing through a pore 27 ~m in diameter and 540 g m long at AP = 1400 Pa (rm~ = 4.7 m s ) . The dotted curve is based on Eq. [ 2 t ] assuming that the particles follow streamlines, while the solid curve is a fit to the measurem e n t including radial migrMjon (see Ref. ( 7 ) ) . The theoretical curves are histograms, where points have been connected with solid lines. ~i' Journal of Colloid and Inter'/hoe Science, Vol. 148, No. I, January 1992
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RAKOTOMALALA 1.0
AL.
In conclusion, Fig. 4 includes the case of very small particles in Poiseuille flow for which / 7 " m i n ~-~ 2 , the case of plug flow for which ~/rmin ~ 1, and two intermediate cases for large particles with and without radial migration. It is evident that these different conditions have a measurable effect on the resulting power spectrum.
0.8 0.6
2~ ~z 0.4
0.2 0.0 0.0
ET
0.2
0.4
0.6
0.8
f,
i .0
RESULTS AND DISCUSSION
Particle Concentration
FIG. 4. The effectof particle size, radial migration, and electro-osmosis on the shape of the theoretical power When the particle size is large enough, the spectrum (see the text). pulse height A V and the voltage/Io in the absence o f particles can be measured directly. sponding to the m i n i m u m transit time and This allows us to test the theoretical predicthe amplitude of the power spectrum have tions o f the previous section. Measurements both been normalized to one. Curves 1 and 3 have been performed with 15-#m diameter illustrate the effect of particle size on the power polystyrene spheres and a pore 70 # m in dispectrum and they have been calculated for ameter and 570 t~m long at room temperature. particle to pore diameter ratios equal to 0.001 We used a solution with a well defined particle (curve 1) and 0.26 (corresponding to 7. l-~tm concentration, corresponding to a mean diameter spheres flowing through a 27-#m di- number of particles per pore volume N = 0.27. ameter pore). The smaller particles can come The conducting solution was 0.15 M NaC1 closer to the wall where the fluid velocity is with 10 wt% sucrose added to match the denless, which results in a larger spread in transit sity of the spheres ( 1.04 g/cm3). This a m o u n t times and a different transit time probability of sucrose increases the viscosity of the soludistribution. The ratio ~/~-mi. approaches 2 tion by a factor 1.33. Using a low particle confor sufficiently small particle sizes and it equals centration enables individual particles to be 1.99 for curve 1 and 1.45 for curve 3. Curve easily detected. The background noise was 4 corresponds to the case where the electro- only a few percent of the pulse height (see osmotic fluid velocity (uniform across the pore Fig. 2). One can determine N by measuring A V, V0, cross-section) is 100 times larger than the maximum Poiseuille fluid velocity at the cen- tv, and using Eq. [ 9], or by measuring V~ms ter of the pore, i.e., for Ve/Vm = 100 and vf and AV and using Eq. [10]. Both methods = 0, a case which becomes important for small give a n u m b e r N -- 0.26 in good agreement pressures and small charged pores. When the with the known value of the concentration. electro-osmotic velocity increases compared to The value of N can also be determined indePoiseuille flow, the spread in transit times is pendently from the slope K o f the measured reduced until eventually all particles flow with particle rate (the number of particles flowing the same velocity in pure electro-osmotic plug through the pore per second) versus the presflow such that ~ /Tmin '~" 1. In Poiseuille flow, sure difference across the pore ( 11 ): radial migration may effect the transit time (,)2 N=32nK~ . [22] probability distribution and therefore also the power spectrum. Curve 2 includes the effect of radial migration and it is based on the mea- The best fit to a straight line passing through sured probability distribution shown in Fig. 3. the origin (negligible electrokinetic effects) Journal q[Col[oid and Inteff?lce Science. Vol. 148, No. 1, January 1992
FLOWING
gives a slope K = 0.108 (s Pa) -1 and N = 0.27, which again is in good agreement with the k n o w n particle concentration of the solution. W h e n the particle size is small, individual pulses cannot be distinguished from the background noise. Yet the knowledge of the volu m e fraction of particles in the pore, F = Nv/ v, where v and v are the particle and pore volume, allows the determination of the particle size and concentration. We first show experimentally that Eq. [ I0 ] is valid. Measurements have been performed with l-tzm diameter polystyrene spheres and a pore 27 # m in diameter and 540 ~zm long. We used solutions with particle concentrations corresponding to a m e a n n u m b e r of particles N in the pore volu m e ranging from 30 to 235; this corresponds to volume fractions ranging from 5 × 10 -5 to 40 × 10 -5 Figure 5 shows the measured V~ms as a function of N for two values of the cell voltage. Each fitted straight line corresponds to one cell voltage. The slope o f the straight line gives the diameter of the particle. In all cases, the value deduced from the slope is consistent with the k n o w n particle diameter. Figure 6 shows the measured Vr~s as a function of the pore voltage. Each fitted straight line corresponds to one particle concentration. The two series o f measurements for the highest particle concentrations show voltage fluctuations that are su-
0.020
t
~
i
99
PARTICLES 150
>~
i
i
i
5
10 pore voltage (V)
15
50
0
0
F i G . 6. M e a s u r e d V~ms(filled circles) as a f u n c t i o n o f the cell v o l t a g e a n d fitted line for t h e s y s t e m d e s c r i b e d in t h e p r e v i o u s figure. E a c h line c o r r e s p o n d s to o n e m e a n
number N, ranging from 30 (lower line) to 235 (upper line).
perlinear at a high applied voltage. This is probably due to increased low frequent noise for these cases that m a y possibly result from heating within the pore. The experimental power spectra have been obtained using the smallest of the two standard cell voltages, for which heating appears to be negligible. The m e a n n u m b e r o f particles N is deduced from the slope o f the straight line. The values obtained are in agreement with the expectations. It should be noted that both particle size and concentration can be obtained experimentally when the volume fraction F of particles is known. Eq. [ 10 ] can be written
2L-
J
20
Fd3"
[23]
The volume of the particle is deduced from the m e a s u r e m e n t of the fluctuations, and eventually the concentration is obtained, using the definition o f F.
0.015
o,olo 0.005
Power Spectrum and Flow Properties 0.000
50
100
150
200
250
N
FlG. 5. Measured V~s (filled circles) as a function of N and fitted line for a solution of 1-~m diameter spheres in a pore 27 ~m in diameter and 540 ~m long. The lower line is for 8.4 V cell voltageand the upper line for 15.5 V.
Figure 7 shows the first part of the measured power spectra obtained with an open circuit, a 3.3-Mft metal film resistor, and an electrolyte filled pore with resistance 1 Mg, all at r o o m temperature, at zero voltage, and amplifier gain setting G = 10 s. At low frequencies, the Journal cfColloid and lnterjace Science, Vok 148, No. 1, January 1992
100
R A K O T O M A L A L A ET AL. 2.0
1.5
2
>
~o
1.0
0.5
0.0
0
200
400
600
f (Hz) FIG. 7. Comparison of power spectra obtained with an open circuit (lower curve), a metal film resistor with R = 3.3 M~, and an electrolyte filled pore with R = 1 M~2 (upper curve). The measured power spectra correspond to current fluctuations at constant voltage and are inversely proportional to R (see Eq. [4]) for Johnson noise.
spectrum is no longer white. In the following, this deviation has been negligible compared to the particle flow noise. The Johnson noise of the pore resistance is approximately equal to that of a metal film resistor of the same resistance, and it can be obtained by subtracting the open circuit power spectrum from the power spectrum when the pore is included in the circuit and then integrating. We find that the Johnson noise of the pore resistance obtained from measured power spectra is equal to 0.9 mV, which is in good agreement with Eq. [ 4 ] when inserting the experimental noise bandwidth of 5 kHz. The experimental power spectra are analyzed in the following way: it is first checked
that the Johnson noise of the pore resistance, measured at zero voltage, is equivalent to that of a metal film resistor of the same resistance and that it is approximately white noise. When the voltage across the pore is non-zero, the noise from the particles merely adds to the Johnson and amplifier noise. The latter are subtracted from the experimental power spectrum, and the resulting power spectrum (only due to particles) is fitted to the theoretical expression (Eq. [15]). Both the particle concentration and the flow properties are o f interest. In the following, the Reynolds n u m b e r Re = oRVo~/71, where o is the fluid density, is ranging from 0.05 to 5. (See detailed values of Re in Table I). Figure 8 shows a measured power spectrum and the fitted theoretical curve (Eq. [ 15 ] ) for the solution o f 15-/zm diameter spheres ( N = 0.27) using a pore 70/zm in diameter and 570 #m long. The experimental power spectrum was computed from 4096 digitized points of the signal with a sampling f r e q u e n c y f = 5 kHz and it was averaged 400 times, which corresponds to roughly 10,000 particles passing through the pore. The pressure drop was Ap = 280 Pa yielding an expected minimum transit time rmin = 4.9 ms. The averaged V ~ measured directly on the time signals was 50.8 mV, which should be approximately equal to the integrated power spectrum. In the fitting procedure, two parameters are used: V r m s and train- We find 52 m V and 4.7 ms, respectively, which is in good agreement with expectations.
TABLE I S u m m a r y of the Measurements Obtained with Different Pore and Particle Sizes D, L (,um)
d(gm)
d/D
N
F(×10 -5)
V~. (mY)
~'ml. (ms)
Re
70, 70, 9, 27,
15 3 0.31 1
0.214 0.043 0.034 0.037
0.27 460 160 30-235
20 300 40 5-40
52 22 41 6-140
4.9 2.8-14.5 3.1-10 5
3.2 1.1-5.5 0.05-0.15 1.5
570 570 100 540
Note. D, L are the pore diameter and length, d is the particle diameter, N is the average n u m b e r o f particles in the pore volume, F is the volume fraction of particles, Vrmsis the m e a n square deviation of the output voltage from its average, rm~, is the m i n i m u m transit time through the pore, a n d Re is the Reynolds n u m b e r of the pore. JourlTal q/Co[lllid and lnle([~tce Science. VoI. 148, No. 1, January 1992
FLOWING PARTICLES
101
0.4
30
0.3
0.2 10 0.1
0
0
50
100
~
,
0£100
150
200
300
f (Hz)
400
f (Hz)
FIG. 8. Power spectrum for a solution of 15-~m diameter spheres flowing through a pore 70 #m in diameter and 570 ~ m long. The dotted curve is the theoretical fit (Eq. [15]). The m i n i m u m transit time Train equals 4.7 ms ( f = 210 Hz).
FIG. 10. The same measurement and fitted curve as that shown in Fig. 9, but for frequencies above 100 Hz: Note the characteristic bump in the spectrum, which occurs close to the frequency corresponding to the minimum transit time.
W h e n the particle size becomes small and the background noise comparatively large, direct measurements of the voltage fluctuations in the signal itself does no longer yield information only about the particles inhibiting the flow. However, Fourier analysis deconvolutes the background white noise from the particle noise. When the particle diameter is reduced to 3 #m, the pulse height is comparable to the background noise and individual pulses can no longer be detected. The measured and fitted power spectrum for 3-~tm diameter spheres are shown in Figs. 9 and 10, below 150 Hz and
above 100 Hz ( A p = 280 Pa). The average number of particles in the pore is now N = 460 (based on the weight percent of particles in the solution). The expected root mean square voltage due to particle flow noise only is 21 mV, while the measured power spectrum yields a value close to 22 mV. The measured power spectrum now contains some excess low frequency noise, the origin of which is not clear. The effect of changing the pressure drop across the pore is illustrated in Figs. 11 and 12. This is a dimensionless representation of
5
1.o
4
0.8
3
I 0.6 ' z
2
0.2 I
1 0
0.4
50
t00
150
f (Hz) FIG. 9. The measured power spectrum for 3-tzm diameter spheres flowing through a pore 70 tzm in diameter and 570 ~m long. The dotted line is the theoretical fit (Eq. [15]). The m i n i m u m transit time Zr.i, equals 5.4 ms ( f = 185 Hz).
0.0 0.0
0.2
0.4
0.6
0.8
1.0
rN FIG. 11. Superposition of three measured power spectra obtained for minimum transit times equal to 2.8, 5.2, and 14,5 ms, for a solution of 3-~m diameter spheres in a pore 70 ~m in diameter and 570 u m long, at low frequencies. The theoretical power spectrum is also shown in the figure. Journal (fColh)id and Intel'[ace Science Vok 148,No. 1, January 1992
102
RAKOTOMALALA
the power spectrum for three pressure drops, corresponding to m i n i m u m transit times equal to 2.8, 5.2, and 14.5 ms. We have plotted S~v(fN) versusfv. The values of V~m~and rr~i, for each case are obtained by fitting to the theoretical power spectrum. The data collapse of the experimental curves and the agreement with the theoretical curve is good, except for low frequencies where some excess noise is measured. Experiments with 1-#m diameter spheres of concentration N ~ 160 in a pore 27 ~tm in diameter and 540 # m long produces power spectra which are nicely fitted to the theoretical expression. When the particle concentration is reduced by a factor four, we observe some low frequency excess noise for this case as well. We have also done measurements using 0.31-#m diameter spheres of concentration N 160 in a pore 9/~m in diameter and 100 # m long. The ratio of the particle diameter to the pore diameter is comparable to that of the previous cases. Figure 13 shows the dimensionless representation of the power spectrum. The m i n i m u m transit times are equal to 3.1, 6.2, and 10 ms. In all cases, the fitted values for V~s are close to the expected noise V ~ = 41 inV. The slowest experiment exhibits some excess low frequency noise, but there seems to be considerably less excess noise for these smaller particles. All the experimental results have been analyzed assuming independent particles. However, further work has to 0.25 0.20 0.15 ~z 0.10 0.05 0.00 0.0
0.5
1.0
1.5
2.0
fN PqG. 12. T h e s a m e m e a s u r e m e n t s a n d t h e o r e t i c a l c u r v e as t h o s e s h o w n in Fig. 11, b u t at h i g h frequencies.
Journal ojColh)id and Mterface Scie~tce, Vol. 148, No. 1, January 1992
ET AL. 1.5
~z~z 1.0
0.5
0.¢ 0.0
0.2
0.4
0.6
' " 0.8
1.0
fN FIG. 13. S u p e r p o s i t i o n o f t h r e e m e a s u r e d p o w e r s p e c t r a
obtained for minimum transit times equal to 3.1, 6.2, and 10 ms, for a solution of 0.31-urn diameter spheres in a pore 9 #m in diameterand 100 ~m long, at low frequencies. The theoretical powerspectrum is also shown in the figure.
be done in order to study the effect o f interactions between particles on the power spectrum. SUMMARY
We have studied the current fluctuations due to the presence of particles in a current carrying pore. Table I summarizes the experimental results obtained with various combinations of pore and particle sizes and well defined particle concentrations. For most of the experiments, the background noise is small compared to the particle flow noise, due to either a large particle concentration or a large particle size. The last case in the table corresponds to the measurements shown in Figs. 5 and 6, for which the volume fraction was varied, producing signal-to-noise ratios (ratio of the measured mean square deviation of the output voltage from its average to that obtained at zero voltage) ranging from 1.2 to 600. Particle size and concentration can be deduced from fluctuation measurements, when the volume fraction of particles in the solution is known. In addition, spectrum analysis o f the signal allows us to determine the characteristic transit time of the particle flow. T w o applications of this method are the m e a s u r e m e n t of molecular weights and the study o f the flow
FLOWING PARTICLES
properties of Newtonian and non-Newtonian fluids. One remaining challenge is low frequency excess noise and the possible appearance of 1/fnoise in the system. ACKNOWLEDGMENTS This work has been supported by NAVF (Norwegian Research Council for Science and the Humanities), NTNF (Royal Norwegian Council for Scientific and Industrial Research), and VISTA (a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a/s). We have greatly benefited from stimulating discussions with Professor C. P. Bean, and we thank Professor R. Brandt for making mica pores.
REFERENCES 1. Verveen, A. A., and DeFelice, L, J., Prog. Biophys. MoL Biol. 28, 189 (1974). 2. Feher, G., and Weissman, M., Proc. Natl. Acad. Sci. U.S.A. 70, 870 (1973). 3. Bean, C. P., and Golibersuch, D. C,, in "Electrical Phenomena at the Biological Membrane Level" (E. Roux, Ed.), p. 311. Elsevier, Amsterdam, 1977.
103
4. Bezrukov, S. M., Drabkin, G. M., and Sibilev, A. I., J. Colloid lnterface Sci. 113, 194 (1986). 5. Bezrukov, S. M., Pustovoit, M. A., Sibilev, A. I., and Drabkin, G. M., Physica B159, 388 (1989). 6. DeBlois, R. W,, and Bean, C. P., Rev. Sci. Instrum. 41, 909 (1970). 7. Berge, L. I., J. ColloidlnterfaceSci. 135, 283 (1990). 8. Fischer, B. E., and Spohr, R., Rev. Mod. Phys. 55, 907 (1983). 9. Maxwell, J. C., "A Treatise on Electricity and Magnetism," 3rd ed., Vol. I. Clarendon, Oxford, 1904. 10. Smythe, W. R., Phys. Fluids 7, 633 (1964). 11. DeBlois, R. W., Bean, C. P., and Wesley, R. K. A., J. Colloid Interface Sci. 61, 323 (1977). 12. Dutta, P., and Horn, P. M., Rev. Mod. Phys. 53, 497 (1981). 13. Nyquist, H., Phys. Rev. 32, 110 (1928). 14. Taylor, G. I., Proc. London Math. Soc., Ser. 2 20, 196 (1920). 15. MacDonald, D. K. C., "Noise and Fluctuations: An Introduction." Wiley, New York, 1962. 16. Goldsmith, H. L., and Mason, S. G., .1. Colloid Sci. 17, 448 (1962). 17. Segrr, G., and Silberberg, A., J. FluidMech. 14, 136 (1962). 18. Berge, L. I., J. FluidMech. 217, 349 (1990).
Join-hal q[Coltoid and lnletfiwe Science, VoL 148, No. 1, January 1992
The resistive pulse technique (Coulter principle) is widely used to count and size small particles suspended in a saline solution by letting them pass one at a time through a constricted current path (a pore). When a particle enters the pore, the momentary change in the resistance of the pore is proportional to the particle volume. As the particle concentration is increased, the particle pulses tend to overlap one another and there is a transition to noise. Even if individual particle pulses cannot be observed, information about particle size and concentration, and the dynamics of the flow, are contained in the fluctuating signal. In addition to the noise from particles that have been added to the conducting solution, there are other noise sources that limit the resolution. These are mainly the amplifier noise, the Johnson noise of the pore resistance, and the ion flow noise. The first two are approximately uniform across the span o f frequencies (white noise) and they give a uniform background in the spectral density 1 To w h o m correspondence should be addressed.
representation. The ion flow noise is similar to the flow noise of particles suspended in the electrolyte. The flow noise is confined to the low frequency part of the power spectrum, where the minimum transit time defines a characteristic upper frequency. A large literature exists on noise theory and measurements. This literature is mostly devoted to physical systems and devices with electrons as charge carriers, while the noise from ionic systems with ions as charge carriers has received less attention. A variety of interesting phenomena can be studied by noise analysis of ionic systems. A review of membrane noise is given by Verveen and DeFelice (1). Feher and Weissman (2) measured the noise from a capillary separating two aqueous solutions of beryllium sulfate and they were able to determine the chemical reaction kinetics. Later, Bean and Golibersuch (3) reported some experiments concerning well characterized macroscopic systems of single pores, macroscopic spheres, and simple ions. They obtained the number of ions in a macroscopic pore from noise measurements. Their work has later been extended for small particle sizes (4) and ions (5).
91
Iournal~fColloMand lnteffhceScience.Vol. 148,No. 1, January 1992
0021-9797/92 $3.00 Copyright© 1992by AcademicPress,Inc. All rightsof reproductionin any formreserved.
92
RAKOTOMALALA ET AL.
In this paper, we give a more comprehensive treatment of the particle flow dynamics and the effect on the power spectrum for a large range of particle sizes. We first briefly present our experimental system. Then, the various noise sources that contribute to the fluctuating signal are discussed. The power spectrum of the noise signal associated with the pressure driven flow of particles through the pore can be modeled using an expression for the particle transit time probability distribution. Our measurements are found to be in good agreement with the theoretical power spectrum, except for some low frequency noise, the origin of which has not yet been established. Also, the particle concentration and the particle size can be obtained from measurements of the fluctuations. MATERIALS AND METHODS
The central part is a Plexiglas cell with two chambers ( ~ 1 cm 3) connected by a pore (6, 7). The complete experimental setup is shown schematically in Fig. 1. The particles to be analyzed are suspended in an electrolytic solution which fills the cell. Particles flow through the pore by pressure drive. Each chamber contains a silver, silver-chloride electrode connected to a constant voltage source. When a particle enters the pore, the resistance between the electrodes increases and consequently the current decreases. The current fluctuations are analyzed using a Keithley Model 427 current amplifier with a voRage output, connected to a D A T A 6000 waveform analyzer (Data Precision Corp., USA), which is interfaced to a personal computer. The waveform analyzer digitizes a portion o f the input signal (a trace) to a chosen n u m b e r of points (~<16 384), each with a preset time duration (>~ 10 ~zs). The cell is placed on a vibration damped table in a brass box that provides shielding against electrical interference. The brass box can be temperature stabilized by circulating water from a temperature controller. The temperature of the cell is measured with a thermistor calibrated against a quartz thermometer. Journal of Colloid and lnteffh6~, Science, Vol. 148, No. 1, January 1992
FIG. 1. Schematic experimental arrangement. The experimental cell connected to a constant voltage source and a current amplifier is the heart of our system. A low-pressure regulator maintains a constant pressure in the res-
ervoir. The signal from the current amplifier is analyzed with a waveform analyzer. The pressure is measured with differential pressure sensors, and the temperature of the cell is measured with a thermistor. The instruments are interfaced to a personal computer. The cell is placed on a vibration damped table in a brass box that provides shielding against electrical interference (not shown). The electrolyte used has been 0.15 M NaC1 in distilled water. The solution was filtered through a 0.22-/zm Millipore filter before monodisperse polystyrene spheres were added (Dynospheres, D y n o Industries, Oslo, Norway). The standard deviation o f the diameter o f the spheres is about 2%. For some o f the measurements using large particles, sucrose was added to match the density o f the particles. Experiments have been carried out using two cylindrical glass capillaries with diameters 27 and 70 # m and resistive lengths 540 and 570 #m, respectively. These pores have been m a d e by drawing down lengths of capillary tubing, epoxying t h e m to thin sheets o f plastic, and then grinding until the desired length is obtained. To m a k e a correction for end effects, the resistive pore length should be used instead o f the geometrical pore length. The resistive pore length exceeds the geometrical pore length by approximately 0.8 times the pore
93
F L O W I N G PARTICLES
diameter (6). Also, a 9-gm equivalent diameter and 100-gin long mica pore has been used. The pore cross section is diamond shaped, and the pore has been made by the etched particle track process (8) by Dr. R. Brandt. The basis for the resistive pulse technique is the increase in resistance which accompanies the presence of a particle inside the pore. For long pores, the electrical pulse from a spherical particle flowing through the pore will be square-like, with a well defined height and width. The relative increase in resistance due to the presence of a sphere inside a long pore is given by Maxwell's first order approximation (6, 9) with a particle size correction factor based on the calculations by Smythe (10). The response of the instrument for spherical particles is (6, 7)
I / D2L A V = G~d3S-~D) ~-1) -1.
[1]
Here A V is the measured voltage change at the output of the current amplifier due to the presence of a sphere with diameter d inside a pore with diameter D and resistive length L. Further, I is the current through the pore (in the absence of particles), G is the current amplifier gain (volts/ampere), and S(d/D) is a size correction factor based on the calculations by Smythe. The size correction factor has been given a simple empirical form, S(d/D) = ( 1 -0.8(d/D)3) -1 ( 11 ). For very small particles, S = 1, and the response may be approximated by AV = IG(d3/D2L); i.e., the voltage increment is proportional to the particle volume divided by the pore volume.
1..5 ,
0.0
i 0
i
i
i
i
i 1.00
i 200
i 300
i 400
500
time (arbitraryunits) P3G. 2. Signals obtained in a pore 70 # m in diameter and 570 ttm long, using solutions of 15 # m diameter spheres at two concentrations ( N = 0.3 and 0.08; upper two signals), a concentrated suspension of 3-ttm diameter spheres ( N = 400), and only electrolyte (the voltages have been scaled differently). The gain is 108 for all cases.
pore 70 ttm in diameter and 570 #m long. The two upper signals were obtained with two different concentrations of 15-~tm diameter spheres, the next signal is for a concentrated suspension of 3-#m diameter spheres, and the bottom signal is the noise obtained for the pure electrolyte. We show that the number density and the minimum transit time may be recovered by Fourier analysis of the noise signal. In the following we present a discussion of the different noise sources that contribute to the measured signal.
The Measured Signal Current fluctuations due to fluctuations in the number of particles in the pore volume are amplified, and the measured signal at the output of the Keithley current amplifier is a voltage,
NOISE SOURCES
Vout = - ( / i n X G),
The resistive pulse technique is usually applied to systems where individual pulses can be detected (6, 7). When the number of particles in the pore becomes sufficiently large, this is no longer possible. Also, when the particle size becomes very small, the response from a single particle is not distinguishable from the background noise. Figure 2 shows examples of these different situations using a
where/in is the current at the input after suppressing the average current through the pore and G (volts/ampere) is the gain of the current amplifier. The relation between the measured voltage fluctuations and the actual current fluctuations is
V 2 s = G212~,
[2]
[31
Journal of Colloid and Inter/hoe Science, Vol. 148,No. 1, January 1992
94
RAKOTOMALALA
where V rm~ 2 -- (V - 12)2 is the mean square deviation of the output voltage from its average and I~ms 2 = ( I - / ) 2 is the mean square deviation of the input current from its average. The bar denotes a time average.
Amplifier Noise The amplifier noise m a y be approximated by white plus 1 / f n o i s e (12). The noise level of the Keithley current amplifier depends on the bandwidth, which varies with the gain and rise time settings. Also, the noise decreases as the input shunt capacitance is reduced. The input shunt capacitance is equal to 60 pF, and a noticeable reduction is not easily achieved with the present setup. The root mean square (rms) noise voltage generated by the amplifier with an open circuit at r o o m temperature is approximately 1.05 m V with G = 108 and rise time setting equal to 0.1 ms (noise bandwidth equal to about 5 k H z ) . This value was obtained by averaging several hundred traces (4096 points per trace). The amplifier noise level is approximately the same as the Johnson noise from a 1-M~ resistor with the same noise bandwidth at r o o m temperature (see below).
Johnson Noise When the electrolytic cell is included in the circuit, the inherent Johnson noise (13) of the pore resistance is an additional noise source which adds to the amplifier noise. The Johnson noise of the pore resistance determines the ultimate limit in resolution. When we apply a constant voltage across a resistance R, the mean square current associated with the fluctuations in resistance is measured as a m e a n square voltage, given by
V2
rms,J
= G2 4kTAf R
'
[4]
where k is the Boltzmann constant, T is the absolute temperature, and A f is the noise bandwidth. The index Y denotes Johnson noise. For uneorrelated noise sources, the mean square noise voltages add and the a m Journal of Colloid and lnte(/hce Science, VoL 148, No. 1, January 1992
ET AL,
plifier noise should preferably be smaller than the Johnson noise of the pore resistance in order to achieve a high resolution. For a 1M r pore resistance, Gms,s = 0.9 m V for G = 108 and A f = 5 kHz at r o o m temperature. Note that the current fluctuations decrease with increasing resistance when the voltage across the pore is constant. F o r small pore resistances, it is more favorable to use a constant current which gives voltage fluctuations proportional to R.
Other Noise Contributions The energy of the fluid in the pore fluctuates around its average energy. This gives rise to voltage fluctuations that m a y be expressed as
(2)
V~
T2k ( I OR] 2 -
c
OT]
(GI)2'
[5]
where c is the heat capacity and R is the pore resistance. The resistance o f the electrolyte in the pore decreases approximately 2% per degree increase in temperature. For our experimental conditions, these energy fluctuations have been small. As an example, V~s = 0.06 m V according to Eq. [ 5 ] due to energy fluctuations in a pore 27 g m in diameter and 540 # m long. The effect o f heating is assumed to be small since there is a constant flow rate through the pore. For small systems, diffusion of ions or particles m a y constitute a significant noise source. The transit time associated with longitudinal diffusion in a pore of length L and diffusion coefficient D is given by (2) L2 rdi~--~ 12D"
[6]
Here L2/12 is the m e a n square distance from the ends of the pore. Often m o r e important, transverse diffusion of ions or small particles in capillary flow m a y modify the transit time probability distribution due to mixing of particles. This becomes i m p o r t a n t when the characteristic diffusion path o f a particle
FLOWING
( D r m i n ) 1/2 is comparable to the radius of the
pore (5). For the large particles (compared to the ions) and long capillaries we have used, the characteristic time associated with longitudinal particle diffusion out of the pore is long and the corresponding frequency small. Also, mixing o f particles due to transverse diffusion has been negligible. For the ions, the characteristic diffusion path might be of the same order of magnitude as the pore radius. With a diffusion coefficient D -~ 10 -5 cm2/s, we obtain (D'rmin) 1/2 = 2.2 # m for a m i n i m u m transit time equal to 5 ms. Thus, transverse diffusion may be an important effect. Longitudinal diffusion becomes negligible for long pores. Using a 100-#m long p o r e , "/'diff ~-~ 0.8 S. This is a very long time compared to typical transit times set by the pressure difference across the pore ( ~ 5 ms) and therefore not important in the frequency regime we are considering.
95
PARTICLES
level in the absence of particles. Note that all voltages are amplified currents and that in our setup V0 is an arbitrary voltage which depends on the current suppression of the amplifier. If there is no correlation between the particles, the probability of having n particles in the pore is given by Poisson's distribution P ( n ) = Nne-U/n!,
[8]
where N is the average number of particles in the pore. Thus, the average voltage is = Vo + N A V ,
[9]
and the mean square deviation of the voltage from its average is V2ms = N ( A V ) 2.
[101
The noise from very small particles may be significant if their number is large. The above results should apply whether N is greater or less than one. Note that measurements of l~ Vo and V~mscan be divided to obtain V-N. The number N is a measure of the particle concentration and it does not depend on the velocity profile or the flow rate. Also, the above relations may be applied to the case where the ions themselves are considered as non-interacting particles. Without any ions in the pore, the current is zero and it will increase as the number of charge carriers increases. The mean square noise voltage is then given by -
THEORY
There is an excess noise to the Johnson noise that arises from fluctuations in the number o f ions in the pore. When particles have been added to the electrolyte, there is also an additional noise from these particles. For both cases, this excess noise is confined to frequencies less than the frequency corresponding to the m i n i m u m transit time through the pore, which makes it possible to separate the flow noise from the background noise for rather small noise levels. We now turn to discussing flow noise in more detail. Statistics o f Uncorrelated Particles
For a particle-containing pore, we assume that the voltage at the output of the current amplifier is given by (see Ref. (3)) V ( n ) = Vo + n A V ,
[7]
where n is the number of particles in the pore, A V is the voltage increment associated with each particle (Eq. [ 1] ), and V0 is the voltage
2 Vrms, i
= G 2 12 Nion s ,
[]1]
where the index i denotes ions, I is the average current, and Nions is the average number of ions inside the pore. This relation has been shown by Bean and Golibersuch (3) to give the correct number of ions in a macroscopic pore. The excess noise due to the fluctuations in the number of ions in the pore covers a frequency range set by the pressure drop across the pore. It comes in addition to the amplifier and Johnson noise and it is similar to the noise from particles which have been added to the electrolyte. Journal of Colloidand Inter'faceScience, VoL 148, No. 1, January 1992
96
RAKOTOMALALA
ET
AL.
Autocorrelation Function
Power Spectrum
The autocorrelation function, C ( r ) , is defined by (14)
The power spectrum S ( f ) is related to the autocorrelation function through the WienerKhintchine relation (15),
C(r) = V(t). V(t + r),
[12]
where the bar indicates a time average as before. If V(t) is taken to be the deviation of the voltage from its average value, then for r equal to zero, the autocorrelation function is simply 2 V rm~ as defined in Eq. [ 10 ] for suspended particles or as defined in Eq. [ 11 ] for the ions. The autocorrelation function is nonzero over a time span where the events are causally related in the voltage signal, determined by the particle transit time through the pore. In plug flow, there is only one transit time and the autocorrelation function is especially simple, consisting of a straight line segment. In Poiseuille flow, the transit time varies continuously between a minimum transit time (rmin) for particles entering on the pore axis and a maximum transit time (rm,x) for particles flowing close to the pore wall (7). The interval of possible transit times depends on the particle size, smaller particles can come closer to the pore wall where the fluid velocity is less, which results in a longer m a x i m u m transit time. For square particle pulses with a spread in transit times, the symmetric autocorrelation function can be modeled by a sum of straight line segments ( r > 0):
l
r ~ r ....
[13]
For times longer than the m a x i m u m transit time (rmax), the autocorrelation function is equal to zero. It measures the fraction of particles remaining in the pore at time r. The sum includes all classes of transit times r~ >~ r. Here Pg denotes the probability of having transit times in a subinterval about r~. A model for the probabilities Pi will be discussed later. Journal o f Colloid and lnte('[ace Science, Vol. 148, No. 1, January 1992
S(f) =
f° C(r)cos(27rfr)dr,
[14]
oo
where f is the frequency. This gives for square particle pulses [ sin r~fr~ \2 7rfr,--- ]l .
S(f)= V~s~Piri!k
[151
The amplitude of the power spectrum is lim S ( f )
=
2
V r m s r- .
[16]
f~0
The average transit time 7 = Zi Pir~, which depends on the particle size, approaches its maximum value 2train for very small particles in pure Poiseuille flow. The integrated power spectrum gives the mean square deviation of the voltage from its average,
f]° s ( f ) d f = V~s. co
[17]
From the above considerations we can estimate the ratio of the particle flow noise to the background noise at zero frequency, A = V ~ , s ( 1 + a2) '
[18]
w h e r e f is the sampling frequency, oz = Vrms,J V~ms,J, and V ~ in the numerator may either be the ion noise (Eq. [11]) or added particle noise (Eq. [ 10] ). The index a denotes the amplifier noise. For the ion case, the above ratio may be expressed as
RI27r f A - 4kTNion~ Af ( 1 + oz2) "
[19]
TO obtain a large ratio A, the value of Vr.... should not be much larger than Vrm~,jand a sampling frequency f two times larger than the frequency bandwidth o f the signal 2xf is desirable. In the following, the normalized power spectrum Su(fu) = S ( f ) / V ~fi, where
97
FLOWING PARTICLES
fN = frmi~, i.e., the amplitude of the power spectrum and the frequency corresponding to the minimum transit time are both normalized to one. Flow Characteristics Particles are driven through the pore by pressure drive which sets up a Poiseuille flow. The probability for having transit times in an interval about ~; is . I~'i+l / 2
Pi =
P('r)dz,
[20]
~ri--l]2
where the probability density may be expressed in the form (7)
P(~-) = c ~ × cl+-~c2x 3 1-~x
a
.
[211
Here C~ is a normalization constant and v f is the electrophoretic particle velocity inside the pore (for charged particles in an electric field). For a pore diameter D and a particle diameter d, Cl = 1 - ( 2 / 3 ) ( d / D ) 2 and c2 = 23.36(1 - C l ) . The m a x i m u m fluid velocity of the parabolic velocity profile is Vm = D2&P/16~L, where the pressure drop across the pore length L is &P, and n is the fluid viscosity. The mini m u m particle transit time is "/'min ~-~ " r 0 / ( C l + V d V m ) , where V~ is the algebraic sum o f v f and v~ (the electro-osmotic fluid velocity in a charged pore). The m i n i m u m transit time for a particle of negligible size in pure Poiseuille flow is r0 = L / V m . The relative radial position is x = 2 r / D , where r is the radial position. The velocity of neutrally buoyant spheres in Poiseuille flow is smaller than the fluid velocity at the center of the sphere (16). This effect diminishes with decreasing particle size and the correction factor (the last term in Eq. [ 21 ]) approaches one. For a particle diameter to pore diameter ratio equal to 0.26, the correction factor varies between 0.81 and 1.04, while for a 10 times smaller particle di-
ameter, the correction factor varies only between 1 and 1.01. If electrophoretic effects are negligible for small particle sizes, then the probability density simplifies to P ( r ) ~ ~-3. For large particles in long pores, the probabilities P~ are not directly deducible from the fluid velocity profile, due to radial migration (motion transverse to streamlines) ( 7, 17, 18). This is illustrated in Fig. 3, which shows a measured (filled circles) particle transit time probability distribution for 7.1-urn diameter spheres transported in Poiseuille flow through a pore 27 t~m in diameter and 540 #m long. The dotted curve is the theoretical probability distribution based on Eq. [ 21 ] and the solid curve is a calculated distribution which includes the effect of radial migration (see Ref. (7)). For large particle sizes, radial migration plays an important role, but the effect is strongly dependent on particle size and it diminishes rapidly when the particles become small. Figure 4 shows four theoretical power spectra illustrating the effect of particle size, radial migration, and electro-osmosis on the shape of the power spectrum. The frequency corre-
0.4
i
i
i
1.5
2.0
2.5
0.2
0.1
0.0 1.0
-"
3.0
FIG. 3. Measured (filled circles) and two theoretical particle transit time distributions for 7.1-1~m diameter spheres flowing through a pore 27 ~m in diameter and 540 g m long at AP = 1400 Pa (rm~ = 4.7 m s ) . The dotted curve is based on Eq. [ 2 t ] assuming that the particles follow streamlines, while the solid curve is a fit to the measurem e n t including radial migrMjon (see Ref. ( 7 ) ) . The theoretical curves are histograms, where points have been connected with solid lines. ~i' Journal of Colloid and Inter'/hoe Science, Vol. 148, No. I, January 1992
98
RAKOTOMALALA 1.0
AL.
In conclusion, Fig. 4 includes the case of very small particles in Poiseuille flow for which / 7 " m i n ~-~ 2 , the case of plug flow for which ~/rmin ~ 1, and two intermediate cases for large particles with and without radial migration. It is evident that these different conditions have a measurable effect on the resulting power spectrum.
0.8 0.6
2~ ~z 0.4
0.2 0.0 0.0
ET
0.2
0.4
0.6
0.8
f,
i .0
RESULTS AND DISCUSSION
Particle Concentration
FIG. 4. The effectof particle size, radial migration, and electro-osmosis on the shape of the theoretical power When the particle size is large enough, the spectrum (see the text). pulse height A V and the voltage/Io in the absence o f particles can be measured directly. sponding to the m i n i m u m transit time and This allows us to test the theoretical predicthe amplitude of the power spectrum have tions o f the previous section. Measurements both been normalized to one. Curves 1 and 3 have been performed with 15-#m diameter illustrate the effect of particle size on the power polystyrene spheres and a pore 70 # m in dispectrum and they have been calculated for ameter and 570 t~m long at room temperature. particle to pore diameter ratios equal to 0.001 We used a solution with a well defined particle (curve 1) and 0.26 (corresponding to 7. l-~tm concentration, corresponding to a mean diameter spheres flowing through a 27-#m di- number of particles per pore volume N = 0.27. ameter pore). The smaller particles can come The conducting solution was 0.15 M NaC1 closer to the wall where the fluid velocity is with 10 wt% sucrose added to match the denless, which results in a larger spread in transit sity of the spheres ( 1.04 g/cm3). This a m o u n t times and a different transit time probability of sucrose increases the viscosity of the soludistribution. The ratio ~/~-mi. approaches 2 tion by a factor 1.33. Using a low particle confor sufficiently small particle sizes and it equals centration enables individual particles to be 1.99 for curve 1 and 1.45 for curve 3. Curve easily detected. The background noise was 4 corresponds to the case where the electro- only a few percent of the pulse height (see osmotic fluid velocity (uniform across the pore Fig. 2). One can determine N by measuring A V, V0, cross-section) is 100 times larger than the maximum Poiseuille fluid velocity at the cen- tv, and using Eq. [ 9], or by measuring V~ms ter of the pore, i.e., for Ve/Vm = 100 and vf and AV and using Eq. [10]. Both methods = 0, a case which becomes important for small give a n u m b e r N -- 0.26 in good agreement pressures and small charged pores. When the with the known value of the concentration. electro-osmotic velocity increases compared to The value of N can also be determined indePoiseuille flow, the spread in transit times is pendently from the slope K o f the measured reduced until eventually all particles flow with particle rate (the number of particles flowing the same velocity in pure electro-osmotic plug through the pore per second) versus the presflow such that ~ /Tmin '~" 1. In Poiseuille flow, sure difference across the pore ( 11 ): radial migration may effect the transit time (,)2 N=32nK~ . [22] probability distribution and therefore also the power spectrum. Curve 2 includes the effect of radial migration and it is based on the mea- The best fit to a straight line passing through sured probability distribution shown in Fig. 3. the origin (negligible electrokinetic effects) Journal q[Col[oid and Inteff?lce Science. Vol. 148, No. 1, January 1992
FLOWING
gives a slope K = 0.108 (s Pa) -1 and N = 0.27, which again is in good agreement with the k n o w n particle concentration of the solution. W h e n the particle size is small, individual pulses cannot be distinguished from the background noise. Yet the knowledge of the volu m e fraction of particles in the pore, F = Nv/ v, where v and v are the particle and pore volume, allows the determination of the particle size and concentration. We first show experimentally that Eq. [ I0 ] is valid. Measurements have been performed with l-tzm diameter polystyrene spheres and a pore 27 # m in diameter and 540 ~zm long. We used solutions with particle concentrations corresponding to a m e a n n u m b e r of particles N in the pore volu m e ranging from 30 to 235; this corresponds to volume fractions ranging from 5 × 10 -5 to 40 × 10 -5 Figure 5 shows the measured V~ms as a function of N for two values of the cell voltage. Each fitted straight line corresponds to one cell voltage. The slope o f the straight line gives the diameter of the particle. In all cases, the value deduced from the slope is consistent with the k n o w n particle diameter. Figure 6 shows the measured Vr~s as a function of the pore voltage. Each fitted straight line corresponds to one particle concentration. The two series o f measurements for the highest particle concentrations show voltage fluctuations that are su-
0.020
t
~
i
99
PARTICLES 150
>~
i
i
i
5
10 pore voltage (V)
15
50
0
0
F i G . 6. M e a s u r e d V~ms(filled circles) as a f u n c t i o n o f the cell v o l t a g e a n d fitted line for t h e s y s t e m d e s c r i b e d in t h e p r e v i o u s figure. E a c h line c o r r e s p o n d s to o n e m e a n
number N, ranging from 30 (lower line) to 235 (upper line).
perlinear at a high applied voltage. This is probably due to increased low frequent noise for these cases that m a y possibly result from heating within the pore. The experimental power spectra have been obtained using the smallest of the two standard cell voltages, for which heating appears to be negligible. The m e a n n u m b e r o f particles N is deduced from the slope o f the straight line. The values obtained are in agreement with the expectations. It should be noted that both particle size and concentration can be obtained experimentally when the volume fraction F of particles is known. Eq. [ 10 ] can be written
2L-
J
20
Fd3"
[23]
The volume of the particle is deduced from the m e a s u r e m e n t of the fluctuations, and eventually the concentration is obtained, using the definition o f F.
0.015
o,olo 0.005
Power Spectrum and Flow Properties 0.000
50
100
150
200
250
N
FlG. 5. Measured V~s (filled circles) as a function of N and fitted line for a solution of 1-~m diameter spheres in a pore 27 ~m in diameter and 540 ~m long. The lower line is for 8.4 V cell voltageand the upper line for 15.5 V.
Figure 7 shows the first part of the measured power spectra obtained with an open circuit, a 3.3-Mft metal film resistor, and an electrolyte filled pore with resistance 1 Mg, all at r o o m temperature, at zero voltage, and amplifier gain setting G = 10 s. At low frequencies, the Journal cfColloid and lnterjace Science, Vok 148, No. 1, January 1992
100
R A K O T O M A L A L A ET AL. 2.0
1.5
2
>
~o
1.0
0.5
0.0
0
200
400
600
f (Hz) FIG. 7. Comparison of power spectra obtained with an open circuit (lower curve), a metal film resistor with R = 3.3 M~, and an electrolyte filled pore with R = 1 M~2 (upper curve). The measured power spectra correspond to current fluctuations at constant voltage and are inversely proportional to R (see Eq. [4]) for Johnson noise.
spectrum is no longer white. In the following, this deviation has been negligible compared to the particle flow noise. The Johnson noise of the pore resistance is approximately equal to that of a metal film resistor of the same resistance, and it can be obtained by subtracting the open circuit power spectrum from the power spectrum when the pore is included in the circuit and then integrating. We find that the Johnson noise of the pore resistance obtained from measured power spectra is equal to 0.9 mV, which is in good agreement with Eq. [ 4 ] when inserting the experimental noise bandwidth of 5 kHz. The experimental power spectra are analyzed in the following way: it is first checked
that the Johnson noise of the pore resistance, measured at zero voltage, is equivalent to that of a metal film resistor of the same resistance and that it is approximately white noise. When the voltage across the pore is non-zero, the noise from the particles merely adds to the Johnson and amplifier noise. The latter are subtracted from the experimental power spectrum, and the resulting power spectrum (only due to particles) is fitted to the theoretical expression (Eq. [15]). Both the particle concentration and the flow properties are o f interest. In the following, the Reynolds n u m b e r Re = oRVo~/71, where o is the fluid density, is ranging from 0.05 to 5. (See detailed values of Re in Table I). Figure 8 shows a measured power spectrum and the fitted theoretical curve (Eq. [ 15 ] ) for the solution o f 15-/zm diameter spheres ( N = 0.27) using a pore 70/zm in diameter and 570 #m long. The experimental power spectrum was computed from 4096 digitized points of the signal with a sampling f r e q u e n c y f = 5 kHz and it was averaged 400 times, which corresponds to roughly 10,000 particles passing through the pore. The pressure drop was Ap = 280 Pa yielding an expected minimum transit time rmin = 4.9 ms. The averaged V ~ measured directly on the time signals was 50.8 mV, which should be approximately equal to the integrated power spectrum. In the fitting procedure, two parameters are used: V r m s and train- We find 52 m V and 4.7 ms, respectively, which is in good agreement with expectations.
TABLE I S u m m a r y of the Measurements Obtained with Different Pore and Particle Sizes D, L (,um)
d(gm)
d/D
N
F(×10 -5)
V~. (mY)
~'ml. (ms)
Re
70, 70, 9, 27,
15 3 0.31 1
0.214 0.043 0.034 0.037
0.27 460 160 30-235
20 300 40 5-40
52 22 41 6-140
4.9 2.8-14.5 3.1-10 5
3.2 1.1-5.5 0.05-0.15 1.5
570 570 100 540
Note. D, L are the pore diameter and length, d is the particle diameter, N is the average n u m b e r o f particles in the pore volume, F is the volume fraction of particles, Vrmsis the m e a n square deviation of the output voltage from its average, rm~, is the m i n i m u m transit time through the pore, a n d Re is the Reynolds n u m b e r of the pore. JourlTal q/Co[lllid and lnle([~tce Science. VoI. 148, No. 1, January 1992
FLOWING PARTICLES
101
0.4
30
0.3
0.2 10 0.1
0
0
50
100
~
,
0£100
150
200
300
f (Hz)
400
f (Hz)
FIG. 8. Power spectrum for a solution of 15-~m diameter spheres flowing through a pore 70 #m in diameter and 570 ~ m long. The dotted curve is the theoretical fit (Eq. [15]). The m i n i m u m transit time Train equals 4.7 ms ( f = 210 Hz).
FIG. 10. The same measurement and fitted curve as that shown in Fig. 9, but for frequencies above 100 Hz: Note the characteristic bump in the spectrum, which occurs close to the frequency corresponding to the minimum transit time.
W h e n the particle size becomes small and the background noise comparatively large, direct measurements of the voltage fluctuations in the signal itself does no longer yield information only about the particles inhibiting the flow. However, Fourier analysis deconvolutes the background white noise from the particle noise. When the particle diameter is reduced to 3 #m, the pulse height is comparable to the background noise and individual pulses can no longer be detected. The measured and fitted power spectrum for 3-~tm diameter spheres are shown in Figs. 9 and 10, below 150 Hz and
above 100 Hz ( A p = 280 Pa). The average number of particles in the pore is now N = 460 (based on the weight percent of particles in the solution). The expected root mean square voltage due to particle flow noise only is 21 mV, while the measured power spectrum yields a value close to 22 mV. The measured power spectrum now contains some excess low frequency noise, the origin of which is not clear. The effect of changing the pressure drop across the pore is illustrated in Figs. 11 and 12. This is a dimensionless representation of
5
1.o
4
0.8
3
I 0.6 ' z
2
0.2 I
1 0
0.4
50
t00
150
f (Hz) FIG. 9. The measured power spectrum for 3-tzm diameter spheres flowing through a pore 70 tzm in diameter and 570 ~m long. The dotted line is the theoretical fit (Eq. [15]). The m i n i m u m transit time Zr.i, equals 5.4 ms ( f = 185 Hz).
0.0 0.0
0.2
0.4
0.6
0.8
1.0
rN FIG. 11. Superposition of three measured power spectra obtained for minimum transit times equal to 2.8, 5.2, and 14,5 ms, for a solution of 3-~m diameter spheres in a pore 70 ~m in diameter and 570 u m long, at low frequencies. The theoretical power spectrum is also shown in the figure. Journal (fColh)id and Intel'[ace Science Vok 148,No. 1, January 1992
102
RAKOTOMALALA
the power spectrum for three pressure drops, corresponding to m i n i m u m transit times equal to 2.8, 5.2, and 14.5 ms. We have plotted S~v(fN) versusfv. The values of V~m~and rr~i, for each case are obtained by fitting to the theoretical power spectrum. The data collapse of the experimental curves and the agreement with the theoretical curve is good, except for low frequencies where some excess noise is measured. Experiments with 1-#m diameter spheres of concentration N ~ 160 in a pore 27 ~tm in diameter and 540 # m long produces power spectra which are nicely fitted to the theoretical expression. When the particle concentration is reduced by a factor four, we observe some low frequency excess noise for this case as well. We have also done measurements using 0.31-#m diameter spheres of concentration N 160 in a pore 9/~m in diameter and 100 # m long. The ratio of the particle diameter to the pore diameter is comparable to that of the previous cases. Figure 13 shows the dimensionless representation of the power spectrum. The m i n i m u m transit times are equal to 3.1, 6.2, and 10 ms. In all cases, the fitted values for V~s are close to the expected noise V ~ = 41 inV. The slowest experiment exhibits some excess low frequency noise, but there seems to be considerably less excess noise for these smaller particles. All the experimental results have been analyzed assuming independent particles. However, further work has to 0.25 0.20 0.15 ~z 0.10 0.05 0.00 0.0
0.5
1.0
1.5
2.0
fN PqG. 12. T h e s a m e m e a s u r e m e n t s a n d t h e o r e t i c a l c u r v e as t h o s e s h o w n in Fig. 11, b u t at h i g h frequencies.
Journal ojColh)id and Mterface Scie~tce, Vol. 148, No. 1, January 1992
ET AL. 1.5
~z~z 1.0
0.5
0.¢ 0.0
0.2
0.4
0.6
' " 0.8
1.0
fN FIG. 13. S u p e r p o s i t i o n o f t h r e e m e a s u r e d p o w e r s p e c t r a
obtained for minimum transit times equal to 3.1, 6.2, and 10 ms, for a solution of 0.31-urn diameter spheres in a pore 9 #m in diameterand 100 ~m long, at low frequencies. The theoretical powerspectrum is also shown in the figure.
be done in order to study the effect o f interactions between particles on the power spectrum. SUMMARY
We have studied the current fluctuations due to the presence of particles in a current carrying pore. Table I summarizes the experimental results obtained with various combinations of pore and particle sizes and well defined particle concentrations. For most of the experiments, the background noise is small compared to the particle flow noise, due to either a large particle concentration or a large particle size. The last case in the table corresponds to the measurements shown in Figs. 5 and 6, for which the volume fraction was varied, producing signal-to-noise ratios (ratio of the measured mean square deviation of the output voltage from its average to that obtained at zero voltage) ranging from 1.2 to 600. Particle size and concentration can be deduced from fluctuation measurements, when the volume fraction of particles in the solution is known. In addition, spectrum analysis o f the signal allows us to determine the characteristic transit time of the particle flow. T w o applications of this method are the m e a s u r e m e n t of molecular weights and the study o f the flow
FLOWING PARTICLES
properties of Newtonian and non-Newtonian fluids. One remaining challenge is low frequency excess noise and the possible appearance of 1/fnoise in the system. ACKNOWLEDGMENTS This work has been supported by NAVF (Norwegian Research Council for Science and the Humanities), NTNF (Royal Norwegian Council for Scientific and Industrial Research), and VISTA (a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a/s). We have greatly benefited from stimulating discussions with Professor C. P. Bean, and we thank Professor R. Brandt for making mica pores.
REFERENCES 1. Verveen, A. A., and DeFelice, L, J., Prog. Biophys. MoL Biol. 28, 189 (1974). 2. Feher, G., and Weissman, M., Proc. Natl. Acad. Sci. U.S.A. 70, 870 (1973). 3. Bean, C. P., and Golibersuch, D. C,, in "Electrical Phenomena at the Biological Membrane Level" (E. Roux, Ed.), p. 311. Elsevier, Amsterdam, 1977.
103
4. Bezrukov, S. M., Drabkin, G. M., and Sibilev, A. I., J. Colloid lnterface Sci. 113, 194 (1986). 5. Bezrukov, S. M., Pustovoit, M. A., Sibilev, A. I., and Drabkin, G. M., Physica B159, 388 (1989). 6. DeBlois, R. W,, and Bean, C. P., Rev. Sci. Instrum. 41, 909 (1970). 7. Berge, L. I., J. ColloidlnterfaceSci. 135, 283 (1990). 8. Fischer, B. E., and Spohr, R., Rev. Mod. Phys. 55, 907 (1983). 9. Maxwell, J. C., "A Treatise on Electricity and Magnetism," 3rd ed., Vol. I. Clarendon, Oxford, 1904. 10. Smythe, W. R., Phys. Fluids 7, 633 (1964). 11. DeBlois, R. W., Bean, C. P., and Wesley, R. K. A., J. Colloid Interface Sci. 61, 323 (1977). 12. Dutta, P., and Horn, P. M., Rev. Mod. Phys. 53, 497 (1981). 13. Nyquist, H., Phys. Rev. 32, 110 (1928). 14. Taylor, G. I., Proc. London Math. Soc., Ser. 2 20, 196 (1920). 15. MacDonald, D. K. C., "Noise and Fluctuations: An Introduction." Wiley, New York, 1962. 16. Goldsmith, H. L., and Mason, S. G., .1. Colloid Sci. 17, 448 (1962). 17. Segrr, G., and Silberberg, A., J. FluidMech. 14, 136 (1962). 18. Berge, L. I., J. FluidMech. 217, 349 (1990).
Join-hal q[Coltoid and lnletfiwe Science, VoL 148, No. 1, January 1992