# (p.414) Appendix B Dielectric Spectra Broadening as the Signature of Dipole–Matrix Interaction

# (p.414) Appendix B Dielectric Spectra Broadening as the Signature of Dipole–Matrix Interaction

The phenomenological Cole–Cole (CC) dispersion function relates the complex dielectric permittivity ${\mathrm{\epsilon}}^{\ast}(\mathrm{\omega})$ to the frequency of the applied field, as:

It has been found experimentally that the parameters $\mathrm{\alpha}$, $\mathrm{\tau}$, and $\underset{\_}{\mathrm{\Delta}}\mathrm{\epsilon}$ are strictly dependent on temperature, sample structure, composition, pressure, and other physical quantities [1, 2]. The fractional Fokker–Planck equation, coupled with the memory function in the Mori–Zwanzig projection formalism, has been found to be a very effective mathematical tool for understanding the fractal nature and the cooperative behavior of the underlying CC relaxation process [1, 3–7]. It was shown that the broadening parameter $\mathrm{\alpha}$, which is controlled by macroscopic physical quantities, reflects the rate of interactions of the dipole relaxation units with their surroundings (see Eq. (1.1.4.19)):

In the dimensionless time interval $\mathrm{\xi}=\mathrm{\tau}/{\mathrm{\tau}}_{c}$ this rate leads to the average number of discrete interactions ${N}_{\mathrm{\tau}}$, which can be described by following recursive fractal model [8]:

where the mass fractal dimension *A* adopts values in the range $0<A\le 1$ and ${\mathrm{\tau}}_{c}$ is a cutoff time that defines the time scaling. According to Eq. (B3), ${N}_{\mathrm{\tau}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{N}_{0}$ if $\mathrm{\xi}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}1$, that is, *N*_{0} represents an average number of elementary relaxation events occurring during the period ${\mathrm{\tau}}_{c}$. The detailed analysis of the different kinds of dependences of $\mathrm{\alpha}$ versus the variable $x=\text{ln}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\tau}$ shows that they can be summarized using one universal function as follows [8]:

where, ${x}_{0}=ln{\mathrm{\tau}}_{c}$ and the parameters of function (B4) are connected to the fractal model (B3) by the relationships:

(p.415)
In general, for a two-component complex systems, where one component is a dipole subsystem [8, 9], $\mathrm{\alpha}$ depends on $\text{ln}\mathrm{\tau}$ as described by Eq. (B4). The morphology, dynamics, and the dielectric properties of the second component (polymers, ions, porous matrix, or second dipole subsystem) are essentially different from the properties of the first, and it may be defined as a matrix. Equation (B4) describes a hyperbolic curve bounded by two asymptotes: the constant *A*, representing the asymptotic value of the parameter $\mathrm{\alpha}$, and the asymptote ${x}_{0}=ln{\mathrm{\tau}}_{c}$, dividing the full plane into two semi planes, $\mathrm{\tau}>{\mathrm{\tau}}_{c}$ and $\mathrm{\tau}<{\mathrm{\tau}}_{c}$ (Figure B1).The various patterns $\mathrm{\alpha}(\text{ln}\mathrm{\tau})$ of complex systems correspond to the different branches presented in Figure B1. Therefore, the model described by Eq. (B3)–(B5) classifies different complex systems in terms of the four monotonic branches. However, the nature of the coupling between the number of the relaxation acts occurring during the characteristic relaxation time and the molecular structure, in which they occur, still remains unclear. One may elucidate this coupling further by exploiting the unutilized parameters $\mathrm{\Delta}\mathrm{\epsilon},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{\epsilon}}_{s}$, and ${\mathrm{\epsilon}}_{\mathrm{\infty}}$ of Eq. (B1). These parameters are related to structural aspects of the relaxation and Froehlich pioneered this approach by associating them with the number of dipoles involved [10]. Using the general relationship for polar dielectrics, he introduced a new function that described the temperature behavior of the polarization of the system under consideration:

(p.416)
where *k* is the Boltzmann constant, *T* is the absolute temperature, *V* is a volume with *N* microscopic cells containing some dipoles or charges, respectively, and ${\mathrm{\epsilon}}_{0}=8.85\times {10}^{-12}\phantom{\rule{thinmathspace}{0ex}}\text{F/m}$ is the permittivity of free space.

The electric dipole moment of this volume is defined as $\mathbf{M}={\sum}_{i\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}1}^{N}{\mathbf{m}}_{i}$, with **m**_{i} as the average dipole moment of the ith cell, and the brackets <…> in Eq. (B6) indicating a statistical averaging over all possible cell configurations. The dipole moment of the macroscopic volume of material polarized by one cell with the dipole moment **m** is denoted by **m**^{*}. As the same underlying thermodynamic variables (temperature, pressure, etc.) drive *B* as well as *x*, it appears to be obvious to consider a relationship between the two parameters. This relationship, when taken together with Eqs. (B3)–(B5), evidently reflects structural and dynamic aspects of the same relaxation. The coordinate axes of this space are $X=lnB,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}Y=ln\mathrm{\tau}$, and $Z=\mathrm{\alpha}$, so any alteration in the complex system is defined by a three-dimensional (3D) trajectory (see, for example, Figure B2).

Whenever water interacts with another dipolar or charged entity, there is a broadening of its dielectric relaxation peak. This we demonstrated for solid matrixes (porous silica glasses) where water played the role of the dipole and the pore surface the role of the matrix [8]. Then the results of main water peak broadening, where the water serves as the solvent are presented in the cases of aqueous solutions of D-glucose and D-fructose (non-ionic solutions) [11], as well as for sodium chloride and potassium chloride aqueous solutions (ionic ones) [12]. The 3D trajectory approach is applied in order to highlight the difference between the dynamics and structure of solutions of salts on the one hand and dipolar solutes on the other hand. This difference is anticipated as resulting from the differing charge–dipole and dipole–dipole interactions. The CC broadening of the main relaxation peak of the water solvent was observed in the nucleotide solutions [13]. Moreover, depending on the nucleotide concentration due to dual (charge/dipole) nature of the nucleotide molecules the both types of dipole–matrix interaction was observed. The 3D trajectory approach is applied in order to highlight the differences between the two types of interaction.