Modeling Dielectric Susceptibilities

In aerosol van der Waals forces, the central role played by the material properties of both the particle and whatever it interacts with is clearly illustrated in the above considerations where the polarizability or electric susceptibility a and the dielectric constant (or permeability) e are used alternatively. Provided that a represents the material s susceptibility, not the molecular polarizability, the usual definition 

Various difficulties with the form of e(w) or the procedure used by NINHAM and PARSEGIAN may arise when conductors or very small particles are examined. In the first case, nonlocal dispersion has not been adequately treated in van der Waals theory in general, but even if the nonlocal effects could be ignored, interband transitions may need to be accommodated. LANDAU and LIFSHITZ [5.50] propose the form e(o)) = 4iria)/a, where a is the conductivity, for the very low-frequency dielectric permeability of conductors. Small particles and clusters also must be treated with caution if they are metallic due to surface-scattering and size quantization effects [5.59]. 

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The alternative approach is based on a non-iterative procedure using the maximum entropy model (MEM) to extract the complex dielectric susceptibility from the intensity measurements. This technique was first proposed 15 years ago (Vartiainen 1992), and recently was used for multiplexed CARS measurements (Petrov et al. 2007, Vartiainen et al. 2006). 

Rather than continue so formally, consider dielectric susceptibilities in terms of illustrative models. Conceptually the simplest picture of a dielectric response is that in an electric circuit. Think about a capacitor as a sandwich of interesting material between two parallel conducting plates (see Fig. L2.22). 

Despite some model differences, dielectric susceptibility can be expressed by the same equation [7] ... 

Let us now turn to a few other informative applications of the pseudopotential model to properties (discussed extensively in Harrison, 1976a). For that purpose it will be most convenient to take the pseudopotential parameters from the LCAO values for V2 and Fj, using Eq. (18-5). We see from Table 18-1 that for V2 this is roughly equivalent to using Empirical Pseudopotential Model parameters. Let us look first at the dielectric susceptibility, which was so important in the development of the LCAO theory. For this, the most convenient form for the susceptibility is Eq. (4-5), which we rewrite in simple form,... 

The infinite continued fraction, Eq. (281), is very convenient for the purpose of calculations so that the complex dielectric susceptibility, Eq. (282), can be readily evaluated for all values of the model parameters r, p/, and a. For a = 1, the anomalous rotational diffusion solution, Eq. (281), coincides with that of Sack [40] for normal rotational diffusion. Moreover, in a few particular cases, Eqs. (281) and (282) can be considerably simplified. In the free rotation limit (( = 0), which corresponds to the continued fraction [Eq. (281)] evaluated at x = 0, that fraction can be expressed (just as for normal rotational diffusion [40]) in terms of the exponential integral function E z) [51] so that the normalized complex susceptibility is... 

Similar heterogeneous model has been used to develop a relaxation function by Chamberlin and Kingsbury (1994), who consider the localized normal modes to be involved in the relaxation process. Localized (domains) regions are assumed to be present between Tg and T. They are described as dynamically correlated domains (DCD). A Gaussian distribution of the domain sizes has been assumed, with each domain characterized by a Debye relaxation time. Expressions for the dielectric susceptibility have been derived and used to fit the experimental susceptibilities of salol, glycerol and many other substances with remarkable agreement over 13 decades of frequency (even when only one adjustable parameter is employed). 

Our review starts with the general formulation of the GGS model in Sect. 2. In the framework of the GGS approach many dynamical quantities of experimental relevance can be expressed through analytical equations. Because of this, in Sect. 3 we outline the derivation of such expressions for the dynamical shear modulus and the viscosity, for the relaxation modulus, for the dielectric susceptibility, and for the displacement of monomers under external forces. Section 4 provides a historical retrospective of the classical Rouse model, while emphasizing its successes and limitations. The next three sections are devoted to the dynamical properties of several classes of polymer networks, ranging from regular and fractal networks to network models which take into account structural heterogeneities encountered in real systems. Sections 8 and 9 discuss dendrimers, dendritic polymers, and hyperbranched polymers. 

The approach presented here was first developed for the dielectric relaxation of regular mesh-like polymer networks built from macromolecules with longitudinal dipole moments [30], and was later applied to disordered polymer networks [31,32]. Its key assiunption, namely the absence of any correlations in the orientations of the dipole moments of the different GGS bonds is obviously rather simphfied. However, it leads, as shown above, to simple analytical expressions for the dielectric susceptibility, a very important dynamical quantity in experimental studies of polymers we can now analyze it in great detail for particular GGS systems of interest. Another advantage of this model arises from the fact that one has a straightforward correspondence between the mechanical and the dielectric relaxation forms. From the expressions for the storage and loss modulus, Eqs. 20 and 21, and from those for the dielectric susceptibility Ae, Eqs. 41 and Eq. 42, one sees readily that [31]... 

The literature on the molecular theory of liquid crystals is enormous and in this chapter we have been able to cover only a small part of it. We have mainly been interested in the models for the nematic-isotropic, nematic-smectic A and smectic A-smectic C phase transitions. The existing theory includes also extensive calculations of the various parameters of the liquid crystal phases Frank elastic constants, dielectric susceptibility, viscosity, flexoelectric coefficients and so... 

The isotropic coefficient and the anisotropic coefficients b(m> and c(m) can have both bulk and surface contributions and depend on crystal symmetry. The linear and nonlinear dielectric constants of the material, as well as the appropriate Fresnel factors at co and 2co, are incorporated into the constants a, b m) and c(m). Table 3.1 shows the susceptibilities contained in each of these constants. The models of Tom... 

In the third period, which ended in 1999 after the book VIG was published, various fluids had been studied strongly polar nonassociated liquids, liquid water, aqueous solutions of electrolytes, and a solution of a nonelectrolyte (dimethyl sulfoxide). Dielectric behavior of water bound by proteins was also studied. The latter studies concern hemoglobin in aqueous solution and humidified collagen, which could also serve as a model of human skin. In these investigations a simplified but effective approach was used, in which the susceptibility % (m) of a complex system was represented as a superposition of the contributions due to several quasi-independent subensembles of molecules moving in different potential wells (VIG, p. 210). (The same approximation is used also in this chapter.) On the basis of a small-amplitude libration approximation used in terms of the cone-confined rotator model (GT, p. 238), the hybrid model was suggested in Refs. 32-34 and in VIG, p. 305. This model was successfully employed in most of our interpretations of the experimental results. Many citations of our works appeared in the literature. 

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