Conductivity and permittivity of nickel-nanoparticle-containing ceramic materials in the vicinity of percolation threshold

Article abstract of DOI:10.1111/j.1551-2916.2006.01126.x

Conductivity and static permittivity of ceramic materials containing nanoparticles of Ni were measured in the vicinity of percolation threshold. It is found that, below this threshold, the experimentally obtained dependences of conductivity and static per

Full text of DOI:10.1111/j.1551-2916.2006.01126.x

J. Am. Ceram. Soc., 89 [9] 2946–2948 (2006)
DOI: 10.1111/j.1551-2916.2006.01126.x
r 2006 The American Ceramic Society
ournal
J
Conductivity and Permittivity of Nickel-Nanoparticle-Containing
Ceramic Materials in the Vicinity of Percolation Threshold
w
Umar Abdurakhmanov, Shamil Sharipov, Yayra Rakhimova, Munira Karabaeva, and Maksudbek
Baydjanov
Physical Faculty, National University of Uzbekistan, Tashkent 100174, Uzbekistan
Conductivity and static permittivity of ceramic materials con-
taining nanoparticles of Ni were measured in the vicinity of per-
colation threshold. It is found that, below this threshold, the
experimentally obtained dependences of conductivity and static
permittivity on the fractional Ni content in these materials are
different from those calculated in the frame of the percolation
theory. The origin of this discrepancy is discussed in terms of the
network hierarchy model proposed recently by Balberg et al. for
composite materials.
Previously, we used the same method to create nickel particles of
2
sizes r30 nm in a polymer matrix. The ceramic material with
high-dispersed nickel particles was prepared by mixing the nick-
el powder with a ceramic in an agate ball mill for 7 h. The nickel
powder used was prepared by thermal decomposition of the
nickel formiate under vacuum at a temperature of 4001C for 5 h.
In both cases, the ceramic had the following identified compo-
2 2 3 2 3
nents: SiO 61.72 wt%, Fe O 3.87 wt%, Al O 12.52 wt%,
CaO 13.48 wt%, MgO 0.94 wt%, Na O 1.78 wt%, K O 0.72
2
2
wt%, and MnO 0.07 wt%. The remaining 4.9% of composition
of the ceramic had various inclusions that are unidentified be-
cause of their small quantity. The desirable values of the frac-
tional Ni content V in the ceramic materials were obtained by
the calculation of the source materials used.
I. Introduction
N recent years, interest in nanoparticle-containing materials
has grown explosively owing to their unique physical prop-
I
To determine the sizes of the high-dispersed particles of the
nickel powder, an electron microscope (BS242E, Tesla, Prague,
Czech Republic) was used. It was found that the sizes of these
particles fell in the range from 1 to 3 mm. The sizes of the nickel
nanoparticles were not determined as the method (X-ray diffrac-
tion at a glancing angle of incidence), which was used by us for
other composite materials (e.g., for composites with nanoparti-
erties, which are substantially different from the properties of
corresponding compact materials. Among these materials, there
are various composites containing metal nanoparticles random-
ly distributed in a dielectric matrix. The application of the clas-
sical percolation theory for description of the electrical
properties of these composites has problems. For example, be-
low the percolation threshold, the behavior of the conductivity
of composites containing the nickel grains embedded in the ma-
2
cles in a polymer matrix ), was not applicable because of the fact
that the densities of Si, Fe, Al, and other components of the ce-
ramic and the density of nickel nanoparticles are closely spaced.
Electric measurements were performed on samples in the
form of tablets 15 mm in diameter and 2 mm in thickness.
The tablets were prepared by pressing the ceramic material
powder under pressure of 200 MPa and by sintering under
vacuum at a temperature of 10001C.
trix of SiO is inexplicable in the frame of this theory. This
2
1
problem has been discussed by Balberg et al. in detail. There, it
has been shown that the conductivity behavior of the Ni–SiO
2
composites can be explained assuming that it is dictated not only
by the nearest-neighbor tunneling as it is suggested by the clas-
sical percolation theory, but by the non-nearest-neighbor tun-
2
neling as well. In our previous paper, we obtained that, for
The conductivity s was determined by measuring the resist-
ance of the samples. To do this, metal electrodes with a diameter
of 13 mm were prepared by vacuum evaporation of aluminum
on the flat surfaces of the samples under study. Two meters were
used: an ohmmeter for the resistance measurements in the range
composites containing the nickel particles of sizes r30 nm em-
bedded in a polymer matrix, below the percolation threshold,
the measured conductivity differs from that calculated in the
frame of the percolation theory. In this paper, we present results
of the study of the behavior the conductivity and static permit-
tivity of composites containing nickel nanoparticles embedded
in a ceramic matrix.
1
13
of 10 –10 O (E6-13A-POB, P8603, St. Petersburg, Russia), and
an ohmmeter for the resistance measurements in the range of
9
17
1
0 –10 O (B7-30-POB, B752, St. Petersburg, Russia). The
values of conductivity obtained exhibited an error of 1%–7%.
The static permittivity e was obtained from data on the fre-
quency dependence of permittivity in the range from 20 to 200
Hz by means of extrapolation of this dependence to zero fre-
quency. To determine the frequency dependence of permittivity,
the measurements of the capacitance were performed on the
samples under study in the specified frequency range. Values of
permittivity were calculated using the formula
II. Experimental Procedure
Two types of composites were used for the investigation. One
was a ceramic material containing nanoscale nickel particles.
The other was a ceramic material containing the high-dispersed
nickel particles. The ceramic material with nickel nanoparticles
was prepared by thermal decomposition of the nickel formiate
mixed preliminary with a ceramic. The mixing was performed
in an agate ball mill for 7 h. Thermal decomposition was
performed under vacuum at a temperature of 4001C for 5 h.
cðoÞh
eðoÞ ¼
(1)
e0S
N. Dudney—contributing editor
where e(o) is the permittivity at a given frequency o, c is the ca-
pacitance of sample, h is the thickness of sample, S is the area of
the above-mentioned electrodes placed on the flat surfaces of the
samples, and e is the permittivity constant. The capacitance of
the samples was measured by means of a bridge of capacitors
(E8-2, Moscow, Russia) using a sine-wave generator (G3-33-POB
0
Manuscript No. 20665. Received June 11, 2005; approved February 5, 2006.
Author to whom correspondence should be addressed. e-mail: abdurakhmanovu@
yahoo.com
w
2
946

September 2006
Communications of the American Ceramic Society
2947
A1333, Moscow, Russia) and a balance indicator (F510, Mash-
pribor, Moscow, Russia). The values of the static permittivity
obtained exhibited an error of 2%.
case of VoV , the agreement between the theoretical and ex-
c
perimental dependences was observed for the ceramic materials
with the high-dispersed nickel particles only. For the ceramic
materials with the nanoscale nickel particles, there was an ad-
ditional contribution to s in the range below V . These results
III. Results and Discussion
c
can be understood in terms of the recently proposed model
(
According to this model, all metal particles in composites, in
1
Balberg et al. ) of electrical conductivity in composites.
Figure 1 shows the experimental dependences of conductivity s
on the fractional Ni content V for both ceramic materials under
study. In this figure, the dependences s on V calculated in the
frame of the percolation theory using the formulas cited below
are also shown.
which the metal particles randomly distributed in a dielectric
matrix, are electrically connected, and the conductivity of these
composites is dictated both by the nearest-neighbor tunneling
and by the non-nearest-neighbor tunneling. The percolation-like
behavior is observed when the contribution of the tunneling be-
tween non-nearest neighbors to the macroscopic conductivity is
negligible. It takes place when the radius b of the particles is
greatly superior to the tunneling range (or tunneling decay) pa-
rameter d. In the case where bꢁd, the tunneling between non-
nearest neighbors makes a contribution to the macroscopic con-
ductivity along with the nearest-neighbor tunneling, and the de-
pendence of the macroscopic conductivity on the fractional
content of metal particles is different from that dictated by the
classical percolation theory. One can see that the above-de-
scribed behavior of the conductivity of the ceramic materials
under study is consistent with these model predictions. In Ba-
3
According to the percolation theory, the conductivity s of
systems containing metal particles randomly distributed in a
dielectric matrix is described by the following formulas:
t
sðVÞ ¼ smðV ꢀ VcÞ for V > Vc
(2)
and
sðVÞ ¼ sdðVc ꢀ VÞꢀ for V < Vc
q
(3)
where s
m
is the conductivity of the metal particles, s
d
is the
is the critical fractional
conductivity of the dielectric matrix, V
c
volume of the metal particles that initiated a first infinite metallic
cluster; the parameters t and q are named as critical indexes.
Using boundary conditions (V 5 0 and V 5 1), formulas 2 and 3
can be written in the following form:
1
lberg et al., the manifestation of these two types of behavior of
the conductivity in composite materials was demonstrated by
studying the Carbon Black–Polymer composites and Ni–SiO2
cermets. A peculiarity of our results is that these two types of
ꢀ
ꢁ
t
conductivity behavior were observed in composites of the same
composition, demonstrating the dependence of the manifesta-
tion of these conductivity behavior types on the size of the metal
particles in these composites.
V ꢀ Vc
sðVÞ ¼ sm
for V > Vc
for V < Vc
(4)
(5)
1
ꢀ Vc
and
1
ꢀ
ꢁꢀ
As is shown in Balberg et al., in composites for which a con-
q
Vc ꢀ V
Vc
tribution to s from the non-nearest neighbor tunneling takes
place, there are two percolation thresholds. One of them is ob-
served at high values of V, and it is the above-defined percolation
sðVÞ ¼ sd
For the ceramic materials under study, the critical fractional
volume V of the Ni particles was derived by differentiation of lg
c
threshold V . Another (the additional percolation threshold Vcd)
was observed at low values of V, and it is a critical fractional
volume of metal particles, which initiates a first infinite cluster of
c
s with respect to V (see the inset in Fig. 1). To determine the
critical index t, the experimental data were presented as an lg
s–lg[(VꢀV )/(1ꢀV )] plot. The value of t is the degree of incli-
tunneling-connected conductors. Fitting the part (for VoV
c
) of
the experimental curve 1 (Fig. 1) for the ceramic material with
the nickel nanoparticles to the functional dependence of Eq. (4)
(denoting, in this equation, the percolation threshold as Vcd and
c
c
nation of this plot. The values of s
m
d
and s were obtained by
extrapolation of this plot to V 5 1 and V 5 0, respectively. It was
found that V 5 0.355 and t 5 2.21 for the ceramic material with
0
0
the critical index as t ), we found that Vcd 5 0.145 and t 5 3.2.
In Fig. 2, the experimental and theoretical dependences of the
static permittivity e on V for the ceramic materials under study
are shown. The experimental dependences are obtained by ex-
trapolation of low-frequency data (for the range between 20 and
200 Hz) to the zero frequency. The theoretical dependences are
calculated with the following formula:
c
the nanoscale nickel particles, and V 5 0.443 and t 5 1.81 for
c
the ceramic material with the high-dispersed nickel particles.
The critical index q was taken to be equal to 1.0, which is true
3
for three-dimensional systems.
As can be seen from Fig. 1, for both types of ceramic mate-
rials under study, the agreement between the theoretical and
c
experimental dependences was observed when V>V . In the
ꢀ
ꢁꢀ
q
Vc ꢀ V
Vc
eðVÞ ¼ ed
;
V < Vc;
(6)
Fig. 1. Comparison of the measured (points) and calculated (solid
curves) dependences of the macroscopic conductivity (s) on the frac-
tional Ni content (V) for the ceramic material containing nanoscale Ni
particles (full points, curve 1) and high-dispersed Ni particles (empty
points, curve 2). In the inset, plots of the dependence of dlgs/dV on V
are shown (the solid line is for the material containing nanoparticles, the
dashed line is for the material containing high-dispersed particles).
Fig. 2. Comparison of the measured (points) and calculated (solid
curves) dependences of the static permittivity (e) on the fractional Ni
content (V) for the ceramic material containing nanoscale Ni particles (full
points, curve 1) and high-dispersed Ni particles (empty points, curve 2).

2
948
Communications of the American Ceramic Society
Vol. 89, No. 9
where e is the permittivity of the ceramic; the index q is just the
the characteristic feature of the V-dependence of e at the addi-
tional percolation threshold Vcd considered above allows one to
conclude that the non-nearest-cluster-included capacitors con-
tribute to the macroscopic capacitance along with the nearest-
cluster-included capacitors. The change of run of curve of de-
pendence of e on V for this composite in the range of Vcd may
be attributed to the process of formation of the continuous
network of tunneling-connected conductors.
d
same as in Eq. (5). This formula, using the boundary condition
V 5 0, was derived from the expression for the static dielectric
permittivity near the threshold, which has the form
3
ed
eðVÞ ¼
;
o ¼ 0
(7)
q
jV ꢀ Vcj
The calculations were performed with the same values of V
c
,
which were obtained from the experimental dependences s on
V, as described above.
IV. Conclusion
As can be seen from Fig. 2, for the ceramic material with the
high-dispersed particles, the experimental dependence of e on V is
well described by the formula Eq. (6). For the ceramic material
with nanoparticles, the experimental dependence of e on V does
not agree with that calculated from this formula and shows an
additional contribution to e. It is remarkable that, for this ceramic
material, the curve of V-dependence for e is similar to that for s.
Based on the qualitative interpretation of the sharp increase
By measuring the dependences of the conductivity s and the
static permittivity e on the fractional Ni content (V), performing
the measurements on two types of Ni-particle-containing ceram-
ic materials with the same compositions, but with the different
sizes of the Ni particles, the influence of the size of these particles
on the form of these V-dependences is demonstrated. It is found
that, for the ceramic material with the Ni nanoparticles, in the
range below the classical percolation threshold, the dependences
of s and e on V are different from that predicted by the classical
percolation theory in the fact that the curve of this dependence
for e is similar to that for s. It is concluded that, in the nickel-
nanoparticle-containing ceramic material under study, the per-
colation-tunneling process, which is considered (Balberg et al.)
as the cause of the low percolation threshold Vcd and determines
the behavior of s in the range below the classical percolation
threshold, also determines the behavior of e in this range.
3
of e near the percolation threshold and the physical view of
1
‘
‘hierarchy’’ of electrically connected networks in composites,
taking into account the fact that, for the corresponding com-
posite materials under study, the curves of V-dependence for e
and s are similar, the following interpretation of the e behavior
can be proposed. In the composite materials under study, as V
increases, the nickel particles form metallic clusters, which are
separated by dielectric material of the matrix. Each pair of the
clusters represents a capacitor. In the case of the composite ma-
terial with relatively large nickel particles (high-dispersed parti-
cles), the contribution of non-nearest-cluster-included capacitors
to the macroscopic capacitance is negligible, and V-dependence
of e is percolation like. This form of dependence resulted from
the fact that the capacitance of the capacitors increases with V
References
1
I. Balberg, D. Azulay, D. Toker, and O. Millo, ‘‘Percolation and Tunneling in
Composite Materials,’’ Int. J. Mod. Phys. B, 18, 2091–121 (2004).
M. K. Abdurakhmanova, A. Kh. Zaynutdinov, A. V. Umarov, U. Ab-
2
(
as a result of the decrease in the clusters’ separation and an
durakhmanov, and M. A. Magrupov, ‘‘Temperature Dependence of Conductiv-
ity in Polymer Composite Materials Based on Polyarilate and High-Dispersed
Nickel,’’ Vysokomol. Soedin. B, 29, 537–9 (1987).
A. L. Efros and B. I. Shklovskii, ‘‘Critical Behaviour of Conductivity and Di-
electric Constant Near the Metal–Non-Metal Transition Threshold,’’ Phys. Stat.
increase of their effective surface) and tends to infinity near the
percolation threshold (as a result of formation in continuous
metallic network). In the case of the composite material with
relatively small nickel particles (nanoparticles), the presence of
3
Sol. B, 76, 475–85 (1976).
&