Fractional rotational diffusion of rigid dipoles in an asymmetrical double-well potential

The longitudinal and transverse components of the complex dielectric susceptibility tensor of an assembly of dipolar molecules rotating in an asymmetric double-well potential are evaluated using a fractional rotational diffusion equation (based on the diffusion limit of a fractal time random walk) for the distribution function of orientations of the molecules on the surface of the unit sphere. The calculation is the fractional analog of the Debye theory of orientational relaxation in the presence of external and mean field potentials (excluding inertial effects). Exact and approximate (based on the exponential separation for normal diffusion of the time scales of the intrawell and overbarrier relaxation processes associated with the bistable potential) solutions for the dielectric dispersion and absorption spectra are obtained. It is shown that a knowledge of the characteristic 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. 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 (e.g., in glass-forming liquids).

Figure 1

Double-well potential $V_0\left(\vartheta \right)∕kT=-{\xi }_V{\cos }^2\vartheta -\xi \cos \vartheta$ for anisotropy (inverse temperature) parameter ${\xi }_V=2$ and various values of the bias field parameter $\xi$ ($\xi =0$ corresponds to symmetric bistable potential).

Figure 2

Angular frequency of maximum loss of Arrhenius process ${\omega }_{\Vert }$ [solid lines: Eq. (33) with exact values of the smallest nonvanishing eigenvalue ${\lambda }_1^{\Vert }$] versus ${\xi }_V$ (inverse temperature) for (a) asymmetry parameter $h=0.2$ and various values of anomalous exponent $\sigma$ and for (b) $\sigma =0.8$ and various values of $h$. Dashed lines: Eq. (33) with ${\lambda }_1^{\Vert }$ from the high barrier asymptote Eq. (43).

Figure 3

Well angular frequency of maximum loss ${\omega }_W$ versus. ${\xi }_V$ for $h=0.2$ and various values of $\sigma$ [solid lines: Eqs. (33, 34) with exact values of the smallest nonvanishing eigenvalue ${\lambda }_1^{\Vert }$, integral relaxation time ${\tau }_{\mathrm{int}}^{\Vert }$, and effective relaxation time ${\tau }_{\mathrm{ef}}^{\Vert }$].

Figure 4

Real and imaginary parts of the longitudinal complex susceptibility ${\chi }_{\Vert }^{\prime }\left(\omega \right)$ and ${\chi }_{\Vert }^{″}\left(\omega \right)$ versus $\omega \tau$ evaluated from the exact matrix continued fraction solution (solid lines) for $\sigma =0.5$, ${\xi }_V=10$ and various values of $\xi$ and compared with those calculated from the approximate Eq. (35) (filled circles) and with the low (dotted lines) and high (dashed lines) frequency asymptotes Eqs. (17, 18), respectively.

Figure 5

The same as in Fig. 4 for $\sigma =0.5$, $\sigma =2$ and various values of ${\xi }_V$.

Figure 6

Real and imaginary parts of the transverse complex susceptibility ${\chi }_{\perp }^{\prime }\left(\omega \right)$ and ${\chi }_{\perp }^{″}\left(\omega \right)$ versus $\omega \tau$ evaluated from the exact matrix continued fraction solution (solid lines) for $\sigma =0.5$, ${\xi }_V=10$, and various values of $\xi$ and compared with those calculated from the approximate Eq. (35) (filled circles) and with the low (dotted lines) and high (dashed lines) frequency asymptotes Eqs. (17, 18), respectively.

Figure 7

The same as in Fig. 6 for $\sigma =0.5$, $\sigma =2$ and various values of ${\xi }_V$.