Cole equations exponent

Figure 4.4 shows Cole—Cole plots for glycine (amino acid), glycylglycine (peptide), and albumin (protein). As a biomolecule becomes more and more complicated and large (amino acid—peptide—protein), the frequency exponent 1-a of the Cole—Cole equation becomes higher (a lower), indicating a broader distribution of time constants. 

The power law-type spectral dependencies of x and x" are well supported by our experiments. While the unpoled sample exhibits an exponent ft 0.2 (not shown), i. e. large poly-dispersivity, the poled sample yields ft 0.67 for both components of x (Figure 15.14 [58]). Obviously the polydispersivity is largely suppressed at low domain wall densities. This seems to show that polydispersivity is less affected by the nonlinearity in the creep regime, v oc Es, than by the mutual wall interactions in the nanodomain regime [46], Very satisfactorily, also the Cole-Cole plot, Equation (15.21), which is another independent test of the ansatz, Equation (15.18), reveals a very similar exponent, ft 0.69 (Figure 15.15 [58]). 

Note The exponent in the HN equation is determined by acc and aoc, which are the exponents of the Cole-Cole and the Davidson-Cole functions respectively and the constants in equation 9.02 have been redesignated as (1 - a) =acc and P =aoc)- P is temperature dependent and the values of P are identified as >9= 0 at Tg and y9= 1 at T. The validity of this approach has been verified in a number of materials and a typical example given by Rault (2000) based on the measurements done in PIBMA (Poly isobutylene methacrylate) by Dhinojwala et al.(1992) using the data of the decay of chromophore orientation in a poled film is shown in Figure 9.05. In the region between Tg and T, >9 is found to vary... 

As q " is strictly independent of the temperature, equation (2) gives in the fast motion (27cfx 1) as well as in the slow motion case (27ifx 1) the refractive index n(T) at the laser wavelength Xq (c.f (9)). In the acoustic relaxation regime D(q ", T) exeeds n(T). In (35) we present different theoretical curves of D(q ", q , T) calculated under the assumption, that the real part of the complex elastic constant c (q, T) can be written in the form c (q, T) = c (T)-Ac/ 1 + 47i (q,T)x (T). For the exponent P<1 this formular describes a Cole davidson function. The relaxation time x was assumed to follow a VFT law. Under these conditions the OADF deviates from n(T) only well above the TGT and... 

It is a common observation that the value of Z for tissue decreases wifli frequency, Y increases with frequency, and e decreases wifli frequency. By choosing a as an exponent in the Z equation (Cole 1940), —a in the Y equation, and 1 — a (equivalent to the loss factor of a capacitor the phase angle of an ideal capacitor is 90°, but die loss angle is 0°) in the e equation (Cole, 1940 Cole—Cole, 1941), the correct frequency dependence is taken care of with the same a value independently of model, a being always positive 1 >a > 0. It is possible to regard the parameter a in several ways ... 

For lumped elements, e.g. resistors, capacitors or combinations of these elements, the differential equations, impedances and VSR are well-known [4]. Distributed elements, i.e. Warburg impedance. Constant Phase Element, or parallel connections like RCPE, also known as ZARC or Cole-Cole element, have non-integer exponents a of the complex frequency s in frequency domain. This corresponds to fractional differential equations in time domain and thus the calculation of the VSR requires fractional calculus, as can be seen in the following derivations. 

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