Polarisability frequency dependence

With polar molecules the value of the dielectric constant is additionally dependent on dipole polarisation and commonly has values between 3.0 and 7.0. The extent of dipole polarisation will depend on frequency, an increase in frequency eventually leading to a reduction in dielectric constant. Power factor-frequency curves will go through a maximum. 

Neufeld, P. and Queenan, E. D., Frequency Dependence of Polarisation Resistance Measured with Square Wave Alternating Potential , Br. Corros. J., 5, 72-75, March (1970) Fontana, M. G., Corrosion Engineering, 3rd edn., McGraw-Hill, pp 194-8 (1986) Dawson, J. L., Callow, L. M., Hlady, K. and Richardson, J. A., Corrosion Rate Determination By Electrochemical Impedance Measurement , Conf. On-Line Surveillance and Monitoring of Process Plant, London, Society of Chemical Industry (1977)... 

As far as explicit approximations for the polarisation functions Tltidq) are concerned only very little is known, even in the static limit. The complete frequency dependence is available for the noninteracting limit ( ). i e. the relativistic generalisation of the Lindhard function [95, 114]. In addition to its vacuum part (A.26) one has... 

Freezing of a dipolar liquid is accompanied by a rapid decrease in its electric permittivity [8-10]. Following solidification, dipole rotation ceases and the electric permittivity is almost equal to n, where n is refractive index, as it arises from deformation polarisation only. Investigation of the dynamics of a confined liquid is possible from the frequency dependences of dielectric properties, which allows both the determination of the phase transition temperature of the adsorbed substance and characteristic relaxation frequencies related to molecular motion in particular phases. 

For g(x) to be proportional to 1/x, it requires the relaxation time to be an exponential function of some random variable such that x = xq exp ( ), where itself has a flat distribution. It means that ( ) = constant, and rt(x) = n ). (d /dx) oc x. If a, the polarisability, is also a function of then it can lead to a sub-linear frequency dependence of a (co). The functional form given for the variation of x can arise from two different relaxation mechanisms. The first is a classical barrier hopping, in which two energetically favourable sites like in a double well potential are separated by a barrier fV and = W/kT. The second mechanism is a phonon assisted quantum tunneling through a barrier, which separates two equilibrium positions, in which case = 2aR, where a is the localization length and R is the separation between the sites. In the first case, by treating JV as independent of R, it has been shown (Poliak and Pike, 1972) that... 

In this section we address the detailed form, and in particular the dispersion behavior, of the molecular response tensors. We note that the frequency dependence of nonlinear polarisabilities and their sum rules have been the subject of a series of incisive works by Bishop and others see, for example, the paper by Bishop and DeKee [42]. In addressing dispersion behavior below, we follow the same general principles, but at the outset we invoke excited-state damping to allow for the incorporation of lineshape. This is a matter that, once its context is established below, we shall return to in the following section. 

DEP force. The frequency dependence and the direction of the DEP force are governed by the real part of the Clausius-Mossotti factor. If the particle is more polarisable than the medium, (Re[/cm] > 0). the particle is attracted to high intensity electric field regions. This is termed as positive dielectrophoresis (pDEP). Conversely, if the particle is less polarisable than the medium, (Re[/cm] < 0), the particle is repelled from high intensity field regions and negative dielectrophoresis (nDEP) occurs. Therefore the real part of the Clausius-Mossotti factor characterizes the frequency dependence of the DEP force, as demonstrated in Fig. 1. 

The transfer of momentum from the beam to an absorbing particle is therefore straightforward compared to other cases, such as transparent particles and atoms. For transparent particles, refraction and induced polarisation must be taken into account. For an atom, the frequency dependence of the absorption and spontaneous emission must be considered, while for an absorbing particle, the absorption can be assumed to be independent of frequency, and inter-atomic collision rates within the particle can be assumed to be high enough to cause deexcitation without re-emission. 

The observation of complex dielectric constant frequency dependence (e = s — ie") shows that, at low frequencies, the different polarisations contribute to a high permittivity c value, beyond that, each kind of polarisation will create one resonance or one relaxation process e decrease and a maximum appears for e . 

Figure 2. Frequency dependence of imaginary part of CNT bundle polarisability.

For the analysis of the reflection spectra, we consider as usual the frequency-dependent complex dielectric tensor s (co). We restrict ourselves here to the case that the direction of polarisation is parallel to the stacking axis, and denote the real and imaginary parts of the complex dielectric function as usual s (jo) = i(co) -i- is2(co). It is related to the complex index of refraction, n = ni -i-in2, via n = The real part of... 

Complementary to the real part a co) of the conductivity is the real part s (co) of the dielectric function. s (co) exhibits a frequency dependence at temperatures between 105 K and 40 K which corresponds to Debye relaxation. The resulting polarisation is described in simplified form by... 

Onaral B, Schwan HP. 1982. Linear and nonlinear properties of platinum electrode polarisation. Part 1 Frequency dependence at very low frequencies. Med Biol Eng Comp 20,299-306. 

Neufeld, P. and Queenan, E. D., Frequency Dependence of Polarisation Resistance Measured with Square Wave Alternating Potential , Br. Corros. J., S, 72-75, March (1970)... 

In this equation the factor, is called the Clausius-Mossotti factor and describes the frequency dependence of the polarisability. Combining Eqs. (1) and (2) leads to the expression for the dipole in terms of the complex permittivity of the particle and the suspending medium (subscript p and m respectively) ... 

However, for real absorbing materials the polarisation does not respond instantaneously to an applied field, which causes dielectric loss. This is expressed by a permittivity that is both complex and frequency dependent. Taking these aspects into consideration a complex index of refraction n can be defined, see (8.3) ... 

There is an important practical distinction between electronic and dipole polarisation whereas the former involves only movement of electrons the latter entails movement of part of or even the whole of the molecule. Molecular movements take a finite time and complete orientation as induced by an alternating current may or may not be possible depending on the frequency of the change of direction of the electric field. Thus at zero frequency the dielectric constant will be at a maximum and this will remain approximately constant until the dipole orientation time is of the same order as the reciprocal of the frequency. Dipole movement will now be limited and the dipole polarisation effect and the dielectric constant will be reduced. As the frequency further increases, the dipole polarisation effect will tend to zero and the dielectric constant will tend to be dependent only on the electronic polarisation Figure 6.3). Where there are two dipole species differing in ease of orientation there will be two points of inflection in the dielectric constant-frequency curve. 

When dipoles are directly attached to the chain their movement will obviously depend on the ability of chain segments to move. Thus the dipole polarisation effect will be much less below the glass transition temperature, than above it Figure 6.4). For this reason unplasticised PVC, poly(ethylene terephthalate) and the bis-phenol A polycarbonates are better high-frequency insulators at room temperature, which is below the glass temperature of each of these polymers, than would be expected in polymers of similar polarity but with the polar groups in the side chains. 

The time factor in stepwise potentiostatic or potentiodynamic polarisation experiments is very important, because large differences can be caused by changes in the scanning rate. Since the steady state depends on the particular system and conditions of exposure, no set rule exists for the magnitude or frequency of potential changes. Chatfield etal. have studied the Ni/H2S04 system and have shown how becomes more passive with increase in sweep rate. 

Impedance Some of the errors arising from the use of linear polarisation resistance led to interest and development in a.c. systems.An early development used a fixed a.c. frequency and a commercial instrument was produced in the UK. Inaccuracies still occurred, however, and were due to the electrode impedance which is fequency dependent. Electrode reactions have a capacitance component, in addition to resistance, resulting in a requirement to measure the impedance. However, the total impedance comprises values for the reaction, solution, diffusion and capacitance. Measurements at different frequency are more reliable, particularly where high solution resistances occur. Simplifications for industrial monitoring have been developed consisting of two measurements, i.e. at a high (10 kHz) and low frequency (0-1 Hz). The high-frequency measurement can identify the... 

---