DIELECTRIC DISPEFWON OF SPHERICAL COLLOIDAL PARTICLES
2407
On the Theory of the Dielectric Dispersion of Spherical Colloidal
Particles in Electrolyte Solution’
by J. M. Schurr Department of Soils and Plant Nutrition, University of California, Berkeley 4, California (Reeeiaed J a n u a r y I S , 1964)
The theory of the low frequency dielectric dispersion of spherical colloidal particles recently presented by Schwarz is shown to rest upon an objectional boundary condition, namely that the free charge transported to the surface of the sphere by normal currents may not respond to tangential electric fields. I n addition, it is pointed out that in constant electric fields the counterion layer of Schware is a nonconductor since the electric and diffusion currents exactly cancel each other. It is found necessary to distinguish between the bound charge current, which is defined to be that which is completely cancelled by the diffusion current (i.e., current due to chemical potential gradients) in the steady state, and true or d.c. current, which persists in accordance with Ohm’g law in the steady state. The true current is recognized as that previously considered for the same problem by O’Konski. The boundary conditions of O’Konski appear to be the most suitable for the true current. The treatments of O’Konski and Schwarz are combined, relying on analogy with the steady state for necessary assumptions. The objectionable boundary condition of Schwaye’s theory is removed as is the steady-state nonconductivity of the counterion layer. The necessary modifications in the expressions for the dielectric increments and dielectric loss increments of the respective authors are indicated. The agreement with experiment appears to be more complete than for either individual theory. The use of dielectrophoretic flow to distinguish between this composite theory and the theory of Schwarz is suggested.
Introduction A theory of the low frequency dielectric dispersion of spherical colloidal particles has been recently presented by Schwarz.2a The theory of Schwarz takes into account the diffusiona,l relaxation of the counterion distribution along tangential concentration gradients (produced by applied electric fields) in a conductive surface layer a t the sphere solution boundary. This theory exhibits remarkable agreement with the findings of Schwan, et a1.,2bfor the frequency dependent part of the lorn frequency surface conductance. A previous theory of O ’ K ~ n s k i which ,~ also takes account of the perturbed charge distribution produced in a conductive surface layer a t the sphere-solution boundary but which neglects the diffusional relaxation of the charge distribution, is unable to account for the low frequency dispersion of the mrface conductance observed by
Schwan, et aZ.,2band Fricke and CurtisS4 However, the theory of O’Konski3 may be appropriate for that part of the surface conductance which is independent of frequency over the region of the low frequency dispersion but which is observed to become frequency dependent a t higher frequencies.*b Critical inspection of the boundary conditions employed by Schwareza reveals (see Appendix A) an unstated assumption implicit in his treatment which is very different from that of O’Kons1L3 This assump(1) This investigation was supported by Public Health Service Predoctoral Fellowship 5-Fl-GM-10, 482-05 from the National Institute of General Medical Sciences. (2) (a) G . Schwara, J . P h y s . Chem., 66, 2636 (1962); (b) H. P. Schwan, G. Schwarz, J. Maczuk. and H. Pauly, ibid., 66, 2626 (1962). (3) C. T. O’Konski, ibid., 64, 605 (1960). (4) H . Fricke and H. J. Curtis, ibid., 41, 729 (1937).
Volume 68, Number 9
September, 1964
J. M. SCHURR
2408
tion of Schwarz is that the charge which is transported to the surface by normal (normal will be used only in the direction sense in this article) currents may not, then, participate in tangential currents by responding to tangential potential gradients., O’Konski assumes, to the contrary, that all of the charge transported to the surface may respond to tangential fields a t the surface. The assumption of O’Konski appears the more reasonable. A steady-state analysis, in the Debye-Huckel approximation, of the effect of varying the potential of an insulating sphere, bearing a surface charge and suspended in a grounded electrolyte (see Appendix B) , shows that increasing the potential leads to a proportionate increase in both the surface charge density and the counterion atmosphere density of charge of opposite sign. If the increase in surface charge density is interpreted as a change in the equilibrium constant for the ion-ion association a t the surface due to the applied potential, it is apparent that all of the charges transported by the applied potential into the counterion layer, which includes the region of ion-ion 3ssociation as well as the diffuse ion atmosphere, should respond to tangential fields in the counterion layer. A further difficulty with the treatment of SchwarzZa is the vanishing of his total current in the counterion layer in the steady state.6 This circumstance arises from the cancellation of ihe electric current by the diffusion current in the steady state. However, it is well known that steady-state distributions‘of ions are good conductors, characterized by t m e or d.c. conductivities. Schwarz has, evidently, neglected the true current in the counterion layer. O’Konski, on the other hand, has considered only the true current in the counterion layer and neglected the displacement current, which is the resultant of the electric and diffusion (furrents due to ions constrained to remain in the vicinity of the spherical colloidal particle. The model of Miles and Robertson,o who permit no macroscopic charge to respond to tangential fields, appears to be a definitely artificial representation of the problem. It is the object of this communication to suggest an alternative approach to the problem combining the treatments of Schwarz and O’Konskj in a way which removes some of the objections to the individual procedures, and which gives a result, a t least qualitatively, in fuller accord with the experimental facts. The notation employed here conforms to that of Schwarz.2a I
K,
= K,
and T h e Journal of Physical Chemistry
+
Kt
= K~
+ iwe,e,
are the complex conductivities, in the presence of an alternating field of angular frequency W , of the bulk electrolyte and the sphere, respectively, where K,, K~ and tu, et are the conductivities and dielectric (relative) constants in the respective media. = 8.85 X f./m. is the absolute dielectric constant of free space. R is the radius of the sphere. uo is the mean surface density of counterions required to neutralize the charge on the sphere. u = uo 8 is the surface density of counterions a t a given point in the presence of an applied field. u is the mechanical (velocity/unit force) mobility of the counterions. are the electric potentials in the bulk solution and in the sphere, respectively. E is the electric field intensity.
+
+%
A Composite Theory for the Effect of an Electric Field on a Sphere with a Counterion Layer I n formulating a suitable niodel the following properties, which have been discussed in the Introduction, are considered to be the most desirable. (1) The counterion region, including the layer of association (of the counterions with the ions on the surface of the sphere) and extending outward to a radius within which there is near neutrality, ought to conduct both true steady-state currents and bound charge currents, which are here defined as those which may be cancelled by opposing currents due to chemical potential gradients. It is thus only the bound charge current which is relaxed by the diffusion current. ( 2 ) True currents across the boundaries of the counterion region are transported through the counterion region as true currents in the steady state. This noninterchangeability of currents is also assumed to obtain in the presence of alternating fields. (3) True currents normal to the boundaries of the counterion region build up no macroscopic surface charge on these boundaries, which are artificial, but may give rise to macroscopic volume charge in the interior of the counterion region. This macroscopic volume charge, if it appears, may respond to tangential electric fields; and since this voluine charge is produced by the divergence of true currents, any tangential currents involved will also be true currents. Although ionic current and charge appear as surface quantities in the formalism, they nevertheless represent real volume quantities in this model. The number of ions participating in the bound current is precisely the number of surface charges on the
~WE,E,
(5) See eq. 50 of ref. 2a. (6) J. B. Miles and H. P. Robertson, Phys. Rea., 40, 583 (1932).
DIELECTRIC DIsPERsl[OS O F
sphere because this is the number of charges which are required for neutralization of the sphere charge and, hence, the number of charges constrained to remain in the vicinity of the sphere. This constraint on location of the counterionic charge is directly responsible for the concentration (chemical potential) gradient of counterions built up in a n applied electric field. Obviously, only those ions so constrained will establish concentration gradients and completely relax the electric current in the steady state. Making use of the definitions and notation of the previous section, we may write the charge transport (continuity) equation for the sum of the bound anld diffusion currents as done by Schwarz beo$
=
eo
e _I $ {sin ba s+ R 2 sin 8 08
8
”)
eo uo - sin 8 kT b8
(1)
where the same assumptions are applied in deriving this equation. It is assumed that true current conduction in the counterion region proceeds according to Ohm’s law (or O’Konski’s two-dimensional modification3) with a conductivity which is independent of the degree of bound charge polarization. The conductivity may be expected to be independent of the bound charge polarization when a R) (r < R)
V2$, = 0
(i% I R )
(-1 dii-)
(8)
+ iwr XoE,
(9)
- eoukT R 88
then we have, using eq. 6 zwr
is = 7 eo2uuoEs = 1 zwr 1
+
~
zw 7
where we have set Xo = eo2uou. The effective conductivity of this bound charge diffusion process is
The corresponding surface dielectric constant is given by Tho
€&*
where at is the surface charge density produced by divergent true currents.
+ iwr
= ___
1
It is evident from the form of eq. 10 and 11 that i, is a complex displacement current and is not to be confused with the true current of eq. 3. V o l u m e 68, N u m b e r 8
September, 196.4
J. M. SCHURR
2410
The electric potentials are determined and found to be
-3K, =
+ K , * Er cose
2K,
r IR
(12)
+
-Er cos e Ki* 2K,
t,ba =
K , ER3 cose K,* r 2
-
+
r 2 R
(13)
where
It is now evident that the dielectric properties of the sphere with its counterion layer are equiyalent to those of a sphere of uniform conductivity
and uniform dielect,ricconstant 2€,* R
Ei*=
t i + - =
e> l / ~eq. , 28 and 29 become determined value of em”(u) or employed a known K ( and used our e_’’ = ( i / w e , ) ~ ~ . I n the former case, ei + € 5 (28’) the 2X/R term has been autoinatically included in tm”(w), and no change in the calculated curve (Fig. 2 of Schwarz) is anticipated. I n the latter case there will be a:change in the curve for Ae”, because in this so that the result of O’Konski is obtained with the case the term 2h/Re,w will contribute to At”. It modification X + X .f Xo. Under these conditions the seems probable that Schwarz has used a measured equations of O’IConski for disks, rods, and ellipsoids value of E,”(w) or K , = K , 2X/R so that the freXo may be applied with the requirement that X + X quency independent part of the surface conductivity in these equations. Furthermore, the comprehensive has been subtracted out of the dielectric loss increment. discussion of O’Konski concerning the interpretation I n this event it is apparent that the present theory of electrical measurements on different materials apgives the same quantitative agreement with the low plies for measurements a t frequencies w >> 1 / ~ . frequency dispersion as that of Schwarz. The dis1/r it is clear that the bound At frequencies w cussion of Schwarz on the relaxation time spectrum charge current is important, if not dominant, xn for r is not affected by the considerations involved in determining the electrical properties of the medium. this composite treatment. The relative magnitudes of X and Xo may be determined Induced Polarization of the Counterion Layer from the magnitude of the low frequency dispersion The dipole moment due to the total surface charge of the excess increment of the complex dielectric condensity may be obtained from eq. 17 and 18 upon stant over that anticipated from simple MaxwellWagner dispersion given in eq. 31. (10) (a) J. C. Naxwell, “A Treatise on Electricity and Magnetism,” The expressions of Schwarz2a may be amended Oxford University Press, London, 1873, article 314; (b) K. W. Wagbeginning with his eq. 43 for the “additional complex ner, Arch. Electrolech., 3, 83 (1914). (27)
))
+
+
+
-
Volzime 68, Number 9
Septemher, 1964
J. M. SCHURR
2412
averaging over the surface of the sphere
Ptot =
4rR3 __ 3
the medium considered above are computed for t,he two models at a distance of 1 cm. from the central wire.
u t + eo5 ( 7 ) (32)
-
It will be noted that, in the low frequency limit w 0, eo$ due to the bound charge is exactly cancelled by the term in u t containing E,*. The expression for the steady-state dipole moment of the free charge, therefore, is
u,
=
9.6 X low5cni./sec. (away from the wire) (39) X lo-’ cm./sec. (toward the wire)
us = -1.5
(40)
The velocity v, computed on the composite model is very small and will be obscured by convection due to joule heating and by diffusion. However, the velocity us computed according to the model of Schwarz is easily large enough to be observed, should that theory be correct.
w-0
I n Schwarz’s treatment the steady-state dipole moment of the free charge is due solely to the bouhd charge and is given by
Acknowledgments. The author wishes to thank R. T . LIerrill for several illuminating discussions on the general problem of establishing boundary conditions, and C. T. O’Konski for originally introducing the author to the problem.
Appendix A Upon evaluating Ptotand P for the following situation: cm., = 0, E $ = 2 4 K , = 0.7 mmho/ R = 1.17 x cm., e, = 78, T = 300°, X = mho, uo = 2 X 1013 ions/cm.2, and u = 1.35 X lo8 c.g.s. units, we obtain
Ptot= 0.9 D./v./m. field strength w-0
P
=
5.4 X lo4 D./v./m. field strength
Schwarz2ahas obtained a solution for
s = vi ( p - ~0Prn)Ej
0
(-41)
v2*.I
=
0
(A21
subject to the conditions
w-0
The large difference in predicted dipole moments of the free charge in this particular case may provide a means of distinguishing between the theory of Schwarz and the present treatment. Observation of the dielectrophoresisll of these particles in nonuniform alternating fields of angular frequency w ‘v lo2
