Fractional rotational Brownian motion in a uniform dc external field

The longitudinal and transverse components of the complex dielectric susceptibility tensor of an assembly of dipolar particles subjected to a dc bias field are evaluated in the context of a fractional noninertial rotational diffusion model. Exact and approximate solutions for the dielectric dispersion and absorption spectra are obtained. It is shown that a knowledge of the effective relaxation times for normal rotational diffusion is sufficient to predict accurately the anomalous dielectric relaxation behavior of the system for all time scales of interest. Simple equations for the characteristic frequencies of the dielectric loss spectra are obtained in terms of the physical model parameters (dimensionless field and fractional exponent). The model explains the anomalous (Cole-Cole like) relaxation of complex dipolar systems, where the anomalous exponent differs from unity (corresponding to the normal dielectric relaxation), i.e., the relaxation process is characterized by a broad distribution of relaxation times.

Figure 1

Integral (${\tau }_{\mathit{int}}^{\gamma }$: solid lines) and effective (${\tau }_{\mathit{ef}}^{\gamma }$: filled circles and asterisks) relaxation times vs $\xi$ for normal rotational diffusion in a dc bias field. Equations (19, 20, 21, 22) have been used in the calculation.

Figure 2

Frequency ${\omega }_c^{\Vert }$ as a function of $\xi$ and $\sigma$.

Figure 3

Frequency ${\omega }_c^{\perp }$ as a function of $\xi$ and $\sigma$.

Figure 4

Dielectric loss spectra ${\alpha }_{\Vert }^{″}\left(\omega \right)$ evaluated from the exact continued fraction solution [Eqs. (8, A4): solid lines] for $\sigma =0.5$ and various values of $\xi$, and compared with those calculated from the approximate Eq. (24) (stars). The low (dotted lines) and high frequency (dashed lines) asymptotes are calculated from Eqs. (13, 14, 19, 20), respectively.

Figure 5

The same as in Fig. 1 for $\xi =5$ and various values of $\sigma$.

Figure 6

Dielectric loss spectra ${\alpha }_{\perp }^{″}\left(\omega \right)$ evaluated from the exact continued fraction solution [Eqs. (8, A9): solid lines] for $\sigma =0.5$ and various values of $\xi$ and compared with those calculated from the approximate Eq. (24) (stars). The low (dotted lines) and high frequency (dashed lines) asymptotes are calculated from Eqs. (13, 21, 14, 22), respectively.

Figure 7

The same as in Fig. 3 for $\xi =5$ and various values of $\sigma$.