Conductivity and permittivity of dispersed systems with penetrable particle-host interphase
A model for the study of the effective quasistatic conductivity and permittivity of dispersed systems with particle-host interphase, within which many-particle polarization and correlation contributions are effectively incorporated, is presented. The structure of the system’s components, including t...
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irk-123456789-1210652017-06-14T03:05:17Z Conductivity and permittivity of dispersed systems with penetrable particle-host interphase Sushko, M.Ya. Semenov, A.K. A model for the study of the effective quasistatic conductivity and permittivity of dispersed systems with particle-host interphase, within which many-particle polarization and correlation contributions are effectively incorporated, is presented. The structure of the system’s components, including the interphase, is taken into account through modelling their low-frequency complex permittivity profiles. The model describes, among other things, a percolation-type behavior of the effective conductivity, accompanied by a considerable increase in the real part of the effective complex permittivity. The percolation threshold location is determined mainly by the thickness of the interphase. The “double” percolation effect is predicted. The results are contrasted with experiment. Запропоновано модель для вивчення ефективних квазiстатичних провiдностi та дiелектричної проникностi дисперсних систем iз вiльно-проникним мiжфазним шаром мiж частинками та середовищем, яка ефективно враховує багаточастинковi поляризацiйнi та кореляцiйнi ефекти. Структура компонентiв системи, включаючи мiжфазний шар, враховується шляхом моделювання профiлiв їх низькочастотної комплексної дiелектричної проникностi. Модель, зокрема, описує перколяцiйно-подiбну поведiнку ефективної провiдностi, що супроводжується суттєвим зростанням дiйсної частини ефективної комплексної проникностi системи. Положення порогу перколяцiї визначається головним чином товщиною мiжфазного шару. Передбачено ефект “подвiйної” перколяцiї. Результати порiвняно з експериментом. 2013 Article Conductivity and permittivity of dispersed systems with penetrable particle-host interphase / M.Ya. Sushko, A.K. Semenov // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13401:1–10. — Бібліогр.: 31 назв. — англ. 1607-324X PACS: 42.25.Dd, 64.60.Ak, 77.22.Ch, 82.70.-y, 83.80.Hj DOI:10.5488/CMP.16.13401 arXiv:1303.5534 http://dspace.nbuv.gov.ua/handle/123456789/121065 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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A model for the study of the effective quasistatic conductivity and permittivity of dispersed systems with particle-host interphase, within which many-particle polarization and correlation contributions are effectively incorporated, is presented. The structure of the system’s components, including the interphase, is taken into account through modelling their low-frequency complex permittivity profiles. The model describes, among other things, a percolation-type behavior of the effective conductivity, accompanied by a considerable increase in the real part of the effective complex permittivity. The percolation threshold location is determined mainly by the thickness of the interphase. The “double” percolation effect is predicted. The results are contrasted with experiment. |
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Sushko, M.Ya. Semenov, A.K. |
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Sushko, M.Ya. Semenov, A.K. Conductivity and permittivity of dispersed systems with penetrable particle-host interphase Condensed Matter Physics |
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Sushko, M.Ya. Semenov, A.K. |
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Sushko, M.Ya. |
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Conductivity and permittivity of dispersed systems with penetrable particle-host interphase |
| title_short |
Conductivity and permittivity of dispersed systems with penetrable particle-host interphase |
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Conductivity and permittivity of dispersed systems with penetrable particle-host interphase |
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Conductivity and permittivity of dispersed systems with penetrable particle-host interphase |
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Conductivity and permittivity of dispersed systems with penetrable particle-host interphase |
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conductivity and permittivity of dispersed systems with penetrable particle-host interphase |
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Інститут фізики конденсованих систем НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/121065 |
| citation_txt |
Conductivity and permittivity of dispersed systems with penetrable particle-host interphase / M.Ya. Sushko, A.K. Semenov // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13401:1–10. — Бібліогр.: 31 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT sushkomya conductivityandpermittivityofdispersedsystemswithpenetrableparticlehostinterphase AT semenovak conductivityandpermittivityofdispersedsystemswithpenetrableparticlehostinterphase |
| first_indexed |
2025-07-08T19:08:02Z |
| last_indexed |
2025-07-08T19:08:02Z |
| _version_ |
1837106917362630656 |
| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 1, 13401: 1–10
DOI: 10.5488/CMP.16.13401
http://www.icmp.lviv.ua/journal
Conductivity and permittivity of dispersed systems
with penetrable particle-host interphase
M.Ya. Sushko, A.K. Semenov
Mechnikov National University, Department of Theoretical Physics, 2 Dvoryanska St., 65026 Odesa, Ukraine
Received July 3, 2012, in final form November 28, 2012
A model for the study of the effective quasistatic conductivity and permittivity of dispersed systems with particle-
host interphase, within which many-particle polarization and correlation contributions are effectively incorpo-
rated, is presented. The structure of the system’s components, including the interphase, is taken into account
through modelling their low-frequency complex permittivity profiles. The model describes, among other things,
a percolation-type behavior of the effective conductivity, accompanied by a considerable increase in the real
part of the effective complex permittivity. The percolation threshold location is determined mainly by the thick-
ness of the interphase. The “double” percolation effect is predicted. The results are contrasted with experiment.
Key words: core-shell particle, dispersion, permittivity, conductivity, percolation
PACS: 42.25.Dd, 64.60.Ak, 77.22.Ch, 82.70.-y, 83.80.Hj
1. Introduction
The studies of disperse systems, nanofluids, and systems of nanoparticles are nontrivial, but impor-
tant for many fields in science and industry. Much attention has been recently focused on the question
how the properties of the particle-host interphase affect the dielectric characteristics of the entire sys-
tem [1–4]. This is an essentially many-particle problem, playing a crucial role, say, in percolation stud-
ies [5–7]. The latter involve (a) various numerical (mainly Monte-Carlo) methods [8, 9], (b) graph [10] and
renormalization group [7] theories, (c) different improvements of the Maxwell-Garnett [11] and Brugge-
man [12] approaches.
It should be noted that the practical realization of methods (a) is labor-consuming and imposes high
computer specifications. Methods (b) provide certain scaling relations, but ignore the physical nature
of dielectric response of an inhomogeneous system. Methods (c) seem more adequate physically, but
are actually one-particle approaches. They can demonstrate a percolation-type behavior of conductiv-
ity [13], but ignore the fractal nature of percolation cluster [14]. Numerous attempts at improving these
approaches are widespread in modern technical literature (see [15]), but are, for most part, poorly justi-
fied physically.
In the present work, the role of the interphase in the formation of a percolation-type behavior of
conductivity is analyzed within an electrodynamic model based upon the method of compact groups of
inhomogeneities [16–18]. The “compact group” means any macroscopic region, in which all distances
between the inhomogeneities are much shorter than the wavelength of probing radiation in the medium.
The method allows one to effectively sum up and average the electric field and displacement vectors
in finely-dispersed systems of particles with complex internal structure and arbitrary permittivities. In
doing so, unnecessary detailed elaboration of the processes of interparticle polarization and correlations
in the system is avoided.
The main idea of the method is that in the long-wave limit, a macroscopically homogeneous and
isotropic particulate system is electrodynamically equivalent to an aggregate of compact groups of par-
ticles. The compact groups include macroscopic numbers of particles and can be viewed as one-point
© M.Ya. Sushko, A.K. Semenov, 2013 13401-1
http://dx.doi.org/10.5488/CMP.16.13401
http://www.icmp.lviv.ua/journal
M.Ya. Sushko, A.K. Semenov
inhomogeneities, with negligibly small particle number fluctuations inside. They are in fact self-similar.
Note that this property is characteristic of a percolation cluster.
Figure 1. The model under consideration.
The model of the disperse system under consid-
eration is depicted in figure 1. Each particle con-
sists of a spherical hard core (black), with diame-
ter d and permittivity ε1. The core is surrounded
by a penetrable concentric shell (gray), with outer
diameter d(1+δ) (δ being the relative thickness of
the shell) and permittivity ε2. The particles are em-
bedded into a host medium (white), with permittiv-
ity ε0. All the permittivities are in general complex.
The volume concentration of the hard-cores is c. The
shells can freely overlap; together, they form the in-
terphase between the hard cores and the host. The
volume concentration φint of the interphase for par-
ticular values c and δ can be calculated by statisti-
cal methods. We take it from Monte Carlo simula-
tions [19, 20].
We expect that, in the main features, our results
should remain valid for macroscopically homoge-
neous and isotropic systems with non-spherical
core-shell particles: the sphericity comes into play
at the points where the explicit form of the func-
tion φint =φint(c,δ) is needed. By contrast, spherical
symmetry of the particles is crucial for most approaches to the problem which often represent different
versions of one- and two-particle approximations and use it either implicitly or through the require-
ment that the shells should be hard. On the other hand, the systems of anisotropic particles are usually
treated as if the latter were solitary. Some examples and further literature on these topics can be found
in [4, 18, 21, 22].
2. General equation for the low-frequency complex permittivity
The structure of the system’s components, including the interphase, is taken into account through
modelling their complex permittivity profiles and the choice of the way of homogenization.
We are interested in the low-frequency value ε = ε′ − i4πσ/ω of the effective complex permittivity
and determine it as the proportionality coefficient in the relation
〈D(r)〉 = 〈ǫ(r)E(r)〉= ε〈E(r)〉, (2.1)
where D(r), E(r), and ǫ(r) are the local values of, respectively, the electric displacement, electric field, and
permittivity in the system, and the angular brackets stand for the statistical averaging. In the present
work, we suggest that this averaging can be reduced to the following two-step procedure.
1. Finding the equilibrium distribution of the particles with penetrable outer shells. This distribu-
tion is characterized by the hard-cores’ volume concentration c and the particles’ effective vol-
ume concentration φ = φ(c,δ) (the latter is the sum of c and the volume concentration of the re-
gions occupied by the shells). Electrodynamically, such a system can be viewed as an aggregate of
non-overlapping (white, black, and gray in figure 1) regions with permittivities εi = ε′
i
− i4πσi /ω,
i = 0,1,2, and volume concentrations 1−φ, c, and φ−c, respectively.
2. Electrodynamic homogenization, by direct integration with respect to its volume [23], of this ag-
gregate within the compact group approach. This suggests that: (a) with respect to a probing field
of long wavelength λ, the aggregate itself is viewed as a set of regions (termed as compact groups)
13401-2
Conductivity and permittivity of dispersed systems
with typical linear sizes much smaller than λ, but yet remaining macroscopic; (b) the macroscopic
dielectric properties of the aggregate are equivalent to those of a macroscopically homogeneous
and isotropic system prepared by embedding the aggregate’s non-overlapping regions into some
fictitious medium, of permittivity ǫ. The permittivity distribution in this fictitious system can be
written as
ǫ(r) = ǫ+δǫ(r), (2.2)
where, in our case,
δǫ(r) = (ε0 −ǫ)Π0(r,Ω0)+
N
∑
a=1
(ε1 −ǫ)Π1(r,Ωa)+
N ′
∑
b=1
(ε2 −ǫ)Π2(r,Ω′
b) (2.3)
is the local permittivity deviation from ǫ due to the presence of a compact group of the aggregate’s
non-overlapping regions at point r. Here, Π0 (r,Ω0), Π1 (r,Ωa ), and Π2
(
r,Ω′
b
)
are the characteristic
functions of the regions Ω0, Ωa , and Ω
′
b
occupied by, respectively, the real host, the ath particle’s
hard core, and the bth connected cluster of the overlapping outer shells:
Πi (r,Ω) =
{
1, r ∈Ω,
0, r ∉Ω.
(2.4)
With themodel permittivity distribution (2.2) and (2.3) proposed, the averaged displacement and field
in formula (2.1) are calculated by the general formulae
〈E〉 =
{
1+
∞
∑
s=1
(
−
1
3ǫ
)s
〈[δǫ(r)]s〉
}
E0 , (2.5)
〈D〉 =
{
ǫ+ǫ
∞
∑
s=1
(
−
1
3ǫ
)s
〈[δǫ(r)]s〉+
∞
∑
s=0
(
−
1
3ǫ
)s
〈[δǫ(r)]s+1〉
}
E0 . (2.6)
In view of the property (2.4), the averages
〈[δε(r)]n〉 =
1
V
∫
V
[δε(r)]n dr
are easy to find:
〈[δǫ(r)]n〉 = (1−φ)(ε0 −ǫ)n +c (ε1 −ǫ)n +
(
φ−c
)
(ε2 −ǫ)n . (2.7)
Substitution of (2.7) into series (2.5), (2.6) and summation of those give
(1−φ)
ε0 −ǫ
2ǫ+ε0
+c
ε1 −ǫ
2ǫ+ε1
+ (φ−c)
ε2 −ǫ
2ǫ+ε2
=
ε−ǫ
2ǫ+ε
. (2.8)
The permittivity ǫ is the only unknown parameter left. In what follows, we suggest that the dielec-
tric properties of the fictitious and effective systems are identical, that is, ǫ = ε. This approximation is
equivalent to the Bruggeman-type homogenization, when the host and dispersed particles are treated
symmetrically. This seems reasonable for the system under study because the shape of the white and
gray regions is extremely complicated. The degree of this complexity increases with c even more.
The other extreme case, ǫ= ε0, corresponds to the Maxwell-Garnett type of homogenization.
3. Main features of the model
In the quasistatic limit and at ǫ= ε, the general equation (2.8) reduces to two real equations, a cubic
one for σ and, once σ is found, a linear for ε′:
(1−φ)
σ0 −σ
2σ+σ0
+c
σ1 −σ
2σ+σ1
+ (φ−c)
σ2 −σ
2σ+σ2
= 0. (3.1)
13401-3
M.Ya. Sushko, A.K. Semenov
(1−φ)
ε′0σ−ε′σ0
(2σ+σ0)2
+c
ε′1σ−ε′σ1
(2σ+σ1)2
+ (φ−c)
ε′2σ−ε′σ2
(2σ+σ2)2
= 0. (3.2)
A general analytical analysis of equation (3.1) can be carried out with the use of Cardano’s formu-
las and is cumbersome [24]. Nonetheless, the major features of the model can be grasped with simple
physical reasoning. For the sake of convenience, we change in equations (3.1) and (3.2) to dimensionless
variables x ≡σ/σ1, y = ε′/ε′0 and xi ≡σi /σ1, yi = ε′
i
/ε′0 (i = 0,1,2). Typically, x0 ≪ 1 and y > 1.
3.1. Percolation-type behavior
In the limit of a non-conducting matrix, x0 → 0, equation (3.1) has three solutions, x = 0 and
x =
3
4
(
c −
1
3
)
+
(
φ−
1
3
−c
)
x2 ±
√
4
3
(
φ−
1
3
)
x2 +
[(
c −
1
3
)
+
(
φ−
1
3
−c
)
x2
]2
. (3.3)
If x2 > 0, a physically meaningful nontrivial solution (that with the positive sign in front of the square
root, see figure 2) appears only under the condition
φ(cc,δ) =
1
3
(3.4)
and is independent of x2. In the case x2 = 0, the well-known value of cc = 1/3 for the two-component
Bruggeman model occurs [13].
Figure 2. Percolation (dashed line, δ = 0) and “double” percolation (solid line, δ = 0.05); x0 = 1 ·10−10 ,
x2 = 5 ·10−5.
The relation (3.4) determines the threshold concentration cc for the effective conductivity. This value
is affected only by the fact of the existence of the interphase (δ, 0), rather than its dielectric and conduc-
tive properties.
Our estimate of cc as a function of the relative thickness δ of the interphase layer is shown in figure 3.
The calculations were performed using the Monte Carlo results [20] (k = (1+δ)−3, ϕ= c/k):
φ = 1− (1−kϕ)exp
[
−(1−k)ϕ
]
exp
{
−
kϕ2
2(1−kϕ)3
[
(8−9k1/3 +k)
− (4+9k1/3 −18k2/3 +5k)kϕ+2(1−k)k2 ϕ2
]
}
. (3.5)
13401-4
Conductivity and permittivity of dispersed systems
Figure 3. Threshold concentration as a function of the relative thickness of the interphase (solid line).
Dashed line: cc ≃ 1
3 (1+δ)−3.
The analysis revealed that for realistic c . 0.5, the relation cc ≃ 1
3
(1+δ)−3 can be used.
In the immediate vicinity of cc (c → cc +0) and for nonzero δ, formula (3.3) takes the form
x ≃
3
4
x2
[
1+
1
3 +c(1− x2)
1
3
−c(1− x2)
]
(
φ−
1
3
)
. (3.6)
Correspondingly, the effective conductivity σ∝ (c−cc)t where the critical exponent t ≃ 1.0. The effective
permittivity ε′, as follows from equation (3.2), increases anomalously as x0 → 0. The latter fact is in accord
with predictions [25].
3.2. Effective critical exponent of conductivity
In practice, both the threshold concentration cc and the critical exponent t are determined by inter-
polating the conductivity experimental data σ=σ(c), obtained for some finite interval c ∈ [c1,c2] near cc
(c1 > cc), with the scaling law σ= A(c −cc)t , A and t being independent of c. Then,
t = log
σ(c2)
σ(c1)
/
log
c2 −cc
c1 −cc
, (3.7)
and this value is expected to be a c-independent constant.
By contrast, the asymptotic formula (3.6) reveals that even small variations of c near cc cause the
expression in the brackets, which is proportional to A, to change considerably. This means that a formal
application of the above procedure and formula (3.7) to a system with conductivity (3.3) will result in an
effective exponent teff sensitive to the parameters c1 and c2 (figure 4). In particular, for a given δ, 0, teff
Figure 4. Effective critical exponent of conductivity as a function of c2 at fixed c1 , δ= 0.1 (cc ≃ 0.251), and
x2 = 5 ·10−5 , calculated with formulas (3.3) and (3.7). From bottom to top, c1 = 0.26, 0.27, and 0.28.
13401-5
M.Ya. Sushko, A.K. Semenov
Figure 5. Effect of the matrix’s conductivity on the
effective permittivity. From top to bottom, x0 = 1 ·
10−6 , 1 · 10−5 , and 1 · 10−4. The other parameters:
y1 = 1.5, y2 = 1, x2 = 0.05, δ= 0.05.
Figure 6. Effect of the interphase thickness on the
effective permittivity. From right to left, δ= 0, 0.05,
and 0.10. The other parameters: y1 = 1.5, y2 = 1,
x0 = 1 ·10−5 , x2 = 0.05.
increases as the interval [c1,c2] (where c2 < 1/3) is: (a) shifted to higher values of c (while its width is
fixed); (b) widened at fixed c1. Also, the threshold concentration found within this procedure is expected
to exceed cc.
Different 3D percolation models [5] and renormalization group calculations [26, 27] give for t esti-
mates of ≈ 1.3÷1.7 and ≈ 1.9, 2.14, respectively. Experimental values of t are usually 1.5÷2.0 and some-
times even twice as much [22]. As evident from figure 4, our theory is capable of reproducing a variety of
these values.
3.3. Effect of the matrix’s conductivity
For real substances, x0 , 0, though x0 can be extremely small. As a result, the percolation-type depen-
dence of x with c changes to a smooth one, with its slope considerably increasing near cc. Simultaneously,
the maximum value of y becomes bounded above and decreases as x0 increases (figure 5). The location
of the maximum shifts to lower concentrations as δ increases (figure 6). Calculations show that it remains
practically independent of x2 and actually equal to cc.
Below the percolation threshold, the effective conductivity is usually approximated by the scaling law
σ= B(cc − c)−s . Given experimental data for some interval c ∈ [c1,c2], c2 < cc, the effective values seff of
the critical exponent s are found as
s =−log
σ(c2)
σ(c1)
/
log
cc −c2
cc −c1
. (3.8)
Figure 7. Effective critical exponent of conductivity below cc as a function of x0 for δ = 0.1 (cc ≃ 0.251)
and x2 = 5 ·10−5 , calculated with formulas and (3.1) and (3.8) at c1 = 0.24 and c2 = 0.25.
13401-6
Conductivity and permittivity of dispersed systems
Our estimates of seff with formulas (3.1) and (3.8) are shown in figure 7. They correlate well with
typical theoretical [26, 27] and experimental [22] values of 0.75 and 0.7÷1.0, respectively.
3.4. Behavior of the permittivity
According to equation (3.2), the effective permittivity is given by
ε′ = x
(1−φ)ε′0 +c
(2x + x0)2
(2x +1)2
ε′1 + (φ−c)
(2x + x0)2
(2x + x2)2
ε′2
(1−φ) x0 +c
(2x + x0)2
(2x +1)2
+ (φ−c)
(2x + x0)2
(2x + x2)2
x2
. (3.9)
For a badly-conducting matrix (x0 → 0) and under the condition x ≪ 1, three particular situations are
of interest to point out to.
1. The system is below the percolation threshold and the conditions x ≪ p
x0, x ≪ p
x0x2, x ≪ x2
(that is, σ≪ p
σ0σ1, σ≪ p
σ0σ2, and σ≪ σ2) are met. Then, the dominant contributions to both
the numerator and the denominator are made by their first terms, and we expect that ε′ ∝ x ∝
(cc −c)−seff .
2. The system is above the threshold and x ≫ p
x0, x ≫ p
x2, x ≫ x2 (σ≫ p
σ0σ1, σ≫ p
σ1σ2, and
σ≫σ2). Now, of significance become the first and the third terms in the numerator, the latter being
almost independent of x, and the second term in the denominator. Correspondingly, the ε′ versus
c dependence is expected to be close to ε′ ∝ x−1 ∝ (c − cc)−teff , with the proportionality constant
slightly dependent on c.
The exponents seff and teff in the two preceding scaling-like laws are independent of the compo-
nents’ permittivities ε′
i
.
3. The system is close to the percolation threshold, x ≫ p
x0, and x ≫ x2 (σ≫ p
σ0σ1, σ≫σ2). Then,
the numerator is almost x-independent, whereas the denominator is mainly contributed to by the
second and the third terms. The dependence of ε′ on x takes the form ε′ ∝ ax/
(
1+bx2
)
, where the
coefficients a and b are easy to recover.
3.5. Applicability to real systems
Figure 8 shows the results of processing with formula (3.9) the experimental data [28] for the effective
permittivity of the composites prepared by embedding spherical Ag particles (amean radius≈ 100 Å ) into
a KCl matrix. The particles were made by evaporating Ag in the presence of argon and oxygen gases so
as to form a thin (according to the authors, of approximately 10 Å , δ≃ 0.10) oxide coating on them. This
coating prevented the particles from cold-welding together, but was thin enough to allow metal-to-metal
contact under high pressure. The composites were prepared by mixing Ag particles and KCl powder and
then compressing the mixture into a solid pellet under high pressure.
It is seen from figure 8 that formula (3.9) not only reproduces data [28] over the entire range of Ag
concentrations investigated, but also gives an estimate of δ≃ 0.14÷0.19, sufficiently close to the expected
one.
The conductivity (resistivity ρ) data for a few Ag–KCl composite samples, prepared in the above way,
are given in [29]. They pertain only to a very narrow vicinity of the percolation threshold, where ρ drops
7 orders of magnitude with a 1% Ag volume concentration increment; the parameters of the KCl matrix
are not specified. As figure 9 reveals, formula (3.1) can reproduce data [29] sufficiently well. Better fits
can be produced by introducing c-dependences for some of the parameters of the model. These facts may
indicate that, in addition to experimental errors, various other factors and phenomena (inaccuracy of the
function φ = φ(c,δ), particles’ size distribution, silver dissolution and local dielectric breakdown in the
KCl matrix, polarization effects, etc.) come into play as cc is approached. The analysis of them goes far
beyond the scope of this paper.
13401-7
M.Ya. Sushko, A.K. Semenov
Figure 8. Effective permittivity data [28] for two series of Ag–KCl composite samples (circles and triangles)
below the percolation threshold and their fits with formula (3.9) at ε′0 = 5.0, δ = 0.186 (solid line) and
ε′0 = 7.0, δ= 0.145 (dashed line). The dotted lines are the scaling-type fits (with cc = 0.20, seff = 0.72 and
cc = 0.22, seff = 0.74, respectively) proposed in [28] to the data for c > 0.11.
3.6. “Double” percolation
For intermediate values of x2 (x0 ≪ x2 ≪ x1), a “double” percolation can be noticeable, that is, a
new increase in x after some levelling off (figure 2); it is accompanied by the appearance of a new peak
in the concentration dependence of the permittivity (figure 10). The physical cause of this phenomenon
is clear — in a concentrated system, the hard cores of particles with penetrable shells begin to contact
intensively to form percolation cluster and add to the effective conductivity.
Evidently, the threshold concentration c ′c for “double” percolation is close to a value of 1/3. In the
region |c − 1
3 |≪ x2 ≪ 1, the c-dependence of the effective conductivity (3.3) is represented by the square-
root law
x =
1
2
(3x2)1/2
[
φ(c,δ)−
1
3
]1/2
+O(x2)=
1
2
[
3x2φ
′(cc,δ)
]1/2
(c −cc)1/2 +O(x2), (3.10)
φ′ being the derivative of φ with respect to c. For concentrations satisfying the condition c − 1
3
≫ x2, it
Figure 9. Effective resistivity data [29] for Ag–KCl composite samples (squares) and the fit to them with
formula (3.1) at σ1 = 6.3 ·107 S/m, x0 = 5 ·10−16 , δ= 0.162 (cc = 0.214), and x2 = 4 ·10−6 .
13401-8
Conductivity and permittivity of dispersed systems
Figure 10. Effective permittivity as “double” percolation occurs; x0 = 1·10−8 , x2 = 5·10−4 , y1 = 1.5, y2 = 1,
δ= 0.05.
becomes linear,
x =
3
2
(
c −
1
3
)
+O(x2), (3.11)
with a considerably greater amplitude as compared to those in formulas (3.6) and (3.10).
As for now, we are unaware of experimental observations of the effect described. Usually, “double”
percolation is associated with a non-monotonous behavior of the conductivity in composites made by
embedding the conducting particles into a two-component matrix (see, for example, [30, 31]).
It can also be shown that a decrease in x can occur at c ∼ c ′c in systems with x2 ≫ 1. Similar effects
were observed in two-phase composite solid electrolytes with a highly conducting interphase layer [22].
4. Concluding remark
The model proposed is interesting in the sense that it is based on rather clear assumptions, incor-
porates many-particle effects in a consistent way, and can be further refined so as to apply to a more
complicated systems, including fluctuation phenomena, etc. For practice, it can serve as a rather flexible
theoretical basis for analysis of dielectric and conductive properties of such complex systems as various
dispersions, colloids, and so-called nanofluids, or for the development of new composite materials.
Acknowledgement
We are grateful to an anonymous Referee for stimulating remarks.
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Провiднiсть та дiелектрична проникнiсть дисперсних
систем iз вiльно-проникним мiжфазним шаром мiж
частинками й середовищем
М.Я. Сушко, А.К. Семенов
Одеський нацiональний унiверситет iменi I. I. Мечникова, вул. Дворянська, 2, 65026 Одеса, Україна
Запропоновано модель для вивчення ефективних квазiстатичних провiдностi та дiелектричної проник-
ностi дисперсних систем iз вiльно-проникним мiжфазним шаром мiж частинками та середовищем, яка
ефективно враховує багаточастинковi поляризацiйнi та кореляцiйнi ефекти. Структура компонентiв сис-
теми, включаючи мiжфазний шар, враховується шляхом моделювання профiлiв їх низькочастотної комп-
лексної дiелектричної проникностi. Модель, зокрема, описує перколяцiйно-подiбну поведiнку ефективної
провiдностi, що супроводжується суттєвим зростанням дiйсної частини ефективної комплексної проник-
ностi системи. Положення порогу перколяцiї визначається головним чином товщиною мiжфазного шару.
Передбачено ефект “подвiйної” перколяцiї. Результати порiвняно з експериментом.
Ключовi слова: структурована частинка, дисперсна система, дiелектрична проникнiсть, провiднiсть,
перколяцiя
13401-10
http://dx.doi.org/10.1002/andp.19354160705
http://dx.doi.org/10.3367/UFNr.0177.200712g.1341
http://dx.doi.org/10.1070/PU2007v050n12ABEH006348
http://dx.doi.org/10.1088/0022-3719/19/9/007
http://dx.doi.org/10.1134/S1063776107080146
http://dx.doi.org/10.1134/S1063784209030165
http://dx.doi.org/10.1088/0022-3727/42/15/155410
http://dx.doi.org/10.1016/0021-9797(85)90246-2
http://dx.doi.org/10.1140/epje/i2003-10006-x
http://dx.doi.org/10.1063/1.447497
http://dx.doi.org/10.1016/0079-6425(93)90004-5
http://dx.doi.org/10.1002/pssb.2220760205
http://dx.doi.org/10.1103/PhysRevB.18.2185
http://dx.doi.org/10.1088/0305-4470/18/11/027
http://dx.doi.org/10.1103/PhysRevLett.46.375
http://dx.doi.org/10.1007/BF00553352
http://dx.doi.org/10.1016/j.eurpolymj.2008.04.013
http://dx.doi.org/10.1016/j.polymer.2006.05.015
Introduction
General equation for the low-frequency complex permittivity
Main features of the model
Percolation-type behavior
Effective critical exponent of conductivity
Effect of the matrix's conductivity
Behavior of the permittivity
Applicability to real systems
``Double'' percolation
Concluding remark
|