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ELECTROKINETIC EQUATIONS FOR GELS AND THE ABSOLUTE MAGNITUDE OF ELECTROKINETIC POTENTIALS J. J. BIKERMAN Research Department, The Metal Boz Company, Ltd., London, England Received December 1, 1941 I. INTRODUCTION
In an extensive series of papers (an incomplete list is given below) Kanamaru and associates (12-18) investigated the electrokinetic behavior of some swelling substances like cellulose hydrate, silk, or glass in water or cellulose nitrate in ethyl alcohol. They found that the apparent electrokinetic potential p , calculated in the usual way from measurements of electrosmosis or streaming potentials, was a function of the degree of swelling. For instance, the streaming potential produced by water flowing through a pad of compressed silk decayed with time, and the uptake of water by silk simultaneously rose.' When the swelling tendency of cellulose hydrate mas reduced by treatment with ethylene glycol, the rate of decay of its r-potential decreased as well. Kanamaru even attempted to correlate quantitatively the rate of the decay of the p-potential with that of the sorption of water vapor. Whatever may be the success of this attempt, the qualitative findings are unquestionable: the apparent {-potential decreases when the solid wall swells. The interpretation of this observation advanced by Kanamaru and collaborators has apparently not yet assumed a final shape. A different interpretation is suggested here. If, as is very generally so, the swollen substance obstructs the hydrodynamic flow of the solution more than the diffusion of ions, the electrokinetic effects must be weakened by swelling independently of the nature of the electrical double layer. In the next section it is shown that membranes of swelling substances offer a much higher resistance to liquid flow than to ionic migration. In section I11 the mathematical implication of this fact for electrokinetic measurements is discussed. Section IV considers some experimental results from this point of view. Section V reviews the absolute magnitude of {-potentials. 11. RESISTANCE OFFERED BY MEMBRANES TO LIQUIDS AND IONS
It has been known since the time of Arrhenius that ions migrate in swollen hydrogels a t approximately the same rate as in water. The literature on this problem has recently been reviewed by Taft and Malm (23) and need not be repeated here. Taft and Malm themselves found, for instance, that the conductivities of 0.1 N potassium chloride in water and in a 4.58 per cent gelatin gel were identical within 0.1 per cent. On the other hand, swollen gels block the flow of liquids almost completely. This property is utilized daily in electrolytic bridges filled with agar agar, in membranes serving for electrodialysis, etc. The magnitude of the resistance 1
The electrical conductivity of the membrane also roBe
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725
offered by membranes to liquids may be illustrated by one example (20). A cellulose sheet, 0.0047 cm. thick, passed at the pressure difference of 1 atm. 0.047 cc. of water per hour per square centimeter of the sheet. If in a pipe of 1 sq. cm. cross section the pressure gradient of 1 atm. per 0.0047 cm. could be maintained, the water flow would be 3.1012cc. per hour, Le., 6 X 10la times as large as that permeating through the membrane. An analogous membrane, according to Morton, did not retard the diffusion of small dyestuff ions which are still much larger than potassium ions, chloride ions, and similar ions usually employed in electrokinetic measurements. Other data on the water permeability of membranes may be consulted, e.g., in a paper by Keys (19). It may be concluded that swollen membranes offer hardly any obstacle to ions but represent a practically impenetrable wall for hydrodynamic liquid flow. 111. ELECTROKINETIC EQUATIONS FOR SWOLLEN MEMBRANES
The last sentence can be restated in the following form: when a gel particle is placed in the way of an electrolytic solution, the cross section available for ions remains almost unchanged, but that open to liquid is reduced by the cross section of the particle. This rule affects the fundamental electrokinetic equations. Electroosmosis The volume v of liquid moving through a capillary (or a system of capillaries) of total cross section SI in unit time is 2)
= -D3 -.YS,
47, D being the dielectric constant of the liquid, 7 its viscosity, and Y the intensity of the applied electric field within the capillary. Except in special cases, neither Y nor SI is capable of direct measurement. Therefore, following Helmholtz, the magnitude of the product Y S I is deduced from the current strength I and the conductivity K in the diaphragm2. If in a conductor the cross section of which is S p a field intensity Y is established,
I
KYSz
(2)
From equations 1 and 2 the equation
results. It is always implicitly assumed that 81 = SI hence
(4)
2 Fairbrother and Mastin (7) first pointed out the effect of aurface conductance on electroiismosis.
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J. J. BIKERMAN
But in swollen membranes Si is less than S2. When water flom through a layer of filter paper or through a pad made of silk threads, the area SI open to it is only that of pores between the paper or silk fibers, whereas S2is of the order of magnitude of the total area of the membrane. It follows that the real transport of liquid which takes place according to equation 3 is much smaller than that which would be expected were equation 5 valid. In other words, the apparent { calculated from equation 5 is SI/& times the actual { calculated from equation 3. Since swelling reduces the cross section of pores (see, e.g., reference 6), Le., SI, without much affecting S2, the ratio S1/Ss decreases when the degree of swelling rises, and { calculated from equation 5 becomes an increasingly smaller fraction of the actual 6. If, therefore, the actual {-potential is not changed by swelling or even slightly increases with it, the apparent {-potential must decrease when the membrane swells; this is the effect observed by Kanamaru. Streaming potential The same reasoning applies to streaming potentials, E. A liquid moving through a capillary (or a capillary system) of cross section SI in a pressure gradient P / L ( L being the length of the capillary) carries with it 11units of electricity per second:
Il
_.-.
4n Lq
s1
This current builds up a counter-acting electromotive force E ; the potential gradient E / L causes an electric current
I2
=
KE
-.s, L
(7)
if S2 is again the cross section of the membrane available for ions and K the specific conductivity within the membranes. In a steady state 11 = 12;hence
is less than 1, the streaming potential is less Since in swollen membranes than it would have been if the usual relation
were true, and { calculated from equation 9 is smaller than the real { in the ratio SI/&. The decrease of this ratio in the course of swelling must reduce the apparent value of I , even if the real remains unchanged or increases slightly.
Cataphoresis If the particles of a sol are non-polarizable conductors, their cataphoretic mobility is 8
Brim (5) first pointed out thst L should include the surface conductance.
ELECTROKINETIC EQUATIONS FOR QELS u1
= 2KOUO/(2KO
+
727
KI)
uo being the mobility of insulating particles under identical conditions, K I the specific conductivity of the particles, and K O that of the medium (10). If K O = K I , the mobility u1 = ~ U O .This lowering of the mobility will usually be less significant than that taking place in electroosmosis, since the cross section of the particles is in any case only a small fraction of the total cross section. The magnitude of sedimentation potentials would presumably also be lower for conducting than for insulating particles. No quantitative statement is attempted here. IV. APPLICATION TO SOME SWELLING SYSTEMS
Equations 3 and 8 should have been used in many other investigations besides those of Kanamaru. Low values of l (about 8 to 15 millivolts) obtained by Briggs‘ (5) for well-soaked filter paper in contact with water, as well as similar values (10 millivolts) for the electroosmosis of hydrochloric acid (pH 1.68) in membranes of a 10 per cent gelatin gel (26), may simply be due to an unjustified application of equations 5 and 9. It is interesting to note that Willey and Hazel (25) found higher values for the electroosmotic potential of the gelatin-acidified water boundary; they measured the movement of water in a glass cell coated with a thin layer of gelatin. In their experiments SI was presumably not very different from SS,and ( values were a t a pH of 2, about three times as high as for membranes used by Zhukov and Yurahenko. Glixelli and Stolzmann (9) determined the electroosmotic flow of diluted hydrochloric acid through gelatin gel membranes. If gels of various concentrations are compared using an invariable concentration of hydrochloric acid (Le., a t different pH values, since some hydrochloric acid is used up for the formation of gelatin chloride) or even more, using a constant pH, the apparent (-potential decreases when the gel concentration increases. For instance, 0.0005N hydrochloric acid gave in 1per cent gelatin = 5 millivolts, and 0.0006 N hydrochloric acid in 4 per cent gelatin gave = 0.7 millivolt. At pH 3.463.54the {-potential of 1 per cent gelatin was 15.9,that of 4 per cent gelatin 11.2, and that of 12 per cent gelatin 6.1 millivolts. The conductivity K of the gels was high; e.g. for 12 per cent gelatin at pH 3.5 (produced by soaking in 0.066N hydrochloric acid), K = 3 X ohm-lcm.-l High conductivity means a high S Z ;SI, on the other hand, was evidently smaller the denser the gel. The ratio &/SZ decreased when the gel concentration rose, and so did the value of the apparent 6-potential. An extreme example seems to be fcund in a paper by Jordan (1 1). He found that the potential produced by water streaming through a benzopurpurin gel was low, although the conductivity of the gel was high; the decrease of the streaming potential caused by a denser packing of the gel particles paralleled
r
r
Briggs has already noticed the decrease of the streaming potential upon the swelling of paper and has suggested a8 an explanation “the variation in the fraction of the e-potential which is extant between the fixed and movable layers of the liquid, the hydrated layer forcing the fixed-movable interface further into the liquid.”
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J. J. BIKERMAN
the reduction in water flow taking place simultaneously; the apparent I-potential was reduced almost to nothing when the gel blocked the water flow almost completely. If the real and S2 are independent of the degree of packing, the ratio “apparent I/&’’ should be constant, or apparent plotted against I/& should be a hyperbola. The curve for benzopurpurin in Fig. 7 of Jordan’s paper seems to conform to this conclusion. If migrating particles of proteins are “torn-off” pieces of gel, as distinct from molecules, the correction suggested in this paper would apply to calculations of the cataphoretic potentials of proteins. V. TEE ABSOLUTE MAGNITUDE OF THE {-POTENTIAL
The effect of the ionic permeability of membranes is a third factor reducing the apparent value of the I-potential as compared with its real value. The first effect to be recognized was that of surface conductance. It is especially important in liquids the bulk conductivity of which is low, that is, in very dilute aqueous solutions, in organic liquids, etc. It has often been observed that the apparent I-potential increased with concentration of added electrolyte, passed through a maximum, and decreased again. This “maximum of the electrokinetic potential” was explained by peculiarities of ion adsorption or by an irregular potential gradient within the double layer; but it disappears when the I-potential is calculated with due regard to surface conductance (1,2,3). Generally speaking, the true I-potential uniformly decreases when the concentration of the added electrolyte increases. The importance of surface conductance is negligible in relatively concentrated solutions of ions (say, over 0.01 N ) . In such solutions the second effect may be strong. Because of the roughness of solid surfaces? a regular movement of liquids past solids cannot start in the immediate vicinity of the solid-liquid bouddary. We may picture the mechanism of this effect thus: The flow of liquid streaming between the elevations on the solid surface is again and again broken by these elevations and is less rapid than that calculated from the hydrodynamic equations postulating an ideally smooth surface. This effect is irrelevant for ordinary flow of liquids’ as long aa the flow is slow enough. But i t is very important for electrokinetics, since a considerable part of the diffuse charge is confined to the region of hindered movement. Just those layers of the liquid which are most strongly charged and, therefore, subjected to most intensive electric forces, are prevented from acquiring the high speed which the electroiismotic theory expects, by mechanical obstacles represented by submicroscopical or microscopical “protuberances” on the solid surface. Freundlich (8),in 1909, and Smoluchowski (22), in 1914, had advocated the existence of an “immobile” layer of liquid adjacent to the solid surface, without at6 Valuable reviews of the roughnese of (chiefly metal) surfaces have recently been published by Schmaltz (21) and by Way (24). 8 A discwion of the effect of roughness on liquid flow may be found, for instance, in Modarn Dwelopmentd in Fluid Dycrmica (edited by 8. Goldstein), pp. 311, 676, Oxford University Press,London (1938).
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tempting to reconcile this view with the common standpoint of hydrodynamics. The contention expressed elsewhere (4) is that there is no “immobile layer” and no “plane of slip” between immobile and mobile liquid, but that the irregular streaming of liquid among the surface asperities retards the flow further away from the interface as if a very thin film of liquid were immovably attached to it. This effect is the more important the higher the ionic concentration of the solution used, since the “thickness of the double layer” is reduced at higher concentrations. The surface of a swollen gel is, presumably, on the average much smoother than that of a hard solid. The effect of liquid being entrapped among surface asperities would be less for gel-liquid interfaces, but in these systems the third effect lowers the apparent l-potentials (it was discussed in sections I to IV). To sum up: Under almost any circumstances7 the electrokinetic effects are weakened by surface conductance at low ionic concentrations, surface roughness a t high ionic concentrations, and gel conductance at all ionic concentrations, as well as by surface roughness on hard solid surfaces and gel conductance on soft solid surfaces. The {-potential calculated in the usual way is almost always less than the real potential difference between tbe liquid in the immediate vicinity of the interface and the liquid in bulk. In many instances the apparent 6potential is only a small fraction of the real one. The apparent {-potential is a function of the surface finish, of the permeability of membranes, etc. All these effects should be eliminated or taken into account before information can be obtained about that of the real potential difference from the behavior of the apparent {-potential. I t is even less convincing when, from measurements of the apparent potential, conclusions are drawn concerning the charge on the solid wall, the adsorption of potential-determining ions, etc. VI. SUMMARY
Since gels hinder the flow of liquids more than the migration of ions, the usual electrokinetic equations cannot be applied to systems incorporating swelling membranes or swollen particles. The {-potentials calculated in the usual way are too low in the ratio SI/&, SI being the area of membrane permeable to liquid, and 82 that permeable to ions. This effect, the surface conductance, and the roughness of hard solid surfaces reduce the magnitude of electrokinetic phenomena under almost any condition. REFEREKCES (1) BIKERMAN, J . J . : Z. physik. Chem. A171, 209 (1934). (2) BIKERMAN, J. J . : Kolloid-Z. 72, 100 (1935). (3) BIKERMAN, J. J . : Tnans. Faraday Soc. 96, 15.4 (1940). (4) BIKERMAN, J. J . : In press. (5) BRICGS, D. R . : J. Phys. Chem. 32, 641 (1928). (6) FAIRBROTHER, F., AND VARLEY, H.: J. Chem. Soc. 1927,1584. (7) FAIRBROTHER, F., AND MASTIN,H . : J. Chem. Soc. 126, 2319 (1924). (8) FREUNDLICH, H. : Kapillarchemie, 2nd edition, p. 341. Akademische Verlagsgesellschaft, Leipsig (1922).
’ Movement of droplets and bubbles requires special consideration.
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(9) GLIXELLI,s., AND STOLZMANN, z.. Roczniki Chem. 11, 690 (1931). (10) HENRY, D.C.: Proc. Roy. SOC. (London) A193, 108 (1931). (11) JORDAN, D.0.: Trans. Faraday SOC. 57, 441 (1941). K.: Kolloid-Z. TI, 351 (1936). (12)KANAMARU, (13) KANAMARU, K., KOBAYASI, T., AND SEKI, M.: Kolloid-Z. 87, 62 (1939). K., AND TAKADA, T.: z. physik. Chem. A M , 179 (1939). (14) KANAMARU, K., AND TAKADA, T.:’Kolloid-Z. 86, 86 (1939). (15) KANAMARU, (17) KANAMARU, K., AND TAKADA, T.: Kolloid-Z. 87, 68 (1939). K., AND TAKADA, T.: Kolloid-Z. 90,315(1940). (17) KANAMARU, K . , TAKADA, T., AND MAEDA,K.: Z. physik. Chem. A181, 278 (1938). (18)KANAMARU, (19) KEYS,A.: Trans. Faraday SOC.93,930(1937). (20) MORTON, T. H.: Trans. Faraday SOC.51,262 (1935). (21) SCHMALTZ, G. : Technische Oberflaechenkunde. Berlin (1936). M.: In Graetz’s Handbuch der Elektrizitdit und Magnetismus, Vol. 2, (22)SMOLUCHOWSKI, p. 393 (1914). (23) TAFT,R., AND MALM,L . E.: J. Phys. Chem. 43,499 (1939). (24) WAY,S.:Proceedings of the Conference on Friction and Surface Finish, Massachusetts Institute of Technology, 1940. (25) WILLEY,A. R.,AND HAZEL,F.: J. Phys. Chem. U ,699 (1937). A. I.: J. Applied Chem. (U. S. S. R.)9, 1739 (1936) (26) ZHUKOV,I. I., AXD YCRZHENKO,
THE SOLUBILITY OF SILVER ACETATE IN AQUEOUS SOLUTIONS OF SOME OTHER ACETATES. T H E FORMATIOS O F DIACETATO-ARGENTATE ION’ F. H. MAcDOUGALL AND MARTIN ALLEN School of Chemistry, Institute of Technology, University of Minnesota, Minneapolis, Minnesota Received February 17, 1948
Previpus investigations carried out in this laboratory have dealt with the solubility and the activity coefficient of silver acetate in the presence of an added strong electrolyte in aqueous solutions (3, 6), in water-alcohol mixtures (4),and in mixtures of water and acetone ( 5 ) . When the added electrolyte was a nitrate of a univalent metal, the observed effect could be accounted for satisfactorily on the basis of the theory of Debye and Hiickel. When the added salt was a nitrate of a bivalent metal, the theory seemed at first to be less successful. The assumption, however, that the second stage of ionization of acetates of bivalent metals is incomplete was found to be capable of explaining this apparent failure of the Debye theory; in fact, it was possible to calculate fairly consistent values for the second dissociation constants of acetates of calcium, strontium, and barium in water, in mixtures of water and ethyl alcohol, and in mixtures of water and acetone ( 5 ) . This paper was abstracted from a thesis presented by Martin Allen to the Faculty of the Graduate School of the University of Minnesota in partial fulfillment of the requirements for the degree of Master of Science, June, 1941.
