Anisotropic rotational diffusion and dielectric relaxation of rigid dipolar particles in a strong external dc field

Dielectric response functions of polar particles (macromolecules) diluted in a nonpolar solvent subjected to a strong external dc electric field are evaluated using the anisotropic noninertial rotational diffusion model. Simple analytic formulas for the longitudinal and transverse components of the dielectric susceptibility and relaxation times are given using the effective relaxation time method. These formulas are tested against numerical solutions of the underlying infinite hierarchy of differential-recurrence equations for statistical moments (ensemble averages of the Wigner $D$ functions) which are obtained by averaging the governing Langevin equation for noninertial rotational Brownian motion over its realizations. The calculations, involving matrix continued fractions, ultimately yield the exact solution of the infinite hierarchy of differential-recurrence relations for the dielectric response functions. In the isotropic rotational diffusion limit, the solution reduces to the known results.

Figure 1

Geometry of the problem.

Figure 2

(Color online) Three-dimensional plots of the effective relaxation times ${\tau }_{\mathrm{eff}}^{\parallel }$, Eq. (23), in comparison with the integral relaxation time ${\tau }_{\mathrm{int}}^{\parallel }$, Eq. (4), vs the field parameter $\xi$ and diffusion anisotropy parameter $\Delta$ ($u_x=0.7$, $u_y=0$, and $\Xi =0$).

Figure 3

(Color online) Three-dimensional plots of ${\tau }_{\mathrm{eff}}^{\parallel }$ and ${\tau }_{\mathrm{int}}^{\parallel }$ vs the field parameter $\xi$ and diffusion anisotropy parameter $\Xi$ ($u_x=0.7$, $u_y=0$, and $\Delta =2$).

Figure 4

(Color online) Real and imaginary parts of the normalized complex susceptibility vs the normalized frequency $\omega {\tau }_D$ for various values of the field parameter $\xi =0$ (free diffusion), 1, 2, and 5 ($u_x=0.4$, $u_y=0.6$, $\Delta =7$, and $\Xi =-1∕2$). Solid lines: matrix continued fraction solution. Stars: the effective relaxation time solution, Eq. (22). Dashed and dotted straight lines: the low- [Eq. (5)] and high- [Eq. (6)] frequency asymptotes, respectively. Dash-dotted lines: isotropic diffusion $(\Delta =\Xi =0)$.

Figure 5

(Color online) Real and imaginary parts of the normalized complex susceptibility vs $\omega {\tau }_D$ for various values of the anisotropy parameter $\Delta =1$, 2, and 5 ($\xi =2$, $u_x=0.4$, $u_y=0.6$, and $\Xi =-1∕2$). Solid lines: matrix continued fraction solution. Stars: the effective relaxation time solution, Eq. (22). Dashed and dotted straight lines: the low- [Eq. (5)] and high- [Eq. (6)] frequency asymptotes, respectively. Dash-dotted lines: isotropic diffusion $(\Delta =\Xi =0)$.

Figure 6

(Color online) Real and imaginary parts of the normalized complex susceptibility vs $\omega {\tau }_D$ for various values of the anisotropy parameter $\Xi =1$, 0, and $-1$ ($\xi =2$, $u_x=0.7$, $u_y=0.0$, and $\Delta =1$). Solid lines: matrix continued fraction solution. Dash-dotted lines: isotropic diffusion $(\Delta =\Xi =0)$.