Fricke’s law

Electrode behavior in the nonlinear region may be studied by electrode polarization impedance Z = R + jX measured as a function of sinusoidal amplitude. The limit current of linearity iL may, for instance, be defined as the amplitude when the values of R or X deviate more than 10% from low current density values. Often 1l is increasing with frequency proportional to f (Schwan s law of nonlinearity) (Onaral and Schwan, 1982 McAdams and Jossinet, 1991a, 1994). m is the constant phase factor (as defined in this book) under the assumption that it is obeying Fricke s law and is frequency independent (Section 9.2.5). When the measuring current is kept limit current of linearity will usually be lower for X than for R. 

According to Fricke s law, there is a correlation between the frequency exponent m and the phase angle (p in many electrolytic systems when experiments show that C = CiP" , then (p = m7r/2. As pointed out by Fricke, m is often also found to be frequency dependent. However, it can be shown that Fricke s law is not in agreement with the Kramers-Kronig transforms ifm is frequency dependent (Daniel, 1967). If we therefore also presuppose m and

complex notation (remembering that j = cos aizH -H jsin aTc/2, cf. Eq. 12.6) ... 

The factor a intervenes both in the constant phase expression (cos aTc/2 - - j sin a7x/2) and the frequency exponent, in accordance with Fricke s law. The dimension of Yq,eF and Gi is siemens [S] and Ri ohm [O]. The values of YcpeP at extreme frequencies are like the general CPE zero DC conductance and no admittance limit at very high frequencies. 

Hugo Fricke showed that, for example, the electrode polarization capacitance often varies as (in this book written as f ), and that there is a basic empirical relationship between the exponent m and the phase angle of the electrode polarization impedance (Fricke s law)

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A simplified form of Fricke s law states that electrode polarization may be characterized by an electrical phase angle 9 which is defined according to the relation... 

In the experimental situation, the polarization capacitance usually does not vary directly as a power of frequency and Fricke s law holds only approximately, as 9 varies from 30° to 45° in the normal experimental situation. Variations in m range from 0.06 to 0.87 (Fricke, 1932) depending upon electrode material and the electrolyte used. For platinized platinum electrodes with physiological saline as the electrolyte. 

As an example of how Fricke s law may be applied, let us suppose that we have made a parallel measurement of electrolyte capacitance over the frequency range from 1 Hz to 100 kHz. We have found a measured value of capacitance for a parallel configuration as depicted in Figure 2.3c (shown simply as C rather than in Figure 2.3c). The experimental plot of versus frequency is shown in Figure 2.8b. Let us assume that the electrodes used have an effective surface area of 100 cm, and that = 160 Q approximately. The variation in R is too small to observe Rp. We would like to estimate Rp by using Fricke s law. 

If we extrapolate the data to infinite frequency, where Cp = 0, then we see that the true sample capacitance Q = 0.001 fiF. If we now plot Cp versus frequency on a double logarithmic basis, we obtain the result shown in Figure 2.8c. The slope of this line is 0.85 and is the coefficient m which appears in the expression of Fricke s law. Normally m varies from 0.3 to 0.5 for platinum electrodes with a saline electrolyte. For the hypothetical sample selected, m = 0.85. 

Figure 2.8. (a) Slope relation for Fricke s law as applied to Cp. (b) Experimental curve for electrolyte capacitance determination, (c) Plot of Cp and Rp after Fricke s law has been applied. 

To find the polarization resistance as a function of frequency, we use Fricke s law as follows ... 

Laws, J.S., Frick, V., and McConnell, J., Hydrogen Gas Pressure Vessel Problems in the M-1 Facilities, NASA CR-1305, NASA, Washington, DC, 1969. 

Fang J, Pilon L (2011) Scaling laws for thermal conductivity of crystalline nanoporous silicon based on molecular dynamics simulations. J Appl Phys 110 064305 Gesele G, Linsmeier J, Drach V, Fricke J, Arens-Fischer R (1997) Temperature-dependent thermal conductivity of porous silicon. J Phys D Appl Phys 30(21) 2911 Gomes S, David L, Lysenko V, Descamps A, Nychyporuk T, Raynaud M (2007) Application of scanning thermal microscopy for thermal conductivity measurements on meso-porous sihcon thin films. J Phys D Appl Phys 40 6677... 

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