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A dipole approximation for a dielectric mixture based on the equal field energy method
Journal of Electrostatics 68 (2010) 116–121
Contents lists available at ScienceDirect
Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
A dipole approximation for a dielectric mixture based on the equal field energy method Fei Lu*, Qizheng Ye College of Electrical and Electronic Engineering, Huazhong University of Science & Technology, Wuhan 430074, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 April 2009 Received in revised form 8 August 2009 Accepted 21 November 2009 Available online 2 December 2009
In this paper, a dipole-energy approximation for calculating the electric field distribution and saturation charge of spheres in an infinitely large dielectric mixture has been carried out. The approximation rests on the assumption that the field energy of mixture calculated using two different methods have the same value. One way is considering that the mixture in a uniform field E0 as a uniform object of effective permittivity 3eff from a macroscopic point of view, where 3eff is seen as the average characteristic parameter of the object. The other way is assuming that the spheres in mixture are in the equivalent external field E00 , and E00 related to the dielectric mismatch and the ratio of the sphere radius to the average distance between neighboring spheres has been obtained. Based on dipole-energy model, the approximate formulas for calculating the maximum field strength and saturation charge of spheres are derived separately. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Dielectric mixture Electric field distribution Saturation charge Field energy Dipole-energy approximation
1. Introduction In recent years, many attempts have been made to provide detailed theoretical descriptions for dielectric mixture, where dielectric spheres of permittivity 3i are embedded in a host material of permittivity 3e. There are many areas of science and technology where the response of merging spheres to applied fields is of interest. Among many approaches to solve this problem, a widely adopted one is to treat the polarized sphere as a dipole, such as the ideal dipole approximation. It assumes that the distance between spheres is so far that the interactions among spheres is negligible, which is sufficiently applied for the calculation of electrical charge in industrial electrostatic precipitation when a small dielectric mismatch between the spheres and the environment [1–4]. However, the ideal dipole approximation is invalid when the dielectric mismatch is large and the spheres are nearly touching, then the contributions of higher-multipole may be significant [5–8]. In the case of merging small particles, and large bodies or surfaces to an electrostatic field, different issues related to the particle charging which is an essential process in many electrostatics applications were investigated [9–12]. Pauthenier and Moreau-Hanot calculated the saturation charge of a spherical particle first [13], the formula rests on the ideal dipole
* Corresponding author. Tel.: þ86 2787542726. E-mail addresses: [email protected] (F. Lu), [email protected] (Q. Ye). 0304-3886/$ – see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2009.11.006
approximation and it will compromise the accuracy when a dielectric mixture with the large dielectric mismatch and the high-volume fraction. Some other approximations adopt the model of non-interacting particles as a point of departure and add the effect of interaction as a perturbation, but this is obviously not an expedient approach for intermediate or strong coupling. Various nonperturbative mean field method [14] and semiphe-nomenological theories [15,16] have attempted to attack the problem with varying degrees of success. In this paper, a simple dipole approximation for calculating the field distribution in dielectric mixture and the saturation charge of spheres has been provided for high-volume fraction particles based on the assumption that the mixture is equivalent to different systems which have equal field energy from multiscale view (the approximation is named dipole-energy approximation in the paper). 2. The ideal dipole model for a dielectric mixture For a dielectric mixture in a uniform external field E0, the ideal dipole approximation rests on the assumption that the field experienced by an arbitrarily selected central sphere is still uniform, and the magnitude of this external field is still E0. That means the sum of the field from all the other spheres is zero and it just needs to consider one sphere in a uniform external field E0 as shown in Fig. 1. Considering a sphere of radius R and permittivity 3i in a uniform field E0. The permittivity of environmental media is 3e. Solving the Laplace equation, we can get
F. Lu, Q. Ye / Journal of Electrostatics 68 (2010) 116–121
117
Fig. 2. A finite-element model of a sphere in a cylindrical volume. Fig. 1. A sphere subjected to a external field E0.
!
E out ¼ r0
3 3e 2E0 R3 E0 cosq þ i $ cosq 3i þ 2 3e r 3
þ q0
E in ¼ z0
E0 sinq þ
!
3i 3e E0 R3 $ sinq ; 3i þ 23e r3
3 3e E ; 3i þ 23e 0
ð1Þ
from all the surrounding spheres is not zero, but it is a uniform field in the parallel direction to E0. So a cell of mixture can be seen as a sphere polarized in a uniform field E00 (E00 ¼ E0 þ Espheres), and the field distribution in a cell is similar to Eqs. (1) and (2) while E0 is replaced by E00 . Due to the dielectric mixture is uniform and infinitely large, the field distributions in each cell are the same and their calculation is limited to a cell.
(2)
where Eout is the outside field of the sphere, and Ein is the inside field of the sphere. Especially, according to Eq. (1), the maximum field strength on the surface of sphere is
Emax ¼
33i E ; 3i þ 2 3e 0
(3)
and it does not depend on the sphere radius. 3. The finite-element method for a dielectric mixture For a dielectric mixture in a uniform external field E0, a cylinder model is proposed for an arbitrarily selected central sphere. If the average distance between two neighboring sphere’s centers is d in the unit of sphere radius R, an accessory cylinder of length d and diameter d outside each sphere can be constructed as shown in Fig. 2. The potential difference between both ends of the cylinder is Ui(Ui ¼ E0d) because the spheres are evenly distributed in the parallel direction to the external field. The potential at the side face is satisfied with the natural boundary condition (v4/vn ¼ 0, the outward surface-normal component of the electric field strength is zero) because the spheres are evenly and randomly distributed in the direction perpendicular to the external field. The boundary condition implicitly accounts for the presence of the surrounding spheres, therefore, only the electric field around a single sphere needs to be analyzed. Now solving the Laplace equation with the finite-element method (FEM) and then get the exact numerical solutions to compare with other models. 4. The dipole-energy model for a dielectric mixture For a dielectric mixture in a uniform field E0, a dipole-energy model is proposed to consider the perturbations due to all the surrounding spheres. The paper assumes that the sum of field Espheres
Fig. 3. Equivalent cells of a dielectric mixture.
118
F. Lu, Q. Ye / Journal of Electrostatics 68 (2010) 116–121
Based on the assumption above, the field energy of dielectric mixture can be calculated using different methods. First, a dielectric mixture in a uniform field E0 is considered as a uniform object of effective permittivity 3eff from a macroscopic point of view. Effective permittivity 3eff is seen as the average characteristic parameter of this object. Selecting a cell of system as shown in Fig. 3 (a) and there is one sphere in a cell, we can get
Wa ¼
1 3 E2 DV; 2 eff 0
(4)
where Wa is the field energy of a cell, DV is the volume of a cell. The paper assumes the cube cell approximates to a spherical cell that the energy integral can be solved easily in the latter paper (it is impossible to calculate the cube volume of integral in the spherical coordinates system), the field energy of a cell is
" 0 2pE02 1 2 02 4 3 3e d=2Þ3 R3 Wb ¼ 3i ð1 kÞ E0 $ pR þ 2 3 3 ! # 5 2 3 R6 R3 3 kR ln : þ k R 2 ðd=2Þ3 ðd=2Þ3
ð9Þ
Finally, the paper assumes that the dielectric mixture has equal field energies of these two equivalent cells, so
Wa ¼ Wb :
(10)
On condition that effective permittivity 3eff is known and precise enough, substituting Eqs. (5) and (9) into Eq. (10), we can get
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3eff 0 E0 ¼ u 3 h 3 3 3 3 iE0 ¼ t 3i ð1 kÞ2 ð2R 1 þ 52k2 2R k 2R þ3e 1 2R ln 2R d d d d d
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3eff u h iE0 ; t 2 3i ð1 kÞ f þ 3e ð1 f Þ 1 þ 52k2 f kf lnf (11)
Wa ¼
1 4 d 3 3eff E02 $ p : 2 3 2
(5)
Second, the field energy of mixture can be obtained by calculating the field energies of spheres and environmental media separately. Based on the assumption above, there is one sphere in a cell and the cell is still in a uniform field that the magnitude is not the primary external field E0, but E00 , as shown in Fig. 3 (b). The field energy of a cell is given as
1 3 2 i
Wb ¼ Wbin þ Wbout ¼
Z
1 2 Ein dVin þ 3e 2
Z
2 Eout dVout ;
(6)
where Wb is the field energy of the cell, Wbin is the field energy of a sphere, Wbout is the field energy of environmental media, Vin is the volume of a sphere, Vout is the environmental media volume of a cell. If E0 is replaced by E00 in Eqs. (1) and (2), we can get (the paper assumes the cube cell approximates to a spherical cell that the energy integral can be solved easily in the spherical coordinates system)
Wbin ¼
1 0 4 3 ð1 kÞ2 E02 $ pR3 ; 2 i 3
where f is the volume fraction of spheres in mixture, with f ¼ (4pR3/3)/(4p(d/2)3/3) ¼ (2R/d)3. The maximum volume fraction of spheres in cube lattice structure is p/6 and it is not very large, therefore, the Maxwell–Garnett model is adoptable to get 3eff accurate enough
3eff ¼ 3e 1 þ
3f ð3i 3e Þ : 3i þ 23e f ð3i 3e Þ
(12)
Now we can calculate the electric field distribution in a dielectric mixture if E0 is replaced by E00 in Eqs. (1) and (2)
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 !2 u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u R3 R3 E ¼ Er2 þ Eq2 ¼ t 1 þ 2k 3 cos2 q þ 1 k 3 sin2 q$E00 : r r (13) Eq. (13) is an analytical formula to calculate the electric field at any point. Clearly it considers the perturbations due to all the other spheres. And the maximum electric field strength on the surface of a sphere is
(7)
6 Dipole
Zp
Z2p
sina da
0 0
E2 ¼ 0 3e 2
dq
Z2p
sinada
0
R3
þ 1k
!2
2 Eout r 2 dr
R
0
Zp
Zd=2
0
dq
Zd=2
2
3 4 1 þ 2kR 3 r
FEM Dipole-energy
4 !2
E/E0
Wbout
1 ¼ 3e 2
5
cos2 q
3R
3 2
sin2 q5r 2 dr;
r3 " ! ! 3 0 2pE02 d 5 R6 3e R3 þ k2 R3 ¼ 3 2 2 ðd=2Þ3 # R3 kR3 ln ðd=2Þ3 where k ¼ (3i 3e)/(3i þ 23e), then
1 0 (8)
0
0.2
0.4
0.6
0.8
1
Z/R Fig. 4. The electric field distributions on the surface of a sphere for 3i/3e ¼ 10, d/R ¼ 2.6.
F. Lu, Q. Ye / Journal of Electrostatics 68 (2010) 116–121
119
6.5
6
Dipole
Dipole FEM
5
FEM
5.5
Dipole-energy
Dipole-energy E/E0
E/E0
4 3
4.5 3.5
2 2.5
1 1.5
0 0
0.2
0.4
0.6
0.8
1
1. 1
1. 2
1. 3
z/R
1
Z/R
Fig. 7. The electric field distributions along the z axis for 3i/3e ¼ 100, d/R ¼ 2.6.
Fig. 5. The electric field distributions on the surface of a sphere for 3i/3e ¼ 100, d/R ¼ 2.6. 0
Emax ¼ð1þ2kÞE0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3eff iE0 : h ¼ð1þ2kÞu t 2 3i ð1kÞ f þ3e ð1f Þ 1þ52k2 f kf lnf
(14)
5. Results and discussion 5.1. The electric field distribution on the surface of a sphere The electric field distributions on the surface of an arbitrary selected sphere are calculated using the ideal dipole approximation, the FEM and the dipole-energy approximation are all shown in Figs. 4 and 5. As seen in these two figures, the calculated results of the dipole-energy approximation are good at the two poles of the sphere. Especially in the area of strong field, the dipole-energy approximation which has more accurate results is better than the ideal dipole approximation, but it is not very good at any point on the surface of the sphere.
Obviously, compared with the results of FEM, the dipole-energy approximation is more accurate than the ideal dipole approximation. 5.3. The maximum electric field strength Figs. 8 and 9 show the calculated results of the maximum electric field strength for different d/R and different permittivity using the three methods. The dipole-energy approximation is better than the ideal dipole approximation to get more accurate results. What is interesting is that the calculated results using the dipole-energy approximation are more accurate when the dielectric mismatch (3i/3e) is large or the high-volume fraction (2R/d), contrary to the ideal dipole approximation. And there is good fit between the calculated results of the minimum field strength of the dipole-energy approximation and those of the FEM, as shown in Figs. 4 and 5. 5.4. The effective permittivity of dielectric mixture According to Eq. (11), it is obvious that the effective permittivity
3eff of dielectric mixture affects the results of dipole-energy 5.2. The electric field distribution outside the sphere The electric field distributions along the z axis are calculated using the three methods, and they are all shown in Figs. 6 and 7.
approximation directly. Although the effective permittivity is usually obtained by Eq. (12) easily, the result of Maxwell–Garnett model is always small when a high-volume fraction or large
25
6.5
Dipole
Dipole FEM
5.5
FEM
20
Dipole-energy Emax/E0
E/E0
Dipole-energy 4.5 3.5
10 5
2.5 1.5
15
0 1
1.1
1. 2
1. 3
z/R Fig. 6. The electric field distributions along the z axis for 3i/3e ¼ 10, d/R ¼ 2.6.
1
1.25
1.5
1.75
2
d/ 2R Fig. 8. The maximum electric field strength on the surface of a sphere for 3i/3e ¼ 10.
120
F. Lu, Q. Ye / Journal of Electrostatics 68 (2010) 116–121
The saturation charge is a function of the sphere parameters and electric field onlyd it does not depend on the volume faction of spheres. Now, using Eq. (14), we can get
100 Dipole 80
FEM 0
Emax/E0
Dipole-energy
Emax ¼ ð1 þ 2kÞE0 ¼
60
Q : 4p3e R2
(17)
Solving this equation results in
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3eff iE0 : h Q ¼ 4p3e ð1þ2kÞR2 u t 2 3i ð1kÞ f þ 3e ð1f Þ 1þ 52k2 f kf lnf
40 20
(18)
0 1
1. 25
1. 5
1. 75
If f/0, then Q/Q0.
2
6. Conclusion
d/2R Fig. 9. The maximum electric field strength on the surface of a sphere for 3i/3e ¼ 100.
dielectric mismatch. As shown in Fig. 10, if 3eff obtained by different methods is placed in Eq. (13), the result of dipole-energy approximation is greatly improved using the more accurate effective permittivity 3eff obtained by iterative technique [17]. So an important characteristic of the dipole-energy approximation is that the more precise effective permittivity 3eff of dielectric mixture we get, the more accurate results can be calculated. Correspondingly, the results of the dipole-energy approximation showed in Figs. 6, 8 and 9 should be improved also. 5.5. The saturation charge of a sphere For a dielectric mixture in a uniform external field, the spheres always become charged while they become polarized (we do not consider the charging mechanism). When the maximum field strength of a sphere cause by the external field is equal to the field strength of the charge Q0 deposited on the sphere, the charging process is ended and Q0 is the saturation charge of a sphere.
Emax ¼ ð1 þ 2kÞE0 ¼
Q0 : 4p3e R2
The research reported in this paper was supported by a Grant from the National Science Foundation of China, Grant No. 50577074.
(16)
References
6.5 IterativeTech
6
FEM
5.5
E/E0
5
MaxwellGarnett
4.5 4 3.5 3 2.5 1
1.1
1.2
Acknowledgements
(15)
Solving this equation results in
Q0 ¼ 4p3e ð1 þ 2kÞR2 E0 :
For a dielectric mixture in a uniform external field E0, a dipoleenergy approximation considering the interactions among spheres is presented in this paper. The dipole-energy approximation is better than the ideal dipole approximation to get more accurate results, especially in the area of strong field, but it is not very good at any point on the surface of sphere. An approximation formula for calculating the maximum field strength is derived. It is suitable for intermediate or strong coupling mixture with closely spaced spheres of high permittivity since it fully considers the interactions of spheres with a uniform equivalent electric field E00 , other than E0. Compared with the result of FEM, The dipole-energy approximation provides an improvement over the previous ideal dipole approximation. Based on the dipole-energy approximation, a formula for calculating saturation charge of spheres is derived for high-volume fraction spheres. What an important characteristic of the dipole-energy approximation is that the more accurate effective permittivity 3eff of dielectric mixture we get, the more accurate results can be calculated.
1.3
z/R Fig. 10. The electric field distributions along the z axis with 3eff obtained by different methods for 3i/3e ¼ 100, d/R ¼ 2.6.
[1] K. Adamiak, Rate of charging of spherical particles by monopolar ions in electric fields. IEEE Trans. Ind. Appl. 38 (2002) 1001–1008. [2] L. Dascalescu, M. Mihailescu, A. Mizuno, The behavior of conductive particles in pulsed corona fields. J. Phys. D. Appl. Phys. 29 (1996) 522–528. [3] L. Dascalescu, D. Rafiroiu, A. Samuila, R. Tobazeon, Charging of insulating spheres on the surface of an electrode affected by monopolar ions. IEEE Trans. Ind. Appl. 34 (1998) 35–42. [4] J.C. Clements, K.K. Yu, Continuum-regime field charging in an electron swarm. IEEE Trans. Ind. Appl. 27 (1991) 1225–1232. [5] M. Washizu, T.B. Jones, Multipolar dielectrophoretic force calculation. J. Electrostat. 33 (1994) 187–198. [6] M. Washizu, T.B. Jones, Dielectrophoretic interaction of two spherical particles calculated by equivalent multipole-moment method. IEEE Trans. Ind. Appl. 32 (1996) 233–242. [7] T.B. Jones, R.D. Miller, Multipolar interactions of dielectric spheres. J. Electrostat. 22 (1989) 231–244. [8] Y. Chen, A.F. Sprecher, H. Conrad, Electrostatic particle-particle interactions in electrorheological fluids. J. Appl. Phys. 70 (1991) 6796–6803. [9] W.B. Smith, J.R. Mcdonald, Development of a theory for the charging of particles by unipolar ions. J. Aerosol Sci. 7 (1976) 151–166. [10] S.Y. Kwon, J.Y. Kim, B.J. Yoon, Continuum field charging of dielectric spheroids. J. Colloid Interf. Sci. 221 (2001) 313–315. [11] A.V. Radchik, A.V. Paley, G.B. Smith, A.V. Vagov, Polarization and resonant absorption in intersecting cylinders and spheres. J. Appl. Phys. 76 (1994) 4827–4835. [12] T.J. Johnson, E. James Davis, An analysis of electrophoresis of concentrated suspensions of colloidal particles. J. Colloid Interf. Sci. 215 (1999) 397–408.
F. Lu, Q. Ye / Journal of Electrostatics 68 (2010) 116–121 [13] M.M. Pauthenier, M. Moreau-Hanot, La charge des particles spheriques dans un champ ionize. J. Phys. Radium 3 (1932) 50–613. [14] K.I. Golden, G. Kalman, Approximation scheme for strongly coupled plasmas: dynamical theory. Phys. Rev. 19 (1979) 2112–2128. [15] N. Studart, O. Hipolito, Electrodynamical properties of two-dimensional classical electron systems. Phys. Rev. A 22 (1980) 2860–2865.
121
[16] E. Trizac, M. Anbouy, L. Bocquet, Analytical estimation of effective charges at saturation in Poisson–Boltzmann cell models. J. Phys. Condens. Matter 15 (2003) s291–s296. [17] W.M. Merrill, R.E. Diaz, M.M. Lore, M.C. Squires, N.G. Alexopoulos, Effective medium theories for artificial materials composed of multiple sizes of spherical inclusions in a host continuum. IEEE Trans. Antennas Propag. 47 (1999) 142–148.
Contents lists available at ScienceDirect
Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
A dipole approximation for a dielectric mixture based on the equal field energy method Fei Lu*, Qizheng Ye College of Electrical and Electronic Engineering, Huazhong University of Science & Technology, Wuhan 430074, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 April 2009 Received in revised form 8 August 2009 Accepted 21 November 2009 Available online 2 December 2009
In this paper, a dipole-energy approximation for calculating the electric field distribution and saturation charge of spheres in an infinitely large dielectric mixture has been carried out. The approximation rests on the assumption that the field energy of mixture calculated using two different methods have the same value. One way is considering that the mixture in a uniform field E0 as a uniform object of effective permittivity 3eff from a macroscopic point of view, where 3eff is seen as the average characteristic parameter of the object. The other way is assuming that the spheres in mixture are in the equivalent external field E00 , and E00 related to the dielectric mismatch and the ratio of the sphere radius to the average distance between neighboring spheres has been obtained. Based on dipole-energy model, the approximate formulas for calculating the maximum field strength and saturation charge of spheres are derived separately. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Dielectric mixture Electric field distribution Saturation charge Field energy Dipole-energy approximation
1. Introduction In recent years, many attempts have been made to provide detailed theoretical descriptions for dielectric mixture, where dielectric spheres of permittivity 3i are embedded in a host material of permittivity 3e. There are many areas of science and technology where the response of merging spheres to applied fields is of interest. Among many approaches to solve this problem, a widely adopted one is to treat the polarized sphere as a dipole, such as the ideal dipole approximation. It assumes that the distance between spheres is so far that the interactions among spheres is negligible, which is sufficiently applied for the calculation of electrical charge in industrial electrostatic precipitation when a small dielectric mismatch between the spheres and the environment [1–4]. However, the ideal dipole approximation is invalid when the dielectric mismatch is large and the spheres are nearly touching, then the contributions of higher-multipole may be significant [5–8]. In the case of merging small particles, and large bodies or surfaces to an electrostatic field, different issues related to the particle charging which is an essential process in many electrostatics applications were investigated [9–12]. Pauthenier and Moreau-Hanot calculated the saturation charge of a spherical particle first [13], the formula rests on the ideal dipole
* Corresponding author. Tel.: þ86 2787542726. E-mail addresses: [email protected] (F. Lu), [email protected] (Q. Ye). 0304-3886/$ – see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2009.11.006
approximation and it will compromise the accuracy when a dielectric mixture with the large dielectric mismatch and the high-volume fraction. Some other approximations adopt the model of non-interacting particles as a point of departure and add the effect of interaction as a perturbation, but this is obviously not an expedient approach for intermediate or strong coupling. Various nonperturbative mean field method [14] and semiphe-nomenological theories [15,16] have attempted to attack the problem with varying degrees of success. In this paper, a simple dipole approximation for calculating the field distribution in dielectric mixture and the saturation charge of spheres has been provided for high-volume fraction particles based on the assumption that the mixture is equivalent to different systems which have equal field energy from multiscale view (the approximation is named dipole-energy approximation in the paper). 2. The ideal dipole model for a dielectric mixture For a dielectric mixture in a uniform external field E0, the ideal dipole approximation rests on the assumption that the field experienced by an arbitrarily selected central sphere is still uniform, and the magnitude of this external field is still E0. That means the sum of the field from all the other spheres is zero and it just needs to consider one sphere in a uniform external field E0 as shown in Fig. 1. Considering a sphere of radius R and permittivity 3i in a uniform field E0. The permittivity of environmental media is 3e. Solving the Laplace equation, we can get
F. Lu, Q. Ye / Journal of Electrostatics 68 (2010) 116–121
117
Fig. 2. A finite-element model of a sphere in a cylindrical volume. Fig. 1. A sphere subjected to a external field E0.
!
E out ¼ r0
3 3e 2E0 R3 E0 cosq þ i $ cosq 3i þ 2 3e r 3
þ q0
E in ¼ z0
E0 sinq þ
!
3i 3e E0 R3 $ sinq ; 3i þ 23e r3
3 3e E ; 3i þ 23e 0
ð1Þ
from all the surrounding spheres is not zero, but it is a uniform field in the parallel direction to E0. So a cell of mixture can be seen as a sphere polarized in a uniform field E00 (E00 ¼ E0 þ Espheres), and the field distribution in a cell is similar to Eqs. (1) and (2) while E0 is replaced by E00 . Due to the dielectric mixture is uniform and infinitely large, the field distributions in each cell are the same and their calculation is limited to a cell.
(2)
where Eout is the outside field of the sphere, and Ein is the inside field of the sphere. Especially, according to Eq. (1), the maximum field strength on the surface of sphere is
Emax ¼
33i E ; 3i þ 2 3e 0
(3)
and it does not depend on the sphere radius. 3. The finite-element method for a dielectric mixture For a dielectric mixture in a uniform external field E0, a cylinder model is proposed for an arbitrarily selected central sphere. If the average distance between two neighboring sphere’s centers is d in the unit of sphere radius R, an accessory cylinder of length d and diameter d outside each sphere can be constructed as shown in Fig. 2. The potential difference between both ends of the cylinder is Ui(Ui ¼ E0d) because the spheres are evenly distributed in the parallel direction to the external field. The potential at the side face is satisfied with the natural boundary condition (v4/vn ¼ 0, the outward surface-normal component of the electric field strength is zero) because the spheres are evenly and randomly distributed in the direction perpendicular to the external field. The boundary condition implicitly accounts for the presence of the surrounding spheres, therefore, only the electric field around a single sphere needs to be analyzed. Now solving the Laplace equation with the finite-element method (FEM) and then get the exact numerical solutions to compare with other models. 4. The dipole-energy model for a dielectric mixture For a dielectric mixture in a uniform field E0, a dipole-energy model is proposed to consider the perturbations due to all the surrounding spheres. The paper assumes that the sum of field Espheres
Fig. 3. Equivalent cells of a dielectric mixture.
118
F. Lu, Q. Ye / Journal of Electrostatics 68 (2010) 116–121
Based on the assumption above, the field energy of dielectric mixture can be calculated using different methods. First, a dielectric mixture in a uniform field E0 is considered as a uniform object of effective permittivity 3eff from a macroscopic point of view. Effective permittivity 3eff is seen as the average characteristic parameter of this object. Selecting a cell of system as shown in Fig. 3 (a) and there is one sphere in a cell, we can get
Wa ¼
1 3 E2 DV; 2 eff 0
(4)
where Wa is the field energy of a cell, DV is the volume of a cell. The paper assumes the cube cell approximates to a spherical cell that the energy integral can be solved easily in the latter paper (it is impossible to calculate the cube volume of integral in the spherical coordinates system), the field energy of a cell is
" 0 2pE02 1 2 02 4 3 3e d=2Þ3 R3 Wb ¼ 3i ð1 kÞ E0 $ pR þ 2 3 3 ! # 5 2 3 R6 R3 3 kR ln : þ k R 2 ðd=2Þ3 ðd=2Þ3
ð9Þ
Finally, the paper assumes that the dielectric mixture has equal field energies of these two equivalent cells, so
Wa ¼ Wb :
(10)
On condition that effective permittivity 3eff is known and precise enough, substituting Eqs. (5) and (9) into Eq. (10), we can get
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3eff 0 E0 ¼ u 3 h 3 3 3 3 iE0 ¼ t 3i ð1 kÞ2 ð2R 1 þ 52k2 2R k 2R þ3e 1 2R ln 2R d d d d d
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3eff u h iE0 ; t 2 3i ð1 kÞ f þ 3e ð1 f Þ 1 þ 52k2 f kf lnf (11)
Wa ¼
1 4 d 3 3eff E02 $ p : 2 3 2
(5)
Second, the field energy of mixture can be obtained by calculating the field energies of spheres and environmental media separately. Based on the assumption above, there is one sphere in a cell and the cell is still in a uniform field that the magnitude is not the primary external field E0, but E00 , as shown in Fig. 3 (b). The field energy of a cell is given as
1 3 2 i
Wb ¼ Wbin þ Wbout ¼
Z
1 2 Ein dVin þ 3e 2
Z
2 Eout dVout ;
(6)
where Wb is the field energy of the cell, Wbin is the field energy of a sphere, Wbout is the field energy of environmental media, Vin is the volume of a sphere, Vout is the environmental media volume of a cell. If E0 is replaced by E00 in Eqs. (1) and (2), we can get (the paper assumes the cube cell approximates to a spherical cell that the energy integral can be solved easily in the spherical coordinates system)
Wbin ¼
1 0 4 3 ð1 kÞ2 E02 $ pR3 ; 2 i 3
where f is the volume fraction of spheres in mixture, with f ¼ (4pR3/3)/(4p(d/2)3/3) ¼ (2R/d)3. The maximum volume fraction of spheres in cube lattice structure is p/6 and it is not very large, therefore, the Maxwell–Garnett model is adoptable to get 3eff accurate enough
3eff ¼ 3e 1 þ
3f ð3i 3e Þ : 3i þ 23e f ð3i 3e Þ
(12)
Now we can calculate the electric field distribution in a dielectric mixture if E0 is replaced by E00 in Eqs. (1) and (2)
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 !2 u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u R3 R3 E ¼ Er2 þ Eq2 ¼ t 1 þ 2k 3 cos2 q þ 1 k 3 sin2 q$E00 : r r (13) Eq. (13) is an analytical formula to calculate the electric field at any point. Clearly it considers the perturbations due to all the other spheres. And the maximum electric field strength on the surface of a sphere is
(7)
6 Dipole
Zp
Z2p
sina da
0 0
E2 ¼ 0 3e 2
dq
Z2p
sinada
0
R3
þ 1k
!2
2 Eout r 2 dr
R
0
Zp
Zd=2
0
dq
Zd=2
2
3 4 1 þ 2kR 3 r
FEM Dipole-energy
4 !2
E/E0
Wbout
1 ¼ 3e 2
5
cos2 q
3R
3 2
sin2 q5r 2 dr;
r3 " ! ! 3 0 2pE02 d 5 R6 3e R3 þ k2 R3 ¼ 3 2 2 ðd=2Þ3 # R3 kR3 ln ðd=2Þ3 where k ¼ (3i 3e)/(3i þ 23e), then
1 0 (8)
0
0.2
0.4
0.6
0.8
1
Z/R Fig. 4. The electric field distributions on the surface of a sphere for 3i/3e ¼ 10, d/R ¼ 2.6.
F. Lu, Q. Ye / Journal of Electrostatics 68 (2010) 116–121
119
6.5
6
Dipole
Dipole FEM
5
FEM
5.5
Dipole-energy
Dipole-energy E/E0
E/E0
4 3
4.5 3.5
2 2.5
1 1.5
0 0
0.2
0.4
0.6
0.8
1
1. 1
1. 2
1. 3
z/R
1
Z/R
Fig. 7. The electric field distributions along the z axis for 3i/3e ¼ 100, d/R ¼ 2.6.
Fig. 5. The electric field distributions on the surface of a sphere for 3i/3e ¼ 100, d/R ¼ 2.6. 0
Emax ¼ð1þ2kÞE0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3eff iE0 : h ¼ð1þ2kÞu t 2 3i ð1kÞ f þ3e ð1f Þ 1þ52k2 f kf lnf
(14)
5. Results and discussion 5.1. The electric field distribution on the surface of a sphere The electric field distributions on the surface of an arbitrary selected sphere are calculated using the ideal dipole approximation, the FEM and the dipole-energy approximation are all shown in Figs. 4 and 5. As seen in these two figures, the calculated results of the dipole-energy approximation are good at the two poles of the sphere. Especially in the area of strong field, the dipole-energy approximation which has more accurate results is better than the ideal dipole approximation, but it is not very good at any point on the surface of the sphere.
Obviously, compared with the results of FEM, the dipole-energy approximation is more accurate than the ideal dipole approximation. 5.3. The maximum electric field strength Figs. 8 and 9 show the calculated results of the maximum electric field strength for different d/R and different permittivity using the three methods. The dipole-energy approximation is better than the ideal dipole approximation to get more accurate results. What is interesting is that the calculated results using the dipole-energy approximation are more accurate when the dielectric mismatch (3i/3e) is large or the high-volume fraction (2R/d), contrary to the ideal dipole approximation. And there is good fit between the calculated results of the minimum field strength of the dipole-energy approximation and those of the FEM, as shown in Figs. 4 and 5. 5.4. The effective permittivity of dielectric mixture According to Eq. (11), it is obvious that the effective permittivity
3eff of dielectric mixture affects the results of dipole-energy 5.2. The electric field distribution outside the sphere The electric field distributions along the z axis are calculated using the three methods, and they are all shown in Figs. 6 and 7.
approximation directly. Although the effective permittivity is usually obtained by Eq. (12) easily, the result of Maxwell–Garnett model is always small when a high-volume fraction or large
25
6.5
Dipole
Dipole FEM
5.5
FEM
20
Dipole-energy Emax/E0
E/E0
Dipole-energy 4.5 3.5
10 5
2.5 1.5
15
0 1
1.1
1. 2
1. 3
z/R Fig. 6. The electric field distributions along the z axis for 3i/3e ¼ 10, d/R ¼ 2.6.
1
1.25
1.5
1.75
2
d/ 2R Fig. 8. The maximum electric field strength on the surface of a sphere for 3i/3e ¼ 10.
120
F. Lu, Q. Ye / Journal of Electrostatics 68 (2010) 116–121
The saturation charge is a function of the sphere parameters and electric field onlyd it does not depend on the volume faction of spheres. Now, using Eq. (14), we can get
100 Dipole 80
FEM 0
Emax/E0
Dipole-energy
Emax ¼ ð1 þ 2kÞE0 ¼
60
Q : 4p3e R2
(17)
Solving this equation results in
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3eff iE0 : h Q ¼ 4p3e ð1þ2kÞR2 u t 2 3i ð1kÞ f þ 3e ð1f Þ 1þ 52k2 f kf lnf
40 20
(18)
0 1
1. 25
1. 5
1. 75
If f/0, then Q/Q0.
2
6. Conclusion
d/2R Fig. 9. The maximum electric field strength on the surface of a sphere for 3i/3e ¼ 100.
dielectric mismatch. As shown in Fig. 10, if 3eff obtained by different methods is placed in Eq. (13), the result of dipole-energy approximation is greatly improved using the more accurate effective permittivity 3eff obtained by iterative technique [17]. So an important characteristic of the dipole-energy approximation is that the more precise effective permittivity 3eff of dielectric mixture we get, the more accurate results can be calculated. Correspondingly, the results of the dipole-energy approximation showed in Figs. 6, 8 and 9 should be improved also. 5.5. The saturation charge of a sphere For a dielectric mixture in a uniform external field, the spheres always become charged while they become polarized (we do not consider the charging mechanism). When the maximum field strength of a sphere cause by the external field is equal to the field strength of the charge Q0 deposited on the sphere, the charging process is ended and Q0 is the saturation charge of a sphere.
Emax ¼ ð1 þ 2kÞE0 ¼
Q0 : 4p3e R2
The research reported in this paper was supported by a Grant from the National Science Foundation of China, Grant No. 50577074.
(16)
References
6.5 IterativeTech
6
FEM
5.5
E/E0
5
MaxwellGarnett
4.5 4 3.5 3 2.5 1
1.1
1.2
Acknowledgements
(15)
Solving this equation results in
Q0 ¼ 4p3e ð1 þ 2kÞR2 E0 :
For a dielectric mixture in a uniform external field E0, a dipoleenergy approximation considering the interactions among spheres is presented in this paper. The dipole-energy approximation is better than the ideal dipole approximation to get more accurate results, especially in the area of strong field, but it is not very good at any point on the surface of sphere. An approximation formula for calculating the maximum field strength is derived. It is suitable for intermediate or strong coupling mixture with closely spaced spheres of high permittivity since it fully considers the interactions of spheres with a uniform equivalent electric field E00 , other than E0. Compared with the result of FEM, The dipole-energy approximation provides an improvement over the previous ideal dipole approximation. Based on the dipole-energy approximation, a formula for calculating saturation charge of spheres is derived for high-volume fraction spheres. What an important characteristic of the dipole-energy approximation is that the more accurate effective permittivity 3eff of dielectric mixture we get, the more accurate results can be calculated.
1.3
z/R Fig. 10. The electric field distributions along the z axis with 3eff obtained by different methods for 3i/3e ¼ 100, d/R ¼ 2.6.
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