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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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