934
HUGOFRICKE
majority of larger ones, was not included because it was difficult t o measure all the smaller particles. The specific surface area of the silica agreed more closely with D,, the “number average” particle size estimated from electroil micrographs. 3. The ‘‘weight average” particle diameter, D,, measured on micrographs, averaged 26% greater than Dn, the “number average’’ diameter; this corresponds to a “weight average” particle
VOl. 57
weight about twice the “number average” particle weight. The authors wish to acknowledge the work of L. A. Dirnberger in conducting the engineering studies and making mathematical calculations, W. M. Heston, Jr., in preparing samples for characterization, R. E. Lawrence in carrying out the analyses, and Dr. C. E. Willoughby of the Chemical Department in preparing the electron micrographs.
THE MAXWELL-WAGNER DISPERSION IN A SUSPENSION OF ELLIPSOIDS BY HUGOFRICKE Walter B. James Laboratory of Biophysics, Biological Laboratory, Cold Spring Harbor, N . Y. Received April 14, 1965
The dispersion equations are first derived for the conductivity and permittivity of a suspension of ellipsoids of vanishing volume concentration, in which the components are characterized by both conductivity and permittivity, by extending to complex admittances an earlier treatment2 of this system for the case of ure conductors. The expressions contain 8 “form factor” which is expressed in terms of elliptic integrals. I t s numerical varues are tabulated. For random orientation of the ellipsoidal axis the suspension is shown t o be electricall equivalent t o a simple resistance-capacity network and to be characterized by three dispersion regions inside each of wtich, it behaves quantitatively like a Debye dipole system. This treatment is subsequently extended t o more concentrated suspensions on basis of the semitheoretical general conductivity equation given in rGf. 2.
The problem can be dealt with in a simple manner A heterogeneous system of conducting dielectrics, in which the ratio of permittivity to conductivity is by making use of the theoretical information aldifferent in the different phases, exhibits Maxwell- ready available on the electric behavior of heterogeWagner (M-W)sp4 dispersion in the frequency zone neous systems composed of pure conductors. By where the field shifts over from its low to high fre- introducing complex variables, the conductivity quency course. Below the dispersion region, where equation for such a system is transformed to the the field is determined by the conductivities of the compIex conductivity equation for the same geomeconstituent phases, the conductivity of the system trical system of conducting dielectrics, from which has its minimal value, referred to in the usual kind equation the dispersion equations for conductivity of mixture formula, while the permittivity is greater and permittivity are thereafter obtained by sepathan the (minimal) value obtained when the field is rating real and imaginary terms. By using this determined by the permittivities. The opposite method, Wagner’s treatment of a suspension of condition, minimal permittivity, increased conduct- spheres could have been simplified since the conivity, is found at frequencies above the dispersion ductivity equation for this system was already given region. by Maxwell. The object of the present paper is to extend this Although the M-W effect is of considerable practical interest, its theoretical treatment does, not treatment to a suspension of ellipsoids. The conappear to have been extended beyond the two ductivity equation for this system was dealt with simple systems considered by Wagner,* viz., a strat- by Fricke,2where references to the earlier literature ified body and a dilute suspension of spheres. The will be found. The equation can be derived rigorbehavior of fibrous systems has been discussed ously only when the volume concentration is low, but qualitatively by different a u t h o r ~ , ~by- ~using sup- reference 2 describes also the derivation of a general posedly equivalent electric circuits. A more gen- conductivity equation, which has been found to eral treatment of suspensions which takes into ac- agree well with the experimental evidence. In count both the effect of particle form and higher dealing with the dispersion problem, we shall therevolume concentrations, has recently become of in- fore first consider the case of low volume concenterest in studies of biological materials in the ultra trations and thereafter extend this treatment to high frequency zone.8-10 higher concentrations by using the general conduct(1) Supported by the U. 9. Office of Naval Research. ivity equation of this earlier work. (2) H.Fricke, Phys. Rev., 24, 575 (1924). Theory.-We shall consider a suspension of (3) J. C. Maxwell, “A Treatise on Electricity and Magnetism,” 2nd homogeneous eIlipsoids (axis 2a 2 2b 2 2c) distribEd., Clarendon Press, Oxford, 1881. p. 398. uted a t random in a homogeneous medium. When (4) K. W. Wagner, Arch. Eleklrotech. 2 , 371 (1914). (5) S. Setch and Y. Toriyama, Insl. Phys. Chem. Research, Tokvo the components are pure conductors, the conducBci. Papers, 3 , 283 (1926). tivities of suspension, suspending and suspended (6) D. DuBois, A. I. E. E., 41, 689 (1922). (7) E. J. Murphy, THISJOURNAL, 33, 200 (1929). phases are called IC, ICI and kz,respectively, expressed (8) B. Rajewaky and H. Sohwan, Naturwissenschaflen, 3 5 , 315 -1 ohm-’ crn.-I. In the complex sysin (9 X loL1) (1948). tem, the corresponding quantities are written Q (9) H. F. Cook, Nature. 168,?47 (1951). ?/a,et,,.., where u and E represent conductivity ,(lo) H. F. Cook, Brdt, J . Apzd. Phus.. 2. 295 (1951).
+
MAXWELL-WAGNER C ISPERSION IN SUSPENSIONS OF ELLIPSOIDS
Dec., 1953
and permittivity, respectively, and n is the frequency in C.P.S. (1) Low Concentration Suspensions.-The suspended ellipsoids are assumed to be so far apart that the local field can be taken to be equal to the external field. Consider first a suspension in which the ellipsoids are arranged with axis 2 a ( a = a, b or c ) parallel with the field. When higher powers of p are neglected, the conductivity equation can then be written2
TABLE I FORM FACTORS x FOR DIFFERENT ELLIPSOIDAL AXIAL RATIOS a/b 1 1 1
2
1
1 1.5 1.5 1.5 1.5 1.5 1.5 2 2 2 2 2 2
- abcLn abcLa dA
+ A) d ( a 2 + A ) ( P + A)@ + A)
La = J,(d
The k's are now replaced by their complex correspondents and real and imaginary terms are separated. For random orientation of the ellipsoidal axis, we obtain
c
=
€1
(4
+ g€1 P
C(1+
- al)(Uz
XaEl)
a/b 3 3 3 3 3 3 4
4 4 4 4 4 6 6 6 6 6 6
(€2
~a),*
(ez
xam1)2
n2
xac1)
xu)
n2
€1) ( € 2
(€2
(QZ
(uz
XQ
2 1.23 0.90 0.57 0.417 0.274 1.61 1.01 0.73 0.478 0.348 0.230 1.42 0.90 0.66 0.432 0.314 0,209
+ n2 4 + + ;i- (1 + + + n2 a + xaei)2 + +4 + xaei) + (1 + + xaci)2 + (a +
+Z~VI)
(1 + x a )
a = u 1 + - -3p u1
e
- 01)(02
xs xb 2 2 2.61 2.61 3.23 3.23 4.49 4.49 5.8 5.8 8.30 8.30 3.291.61 4.33 2.17 5.3 2.74 7.5 3.88 9.6 5.1 13.9 7.3 4.79 1.42 6.3 1.96 7.9 2.50 11.0 3.63 14.2 4.76 20.7 6.9
b/c
1 1.5 2 3 4 6
€1) ( € 2
11.1 13.7 19.5 25.1 36.6 1 12.1 1.5 16.5 2 20.8 3 29.6 4 38.4 6 56 1 21.6 1.5 30.1 2 38.8 3 55 4 70 6 103
Za)ar*
(: - 9
Xb
XQ
1.24 1.24 1.77 0.80 2.29 0.59 3.35 0.389 4.45 0.285 6.5 0.190 1.161.16 1.69 0.74 2.19 0.56 3.22 0.366 4.29 0.273 6.4 0.181 1.09 1.09 1.61 0.71 2.10 0.53 3.10 0.355 4.22 0.259 6.2 0.174
(5)
(2 - 2)
n2
(cz
(a)
xa 8.2
The equations obtained for and e show that a suspension of ellipsoids having one of the principal axes parallel with the field, is characterized by a single Debye type dispersion zone. When the axes are oriented at random, there are three such dispersion zones, the electric behavior of the suspension being the same as that of an even mixture of the three principal orientations of the ellibsoids. It will readily be verified also, that a suspension
For random orientation of the ellipsoidal axis
(UZ
b/c 1 1.5 2 3 4 6 1 1.5 2 3 4 6 1 1.5 2 3 4 6
1
where p is the fractional volume of the suspended ellipsoids and =
935
(6)
2ae1)'
By removing the summation sign and replacing p / 3 by p , we obtain the dispersion equations (5a
and 6a) for a suspension of ellipsoids having axis 2a parallel with the external field. For spheres 5 = 2 and the expressions are then the same as those given by Wagner. The values of L a were calculated before2 only for ellipsoids of rotation. The general solution is obtained by substituting X = (az - cz))Izz- a2and X = (b2z2- c2)/(1 - z2) in Laand L,, respectively, whereby these integrals are reduced to elliptic integrals of first and second kind abcL. =
2abc -(F(arc cos ( a z - bz)l/aZ - cz
2aa abcL. = G~ (a2
-
-~
2abc 2
Modulus k =
)
d
i
Z
T
(? I C ) (
~
-
4-
-
The values of zn for different axial ratios are recorded in Table I and Fig. 1. For cylinders arranged with their axis a t right angle to the electric field, 2 = 1.
A X I A L RATIO.
Fig. 1,-Form a = =
.[ .[
factors xa for ellipsoids of rotation, half axis a 2 b 2 c. ,,,graph 11, right scale. "= x xc, graph IV, left scale. xa, graph 111, right scale. x,, = xc, graph I, left scale.
of randomly oriented ellipsoids is electrically equivalent to the diagram in Fig. 2 where RI-'
40)
(10)
HITGO FRICKE
936
Vu]. 57
A suspension of randomly oriented ellipsoids behaves as an even mixture of three different Debye dipole species. Below and above all dispersion regions, the conductivities and permittivities of a suspension are
Fig. 2.-Electrical
diagram of suspension of ellipsoids.
The quantities u(O), ~ ( 0 )and u( a), e( a) are the values of u and e at frequencies below and above all dispersion regions, respectively. The spectral position of the dispersion region, which is associated with ellipsoids having axis 2a parallel ivith the field, can be defined by frequency
which is the frequency a t which u and E are the arithmetical means of their respective values above and below the region. When u2/ul > e2/e1, the value of ma decreases when xa increases. The opposite is the case when u2/u1 < e2/e1. Since xa, 2 Xb 2 xo, the dispersion regions are therefore placed in the spectrum as follows. When uZ/u1 > e2/el:nRI nb S no (15) ul/ul < e 2 / e I : n , 2 n b 2 n, (16) If the electrical characteristics of the phases are the same in a suspension of parallel ellipsoids of I'Otation and in a suspension of spheres, it will also be seen that, when u2/u1 > e2/e1, the dispersion region in the former system always lies below that in the latter, when the field is parallel with a major axis and above it when the field is parallel with a minor axis, while the opposite is the case when u2/ul < d E 1 .
If nl and are frequencies below and above a dispersion region, respectively, the apparent dielectric loss factor, associated wit,h t,hia region and measured at the frequency n, may he defined by t;tn 6 =
U(II) ~
~
- u(n,) ~(17)
( n/2)4 n )
It will be verified easily, then, that which is the relation characteristic of a Debye system." Inside a dispersion region, an ellipsoidal suspension behaves therefore dielectrically as a Debye system with the relaxation time
(11) P. Debye, "Polar Molecules," Chemical Catalog Company,
New Ybrk, N. Y.,1929,pp. 92-94.
From equation 21, me obtain the permittivity of the suspension when the frequency is so low that the field is determined by the conductivities of the phases. From equation 22 we obtain the conductivity when the frequency is so high that the field is determined by the permittivities. The dispersion in conductivity u( a) - a(0) and permittivity ~ ( 0 ) E( a ) over the whole dispersion region is given by CR,-l and C47rCaof equations (4
(a)
12 and 13. Both quantities are always positive and exhibit, when plotted against u2/e2,a minimum at U Z / E Z = q / e l , as could be predicted since the lines of electric force through a heterogeneous system composed of pure conductors or pure dielectrics are distributed in such a manner that the energy consumed or stored is minimum. When the difference between uz/e2 and q / e l is only moderate, the dispersion in conductivity and permittivity of a suspension is therefore relatively small. The dispersion equations give us the means of calculating the electric characteristics of the suspended phase, when the other quantities have been measured. When the ellipsoids are parallel (including sphere and cylinder) solution of equations 5a and 6a with respect to u2and e2 leads to equations 29 and 30 of the following section. When the axes are distributed at random, the calculation is generally best carried out by trial and error, except when t'he frequency is so high or so low that equations 22 and 23 or 20 and 21, respectively, are valid. When the frequency is very high, for example, the value of €2 can be obt'ained, from equation 23, by solving a cubic (quadratic in the case of ellipsoids of rotation) algebraic equation and u2 can thereafter be obtained from equation 22. Higher Volume Concentrations.-The following treatment deals with suspensions in which the suspended particles are all alike and distributed at random in space. Under these conditions, the conductivity equation for the suspension was derived in ref. 2 on the assumption that the effective internal field acting upon a suspended particle is equal t o the average field in the suspending
Dec., 19Fj3
A(U*UI)B(UZU1)
u =
937
MAXWELL-WAQNER DISPERSION IN SUSPENSIONS OF ELLIPSOIDS
+ na S A
+ n2 5 + 1)2
(€,€,)B(€2€1)
(za
Pel2
n2
01
[B(u2u1)12
+ ';z [B(€Z€l)l~
k=k*+
(25)
(ki
- kz)(l - P )
1+:xk* (3 '
712
[B(u2u1)12
(z - z)
+f
[B(€2€1)12
A closer examination of these equations will show that they represent the electric behavior of the same two model systems-an electrical circuit of the form shown in Fig. 2 (with one CaRabranch) and a Debye dipole system-which were found to represent the behavior of a low concentration SUSpension. The dispersion in conductivity and permittivity can be written in the forms
(31)
sion equations are rather complicated, we shall set down only the expressions for u and B a t frequencies above all dispersion regions. Analogous expressions are obtained a t frequencies below the dispersion regions.
As in the case of the low concentration suspension, the expressions contain the difference between a z / q and eZ/el in its squared form, showing again (12) H.Fricke and S. Morse, Phys. Reo., 25, 361 (1925). (13) H.Fricke, Physics, 1, 106 (1931). (14) E. Ponder, J . Phvaiol., 85, 439 (1935). (15) E. E'. Burton and L. G . Turnbull, Proc. Roy. SOC.(London), A158,182 (1937). (16) S. Velick and M. Gorin, J . Gen. Physiol., 2 3 , 753 (1940).
The calculation of az and e2 from these equations can be carried out in a manner similar to that in the low concentration case.
