I. Multipole expansions - CiteSeerX

Dielectric function for a material containing hyperspherical inclusions to O(c2 ) I. Multipole expansions By T. C. C h o y1 , A r i s Alexopoulos1 a n d M. F. T h o r p e2 1

Department of Physics, Monash University, Clayton, Victoria 3168, Australia 2 Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA Received 16 October 1997; accepted 3 April 1998

By averaging over pairs of hyperspheres, we have obtained the dielectric function for a binary mixture containing hyperspherical inclusions up to order c2 , where c is the volume fraction of inclusions. The method used is based on multipole expansions for the potential of two spheres in a uniform field and is a generalization of the method of Jeffrey to d-dimensional space. Numerical results are presented for the second-order coefficient κ in the low-c expansion of the dielectric constant for arbitrary d; these verify earlier known results, as well as showing the dependence of κ on dimensionality, which is particularly simple as d → 1 and as d → ∞. Keywords: dielectric inclusions; multipole expansions; images; hyperspheres; two inclusions; concentration of inclusions

1. Introduction Ever since the time of Maxwell (1873), the study of the properties of random mixtures, or suspensions containing a low-volume fraction of inclusions, has been of much interest. The problems concerned are applicable to the study of electrical conduction, thermal conduction, electric permittivity, magnetic permeability and others, by virtue of the universality of Laplace’s equation. In this paper, we will concentrate on the case of dielectric inclusions; other cases being easily obtainable by renaming the symbols. Maxwell (1873) provided the exact first-order coefficient to O(c) for a system of spherical inclusions. Other similar problems to O(c), not within the universal class, have also attracted much interest, for example the O(c) coefficient for the viscosity of a suspension containing a system of hard spheres was found by Einstein (1906). More recent exact results for the O(c) coefficient of electrical conductivity for inclusions of other shapes in two dimensions were obtained by Thorpe (1992), using conformal mapping—a method favoured by Maxwell. It took nearly 100 years before any serious quantitative studies were made of the second-order O(c2 ) coefficient, the original work being done by Batchelor (1972), and Batchelor & Green (1972), who unfortunately undertook initially to study the more complex fluid suspensions problem. To this day, few exact results are known (Batchelor 1974, 1977). We remark here that the second-order coefficient we are interested in is related to the so-called Huggins coefficient κH for fluid suspensions, i.e. the expansion of the viscosity η is given by η = η0 (1 + [η]c + κH ([η]c)2 + . . . ), Proc. R. Soc. Lond. A (1998) 454, 1973–1992 Printed in Great Britain

1973

c 1998 The Royal Society

TEX Paper

1974

T. C. Choy and others

where [η] is the first-order coefficient. Unfortunately, no established name is given to the corresponding coefficient for dielectric systems in the literature. We will define it to be κ, the quantity of central focus in this paper, given by the low-volume fraction expansion as (1.1)  = 0 (1 + []c + κc2 + . . . ). Here [] the first-order coefficient, as given by Maxwell (1873), is for a d-dimensional sphere:   1 − 0 [] = dβ ≡ d , (1.2) 1 + (d − 1)0 where β, the expression in brackets, is proportional to the polarizability associated with an isolated spherical inclusion with dielectric constant 1 in the host medium with dielectric constant 0 . Of particular interest to us here is the work of Jeffrey (1973), who essentially transferred the Batchelor–Green multipole expansion formalism to the more tractable problem of dielectric inclusions, which remains a classic work to this day. Binns & Lawrenson (1973) and more recently Djordjevi´c et al . (1996) have also studied this problem, but in two dimensions, where the method of images simplifies, leading to an infinite series with much better convergence properties than the multipole expansion method. Such a closed-form solution for all images does not exist for higher dimensions, as we will see in the companion paper (Choy et al . 1998), hereafter referred to as paper II (this issue). Nonetheless, the image method is still very useful and it seems that a direct link between the method of images and the multipole expansions would provide additional insight, as well as a means to probe the poor convergence properties of the latter. The equivalence of the two methods will be demonstrated in paper II. The history of this problem seems to be plagued by a misunderstood conditionally convergent integral in the final expression for the dielectric function; a difficulty first encountered by Lord Rayleigh (1892) in his famous regular array of spheres problem. Batchelor (1972) and Jeffrey (1973) appear to favour a renormalization procedure to treat this problematic integral, while Felderhof et al . (1982) rederive their result, bypassing this difficulty via a virial series expansion analogous to those done in statistical mechanics. In this paper we will show that this apparent difficulty can be avoided following the earlier method of first integrating over orientations (Djordjevi´c et al . 1996). The troublesome divergent dipole integral term is removed naturally in this approach, and as part of the motivation for this work we will rederive and show complete agreement with Jeffrey (1973) for d = 3 and Djordjevi´c et al . (1996) for d = 2. In this first paper, we will generalize Jeffrey’s solution to arbitrary integer dimensions d using some known and some newly derived properties of d-dimensional spherical harmonics. In this way we will see how all previous results for d = 1, 2, 3, can be developed in a general d-framework. In the companion paper II, we will also derive results via a different method, i.e. the method of images; albeit through a considerably more complex procedure than in the two-dimensional case of Djordjevi´c et al . (1996). This method has the advantage of more rapid convergence than that of Jeffrey (1973) and can be shown to be equivalent to the latter via an order-by-order expansion for the polarization as a function of ξ = a/R, where a is the sphere radius and R the separation between the two spheres. It is, however, computationally much Proc. R. Soc. Lond. A (1998)

Dielectric function to O(c2 ) for hyperspheres

1975

more involved and less direct for d > 2. Another motivation for this work is to derive and collect the results for arbitrary d for future studies of effective medium theories that seek to improve the poor value of κ in the latter theories, particularly in the perfectly conducting limit. The arrangement of this paper is as follows. In § 2 we derive the general multipole expansion for arbitrary d and show how Jeffrey’s (1973) solution can be generalized. Various mathematical results and theorems regarding d-dimensional spherical harmonics will be obtained along the way, which we have included in the appendices. In § 3 we show the results obtained for dimension d = 1, 2, 3, 4. In § 4, we discuss the results obtained for higher dimensions and study the trend of κ versus dimensionality. In § 5 we will conclude by summarizing the present work.

2. The d-dimensional Jeffrey solution The classical problem of two spheres of radii a1,2 at a distance apart of R in a uniform electric field E has been studied by a variety of methods in the past. Among these are the method of bispherical coordinates (Jeffery 1912; Moons & Spencer 1988; Morse & Feshbach 1953), the method of images (Binns & Lawrenson 1973; Landau et al . 1984) and the twin spherical harmonics expansion (Ross 1968; Jeffrey 1973). Although Laplace’s equation is separable in bispherical coordinates, this method turns out to be less well suited to the problem of two spheres, as the boundary conditions lead to a set of four simultaneous difference equations for which there are no known solutions (Davis 1964). In fact the twin spherical harmonics expansion (Ross 1968) turns out to be the most appropriate method as long as a method for transferring spherical harmonics, as obtained by Hobson (1931) for d = 3, is generalizable. A major step is therefore to generalize Hobson’s theorem to arbitrary d, which we develop in Appendices A and B of this work. Before proceeding we briefly mention the method of images. The work of Djordjevi´c et al . (1996) seems to show that this is a promising approach. Unfortunately, further studies (see paper II) reveal that the general d-dimensional image method is not as mathematically and computationally convenient as the multipole series, except in appropriate limits; in particular when the inclusions are perfect conductors or holes. The image approach is very effective in the weak scattering limit. The dimension d = 2 is in fact quite special, and that in general multiple dipole images lead to line dipoles and line charges that rapidly become intractable. Nonetheless, with modern algebraic manipulation packages some very useful results are obtainable, and convergence is surprisingly fast in all cases. This we will discuss in paper II. The coordinate system we choose is that of figure 1 as in Jeffrey (1973), but now we visualize two hyper spheres as embedded in a d-dimensional space. (a) Parallel case We first consider the case where the electric field E0 is parallel to R, the vector joining the centres of the two spheres, which we take as the z-axis. The twin spherical harmonic expansion for the potential inside each sphere is given by  n ∞ X ri E0 dn(i) Cnp/2 (cos θi ), (2.1) Φ(i) = E0 z + a i n=0 Proc. R. Soc. Lond. A (1998)

1976

T. C. Choy and others

r2

r1 θ2

2 θ1

R

1 Figure 1. Twin spherical coordinates. We note that in d-dimensions there is a common azimuthal angle φ, around the line joining the two centres, and p = d−2 polar angles. The two polar angles shown, θ1 and θ2 , have a special significance by virtue of the axial symmetry.

where i = 1, 2 labels the spheres, and the electric field is taken to be in the direction of the centres, 1 to 2, along the vector R. For the potential outside the spheres,    n  n ∞ X a1 a2 E0 gn(1) Cnp/2 (cos θ1 ) + gn(2) Cnp/2 (cos θ2 ) . (2.2) Φ = E0 z + r1 r2 n=0 p/2

In the above equations, the Cn (cos θ) are Gegenbauer polynomials in the standard notation of Erd´elyi et al . (1953), but we will also refer to them as the generalized pLegendre polynomial of order n (see Hochstadt 1971) and use the notation Pn (cos θ) (i) (i) for familiarity and simplicity. The coefficients dn and gn are to be found by the boundary conditions on the surface of each sphere, which require the continuity of the potential and its derivative normal to the surface, weighted with the appropriate dielectric constant. In order to achieve this goal, a key formula for the shifting of the spherical harmonics from one centre to the other has to be obtained. The proof of this formula is somewhat involved and the details are given in Appendix A. It is a generalization of a theorem by Hobson (1931) and here we state the result:  n+p X  s  n+p ∞  ai ai p + s + n − 1 r3−i Pn (cos θi ) = Ps (cos θ3−i ). (2.3) p+s−1 ri R Rs s=0 Using this formula in the boundary conditions and letting αi = i /0 , we have the two equations s+p  n  ∞  X a3−i ai p+s+n−1 gs(3−i) , (2.4) dn(i) = gn(i) + p+n−1 R R s=0 and αi dn(i) + (1 + pn)gn(i) −

s+p  n  ∞  X a3−i ai p+s+n−1 s=0

Proc. R. Soc. Lond. A (1998)

p+n−1

R

R

gs(3−i)

1 = (−1)i (αi − 1)ai δ1,n , p

(2.5)

Dielectric function to O(c2 ) for hyperspheres

1977

where we have used the property of the p-Legendre polynomial: P1 (x) = px in (i) (i) deriving the latter formula. The solution for the coefficients dn and gn in (2.4) and (2.5) determine the potentials (2.1) and (2.2) uniquely. We now turn to the perpendicular case. (b) Perpendicular case In defining the z-axis as the axis along the centre of the two spheres, the case of the electric field perpendicular to this axis admits d − 1 possible directions. It is clear that these d − 1 cases are degenerate because of the axial symmetry. Let us first set up the notation for the generalized associated p-Legendre polynomials as follows: Pnm (cos θ) = (−1)m sinm θ

dm Pn (cos θ). d(cos θ)m

(2.6)

Now the spherical harmonic expansion for the perpendicular case is dictated by the direction of the field in the following way. Recall that the polar coordinates in d = p + 2-dimensional space are related to the Cartesian coordinates by x(0) = r cos θ(1) x(1) = r sin θ(1) cos θ(2) x(2) = r sin θ(1) sin θ(2) cos θ(3) .... Then the multipole expansion we seek is   n  ∞ X x(j) (i) (i) ri 1 Φ = E0 x(j) + E0 dn Pn (cos θi,(j) ) . ai r sin θ(1) n=1

(2.7)

(2.8)

Without loss of generality we consider the most convenient choice with the perpendicular field along the first orthogonal axis x(1) (counting from zero), where x(0) = z. Thus (2.8) becomes  n ∞ X ri E0 dn(i) Pn1 (cos θi,(2) ) cos θi,(2) . (2.9) Φ(i) = E0 x(1) + a i n=1 This choice of axes is identical to that of Jeffrey (1973). The other choices can be treated in a similar way since the additional angles are superfluous when we come to match the boundary conditions, as they only require the continuity of the potential and its derivative along the radial rˆi directions. However, the calculation for the flux integrals for these cases are non-trivial. They require careful consideration in order to verify the flux formula for the perpendicular case in general. We postpone these discussions to the next subsections. The potential outside the spheres is written in a similar way:   n ∞ X (1) a1 Φ = E0 x(1) + E0 gn Pn1 (cos θ1,(1) ) cos θ1,(2) r 1 n=1  n  (2) a2 1 + gn Pn (cos θ2,(1) ) cos θ2,(2) . (2.10) r2 Proc. R. Soc. Lond. A (1998)

1978

T. C. Choy and others

The fundamental shift formula for this case is cumbersome to prove (see Appendix B). The result is given by the formula  n+p X  s  n+p ∞  ai ai p + s + n − 1 r3−i Pn1 (cos θi ) = Ps1 (cos θ3−i ). (2.11) s p+s ri R R s=1 Using this result we can again match boundary conditions to obtain the analogous equations to (2.4) and (2.5) as s+p  n  ∞  X a3−i ai p+s+n−1 (i) (i) gs(3−i) , (2.12) dn = gn + p+n R R s=1 and αi dn(i)

+ (1 +

pn)gn(i)

−

s+p  n  ∞  X a3−i ai p+s+n−1 s=1

p+n

R

R

gs(3−i)

1 = (αi − 1)ai δ1,n . (2.13) p

In deriving the latter result, we have used the property of the associated p-Legendre polynomial: P11 (cos θ) = −p sin θ. Note that by virtue of the degeneracy mentioned (i) (i) above, all the dn and gn are identical for any perpendicular direction of the field. To complete the solution we now have to evaluate the flux integral for each sphere given by Z (i) (2.14) Sk = 0 (αi − 1) ∇Φ(i) dV, where the integral is over the volume of the sphere (Jeffrey 1973). We recall that the d-dimensional gradient operator is given by (Erd´elyi et al . 1953) ∇ = rˆ

1 ∂ 1 ∂ 1 ∂ ∂ + θˆ(1) + θˆ(2) + θˆ(3) ∂r r ∂θ(1) r sin θ(1) ∂θ(2) r sin θ(1) sin θ(2) ∂θ(3) 1 ∂ . (2.15) + · · · + φˆ r sin θ(1) sin θ(2) . . . sin θ(p) ∂φ

(i) Parallel flux integral It is necessary to consider the parallel and perpendicular cases separately. The parallel case is the simpler case as here the potential depends only on r and θ. Then Z

∇Φ(i) dV =

ωp adi E0 zˆ + d

Z X ∞ n=1

 nrin−1 Pn (cos θi ) dV rˆ ani  n−1  Z X ∞ (i) ri + E0 dn Pn1 (cos θi ) dV θˆi , n a i n=1

E0 dn(i)



(2.16)

where ωp = 2π 1+p/2 /Γ(1 + 12 p) is the p-solid angle. Using the d-dimensional volume element, we note that only the projections of all unit vectors onto the zˆ direction Proc. R. Soc. Lond. A (1998)

Dielectric function to O(c2 ) for hyperspheres

1979

survive the angular integrations. Then we can obtain Z ωp adi ∇Φ(i) dV = E0 zˆ d Z  ∞ X nap+1 i + ωp−1 sinp θi cos θi Pn (cos θi ) dθi zˆ E0 dn(i) n + p + 1 n=1 Z  ∞ X nap+1 (i) i sinp θi sin θi Pn1 (cos θi ) dθi zˆ. (2.17) − ωp−1 E0 dn n + p + 1 n=1 Using the orthogonality properties of the p-Legendre polynomials, we see that only the n = 1 term survives, with the result  p+1  Z ωp adi (i) ai (i) E0 + ωp−1 E0 d1 ∇Φ dV = d d Z  Z p p 2 2 × sin θi p cos θi dθi + sin θi p sin θi dθi zˆ  (i)  d1 ωp adi E0 zˆ 1 + p . = d ai

(2.18)

From the parity of the polar angles in figure 1, we obtain the final result for the parallel flux,  (i)  (i) d 3−i d1 z 1 + (−1) p , (2.19) Sk = υi E0 0 (αi − 1)ˆ ai where the volume of the sphere i is denoted by υid = ωp adi /d. (ii) Perpendicular flux integral The flux integral for the perpendicular case is quite troublesome to obtain. By using the results obtained for the case where the field is along x(1) , all other cases follow by symmetry arguments. That is all perpendicular directions are equivalent. Let us consider sphere 1 to start with. From (2.9), the gradient of Φ(1) , say, is given by  n−1  ∞ X nr1 ˆ(1) + E0 d n Pn1 (cos θ(1) ) cos θ(2) rˆ ∇Φ(1) = E0 x n a 1 n=1   ∞ X r1n−1 ∂Pn1 (cos θ(1) ) + E0 d n cos θ(2) θˆ(1) n a ∂θ (1) 1 n=1  n−1  ∞ X sin θ(2) ˆ r − E0 dn 1 n Pn1 (cos θ(1) ) (2.20) θ(2) , a1 sin θ(1) n=1 where all quantities are understood to refer to sphere 1. Here again the integral R dV Rprojects out only the x ˆ(1) terms due to the orthogonality of integrals of the type sinp θ cos θ dθ = 0. This can be seen easily by writing down the unit vectors rˆ, θˆ(1) , θˆ(2) , . . . etc., in the original Cartesian basis. The result with i = 1 or 2, looks Proc. R. Soc. Lond. A (1998)

1980

T. C. Choy and others

somewhat like (2.16): Z ωp adi E0 x ˆ(1) ∇Φ(1) dV = d  n−1  Z X ∞ (i) nri + E0 dn ˆ(1) Pn1 (cos θi ) sin θi,(1) cos2 θi,(2) dV x n a i n=1  n−1  1 Z X ∞ r ∂Pn (cos θi ) + E0 dn(i) i n cos θi,(1) cos2 θi,(2) dV x ˆ(1) a ∂θ i,(1) i n=1  n−1  1 Z X ∞ Pn (cos θi ) (i) ri + E0 dn sin2 θi,(2) dV x ˆ(1) , (2.21) n a sin θ i,(1) i n=1 but there are enough differences to need more work. We first perform the θ(2) integration and, using the results Z Z 1 sinp−1 θ(2) cos2 θ(2) dθ(2) = sinp−1 θ(2) dθ(2) , (2.22) p+1 and

Z

p+1

sin

θ(2) dθ(2)

p = p+1

Z

sinp−1 θ(2) dθ(2) ,

(2.23)

R the factors sinp−1 θ(2) dθ(2) can now be reabsorbed back into the volume integral, so that (2.21) becomes  n−1  Z Z X ∞ 1 nri ∇Φ(1) dV = υ d E0 x ˆ(1) + E0 dn(i) ˆ(1) Pn1 (cos θi ) sin θi,(1) dV x p + 1 n=1 ani  n−1  1 Z X ∞ 1 ∂Pn (cos θi ) (i) ri + E0 dn cos θi,(1) dV x ˆ(1) n p + 1 n=1 ai ∂θi,(1)  n−1  1 Z X ∞ p r Pn (cos θi ) + E0 dn(i) i n dV x ˆ(1) . (2.24) p + 1 n=1 ai sin θi,(1) Upon grouping the terms, we find that the integrand is an expression containing the p-Legendre polynomials of the form p ∂Pn1 + P1 ∂θ(1) sin θ(1) n   p ∂ 2 Pn ∂Pn + cos2 θ(1) 2 , = n sin θ(1) + sin θ(1) ∂θ(1) ∂θ(1)

n sin θ(1) Pn1 + cos θ(1)

(2.25) (2.26)

where we have suppressed the arguments of the Legendre polynomials for convenience. Further progress requires the use of the p-Legendre differential equation. This is given by   1 d p dPn sin θ + n(n + p)Pn = 0 (2.27) sinp θ dθ dθ Proc. R. Soc. Lond. A (1998)

Dielectric function to O(c2 ) for hyperspheres

1981

(see Appendix A). Using this expression, (2.26) can be integrated, and again employing the orthogonality properties of the p-Legendre functions, we obtain   Z Z dPn − n cos θ(1) Pn dθ(1) = −p(1 + p) sinp θ(1) dθ(1) . (n + p) sinp θ(1) sin θ(1) dθ(1) (2.28) The latter, we see again, can be reabsorbed back into the volume integral. Similar considerations apply to sphere 2, which has no parity difference in this case. Thus for the perpendicular flux in the direction x ˆ(1) , we have  (i)  d1 (i) d S⊥ = υi E0 0 (αi − 1)ˆ x(1) 1 − p . (2.29) ai This result is primary to all subsequent derivations of the flux integrals in arbitrary perpendicular directions, which also reduce to this case, as expected from symmetry considerations (see Appendix C). Hence for the field in an arbitrary perpendicular axis x ˆ(j) , where j = 1, 2, 3, . . . , (p + 1), the flux integral takes the form  (i)  d1 (i) d S⊥ = υi E0 0 (αi − 1)ˆ x(j) 1 − p . (2.30) ai Using these results, we can now consider the dielectric function to O(c2 ) in the next section.

3. Symmetrical bispheres The above results complete the generalization of the Jeffrey solution to d dimensions. For the remainder of this paper we will concentrate on the symmetrical case, i.e. a1 = a2 = a and 1 = 2 so that αi = α = 1 /0 . In this case we can easily eliminate dm,n from (2.4), (2.5), (2.12) and (2.13), and by symmetry, we can replace (1) (2) = (−1)(m−1) gm,n = agm,n , gm,n

(3.1)

where following Jeffrey (1973), we have used the notation gm,n , with m = 0, 1 for the parallel, perpendicular case, respectively. Since gm,0 = 0, as there are no point charges within the spheres, we obtain by straightforward manipulations the overall formula   n+p+s ∞  X a n+p+s−1 (m−1) pgm,n + βn = βn δ1,n . (3.2) pgm,s (−1) n+p+m−1 R s=1 In equation (3.2), the βn is a generalized polarizability given by βn =

n(α − 1) . nα + n + p

(3.3)

Using the expression for dm,n obtained above, we can show that the formula for the flux integrals (2.19) and (2.30) is Sm = υ d Em 0 (p + 2)(−1)m−1 pgm,1 ,

(3.4)

where m = 0, 1 for the parallel and perpendicular cases as before. Notice that the quantity pgm,1 is most important, but to obtain the latter we need to solve the Proc. R. Soc. Lond. A (1998)

1982

T. C. Choy and others κ

2

1

-3

-2

-1

1

2

3

log 10 (α ) Figure 2. Graph of κ versus log10 (α) for d = 2. The dashed curve is the result of Djordjevi´c et al . (1996), which is convergent on the scale of this graph, and is shown here for comparison. Here α = i /0 is the ratio of the dielectric constant of the inclusions to that of the host material.

system of equations (3.2). This has to be achieved by expanding the gm,n as a power series in ξ = a/R. The result of this expansion can be written in the form    ∞ X as E·R d R (3.5) As E − Bs S(R) = υ dβ1 0 E − dβ1 0 Rs R2 s=d

(Batchelor 1972; Jeffrey 1973), where β1 = β and we have here explicitly emphasized the R dependence in S. The first few coefficients in the above expansion are readily obtained: Ad = β1 , −β12 ,

A2d = Bd = dβ1 , B2d = dpβ12 ,

Ad+1 = Ad+2 = · · · = A2d−1 = 0, A2d+1 = 0, A2d+2 = −dβ1 β2 . . . , Bd+1 = Bd+2 = · · · = B2d−1 = 0,

B2d+1 = 0,

B2d+2 = 12 d(d + 1)pβ1 β2 . . . ,

(3.6)

and so on. The structure of (3.5) deserves some comment. It has the form of a dipole flux field, the first term being the trivial single-sphere polarizability (1.2) and the rest are due to two-sphere interactions. The first term of the latter is the bare dipole field from the second sphere and it corresponds to the first image correction in image theory (see paper II). All subsequent terms can be viewed as corrections due to multiple images. This is particularly obvious in two dimensions where all the Bs vanish for s > 2. The resultant series can be put into one-to-one correspondence with the continued fraction expansion results of Djordjevi´c et al . (1996). The relation to image theory for d > 2 is more complicated and will be discussed in paper II. We proceed by performing the angular averages first, a necessary order of procedure, as the leading first-image dipole term diverges. The justification for this is that Proc. R. Soc. Lond. A (1998)

Dielectric function to O(c2 ) for hyperspheres

1983

Table 1. Perfectly conducting inclusions (The coefficient κ to seven significant figures, is tabulated for d = 2, 3, 4 for the perfect conductor α → ∞. The series (3.9) is evaluated from (3.5) up to s = n + 1, where n is given by the first column, and the extrapolated value for n → ∞ is also given. The slow convergence towards the known results of Djordjevi´c et al . (1996) for d = 2 and Jeffrey (1973) for d = 3 is particularly apparent. We have not included the results for d = 1 which would be unity for the whole column.) n

d=2

d=3

d=4

20 40 60 80 100 120 140 160 180 200 220

2.722194 2.736795 2.740489 2.742055 2.742884 2.743385 2.743715 2.743945 2.744114 2.744241 2.744340

4.370400 4.454318 4.477818 4.488535 4.494546 4.498351 4.500958 4.502846 4.504271 4.505381 4.506268

5.566036 5.692201 5.724835 5.738464 5.745670 5.750006 5.752853 5.754840 5.756291 5.757389 5.758244

extrapolated known

2.7450 2.744989

4.512 4.51

5.764

it reproduces the results of Batchelor (1972) and Jeffrey (1973) and should thus be equivalent to their more complicated renormalization procedure. Alternatively, one can use the approach of Lord Rayleigh (1892) by considering a finite sample L which is taken to ∞ at the end of the calculation. In performing the angular average, we observe that the field E and the vector R together define a hypersurface in d-space. In view of the degeneracy of this surface, we can for convenience choose the simplest case for the perpendicular field; the other cases correspond to a rotation about the symmetry axis. Thus the average flux is given by Z ∞ Z S¯ = n Rd−1 dR dΩp (cos2 θSk + sin2 θS⊥ ), (3.7) 2a

where n is the number of spheres per unit volume; θ is understood to be the first polar angle θ(1) and dΩp is an element of solid angle. We remark here that this averaging procedure corresponds to a well-stirred suspension in which the second sphere is allowed to occupy all positions with equal probability, subject only to a hard-sphere constraint. This is equivalent to saying that the usual pair correlation function is unity, unless the centres of the hyperspheres are closer than the sum of the two radii, in which case the pair correlation function is zero. Without additional input from the process of manufacture or microstructural information on the sample, this seems to be a reasonable assumption and is used widely. After performing the angular integrals, the divergent term is eliminated and we have the net result X Bs − dAs . (3.8) S¯ = υ d dβ0 c (s − d)2s−d s=2d

Proc. R. Soc. Lond. A (1998)

1984

T. C. Choy and others κ 4

2

-3

-2

-1

1

2

3

log10 (α) Figure 3. Graph of κ versus log10 (α) for d = 3, compared with the results of Jeffrey (1973), shown as dots. Here α = i /0 is the ratio of the dielectric constant of the inclusions to that of the host material.

This expression corresponds to the averaged dipole moment P¯ 0 of Djordjevi´c et al . (1996). Following similar arguments using a Clausius–Mossotti-type formula, we now have the d-dimensional dielectric function to O(c2 ) as (1.1) with κ given by  X Bs − dAs  κ = dβ β + . (3.9) (s − d)2s−d s=2d

In two dimensions, all βn are equal to β and we have shown computationally using symbolic manipulation via Mathematica 3.0, that the sum of the series (3.9) corresponds with Djordjevi´c et al . (1996), up to 100 terms and more. In three dimensions we naturally reproduce the results of Jeffrey (1973). In one dimension, κ = β 2 is an exact result (the conductances which are equivalent to the dielectric constants add in parallel to give /0 = (1 − cβ)−1 )), and all βn collapse to 1 − (1/α) and only the parallel case with g1 = β survives the limit. Thus we have verified our general d solution against all known limits. In figure 2 we have plotted κ versus log10 (α) for d = 2. We have also plotted the results of Djordjevi´c et al . (1996) on the same graph, which is a series in β to O(β 37 ). The latter has a much faster convergence than our series (3.9) obtained from (3.5) to O(ξ 21 ), before averaging, which is shown for comparison. Of particular interest is the case of the perfect conductor, where α → ∞ and thus β → 1. In table 1 we have summarized some results of κ to show the convergence property of the multipole expansion. The last row in table 1 are the known results for d = 2 of Djordjevi´c et al . (1996) and for d = 3 Jeffrey (1973). Jeffrey presumably and admirably obtained the expansion (3.9) to over 100 terms by hand and our results are in agreement with his to three significant figures. Our program, which iterates (3.2) by symbolic manipulation using Mathematica 3.0, runs for about two days on a Silicon Graphics workstation Proc. R. Soc. Lond. A (1998)

Dielectric function to O(c2 ) for hyperspheres

1985

Table 2. Hole inclusions (The coefficient κ to seven significant figures, is tabulated for d = 2, 3, 4 in the limit of holes where α → 0. The series (3.9) is evaluated from (3.5) up to s = n + 1, where n is given by the first column, and the extrapolated value for n → ∞ is also given. Note the much faster convergence towards the known results of Djordjevi´c et al . (1996) and Jeffrey (1973) for d = 2 and 3, respectively. We have not included the results for d = 1 which would be infinity for the whole column.) n

d=2

d=3

d=4

20 40 60 80 100 120

1.277806 1.263205 1.259511 1.257945 1.257116 1.256615

0.590254 0.588277 0.588023 0.587937 0.587901 0.587883

0.381904 0.381573 0.381476 0.381456 0.381447 0.381444

extrapolated known

1.2550 1.255284

0.5878 0.588

0.3814

for the n = 160 value. Notice that the series clearly has not yet converged to three significant figures even for n = 220, but a n−3/2 plot of the data extrapolates to κ = 4.512, consistent with Jeffrey’s value of 4.51. Figure 3 shows the plot for d = 3 to be compared with figure 2 of Jeffrey (1973). We also observe the slower convergence as the dimensionality d increases. Before concluding this section we tabulate the results in the limit for holes, where α → 0. The two-dimensional case is unique here, as all βn → −1, and there is a duality relation: κ − 2 for holes, is equal to 2 − κ for perfectly conducting inclusions (see (3.9)), and F (β) is an odd function of β as given by Djordjevi´c et al . (1996). Thus the sum of the κ for holes and perfectly conducting inclusions is 4 in two dimensions, as can be seen from tables 1 and 2. This duality property does not hold in other dimensions. These results are summarized in table 2.

4. Higher dimensions Included in tables 1 and 2 are the results for four dimensions. Again convergence is better at the holes’ limit and poorer at the perfect conducting limit. In fact, for the latter, we can see that the convergence for d = 4 is even slower than for d = 3. In figure 4 we have also plotted the results for several dimensions. This raises the interesting question of the large-d limit. The equations (3.2) are somewhat difficult to study in the limit d → ∞. One may think that the interaction between the spheres will vanish, and this is essentially what happens, as is shown in paper II. Here and in paper II we will define the function Fd (β) via κ = dβ 2 + β 3 Fd (β),

(4.1)

so that from (3.9) we have Fd (β) =

d X Bs − dAs . β2 (s − d)2s−d s=2d

Proc. R. Soc. Lond. A (1998)

(4.2)

1986

T. C. Choy and others κ d=1

6 d=4 d=3 4 d=2 2

d=2

d=1

d=3 d=4 -3

-2

-1

1

2

3

log10 (α) Figure 4. Graph of κ versus log10 (α) for d = 1, 2, 3, 4. Here α = i /0 is the ratio of the dielectric constant of the inclusions to that of the host material.

Our algorithm for computing the coefficients becomes increasingly inefficient as d increases, and thus it is difficult to answer the question concerning the behaviour of Fd (β) as d increases, but we show in paper II that the simple law of mixtures is recovered as d → ∞. This is equivalent to saying that κ → 0 as d → ∞, which is clear from figure 4 for α < 1. For α > 1, the rise in κ moves out to higher and higher α to the right in figure 4, as the dimension d increases. Hence κ → 0 for all d as d → ∞. We discuss this limit more extensively in paper II.

5. Conclusions We have studied the dielectric function for the bispherical system in arbitrary integer dimensions d. We have shown that Jeffrey’s (1973) solution can be generalized once the required mathematical results regarding d-dimensional spherical harmonics are obtained as given in the appendices. Our results are in agreement with the previous work of Jeffrey (1973), for d = 3, and Djordjevi´c et al . (1996), for d = 2. We have shown that the limiting behaviour is simple as both d → 1 and d → ∞. Our approach allows results to be obtained for a general d; albeit as an infinite series that converges somewhat slowly. Nevertheless, the d dependence is shown explicity as d appears as a parameter throughout this work. The multipole approach used in this paper will be compared with the image method in the companion paper II. This work was initiated during a visit to Monash University in the Autumn of 1996 by M.F.T., who thanks Monash University for its hospitality. T.C.C. thanks Michigan State University for its hospitality during a visit in the Spring of 1997, when this work was completed. This work was partly supported by NSF grant CHE 9633798. Proc. R. Soc. Lond. A (1998)

Dielectric function to O(c2 ) for hyperspheres

O'

1987

θ' θ"

r'

ρ

r

θ O

Figure 5. Reoriented twin spherical coordinates. The two polar angles θ and θ 0 are equivalent to θ1 and θ2 of figure 1.

Appendix A. Spherical harmonics shift formula (2.3) Shift formulae involving spherical harmonics find applications in many areas of theoretical physics (Lord Rayleigh 1892; Kohn & Rostocker 1954). In this appendix we will be concerned with a classic formula, attributed to Hobson (1931), which we will prove generalizes to arbitrary integer dimensions d. We begin with the geometrical configuration of figure 5. With the usual definition whereby z = x0 , x1 , x2 , . . . , xd−1 are the Cartesian coordinates, by Taylor’s theorem the electrostatic potential in d dimensions can be expanded as   ∞ X 1 (−1)` ` ∂ ` 1 ρ = `! ∂z ` rp [(z − ρ)2 + x21 + x22 + · · · + xd−1 ]p/2 `=0 −p/2  z ρ ρ2 1 + = p 1−2 r rr r  `  ∞ 1 X ρ z = p P` r r r `=0

∞ X ρ` = P` (cos θ). r`+p

(A 1)

`=0

We must remember that the P` here are the p-Legendre polynomials that are defined in terms of the Gegenbauer polynomials (see comments after equation (2.2)). ThereProc. R. Soc. Lond. A (1998)

1988

T. C. Choy and others

fore, (−1)` ∂ ` P` (cos θ) = r`+p `! ∂z `



 1 . rp

(A 2)

Then by (A 1) we have the following relation:   ∂` ∂` 1 1 = ∂z ` rp ∂z 0` [x21 + x22 + · · · + (z 0 + ρ)2 ]p/2 ∂` 1 ` 02 0 ∂ρ [r + 2z ρ + ρ2 ]p/2 ∞ ∂ ` X r0s = ` Ps (cos θ00 ) ∂ρ s=0 ρp+s =

= (−1)

`

∞ X (p + s + ` − 1)! s=0

(p + s − 1)!

r0s ρp+s+`

Ps (cos θ00 ).

(A 3)

Hence by (A 2) we find that

 ∞  P` (cos θ) X p + s + ` − 1 r0s = P (cos θ00 ), p+s+` s p+s−1 r`+p ρ s=0

(A 4)

thus proving the first shift formula (2.3) in the text.

Appendix B. Spherical harmonics shift formula (2.11) The shift formula (2.11) involves a few more steps to prove than does (2.3). We first define complex coordinates: ζ = x + iy, and η = x − iy. The main step requires the proof of the following properties of the raising operator:   ∂ m P`−m (µ) 1 1 1 ∂m + = m `+p P`m (µ) cos mφ. (B 1) 2 ∂ζ m ∂η m r`−m+p 2 r Here µ = cos θ(1) = cos θ and φ is equivalent to θ(2) (cf. (2.7)) and the triplet (x, y, z) ≡ (x1 , x2 , x0 ) now defines a three-dimensional subspace. This is an invariant subspace in the sense that the triplet can be rotated arbitrarily without affecting our results. We also need   ∂m 1 1 ∂m m + (µ) cos mφ. (B 2) r` P` (µ) = m r`−m P`−m 2 ∂ζ m ∂η m 2 We suppress the subscript (1) of the appropriate angles for convenience as in figure 5. We sketch the proof of these properties in a moment. First we will rewrite (A 4) by changing ` → ` − m so that  ∞  r0s P`−m (µ) X p + s + ` − m − 1 = P (cos θ00 ). (B 3) p+s+`−m s p+s−1 r`−m+p ρ s=0 Now by applying equations (B 1) and (B 2) we have  ∞  P`m (µ) X p + s + ` − m − 1 r0s m = Ps−m (cos θ00 ), `+p p+s+`−m p+s−1 r ρ s=0

Proc. R. Soc. Lond. A (1998)

(B 4)

Dielectric function to O(c2 ) for hyperspheres

1989

 ∞  X r0s P`m (µ) p+s+`−1 = P m (cos θ00 ), p + s + m − 1 ρp+s+` s r`+p s=m

(B 5)

and the equation

thus proving the shift formula (2.11). It now remains to prove the relations (B 1) and (B 2). We only need to prove these relations for the case m = 1 as the general m case (not needed in our work) follows a similar line which we will not present here. We start from     ∂ P`−1 (µ) cos θ ∂ P`−1 (µ) ∂ ∂ iφ +i + = e sin θ ∂x ∂y rp+`−1 ∂r r ∂θ rp+`−1   P`−1 (µ) cos θ 1 iφ = e −(p + ` − 1) sin θ p+` + p+` P`−1 (µ) , (B 6) r r where we have defined x = x1 and y = x2 as above and use is made of the properties of the gradient operator (2.15) in the invariant subspace. Recalling the definition of the p-Legendre polynomials: P`1 (µ) = − sin θP`0 (µ), where the prime denotes derivative with respect to µ, we now have   ∂ P`−1 (µ) eiφ sin θ ∂ 0 +i = − [(p + ` − 1)P`−1 (µ) + µP`−1 (µ)]. (B 7) ∂x ∂y rp+`−1 rp+` Now from the generating function of our p-Legendre polynomials: ∞

G(t, µ) =

X 1 = P` (µ)t` , 2 p/2 (1 − 2µt + t ) 0

(B 8)

we can derive numerous properties. Among them is the p-Legendre differential equation (1 − µ2 )P`00 − (p + 1)µP`0 + `(` + p)P` = 0

(B 9)

and the relation 0 , P`0 = (p + ` − 1)P`−1 + µP`−1

(B 10)

where we have suppressed the arguments of the Legendre functions for convenience. Using the latter relation in (B 7) we finally obtain   ∂ P`−1 (µ) eiφ P`1 (µ) ∂ +i = . (B 11) ∂x ∂y rp+`−1 rp+` The extension of this theorem to m > 1 uses the same techniques. A final comment here is that our results are obviously connected with properties of the orthogonal group O(p + 2) and that there may be shorter derivations than those presented here which use group theory methods.

Appendix C. Perpendicular flux formula (2.30) In this appendix we demonstrate that the flux formula (2.30) can be deduced for any arbitrary perpendicular direction of the field using the essential results contained in § 2 b (ii). For this purpose, we will choose another perpendicular axis for the field. Since we have previously dealt with the axis x(1) , let us now consider x(2) . We will Proc. R. Soc. Lond. A (1998)

1990

T. C. Choy and others

show that (2.30) also holds for this case. Thereafter the general result for x(n) , where n = 3, 4, . . . , (d − 1), can be obtained likewise. For a perpendicular field in the x(2) -direction, equation (2.8) now takes the form (i)

Φ

= E0 x(2) +

∞ X

 E0 d n

n=1

r1 a1

n

Pn1 (cos θ(1) ) sin θ(2) cos θ(3) ,

(C 1)

where all angles and other quantities that refer to sphere 1 have their subscripts suppressed for convenience. We also suppress the arguments for the Legendre functions. Now the gradient of this potential again looks like (2.20): ∇Φ

(1)

 nr1n−1 = E0 x ˆ(2) + E0 d n Pn1 (cos θ(1) ) sin θ(2) cos θ(3) rˆ n a 1 n=1  n−1  1 ∞ X ∂Pn (cos θ(1) ) r + E0 d n 1 n sin θ(2) cos θ(3) θˆ(1) a ∂θ (1) 1 n=1  n−1  ∞ X cos θ(2) cos θ(3) ˆ r + E0 dn 1 n Pn1 (cos θ(1) ) θ(2) a1 sin θ(1) n=1  n−1  ∞ X sin θ(3) ˆ r − E0 dn 1 n Pn1 (cos θ(1) ) θ(3) . a1 sin θ(1) n=1 ∞ X

Once again the integral Z

∇Φ(1) dV =

R



(C 2)

dV projects out only the x ˆ(2) terms and we have

ωp adi E0 x ˆ(2) d  n−1  Z X ∞ nri + E0 dn(i) ˆ(2) Pn1 sin θ(1) sin2 θ(2) cos2 θ(3) dV x n a i n=1  n−1  Z X ∞ ∂Pn1 (i) ri + E0 dn cos θ(1) sin2 θ(2) cos2 θ(3) dV x ˆ(2) n a ∂θ (1) i n=1  n−1  Z X ∞ r Pn1 + E0 dn(i) i n cos2 θ(2) cos2 θ(3) dV x ˆ(2) a sin θ (1) i n=1  n−1  Z X ∞ Pn1 (i) ri + E0 dn sin2 θ(3) dV x ˆ(2) . (C 3) n a sin θ (1) i n=1

We have to perform the θ(2) and θ(3) integrals at this stage. We use the formulae Z Z (p − 1) sinp θ(3) dθ(3) = sinp−2 θ(3) dθ(3) (C 4) p and

Z

sinp−2 θ(3) cos2 θ(3) dθ(3) =

Proc. R. Soc. Lond. A (1998)

1 p

Z

sinp−2 θ(3) dθ(3) .

(C 5)

Dielectric function to O(c2 ) for hyperspheres

1991

Reabsorbing the integrals over θ(3) back into dV we find the similar form to (2.24):  n−1  Z Z ∞ 1 X nr1 ∇Φ(1) dV = υ d E0 x ˆ(2) + E0 d n ˆ(2) Pn1 (sin θ1 ) sin2 θ(2) dV x p n=1 an1  n−1  Z ∞ 1 X r ∂Pn1 + E0 d n 1 n cos θ(1) sin2 θ(2) dV x ˆ(2) p n=1 a1 ∂θ(1)  n−1  Z ∞ 1 X r Pn1 + E0 d n 1 n cos2 θ(2) dV x ˆ(2) p n=1 a1 sin θ(1)  n−1  Z ∞ p−1 X r Pn1 + E0 d n 1 n dV x ˆ(2) . (C 6) p a1 sin θ(1) n=1 Now the θ2 integrals can be done in a similar way to § 2 b (ii), with the use of (2.22) and (2.23). Thus,  n−1  Z Z X ∞ 1 nr1 (1) d ∇Φ dV = υ E0 x ˆ(2) + E0 d n ˆ(2) Pn1 (sin θ1 ) dV x p + 1 n=1 an1  n−1  Z X ∞ 1 r ∂Pn1 + E0 d n 1 n cos θ(1) dV x ˆ(2) p + 1 n=1 a1 ∂θ(1)  n−1  Z X ∞ 1 r Pn1 + E0 d n 1 n dV x ˆ(2) p(p + 1) n=1 a1 sin θ(1)  n−1  Z ∞ p−1 X r Pn1 + E0 d n 1 n dV x ˆ(2) . (C 7) p a1 sin θ(1) n=1 Upon grouping the last two terms, we are back at (2.24) and the rest of the derivation follows as before. We can see that this reduction process will carry through in a similar way for any arbitrary perpendicular direction x(n) , where n = 3, 4, . . . , (d − 1). This completes the proof of (2.30).

References Batchelor, G. K. 1972 J. Fluid Mech. 52, 245–268. Batchelor, G. K. 1974 A. Rev. Fluid. Mech. 6, 227–255. Batchelor, G. K. 1977 J. Fluid Mech. 83, 97–117. Batchelor, G. K. & Green, J. T. 1972 J. Fluid Mech. 56, 401–427. Binns, K. J. & Lawrenson, P. J. 1973 Electric and magnetic field problems, pp. 48–53. Oxford: Pergamon. Choy, T. C., Alexopoulos, A. & Thorpe, M. F. 1998 Proc. R. Soc. Lond. A 454, 1993–2013. (Following paper.) Davis, M. H. 1964 Q. J. Mech. App. Math. 17, part 4, 499–511. Djordjevi´c, B. R., Hetherington, J. H. & Thorpe, M. F. 1996 Phys. Rev. B 53, 14 862–14 871. Einstein, A. 1906 A. Phys. 19, 289–306. Einstein, A. 1911 A. Phys. 34, 591–592. Erd´elyi, A. 1953 Higher transcendental functions, vol. 2, pp. 232–261. New York: McGraw-Hill. Felderhof, B. U., Ford, G. W. & Cohen, E. G. D. 1982 J. Stat. Phys. 28, 649–672. Proc. R. Soc. Lond. A (1998)

1992

T. C. Choy and others

Hobson, E. W. 1931 Spherical and ellipsoidal harmonics. Cambridge University Press. Hochstadt, H. 1971 The functions of mathematical physics, pp. 168–189. New York: WileyInterscience. Jeffery, G. B. 1912 Proc. R. Soc. Lond. A 87, 109–120. Jeffrey, D. J. 1973 Proc. R. Soc. Lond. A 335, 355–367. Kohn, W. & Rostocker, N. 1954 Phys. Rev. 94, 1111–1120. Landau, L. D., Lifshitz, E. M. & Pitaevskii, L. P. 1984 Electrodynamics of continuous media. London: Pergamon. Lord Rayleigh, 1892 Phil. Mag. 34, 481–502. Maxwell, J. C. 1873 In Electricity and magnetism, 1st edn, Oxford: Clarendon. Moons, P. & Spencer, D. E. 1988 Field theory handbook: including coordinate systems, differential equations, and their solutions. Berlin: Springer. Morse, P. M. & Feshbach, H. 1953 Methods of theoretical physics, vol. 1. New York: McGraw-Hill. Ross, D. K. 1968 Aust. J. Phys. 21, 817–822. Thorpe, M. F. 1992 Proc. R. Soc. Lond. A 437, 215–227.

Proc. R. Soc. Lond. A (1998)