Dielectric constant relaxation function parameters

Relaxation Function Parameters and Their Confidence Limits for Representing the Complex Dielectric Constant for Various Values of C and L = 8... 

As a factor related to chain motion, one may also consider polymer elastic modulus. As indicated in Properties of Polymers (39), at one time it was hoped that mechanical studies of polymers could be entirely replaced by electrical measurements. There are indeed close similarities between the general shapes and temperature-dependences of the mechanical and dielectric loss curves, but the quantitative connection between these phenomena is not as simple as was origindly believed. Electrical measurements constitute a useful addition to, but not a substitute for, mechanical studies. However, the elastic modulus and the dielectric constant are related to similar physic behavior. These parameters are response functions obtained by stimulating a material and measuring the subsequent relaxation phenomena. Therefore, similar to the dielectric constant, the elastic modulus depends not only on chain motion but also on free space in the polymer matrix. In fact, relaxation phenomenon of... 

Where As= Ss- s, Sao and Ss are the unrelaxed and relaxed values of the dielectric constant, Thn is a characteristic relaxation time, and b and c are shape parameters (0 < Z>, c < 1) which describe the symmetric and the asymmetric broadening of the equivalent relaxation time distribution function,... 

Circular plots aid in studying polarization effects as function of frequency (52). If a single dielectric relaxation time is involved, the plot of the loss index against the dielectric constant (for fi equencies fi om just above and below those for which polarization occurs) is a semicircle with its center on the dielectric constant axis (Fig. 30a). If more than one dielectric relaxation time is involved, it is still possible to plot the loss index against the dielectric constant axis (Fig. 30b). A new parameter is required, the storage coefficient, a, which is the complement... 

Here e , is the high frequ y limit of s, So is the static dielectric constant (low frequency limit of s ). So - Soo = A is the dielectric increment, fR is the relaxation frequency, a is the Cole-Cole distribution parameter, and P is the asymmetry parameter. The relaxation frequency is related to the relaxation time by fa = (27It) A simple exponential decay of P (oc,P = 0) is characterised by a single relaxation time (Debye-process [1]), P = 0 and 1 < a < 0 describe a Cole-Cole-relaxation [2] with a symmetrical distribution function of t whereas the Havriliak-Negami equation (EQN (4)) is used for an asymmetric distribution of x [3]. The symmetry can be readily seen by plotting s versus s" as the so-called Cole-Cole plot [4-6]. 

Independently of the specific dielectric technique used, the results of dielectric measurements are usually analyzed in the form of complex dielectric permittivity e(o ) = e w) — ie (w) at constant temperature by fitting empirical relaxation functions to e(o ). In the examples to be given later in this chapter, the two-shape-parameter Havriliak-Negami (HN) expression [22]... 

In the standard description of the dielectric properties of the chiral tilted smectics worked out by Carlssonet al. [152], four independent modes are predicted. In the smectic C the collective excitations are the soft mode and the Goldstone mode. In the SmA phase the only collective relaxation is the soft mode. Two high frequency modes are connected to noncollective fluctuations of the polarization predicted by the theory. These two modes become a single noncollective mode in the smectic A phase. There is no consensus [153] as yet as to whether these polarization modes really exist. Investigations of the temperature dependence of the relaxation frequency for the rotation around the long axis show that it is a single Cole-Cole relaxation on both sides of the phase transition between smectic A and smectic C [154]. The distribution parameter a of the Cole-Cole function is temperature-dependent and increases linearly (a=a-pT+bj) with temperature. The proportionality constant uj increases abruptly at the smectic A to SmC transition. This fact points to the complexity of the relaxations in the smectic C phase. 

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