Simplified Calculation of Electrophoretic Mobility of Non-Spherical Particles When the Electrical Double Layer is Very Extended L E E B. H A R R I S Research Laboratories, Xerox Corporation, Rochester, New Yorlc
Received May 4, 1970 Experiments in our laboratories (1) have shown t h a t some dispersions in nonpolar solvents exhibit very large differences in electrophoretic mobility (sometimes as much as an order of magnitude) from particle to particle. Naturally, the question arises as to why this is so. Generally, since the ion concentration is low in these liquids, the diffuse part of the electrical double layer is very extended and does not contribute appreciably to the drag (i.e., the "electrophoretic drag" and "relaxation effect" (2) are negligible). Therefore, to calculate the mobility of a particle of charge Q, one may simply balance the force EQ, due to the applied electric field E, against the opposing force due to ordinary hydrodynamic drag. Using this approximation, which greatly simplifies the calculations, we can investigate whether the observed large variations in mobility can be caused by variations in shape. The answer is no, as is shown b y the following calculations for el]ipsoidal particles} * No calculations of electrophoretic mobility have been carried out for shapes other than spheres and cylinders (2). Henry (3) computed the mobility of an infinitely long cylinder moving either parallel or at right angles to its axis of symmetry. However, when the diffuse part of the electrical double layer is very thick, the equation derived by Henry for motion perpendicular to the cylinder axis is unsatisfactory for the following reasons: First, as Henry pointed out, his method of solution is valid only if the potential drops off more rapidly than any inverse power of distance from the surface. Second, for an infinite cylinder with a very thick diffuse layer, the surface potential is not well defined, since the electric field is inversely
If the surface of the particle is assumed to be charged to a definite surface potential V~ (this is the basic assumption underlying the concept of the "zeta-potential"), then if the diffuse counter-charge layer is very extended, so that it does not contribute appreciably to V~, the total charge Q is determined by the capacitance C of the ellipsoid through the equation Q = V.~C,
where, following Jeans (5), C is given b y C/K = 2/x,
x = f0~ a (d~ x)'
and A(X) = %/ial = q- X)(a== q- )0(aa 2 q- X).
In Eqs. - a l , a2, aa are the three semiaxes of the ellipsoid, X is the "radial" coordinate in confocal ellipsoidal coordinates, K is the dielectric constant of the surrounding liquid, and electrical units are unrationalized esu. For arbitrary orientation of the ellipsoid with respect to its velocity vector, the drag
proportional to distance. Third, the hydrodynamic problem cannot be solved without partially including the inertial terms in the flow equations (4). The resulting expression for the drag coefficient includes the Reynolds number as a parameter, with the result that the drag is not proportional to velocity (4). Journal of Colloidand In~erfaveScience,Vol. 34, No. 2, October1970 322
CALCULATION OF ELECTROPHORETIC MOBILITY OF NONSPHERICAL PARTICLES 323 force F is not necessarily along the direction of travel; for very low Reynolds' number (i.e., Stokes' flow), the drag is given (6) b y P = -~D-u,
by an equation similar to  regardless of orientation. 2 Now for some numerical results. Happel and Brenner (7) computed from Eqs.  and  the components of the drag tensor for flow parallel and perpendicular to the axis of symmetry of an ellipsoid of revolution. Analogous to Eq. , they express their results in terms of an equivalent Stokes' radius R ; , defined by the relation 67rR~ = D ~ (cf. Eqs.  and ). Then
where ~ is the viscosity of the liquid, D is a symmetric tensor of second rank, and u is the translational velocity of the particle in the liquid. If the coordinate axes are chosen to lie along the principal axes of the ellipsoid, D is diagnonal, its three nonzero elements being given by (4, 6) D . = 16~/(x + a ~ ) ,
where F~ and u~ are the components of drag and velocity, respectively, along the ith principal axis. Again balancing drag against electric force, we obtain
where x is defined b y Eqs.  and , and
(1 + x/a,~)~(x)"
F~ = -67r~R~u~ ,
For a sphere (a~ = a), Eq.  reduces to IS]
C = Ka,
According to Jeans (5), for an ellipsoid of revolution, Eqs. - yield
and Eq.  reduces to the usual formula for Stokes' drag, namely, F = --6~r~au.
(C/K)/R . . . .
However, in the more general ease of an ellipsoid with unequal semiaxes, the same procedure (with the use of Eqs. -) yields
for the mobility ~ when the electric field is applied along the ith principal axis of the ellipsoid. Now we can deduce a very interesting result. Since (from Eqs.  and ) 0 < a~a~2/X < 1,
we find from  and  that no matter w h a t the values of the a~ ,
< ~/~0 < ~ ,
where ~0 is defined b y Eq. . The mobility for any arbitrary direction of appIied field must lie between the largest and smallest of the ~ and is therefore also restricted
Balancing the electrical force and the drag force then yields the usual formula for the mobility of a sphere with very extended double layer, namely, = K V , / 6 ~ r V =-- ~ o .
= 2 e/sin-le
. . . . = e/ln[(1 + e)/(1 -
e)] (a > c).
2 In addition to the change in equilibrium drag coefficient, rotation of the particle in the applied field may also affect its mobility due to (i) change in the amount of surface charge caused by the change in electrostatic configuration, (ii) energy dissipated by rotation. Effect (i) is negligible if either (a) the potential drop across the particle, due to the applied field, is smalI compared to the /'-potential; or (b) the field is applied for so short a time during measurement that no change in the amount of surface charge can occur. Condition (a) is met for submicron particles in low-field measurements (< 10V/era). Condition (b) is met in measurements of the type described in ref. 1, since the relaxation time for charge redistribution in the liquid is p e >0.2 see; but even in these measurements, order-of-magnitude variations of electrophoretic mobility are observed from particle to particle (unpublished data). Effect (ii) is eliminated in the experiments cited above by the observation that the particles, many of which have a discernible characteristic shape, do not rotate during the time the field is applied.
Journal of Colloid and Interface Science, Vol. 34, No. 2, October 1970
VALUES OF A s AND M i AS A FUNCTION OF L E N G T H - T o - D I A M E T E R
Here A is the ratio of equivalent Stokes' radius of an ellipsoidal particle to that of a circumscribed sphere, and ( C / K ) / R ~ is the ratio of the capacitance of the ellipsoid to that of the circumscribed sphere. M is the ratio of the mobility of the particle to that of any sphere at the same surface potential. Values apply to an ellipsoid of revolution. The length is measured along the axis of revolution and the diameter, perpendicular to the axis of revolution. Length diameter
0 0.01 0.1 0.2 0.3 0.5 0.7 1.0 1.5 2.0 5.0 10.0 20.0 50.0 500.0
Flow .l_ t o axis of symmetry A .L
Flow  to axis of symmetry A t*
M , = ~,,/~
8/3~ 0.849 0.852 0.862 0.874 0.905 0.942 1.000 0.735 0.602 0.357 0.265 0.290 0.162 0.106
3/4 0.755 0.794 0.831 0.862 0.914 0.954 1.000 1.05 1.09 1.20 1.26 1.30 1.34 1.36
2/w 1 0.641i 0.677r/ 0.7171 0.753 t 0.836| 0.896J 1.000 0.772 0.656 0.428 0.334 0.272 0.21' 0.14,
M ~ = # ~./~o
16/9~ 0.571 0.613 0.660 0.705 0.793 0.878 1.000 0.796 0.690 0.474 0.381 0.318 0.261 0.180 0
9/8 1.12 1.10 1.09 1.07 1.04 1.02 1.000 0.973 0.954 0.901 0.872 0.851 0.831 0.805 3/4
and total charge may be computed from
Here e is the eccentricity; a and c are the semiaxes along the axis of symmetry and perpendicular to it, respectively; and R~a~ is the larger of a and c. The quantities M~ ---- its/#0
V~ = 6 ~ m / K M , ;
Q = 67r~p~Rr~a~A~.
A~ ~- R~/R .... are listed in Table I for various values of the ratio of length to diameter. (It can be shown from Eqs.  to  t h a t the mobility depends only on shape and not on size.) The length is measured along the axis of symmetry and the diameter perpendicular to it; the ellipsoid is oblate (prelate) when this ratio is less than (greater than) one. The values of At are based directly on values given b y Happel and Brenner (7); and the values of M~ are computed by dividing ( C / K ) / R m ~ (from  or ) b y A~ (cf. Eq. ). (C/K)/Rm~x is just the ratio of the capacitance of the ellipsoid to t h a t of a sphere that will just contain it; values of this ratio are shown in the last column of the table. In terms of A~, M s , and the measured mobility m , the surface potential
From the tabulated results, it is apparent that (a) as predicted by Eq. , the mobility, even of a very nonspherical particle, is not greatly different from that of a sphere at the same surface potential, and (b) the charge corresponding to a given mobility is a relatively weak function of (length/diameter) for oblate (dislike) particles and a relatively strong function of (length/diameter) for prelate (needlelike) particles. As shape becomes more needlelike, the charge corresponding to a given mobility becomes small because R~ becomes small (cf. Eqs.  and ). These results lead one to expect at most a two-to-one ratio of the mobilities of different particles owing to differences in shape alone, providing the charge is determined by a definite surface potential and the double layer is very extended. Because Eq.  stems from the similarity of the partial
Journal of Colloid and Interfaze Science, Yol. 34, i~'o. 2, October 1970
CALCULATION OF ELECTROPHORETIC MOBILITY OF NONSPHERICAL PARTICLES~325 differential equations governing the electrical potential and the velocity potential in Stokes' flow (4, 5), the qualitative picture presented here is probably not limited to ellipsoidal particles. We conclude then that when dispersions in nonpolar liquids display large differences in electrophoretic mobility from particle to particle, these differences are not due to differences in size or shape but indicate either (a) that the electrochemical surface properties of the dispersed material vary widely from particle to particle or (b) that the system is not in electrochemical equilibrium characterized by a definite surface potential.
lournal of Colloid and Inter/ace Science,
Vol. 34, No. 2, October1970
REFERENCES 1. HARRIS, LEE B., Rev. Sci. Instrum. 40, 905 (1969). 2. OVER/]EEK, J. TH. G., AND WIERSERVA.,P. i . , in M. Bier, Ed., "Electrophoresis," Vol 2, p
1. Academic Press, New York, 1967. 3. HENRY,D. C., Proc. Roy. Soc. Set. A 133,106 (1931). 4. LAMB, H., "Hydrodynamics," 6th ed., p 614. Dover, New York, 1945. 5. JEANS, J. H., "The Mathematical Theory of Electricity and Magnetism," 5th ed. Cambridge Univ. Press, Cambridge, 1925. 6. BRENNER, H., Advan. Chem. Eng. 6,287 (1966). 7. HAPPEL, J., AND BRENNER, H., "Low Reynolds Number Hydrodynamics." Prentice-Hall, Englewood Cliffs, New Jersey, 1965.