The THEORY qf ELECTROKINETIC PHENOMENA WILLIAM HORWITZ* University of Minnesota, Minneapolis, Minnesota
Electrokinetics, the subject dealing with the relationships of the electromotive force and the mechanical motion at an interface of a solid and a liquid, has acquired a prominen1 place in the literature of physiology as well as colloidal and biological chemistry. The practice1 application of the technics of electrokinetics, especially i n physiology, has g r m sleadily, but the theory has remained rather dormant. The electrokinetic potential at a n interface is due to the formation of a n electric double layer which arises from the unepunl distribution of the potential determining ions which in turn is based upon lattice forces. The condenser-like theory of the structure of the double layer of Helmholtz has been abandoned i n fawor of the diffuse double layer of Gouy or the combination of the latter two as presented by Stern. The theory of Debye and Huckel i s as important in the study of the structure of the double layer as in the theory of electrolytes. Newer developments that are likely to influence this field are the a p p l k t i o n of statistical mechanics to the double layer and studies on the hydration of ions.
under the influence of an external electromotive force (designated by such various terms as electroosmosis, endosmosis, electroendosmosis, and so forth). (2) The movement of particles under the influence of an external electromotive force (cataphoresis, electrophoresis, and so forth). (3) The peneration of an electromotive force bv the s t r e k i n g 2 a liquid through a capillary (streaming potential). (4) The generation of an electromotive force by the fall of particles through a liquid (sedimentation potential). Before defining these phenomena more precisely, it will be well to consider them descriptively.
(I) The first observations of electroosmosis and electrophoresis were the results of attempts to repeat or modify the experiments of Nicholson and Carlisle (1800) on the decomposition of water by an electric current. Reuss (1808) placed some powdered quartz a t the bottom of a HISTORICAL
C
OLLOID chemistry concerns itself with the study
of matter in a state of subdivision larger than the ordinary molecule but h e r than the particles of a fine suspension: between material in true solntion and matter in mass. Colloidal properties are those of averylarge surface concentrated in a very small volume. The forces playing the most important rble in this realm are essentially kinetic and electrical in nature. The actions of a colloidal particle as a particle are kinetic; its actions as a surface are electrical. Kinetic properties of colloids are Brownian movement, diffusion, osmotic pressure, dialysis, and the like. Electrical properties ultimately include adsorption, interfacial tension, stability, mutual precipitation, flocculation, adhesion, and the behavior of colloidal systems under the iduence of electromotive forces. The purpose of the following discussion is to study the behavior and influence of the internally present potential difference-electrokinetic potential-of colloidal systems. Some examples of the type of phenomena under consideration are: (1) The streaming of liquids through a capillary
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U-tube and found that on the application of an electric current, water rose about nine inches in the arm containing the negative electrode. The quartz powder had acted as a diaphragm with a great many small capillaries and thus showed the phenomenon of electroosmosis. In a second experiment, Reuss used a block
of clay covered with sand into which were sunk two glass tubes containing electrodes. With the current on, he observed no transfer of water, but rather clay particles diffusing to the positive poles. This move-
it; it is, however, p,roportional to the pressure." In this case also, addibou of foreign material influenced the phenomenon; alcohol raised the potential while salts lowered it. In addition, it was shown that practically every substance from to paper was charged when in contact with a liquid, especially an aqueous media, and in electroosmosis the water could travel in either direction, depending upon the nature of the solid used. Quincke attempted to explain these observations by assuming that the glass or capillary wall became negatively charged and that the water next to i t became oppositely charged. Between the two surfaces will exist a difference of potential determined by the constitution of the surfaces involved. Since the velocity of the water or the particle depends upon the potential, measurements of the velocity give something that may be related to the chemical composition of the components of the adjacent molecules of the interface. The reciprocal phenomenon of electrophoresis-sedimentation potential-the production of an electromotive force by the mechanical movement of particleswas finallyobserved by Dorn (1878) who allowed glass beads or sand to fall freely between platinum electrodes in a rotating tube. The particles, he found, were negative to the liquid.
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diaphragm was proportional to the current, but was independent of the area or thickness of the plate. The converse effect of electroosmosisthe streaming of a liquid under pressure through a diaphragm producing an electromotive f o r c e w a s first demonstrated by Quincke (1850) in the apparatus of Figure 3. He proved, "The electromotive force produced by the streaming of water under a given pressure through a clay plate is independent of the size and thickness of the diaphragm, of the amount of water forced through
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ment of particles demonstrated electrophoresis. The simple apparatus is shown in the first two figures. It was shown by Napier (1846) that acids and salts diminished the transport of water by electroosmosis. The first quantitative experimentation, however, was done by Wiedemann (1852) whose observations proved that the amount of liquid transported through a porous
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CITARACTERISTICS EmcmoKINETIC SYSTEMS
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FIGURE4.-AnA poteutial difference exists a t OF the boundary of two phases. At C H A no E s I N A a solid-liquid interface this potential difference was originally assumed to lie in a rigid electric double layer much like that assigned to a flat plate condenser as shown in Figure 4. The double layer was treated mathematically as a condenser using the conventional formulas derived in electrostatics. The potential across the double layer was called the electrokinetic potential, or, more simply, the zeta (t) potential. The previously described phenomena-electroosmosis, electrophoresis, streaming potential, and sedimentation poteutial-which can be explained by means of the double layer and which measure the f-potential across it by means of formulas that will be subsequently developedare called electrokinetic phenomena. The above terms can now be defined more precisely. The characteristic feature of electrokmetic phenomena is that one phase is displaced mechanically along the boundary of another phase accompanied by a coexisting electromotive force. One of the phases must be a liquid whiletheotberisgenerallyasolid. Four cases can arise (2).
(A) An external electromotive force produces motion: (1) Of a liquid against a solid-dectroosmosis. (2) Of solid particles against a liquid-electrophoresis. (B) An external motion produces an electromotive force : (3) The liquid is moved relative to the solid surface -streaming potential. (4) The solid body is moved relative to the liquidsedimentation potential. The classification can also be made on the basis of the moving phase. The liquid moves in electroosmosis and streaming potential while the solid moves in electrophoresis and sedimentation potential. These schemes show clearly the interrelationships of these phenomena. MATHEMATICAL
TREATMENT OF PHENOMENA
ELECTROKINETIC
Electroosmosis.-Every interface is the seat of an electric field which may arise from the difference in "solubilities" of the electrons in the two phases or from adsorption phenomena. In the case of metal interfaces, the potential developed is used to measure temperature differences in the ordinary thermocouple. At solid-liquid interfaces, this field is easily demonstrated by the application of a potential parallel to the interface. There occurs a sizable displacement of the liquid relative to the solid wall, providing that the surface of the solid is sufficiently large in relation to the volume of the liquid. This occurs in a system of capillaries. This displacement of the liquid (in most cases, water) is not due to mere transference since one hundred grams of water can be moved in the absence of electrolytes by 0.001 Faraday (3). It has been mentioned that Quincke suggested the electric double layer as a means of explaining electrokinetic phenomena, and, in particular, electroosmosis. If the positive layer is in the liquid and the negative layer is firmly attached to the wall of the capillary, a potential gradient parallel to the wall will displace the positively charged layer along the rigid negative layer dragging the rest of the liquid along with i t by the force of friction as is shown diagrammatically in Figure 5. Sir W. Thomson, in 1860, drew an analogy between the potential across a polarized metal electrode and that across a charged condenser. This concept was extended by Varley (4), who measured the capacity of a platinum electrode in dilute acid. But it was Helmholtz (5) who originally treated the properties of the double layer of two-phase boundaries in tangential motion to one another from a mathematical point of view by combining the laws of electrostatics with the laws of hydrodynamics. The aeneral form of the equations he developed are still in i s e although modified in several minor respects. In his treatment, Helmholtz made the following assumptions (6). (1) The liquid is oppositely charged to the rigid wall,
forming, along the wall, an electric double layer. The thickness of the double layer is extremely small, but not vanishingly s o a b o u t molecular dimensions. (3) The layer of water molecules in contact with the wall is not movable (called the rigid layer). It is fixed to the wall regardless of the mechanical force impressed-there is no slip. The rest of the molecules in the liquid, near the wall and in the double layer (the strongly adsorbed layer), are movable and are subject to the ordinary laws of friction for normal liquids. Only laminar flow of liquid can occur. (4) (5) The external difference of potential is simply superimposed upon the difference of potential in the double layer itself. (6) The wall is an insulator and the contained liquid is an electrolytic conductor. If u is the charge density on the wall, 6 the mean electrical thickness of the double layer, and D the dielectric constant of the media between the two layers, (2)
then the potential across the layers, the Helmholtz theory will be (7) :
r, on the basis of
This general equation will be referred to repeatedly. Helmholtz, and later Smoluchowski, derived the electroosmotic equations by combining hydrodynamic and Poisson's equations. Pemn (8) used the definitions of electrical force and viscosity as the origin. More recently, Bikerman (9) has derived all of the electrokinetic equations on the basis of the more acceptable theory of the diffuse doublelayer. The following derivation given by Porter (10)is independent of the particular distribution of charges in the double layer. Consider the stream line flow of liquid through a narrow capillary, aa, of cross-sectional area A as in Figure 6 to which a longitudinal voltage, Vl - V2,is applied so as to be uniform across the cross-section. If there is a charge, q, anywhere in the liquid, the electrical force, dF,, acting on that charge is: dV
dF. = - p z
select a cylinder of unit length and of radius r enclosing the charge; the total normal induction over the cylinder, by Gauss's law, is (4irq)/Dwhere D is the di-
electric constant of the media in which this charge is If the experiment is conducted in a U-tube, rather immersed. The curved area of this cylinder is Zar, than in a horizontal one, a hydrostatic head is built up and the electric force outward from the curved surface that gives rise to a flow of liquid in the opposite direction is - (d V/dr). Hence: t o that caused by the electroosmotic flow. For the simple case of the ordinary capillary, Poiseuille's law (2) for the flow of liquids in tubes can be combined with = equation (6) t o give: If u is the velocity of flow after a steady state has been f = Pr2r'/2DRIi (8) reached, the electrical force moving the water is just slightly greater than the frictional force, F,, opposing where P is the pressure gradient, r the radius of the capillary, and I the length of the capillary. that motion. Or: The assumptions made in this proof are the same as F a = F, - d l $ those made by Helmholtz. The same equation is deand rived but the method is simpler, clearer, and more dF. = dF, - d2P, (3)
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The diiereutial of the second order can be neglected. The force of friction against the area, from the definition of viscosity, is, where n is the coe5cient of viscosity:
Substituting in (3) :
Eliminating q from equations (2) and (4) :
From Ohm's law, the electric current across the section A with a specific resistance, R, is given by: I = -(A/R).(dV/dx)
Therefore :
It is to be noted that this equation is independent of the particular distribution of the electric charges. If Vmand u, are the potential and velocity of a point on the tube axis and w, and 0 the corresponding values for the tube wall: If the charges in the liquid that form a part of the double layer are in close proximity to the tube wall, there will be no charge in the bulk of the liquid, that is, for g = 0, du/dr = 0 (from equation (4)). In other words, the bulk of the liquid has the same velocity. The total liquid flow, U,will consequently be 4. Therefore:
The electrokinetic potential, f , is the drop of potential across the double layer and is of the order of magnitude of ten to one hundred millivolts. The amount of liquid transported depends upon the nature of the liquid (D,R, and q ) and the current, I, but is independent of the dimensions of the tube or diaphragm.
general. In addition, the analogy to a condenser and the assumption of a fictitious electrical center of a double layer are unnecessary. Eleclrofihoresis.-The phenomena of electrophoresis can be discussed from two points of view-from the theory of ionic migration or from a standpoint similar to that used for electroosmosis. From the first point of view, coUoidal particles behave as individual ions of high molecular weight and charge (called "Gegenionen") and show a migration rate in an electric field that approximates that of the common ions, except hydrogen and hydroxyl. Size has little effect upon the rate since a large radius corresponds to a large surface with a larger charge which counteracts the effect of the increased viscosity. Colloids with a positive charxe migrate to the cathode while those with a negative charg' go to the anode. This migration occurs in pure water or in solutions of electrolytes. There are some colloids that do not move in an electric field when suspended in pure water, but a great many of these are positive in acid solution and negative in basic ones. From the second point of view, a double layer arises on the surface of the particles. An electric field will tend to displace the negatively charged particle, as in Figure 7, with respect to the positively charged liquid. The particle is more free to move than the liquid, which is retarded by its own viscosity, and consequently exhibits a migration. Considering the particles as "Gegenionen," an electric field acts upon them with a force, F., which is equal to the product of the field strength, E, and the charge, ze, where z is the number of elementary charges, e. F. = eeE
The frictional force acting upon a spherical particle, F,, is, from Stokes's law, F,
=
-6mp
where r is the radius of the particle and c its velocity. At the steady state the two forces balance, so:
But ze is the charge, p, and the potential of a sphere in a medium with a dielectric constant, D, is: Therefore equation (9) becomes:
The electrophoretic velocity is proportional to p/r. If g is directly proportional to r, c will be independeut of the size of the particle. The assumptions in this proof are: spherical form. a rigidly attached layer of water that produces friction with the main body of the liquid, no orientation of the water by the electric field of the "Gegenionen," and no disturbing fields as would arise from the presence of other ions. Helmholtz did not derive an equation for electrophoresis but suggested a method and also pointed out that the assumptions made in the consideration of electroosmosis were necessary in this case as well. Smoluchowski later derived the equation that is used a t the present time (11). It is readily seen, however, that an independent derivation of the electrophoretic equation is unnecessary since electrophoresis is the negative of electroosmosis in the first case the solid moves, while in the second the liquid moves. Many independent derivations are to be found in the literature. Porter (10) finally amves a t the equation: =
-4rnolDRj
(11)
where v is the electrophoretic velocity and j the current density. In this case the area enters into the current term instead of the flow term as in electroosmosis. Equation (11) solved for u and equation (10) are equivalent, having the same dimensions, but differ in that equation (10) has the numerical factor '/o while equation (11) has the factor l/4. Debye and Hiickel (12) in their derivation of the formula describing electrophoresis amve a t an expression for the total force of the system: k = -Am A2DE.t
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where A, and APhave the dimensions of length and, in general, are not known. At equilibrium, k = 0, so:
AI/A, is a dimensionless quantity which dependsupon
the shape of the particle under consideration. For a sphere it is rand for a cylinder it is l/,?r. Experimentally it has been found that the migration velocity does not vary with the shape of the particle (13). An examination of the assumptions used explains the difEculty. Those made by Smoluchowski were (14) : (1) The presence of the particle produces a distortion of the electric field in such a way that the electric current passes tangentially a l o n ~the surface of the sphere. (2) The double layer is so thin that the electric field can be considered to be ~arallelto the double layer in the entire range df the latter. (3) The electric field does not deform the double layer. The key to the discrepancy of the numerical factor lies in assumption (3). Henry has shown (15) that if
the thickness of the double layer is so large that the greatest part of it is so far away from the particle that the distortion of the field is negligible there, then the factor is valid. For this case he calculates that the thickness of the double layer must be about six hundred times as large as the radius of the particle. For values between one and six hundred, the constant increases from four to six. Therefore, both results are correct for limiting cases. For very thick double layers the factor may be even larger than 6 due to the formation of an asymmetric double layer (16). Streaming Potatiel.-If, instead of applying a potential to the electrodes of the electroosmotic apparatus and allowing the liquid to stream through spontaneously, the liquid is forced through the diaphragm, it is found that there exists a potential difference across the electrodes. The movable part of the double layer is being forced to travel between the two electrodes, thus setting up the potential difference. This streaming
potential is most easily calculated by the method presented by MacDougall (17). The streaming potential set up is sufficient to send an equal current in the opposite direction. The velocity, c, of the liquid in the capillary of radius r varies with the distance, x, from the axis. This variation is xivet1 by the expression (18): . . " -
AA , - .-2),,/ -,,"
P~*P
, ,A",
where P is the Pressure difference and 1 the length of the capillary. Poiseuille's equation states that the total flow of liquid, U, is: U = sr2-rZP/8nl The average value of c is: u = U/tirP= PrS/%l The velocity of the movable part of the double layer, a,can be found if (r - 6) is set equal to x in equation (13) where 6 is the thickness of the double layer. Since 6 is small in comparison to r, can be neglected and the following is obtained: cs = P6r/Zrll = 46u/r The current I from the motion of that charged layer is: u being the charge density. The equal c u m a t in the opposite direction set up by the streaming potential is given by the equation: I = sr2kS/I
where k is the specific conductance of the liquid in the capillary. Equating the two currents and solving for S, the streaming potential: -.V = -PmA/rb -",.,,. Substituting into the fundamental equation (1): S = P
