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Modelling the, 0.6 – 0.7, power law of permittivity and
admittance frequency responses in random R-C networks.
Baruch Vainasa
a
Soreq Nuclear Research Center, Yavne 81800. Israel
(arXiv, dated Sept 20, 2005)
Abstract
The dielectric response of complex materials is characterized, in many cases, by a
similar power law frequency dependence of both the real and the imaginary parts of
their complex dielectric constant. In the admittance representation, this power law is
often shown as the constant phase angle (CPA) response. Apparently, the power that
characterizes many different systems, when expressed as the frequency dispersion of
conductivity (the real part of admittance) is often found to be in the range of, 0.6-0.7,
or having frequency independent, constant phase angles (CPA) of about 54 - 63 deg.
The model suggested here is based on series-parallel mixing of resistors’ and
capacitors’ responses in a random R-C network. A geometric mean evaluation of the
effective resistivity of conductors having a uniform distribution of resistivity is used.
In contrast to models based on percolation arguments, the model suggested here can
be applied to both 2D and 3D systems.
Introduction
It has been shown that the frequency response of random R-C networks having a
binary composition of elements can model the response of systems composed of
conductive and capacitive spatial domains. For example, 1 Kohm resistors and 1 nF
capacitors positioned at random locations on square R-C networks (Fig. 1) have a
well defined frequency region (Fig. 2) where the phase angle between the applied AC
current and the corresponding voltage drop has a constant value which is independent
of frequency. This phenomenon is known as the constant phase angle (CPA) response
in electrochemistry [1]. It can also be expressed in terms of the power law exponent
appearing in the phenomenological Cole-Davidson expressions for the complex
dielectric response, HZ. Apparently, many systems [2, 3] at ambient temperatures
can be characterized by the exponent, 1>E>0, in the Cole-Davidson expression, eq. (1)
below, or the exponent, D E, in the admittance representation.
This power law characteristic is often referred to as the “universal power law”
frequency response [4, 5].
HZviZWEeq. (1)
where ZWand, i, are the angular frequency, the time constant, and 1 ,
respectively.
Pure Debye response is characterized by, E = 1. It corresponds to a frequency
response of a single, series R-C element [5]. At high frequencies, Z!!W , the
capacitance part of this circuit is, effectively, shortened. The resistive component is
then responding to the AC signal with a 00 phase difference between the applied AC
current and the resulting voltage drop. Using the complex permittivity representation,
the high frequency response is expressed by separating the real part, proportional to
1/(1+(ZW)2), and imaginary part, proportional to ZW/(1+(ZW)2), of eq. (1), and noting
that the real part of,H(pure capacitance), becomes negligible, while the imaginary
(loss) part becomes inversely proportional to frequency, which corresponds to a
constant conductivity in the admittance representation.
At very low frequencies, ZW , the phase angle is 900 as the capacitor has the
dominant impedance to the flow of the applied AC current. This capacitive response is
expressed through the finite and constant real part of, H,while the imaginary, loss
part, in eq. (1) becomes insignificant. It should be noted that, for the simple R-C
Debye element, there is a sharp transition between the phase angle of 900 at low
frequencies to 00 phase angle at high frequencies, at the 1/W characteristic frequency. In
contrast to this characteristics of the phase angle change with frequency for the ColeDavidson response with, 1>E>0, shows the characteristic CPA plateau, as shown in
fig. (2).
Many heterogeneous systems are characterized by the universal power law response
with, 1-E | 0.6 - 0.7 [2, 3]. Such systems can not be represented by simple
series/parallel R-C equivalent circuits. We have suggested [6, 7] the use of a mixture
of series and parallel R-C connections, which are inherent to arrays of resistors and
capacitors placed at random. The power law characteristics, and the CPA, were shown
to be the result of the logarithmic, series-parallel mixing of resistors’ and capacitors’
responses. The reasoning behind the series-parallel mixing was based on the “inbetween” the arithmetic and the hyperbolic averaging expression, to model the mixed
connectivity characteristics in random RC networks.
In a network having equal numbers of resistors and capacitors, the frequency was
shown [7] to follow eq. (1) with, E = 1/2, corresponding to a 450 CPA and power
laws of 1/Z for both the real (H’) and the imaginary (H’’) parts of the complex
dielectric constant (H) at high frequencies.
At intermediate values of the ratio between the number of capacitors and resistors in
the network, 1-E was proportional to this ratio. For example, a simulation of a
network containing 60% capacitors and 40% resistors shows [7, Fig. 3] a 540 phase
plateau corresponding to E = 0.4 power in the Cole-Davidson expression .
It should be noted that the phase angle referred to in Fig. 2 below is the currentvoltage phase angle, corresponding to the admittance representation. It is related to the
conductivity power law exponent, or the logarithmic mixing power Dapplied to
spatially disordered, series-parallel R-C netsby D E>@. The admittance phase
angle is then proportional to D by, DS, and D itself is given by the fraction of the
dielectric phase in the binary mixture.
As noted above, the power law admittance response, in many experimental systems,
stretches over a wide range of frequencies with, D = 1-E | 0.6 - 0.7. While it can be
argued that such values of, E orD, can characterize 3D systems at the percolation
threshold for conductivity (Pc | 1/3), it is not clear why so many experimental
systems should be “pinned” to the Pc composition in 3D. In particular, this
percolation argument can not explain the ac response of 2D ionic conductors that,
nevertheless, exhibit a universal, D | 0.6 - 0.7, power law rather than the expected,D
= 1/2, exponent.
In the present work we are suggesting an explanation to the apparent universal value
of D E, the validity of which will not be limited to 3D systems at the percolation
threshold composition.
The model
Let us first derive an expression for the equivalent resistance of a series-parallel
resistive network, in which the individual resistors obey a uniform distribution of
values. It had been argued before [9] that the equivalent value of a series-parallel
network of elements should correspond to the geometric mean of the values of all its
elements. The argument is based on the fact that the two limiting cases, the purely
series and purely parallel connections, are related to arithmetic and hyperbolic means,
which are characterized by the exponents, 1, and, -1, respectively. The series-parallel
mixture corresponds to the “intermediate”, geometric mean.
Given the argument above, the equivalent resistance of series-parallel random resistor
network the individual resistances of which are uniformly distributed between a
minimal resistivity, R0, and a maximal resistivity, R1, is the geometric mean, Rg,
given by,
ln R g 1
1
ln( x)dx
R1 R0 R³0
R
R1
R0
·
§1
ln¨ R1 R1 R0 R0 R1 R0
e
¹̧
©
1
R1 lnR1 R0 lnR0 1
R1 R0
eq. (2)
where, e, is the base of natural logarithms.
It should be noted that, following Hashin-Shtrikman model, the geometric mean
derivation in the form of eq. (2) is usually applied to the conductivities of the
elements, rather than to their resistivity values [9]. However, since the addition of
resistors and conductors, in the two extreme cases of parallel and series connections,
apply (in an opposite way) to the series-parallel mixing discussed above, this mixing
can be used in the case of calculating the equivalent resistance of a collection of
separate resistors, connected in a network.
It has also been shown that the series-parallel argument leading to the derivation of
the geometric mean applies both to 2D and 3D systems [9], which is a significant
result in the context of any model trying to explain why the 0.6-0.7 power law appears
to apply in so many different systems. As been mentioned in the introduction, in
percolation-based models the 3D case suggest the “pinning” to the Pc composition,
while percolation in 2D can not explain the 0.6-0.7 power law, even in the case of
“pinning”.
Eq. (2) is an extension of the well known result that the geometric mean of a
continuous variable distributed uniformly within an interval {0, 1} is, 1/e | 0.37.
Suppose now that a set of resistors spans over a wide range of values. Let the low
resistivity limit be 100 ohm and the high resistivity limit be 100 Kohm.
Given that R1 >> R0, then, according to eq. (2), R1, will dominate the expression,
reducing the expression for the geometric mean to, (1/e)R1. Note that an arithmetic
mean would be, (1/2)R1, given that the low (relative to R1) value of R0, can be
ignored in the expression, (R1 – R0)/2. This is the result of the geometric mean “bias”
towards small values.
The lower resistivity limit in this case, R0 = 100 Ohm, is three orders of magnitude
lower than, R1. We therefore approach a limiting case of a uniform distribution of
resistances between ~0 and 100 Kohm. The equivalent resistance of a random resistor
network constructed from resistors of this distribution is then, Rg = 37 Kohm, as
argued above.
Using a statistical reasoning, lower values of resistances (higher conductivities)
signify a higher probability of finding a unit conductor. By this reasoning, an
equivalent resistance of, say, 50 Kohm in the uniform range of components’ values
between ~ 0 and 100 Kohm (extreme values representing a binary-like split between
unit value conductors vs. no conductors) would correspond to a probability of 0.5 for
the bonds in the network to be occupied by unit conductors.
It is therefore suggested that it can be possible to map the position of the effective
resistance value, Rg, in the continuous range of all resistivities between, R0, and, R1,
into the probability of finding/not finding a unit conductor at a random location in the
network. It characterizes an intermediate situation between the two limiting extremes,
an entirely “nonconductive” network made of 100 Kohm resistors and, entirely
conductive, all 100 Ohm elements network.
Suppose that Rg turns out to be 100 Kohm. It is a reference state for 100% “holes” and
0% conductors. Let Ph be the probability of finding a “hole” at a random position on
the net. Then, Pcd = 1 - Ph, is the probability of finding a conductor at a random
position on the net. For the 100 Kohm network, Ph = 1, and, Pcd = 0.
For a uniform distribution of resistances in the range between, R1 and R0 (where R1 >>
R0) it is argued that the RC network considered here, can be represented by a partially
populated network for which, Pcd = 1 - Ph = 1 - 37/100 = 0.63, fraction of bonds are
conductive. The rest of bonds are not conductive.
In other words the geometric mean averaging has “biased” for low resistivity values
(hence, for high conductivity values) relative to a possible arithmetic averaging that
would have resulted in a 0.5 fraction of bonds being conductive.
To test the probabilistic model suggested above, R-C networks containing 50%
resistors and 50% capacitors had been constructed. All capacitors are 1 nF. For
resistors, a uniform distribution of resistors from 100 ohm to 100 Kohm has been used
(see Fig. 1). This choice reflects the natural expectation for having a similar number
of grains and gaps in granular or heterogeneous material, while the uniform
distribution of resistivities could be rationalized as an expression of a random grain
size distribution.
According to the argument above, a fraction, 0.5*Pcd = 0.315, of all of the links in the
network are conductive, as the “conductor – no conductor” transformation of the
resistors’ uniform distribution has reduced by the factor Pcd = 0.63 the maximal
number of conductors (½ the total number of links in the network). On the other
hand, according to the present model, capacitors always occupy 0.5 of network’s
links. Therefore, the fraction of capacitive bonds, relative to all “used” bonds (by
capacitors and conductors) is, 0.5 / (0.5 + 0.315) = 0.61. This fraction should then be
expressed as a 550 (0.61*S/2) phase in the CPA region.
The results below (Fig. 2) clearly confirm the model suggested here.
1n
C94
1n
C92
1n
C93
1n
R99 C20
20K
1n
C95C89
1n
68K
100
R92
R90
C98
1n
1n
100
R93
33K
R95
1n
C66
1n
C100
R100C87
10K
1n
R98 C88
50K
68K
R91
R53
22K
1n
C96C90
1n
100K
33K
R94
R89
C99
1n
1n
50K
R96
1n
15K
R28
R97
33K
C91
R74 C68
10K
100
33K
20K
R76
R70
68K
R68
C69
1n
1n
R75
R78
100K
R41 C26
100
100K
33K
R65
C73
1n
1n
R20
C47
1n
R72 C61
33K
R35
C32
1n
15K
1n
C70C64
1n
20K
15K
R51
R33
68K
R30
C65
1n
1n
100
100K
R2
C72
1n
C31C21
1n
15K
47K
R4
R21
C30
1n
R73
22K
47K
R3
C71
1n
1n
C78C63
1n
1n
47K
R77
R26
33K
10K
R37
R36
68K
C34C40
1n
R27 C75
10K
1n
1n
R39 C25
15K
R40
C53
1n
1n
R50 C22
100K
R59 C57
15K
R8
C14
1n
50K
20K
R38
C41
1n
100K
10K
R64 C36
22K
R88
R24
50K
R43
C35
1n
1n
R13 C16
22K
R25 C55
15K
100K
C52 C2
1n
1n
C51C23
1n
1n
R46
C42
1n
1n
R69 C12
100
R48 C29
68K
33K
R11
R9
68K
1n
68K
R58
C18
1n
1n
1n
R44
C11
1n
1n
R45
C17
1n
R18 C37
100K
C49 C1
1n
R57
R12
33K
C54 C8
1n
C33C44
1n
R17
R67
100K
10K
R52
R32
100
1n
R83 C13
33K
15K
1n
68K
R80
C81
R85
1n
100
R54
C84
1n
1n
1n
R82 C60
47K
C27
100K
R79
R31
33K
R6
15K
10K
R16
R34
100
R55 C97
50K
R86 C48
10K
C79
22K
R101
C3
1n
1n
1n
22K
R19
C85
1n
1n
R62
C5
1n
1n
C28 C7
1n
R60
R23
47K
1n
C50C45
1n
100
R5
C39 C6
1n
100
100
47K
C9
1n
R49
R56
50K
1n
R42
R15
15K
100K
68K
R22
R47
100K
1n
1n
10K
C43C38
1n
1n
C19C46
1n
100
R63 C58
20K
1n
100
10K
1n
R61 C76
100K
R29
C83
1n
1n
R84 C24
10
1n
C59C67
1n
1n
10K
1n
R71 C62
100
1n
R10 C15
10K
1n
C74C77
1n
1n
100
R14
C86
1n
VOUT
100G
R1
V1
AC 50
1n
C4 C56
1n
68K
R7
R66
100K
1n
C80C10
1n
1n
10K
100
C82
R81
R87
Fig. 1: Constant amplitude ac simulation of a random RC network having a uniform
distribution of resistor values between 100 and 100000 ohms.
1: ph(:vout)
-10
-20
-30
-40
-50
-60
-70
-80
100m 1
10
100
1K
10K 100K 1M 10M 100M 1G
Frequency/Hertz
Fig. 2: The phase angle measured at the terminal “Vout” in Fig. 1 relative to the ac
input to the random RC network.
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[2] Sidebottom D L , Green P F and Brow R K 1995 J. Non-Cryst. Solids 183 151
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[4] Jonscher A K 1996 Universal Relaxation Law (London: Chelsea Dielectric)
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[6] Vainas B, Almond D P, Lou J and Stevens R 1999 Solid State Ionics 126 65
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[9] Piggott A R and Elsworth D 1992 J. Geophys. Res. 97 2085