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Cole-Davidson function, equation

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... [Pg.86]

Where M2 is the second moment of the NMR lineshape, J the spectral density function, with (Dq the Larmor frequency, and (0i the frequency of the spin-locking field. The spectral density can be written in terms of the molecular correlation time, x, and the overall shape of the Tjp - temperature dispersion and the relatively shallow minima arc due to the correlation time distribution, although the location of the minimum is unaffected by this distribution. We have examined several models for the distribution, all of which give essentially the same results. One of the more simple is the Cole-Davidson function (75), which has also been applied to the analysis of dielectric relaxations. The relevant expression for the spectral density in this case is given by Equation 4. [Pg.256]

In the particular application to dielectric relaxation, fit) is the aftereffect function following the removal of a constant field [8]. The solution of Eq. (93) rendered in the frequency domain yields the Cole-Davidson equation [Eq. (10)] [28],... [Pg.314]

There were several attempts to generalize the Debye function like the Cole/Cole formula (Cole and Cole 1941) (symmetric broadened relaxation function), the Cole/Davidson equation (Davidson and Cole 1950, 1951), or the Fuoss/Kirkwood model (asymmetric broadened relaxation function) (Fuoss and Kirkwood 1941). The most general formula is the model function of Havriliak and Negami (HN function) (Havriliak and Negami 1966,1967 Havriliak 1997) which reads... [Pg.1311]

A common approach to model the dielectric response, typically used for impedance spectroscopy, is based on equivalent circuits consisting of a number of resistors, capacitors, constant phase elements, and others. Alternatively, the dielectric response can be modeled by a set of model relaxation functions like the Debye function or more generalized (semiempirical) Cole-Cole, Cole-Davidson, or Dissado-Hill equation (Kremer and Schonhals 2002). [Pg.599]

The memory effect is described by the integral term of Eq. (10-22), which is absent in the standard approach. Equation (10-22) holds for a liquid crystal with a Debye type of relaxation but it can be easily modified or generalized to other relaxation models (such as Cole-Davidson, Havriliak-Negami and other models with different functional forms of a(t — f) [5, 17]). [Pg.234]

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... [Pg.385]

Other models have been proposed which follow the outlines of the equations already discussed. Equations with parameters that vary as a function of temperature, sunlight, and nutrient concentration have been presented by Davidson and Clymer (9) and simulated by Cole (10). A set of equations which model the population of phytoplankton, zooplankton, and a species of fish in a large lake have been presented by Parker (II). The application of the techniques of phytoplankton modeling to the problem of eutrophication in rivers and estuaries has been proposed by Chen (12). The interrelations between the nitrogen cycle and the phytoplankton population in the Potomac Estuary has been investigated using a feed-forward-feed-back model of the dependent variables, which interact linearly following first order kinetics (13). [Pg.144]

Equations (1.23a), (1.23b) and (1.23c) are, respectively, Cole-Cole (C-C) (0Davidson-Cole (D-C) (0Havriliak-Negami (0empirical laws. The calculations of permittivity on the base of Eq. (1.22) with relaxation function corresponding to KWW law (see Eq. 1.20) yield Eq. (1.23c) with y8 = a - [30]. Expression (1.23c) delivers pretty good description of experimental data obtained by dielectric spectroscopy, radiospectroscopy and quasielastic neutron scattering. It can be shown, that the physical mechanism, underlying the expressions (1.23) is the distribution of relaxation times in a system. Namely, Equation (1.23) can be derived by the averaging of simple Debye response (1.21) with properly tailored distribution function of relaxation times F(x) ... [Pg.21]

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. [Pg.149]


See other pages where Cole-Davidson function, equation is mentioned: [Pg.290]    [Pg.291]    [Pg.293]    [Pg.312]    [Pg.313]    [Pg.420]    [Pg.32]    [Pg.606]    [Pg.278]    [Pg.278]    [Pg.939]    [Pg.607]   
See also in sourсe #XX -- [ Pg.104 ]




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