Symmetrical distribution of relaxation times

Perfect Circular Arcs, Cole-Compatible Distribution of Relaxation Times 

The characteristic time constant Tc in the form of tz and xyof Eqs (9.26) and (9.29) deserves some explanation. A two-component RC circuit with ideal components has a time constant t = RC. A step excitation results in an exponential response. With a CPE, the response will not be exponential. In Section 9.2.5, the time constant was introduced simply as a frequency scale factor. However, the characteristic time constant Tc may be regarded as a mean time constant because of a DRT. When transforming the Colez impedance Z to the Coley admittance Y or vice versa, it can be shown that the as of the two Cole Eqs 9.26 and 9.29 are invariant, but the characteristic time constants Tc are not. In fact  

It is possible to determine the relationship between the DRT and the two parameters a and T in the Cole—Cole equation (Cole and Cole, 1941)  

Other distributions than the Cole—Cole of Eq. 9.40 are possible for instance, a constant function extending over a limited range of time constants. Such different distributions result in Wessel diagram loci surprisingly similar to circular arcs, 

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In the present work and are not readily determinable, particularly since the chemical nature of the polymer Is changing. However, for a given relaxation, if it can be assumed that there Is a symmetrical distribution of relaxation times and Cq can be related by the Kramers-Kronig equationCiS), or by the following alternative expressionCH),... 

When a is dose to unity this again reduces to Debye s model and for a smaller than unity an asymmetric diagram is obtained. The Cole-Cole diagram arise from symmetrical distribution of relaxation times whereas the Cole-Davidson diagram is obtained from a series of relaxation mechanisms of decreasing importance extending to the high-frequency side of the main dispersion. 

In real systems, when a symmetric distribution of relaxation times exists, the complex permittivity is described by the fimction ... 

The expression (1.24) allows obtaining the distribution function of relaxation times for all empirical laws (1.23). In Fig. 1.9, we show the relaxation time distribution functions, obtained in Ref. [31] with the help of Eq. (1.24). The distribution functions have been obtained for the laws of Cole-Cole k = 0.2), Davidson-Cole (P = 0.6) and Havriliak-Nagami at a = 0.42 when it corresponds to KWW law. It is seen that only C-C law leads to symmetric dishibution function while DC and KWW laws correspond to essentially asymmetric one. The physical mechanisms responsible for different forms of distribution functions in the disordered ferroelechics had been considered in Ref. [32]. It has been shown that random electric field in the disordered systems alters the relaxational barriers so that the distribution of the field results in the barriers distribution, which in turn generates the distribution of relaxation times. Nonlinear contributions of random field are responsible for the functions asymmetry, while the linear contribution gives only symmetric C-C function. 

Comparison between Cole-Cole plots for the Debye, Cole-Cole and Davidson-Cole equations is made in Fig. 4.2. The arc corresponding to the Cole-Cole equation is symmetrical and forms a portion of a circle, the centre of which is below the 6 -axis. The corresponding distribution function of relaxation times is symmetric, although there is no closed expression for F t) which would give the Cole-Cole equation. The Davidson-Cole arc is a skewed one, and reflects strongly asymmetric distribution of relaxation times. The distribution is peaked at the critical relaxation time Tq, with a decaying tail of shorter relaxation times. There is an exact expression for the autocorrelation function leading to the Davidson-Cole equation. 

Assumption b) is known to be a good approximation for small molecules. However studies of polymers and glass forming materials by dielectric and mechanical loss methods have frequently been interpreted by assuming that molecular motions are best described by a distribution of correlation times. This has resulted in the formulation of a number of well-known distribution functions such as the Cole-Cole (symmetric) and Cole-Davidson (asymmetric) functions, which have been used to fit dielectric data. It is reasonable to suppose that magnetic relaxation times are also subject to the possible presence of distributions, and a number of modifications of Eq, (4) have been made [16 —i 9] on this basis. 

As one can see in Figure 14 the relaxation spectra of the isotropic phase of all substances studied have maxima of losses above 100 MHz, so our HP setup can cover only a low-frequency part of the absorption bands. Therefore, to obtain the relaxation times vs. T or p we had to extrapolate the measured spectra to higher frequencies in order to find the critical frequency / = 1/(27tt ). According to Parneix et al. the Cole-Davidson skewed arcs should be used for that purpose. However, the recent measurements carried out by Gestblom and co-workers with the use of the TDS method have shown that the spectra of the isotropic phase of 5CB and 5PCH could be well described by the Cole-Cole equation with a symmetric distribution of the relaxation times. 

The dynamic friction term gives the correlation between the Brownian displacement in each time interval and the driven motions of the same particle at later times. As was first shown by Mazo(33), these correlations are nonzero and serve to retard diffusion. Correlations between Brownian and driven motions arise because each particle in solution is surrounded by its radial distribution function. The particle s Brownian displacements momentarily carry it off-center relative to the spherically symmetric distribution of neighboring particles. Until the radial distribution relaxes to the new location of the particle, which does not occur instantaneously, the neighboring particles tend to drive the particle of interest back to its initial location. The contributions of dynamic friction to D (34), Ds(35), the drag coefficient fo for motion at constant velocity(36), and r (37) have been obtained. 

These nuclei (and they form by far the majority of the NMR-active nuclei ) are subject to relaxation mechanisms which involve interactions with the quadrupole moment. The relaxation times Tj and T2 (T2 is a second relaxation variable called the spin-spin relaxation time) of such nuclei are very short, so that very broad NMR lines are normally observed. The relaxation times, and the linewidths, depend on the symmetry of the electronic environment. If the charge distribution is spherically symmetrical the lines are sharp, but if it is ellipsoidal they are broad. 

Here r and v are respectively the electron position and velocity, r = —(e2 /em)(r/r3) is the acceleration in the coulombic field of the positive ion and q = /3kBT/m. The mobility of the quasi-free electron is related to / and the relaxation time T by p = e/m/3 = et/m, so that fi = T l. In the spherically symmetrical situation, a density function n(vr, vt, t) may be defined such that n dr dvr dvt = W dr dv here, vr and vt and are respectively the radical and normal velocities. Expectation values of all dynamical variables are obtained from integration over n. Since the electron experiences only radical force (other than random interactions), it is reasonable to expect that its motion in the v space is basically a free Brownian motion only weakly coupled to r and vr by the centrifugal force. The correlations1, K(r, v,2) and fc(vr, v(2) are then neglected. Another condition, cr(r)2 (r)2, implying that the electron distribution is not too much delocalized on r, is verified a posteriori. Following Chandrasekhar (1943), the density function may now be written as an uncoupled product, n = gh, where... 

In the case of the symmetry of the potential profile with respect to the coordinate of the initial distribution x0 (which may be taken as x0 = 0) and of the symmetrical decision interval [—d, I d from (5.96), one can obtain the following expression for the relaxation time ... 

If there is the symmetrical potential profile initial distribution is located at the origin (i.e., xo = 0), then all results concerning the relaxation times will be the same as for the potential profile in which the reflecting wall is at x = 0 and at xo = +0. This coincidence of the relaxation times may be proven in a common case taking into account that the probability current at x = 0 is equal to zero at any instant of time. 

Interdependence Between Relaxation and Decay Times. Consider the potential profile of type I which is symmetrical with respect to x = d > 0 and where the initial distribution is located at xq = 0 and the decision interval is [—00, d] (see Appendix). The relaxation time that follows from (5.97) is... 

No attempt is made to summarize conductivity data. Conductivity increases similarly in several major steps symmetrical to the changes of the dielectric constant. These changes are in accord with the theoretical demand that the ratio of capacitance and conductance changes for each relaxation mechanism is given by its time constant, or, in the case of distributions of time constants, by an appropriate average time constant and the Kramers-Kronig relations. 

When a central ion moves in an electric field the ionic cloud surrounding the ion is permanently formed. This requires a certain time called the relaxation time. Therefore, as illustrated in Fig. 6-1, the charge density around the central ion is no longer symmetrical, but is lower in front of the central ion than behind it. This dissymmetry in charge distribution leads to an electrostatic deceleration of the central ion which reduces the ion mobility. 

Treating the problem as one of rotary Brownian movement, Kirkwood and Fuoss32 were able to calculate the distribution functions /x m and F (r) of Eqs. 22 and 29. In their Gase F(t) was a symmetrical function and they identified the average relaxation time with the value rm corresponding to the maximum in the loss curve. Unfortunately their theory is incompatible with existing experimental data on dilute solutions, since it specifies that rm should be proportional to the degree of polymerization. 

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