Electrodynamic properties of matrix dispersed systems with two-layer inclusions
Theoretical approach for calculation of the effective dielectric permittivity (ε) of matrix dispersed systems (MDS) that consist of a dielectric matrix with randomly arranged two-layer spherical inclusions of different radiuses has been proposed and departures from the Maxwell-Garnet fo...
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Chuiko Institute of Surface Chemistry National Academy of Sciences of Ukraine
2002
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| author | Grechko, L. G. Mal'nev, V. N. Shkoda, N. G. Shostak, S. V. |
| author_facet | Grechko, L. G. Mal'nev, V. N. Shkoda, N. G. Shostak, S. V. |
| author_institution_txt_mv | [
{
"author": "L. G. Grechko",
"institution": "Інститут хімії поверхні НАН України"
},
{
"author": "V. N. Mal'nev",
"institution": "Taras Shevchenko Kyiv National University"
},
{
"author": "N. G. Shkoda",
"institution": "Інститут хімії поверхні НАН України"
},
{
"author": "S. V. Shostak",
"institution": "Taras Shevchenko Kyiv National University"
}
] |
| author_sort | Grechko, L. G. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-11-27T09:42:19Z |
| description | Theoretical approach for calculation of the effective dielectric permittivity (ε) of matrix dispersed systems (MDS) that consist of a dielectric matrix with randomly arranged two-layer spherical inclusions of different radiuses has been proposed and departures from the Maxwell-Garnet formula due to increasing of an inclusion volume fraction are studied. It is shown that the effects of direct dipole-dipole interactions between inclusions become important in this case. In the electrostatic approximation we have exactly solved the problem of a response of this N–particle system on the external electric field and obtained corrections to the Maxwell-Garnet formula for ε with account of the pair dipole-dipole interaction between inclusions. |
| first_indexed | 2025-07-22T19:30:09Z |
| format | Article |
| fulltext |
89
ELECTRODYNAMIC PROPERTIES OF MATRIX
DISPERSED SYSTEMS WITH TWO-LAYER INCLUSIONS
L.G. Grechko1, V.N. Mal’nev2, N.G. Shkodа1, and S.V. Shostak2
1Institute of Surface Chemistry, National Academy of Sciences
Gen. Naumov Str. 17, 03680 Kyiv-164, UKRAINE
2Taras Shevchenko Kyiv National University, Glushkov Prospect 6, 03143 Кyiv, UKRAINE
Abstract
Theoretical approach for calculation of the effective dielectric permittivity ( )e~ of
matrix dispersed systems (MDS) that consist of a dielectric matrix with randomly arranged
two-layer spherical inclusions of different radiuses has been proposed and departures from the
Maxwell-Garnet formula due to increasing of an inclusion volume fraction are studied. It is
shown that the effects of direct dipole-dipole interactions between inclusions become
important in this case. In the electrostatic approximation we have exactly solved the problem
of a response of this N–particle system on the external electric field and obtained corrections
to the Maxwell-Garnet formula for e~ with account of the pair dipole-dipole interaction
between inclusions.
Introduction
An increasing interest to study of interaction of an electromagnetic radiation and
matrix dispersed systems (MDS) displayed recently is associated first of all with the fact that
these systems have some features that are absent in corresponding continuous media [1-11].
The simplest and the most efficient method of calculation of the electrodynamics
characteristics of MDS is the method of an effective dielectric permittivity ( )e~ and an
effective conductivity ( )s~ . A detailed review of some methods of calculation of these
quantities for different MDS are given in [1-4]. It is necessary to note that MDS constituted of
a dielectric matrix with randomly embedded inclusions of different physicochemical nature
and shape (spheres, ellipsoids, cylinders, and so on) are ones of the best studied. It is clear that
these MDS simulate good enough real systems including soils, rocks, and even biological live
objects such as suspensions of cells. Moreover, MDS and statistical blends are the basement
for creation of different composite materials with the beforehand given electrodynamics
properties.
In this paper, we performed calculations of e~ for МDS with two-layer spherical
inclusions of different size. With the help of the method of expansion of polarizability of
these systems with respect to group of particles (inclusions) [5-7] and the theorem of spherical
harmonics [8-9], in the first section we solved the problem of obtaining of electric potential at
an arbitrary point of the matrix consisting of N inclusions in the external electric field
ti
oeE w-
r
~ (in the electrostatic approximation when a wave length of the field
w
pl c2~ is much
larger than a typical size of particles and an average distance between them). We obtained the
relations that allow one to perform calculation of the potential of electric field with account of
any order of the multipole interaction between inclusions. In the second section, we obtained
the expression e~ of our МDS with account of only pair interaction between inclusions.
90
1. N–particle problem
We consider a system of N particles in a volume V and in the homogeneous varying
with time electric field ( ) ( ) tieEtrE ww -=
rrrr
00 , . In the electrostatic approach, the field ( )rE rr at
any point of the system may be obtained from the relation ( ) ( )rgradrE j-=
rr , where the
potential ( )rrj satisfies the equation
0=Dj (1)
The inclusions are two-layer spherical particles of an external radius ib , and radius of
the inclusion nucleus is ia ; ( )we )(
1
i , ( )we )(
2
i , 0e are the dielectric permittivities of the
inclusion nucleus, shell and the matrix respectively (Fig.).
X
Y
Z
O
j
q
eee 0
0
0
E
0
P
1 2
( j )
r
rr
r
r
r r
ji
si
si
(i)
(j) (j)
- i
b j
aj
i,j=1,N
ai
bi
Fig. Mutual disposition of balls.
Taking into account of the problem symmetry, may present a solution of (1) with a
center at the i-th particle in the following form [12]:
for the particle nucleus
( )ilm
l
ml
i
i
lm
I
i rrYrrAE €€
,
)(
0
)( rrrr
---= åj , (2)
for the particle shell
91
( )å -
ú
ú
û
ù
ê
ê
ë
é
-
+--= +
ml
ilml
i
i
lm
l
i
i
lm
II
i rrY
rr
DrrCE
,
1
)(
)(
0
)( €€€€ rr
rr
rrj , (3)
and for the matrix
( ) ( ) ( )--
-
----= å å +
lm lm
ilml
i
i
lm
ilm
l
i
i
lm
III rrY
rr
BErrYrrdEr €€€€ 1
)(
0
)(
0
)( rr
rr
rrrrrrj
( )å
¹
+ -
-
-
q ,
q ,1
)(
,
0
€€
p
ij
ipp
i
i
qp rrY
rr
B
E rr
rr , (4)
where ( )iqp rrY €€
,
rr
- are spherical functions [13],
i
i
i rr
rrrr rr
rr
rr
-
-
=- €€ . In equation (4) the first term
is the potential of the external field expanded with respect to sherical harmonics with a
centered ir
r and in the observation point Р; the second term is the potential created by the i-th
particle due to its polarization at the point Р, the third term is sum of potentials that created by
rest (N-1) particles at Р. While writing expressions (2)-(4), we took into account regularity of
the potential )( I
ij at the center of the i-th sphere and at infinity ( )rIII r)(j . Unknown constant
)()()()( ,,, i
lm
i
lm
i
lm
i
lm DCBA , may be obtained requiring solutions (2)-(4) must satisfy the boundary
conditions [13]
( ) ( )
si
sisiis rr
II
i
i
rr
j
ii
i
rr
II
irr
I
i gradngradn
rr
rrrrrr
rr
==== ×=×=
)()(
2
)()()()( ;
jejejj (5)
( ) ( ) ''
'
' ; )(
0
)()(
2
)()(
sisi
si
si rr
III
irr
I
ii
i
rr
II
rr
II
i gradngradn rrrr
rr
rr
rr
====
×=×= jejejj
where vectors ', sisi rr rr are shown at Fig., vectors isi rr rr - and isi rr rr -' are parallel to the
external normal inr ; iisi arr =- rr , iiSi brr =-
rr ' .
Since the spherical harmonics in the third term in (4) are centered at the j-center, we
could not meet conditions (5) directly. It is necessary to reduce everything to the i-th center.
For this we may use the addition theorems of spherical harmonics. One of these theorems at a
domain iji rrrr rrrr
-<- gives [8-9]:
( ) ( ) ( )iml
l
iij
ml
lm
mll
j
jlm rrYrrrra
rr
rrY €€ ''
'
''
''1
rrrrrr
rr
rr
---=
-
-
å+ , (6)
where ( ) ( )( ) ( )
( )( )
lm
ml
ml
ll
ij
ijmmll
ij
lm
ml K
lll
l
rr
rrY
rra '''
''
''
122 12
124
1
€€
''1
,
*
ú
ú
û
ù
ê
ê
ë
é
+++
+
-
-
-
- +
++
-+ p
rr
rr
rr
( ) ( )
( ) ( ) ( ) ( )
2/1
''''
''''
!!!!
!!''
''
ú
ú
ú
û
ù
ê
ê
ê
ë
é
-+-+
-++-++
==
mlmlmlml
mmllmmll
KK ml
lm
lm
ml (7)
Substituting (2)-(4) and (6) in (5), we obtain:
92
( )
( ) ( )
ï
ï
ï
ï
ï
î
ï
ï
ï
ï
ï
í
ì
ú
û
ù
ê
ë
é +
×+=×
+
-
+=+
×
+
×-=-
=-
++
++
+
+
;11
;
;1
;
12
)(
)(
)(
2
0
12
)(
)(
12
)(
)(
12
)(
)(
12
)(
)(
1
)(
2)(
)(
1
)(
2)(
12
)(
)()(
l
l
b
BN
b
D
l
lC
b
B
N
b
D
C
a
D
l
lCA
a
DCA
l
i
i
lmi
lmil
i
i
lmi
lm
l
i
i
lmi
lml
i
i
lmi
lm
l
i
i
lm
i
i
i
lmi
i
i
lm
l
i
i
lmi
lm
i
lm
e
e
e
e
e
e
(8)
where
( )å
¹
-+=
ij
ml
ij
ml
lm
j
ml
i
lm
i
lm rraBdN
''
''
''
)()()( rr
. (9)
From (8) and (9), we may obtain the system of algebraic equations for the coefficients )(i
emB
( )åå å
¹
¥
= -=¢
¢¢¢
¢¢ -=-+
N
ij l
l
lm
i
lmij
ml
lm
j
ml
i
l
i
lm drraB
B
1
)()(
)(
)(
'
rr
a
, (10)
where )(i
lmd are coefficients of expansion of the external electric field with respect spherical
functions that are centered at the i-th inclusion, аnd
( )[ ] ( ) ( ) ( )[ ]{ }
( )[ ] ( )[ ] ( )( ) 12)(
2
)(
2
)(
1
)(
20
)(
1
)(
2
12
0
)(
2
)(
2
)(
10
)(
2
)(
1
)(
2
12
)(
)1(11
112
+
++
--++++×++
++×-+-×++
=
l
i
iiiiii
l
i
iiiiiil
ii
l
qllllll
qlllllb
eeeeeeee
eeeeeeee
a , (11)
i
i
i
b
a
q = , ( )ij
ml
lm rra rr
-
''
are given by (7). After obtaining )(i
lmB , we may easily get coefficients
i
lm
i
lm
i
lm DCA ,, from (8).
Relations (7)–(8) and (10)–(11) completely solve the problem of electric response of
the system of N two-layer spheres on the field 0E
r
accounting of the direct multipole
interaction between particles. Keeping in (10) terms of the order 1' == ll , we take into
account only the dipole-dipole interaction between them. Picking up terms with 2' == ll , we
may take into account the quadrupole interaction, and so on.
It is worth noting that )(i
la is the polarization of the і-th particle of the order l in the
field 0E
r
. At 1=l (dipole polarization) formula (11) transforms into formula (5.36) [2], and аt
,0=iq )(i
la coincides with the l order multipole polarization of the i-th sphere–balls [6].
2. Effective dielectric permittivity of MDS with two-layer inclusions
Comparing relations (10) and (11) with corresponding formulas of papers [7, 10, 11]
(for example, formulas (10) аnd (11) from [10]) we may note that the system of equations for
)(i
lmB in the case of MDS with two-layer inclusions differ from equations of MDS for one-layer
inclusions by the magnitude of the multipole polarization of the l-th order ( )(i
la ) of a
particular inclusion. In other words, to obtain e~ of the MDS systems under consideration one
93
may use directly by formulas of papers [7, 10, 11] changing relation )(i
la with (11). For
example, by using formula (1) of paper [11], we may straight forward obtain e~ for MDS with
two-layer spherical inclusions accounting the pair dipole interaction between them:
( ) ( ) ( )[ ]kpkpkp
II
kpab
pk
kpkppk
k
k
kk
k
k
RRdRRRnn
nn
^
¥
+×
´
÷
ø
ö
ç
è
æ
-=
-
+
å ò
åå
bbf
aapee
ee
2
1
3
4
1
~
2~
, 0
2
2
)(
1
)(
1
0
0
. (12)
Here coefficients )(
1
ia are given by relation (11) at l =1, pkkp rrR
rr
-= ,
V
Nn K
K = is a density
number of the k-th type of particles, ( )kpRf is the two-particle distribution function of
inclusions that was chosen in the form:
( )
ïî
ï
í
ì
+³
+<
=
.,1
,,0
bbRif
bbRif
R
kkp
pkkp
kpf (13)
In the case of pair dipole interaction ( 1' == ll ), the coefficients II
ijb і ^
ijb take the form:
( ) 3
)(
1
)(
1)(
1
)(
10 2
ij
ji
i
ij
iII
ij R
R aa
acb --= , (14)
( ) 3
)()(
1)(
1
)(
11
ij
j
j
i
i
ij
i
ij R
R
aa
acb +-=^ .
Expressions of ( )ij
i R)(
10c and ( )ij
i R)(
11c (at 1' == ll ) may be obtained from (10)
( )
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ë
é
D+÷
÷
ø
ö
ç
ç
è
æ
D-÷
÷
ø
ö
ç
ç
è
æ
+
D-÷
÷
ø
ö
ç
ç
è
æ
D+÷
÷
ø
ö
ç
ç
è
æ
=
32
3
2
1
2
3
2
1
32
3
2
1
2
3
2
1
3)(
10
21212
1
iij
ji
ij
j
i
iij
ji
ij
j
i
iij
i
BB
B
B
BB
B
B
bR
dd
c , (15)
( )
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ë
é
D+÷
÷
ø
ö
ç
ç
è
æ
D-÷
÷
ø
ö
ç
ç
è
æ
+
D-÷
÷
ø
ö
ç
ç
è
æ
D+÷
÷
ø
ö
ç
ç
è
æ
=
32
3
2
1
2
3
2
1
32
3
2
1
2
3
2
1
3)(
10
112
1
iij
ji
ij
j
i
iij
ji
ij
j
i
iij
i
BB
B
B
BB
B
B
bR
dd
c ,
where ,
i
j
ij b
b
=D
ij
i
i R
b
=d and 3
)(
1
i
i
i b
B a
= .
Taking into account the explicit form of ( )ijRf (13) and relations (14)-(15) and using
(12), we obtain the formula of effective dielectric permittivity e~ of МDS with two-layer
spherical inclusions in the approximation of dipole-dipole interaction between them:
94
( ) ( )
( ) ( )
( ) ( )
( ) ( )
.
2
ln1
2
ln
1
3
1
3
4
1
~
2~
2
1)(
1
)(
1
3
2
1)(
1
)(
1
32
1
)(
1
)(
1
3
3
2
1)(
1
)(
1
3
2
1)(
1
)(
1
3
2
1
)(
1
)(
1
3
3
)1()1(
,
2
)(
1
)(
1
0
0
ïþ
ï
ý
ü
ú
ú
û
ù
++
-+
ú
ú
û
ù
ê
ê
ë
é
÷÷
ø
ö
çç
è
æ
×-+
ú
ú
û
ù
-+
++
´
ïî
ï
í
ì
´
ú
ú
û
ù
ê
ê
ë
é
÷÷
ø
ö
çç
è
æ
×+×´×××
÷
ø
ö
ç
è
æ
-=
-
+ å
åå
pk
Pk
pk
Pk
k
p
p
k
pk
Pk
pk
Pk
k
p
p
k
pkp
pk
k
k
k
kk
k
k
bb
bb
b
b
bb
bb
b
bnn
nn
aa
aa
a
a
aa
aa
a
a
aa
aa
pee
ee
(16)
Neglecting the second term in the right hand side of (16), we obtain the
Maxwell-Garnet relation for MDS with two-layer inclusions which represent mixture of
particles with different sizes and different electrodynamics characteristics (different )(
2,1
ie ). In
the case of identical particles, relation (16) gives:
,
28
8ln
3
21
31~
0
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
-
+
--
+=
B
BfBfB
fB
ee (17)
where 2
)(
2
)(
21
)(
1
)(
1 ;;; eeeeee ======== jiji
jiji bbbaaa
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) 3
02212012
3
02210212
3
1
22 2
2 2
q
q
b
B
eeeeeeee
eeeeeeeea
+-+-+
+-+-+
==
b
aq = ; nbf 3
3
4
p= , n is a density number of inclusions. At 1<<fB , (17) gives
( ) ...;3
28
8lnln231~
20 +úû
ù
êë
é -
-
+
--=
B
BfBfB
e
e
(18)
or
...
28
8ln2331
~
22
0
+úû
ù
êë
é
-
+
+++=
B
BBffB
e
e . (19)
Formula (18) coincides with formula (18) of paper [14], аnd formula (19) coincides with
formulas (5) and (7) of paper [6], provided that ( )00 ®= aq .
Conclusion
The results obtained in this paper allow us to make some general conclusions. The
form of system of equations (10)-(11) shows that the problem of obtaining the coefficients
)(i
emB and the effective dielectric permittivity e~ of МDS with complex spherical inclusions
(one-layer of different radius, two-layer, inhomogeneous, and so on) is completely equivalent
to the problem of obtaining these coefficients and e~ of МDS with continuous spherical
inclusions but with a new magnitude of the multipole polarization of an individual inclusion
)(i
la . This is a result of the boundary conditions (8), since corrections due to the multipole
interaction (coefficients )( j
lmB in (18) are added to the interaction coefficients of inclusions
95
with the external field )(i
lmd , formula (9). An analogous picture takes place for MDS with
ellipsoidal inclusions. We would like to stress that the above-developed calculation method of
effective dielectric permittivity e~ for similar МDS enables one to take into account even the
higher term of mulipole interaction between inclusions. However, this requires knowledge of
the many-particle (three, four, and so on) statistical distribution functions of inclusions in a
matrix. Unfortunately, exact expressions of these functions are unknown at present with the
exception of the two-particle function [15].
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L.G. Grechko1, V.N. Mal’nev2, N.G. Shkodа1, and S.V. Shostak2
Abstract
Introduction
Conclusion
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| id | oai:ojs.pkp.sfu.ca:article-86 |
| institution | Surface |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-22T19:30:09Z |
| publishDate | 2002 |
| publisher | Chuiko Institute of Surface Chemistry National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | surfacezbircomua/33/625ca3365d3d09a4890615a4c336a233.pdf |
| spelling | oai:ojs.pkp.sfu.ca:article-862018-11-27T09:42:19Z Electrodynamic properties of matrix dispersed systems with two-layer inclusions Electrodynamic properties of matrix dispersed systems with two-layer inclusions Electrodynamic properties of matrix dispersed systems with two-layer inclusions Grechko, L. G. Mal'nev, V. N. Shkoda, N. G. Shostak, S. V. Theoretical approach for calculation of the effective dielectric permittivity (ε)&nbsp;of matrix dispersed systems (MDS) that consist of a dielectric matrix with randomly arranged two-layer spherical inclusions of different radiuses has been proposed and departures from the Maxwell-Garnet formula due to increasing of an inclusion volume fraction are studied. It is shown that the effects of direct dipole-dipole interactions between inclusions become important in this case. In the electrostatic approximation we have exactly solved the problem of a response of this N–particle system on the external electric field and obtained corrections to the Maxwell-Garnet formula for ε&nbsp;with account of the pair dipole-dipole interaction between inclusions. Theoretical approach for calculation of the effective dielectric permittivity (ε)&nbsp;of matrix dispersed systems (MDS) that consist of a dielectric matrix with randomly arranged two-layer spherical inclusions of different radiuses has been proposed and departures from the Maxwell-Garnet formula due to increasing of an inclusion volume fraction are studied. It is shown that the effects of direct dipole-dipole interactions between inclusions become important in this case. In the electrostatic approximation we have exactly solved the problem of a response of this N–particle system on the external electric field and obtained corrections to the Maxwell-Garnet formula for ε&nbsp;with account of the pair dipole-dipole interaction between inclusions. Theoretical approach for calculation of the effective dielectric permittivity (ε)&nbsp;of matrix dispersed systems (MDS) that consist of a dielectric matrix with randomly arranged two-layer spherical inclusions of different radiuses has been proposed and departures from the Maxwell-Garnet formula due to increasing of an inclusion volume fraction are studied. It is shown that the effects of direct dipole-dipole interactions between inclusions become important in this case. In the electrostatic approximation we have exactly solved the problem of a response of this N–particle system on the external electric field and obtained corrections to the Maxwell-Garnet formula for ε&nbsp;with account of the pair dipole-dipole interaction between inclusions. Chuiko Institute of Surface Chemistry National Academy of Sciences of Ukraine 2002-06-12 Article Article application/pdf https://surfacezbir.com.ua/index.php/surface/article/view/86 Surface; No. 7-8 (2002): Chemistry, Physics and Technology of Surface; 89-95 Поверхность; № 7-8 (2002): Химия, физика и технология поверхности; 89-95 Поверхня; № 7-8 (2002): Хімія, фізика та технологія поверхні; 89-95 3154-8091 3154-8083 en https://surfacezbir.com.ua/index.php/surface/article/view/86/85 Авторське право (c) 2002 L.G. Grechko, V.N. Mal’nev, N.G. Shkodа, S.V. Shostak |
| spellingShingle | Grechko, L. G. Mal'nev, V. N. Shkoda, N. G. Shostak, S. V. Electrodynamic properties of matrix dispersed systems with two-layer inclusions |
| title | Electrodynamic properties of matrix dispersed systems with two-layer inclusions |
| title_alt | Electrodynamic properties of matrix dispersed systems with two-layer inclusions Electrodynamic properties of matrix dispersed systems with two-layer inclusions |
| title_full | Electrodynamic properties of matrix dispersed systems with two-layer inclusions |
| title_fullStr | Electrodynamic properties of matrix dispersed systems with two-layer inclusions |
| title_full_unstemmed | Electrodynamic properties of matrix dispersed systems with two-layer inclusions |
| title_short | Electrodynamic properties of matrix dispersed systems with two-layer inclusions |
| title_sort | electrodynamic properties of matrix dispersed systems with two-layer inclusions |
| url | https://surfacezbir.com.ua/index.php/surface/article/view/86 |
| work_keys_str_mv | AT grechkolg electrodynamicpropertiesofmatrixdispersedsystemswithtwolayerinclusions AT malnevvn electrodynamicpropertiesofmatrixdispersedsystemswithtwolayerinclusions AT shkodang electrodynamicpropertiesofmatrixdispersedsystemswithtwolayerinclusions AT shostaksv electrodynamicpropertiesofmatrixdispersedsystemswithtwolayerinclusions |