June. 1952
PHENOMENOLOGY OF ELECTRO-OSMOSIS AND STREAMING POTENTIAL
775
THE PHENOMENOLOGY OF ELECTRO-OSMOSIS AND STREAMING POTENTIAL1 BY PHILIP B. LORENZ Petroleum Experiment Station, Bureau of Mines, Bartlesville, Oklahoma Received October $2, 1961
All electrokinetic processes in a porous solid can be compactly described by two equations, with the four variables current,, flow, pressure and voltage. Only three parameters are necessary: electrical conductance, permeability and the newlydefined electropermeability. Electropermeability can be expressed in terms of the “r-r)otential” but this is of doubtful value i n electrokinetics. The three parameters are not completely independent, but are subject to the restriction that a certain quotient must be less than 1. The same quotient is an explicit expression for the electroviscous effect, and an analogous eleclroosmotic cmductunce efect. Electrical conductance and permeability vary somewhat, and surface conductance can have very different values, depending on the conditions of measurement.
In connection with a study of the influence of surface forces on the flow of liquids through porous media, an investigation was undertaken of electrokinetic effects which influence this flow. It was found that previous work in this field could be more readily analyzed by developing a new formulation of electrokinetic relations. This has also been useful in predicting and interpreting new results. Electroviscosity and surface conductance, as well as the standard relations for electro-osmosis and streaming potential, are treated in the same unified development. The usual approximations that result from our incomplete knowledge of the electric double layer are avoided by basing the entire treatment on established experimental laws. Basic Equations The laws to start with are as follows.2 When an electric field is applied to a liquid in capillaries, the rate of flow is proportional to the electric current or to the applied potential (Wiedemann’s first law). If instead this flow is prevented, the electro-osmotic pressure is likewise proportional to the current or potential (Wiedemann’s second law). On the other hand, when a liquid is forced through capillaries, the streaming potential or streaming current, depending on which is measured, is proportional to the pressure difference applied (Quincke’s law). I n recent times most of these classical laws have been confirmed quantitatively for a variety of solids and liquids, and for porous plugs as well as single circular ~apillaries.~-llNon-linear relations have been reported from time to time6J2-17 but (1) Presented at the Boston Meeting of the American Chemical Society, April 4, 1951. (2) M. V. Smoluohowski in Graetz “Handbuch der Elektrizitilt und des Magnetismus,” Vol. 2,p . 366, Barth, Leispig, 1921. (3) A. S. Buchanan and E. Heymann, Proc. R a g . Soc. (London). A196, 150 (1948). ( 4 ) A. S. Buohanan and E. Heymann, J . Colloid Sei.,4, 157 (1949). (5) H. B. Bull, KoEloid-Z., 66,20 (1934). (6) H. P. Dakin, F. Fairbrother and A. E. Stubbs, J . Chem. Sac., 1229 (1935). (7) K.Kanamaru and T. Takada, Kolloid-Z., 86,86 (1939). (8) K. Kanamaru, T. Takada and K. Aikawa, ibid., 83, 288, 294 (1938). 43, 641 (9) M. A. Lauffer and R. A. Gortner, THISJOURNAL, (1938). (10) H.Reichardt, Z. p h y s i k . Chem., A164, 337 (1931). (11) J. Veliaek and A. Varjicek, ibid., 8171,281 (1935). 36, 111 (1932). (12) H.B. Bull and R. A. Gortner, TEISJOURNAL, (13) G. Ettiaoh and A. Zwanzig, Z . phyaik. Chem., Al47, 151 (1930). (14) F. Fairbrother and M. Balkin, J . Chem. SOC.,389 (1931), (16) 0.Kohler, Z. physik. Chom., AX67, 113 (1931).
some have been found to be in and others have been attributed to dimensional changes in the capillaries12s18or high-voltage effects similar to the Wien effect in electrolytic conduction.6 The present discussion applies only to those systems which do obey the classical laws, together with Ohm’s law and Darcy’s law. l9 We postulate that when transport of liquid and transport of electric charge take place by simultaneous electrical and hydrodynamic processes, they combine by simple addition, as I = CiiE
v=
C2lE
+ Ci2P + C22P
(la) (,lb)
where I = electric current through capillary system (taken as positive when it passes from side 1 to side 2) ; V = volume rate of fluid flow through system (in same sense); E = electric potential difference across system (taken as positive when side 1 is positive); P = pressure difference across system (taken as positive when it is high on side 1); and the C’s are experimental coefficients. Using different coefficients we might write instead I = CiE V = CII
+ CvV + CPP
(2%) (2b)
If hydrddynamic and electrical processes did not combine additively, there should be additional terms in the equations. However, the type of combination proposed explicitly in equations 1 and 2 has been implicit in many discussions in the literature of electrokinetics. There is experimental evidence against additional terms in the work of Manegold and Solf,20 who reported that electroosmotic pressure in the presence of a n applied pressure difference is essentially identical with that obtained with a potential alone. Evaluation of Cofficients The coefficients in equations 1 and 2 are related as
cv
=
c12/c22
CI = CPl/Cll Cl = ClI(1 - clPczl/cllczP~ CP = CZ(1 - cl2czl/cl~c22~
(3a) (3b) (3c) (3d)
(16) W. hlcK. Martin and R . A. Gortner. J . Phys. Chem.. 34, 1509 (1930). (17) N. SchBnfeldt, Wiss. Verlifent. Siemens-KonzeTn, 12, 39 (1933). (18) Cf. N. A. Krylov, C. A . , 32,4039 (1938). (19) M.Muskat, “The Flow of Homogeneous Fluids through Porous Media,” McGraw-Hill Book Go., Inc., New York, N. Y., 1937, p. 71. (20) E. Manegold and K,Solf, Kolloid-Z,, 86, 273 (1931).
PHILIP B. LORENZ
776 Let us define
Q = c,,c,,/c,,cz2
(4)
Since CI and Cll (or Czand CZZ)must both have the same sign, it is immediately obvious that Q cannot exceed unity. This restriction on the relative values of the coefficients should be borne in mind in assigning theoretical values to them. To relate the coefficients of equations 1 to experiment, we remember that the standard electrokinetic formulas are relations between two of the experimental quantities I, V , E and P. Since there are four variables and only two equations, we need a third independent equation before we can arrive a t a definite relation between two of the variables. This third equation generally specifies that one of the variables is held equal to zero. For example, the relation between current and voltage has a definite meaning only under some specified condition, such as P = 0. We see that C11 is the conductance, or reciprocal resistance, when there is no pressure difference. Similarly, CZZis the gross permeability when there is no potential difference. However, other conditions than those mentioned may be specified. For example, we may also define Cll by specifying any constant pressure difference
cn
=
(M/dE)P
(5)
It is clear that the conductance, dI/dE, can have any number of values in addition to C11 and C1, and the permeability, dV/dP, any number of values in addition to Cz2 and Czl depending on experimental conditions. I n what follows, the subscript will be used to indicate a variable being held equal to zero. Streaming potential is measured when there is no net electric current, and electro-osmotic flow when there is no pressure gradient. Under these conditions we get (E/Ph
-CdCii (V/Z)P = C2dClI
(6)
(7)
The minus sign occurs in equation 6 because the potential and pressure gradients act in opposite directions, something which is not ordinarily considered. We are now in a position to introduce an additional experimental law of electrokinetics. This law, established by Saxen (cf. ref. 2), and recently checked with accuracy,21.22can be written
Vol. 56
The quantity Ce defined in equation 9 may he called the “electro-permeability.” Together with electrical conductance and hydrodynamic permeability i t is a characteristic parameter of electrokinetic formulas.
Comparison with Usual Formulas If we compare the above formulas with those ordinarily given in textbook^,^^^^^ the values of the coefficients are: (a) For a circular capillary, radius r, length L CI1
= d K / L ; Czz = ar4/87L; C, = -rar2/47L
(10)
(b) For a porous plug, area A , length L, specific permeability k , conductivity cell constant K C11
= K / K ; CZZ= kA/qL; C, = -ra/4rqK
(11)
I n these formulas 7 , K and e are, respectively, the viscosity, specific conductivity and dielectric constant of the liquid in the capillaries, and I is the electrokinetic potential. It has been s t r e s ~ e d ~that ~ - ~e ~is unknown near the interface and only the product {e has any real significance. This product has been related to the electric moment of the double layer. However, even the electric moment lacks significance since the viscosity near the interface is also unknown. Only the combination of variables re/7 5 w = 4?rCeK0is capable of evaluation by purely electrokinetic measurements. This quantity has the dimensions of electric charge multiplied by time and divided by mass, which has no such straightforward interpretation as electric moment. It would be well to refer to w simply by the noncommittal term “electrokinetic coefficient.” For a given liquid it is proportional to I and has the same functional dependence on electrolyte concentration, There is less advantage than might first appear in replacing the convenient parameter C , by a more detailed expression. The relation of C e to the actual potential a t the solid surface is of great interest in other fields besides electrokinetics but even the more-refined formulas give this relation only approximately. This is further brought out by evaluating Q (equation 4) with the aid of equations 10 and 11 in turn (a) for circular capillaries Q = .?2c2/2n2r%K (12) (b) for porous plugs Q = (rze2/16n2k7~) X L / A K (13)
We have seen that Q is necessarily smaller than 1, but it is not obvious from these expressions. A dependence of apparent I on K has been Comparing equations 6 , 7 and 8, we see that but this new requirement places a definite restricc 1 2 = C2I = c. (9) tion on the relative values of conductivity and the This relation, in a slightly different form, has been electrokinetic potential that has not previously been demonstrated theoretically by O n ~ a g e r ~as~ Ja~ recognized. general consequence of the principle of microscopic Electrokinetic Formulas reversibility for systems where there is coupling Since there are four quantities to be taken two of two different transport processes. It should be valid even if the regions of mechanical and electrical at a time, while a third is held equal to zero, there transport do not completely overlap,25 which are 24 possible permutations. However, half of would merely contribute extra terms to Cll or CZZ. these are reciprocals, so we can expect twelve -(E/P)i = (V/Z)P
(8)
(21) A. J. Rutgers and M. de Srnet, Trans. Faraday BOG, 48, 102 (1947). (22) P. W. 0. Wijga, Thesis, Utreaht, 1946. (23) L. Onsager, Phy8. Rm.,81, 406 (1931). (24) L. Onsager. ibid., 88, 2265 (1931). (25) J. J. Bikerman, THISJOURNAL, 46, 724 (1942).
(26) 8. Glasstone, “Textbook of Physical Cherniatry,” D. van Nostrand Co., h a . , New York, N. Y., 1940, pp. 878, 1198, 1199. (27) H.B. Bull and R. A. Gortner, Physicr, 9, 21 (1932). (28) E. A. Guggenheim, Tram. Faraday Soc., 86, 139 (1940). (29) L. A. Wood, J . Am. Chem. Soc., 88, 432 (1946). (30) A. J. Rutgers, Trans. Faraday SOC.,86, 69 (1940).
June, 1952
,
PHENOMENOLOGY OF ELECTRO-OSMOSIS AND STREAMING POTENTIAL
777
ured. The usual condition of measurement is zero current, equation 17. It has long been recognized that, in this case, streaming potential has a retarding effect on the flow. This is called electrokinetic blocking35or eleclroviscosiiy.36 In comparison with equation 16, the fractional decrease in permeability From equations 1 From eoustions 2 when a streaming potential develops is Q, which is therefore an exact expression for the relative electroviscous effect. Four formal interpretations of the electroviscous Permeability 1 = (16) effect have been suggested, each of which can be 1 ( V / P ) i = CZ = C B ( ~- Q ) (17) expressed in the present notation. The first is that the permeability of the capillaries decreases, either through rigidification of surface layers by electrical forces“ or increased viscosity of the liquid in the ~ a p i l l a r i e s . If ~ ~we denote the decrease in permeability by CL, V = ((A2 - C$P, and C;, = QCZ. (22) The second is that of A b r a m s ~ nwho , ~ ~calculated a correction to Poiseuille’s law, assuming a partial backward flow of liauid bv electro-osmosis. If V’ represents this backhow, f’ = Cs2P - V’, and ‘V’ ( P / E ) v = -C,/Cx (24) Flectro= [&/(I - Q)lV. (P/Z)V = -Cr/Cz osmotic Third, B u ~ !suggested ~~ making the same calcu= -Ce/Ci1C2*(l - (3) pressure lation with an electro-osmotic back-pressure. If P’ = -ce/(cilczz - C,Z) (25) represents this back-pressure, V CZZ(P- P’), Especial attention is directed to formula 25. An and P’ = QP. equa,tion for (P/I)v has always been derived from Fourth, Bull and Moyer40 pointed out that the equation 24, or vice versa,2~20~31 by use of the in- reduction of flow is given by the proportion of correct relation ( P / l ) v = (P/E)v(E/l)p, and the mechanical energy expended in driving the streamterm Ce2fails to appear in the denominator. ing current against the streaming potential. If On examination, it appears that there are three we write i for this streaming current (which is other cross relations like SaxBn’s law (equation 8) carried along the wall, and compensated by a between streaming and electro-osmotic effects current back-conducted over the whole area), V [ l - (-Ei/PV)r] = Cn(1 - Q)P. According to ( ~ / P )= E (17/E)p (28) equations 4, 19 and 22, Q = -(E/P]I(I/V)E. (I/V)E -(P/E)\’ (27) It follows that (i/V)r is equal to the ordinary (E/V)I = ( P m v (28) steaming current ( I / V ) E . The original SaxBn’s law could not be tested accuNot all the previous treatments gave correct rately in concentrated electrolyte solution, where the results. In a table of calculated values in Abramvalue of Cll in the denominator would become in- son’s paper3*there is one entry that would require creasingly larger. However, the cross relations of a negative rate of flow in very small capillaries, equations 26 and 27 would not be subject to this because he used expressions that permit Q to be disad~antage.~~ greater than 1. Elton’s detailed calculation^^^*^^ In a recent paper by Mazur and OverbeelP,34 introduce an incorrect factor of 2/3. These the eight formulas for streaming and electro-osmotic examples are cited to illustrate the value of a effects (equations 18 through 25) are given in terms simple phenomenological approach. of resistance coefficients, R, rather than conductance coefficients, C, following the usage of Onsager. Electro-osmotic Conductance Effect and Surface Conductance Let Rip. = Rzl = Re; then Re2/Rl1R22= Q, and the two notations are related as Neither of the two electrokinetic conductances given by equations 14 and 15 is equal to the ordiRiiCii = RPPCZZ = 1/(1 - Q ) nary conductance, Go,measured in the absence of all R.C. = -&/(I - &) electrokinetic effects. The difference (CIL - Co) is the so-called surface conductance. The differElectroviscous Effect ence (C11 - CJ is an electro-osmotic conductance We have seen that the value of the permeability analogous to the electroviscous effect. It has depends on the conditions under which it is meas- efect not been specifically recognized, but could be ob(31) D. Maturo, Reu. facullad qufm. ind. agr. (Uniu. nacl. Zitoral, served by comparing conductance during flow under Santa F6> Argenlina), 6 , 42 (1937). constant pressure (cf. equation 5) with conductance (32) Equations 8, 26,27 and 28, which are reminiscent of the bmilseparate electrokinetic relations. Six of these are readily derived from equations 1, making use of equation 9, the other six are correspondingly dcrived from equations 2, making use of equations 3, 4 arid 9:
*
iar “Maxwell relations” of thermodynamics, were given in the equivalent differential form by Maturo,al based on an “equilibrium state” where electro-osmosis supposedly oocurs without thermodynamically irreversible e5ects. Using equation 8 and Wiedemann’s first law, he derived equations equivalent to la and 2b. (33) P. Maaur and J. Th. G. Overbeek. Rcc. trow.. chirn., TO, 83 (19.51). (34) Maeur and Overbeek’a paper oame to the author’r attention after the completion of the prenent paper.
(35) G. H.Bishop, F. Urban and H. L. White, THISJOURNAL, S I , 137 (1931). (36) 0.A. H. Elton, Proc. Roy. SOC.(London), A194,259 (1948). (37) H.L. White, B. Monaghan and F. Urban, J . Uen. Physiol.. 18, 515 (1935). (38) H.A. Abramaon, Sbid., ill, 279 (1932). (39) H.B. Bull, KoZZoid-Z., 80, 130 (1932). (40) H.B. Bull and L. 8. Moyer, TRIEJOWRNAL, 40, 9 (1036). (41) G. A. H. Elton, Proc. Roy. SOC.(London), A l W , 581 (1949).
ARNOLDE. REIF
778 when net flow is prevented. fractional difference
Vol. 56
factor which depends on the structure of the double layer. I n general, Cu = Co(l QoF X) = Co(1 (C11 - C,)/C,l = Q (29) X)/(1 - Q F ) where QoF is the electro-osmotic Surface conductance is considered to arise from term and X represents the other two. F is an the increase of electrolyte concentration and integral depending on the viscosity and potential altered ionic mobility near the wall, and an electro- distribution assumed in the double layer. It can osmotic term, due to the motion of the liquid be evaluated explicitly from the formulas of the carrying a surplus of charges of one sign. To evalu- authors mentioned. ate surface conductance theoretically, it is necesThe relative surface conductance takes the form sary to make a detailed analysis of the electric (C11 - CO)/CO = &OF ;t x (32) double layer. This has been done for single capillaries by B i k e ~ - m a nReichardtlo~~~ ,~~ and Kan- It must be measured with a constant pressure e k ~ .All~ three ~ authors obtained fundamentally difference. If it were measured with zero flow, the same expression for the electro-osmotic term,45 it would be which reduces to the Smoluchowski expression4' (CI - CO)/CO = & d F - 1) x (33) when a number of more or less customary assumpThis quantity would have a very different value tions are made and might even be negative. Still other values of (Cll - Co)/C0 = r 2 ~ 2 / 1 6 ~ 2 6 r ~ ~ o (30) surface conductance could be measured, correwhere 6 is the average thickness of the double layer, sponding with the variety of possible conductances and KO is the ordinary specific conductivity of the previously discussed. Perhaps if these possibilities liquid. The similgrity to equation 12will be noted. had been realized, there might be fewer discrepIf we define QO = Ce2/CoCz2, equation 30 becomes ancies between surface conductance values measured a t different laboratories. (GI - CO)/CO= &o(r/86) (31) The relation between Q and Qois Comparison of equations 29 and 31 shows that the electro-osmotic term of surface conductance differs Qo = Q(1 X)/(1 - &F) from the electro-osmotic conductance effect by a It follows that Q0 < (1 X)/(1 - Q F ) < (1 (42) J. J. Bikerman, 2.physik. Chew., 8163, 378 (1932) w/(1 - F ) , which amounts to an explicit restric(43) H. Reichardt, ibid., A l 6 6 , 433 (1933). tion on relative values of s", surface conductance, (44) 9. Kaneko, J . Chem. SOC.J a p a n , 66, 600 (1935). and capillary radius. (45) Numerical errors must be corrected in Reichardt's formula 15a43 and Kaneko's formula 35. Bikerman's formula 37 omits the electroThe author is grateful to Dr. A. S. Coolidge, of osmotic term from conductance in the expression for streaming potenHarvard University, for reading the original tial, but ite presence is clearly required by formula 22 of the present manuscript and pointing out the relation of Onpaper. sager's work to this paper. (46) Reference 37, formula 43.
I
We have for the
+
+
+
+
+
+
+
-
A STUDY O F THE REACTIONS OF CARBON MONOXIDE WITH COKE' BYARNOLDE. RE IF^ Coal Research Laboratory, Carnegie Institute of Technology, Pittsburgh I S , Pennsylvania Received October 8, 1961
The reaction between carbon monoxide and a degassed high temperature coke has been invest,igated in the temperature range from 700 to 1000° and in the pressure range from 1 to 63 cm. The equations CO e (CO) and CO (CO) + COz C (1)where (CO) represents a molecule chemically bonded to the carbon surface are pro osed for the initial reactions observed under these conditions. Rate constants and temperature coefficients that described tiese equations have been evaluated. The data indicate that the tendency for carbon dioxide formed during this reaction to revert to carbon monoxide increases as the carbon dioxide-carbon monoxide equilibrium composition is approached.
+
Introduction The trend in use of liquid or gaseous fuels in preference to solid fuels has given impetus to research directed toward gasification of solid fuels.%4 The gasification of carbonaceous fuels by oxygen, carbon dioxide or steam has long been known to be retarded by carbon monoxide and hydrogen, products (1) Abstracted from a dissertation by the author, Coal Research Laboratory Fellow in the Department of Chemistry, Carnegie Institute of Technology, submitted in partial fulfillment of the requirements for the degree of Doctor of Science, June, 1950. (2) McArdle Memorial Laboratory, Medical School, University of Wisconsin, Madison, Wisconsin. (3) J. J. Morgan, "Chemistry of Coal Utilization," H. H. Lowry, Editor, John Wiley and Sons, h a . , New York, N. Y., 1945, pp. 1693-
1709. ( 4 ) B, J , C , vnn der Hoeven, %'bid', pp, 1586-1636,
+
of the gasification reactions. However, lack of agreement exists between various authorities as to the mechanism of the retardation reactions. This mechanism is usually studied by flowing carbon dbxide-carbon monoxide mixtures, O r steam-hydrogen mixtures, through a bed of heated carbonaceous fuel under controlled conditions. Another approach is to study the direct interaction Of carbon monoxide Or Of hydrogen with such materials. While the reaction between hydrogen and low-ash carbons has been carefully investigated by Barrer and Rideal,6,6 there is little information on (5) R. M. Barrer, Proc. Roy. Boc. ( L o n d o n ) , A149, 253 (1935); J . Chsm. Soc., 1256 (1936); 1261 (1936). (6) R. M. Barrer and E. K,Rideall Proc, Roy, 8001(London), AX491 231 (isar3,
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