ELECTROKINETICS.
XXI. ELECTROKINETIC THEORY
STREAMING POTENTIAL AND
THE
ELECTROOSMOTIC COUNTEREFFECT^
MAX A . LAUFFERZ A N D ROSS AIKEN GORTNER Division of Agricultural Biochemistry, University of Minnesota, St. Paul, Minnesota Received January 19, 1959 INTRODUCTION
In preceding publications of this series (19, 21, 22) it was assumed that the theory of Helmholtz (18) could be applied to streaming potential studies involving organic liquids and various solids. In this paper the justification of such an assumption will be attempted. Helmholtz (18) postulated that, a t certain liquid-solid interfaces, the solid component assumes an electric charge of one sign and the liquid assumes a charge of equal magnitude but of opposite sign. These charges were pictured as being arranged in two adjacent layers, one of which was fixed to the solid and the other of which was in the body of the liquid. The potential assumed to exist between these two layers of opposite charge is known as the electrokinetic or the zeta potential. If the solid is held rigidly and the liquid is forced past it, the liquid will tend to carry its charges with it. This will cause a potential gradient to be set up in the direction of streaming, the magnitude of which will depend upon the rate a t which charges are transported by the moving liquid and upon the resistance to the flow of an electric current in the direction opposite to that of the streaming. This potential gradient is known as the streaming potential gradient. The total potential built up in the direction of streaming in this manner is the streaming potential. Helmholtz expressed his theory in mathematical form as follows:
1 Paper No. 1686, Journal Series, Minnesota Agricultural Experiment Station. This paper is greatly condensed from a thesis presented by Max A. Lauffer, Jr., t o the Graduate School of the University of Minnesota in partial fulfillment of the requirements for the degree of Doctor of Philosophy, June, 1937. The thesis, containing detailed tabular data, is on file in the Library of the University of Minnesota. 2 Present address: The Rockefeller Institute for Medical Research, Department of Animal and Plant Pathology, Princeton, S e w Jersey. 721
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MAX A. LAUFFER AND ROB8 AIKEN OORTNER b
where H is streaming potential, P is pressure of streaming, c is specific inductive capacity of the liquid, < is the electrokinetic potential, 7 is the coe5cient of viscosity, and K is the specific conductivity of the liquid. The related phenomena of electroosmosis and electrophoresis were also explained in terms of the Helmholtz double layer theory, and the equations involving zeta were derived for each. The applicability of the Helmholtz theory, as later modified by Guoy (17) and Briggs (5) and derived by Bikerman (4), to systems composed in part of aqueous solutions of electrolytes has been tested by several means. If this theory is adequate, the zeta potential values for a given interface should be identical when calculated from electrophoretic, electroosmotic, and streaming potential measurements. Saxen (Briggs (5, 7)) showed that, if zeta from streaming potential is the same as that from electroosmosis, electroosmotic velocity current
is equal t o
streaming potential pressure
a relationship which he found to hold approximately for a clay diaphragm and solutions of electrolytes. Briggs (6, 7) measured the zeta potential of protein adsorbed on quartz powder against electrolytes a t various p H values by the streaming potential method. The values obtained agree only fairly well with those obtained cataphoretically on a similar system by Abramson (1) and on an identical one by AbramsonandGrossman (3). Bull (10) determined the zeta potentials of Pyrex glass coated with egg albumin and Bacto peptone in contact with solutions of the respective proteins a t various values of pH, using streaming potential, electroosmotic, and electrophoretic measurements. The results of a large number of determinations indicate that the values of zeta calculated from the three types of data are identical. Monaghan, White, and Urban (23) found identical isoelectric points for gelatin on Pyrex glass (pH of 4.75) by streaming potential, cataphoresis, and electroosmosis. DuBois and Roberts (14) measured both streaming potentials and electroosmotic velocities through a glass slit of known dimensions. Using 10-g normal electrolytes, they found that electroosmotic and streaming potential studies gave the same value for the zeta potential. Moyer and Abramson (25) found agreement between electroosmotic and electrophoretic studies on several protein-coated surfaces, even in very dilute solutions of electrolytes, and Moyer (24) found the same relationship to hold for very dilute solutions of protein even in the absence of electrolytes. I n spite of the inconclusive nature of many of the earlier studies on this question, this more recent evidence leads to the conclusion that a t interfaces, one of whose phases is an aqueous solution of electrolytes, the value calculated for the electrokinetic potential is independent of the method of evaluating.
ELECTROKINETICS.
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723
Another method of testing the validity of the theory of Helmholtz as applied to streaming potential is to observe whether or not the streaming potential-streaming pressure ratio is a constant for a given system for all values of pressure (see equation 1). Ettisch and Zwanzig (15, 16) found an increase in this ratio for increasing values of streaming pressure. Bull (9) repeated their work and found the ratio to be constant in most cases. Even in those few cases in which Bull found the ratio to vary, the potential was a linear function of the pressure. Furthermore, Bull and Gortner (11) found the potential-pressure ratio to be constant for IO-'N sodium chloride in a cellulose diaphragm. It would seem that this second test of Helmholtz's theory yields favorable conclusions. Unfortunately, there are no data on the identity of zeta potentials calculated from cataphoretic, electroosmotic, and streaming potential measurements for systems composed of purified organic liquids and solids. However, some data on pressure-potential ratios are available. Although Martin and Gortner (22) reported that streaming potential-pressure ratios were not constants in various alcohol-cellulose systems, Lauffer and Gortner (21) have recently pointed out that the data of Martin and Gortner show that the streaming potential is a linear function of the streaming pressure. Constancy of potential-pressure ratios demands that streaming potential-pressure graphs not only be straight lines, but also that they pass through the origin of coordinates. It was suggested (21) that the lines obtained by plotting these data failed to pass through the origin because the potentials recorded represented streaming potentials increased or diminished by constant potentials originating elsewhere in the electrical circuit. Jensen and Gortner (19) found pressure-potential ratios to be constant for several acid-aluminum oxide and ester-aluminum oxide systems, and Lauffer and Gortner (21) found further that these ratios were constants in a considerable number of alcohol-aluminum oxide, organic acid-cellulose, and ester-cellulose systems. It is seen, then, that one essential requirement of the double layer theory of Helmholtz is fulfilled by organic liquid-solid systems. Bikerman (4) has recently derived an equation for streaming potential, taking into account the diffuse double layer theory of Guoy (17). His equation, somewhat simplified, is
H = 2rL(K+K,)
e t) 2
-
where K , i s the specific conductivity of the surface, u is the reciprocal of the mean thickness of the double layer, L is the radius of the capillary, and the other terms have the same meaning as before. When the capillary radius is large compared to the thickness of the double layer (d is great), this equation reduces to that of Helmholtz, as modified by Briggs (5).
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MAX A. LAUFFER AND ROSS AIKEN GORTNER
This question of capillary radius, as related to streaming potential, has recently been the subject of much discussion. The potentials derived by streaming liquids through very small pores have been studied by White, Urban, and Krick (28) and by White, Monaghan, and Urban (29). They found that the streaming potential-pressure ratio decreases with decreasing capillary diameter. This they believed to be due to increased surface conductivity in small capillaries. Bull (€9,Reichardt (27), Komagata (20), Bull and Moyer (13), and Moyer and Bull (26) have discussed the measurement of &earning potentials in small capillaries from a theoretical point of view. In this connection there appear to be three factors that must be considered. The streaming potential equation of Helmholtz and Smoluchowski is derived by considering the double layer as being made up of two parallel plates. Actually it consists of two coaxial cylinders. When the radii of the cylinders are small compared to the differences in radii, a large error results, owing to this consideration. Furthermore, in very small capillaries there is an increase in viscosity, dependent upon the radius. This effect becomes pronounced in pores smaller than cm. Finally, there is the effect of electroosmotic counter pressure, which, according to the treatment of Bull (8), is regarded as being a function of capillary size, specific conductivity, and electric moment of the double layer. It is with the consideration of this effect that the experimental portion of this paper is concerned. EXPERIMENTAL
In the pursuit of the study previously reported by the authors (21) an important innovation in experimental technique was employed. It will be remembered that the streaming potential for any system is inversely proportional to the specific conductivity of the system (see equation 1). Many of the organic liquid systems studied have very low conductivities, and therefore the streaming potentials are very high. Two experimental difficulties are encountered in such systems: (1) the accurate measurement of the very low conductivities and (2) the measurement of the high values of streaming potential. I n order to obviate these difficulties, a short circuit of greater conductivity (about 10-7 ohms-') was placed across the terminals of each of the streaming potential systems. It will be shown in the discussion that, as a natural consequence of the simple theory for streaming potential, this siniplification of experimental method should not alter the value of the electrokinetic functions calculated for the several systems. I n order to test the theory, streaming potentials were measured on both shunted and unshunted cells for several ester-cellulose systems. Ester systems were chosen for this test because their conductivities in the diaphragms are considerably lower than that of the shunt, but high enough to be measured with confidence. The conductivities of the organic acid
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systems used in the previous study were too low to be measured reproducibly, whereas the alcohol-aluminum oxide systems included in that work had conductivities of the same order of magnitude as that of the shunt, and hence were useless as a test of the theory. The details of the experimental procedure, as well as a discussion of the electrokinetic functions calculated from the data, are found in the previous publication (21). PRESENTATION AND DISCUS810N OF DATA
In a given diaphragm the rate a t which charges are carried by the streaming liquid to the receiving electrode is determined by the rate of flow of the liquid through the diaphragm. The rate of flow is proportional to pressure. Therefore the current that must be conducted from the receiving electrode to the other one a t the equilibrium potential is directly proportional to the effective streaming pressure. Since H = RZ, where H is the potential, R is the resistance, and Z is the rate of flow of electric current, H = CRP, where Cis a proportionalityfactor and Pis the streaming pressure. Therefore H/RP = C, or, where K is the conductivity between the electrodes, H K / P = C for any given diaphragm-liquid system, regardless of the value of the conductivity between the electrodes. Therefore H K / P = H'K'/P', where H is the streaming potential generated by streaming a liquid under an effective pressure P through a diaphragm of conductivity K , and H' and P' are the streaming potential and the effective streaming pressure for the same diaphragm, the electrodes of which have been shunted through a resistor whose conductivity is K' (K' is much greater than K ) . In the equation, q.d. = qHK./P K, is the specific conductivity of the liquid in the diaphragm. represent the cell constant, the equation becomes
(2)
Letting A
q.d. = 7HKA/P = qH'K'A/P' = qH'K,'/P'
Therefore, one should be entirely justified in measuring the electrokinetic function q.d. in a cell whose electrodes are shunted through a known resistance, low compared to that of the diaphragm itself. Indeed, as long as the conductivity of the system, shunt in parallel with diaphragm, is known, no restrictions other than practical ones are imposed upon the magnitude of the conductivity of the shunt that should be used. It should be emphasized that the value of the cell constant must be measured, even though a shunt is used across the terminals of the diaphragm. This does not appear to be unreasonable when one realizes that the cell constant is really a measure of the dimensions of the diaphragm, and hence of the
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MAX A. LAUFFER AND ROSS N K E N GORTNER
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ELECTROKINETICS. XXI
727
number of charges that can be carried to one electrode a t a given pressure. Therefore, in order to be able to obtain comparable results from one diaphragm to another for the same system, the cell ronstant must be measured. I n table 1 we find a comparison of the values of q.d. measured with no shunt with those of (q.d.)’ measured while using the 11-megohm shunt for a series of ester-cellulose systems. Instead of q.d. being equal to (q.d.)’ for each ester system, as our reasoning would lead us to expect, (q.d.)’ is actually greater than q.d. in every case. Abramson (2) and Bull (8) pointed out independently that, when a potential is generated across the ends of a diaphragm by streaming a liquid through it, an electroosmotic counter pressure is set up, which tends to cause liquid to flow in the direction opposite to that of the applied pressure.s In other words, there is a tendency for the effective streaming pressure to be less than the pressure imposed across the diaphragm. This effect is proportional to the potential drop across the diaphragm and hence, for a fixed applied pressure, should be greater when no shunt is placed across the ends of the diaphragm than it is when the electrodes of the same diaphragm are shunted. In other words, on the unshunted cell the applied pressure overestimates the effective streaming pressure more than it does on the shunted cell, and therefore q.d. is underestimated to a greater extent than (q,d.)’. Hence (q.d.)’ should be expected to be greater than q.d. for all systems. Following the method of Bull, this subjert may be treated mathematically for the cast. of a capillary in the following manner:
P. = P
- P,
where P, is the effective streaming pressure, P is the applied streaming pressure, and P , is the electroosmotic back pressure. From the equation for streaming potential, P, = qHK,/q.d. (3) and from the equation for electroosmosis,
P, = 8Hq.d./r2
(4)
where r is the radius of thc capillary. Combining these three equations, we get P = qHK,/q.d. 8Hq.d./rz Dividing by P , this becomes P / P , = qK,r2/8(q.d.)2 1 (5)
+
+
Bull considered the electro6smotic counter effect in relation to the error i t would cause in the measurement of the streaming potentials in systems with small capillaries, whereas Abramson, proceeding in an essentially identical manner, was concerned with an explanation for the anomalous viscosities shown by very dilute solutions of electrolytes.
728
MAX A. LAUFFER AND ROSS AIKEN GORTNER
From this equation it is apparent that, for systems in which K. and T are small and q.d. is large, the electroosmotic back pressure effect becomes appreciable, but when K. or T is large, or q.d. small, this effect vanishes. For fixed values of T and q.d. the back pressure effect becomes important with very small values of K,, but vanishes when K , is increased sufficiently. Data for several studies on the system ethyl acetate-cellulose are summarized in table 1. The first two values of q.d. for this system were measured on unshunted ethyl acetate-cellulose diaphragms of different conductivities. The other values were obtained by making measurements on the second diaphragm shunted through loo-, 50-, 11-, and 1-megohm resistors. Let us assume that the value of q.d. measured using the 11megohm resistor as a shunt is the true value for q.d. for any system, ethyl acetate-cellulose, and that all smaller values of q.d. for this system are due t o deviations resulting from electroosmotic back pressure. We shall designate this value of q.d. by the symbol (q.d.)‘. Then (q.d.)’ is the real value of the electric moment and is given by the expression
(qd.)’ = qHK,/(P - P,) remembering that P , is assumed to be negligible in the diaphragm shunted through 11 megohms. The apparent value for the electric moment is given by the equation g.d. = qHK./P
With any given diaphragm for a fixed value of P ,
Hence from the ratio of q.d., calculated for any ethyl hcetate-cellulose system, to ( q d ) ’ , the value calculated for the diaphragm shunted through 11 megohms, we can calculate the ratio ( P e / P )of the electroosmotic back pressure to the applied pressure. Values of that ratio are tabulated in table 1. The log of qKd/(q.d.)’2for each ethyl acetate-cellulose system is readily calculated from the data found in table 1. These values also rlK* appear in the table. I n figure 1 we find P,/P plotted against log .-
.
for the studies on ethyl acetate. The points represent values calculated in the manner just described, and the smooth curve is the theoretical curve of figure 2, which has been displaced along the horizontal axis. Figure 2 shows the theoretical relationship of P J P to
ELECTROKINETICS.
729
XXI
It was plotted by assigning arbitrary values to the latter expression and calculating P e / P by substituting this into equation 5. The horizontal displacement of the theoretical curve necessary to make it fit the points of S 4
t3 .2
.I 0
IO
I1
13
I2
Lo$
19
%Q K 1
FIG. 1. Diagram showing the relationship between P J P and log ---T~. (q.d.1
The
points are calculated from the data on the system ethyl acetate-cellulose, and the smooth curve is that of figure 2 displaced along the horizontal axis an amount equal to -log +/8.
1
2
Lop
6' + Log
y
3
4
FIG.2. Theoretical curve plotted from equation 5 showing the relationship ber) 8
tween PJP and log -
K"1 + log -yz . (q.d.1
figure 1 is taken to be a measure of log r2/8,a term found in the equation for the curve of figure 2 but not in that of figure 1. Log r 2 / 8 then becomes cm. = 0.274 micron. - 10.030. r is equal to 2.74 X
730
MAX A. LAUFFER ,AND ROSS AIKEN QORTNER
The evaluation of r a n d (q.d.)’, the true electric moment, may be arrived at somewhat more directly by the following m e t h ~ d According :~ to the considerations of Bull, P = P, P,. By combining this equation with equations 3 and 4 and using the ideal value of the electric moment, (qd.)’, one obtains the relationship:
+
p
=7HK2 (q.d.)’
qHKa
(q.d.)’ = -
P
+
8(q.d.)’H r2 ~
+ S(q.d.)”H Pr2
But
(q.d.)’ = q.d.
H + 8(q.d.)” r2 P ~
If q.d. is plotted against H I P , a straight line of intercept (q.d.)’ and of slope -8(q.d.)’2/r2 will be obtained. Expressed in practical units ( H in millivolts, P in cm. of mercury), io6q.d. = 10yq.d.y
- 2.01
x
10-3(q.d.)’2p
x
HIP
We find that, evaluated by this method, r is 0.277 micron and (q.d.)’ has a value of 5.72 X I n these considerations r cannot be taken to mean more than the effective mean radius of the pores in the diaphragm, if we regard the diaphragm as being a bundle of capillary tubes. Moyer and Bull (26) calculated r for a cellulose diaphragm filled with a dilute sodium chloride solution, using an entirely different method. They obtained a value for r of 0.86 micron. Their value is subject to the same limitations as is this one. It is a t least a remarkable coincidence that they are of the same order of magnitude. This method of approach is the first attempt of which the writers are cognizant to verify quantitatively by means of streaming potential the electroosmotic back pressure effect postulated by Abramson and by Bull. The fact that the data on these ester-cellulose systems may be treated successfully by the method of Bull is significant, not only because it constitutes evidence in favor of the electroosmotic back pressure effect as a reality, but also because it constitutes a new justification of the applicability of Helmholtz’s theory to electrokinetic studies involving purified organic liquids. The considerations of Bull and of Abramson are straightforward deductions from the Helmholtz theory as applied to streaming 4 This solution is an adaptation of one proposed t o the authors by Professor F. H. MaoDougall, t o whom the authors are very grateful.
ELECTROKINETICS.
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73 1
potential and to electroosmosis. They involve no assumptions not inherent in that theory. Therefore, if the Helmholtz theory is applicable to systems such as those under our consideration, the electroosmotic back pressure effect must exhibit itself in systems with conductivities as low as those encountered and with mean pore radii of the order of magnitude of that determined for a somewhat similar system by Moyer and Bull. The fact that the observations fit the theory in this case shows that a second essential requirement of the theory of Helmholtz is fulfilled by organic liquid-solid systems. If the value €or pore size just calculated is at all near the actual value in the diaphragm, it is entirely possible that these pores are below the critical range for the validity of the assumption that the double layer consists of two parallel plates, made in the derivation of the equation used in the calculation of q.d. Considerations of this subject must await more definite knowledge concerning the nature of the double layer at organic liquid-solid interfaces. SUMMARY AND CONCLUSIONS
The electroosmotic counter pressure theory has been investigated qualitatively and quantitatively. The results of this study can be rationalized with a high degree of success in terms of the counter pressure theory. This constitutes a second justification for applying the theory of Helmholtz to streaming potential studies involving organic liquids. The original justificat,ion is the fact that streaming potentials of such systems have been shown to be directly proport'ional to the pressures of streaming, as the Helmholtz equation demands. The effective mean pore size of a cellulose diaphragm containing ethyl acetate has been calculated in accordance with the bark pressure theory and found to have the valw of 0.27 micron. REFERENCES (1) ABRAMSON, H.A , : J. Am. Chem. SOC.60,391 (1928). H.A.: J. Gen. Physiol. 16, 279 (1932). (2) ABRAMSON, H.A., AND GROSSMAN, E. B . : J. Gen. Physiol. 14, 563 (1931). (3) ABRAMSON, (4) BIKERMAX, J. J.: 2 . physik. Chem. Al63, 378 (1933). (5) BRIQGS,D.R . : J. Phys. Chem. 32, 641 (1928). (6) BRIGGS,D.R . : J. Am. Chem. SOC. 60,2358 (1928). (7) BRIGGS,D.R . : Cold Spring Harbor Symposia Quant. Biol. 1, 14 (1933). (8) BULL,H.B . : Kolloid-2. 60,130 (1932). (9) BULL,H.B . : Kolloid-2. 66, 20 (1934). (10) BULL,H. B . : J. Phys. Chem. 39, 577 (1935). R . A . : J. Phys. Chem. 36,456 (1931). (11) BULL,H.B . , AND GORTNER, (12) BULL,H.B . , AND GORTNER, R . A . : Physics 2, 21 (1932). (13) BULL,H.B . , AND MOYER,L. S.: J. Phys. Chem. 40, 9 (1936). (14) DUBOIS,R . , AND ROBERTS,A. H . : J. Phys. Chem. 40,543 (1936). (15) ETTISCH,G., AND ZWANZIG,A . : 2. physik. Chem. A147, 151 (1930). A.: Z. physik. Chem. A160, 385 (1932). (16) ETTISCH,G.,AND ZWANZIG,
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MAX A. LAUFFER AND ROSS AIKEX GORTNEIt
(17) GUOY,M. D . : J. phys. radium 141 9,457 (1910). (18) HELMHOLTZ, H.:Wied. Ann. 7, 337 (1879). R. A , : J. Phys. Chem. 36,3138 (1932). (19) JENSEN,0.G.,AND GORTNER, (20)KOMAGATA, S.: Researches Electrotech. Lab. (Tokyo), No. 362 (1934). (21) LAUFFER,M. A , , AND GORTNER, R. A , : J. Phys. Chem. 42,641 (1938). (22) MARTIN, W. M., AND GORTNER, R. A.: J. Phys. Chem. 34, 1509 (1930). (23) MONAGHAN, B.,WHITE,H . L., AND URBAX,F . : J. Phys. Chem. 39,585 (1935). (24) MOYER,L. S.: J. Gen. Physiol. 4!Z, 391 (1938). L. S.,A N D ABRAMSON, H . A . : J . Gen. Physiol. 19, 727 (1936). (25) MOYER, (26) MOYER,L.S.,A N D BULL,H. B.: J . Gen. Physiol. 19,239 (1935). H.:Z.physik. Chem. Al66,433 (1933). (27) REICHARDT, (28) WHITE,H . I,,,URBAN,F.,AND KRICK,E. T.: J . Phys. Chem. 36, 120 (1932). (29) WHITE,H . L.,MOKAGHAN, B., AND UHBAX-, F . : J. Phys. Chem. 40,207 (1936).
