R893 I Philips Res. Repts 30, 83*-90*, 1~75 Issue in honour of C. J. Bouwkamp EFFECTIVE CONDUCTIVITY, DIELECTRIC CONSTANT AND PERMEABILITY OF A DILUTE SUSPENSION*) by Joseph B. KELLER Courant Institute of Mathematical Sciences, New York University New York, U.S.A. 1. Introduetion. (Received November 6, 1974) Let us consider a medium of dielectric constant So, magnetic permeability {to and electrical conductivity ad containing a suspension of particles of different electromagnetic properties. We wish to calculate s", {t* and a*, the effective or bulk properties of the composite medium for static electromagnetic fields. We shall do so provided that the concentration of particles is so small that electromagnetic interaction between them is negligible. Thus our result will be valid only up to and including terms of first order in the concentration. However, no restrietion will be placed upon the electromagnetic properties of the particles, so that they may be very different from those of the surrounding medium. Furthermore all the parameters ofthe medium and the particles may be tensors. The method to be used is that given by Landau and Lifshitz 1) for calculating a*. An extension to higher concentrations is given in sec. 5. Corresponding results for high concentrations of perfectly conducting spheres or cylinders arranged in a cubic or square lattice have been given by the author 2) and numerical results for arbitrary concentrations by Keller and Sachs 3). When the media inside and outside the cylinders in a square lattice are interchanged, the effective conductivity changes in a manner described by the author 4). 2. Derivation of general expressions for e* and a* We begin by considering a large region R of the composite medium having volume Vand containing N particles. Let the region R be immersed in a uniform static electric field Eo and let the resulting electric field in R be E(x). Then the displacement D(x) and the current J(x) are given by and D = e(x)e (1) J = c(x) E. (2) Here e(x) and a(x) are the dielectric constant and conductivity at the point x *) Research supported in part by the Office of Naval Research under Contract No. N00014-67-A-0467-0015.
84* JOSEPH B. KELLER in the composite medium. Let J= V-1 f f(x) dx R denote the average over R of any functionf(x). Then we define 8* and a* by the relations D= B*E (3) and To determine B* we note that J= a*e. (4) n- BO.E = V-I f CD - BO E) dx R = V-I f (B - BO) E dx R N = V-I If(B-Bo)Edx. (5) '=1 RI In (5) we have used the fact that B = Bo except within the regions Rio i = 1, o, N, occupied by the particles. Now if (3) is used for D, we can write (5) as (B* - Bo) E = N V-I ( f (B - 80) E dx ). (6) Here N (g,) = N-1 L s. '=1 denotes the average of a function over the particles. Let us define p, the average polarizability tensor of a particle, by the equation RI p E = ( f (B - Bo) E dx). (7) RI Then (6) and (7) yield the result B* = 80 + N V-I p. (8) In exactly the same way we find a* = ao + N V- 1 q. (9)
EFFECTIVE CONDUCTIVITY, DIELECTRIC CONSTANT, PERMEABILITY OF A SUSPENSION 85* Here q, the average conductivity tensor of a particle, is defined by (10) If the region R is immersed in a uniform static magnetic field Ho, we denote by H(x) the resulting magnetic field. Then the magnetic induction B(x) is given by B = lax) Hand /h* is defined by B = fl-* H. Proceeding as above we find fl-* = /ho+ N V-,l m. (11) Here the average magnetizability tensor m of a particle is defined by (12) The results (8), (9) and (11) for e*, a* and fl-* together with the definitions (7), (10) and (12) of p, q and m are our general results. They show that the difference between any effective property of the composite medium and the corresponding property of the original medium is proportional to the number density of suspended particles. lts coefficient depends upon the entire particle distribution in general. However, it is characteristic of a single particle when the number density is small enough. This is the case to be considered in the next two sections. 3. Determination of p, q and m To determine pand q we must consider a medium of properties eo, ao and fl-o with a uniform applied electric field Eo and find the resulting field E(x). To do so we let a(x) and e(x) denote the conductivity and dielectric constant of the particle at the point x, if x is in RI, and of the surrounding medium if x is outside RI. Since 'V J = 0 for a static field, it follows from (2) that 'V (a E) = o. In addition 'V xe = 0 so E = 'V cp where the potential cp is a scalar. Therefore 'V (a 'V cp) = O. (13) We recall that the tangential component of E and the normal component of J are continuous across surfaces of discontinuity of the medium (i.e. of o(x) and e(x)). This implies that cp and the normal component of a 'V cp are continuous across discontinuity surfaces. At infinity cp must tend to the external potential Eo. x. Equation (13), these continuity conditions and the condition at infinity suffice to determine cp, and therefore E, provided a(x) =1= O. Then (7)
86* JOSEPH B. KELLER and (10) yieldp and q. The additional Maxwell equation \l (e E) = e determines the charge density e(x). If o(x) = 0 within any region Ro, (13) is identically satisfied there. In such a region e(x) can be prescribed arbitrarily and the equation \l (e \l cp) = e(x) must hold there. When e(x) = 0 in Ro this becomes \1. (e \l cp) = O. (14) Equation (14) in Ro and (13) outside Ro, together with the continuity conditions and the condition at infinity, serve to determine cp and therefore E. Then (7) and (10) yield pand q. If c(x) = 0 and e(x) = 0 everywhere, then Ro is the entire space and (I4) holds everywhere. To determine m we consider a single particle in a medium of properties eo, <io and P,o with a uniform applied magnetic field Ho. To find H(x) we utilize the static equation \1 xh = 0 to write H = \l1p where 1p is a scalar. Then the equation \l B = 0 yields \1. p,h = 0 from which we obtain \1. (p, \l1p) = o. (15) The continuity of the tangential component of H and the normal component of B at discontinuities implies that 1p and the normal component of p, \l1p are continuous. At infinity 1p must tend to Ho. x. These conditions and (15) determine 1p and thus H(x). Then (I2) yields m. From the above considerations the following simple theorems result at once. Theorem 1. If c(x) = (X e(x), where (X is a constant, then q = (Xpand <i* = (X e*. Proof Comparison of (7) and (IO) yields the first part, while (8) and (9) yield the second part. Theorem 2. If <i(x) = (X p,(x), where (X is a constant, then q = (X m and <i* = (X p,*. Proof Comparison of (13) and (15) shows that cp= 1p if Eo = Ho. Then (10) and (12) show that q = (X m. Finally (9) and (11) yield <i* = (X p,*. Theorem 3. Let el(x) and <il(x) be the properties of a particle in a medium and e2(x), <iz(x) those of a different type of particle in another medium. Let <il(x) = 0 and <iz(x) = (X el(x). Then q2 = (XPI and <i2* = (X e l *. Proof In medium one, CPl satisfies (14) with e = el' while in medium two CP2 satisfies (13) with <i = <i2' By the hypothesis these equations are identical and therefore so are the solutions. Thus CPl = CP2 and El = E2' so (7) and (10) yield q2 = (XPI while (8) and (9) yield <i2* = (X el *.
EFFECTIVE CONDUCTIVITY, DIELECTRIC CONSTANT, PERMEABILITY OF A SUSPENSION 87* 4. Spherically symmetric particles Let us assume that a(x) is a scalar depending only upon r, the distance from the centre of a particle. Then (13) becomes Let us choose the direction of the applied field Eo to be the z-axis and let the magnitude of Eo be unity. Then rp must tend to z as r becomes infinite. In polar coordinates r, 8, '1jJ with the z-axis as the polar axis, we have z = r cos 8, so (16) cp '" r cos 8 at r = 00. (17) To solve (16) and (17) we set cp = fer) cos e. Then (16) and (17) become I" + (2 r- 1 + a- 1 ar)f' - 2 r=? 1= 0, (18) at r = 00. At discontinuities of a(r), if any, the continuity of cp and a rpr require that I and af' be continuous. In terms oflwe can evaluate q from (10). By symmetry it is clear that only the z-component of the integral in (10) is not zero. Then since the z-component of E is rpz = rpr cos 8 - r- 1 rpo sin 8, (10) yields (19) 0() " q = 2:n;f f [a(r) - ao] [I' cos- 8 +,.-1 Isin 2 8] sin 8 d8 r 2 dr. (20) o 0 Integration with respect to 8 gives 4:n; foo q =3 [a(r) - ao] [r 2 1'(r) + 2 rfer)] dr. o (21) If e(x) is also a scalar function of r, (7) shows that p is a scalar given by 4:n; foo p =3 [eer) - eo] [,.21'(") + 2 rfer)] dr. o (22) Let us now consider some examples. Example I. A conducting sphere in a conducting medium Consider a spherical particle of radius a and constant conductivity a1 in a medium of conductivity ao. From (18), (19) and the continuity conditions we
88* JOSEPH B. KELLER have and for fer) = I' + a 3 (ao - al) (2ao + al)-l 1'-2 for r~a. Thus (21) and (22) yield 4n a 3 3ao (al - ao) q=-- (23) 3 2ao+al and a 12 n ao f p = ree/') - eo] 1'2 dr. (24) 2ao + al o If eer) is a constant el inside the particle, (24) becomes 4n a 3 3ao (el - eo) p=------- 3 2ao + al If ft = ftl for I' < a and ft = fto for /' > a, where ftl and fto are constants, a similar calculation, or the use of theorem 2, yields 4n a 3 3fto (ftl - fto) m=------- 3 2fto + ftl The result for a* when (23) is used in (9) was obtained by Landau and Lifshitz 1). Example Il. A conducting particle in a non-conducting medium Suppose that a spherical particle of radius a and constant conductivity a l is surrounded by a medium of conductivity ao = 0 and constant dielectric constant eo. Then (13) holds for I' < a and (14) holds for /' > a. At the interface cp and a CPr must be continuous. We may still seek cp in the form cp = f(/') cos (J. Then (13) shows thatfsatisfies (18) for /' < a, and since al is constant, alr = O. For I' > a, (14) shows that f satisfies (18) with a replaced by eo, and eo r = O. From these equations and continuity conditions, as well as (19), it follows that f(/') = 0 for I' < a and j'(r) = I' - a 3 /,-2 for I' > a. Then (21) and (22) yield q = 0 and p = O. Then (8) becomes e* = eo + oen V- l ) while (9) becomes a* = 0 + oen V- l ). The results e* = eo and a* = 0 are exact in this case for all values of N V-l provided that particles in contact do not form a path through the entire region. This follows because, from (13), cp is harmonic inside the particle, while from the continuity of a cp" CPr = 0 at the surface. Thus cp = constant inside the particle so E = 0 there. Then (5) shows that e* = eo and similarly a* = O. (25) (26)
EFFECTIVE CONDUCTIVITY. DIELECTRIC CONSTANT. PERMEABILITY OP A SUSPENSION 89* These results are independent of the shapes of the particles and are also true when a is not constant inside them, provided it is not zero. We state this result as theorem 4. Theorem 4. For perfectly conducting particles in a non-conducting medium, s* = So and (f* = 0 provided that particles in contact do not form a path across the entire region. The conclusion that s* = So for conducting particles in a non-conducting medium is in disagreement with the well-known results for this case. This is because those results are based upon a different definition of s* from that which we have used. It suggests that our definition may not be so useful in this case. However, we can recover the usual results by putting a = 0, considering the particle to have dielectric constant SI' and then letting SI tend to infinity. This is shown in sec. 4, example IV. Example Ill. A conducting sphere surrounded by a spherical layer in a conducting medium Let us consider a spherical particle of radius a having conductivity (f2 for o < r < b and conductivity o1 for b < r < a, in a medium of conductivity (fo, where (fo, (fl and (f2 are constants. The solution of (18) and (19) for this case, subject to continuity off and a f', is 3(fl (a)3 f= - yr (f2-(ft b ' 0< r < b; (27) b < r < a; (28) (29) where v = 3(fo ((f2 -(fl{2 ((f2 By using this solution in (21) we obtain - (fl)((f1 - (fo) + (2(fo + (fl)(2(fl + (f2) (~rj-l. (30) If S = S2 for 0 < r < b, S = SI for b < r < a and S = So for r > a, where
90* JOSEPH B. KELLER So, SI and S2 are constants, (22) yields If # = #2' #1 and #0 in the regions 0 < r < b, b < r < a and r > a respectively, where the #1 are constants, then m is ~givenby the right side of (31) with al replaced by #1' Example IV. A non-conducting sphere in a non-conducting medium Now we consider a spherical particle of radius a with dielectric constant el in a medium of dielectric constant So, with a = 0 both inside and outside the particle. By applying theorem 3 to the result (23) we obtain 4n a 3 3eo (SI - eo) p=--.. 3 2so + SI (33) As SI/SO ~ 00 we see that p ~ 4n a 3 So, which is the polarizability of a conducting sphere. 5. Extension to higher concentrations The results (8), (9) and (11) can be extended to higher concentrations by supposing that each particle is surrounded by the effective medium, rather than by the original medium. Thus in (8) p = peso) depends upon eo, so we replace it by p(s*). Then (8) becomes the following equation, which is to be solved for s": S* = So + N V-I p(s*). (34) Similarly, (9) and (11) become equations to be solved for a* and #* respectively. As an example, for non-conducting spheres in a non-conducting medium, we use (33) in (34) to get 4n a 3 N 3s* (SI - s") s* = So +. 3V 2s* + SI (35) This is a quadratic equation for e*, the positive root of which is to be used. REFERENCES 1) L. Landau and E. M. Lifshitz, Electrodynamics of continuous media, Pergamon Press, London, 1960. 2) J. B. Keller, J. appl. Phys. 34, 991. 1963. 3) H. B. Keller and D. Sachs, J. appl. Phys. 35, 537-538, 1964. 4) J. B. Ke lle r, J. math. Phys, 4. 548-549, 1964.