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Fractal characterization of the dipole moments of dielectric particle chains
Journal of
ELECTROSTATICS Journal of Electrostatics 44 (1998) 47-5 1
Fractal characterization
of the dipole moments of dielectric particle chains
Zehui Jianga, Xiaodong Heb, Jiecai Hanb, Boming Zhangb and Shanyi Dub ‘Departmentof Applied physics, HarbinInstituteof Technology, Harbin 150001, P. R. China bResearch Lab of Composite Materials, Harbin Institute of Technology, Harbin 150001, P. R. China
Abstract A simple self-similar fractal model is presented for obtaining the dipole moments of dielectric particle chains subjected to uniform electric field. The chains are replaced by equivalent spheres, and the effective radii of these spheres are determined from a fractal generating process. The dipole moments are determined in terms of the effective radii and are expressed as simple functions of the fractal dimension and the number of the particles in chain. The computed results for the longitudinal dipole moments are consistent with the previous calculations of the linear multipole expansion. The many-sphere nature of the moments of chains is well understood. Keywords:
dipole moment, particle chain, multipole interaction, fractal
1. Introduction The determination of the dipole moments of particle chains immersed in uniform electric field is a problem of recurring interest and one of considerable theoretical and practical importance. The calculation of the dipole moments in general is a difficult task, because the strong multiple interactions among the particles must be taken into account, which may occur when the particles are closely spaced. Useful simple expressions for long chains are rare. In recent decades a great amount of work has been done on the dipole moments of dielectric particle chains. Because of the complexity of the problem, most investigations concentrate on two particle systems. The dipole moments of two identical uncharged dielectric spheres [l-6], two unequal dielectric spheres [7], two intersecting spheres [8], and two intersecting cardioidal particles [9] have been evaluated. Very recently Jones and coworkers [lo] calculated the longitudinal dipole moments of chains of more than two spheres employing linear multipole expansions. To improve the accuracy of the calculation, a large number of multipolar terms are necessary. When the relative permittivity of the spheres is high, this method encounters the convergence difficulty and becomes impractical. The other methods suitable for conducting particle chains, such as the image method [ 111 and the nearest0304-3886/98/$19.00 0 Elsevier Science B.V. All rights reserved. PII SO304-3886(98)00021-7
48
Z. Jiang et al. /Journal
of Electrostatics
44 (1998) 47-51
neighbor approximation [ 121, are difficult applied to the dielectric particle chains. In a recent paper [13] we develop a self-similar fractal model for obtaining the dipole moments of equally spaced conducting spheres subjected to uniform electric field. The dipole moments are determined iteratively in terms of those of conducting particle pair (the generator). The contributions of higher multipoles are estimated indirectly. The approximate fractal characters of the moments are found. Here we extend this model to the dielectric particle chains, in which the spheres are in contact. Simple formulas are derived. The computed results are compared with the previous calculations of Jones et al [lo]. Our results are acceptable. 2. Theory The core of the problem of calculating the dipole moments of particle chains lies in the determination of the higher multipole interactions between the particles in chain. To estimate these interactions, however, is a laborious work. A convenient approach is to calculate the dipole moments iteratively in terms of those of sphere pair, which have been known analytically. The dipole moments of two identical uncharged spheres with arbitrary separation have been evaluated by Poladian [4], who used the image method. Following his treatment, the longitudinal and transverse dipole moments of a touching pair may be written as p;(r)
= &(r)p,
PTW = a%)Po 2r -4m,,, p”=3-r
4&(r) = -r [ = -
2Li,(-7) 7
[14]. p.
(1)
PO>
di, (z) PO?
R’E ,
where, r = (E - &,,,)I(& + E,,,) , is the scaled ‘dielectric contrast. the dielectric constant of the spheres and matrix. Li, (7) functions
1
2Li,($
E ,
E, are, respectively,
are the polylogarithm
is the dipole moment of a single isolate sphere of radius R in a
constant field E. In eqs. (1) and (2), we separate the dipole moments into two parts; p. and relative polarization (a. L(r) and a’(z) ). p. represents the contribution of the size of the spheres, while
a L(7) and ar (r)
represent the contributions
of the
multipole interactions between the two spheres. Two dielectric spheres are related together by the mutual interactions between them. We assume that they may be treated as one body with the same dipole moment, i.e. a larger single equivalent sphere of dielectric constant E (the dashed sphere in Fig. 1). This is the first stage of the iteration. The two solid spheres are the generator. For the longitudinal case, the radius of the equivalent sphere is R, = (a” (r))“3 R . At the second stage, a new pair is constructed with two dashed spheres, containing in total N = 22 original spheres. Replacing this new pair with a single sphere of dielectric sphere is obtained, constant E, the effective radius for this equivalent R, = (aL(r))“‘R,
= (aL(r))2’3 R . This process is repeated recursively.
The effective
Z. Jiang et al. /Journal of Electrostatics 44 (1998) 47-51
radius
for
the
k-th
stage,
R, = (cz”(.~))“~R,_, = (a”(r))“”
containing R. Substituting
moment for the N = 2 Ir sphere chain, arbitrary N, the longitudinal
in
total
N = 2’
49
spheres,
is
it into eq. (3), we get the dipole
pk (z) = (a L (z))’ pO. Extend this formula to
dipole moment may be written as
p;(T)
=(aL(r))‘nN”n2p0.
(4)
In a similar fashion, the transverse dipole moment may be obtained, p;(7)
= (,T(r))‘nN”n2p0.
(5)
FIG. 1. First two stages of the iteration. A pair of spheres (the generator) is replaced by a single equivalent sphere, which is used to construct a larger pair in a self-similar manner. The procedure is repeated infinitely, and a self-similar fractal is built.
The iterative procedure described above, actually, constructs a self-similar fractal for the radii of the equivalent spheres. The similarity dimension of this fiactal is for the longitudinal
D/” =31n2/lnaL(r),
(7)
case, and 0,’ =31n2/lnar(r),
(8)
for the transverse case. Equs. (4) and (5) may be rewritten as p;(r) = N3’D4p, = Nin&r)/rn2P0, p;(r)
(9)
= N3’D:p, = Nrna’(r)/in2p0.
(10)
3. Discussion Computed
normalized
dipole
moments
per particle
( pN (7) / Np,)
versus
the
number of the particles in a chain are shown in Fig. 2 and Fig. 3. For the purpose of comparison with the previous calculations of Jones and co-workers in ref. [lo], the results for a range of relative permittivity values, E /~,,,=2, 3, 4, and 10 (correspondingly, 2-l/3, 0.5, 0.6, and 9/11), are depicted in Fig. 2. In ref. [lo], a calculation for the longitudinal dipole moments was performed employing a dipole approximation. Our results are about 1-2 times larger than the previous ones (Fig. 5 in ref. [lo]) in the range of N=2-7. This is due to the fact that the higher-order terms are neglected in the dipole approximation. We also notice that the transverse dipole moment per particle approaches to zero when the length of the chain becomes infinite. This may be attribute to the repulsive interactions among the particles.
50
Z. Jiang et al. /Journal
of Electrostatics
44 (1998) 47-51
0.95 0.9 0.65 0.6 0.75 z 0.7 R i 0.65 ? 8 0.6 3 J
.I
6
11
16
21
26
31
36
1
NUMBER OF PARTICLES IN CHAIN, N
FIG. 2. Normalized moment
per
longitudinal
particle
(pi(t)
11
16
21
26
31
36
NUMBER OF PARTICLES IN CHAIN, P
FIG.
dipole
3.
moment
/ Np,)
6
Normalized per
particle
transverse
dipole
( p; (I) / Np, )
versus the number of the particles in a
versus the number of the particles in a
chain, for z=O.l, 0.33, 0.5, 0.6, 0.82, 0.9,
chain, for r=O.l,
1.0.
1.0.
When
r takes a very small value,
0.2, 0.4, 0.6, 0.8, 0.9,
a L(7) - a T(7) -2, and pk (r) 7~;
(I) -
Np, . In
this case, only the dipoles are induced in each sphere, and the contributions of the higher-order multipoles may be ignored. When r approaches to one, that is the case of conducting spheres aL(r) = 4<(3) and a’(r) = 1.5<(3), (E--)=)), c(3) = 1.202OA is the Riemarm zeta-function
of argument of 3 [ 151. In this case, the
multipole interactions became very strong, p;(5) = N*46(3)‘*zpo = N2.r”po and p;(T)
and eqs. (8) and (9) reduce to = Ntni%(3)/in2po = N”.g~opo, Which
have been given in ref. [ 131. 4. Conclusion The fkxtal model developed here provides a simple method of determining the dipole moments of particle chains, which rely on the relative premittivity of the spheres, the number of the particles in chain, and the orientation of the applied field. In this model, all higher multipole interactions are relegated to the size of equivalent sphere and are brought in the next stage of the iteration. One advantage of this model is it is not limited in the relative permittivity of the spheres. We have compared our results with the previous calculations of other method. It is found that our results are acceptable. References 1. A. Goyette and A. Navon, Two dielectric spheres in an electric field, Phys. Rev. B., 13 (1976) 4320-4327.
Z. Jiang et al. /Journal
of Electrostatics
44 (1998) 47-51
51
2. A. V. Paley, A. V. Radchik, and G. B. Smith, Quasistatic optical response of pairs of touching spheres with arbitrary dielectric permeability, J. Appl. Phys. 73 (1993) 3446-3453. 3. L. Poladian, General theory of electrical images in sphere pairs, Q. Jl Mech. Appl. Math. 41 (1988) 395417. 4. L. Poladian, Long-wavelength absorption in composites, Phys. Rev. B., 44 (1991) 2092-2107. 5. M. Sancho, G. Martines, and M. Llamas, Multipole interaction between dielectric particles, J. Electrostat. 21 (1988) 135-144. 6. R. D. Stoy, Interactive dipole model for two-sphere system, J. Electrostat. 33 (1994) 385-392. 7. I. V. Lindell, J. C.-E. Sten, and K. I. Nikoskinen, Electrostatic Image method for the interaction of two dielectric spheres, Radio Sci. 28 (1993) 319-329. 8. A. V. Radchik, A. V. Paley, G. B. Smith, and A. V. Vagov, Polarization and resonant absorption in intersecting cylinders and spheres, J. Appl. Phys. 76 (1994) 4827-4835. 9. A. V. Radchik, A. V. Paley, and G. B. Smith, “Invisibility” in certain intersecting particles and arrays of such particles in a solid host, J. Appl. Phys. 79 (1996) 26132621. 10.T. B. Jones, R. D. Miller, K. S. Robison, and W. Y. Fowlkes, Multipole interactions of dielectric spheres, J. Electrostat. 22 (1989) 23 l-244. 11 .T. B. Jones, Dipole moments of conducting particle chains, J. Appl. Phys. 60 (1986) 2226-2236; Addendum, 61 (1987) 2416-2417. 12.R. J. Meyer, Nearest-neighbor approximation for the dipole moment of conductingparticle chains, J. Electrostat. 33 (1994) 133-146. 13.Zehui Jiang, Xueru Zhang, Jiecai Han, and Shanyi Du, Fractal model of the dipole moments of conducting particle chains, J. Electrostat. 39 (1997) 297-304. 14.L. Lewin, Polylogarithms and associated functions, New York, Elsevier North Holland Inc. 198 1. 15.M. Abamowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.
ELECTROSTATICS Journal of Electrostatics 44 (1998) 47-5 1
Fractal characterization
of the dipole moments of dielectric particle chains
Zehui Jianga, Xiaodong Heb, Jiecai Hanb, Boming Zhangb and Shanyi Dub ‘Departmentof Applied physics, HarbinInstituteof Technology, Harbin 150001, P. R. China bResearch Lab of Composite Materials, Harbin Institute of Technology, Harbin 150001, P. R. China
Abstract A simple self-similar fractal model is presented for obtaining the dipole moments of dielectric particle chains subjected to uniform electric field. The chains are replaced by equivalent spheres, and the effective radii of these spheres are determined from a fractal generating process. The dipole moments are determined in terms of the effective radii and are expressed as simple functions of the fractal dimension and the number of the particles in chain. The computed results for the longitudinal dipole moments are consistent with the previous calculations of the linear multipole expansion. The many-sphere nature of the moments of chains is well understood. Keywords:
dipole moment, particle chain, multipole interaction, fractal
1. Introduction The determination of the dipole moments of particle chains immersed in uniform electric field is a problem of recurring interest and one of considerable theoretical and practical importance. The calculation of the dipole moments in general is a difficult task, because the strong multiple interactions among the particles must be taken into account, which may occur when the particles are closely spaced. Useful simple expressions for long chains are rare. In recent decades a great amount of work has been done on the dipole moments of dielectric particle chains. Because of the complexity of the problem, most investigations concentrate on two particle systems. The dipole moments of two identical uncharged dielectric spheres [l-6], two unequal dielectric spheres [7], two intersecting spheres [8], and two intersecting cardioidal particles [9] have been evaluated. Very recently Jones and coworkers [lo] calculated the longitudinal dipole moments of chains of more than two spheres employing linear multipole expansions. To improve the accuracy of the calculation, a large number of multipolar terms are necessary. When the relative permittivity of the spheres is high, this method encounters the convergence difficulty and becomes impractical. The other methods suitable for conducting particle chains, such as the image method [ 111 and the nearest0304-3886/98/$19.00 0 Elsevier Science B.V. All rights reserved. PII SO304-3886(98)00021-7
48
Z. Jiang et al. /Journal
of Electrostatics
44 (1998) 47-51
neighbor approximation [ 121, are difficult applied to the dielectric particle chains. In a recent paper [13] we develop a self-similar fractal model for obtaining the dipole moments of equally spaced conducting spheres subjected to uniform electric field. The dipole moments are determined iteratively in terms of those of conducting particle pair (the generator). The contributions of higher multipoles are estimated indirectly. The approximate fractal characters of the moments are found. Here we extend this model to the dielectric particle chains, in which the spheres are in contact. Simple formulas are derived. The computed results are compared with the previous calculations of Jones et al [lo]. Our results are acceptable. 2. Theory The core of the problem of calculating the dipole moments of particle chains lies in the determination of the higher multipole interactions between the particles in chain. To estimate these interactions, however, is a laborious work. A convenient approach is to calculate the dipole moments iteratively in terms of those of sphere pair, which have been known analytically. The dipole moments of two identical uncharged spheres with arbitrary separation have been evaluated by Poladian [4], who used the image method. Following his treatment, the longitudinal and transverse dipole moments of a touching pair may be written as p;(r)
= &(r)p,
PTW = a%)Po 2r -4m,,, p”=3-r
4&(r) = -r [ = -
2Li,(-7) 7
[14]. p.
(1)
PO>
di, (z) PO?
R’E ,
where, r = (E - &,,,)I(& + E,,,) , is the scaled ‘dielectric contrast. the dielectric constant of the spheres and matrix. Li, (7) functions
1
2Li,($
E ,
E, are, respectively,
are the polylogarithm
is the dipole moment of a single isolate sphere of radius R in a
constant field E. In eqs. (1) and (2), we separate the dipole moments into two parts; p. and relative polarization (a. L(r) and a’(z) ). p. represents the contribution of the size of the spheres, while
a L(7) and ar (r)
represent the contributions
of the
multipole interactions between the two spheres. Two dielectric spheres are related together by the mutual interactions between them. We assume that they may be treated as one body with the same dipole moment, i.e. a larger single equivalent sphere of dielectric constant E (the dashed sphere in Fig. 1). This is the first stage of the iteration. The two solid spheres are the generator. For the longitudinal case, the radius of the equivalent sphere is R, = (a” (r))“3 R . At the second stage, a new pair is constructed with two dashed spheres, containing in total N = 22 original spheres. Replacing this new pair with a single sphere of dielectric sphere is obtained, constant E, the effective radius for this equivalent R, = (aL(r))“‘R,
= (aL(r))2’3 R . This process is repeated recursively.
The effective
Z. Jiang et al. /Journal of Electrostatics 44 (1998) 47-51
radius
for
the
k-th
stage,
R, = (cz”(.~))“~R,_, = (a”(r))“”
containing R. Substituting
moment for the N = 2 Ir sphere chain, arbitrary N, the longitudinal
in
total
N = 2’
49
spheres,
is
it into eq. (3), we get the dipole
pk (z) = (a L (z))’ pO. Extend this formula to
dipole moment may be written as
p;(T)
=(aL(r))‘nN”n2p0.
(4)
In a similar fashion, the transverse dipole moment may be obtained, p;(7)
= (,T(r))‘nN”n2p0.
(5)
FIG. 1. First two stages of the iteration. A pair of spheres (the generator) is replaced by a single equivalent sphere, which is used to construct a larger pair in a self-similar manner. The procedure is repeated infinitely, and a self-similar fractal is built.
The iterative procedure described above, actually, constructs a self-similar fractal for the radii of the equivalent spheres. The similarity dimension of this fiactal is for the longitudinal
D/” =31n2/lnaL(r),
(7)
case, and 0,’ =31n2/lnar(r),
(8)
for the transverse case. Equs. (4) and (5) may be rewritten as p;(r) = N3’D4p, = Nin&r)/rn2P0, p;(r)
(9)
= N3’D:p, = Nrna’(r)/in2p0.
(10)
3. Discussion Computed
normalized
dipole
moments
per particle
( pN (7) / Np,)
versus
the
number of the particles in a chain are shown in Fig. 2 and Fig. 3. For the purpose of comparison with the previous calculations of Jones and co-workers in ref. [lo], the results for a range of relative permittivity values, E /~,,,=2, 3, 4, and 10 (correspondingly, 2-l/3, 0.5, 0.6, and 9/11), are depicted in Fig. 2. In ref. [lo], a calculation for the longitudinal dipole moments was performed employing a dipole approximation. Our results are about 1-2 times larger than the previous ones (Fig. 5 in ref. [lo]) in the range of N=2-7. This is due to the fact that the higher-order terms are neglected in the dipole approximation. We also notice that the transverse dipole moment per particle approaches to zero when the length of the chain becomes infinite. This may be attribute to the repulsive interactions among the particles.
50
Z. Jiang et al. /Journal
of Electrostatics
44 (1998) 47-51
0.95 0.9 0.65 0.6 0.75 z 0.7 R i 0.65 ? 8 0.6 3 J
.I
6
11
16
21
26
31
36
1
NUMBER OF PARTICLES IN CHAIN, N
FIG. 2. Normalized moment
per
longitudinal
particle
(pi(t)
11
16
21
26
31
36
NUMBER OF PARTICLES IN CHAIN, P
FIG.
dipole
3.
moment
/ Np,)
6
Normalized per
particle
transverse
dipole
( p; (I) / Np, )
versus the number of the particles in a
versus the number of the particles in a
chain, for z=O.l, 0.33, 0.5, 0.6, 0.82, 0.9,
chain, for r=O.l,
1.0.
1.0.
When
r takes a very small value,
0.2, 0.4, 0.6, 0.8, 0.9,
a L(7) - a T(7) -2, and pk (r) 7~;
(I) -
Np, . In
this case, only the dipoles are induced in each sphere, and the contributions of the higher-order multipoles may be ignored. When r approaches to one, that is the case of conducting spheres aL(r) = 4<(3) and a’(r) = 1.5<(3), (E--)=)), c(3) = 1.202OA is the Riemarm zeta-function
of argument of 3 [ 151. In this case, the
multipole interactions became very strong, p;(5) = N*46(3)‘*zpo = N2.r”po and p;(T)
and eqs. (8) and (9) reduce to = Ntni%(3)/in2po = N”.g~opo, Which
have been given in ref. [ 131. 4. Conclusion The fkxtal model developed here provides a simple method of determining the dipole moments of particle chains, which rely on the relative premittivity of the spheres, the number of the particles in chain, and the orientation of the applied field. In this model, all higher multipole interactions are relegated to the size of equivalent sphere and are brought in the next stage of the iteration. One advantage of this model is it is not limited in the relative permittivity of the spheres. We have compared our results with the previous calculations of other method. It is found that our results are acceptable. References 1. A. Goyette and A. Navon, Two dielectric spheres in an electric field, Phys. Rev. B., 13 (1976) 4320-4327.
Z. Jiang et al. /Journal
of Electrostatics
44 (1998) 47-51
51
2. A. V. Paley, A. V. Radchik, and G. B. Smith, Quasistatic optical response of pairs of touching spheres with arbitrary dielectric permeability, J. Appl. Phys. 73 (1993) 3446-3453. 3. L. Poladian, General theory of electrical images in sphere pairs, Q. Jl Mech. Appl. Math. 41 (1988) 395417. 4. L. Poladian, Long-wavelength absorption in composites, Phys. Rev. B., 44 (1991) 2092-2107. 5. M. Sancho, G. Martines, and M. Llamas, Multipole interaction between dielectric particles, J. Electrostat. 21 (1988) 135-144. 6. R. D. Stoy, Interactive dipole model for two-sphere system, J. Electrostat. 33 (1994) 385-392. 7. I. V. Lindell, J. C.-E. Sten, and K. I. Nikoskinen, Electrostatic Image method for the interaction of two dielectric spheres, Radio Sci. 28 (1993) 319-329. 8. A. V. Radchik, A. V. Paley, G. B. Smith, and A. V. Vagov, Polarization and resonant absorption in intersecting cylinders and spheres, J. Appl. Phys. 76 (1994) 4827-4835. 9. A. V. Radchik, A. V. Paley, and G. B. Smith, “Invisibility” in certain intersecting particles and arrays of such particles in a solid host, J. Appl. Phys. 79 (1996) 26132621. 10.T. B. Jones, R. D. Miller, K. S. Robison, and W. Y. Fowlkes, Multipole interactions of dielectric spheres, J. Electrostat. 22 (1989) 23 l-244. 11 .T. B. Jones, Dipole moments of conducting particle chains, J. Appl. Phys. 60 (1986) 2226-2236; Addendum, 61 (1987) 2416-2417. 12.R. J. Meyer, Nearest-neighbor approximation for the dipole moment of conductingparticle chains, J. Electrostat. 33 (1994) 133-146. 13.Zehui Jiang, Xueru Zhang, Jiecai Han, and Shanyi Du, Fractal model of the dipole moments of conducting particle chains, J. Electrostat. 39 (1997) 297-304. 14.L. Lewin, Polylogarithms and associated functions, New York, Elsevier North Holland Inc. 198 1. 15.M. Abamowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.