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Equivalent medium theory of layered sphere particle with anisotropic shells
Journal of Quantitative Spectroscopy & Radiative Transfer 179 (2016) 165–169
Contents lists available at ScienceDirect
Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
Equivalent medium theory of layered sphere particle with anisotropic shells Li Xingcaia,n, Wang Minzhongb,n, Zhang Beidouc a School of Physics & Electrical Information Engineering, Ningxia Key Laboratory of Intelligent Sensing for the Desert Information, Ningxia University, Yinchuan 750021, China b Urumqi Institute of Desert Meteorology, China Meteorological Administration, Urumqi, Xinjiang 830002, China c Key Laboratory for Semi-Arid Climate Change, Ministry of Education, College of Atmospheric Science, Lanzhou University, Lanzhou 730000, China
a r t i c l e i n f o
abstract
Article history: Received 3 December 2015 Received in revised form 3 March 2016 Accepted 3 March 2016 Available online 15 March 2016
Researches on the optical properties of small particle have been widely concerned in the atmospheric science, astronomy, astrophysics, biology and medical science. This paper provides an equivalent dielectric theory for the functional graded particle with anisotropic shells, in which inhomogeneous and anisotropic particle was equivalently transformed into a new kind of homogeneous, continuous and isotropic sphere with same size but different permittivity, and then greatly simplify the calculation process of particle's optical property. Meanwhile, the paper also discusses whether the charge on the particle can change the expression of its equivalent permittivity or not. These results proposed in this paper can be used to simulate the electrical, optical properties of layered sphere, it also meet the research requirement in the design of functional graded particles in different subjects. & 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Charged Layered Anisotropy Equivalent medium
1. Introduction Estimation and simulation of the optical properties for small particles became a common research subject in the atmospheric science, astronomy, astrophysics, biology, medical science and the chemistry. Because of its complexities in particle shapes and structures [1,2], several methods to resolve the problem have been suggested, for example, Rayleigh approximation [3], Lorenz–Mie theory [3,4], Nlayered Mie theory [5–8], DDA [9,10], T-matrix method [11–16], Separation of Variables Method [2,17], FDTD [18,19], Deby series solution [20–22] etc. Besides, because of its clear physical significance and concision, equivalent medium theory has been obtained much attention, and it has been n
Corresponding authors. E-mail addresses: [email protected], [email protected] (M. Wang).
used to do some numerical simulation works on the optical and electrical properties of intermixed random medium [23,24]. With the help of the equivalent medium theory, Videen et al. discussed the electromagnetic scattering of sphere with an absorptive core [25]. Kolokolova et al. simulated the electromagnetic scattering property of discontinuous particle, and compared it with the experiment results [26]. They found that the equivalent medium theory does not work for calculations of back-scattering characteristics of large particles. Li et al. proposed the equivalent permittivity of a multilayered isotropic sphere, and discussed its application in the calculation of particles' electromagnetic scattering [27]. They found the equivalent permittivity can be used when the Rayleigh hypothesis is meet. Liu et al. compared the similarities and differences of four major theories on the equivalent medium, such as the Bruggeman theory, the Maxwell-Garnett theory, and two different Wiener bounds, and analyzed their scopes of application in
http://dx.doi.org/10.1016/j.jqsrt.2016.03.008 0022-4073/& 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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X. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 179 (2016) 165–169
calculating the electromagnetic scattering properties of inhomogeneous medium particles [28]. However, these models mentioned above are only applicable to the isotropic medium. Meanwhile, functional graded materials make it easy to achieve invisibility at specific frequency ranges, which is attracting more and more attentions from scientific researchers [29,30]. The anisotropic shell can significantly change the electric field distribution inside the particle, so as to realize some interesting features [29]. Numerical simulations of the electromagnetic field both inside and outside of the particle with different structural can give an important guidance to the structural designs, and the equivalent medium theory can significantly simplify related calculations and meet the accurate requirement of engineering design. There are quite a few researches on the optical properties of particles with multilayered structures or some anisotropic shells, but there is still a lack of detailed and systematic research on its equivalent permittivity. Therefore, this paper devoted to the study of equivalent permittivity of these kinds of particles with similar structures. In addition, we also discussed the influences of the surface charge on the particle's equivalent permittivity.
2. Basic theories and models 2.1. Equivalent permittivity of homogeneous particle with some anisotropic shells Assuming there's a core–shell structured spherical particle with an anisotropic coating, and the core is an isotropic medium, its permittivity is ϵ0 and the radius is r 0 . The shell is an anisotropic material, its outer radius is r 1 , and its permittivity can be expressed with the spherical coordinate as follows, 2 3 εr 0 0 6 7 ð1Þ εs ¼ 4 0 εθ 0 5 0 0 εφ The background medium around the particle is isotropic and its permittivity is εh . Considering the electrostatic field component of the incident wave is Ein ¼ E0 x. With the spherical coordinates ðr; θ; φÞ, the electric potentials both inside of the particle and outside of it can be expressed as [29,30], ϕc ¼ Ec r cos θ
r o r0
ϕs ¼ A1 r t1 þ A2 r t2 cos θ
r 0 or o r 1
Through Eqs. (2), (3) we can obtain Ec ¼
3εr εh ðt 2 t 1 ÞE0 ðεr t 2 ε0 Þa1 ðεr t 1 ε0 Þa2
A1 ¼
ðεr t 2 ε0 ÞEc εr ðt 2 t 1 Þr 0t1 1
A2 ¼
ð4Þ ðεr t 1 ε0 ÞEc
εr ðt 2 t 1 Þr 0t2 1
D¼
ε~ εh 3 r ε~ þ 2εh 0
Here a1;2 ¼ εr t 1;2 þ2εh λt1;2 1 ; λ ¼ r 1 =r 0 ; a1;2 ¼ εr t 1;2 þ 2εh λt 1;2 1 and ε~ ¼
ðεr t 2 ε0 Þt 1 λt 1 t 2 ðεr t 1 ε0 Þt 2 εr ðεr t 2 ε0 Þλt 1 t 2 ðεr t 1 ε0 Þ
Compared (2c) with the expression of outer potential for a homogeneous sphere in a uniform electric field, ε~ can be considered as the equivalent permittivity of spherical particle with an anisotropic coating. As shown in the above equation, the coefficient before E0 in Ec is a constant, which related to the core–shell radius ratio and the permittivity of the particle, which means that the core–shell geometric parameters and the electrical properties of shell all can change the electric field in the particle. Based on the new parameter expression ε~ , the equivalent dielectric constant of the coated sphere with an anisotropic shell, and the method proposed by Li et al. [27], we can expand this result to other conditions, and then obtain the equivalent parameter of any particle with some types of special structures. Those results are discussed as follows. 1) N-layered core–shell particle with an anisotropic shell and N 1 isotropic shells For this kind of special particle, supposed the radius for the core and shells are r 0 ; r 1 ; r 2 ; r 3 ; ⋯r N , the permittivity is ε0 ; ε1 ; ε2 ; ε3 ; ⋯εN , respectively. Here the dielectric constant ε1 for the first shell have a similar expression with the Eq. (1). In order to obtain the equivalent permittivity of this multi-layered particle, we can set the core and the first shell as a new core, whose dielectric constant can be obtained through the Eq. (5). With a similar operation with reference [27] we can obtain its permittivity, β þ 2g n þ 2δn ð βn 1 g n Þ signed as ε~ n ¼ n 1 εn - β n εn ð6Þ βn 1 þ 2g n δn ðβn 1 g n Þ
ð2aÞ
here
ð2bÞ
δn ¼ r 3n 1 =r 3n ; g n ¼ εn =~ε n 1 ; β1 ¼ 1; ε~ 1 ¼ ε~ ; βn 1 þ2g n þ 2δn ðβn 1 g n Þ βn ¼ ; n ¼ 2; 3; ……N: βn 1 þ 2g n δn ðβn 1 g n Þ
ð2cÞ ϕh ¼ r þ D=r 2 E0 cos θ r 4 r 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here t 1;2 ¼ 1 7 1 þ 8εθ =εr =2, E0 is the intensity of electrostatic field, and A1 ; A2 ; Ec ; D; are undetermined coefficients, which can be determined through the corresponding boundary conditions, 8 ∂ϕc ∂ϕ > > ¼ εr s < r ¼ r 0 ϕc ¼ ϕs ε0 ∂r ∂r ð3Þ ∂ϕh ∂ϕs > > ¼ ε ϕ ¼ ϕ ε r ¼ r : r 1 h s h ∂r ∂r
ð5Þ
The parameter Ec in the electric potential of the corezone can be expressed as, Ec ¼
3εr εh ðt 2 t 1 ÞAE0 ðεr t 2 εc Þa1 ðεr t 1 εc Þa2
ð7Þ
9εh εi ðε þ2ε Þð ε g þ2ε i1 i Þ þ2δi ðεi εh Þðεg i 1 εi Þ h i¼2 i
ð8Þ
N
A¼ ∏
X. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 179 (2016) 165–169
The above equation reflects the effect of shells on particle's inner potential. 2) N-layered core–shell particle with N 1 anisotropic shells For this kind of particle, supposed the radius for the core and those of the shells for the particle are r 0 ; r 1 ; r 2 ; r 3 ; ⋯r N , the permittivity for the shell is εir ; εiθ ; i ¼ 2; 3; ⋯N, respectively. The dielectric constant of the core zone is ε0 . In order to obtain the equivalent permittivity of this particle, we firstly set ε1r ¼ εr ; ε1θ ¼ εθ , and take the core and the first shell as a new core, whose equivalent permittivity is showed in the Eq. (6), then with a similar operation with the reference [27], we can obtain the permittivity of this particle, n n n tn1 tn2 n n n εr t 2 εg εr t 1 εg n 1 t 1 λn n 1 t2 n ε~ n ¼ ð9Þ n n εr t1 t2 εnr t n2 εg εnr t n1 εg n 1 λn n1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi here n ¼ 2; 3; ⋯N; t i1;2 ¼ 1 7 1 þ8εiθ =εir =2; εe1 ¼ ε~ ; λn ¼ r n =r n 1 : The expansion coefficient of electric potential in the core zone can be expressed as follow: ( ) N 3εnr εh ðt n2 t n1 Þ n n n E0 ð10Þ Ec ¼ ∏ n n n n 1 a1 εr t 1 εg n 1 a2 n ¼ 2 εr t 2 εg n Here an1;2 ¼ εnr t n1;2 þ 2εh λn t 1;2 1 .
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The physical parameters in here are same as described in the previous context. 4) Core–shell particle with a continuous variation anisotropic shell If the particle's core is an isotropic medium and its permittivity and radius are ε0 and r 0 . The particle's shell is a kind of functional graded material with anisotropy, and its outer radius is R, its permittivity can be expressed as: 2 3 εr ðrÞ 0 0 60 7 εθ ðrÞ 0 εs ðrÞ ¼ 4 5 0 0 εφ ðrÞ For this condition, it is hard to obtain an analytical expressions of the electric potential for the particle's inner and outer zone. Dong et al. [30] used an anisotropic differential effective dipole approximation to investigate the electric properties of graded spherical particles with anisotropic materials, but it is too complex. However, we can choose proper step length and make the functional-graded anisotropic material manually stratification. Assuming the shell is divided into N layers [7], each of which has an outer radius r i ¼ R Nr0 i þ r 0 ; i ¼ 1; 2; 3⋯N, the corresponding permittivity is, 2 3 εir ðr i Þ 0 0 6 7 7 0 εiθ ðr i Þ 0 εis ðr i Þ ¼ 6 4 5 0 0 εiφ ðr i Þ
3) N-layered core–shell particle, and the properties of shells periodic changed For the particle, supposed its core is an isotropous medium, its permittivity and radius are ε0 and r 0 , and the odd layers are anisotropic media, their corresponding permittivity and radius are εir ; εiθ ; r i ; i ¼ 1; 3; 5⋯, while the even layers are isotropous media, of which the permittivity and radius is εir ¼ εiθ ; r i ; i ¼ 2; 4; 6⋯ respectively. Then its equivalent permittivity is, β2n 1 þ 2g 2n þ 2δ2n ðβ2n 1 g 2n Þ signed as ε2n - β2n ε2n εf 2n ¼ β2n 1 þ 2g 2n δ2n ðβ2n 1 g 2n Þ
Through this operation, the permittivity and radius of each shell are constant. Therefore, we can directly use above results to obtain the particle's equivalent permittivity, and then investigate its electrical or optical properties through the Rayleigh approximation, the Mie theory, or the electrostatic. However, the calculation precision based on this method depend upon the thickness of each shell. Accordingly, it needs to control the relative error in an actual calculation.
ð11Þ
Most particles under natural conditions are charged [31,32]. For example, the sand or dust particles, which all carry much electrostatic charge because of its mutual contacts and collides during moving. Zheng et al. carried out a detailed and systematic study on the electrification phenomenon of wind-blown sand, and discussed its effect on the movement of wind-blown sand and the evolution of land form of desert, and on the microwave propagation [33], whose works renew the extant theories of the physics
ε~ 2n þ 1 ¼
n t2n 2n t 2n 2n 2n 1 2 ε2n ε2n 2n t 1 λ2n 2n t 2 2n þ 1 r t 2 εf r t 1 εf 2n 2n εr t1 t2 2n 2n ε2n ε2n 2n λ2n 2n r t 2 εf r t 1 εf ð12Þ
The expansion coefficient of the inner potential in core area is
Ec ¼
8 > > > > > > > > > > <
2.2. Permittivity of partially charged particle with anisotropic coatings
9 > > > > > > > > > > =
N 3εnr εh ðt n2 t n1 Þ 9εh εi n n n E ∏ n n t εg an ðε þ 2ε Þðεg þ2ε Þ þ2δ ðε ε Þðεg ε Þ> 0 > g ε t ε ε a i i 1 i i i i1 i > h h n 1 n 1 > r 2 r 1 1 2 > > > > n ¼ 2k þ 1 > > > > > > > > i ¼ 2k > > > > ; : k ¼ 1; 2; 3⋯
ð13Þ
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of wind-blown sand and make it to be more reasonable and sophisticated. Bohren and Huffman are the first one who made theoretical analysis on the influence of electric charge on the particle's optical properties [34], but they did not make any numerical analysis. In order to explain the phenomenon that the actual measurements of microwave attenuation in sand storms are much bigger than the theoretically estimated values, and considering the charge on particle surface, Zhou et al. used Rayleigh approximation to discuss the influences of electric charges on particles' optical properties [35], and their studies firstly revealed the important impact of electric charges on the particle's optical properties. After that, many scholars carried out some numerical studies on the electromagnetic scatterings of the isotropic charged sphere with different material components [5,36–43]. However, there are still few of researches on the electromagnetic scattering of charged anisotropic particle [42]. In view of the diversity of particle properties in the nature, and the concision of using equivalent medium theory to compute particle's optical properties, in this section, we devoted to study the equivalent permittivity of partially charged sphere particles with anisotropic shells. Regarding to a small-sized coated sphere with local electrification, its structure and the physical parameters are the same as them described in Section 2.1, but its surface is partially charged, and the charge uniformly distributes on a spherical crown with an angle 2θ0 . According to existed reference [35], the electric potential both inside of the charged particle and outside of it should be expressed as follows, ϕc ¼ Ec r cos θ þ A
r o r0
ð14aÞ
C ϕs ¼ B þ þ A1 r t1 þA2 r t2 cos θ r ϕh ¼
D þ r þE=r 2 E0 cos θ r
r 0 o r or 1
ð14bÞ
r 4 r1
Compared with the results showed in Ref. [35], we can find that the parameter ε~ q in the Eq. (16) can be regard as the equivalent permittivity of the partially charged core– shell particle with a anisotropic shell. In addition, the parameter ε~ q is exactly the same as Eq. (5) in this paper. Therefore, we can believe that the surface charge on particle does not change the expression of equivalent permittivity for the inhomogeneous particle or the anisotropic particle. So we can speculate that, for a spherical particle with multilayered structure, its equivalent permittivity can be calculated via Eqs. (5), (6), (9), (11), (12), no matter its surface is charged or not, even if its shells are anisotropic medium.
3. Conclusion This paper, based on the method proposed in the reference [27], proposed the equivalent permittivity of layered spherical particle with different forms of coatings under the Rayleigh hypothesis, and deduced its iteration relations of the corresponding equivalent permittivity while those particles have some isotropic coatings, or some anisotropic coatings, and even them have certain intervals. In addition, we give the expansion coefficients of the electric potentials in the core area of particle. Meanwhile, the paper also discussed whether the charges distributed on the particle surface change the expression of equivalent permittivity of layered particle or not. However, those theoretical expressions are just suitable to the condition that the particle size is much smaller than the wavelength of the incident wave. The study, nevertheless, can be used to simulate the electrical, optical properties of layered spheres and those for the functional graded material in different subjects.
ð14cÞ Acknowledgment
The corresponding boundary conditions are [42]:
8 < r ¼ r0
ϕc ¼ ϕs
: r ¼ r1
ϕs ¼ ϕh
∂ϕs c εc ∂ϕ ∂r ¼ εr ∂r
ð15Þ
∂ϕs h εh ∂ϕ ∂r εr ∂r ¼ σH ðθ θ0 Þ
Here A; B; C; D; E are undetermined coefficients, σ is the charge density on particle surface, H ðθ θ0 Þ is Heaviside function, and 2θ0 is the charge distribution angle on particle surface. The undetermined coefficients can be determined through the above equations and the boundary conditions: 3εr εh ðt 2 t 1 ÞE0 ðεr t 2 εc Þa1 ðεr t 1 εc Þa2 ðεr t 2 εc ÞEc ðεr t 1 εc ÞEc A1 ¼ A2 ¼ εr ðt 2 t 1 Þr 0t1 1 εr ðt 2 t 1 Þr 0t 2 1 Ec ¼
E¼
εeq εh 3 r εeq þ 2εh s
σ σ H ðθ θ0 Þ r 1 D ¼ H ðθ θ0 Þ r 21 εh εh Here a1;2 ¼ εr t 1;2 þ2εh λt1;2 1 ; λ ¼ r 1 =r 0 , and
A¼B¼
εeq ¼
ðεr t 2 εc Þt 1 λt 1 t2 ðεr t 1 εc Þt 2 εr ðεr t 2 εc Þλt 1 t2 ðεr t 1 εc Þ
ð16Þ
We acknowledge support from the National Natural Science Foundation of China (11302111, 11562017, 41305035, 41305027), the Program for International S&T Cooperation Projects of China (2011DFA11780), the opening project of Key Laboratory of Mechanics on Disaster and Environment in Western (Lanzhou University), Ministry of Education, the opening project of State Key Laboratory of Advanced Optical Communication Systems and Networks (Shanghai Jiao Tong University, 2013GZKF031306).
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Contents lists available at ScienceDirect
Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
Equivalent medium theory of layered sphere particle with anisotropic shells Li Xingcaia,n, Wang Minzhongb,n, Zhang Beidouc a School of Physics & Electrical Information Engineering, Ningxia Key Laboratory of Intelligent Sensing for the Desert Information, Ningxia University, Yinchuan 750021, China b Urumqi Institute of Desert Meteorology, China Meteorological Administration, Urumqi, Xinjiang 830002, China c Key Laboratory for Semi-Arid Climate Change, Ministry of Education, College of Atmospheric Science, Lanzhou University, Lanzhou 730000, China
a r t i c l e i n f o
abstract
Article history: Received 3 December 2015 Received in revised form 3 March 2016 Accepted 3 March 2016 Available online 15 March 2016
Researches on the optical properties of small particle have been widely concerned in the atmospheric science, astronomy, astrophysics, biology and medical science. This paper provides an equivalent dielectric theory for the functional graded particle with anisotropic shells, in which inhomogeneous and anisotropic particle was equivalently transformed into a new kind of homogeneous, continuous and isotropic sphere with same size but different permittivity, and then greatly simplify the calculation process of particle's optical property. Meanwhile, the paper also discusses whether the charge on the particle can change the expression of its equivalent permittivity or not. These results proposed in this paper can be used to simulate the electrical, optical properties of layered sphere, it also meet the research requirement in the design of functional graded particles in different subjects. & 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Charged Layered Anisotropy Equivalent medium
1. Introduction Estimation and simulation of the optical properties for small particles became a common research subject in the atmospheric science, astronomy, astrophysics, biology, medical science and the chemistry. Because of its complexities in particle shapes and structures [1,2], several methods to resolve the problem have been suggested, for example, Rayleigh approximation [3], Lorenz–Mie theory [3,4], Nlayered Mie theory [5–8], DDA [9,10], T-matrix method [11–16], Separation of Variables Method [2,17], FDTD [18,19], Deby series solution [20–22] etc. Besides, because of its clear physical significance and concision, equivalent medium theory has been obtained much attention, and it has been n
Corresponding authors. E-mail addresses: [email protected], [email protected] (M. Wang).
used to do some numerical simulation works on the optical and electrical properties of intermixed random medium [23,24]. With the help of the equivalent medium theory, Videen et al. discussed the electromagnetic scattering of sphere with an absorptive core [25]. Kolokolova et al. simulated the electromagnetic scattering property of discontinuous particle, and compared it with the experiment results [26]. They found that the equivalent medium theory does not work for calculations of back-scattering characteristics of large particles. Li et al. proposed the equivalent permittivity of a multilayered isotropic sphere, and discussed its application in the calculation of particles' electromagnetic scattering [27]. They found the equivalent permittivity can be used when the Rayleigh hypothesis is meet. Liu et al. compared the similarities and differences of four major theories on the equivalent medium, such as the Bruggeman theory, the Maxwell-Garnett theory, and two different Wiener bounds, and analyzed their scopes of application in
http://dx.doi.org/10.1016/j.jqsrt.2016.03.008 0022-4073/& 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
166
X. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 179 (2016) 165–169
calculating the electromagnetic scattering properties of inhomogeneous medium particles [28]. However, these models mentioned above are only applicable to the isotropic medium. Meanwhile, functional graded materials make it easy to achieve invisibility at specific frequency ranges, which is attracting more and more attentions from scientific researchers [29,30]. The anisotropic shell can significantly change the electric field distribution inside the particle, so as to realize some interesting features [29]. Numerical simulations of the electromagnetic field both inside and outside of the particle with different structural can give an important guidance to the structural designs, and the equivalent medium theory can significantly simplify related calculations and meet the accurate requirement of engineering design. There are quite a few researches on the optical properties of particles with multilayered structures or some anisotropic shells, but there is still a lack of detailed and systematic research on its equivalent permittivity. Therefore, this paper devoted to the study of equivalent permittivity of these kinds of particles with similar structures. In addition, we also discussed the influences of the surface charge on the particle's equivalent permittivity.
2. Basic theories and models 2.1. Equivalent permittivity of homogeneous particle with some anisotropic shells Assuming there's a core–shell structured spherical particle with an anisotropic coating, and the core is an isotropic medium, its permittivity is ϵ0 and the radius is r 0 . The shell is an anisotropic material, its outer radius is r 1 , and its permittivity can be expressed with the spherical coordinate as follows, 2 3 εr 0 0 6 7 ð1Þ εs ¼ 4 0 εθ 0 5 0 0 εφ The background medium around the particle is isotropic and its permittivity is εh . Considering the electrostatic field component of the incident wave is Ein ¼ E0 x. With the spherical coordinates ðr; θ; φÞ, the electric potentials both inside of the particle and outside of it can be expressed as [29,30], ϕc ¼ Ec r cos θ
r o r0
ϕs ¼ A1 r t1 þ A2 r t2 cos θ
r 0 or o r 1
Through Eqs. (2), (3) we can obtain Ec ¼
3εr εh ðt 2 t 1 ÞE0 ðεr t 2 ε0 Þa1 ðεr t 1 ε0 Þa2
A1 ¼
ðεr t 2 ε0 ÞEc εr ðt 2 t 1 Þr 0t1 1
A2 ¼
ð4Þ ðεr t 1 ε0 ÞEc
εr ðt 2 t 1 Þr 0t2 1
D¼
ε~ εh 3 r ε~ þ 2εh 0
Here a1;2 ¼ εr t 1;2 þ2εh λt1;2 1 ; λ ¼ r 1 =r 0 ; a1;2 ¼ εr t 1;2 þ 2εh λt 1;2 1 and ε~ ¼
ðεr t 2 ε0 Þt 1 λt 1 t 2 ðεr t 1 ε0 Þt 2 εr ðεr t 2 ε0 Þλt 1 t 2 ðεr t 1 ε0 Þ
Compared (2c) with the expression of outer potential for a homogeneous sphere in a uniform electric field, ε~ can be considered as the equivalent permittivity of spherical particle with an anisotropic coating. As shown in the above equation, the coefficient before E0 in Ec is a constant, which related to the core–shell radius ratio and the permittivity of the particle, which means that the core–shell geometric parameters and the electrical properties of shell all can change the electric field in the particle. Based on the new parameter expression ε~ , the equivalent dielectric constant of the coated sphere with an anisotropic shell, and the method proposed by Li et al. [27], we can expand this result to other conditions, and then obtain the equivalent parameter of any particle with some types of special structures. Those results are discussed as follows. 1) N-layered core–shell particle with an anisotropic shell and N 1 isotropic shells For this kind of special particle, supposed the radius for the core and shells are r 0 ; r 1 ; r 2 ; r 3 ; ⋯r N , the permittivity is ε0 ; ε1 ; ε2 ; ε3 ; ⋯εN , respectively. Here the dielectric constant ε1 for the first shell have a similar expression with the Eq. (1). In order to obtain the equivalent permittivity of this multi-layered particle, we can set the core and the first shell as a new core, whose dielectric constant can be obtained through the Eq. (5). With a similar operation with reference [27] we can obtain its permittivity, β þ 2g n þ 2δn ð βn 1 g n Þ signed as ε~ n ¼ n 1 εn - β n εn ð6Þ βn 1 þ 2g n δn ðβn 1 g n Þ
ð2aÞ
here
ð2bÞ
δn ¼ r 3n 1 =r 3n ; g n ¼ εn =~ε n 1 ; β1 ¼ 1; ε~ 1 ¼ ε~ ; βn 1 þ2g n þ 2δn ðβn 1 g n Þ βn ¼ ; n ¼ 2; 3; ……N: βn 1 þ 2g n δn ðβn 1 g n Þ
ð2cÞ ϕh ¼ r þ D=r 2 E0 cos θ r 4 r 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here t 1;2 ¼ 1 7 1 þ 8εθ =εr =2, E0 is the intensity of electrostatic field, and A1 ; A2 ; Ec ; D; are undetermined coefficients, which can be determined through the corresponding boundary conditions, 8 ∂ϕc ∂ϕ > > ¼ εr s < r ¼ r 0 ϕc ¼ ϕs ε0 ∂r ∂r ð3Þ ∂ϕh ∂ϕs > > ¼ ε ϕ ¼ ϕ ε r ¼ r : r 1 h s h ∂r ∂r
ð5Þ
The parameter Ec in the electric potential of the corezone can be expressed as, Ec ¼
3εr εh ðt 2 t 1 ÞAE0 ðεr t 2 εc Þa1 ðεr t 1 εc Þa2
ð7Þ
9εh εi ðε þ2ε Þð ε g þ2ε i1 i Þ þ2δi ðεi εh Þðεg i 1 εi Þ h i¼2 i
ð8Þ
N
A¼ ∏
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The above equation reflects the effect of shells on particle's inner potential. 2) N-layered core–shell particle with N 1 anisotropic shells For this kind of particle, supposed the radius for the core and those of the shells for the particle are r 0 ; r 1 ; r 2 ; r 3 ; ⋯r N , the permittivity for the shell is εir ; εiθ ; i ¼ 2; 3; ⋯N, respectively. The dielectric constant of the core zone is ε0 . In order to obtain the equivalent permittivity of this particle, we firstly set ε1r ¼ εr ; ε1θ ¼ εθ , and take the core and the first shell as a new core, whose equivalent permittivity is showed in the Eq. (6), then with a similar operation with the reference [27], we can obtain the permittivity of this particle, n n n tn1 tn2 n n n εr t 2 εg εr t 1 εg n 1 t 1 λn n 1 t2 n ε~ n ¼ ð9Þ n n εr t1 t2 εnr t n2 εg εnr t n1 εg n 1 λn n1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi here n ¼ 2; 3; ⋯N; t i1;2 ¼ 1 7 1 þ8εiθ =εir =2; εe1 ¼ ε~ ; λn ¼ r n =r n 1 : The expansion coefficient of electric potential in the core zone can be expressed as follow: ( ) N 3εnr εh ðt n2 t n1 Þ n n n E0 ð10Þ Ec ¼ ∏ n n n n 1 a1 εr t 1 εg n 1 a2 n ¼ 2 εr t 2 εg n Here an1;2 ¼ εnr t n1;2 þ 2εh λn t 1;2 1 .
167
The physical parameters in here are same as described in the previous context. 4) Core–shell particle with a continuous variation anisotropic shell If the particle's core is an isotropic medium and its permittivity and radius are ε0 and r 0 . The particle's shell is a kind of functional graded material with anisotropy, and its outer radius is R, its permittivity can be expressed as: 2 3 εr ðrÞ 0 0 60 7 εθ ðrÞ 0 εs ðrÞ ¼ 4 5 0 0 εφ ðrÞ For this condition, it is hard to obtain an analytical expressions of the electric potential for the particle's inner and outer zone. Dong et al. [30] used an anisotropic differential effective dipole approximation to investigate the electric properties of graded spherical particles with anisotropic materials, but it is too complex. However, we can choose proper step length and make the functional-graded anisotropic material manually stratification. Assuming the shell is divided into N layers [7], each of which has an outer radius r i ¼ R Nr0 i þ r 0 ; i ¼ 1; 2; 3⋯N, the corresponding permittivity is, 2 3 εir ðr i Þ 0 0 6 7 7 0 εiθ ðr i Þ 0 εis ðr i Þ ¼ 6 4 5 0 0 εiφ ðr i Þ
3) N-layered core–shell particle, and the properties of shells periodic changed For the particle, supposed its core is an isotropous medium, its permittivity and radius are ε0 and r 0 , and the odd layers are anisotropic media, their corresponding permittivity and radius are εir ; εiθ ; r i ; i ¼ 1; 3; 5⋯, while the even layers are isotropous media, of which the permittivity and radius is εir ¼ εiθ ; r i ; i ¼ 2; 4; 6⋯ respectively. Then its equivalent permittivity is, β2n 1 þ 2g 2n þ 2δ2n ðβ2n 1 g 2n Þ signed as ε2n - β2n ε2n εf 2n ¼ β2n 1 þ 2g 2n δ2n ðβ2n 1 g 2n Þ
Through this operation, the permittivity and radius of each shell are constant. Therefore, we can directly use above results to obtain the particle's equivalent permittivity, and then investigate its electrical or optical properties through the Rayleigh approximation, the Mie theory, or the electrostatic. However, the calculation precision based on this method depend upon the thickness of each shell. Accordingly, it needs to control the relative error in an actual calculation.
ð11Þ
Most particles under natural conditions are charged [31,32]. For example, the sand or dust particles, which all carry much electrostatic charge because of its mutual contacts and collides during moving. Zheng et al. carried out a detailed and systematic study on the electrification phenomenon of wind-blown sand, and discussed its effect on the movement of wind-blown sand and the evolution of land form of desert, and on the microwave propagation [33], whose works renew the extant theories of the physics
ε~ 2n þ 1 ¼
n t2n 2n t 2n 2n 2n 1 2 ε2n ε2n 2n t 1 λ2n 2n t 2 2n þ 1 r t 2 εf r t 1 εf 2n 2n εr t1 t2 2n 2n ε2n ε2n 2n λ2n 2n r t 2 εf r t 1 εf ð12Þ
The expansion coefficient of the inner potential in core area is
Ec ¼
8 > > > > > > > > > > <
2.2. Permittivity of partially charged particle with anisotropic coatings
9 > > > > > > > > > > =
N 3εnr εh ðt n2 t n1 Þ 9εh εi n n n E ∏ n n t εg an ðε þ 2ε Þðεg þ2ε Þ þ2δ ðε ε Þðεg ε Þ> 0 > g ε t ε ε a i i 1 i i i i1 i > h h n 1 n 1 > r 2 r 1 1 2 > > > > n ¼ 2k þ 1 > > > > > > > > i ¼ 2k > > > > ; : k ¼ 1; 2; 3⋯
ð13Þ
168
X. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 179 (2016) 165–169
of wind-blown sand and make it to be more reasonable and sophisticated. Bohren and Huffman are the first one who made theoretical analysis on the influence of electric charge on the particle's optical properties [34], but they did not make any numerical analysis. In order to explain the phenomenon that the actual measurements of microwave attenuation in sand storms are much bigger than the theoretically estimated values, and considering the charge on particle surface, Zhou et al. used Rayleigh approximation to discuss the influences of electric charges on particles' optical properties [35], and their studies firstly revealed the important impact of electric charges on the particle's optical properties. After that, many scholars carried out some numerical studies on the electromagnetic scatterings of the isotropic charged sphere with different material components [5,36–43]. However, there are still few of researches on the electromagnetic scattering of charged anisotropic particle [42]. In view of the diversity of particle properties in the nature, and the concision of using equivalent medium theory to compute particle's optical properties, in this section, we devoted to study the equivalent permittivity of partially charged sphere particles with anisotropic shells. Regarding to a small-sized coated sphere with local electrification, its structure and the physical parameters are the same as them described in Section 2.1, but its surface is partially charged, and the charge uniformly distributes on a spherical crown with an angle 2θ0 . According to existed reference [35], the electric potential both inside of the charged particle and outside of it should be expressed as follows, ϕc ¼ Ec r cos θ þ A
r o r0
ð14aÞ
C ϕs ¼ B þ þ A1 r t1 þA2 r t2 cos θ r ϕh ¼
D þ r þE=r 2 E0 cos θ r
r 0 o r or 1
ð14bÞ
r 4 r1
Compared with the results showed in Ref. [35], we can find that the parameter ε~ q in the Eq. (16) can be regard as the equivalent permittivity of the partially charged core– shell particle with a anisotropic shell. In addition, the parameter ε~ q is exactly the same as Eq. (5) in this paper. Therefore, we can believe that the surface charge on particle does not change the expression of equivalent permittivity for the inhomogeneous particle or the anisotropic particle. So we can speculate that, for a spherical particle with multilayered structure, its equivalent permittivity can be calculated via Eqs. (5), (6), (9), (11), (12), no matter its surface is charged or not, even if its shells are anisotropic medium.
3. Conclusion This paper, based on the method proposed in the reference [27], proposed the equivalent permittivity of layered spherical particle with different forms of coatings under the Rayleigh hypothesis, and deduced its iteration relations of the corresponding equivalent permittivity while those particles have some isotropic coatings, or some anisotropic coatings, and even them have certain intervals. In addition, we give the expansion coefficients of the electric potentials in the core area of particle. Meanwhile, the paper also discussed whether the charges distributed on the particle surface change the expression of equivalent permittivity of layered particle or not. However, those theoretical expressions are just suitable to the condition that the particle size is much smaller than the wavelength of the incident wave. The study, nevertheless, can be used to simulate the electrical, optical properties of layered spheres and those for the functional graded material in different subjects.
ð14cÞ Acknowledgment
The corresponding boundary conditions are [42]:
8 < r ¼ r0
ϕc ¼ ϕs
: r ¼ r1
ϕs ¼ ϕh
∂ϕs c εc ∂ϕ ∂r ¼ εr ∂r
ð15Þ
∂ϕs h εh ∂ϕ ∂r εr ∂r ¼ σH ðθ θ0 Þ
Here A; B; C; D; E are undetermined coefficients, σ is the charge density on particle surface, H ðθ θ0 Þ is Heaviside function, and 2θ0 is the charge distribution angle on particle surface. The undetermined coefficients can be determined through the above equations and the boundary conditions: 3εr εh ðt 2 t 1 ÞE0 ðεr t 2 εc Þa1 ðεr t 1 εc Þa2 ðεr t 2 εc ÞEc ðεr t 1 εc ÞEc A1 ¼ A2 ¼ εr ðt 2 t 1 Þr 0t1 1 εr ðt 2 t 1 Þr 0t 2 1 Ec ¼
E¼
εeq εh 3 r εeq þ 2εh s
σ σ H ðθ θ0 Þ r 1 D ¼ H ðθ θ0 Þ r 21 εh εh Here a1;2 ¼ εr t 1;2 þ2εh λt1;2 1 ; λ ¼ r 1 =r 0 , and
A¼B¼
εeq ¼
ðεr t 2 εc Þt 1 λt 1 t2 ðεr t 1 εc Þt 2 εr ðεr t 2 εc Þλt 1 t2 ðεr t 1 εc Þ
ð16Þ
We acknowledge support from the National Natural Science Foundation of China (11302111, 11562017, 41305035, 41305027), the Program for International S&T Cooperation Projects of China (2011DFA11780), the opening project of Key Laboratory of Mechanics on Disaster and Environment in Western (Lanzhou University), Ministry of Education, the opening project of State Key Laboratory of Advanced Optical Communication Systems and Networks (Shanghai Jiao Tong University, 2013GZKF031306).
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