ELECTROPHORESIS AKD THE DIFFUSE IONIC LAYER BY MELVIN MOONEY*
Introduction I n a previous publication' the author has reported some electrophoresis measurements according to which the electrophoretic mobility of an oil drop suspended in distilled water increases with the diameter of the drop. A few measurements with dilute solutions of electrolytes indicated that electrolytes decrease this variation in mobility with diameter.
FIG.I Cataphoretic mobility of Red Oil and benzyl chloride measured in distilled water.
The nature of the mobility-diameter curve in distilled water is shown in Fig. I . Xlty2 has reported that the mobility curve3 of air bubbles in distilled water passes through a maximum at a diameter of about 0.2 mm; and he suggested that, if the author's measurements had been extended t o larger diameters, there would have been found a maximum in the mobility curve for oil-drops also. The measurements plotted in Fig. I extend to diameters of nearly 0.j mm; but they indicate only an asymptotic approach of the mobility to an upper limit. * Formerly Sational Research Fellow in Physics. hlooney: Phys. Rev., 23, 396 (1924). Proc. Roy. Soc., 112A, 2 3 j (1926). 3 T h etheory that will be outlined in the present paper cannot be tested withthesedata of Alty's, for tn'o reasons; first, because his gas bubbles were not in equilibrium but were dissolving and decreasing continuously in diameter, and second, because all of his observed velocities require a correction for the electro-osmotic mobility, which was not determined.
33 2
X E L V I S MOONEY
Since the publication of the article just referred to, the effects of electrolytes on the mobility-diameter curve have been investigated in more detail; and a theory purporting to explain the experimental results has been developed. It is the purpose of the present paper to give a brief outline of the theory and to compare it with some of the experimental data. More details of the theory and the experiments will be given in a series of papers to be published elsewhere. Current Theory Up to the present time there have been published two theoretical formulas for electrophoretic mobility: the Helmholtz formulaJ4later modified by Lamb” and by Smoluchowski;6 and the Debye-Huckel formula.’ Since neither formula suggests or explains any variation in mobility with particle size, let us consider the necessary assumptions underlying these formulas. They are : A-HELMHOLTZ I . The particle is rigid and nonconducting. 2 . D, the dielectric constant and 7, the coefficient of viscosity, have the same values within the electric double-layer as in the liquid in bulk. 3. The charge distribution in the double-layer is unaffected by the impressed field or by the relative motion of the particle and the suspension medium. 4. The thickness of the double-
layer is small in comparison with the radii of curvature of the particle surface.
B-DEDYE-HUCKEL I. The particle is rigid and spherical. 2 . Identical with A z .
3. The spherical symmetry of the
charge distribution in the double-layer is unaffected by the impressed field or by the relative motion of the particle and the suspension medium. 4. The thickness of the doublelayer is large in comparisorl with the radius of the particle.
The formulas obtained on these assumptions are:
in which fie is the electrophoretic mobility, v the velocity, E the impressed field, D the dielectric constant, 7 the coefficient of viscosity of the liquid, and The difference between the assumption -44 (o the potential at the surface. and Bq accounts for the difference between the two resulting formulas. 4 Helmholtz: Ann. Physik, 7,337 (18j9); Mem. LondonPhgs. SOC.(1888); Ges. Abhandl. Kenntnis Kohle, 1, 8j5 (1922). Lamb: Brit. Ass. Adv. Sci. Repts., 495 (1887); Phil. Mag., 25, j2 ( 1 8 8 8 ) . Smoluchowski: Bull. intern. acad. sei. Cracovie, 182 (1903). Debye and Hili k c l : Physik. Z., 25, 49, 204 (1924).
’
ELECTROPHORESIS .4ND THE DIFFUSE IOSIC LhYER
333
The assumption B4 is not mentioned in the Debye-Huckel articles; and indeed no one seems to realize that such an assumption is involved. In fact, however, this condition is necessary to the validity of the Debye-Huckel analysis, the reason being that those authors neglected the electric polarization of the particle under the influence of the impressed field. I n doing so they necessarily neglected also the interaction between the polarization charges and the mobile charge of the double-layer; and this interaction would produce an appreciable effect on the mobility if an appreciable fraction of the mobile charge were situated close to the particle. Assumption A z , or Bz, concerning the constancy of D and 7, has been subjected to criticism,a and it is certainly reasonable to expect some change in the values of these “constantsJ’ due to the special forces within one or a few molecular diameters from a surface. However, if we consider the tendency of the mobile ions to diffuse away from the surface, we arrive at a picture, first drawn for us by G O U in ~ , which ~ the thickness of the double layer often far trancends the limits of these surface forces. Furthermore, present day knowledge concerning the nature of adsorbed layers leads us to reject as very improbable any surface slip such as was considered by both Helmholtz and Lamb.
Theory of Electrophoresis based on the Diffuse Layer I n the author’s attempts to improve the theory of electrophoresis, assumption hz has been retained and assumptions A3 and A4, B3 and €34 have been rejected as being much less acceptable. The rejected assumptions are replaced with a quantitative theory of the charge distribution in the outer, mobile part of the double-layer, hereafter termed the ionic atmosphere or the diffuse layer. Specifically, it is assumed: a. The particle is spherical, non-conducting, or surrounded by a nonconducting surface, and is riad-at least to the stresses involved in electrophoresis. The liquid in which the particle is suspended contains in solution a single di-ionic electrolyte. b. The ions in the diffuse layer move in accordance with established physical laws, under the influence of I , the velocity of the water; 2 , the local electric field; 3, diffusion. c. D, 7 , and also ml and m2, the absolute mobilities of the positive and negative ions, respectively, are constant up to the spherical surface enclosing the particle and any rigidly adsorbed material. As in previous analyses of electropharesis, the inertia terms in the hydrodynamic equations are neglected in comparison with the viscosity terms. The formulas obtained are consequently valid only for small velocities. I t is obvious that, on the basis of assumption b, the mean thickness of the Helmholtz double-layer will vary with the concentration and valence of the electrolytic ions, the conditions reducing to Gouy’s theory in the special Harkins: Colloid Symposium Monograph, 6,1 7 (1928). Compt. rend., 149, 654 (1909);J. Phys., 0, 457 (1910).
334
NELVIN MOONEY
simple case of an infinite sphere (plane surface) with zero impressed field. The mobility of a finite spherical particle will be shown to depend theoretically upon the ratio of its radius to the mean thickness of the diffuse layer. The system of differential equations representing my assumptions is too complicated to allow any hope for a complete solution; but I have succeeded in obtaining the first two terms of the solution in the form of a power series in x,the surface curvature:
in which x is the surface curvature, or I/r, r being the radius of the sphere; F1 is a known function of cp, and the F,J’s are undetermined functions of cp. The first term in this formula is the familiar Helmholtz mobility formula. The second term gives the first order correction for the curvature of a finite sphere. The remaining terms all involve higher powers of x and E. For purposes of comparison with experiment it is convenient to differentiate equation ( 2 ) with respect to x, thus obtaining an expression for SOthe limiting slope at zero curvature,
in which I+X In-.I-x
XI
= 4w-
I I
-
dx - 8w In ( I x 3(1+w)
+ w ) - 3(1--w) In 8W
~
(I
- w)
w2
+ w?
w = sinh-cp
2A
t = v - DkT 8nneP ,uo is the limiting (extrapolated) mobility at zero curvature, k is the Boltzmann constant, T the absolute temperature, n the number of ions of one kind per ml of solution, v the valence of the ions, and e is the electronic charge. The definitions of the other terms are the same as have already been given. The term A s does not occur in these equations if it is assumed that the surface potential, ‘p, is independent of x;but it does occur if it is assumed that it is u, the surface charge, that is independent of x. I t can be argued on very reasonable grounds that these assumptions represent the two extreme possibilities and that the actual situation lies somewhere between the two. Fortunately, X a is found to be small in comparison with the other terms in
ELECTROPHORESIS AND THE DIFFUSE IOSIC L.4YER
335
equation (3) ; and in view of the present inaccuracy in electrophoresis measurements, we do not need to concern ourselves further with the exact variation of 9 or u with x. In testing equation (3) with experimental data, the value of cp is calculated from p o by means of the Helmholtz formula. w and the A’s are thereby determined; and the right member of equation (3) is evaluated by inserting the proper values of the parameters. Experimental Procedure The electrophoretic mobilities of oil drops suspended in various solutions of electrolytes were determined, the microscopic method being used. The electrophoresis cells employed were thin-walled, cylindrical glass U-tubes immersed in a small water bath. The oil phase in the suspensions was always quite small, I per cent or less. The suspensions were formed in some cases by shaking a few drops of oil with 100 or 2 0 0 ml of water or aqueous solution in a separatory funnel; in other cases by squirting into the water or solution a fine stream of oil under high pressure. The following oils were used, either singly or combined in mixtures of the same density as the solution used:-phenyl chloride, phenyl bromide, anisol, anethol, Stanolind and Red Oil, the last two being commercial paraffin hydrocarbon oils manufactured by the Standard Oil Company. Measurements at room temperature were made without any thermostatic control. The data thus obtained were recalculated to the standard basis of zo°C on the assumption that all mobilities would merely be changed inversely as the viscosities of the water a t the two temperatures considered. The precision attained in this work was better than that in most measurements of electrophoresis, the mean error being I per cent or less under favorable conditions. Nevertheless, this precision is obviously not at all what it should be for testing a theory which predicts only the limiting slope of the mobility curve at zero curvature; that is, at infinite radius. A still greater hindrance to accuracy in determining the limiting slope was found to lie in the suspensions themselves. They were difficult to reproduce; and in the more concentrated electrolyte solutions they seldom gave smooth mobility curves. Generally many drops of the same diameter were found with widely different mobilities. Consequently, all that can be done with the experimental data is to determine very roughly the limiting slope of the mobility curve and see whether or not the theoretically predicted slope is correct in its order of magnitude. Comparison of Experimental Results with Theory I n Fig. z are shown some electrophoretic mobilities, plotted against curvature, in various concentrations of KOH. Extrapolation to zero curvature gives us PO,from which we calculate the limiting slope of the mobility curve as x approltches 0. These theoretical slopes and the values assumed for po are indicated by the straight lines in the figure. I t is seen that the theoretical
336
MELVIN hlOONEY
h I 0 1
+
M
0 '
L
,
N
0
4 d
I
io
0
O
FIG.2 Cataphoretic mobilities in cmjsecj volt/cm, measured in KOH.
F
R
1o->
0.S
16
24
CURVATURE
-
32
40
p - 1
FIG.3 Cataphoretic mobilities in cm/sec/ voltjcm, measured in KCl.
l
0
0
16 24 CURVATURE
08
32-, 40
FIG.4 Cataphoretic mobilities in cm/sec/ volt/cm, measured in KC1.
0
08
1.6 &4 32 CURVATURE
40
FIG.5 Cataphoretic mobilities in cm/sec/ volt/cm, measured in CuSOa.
ELECTROPHORESIS A S D THE DIFFUSE IONIC LAYER
33 7
slope rapidly approaches zero as the concentration increases and is in rough agreement at all concentrations with the experimental slope. Figure 3 shows a series of measurements made in KC1 solutions. The agreement between theory and experiment is satisfactory with concentrations up to IO-^ molar; but at the higher concentrations definite disagreement appears. The difference between the theoretical and experimental slopes in IO-' molar KC1 is not large; but their ratio appears to differ considerably from unity. I n the next series, Fig. 4, with HC1 as the electrolyte, the agreement between theory and experiment is still good with the dilute solutions; but with the concentrated solutions the disagreement is more pronounced than with concentrated KC1. I n Fig. j are shown the results obtained with CuS04, a salt with bivalent ions; and the agreement here seems to be good throughout the entire series. The calculations in this case are based on the assumption that the C U S O ~ is partially ionized but not hydrolyzed at all. Fig. 6 shows some cataphoretic mobilities a t 7j"C. The theory agrees with the experimental results except in the case of 0.5 X IO-^ molar CuSO4. To summarize these results, it can be said that the theoretical slope of the mo0 08 I6 26 32 40 bility-curvature curve agrees fairly well CURVATURE -r' with the experimental data for dilute FIG.6 solutions up to concentrations of I O + Cataphoretic mobilities in cm,!sec 1 volt/cm, measured at 7jOC. molar and in some cases for the more concentrated solutions. Assuming that such agreement as is obtained is not accidental, but results from the essential soundness of the theory, we have yet to explain the lack of agreement for concentrated neutral and acid solution. Early in this program of research the question was examined as to how soon equilibrium is established between the surface charge and the ionic atmosphere surrounding the drop. Suspensions of oil in distilled water, in IO-^ molar KOH and HCl and in weaker solutions of KOH, were examined when formed and also after ageing for several days; but no significant changes in electrophoretic mobilities were found. Extended ageing experiments with more concentrated solutions were attempted but failed, because of the instability of the oil suspensions. On the basis of the results in dilute solutions, it was assumed that in all cases an oil-water interface attains its equilibrium surface charge practically as soon as it is formed. However, certain other experimental results in addition to those already shown led to a re-examination of the question of equilibrium. Greater stability of the suspensions was attained by using mixtures of oils having
338
RlELYIN MOONEY
practically the same densities as the electrolytic solutions in which the oils were suspended. Fig. 7 shows some data obtained with IO-^ molar HC1. The “hot” suspensions were made by atomizing the oil in boiling hot solution. The suspension was then allowed to cool slowly-in an hour or two-to room temperature before filling the electrophoresis cell. I t is obvious that the cold suspension on the second day had changed from what it was when fresh, but had not yet reached the state of the “hot” suspension, which appears to be more stable. -4series of such experiments indicated that. in all cases in which K?
m
N
N
-
a ?
0
” 0
>c +@
-
1 0 2 0 3 0 4 0 ~
0
COLD FRESH
- m
’f
UI
c;’
=.O?
v 0
0 3 +
10 20 30 40 50 C O L D 1 DAY OLD
rl
0 10 PO 30 40 50 ‘0 10 20 30 40 50 H O T FRESH HOT 1 DAY O L D DIAMETCK -
FIG.7 Cataphoretic mobilities in ,,cm /sec/volt/cm, measured in 10- molar HC1. Hot” indicates a suspension formed in boiling hot solution and cooled slowly to room temperature. “Cold” indicates a suspension formed a t room temperature.
-0
10
20
DlA,VETER
-
30 fl
40
FIG.8 Cataphoretic mobilities in c m / sec/volt/cm, measured in IO-’ molar HC1. “Shaken” indicates suspension formed by shaking oil with the acid solution.“Byparts” indicates suspension formed by shaking oil with water and then adding the acid.
the theoretical slope of the mobility-curvature curve departed notably from the experimental curve, the suspension was not in its equilibrium state. Logically the next step in the experimental procedure would be to produce suspensions that are in equilibrium; but this seems to be very difficult to do, at least if we assume as one criterion of equilibrium that all drops of the same diameter must have the same electrophoretic mobility. Since the oil drops coalesce, extended ageing cannot be used; and no method of preparing the fresh suspensions was found which gave reproducible, smooth and stable mobility curves. This difficulty is much greater with acid than with neutral or alkaline solutions. It is illustrated in Fig. 8, which shows the different results obtained, when the suspension is formed by shaking the oil in a IO-’ molar solution of HCl, in the usual manner, and when it is formed “by parts”, by shaking the oil in distilled water and adding the acid afterwards. The latter method is the only one that gives a definite curve, but ageing and other tests show that the suspension thus formed is further removed from equilibrium than the one formed by the first method.
ELECTROPHORESIS A S D THE DIFFUSE I O S I C LAYER
339
Anomalous Effects Considerable effort was spent in trying to determine the disturbing factor that was interfering with the establishment of equilibrium. In addition to the vffect just mentioned, depending upon the method of preparing the suspension, F, variety of other anomalous effects xere thus discovered. Either ageing or temporary heating may sometimes increase and sometimes decrease the mobility. There are both permanent and temporary changes in mobility with the duration of the impressed field; and the mobility sometimes varies with the intensity of the impressed field. By coating the wall of the electrophoresiscell with the oil used in the suspension, the ratio of electrophoretic mobility of the large drops to the electro-osmotic mobility was determined. This ratio was found to vary considerably. For suspensions which were normal in other respects, it was I , which is the theoretical ratio on the basis of the Helmholtz theory. For suspensions which were abnormal in other respects, the ratio was usually greater than I , the largest observed ratio being 2 . All these and several other anomalous effects taken together can scarcely be said to establish the underlying cause of the anomalies. revertheless, they all seem to indicate or allow an explanation based on the following “slow motion” picture of the formation of the Helmholtz double-layer: I. -4s soon as an oil-water surface is formed, adsorption of ions from the water begins. In general, ions of opposite sign will not be adsorbed equally; and a net surface charge will result. 2. The surface charge attracts ions of the opposite sign and repels ions of the same sign and thereby causes the development of a diffuse ionic layer in the water near the surface. 3 . The surface forces, including that due to the surface charge, generally cause strong adsorption of water and electrolyte molecules. These adsorbed molecules sometimes form a semi-rigid crust or plastic layer between the surface and the diffuse layer. The plastic layer mag be many molecules deep and its growth may occupy seconds, minutes or even hours; but as soon as it begins to form, it interferes with the further adsorption of ions by the surface and hence tends to stabilize the surface charge before true equilibrium has been reached. In order to make this picture consistent with experimental results, it is necessary further to postulate that the plast’ic layer has a volume charge of the same sign as that of the diffuse layer, that it can be more or less disrupted by mechanical or electric forces, and that in electrokinetic processes it is more easily dislodged from a finite sphere than from a plane surface. The first two elements of this picture, the adsorbed surface charge and the diffuse ionic layer, have already been depicted in many of the recent discussions of electro-kinetic phenomena. The third element, also, the layer of adsorbed molecules, enters into some of these discussions;l0I1land its existence is further indicated by a large number of phenomena other than electrol1
Gyemant: 2. Physik, 17, 190 (1924); “Kolloidphysik,” (1925). Stern: 2. Elektrochemie, 30, 508 (1924).
3 40
MELVIN WOONEY
kinetic, such as the hydration of colloids, precipitation under conditions that permit peptization, thixotropy, plastic oil-water surfaces which the author has observed in the presence of FeC13,and the lack of adhesionL2in some cases between the particles of a suspension and a glass surface. There is thus plenty of precedent for postulating this third element in the picture of the doublelayer. The novel features suggested here are the susceptibility of this molecular or poly-molecular layer to external forces and its interference with ionic diffusion and interchange between the surface and the diffuse ionic layer. Surface Charge Preliminary to analyzing the diffuse ionic layer around a sphere, it vias necessary, among other things, to find the relationship between the electrokinetic potential and the surface charge of a plane surface. This relationship is
I n Fig. 9 are plotted on a 1ogariti.mic scale values of u calculated from PO, the limiting electrophoretic mobilities of oil drops. Surface Conductivity As a consequence of the tangential motion of the mobile ions in the doublelayer, there will be an increment in the conductivity of the liquid near a surface. The value of this increment per unit area of a plane surface, or the specific surface conductivity, K , will be expressed by an integral of the form
or
in which p1 and - p 2 are the charge densities due to the positive and negative ions, respectively; p , is the charge density outside the diffuse layer due to the positive ions; w is the velocity of the water under unit field; ml and m2 are the absolute mobilities of the positive and negative ions, respectively; and x is the distance from the surface. The term, ( p l - p z ) w , expressing the current due to the net charge carried by the water, is the only term included in Smoluchowski’s formula for surface cond~ctivity.~3The other two terms in the integrand of equation (6) represent the current due to the motion of the ions with respect to the water. l2
l3
Buzigh: Kolloid-Z., 51, r o j (1930) Smoluchowski: Physik. Z., 6,529 (r9oj).
ELECTROPHORESIS AXD THE DIFFUSE IOXIC LAYER
341
The indicated integration can be accomplished without difficulty in the case of a di-ionic electrolyte, yielding
in which p(,is now the electro-osmotic mobility. The rest of the notation is the same as in previous Fections. We can test this formula with the measurements of absolute conductivity of plane surfaces recently reported by 1IcBain and Peaker." They found ti = 4.3 X IO-^ mhos for polished glass in contact with 10-3molarIi('lat zjO(',andti = 3 . j X IO-^ mhos for a stearic acid film on distilled n'ater. For the electro-kinetic potential of glass agziinst the salt solution we can use the figure 0 . 8 millivolt, reported by Powis.'j Equation (; then gives us ti = 8 . j X ~ o - l ~ m h o which s, is only 0 . 0 2 of the experimental value. S o figure is available for the electro-kinetic potential of stearic acid in water; but from the order of magnitude of all such potentials we knoiv that in this case also equation i j ) will give a value for K many times too small. From these discrepancies we must conclude that there are other factors contributing t o the surface conductivity in addition to xhat we have considered. \\-hat they may be can only be surmised; but it is noteworthy in this connection FIG 9 that RancelinI6found that a glass surface Surface charge expressed as the numadsorbs 11 X IO-^ g 'cmy of S a C l from her of elementary charges per cm?. a 10-4 molar solution. If the surface was plane, this adsorption figure corresponds t o 1 . 1 molecular layers with the molecules normal to the surface; that is, 1.1 double ionic layers. If we assume that these malecules are oriented, some with their chlorine atoms and some with their sodium atoms towards the glass, that some of them are completely ionized, and that many of them are partially ionized in such a way that the ion on the water side of the surface is free to move parallel to the surface, but not normal to it-then we have a picture which meets all the requirements of the theory of the diffuse layer and of the experimental measurements of surface conductivity and of electro-osmotic mobility. 1
Proc. Roy. Soc., 125A,394 (1929). Z. physik Chem., 89, 91 (1915). 16 J. Chim. phys., 22, j r 8 (19zj). ls
342
MELVIN MOOKEY
The conductivity of the stearic acid-water surface presents a different problem; for in this case we know that practically all of the acid molecules are turned the same way, with the -COOH group towards the water. I n order to explain the large surface conductivity in this case, it is necessary to assume eit’her that impurities in the water are adsorbed and behave like the NaCl in the preceding picture or that the ionization of the water itself is increased in the neighborhood of the surface. The plastic adsorbed layer, postulated in section 6, has not entered into this discussion of surface conductivity, because KaC1 or KC1 are not very active in forming a plastic layer. Among the electrolytes which were used in the author’s work the most act’ive in this respect were FeC13, Z T ( N O ~ )and ~, Th(N03),, all of which are salts with poly-valent cations. Recently Briggs” has measured the relative surface conductance of cellulose fibers (filter paper) in several electrolytic solutions; and he found that the surface conductance is increased by HC1 and KCl, but very little affected by ThCla. We cannot avoid the suggestion here that the activity of Th(NOa)4 in building up a plastic adsorbed layer and the inactivity of ThC14 in promoting surface conductance are closely associated phenomena.
Discussion The present state of the theory and general understanding of electrokinetic phenomena is deplorably chaotic. Some writers are using the DebyeHuckel formula for calculating electro-kinetic potentials, while others stick to the Helmholtz formula. Those who are trying to determine experimentally which of the two formulas is “right” do not consider the difference in the derivations of the formulas or discuss adequately the significance of an experimental discrimination. Recently Than'* has questioned whether the electrokinetic potentials as determined by electrophoresis and by streaming potential are essentially the same thing, as has always been assumed; and both H a r k i d and PllcBainlg have argued that the “electro-kinetic potential” does not exist or mean anything, anyhow. In this rather extended discussion an attempt will be made to clarify and settle some of these cloudy questions. The original Helmholtz conception of a very thin electric double-layer of atomic dimensions is frequently criticized, usually on the basis of Gouy’s theory of the diffuse ionic layer; and sometimes the conclusion is drawn that the Helmholtz analysis of electro-kinetic phenomena must be discarded. However, it has been shown in this paper that when we make a quantitative analysis of the effect of the diffuse layer, considering also its distortion during electrophoresis, we find that the first approximation to the electrophoretic mobility of a large sphere is the familiar Helmholtz formula. It’ can likewise be shown that the Helmholtz analysis is valid for any surface in water or aqueous solutions, so long as the radii of curvature are large in comparison with the thickness of the double-layer. 17 l8
Colloid Symposium Monograph, 6,11 (1928). Z. physik. Chem., 147, 147 (1930). J. Phys. Chem., 28, 706 (1924).
ELECTROPHORESIS AND THE DIFFUSE IONIC LAYER
3 43
Furthermore, barring mathematical blunders, there can be no question concerning the fundamental validity of the theory of the diffuse ionic layer as it has been outlined in this paper: for the theory is based upon well-established physical laws which have their origin and proof in lines of experiment entirely outside of electro-kinetic phenomena. The only assumption ad hoc is that the moving particle has an electric charge, and that is really a conclusion necessitated by the visible motion of the suspended particle. Consequently, even if the theory is found to disagree completely with the data obtained under any particular experimental conditions, we cannot conclude that the theory is false. JVe have only proved that it is incomplete or that incorrect values have been assigned to the parameters involved. As for the incompleteness, a number of experimental results in electrophoresis have already been interpreted in this paper as being due to the strange behavior and interference of a plastic or semi-solid adsorbed crust which was not contemplated in the theory. Concerning the values of the parameters close to a surface, we do not have entirely satisfactory information; but it is yet to be proved that the values there are much different from what they are in the liquid in bulk. Kallmann and DorschZ0have measured the dielectric constant of films of liquids about 1p0 thick; and they found no perceptible difference from the normal values. No variation in dielectric constant less than their mean error, about 0.2 per cent, could have any important effect on the charge distribution in a diffuse layer that extends several mp from the surface. Concerning the viscosity near a surface, if there is any deviation at all from the normal value, current theories of surface structure would lead us to expect an increase, associated with more or less molecular orientation. Any electro-kinetic phenomenon would tend to dist,urb this special orientation of the molecules, if it exists; and we would anticipate some lack of proportionality between cause and electrokinetic effect. For example, Ettisch and Zwanzig?' found a variation in streaming electro-kinetic potential with pressure when forcing mixtures of methanol and water containing a trace of KC1 through glass capillaries; and a variation in cataphoretic mobility with impressed field in some of my own measurements has already been mentioned. But in the absence of such abnormal effects there is no reason for doubting that in dilute and moderately concentrated solutions, the viscosity is constant throughout the greater part of the double-layer. In view of these considerations the conclusion seems justified that, with limitations of the kind that have been discussed, the Helmholtz analysis of electro-kinetic phenomena is applicable whenever the curvature of the surface involved is small in the sense previously indicated. Unfortunately, the situation is quite different in regard to surfaces of large curvature. ?'he system of differential equations resulting from the author's set of assumptions can probably be solved for the mobility as a power series in p, the electro-kinetic potential; and such a solution would be $0
21
Z. physik. Chem., 126, 305 (1927). Z. physik. Chem., 147, I jI (1930).
344
MELVIN MOONEP
valid for a sphere of any size, provided that (o were sufficiently small. But the development’ of this solution still remains for the future.22 The DebyeHuckel mobility formula can be used for a small sphere in a very dilute solution; but the assumptions B3 and B4 of section z are such severe restrictions that the applicability of the formula is limited to very dilute solutions and to ultramicroscopic micelles, composed of only a few molecules and carrying only a few elementary charges. In between the regions covered by the Helmholtz and the Debye-Huckel theories there is a large region in which the radii of surface curvature are of the same order of magnitude as the Gouy diffuse layer. This region is a “no-man’s land,” not yet belonging t o either theory. Obviously, ignoring these limitations of the theories will lead to false deductions and to confusion. This fact is illustrated in Thon’s suggestion that electrophoresis and electro-osmosis are different in some fundamental way not yet understood. The indiscriminate application of Stokes’ law and the treatment of all colloids as electrolytes are violations of the restrictions upon the Debye-Huckel electrophoresis theory. Some of the discrepancies that result in this case have already been discussed by Pauli and Yalk6.23
Summary The theory of the diffuse ionic layer and its influence on the electrcphoresis of a sphere is outlined and a theoretical formula for the limiting slope of the mobility-curvature curve is given. 2. Experimental data are reported which are in rough agreement with the theory as applied to dilute aqueous solutions. In concentrated solutions there is definite disagreement in some cases. 3 . Formulas are given for surface charge and surface conductivity. The latter formula does not agree with measured values. 4. Yarious anomalous effects in cataphoresis suggest the existence of a plastic adsorbed layer which seems to have a profound effect on the electrokinetic potential in concentrated acid and neutral solutions. 5. I t is pointed out that the validity of the Debye-Huckel cataphoresis formula is limited to conditions in which the colloidal particle is much smaller than its diffuse ionic layer. The experimental work reported in this paper was done a t the Ryerson Physical Laboratory of the University of Chicago. I am greatly indebted to Professor A. C. Lunn and orher members of the faculty for their assistance and interest. I feel quite incapable of expressing my gratitude to the Cniversity for the facilities placed at my disposal, especially as an appreciable part of the work was done during my spare time after I was no longer officially connected with the Laboratory, and at a time when space was very much in demand for full-time research workers. To the Kational Research Council, I am greatly indebted for their long-continued support in a baffling problem. I.
Ryervon Physical Laboratory, Unioersity of Chicago, Chicago. 112. 22 The corresponding analysis of electro-osmosis in a straight capillary tube of small radius has recently been published by Komagata: Bull. Chem. Soc. Japan, 4,2j5 (1929). 23 “Elektrochemie der Kolloide,” 268 (1929).
