Adiabatic change in the Smoluchowski equation: Orientational diffusion of polar particles

Spatio-angular fluorescence microscopy III. Constrained angular diffusion, polarized excitation, and high-NA imaging

We investigate rotational diffusion of fluorescent molecules in angular potential wells, the excitation and subsequent emissions from these diffusing molecules, and the imaging of these emissions with high-NA aplanatic optical microscopes. Although dipole emissions only transmit six low-frequency angular components, we show that angular structured illumination can alias higher-frequency angular components into the passband of the imaging system. We show that the number of measurable angular components is limited by the relationships between three time scales: the rotational diffusion time, the fluorescence decay time, and the acquisition time. We demonstrate our model by simulating a numerical phantom in the limits of fast angular diffusion, slow angular diffusion, and weak potentials.

Vibrational Sum Frequency Generation Spectroscopy Measurement of the Rotational Barrier of Methyl Groups on Methyl-Terminated Silicon(111) Surfaces

The methyl-terminated Si(111) surface possesses a 3-fold in-plane symmetry, with the methyl groups oriented perpendicular to the substrate. The propeller-like rotation of the methyl groups is hindered at room temperature and proceeds via 120° jumps between three isoenergetic minima in registry with the crystalline Si substrate. We have used line-shape analysis of polarization-selected vibrational sum frequency generation spectroscopy to determine the rotational relaxation rate of the surface methyl groups and have measured the temperature dependence of the relaxation rate between 20 and 120 °C. By fitting the measured rate to an Arrhenius dependence, we extracted an activation energy (the rotational barrier) of 830 ± 360 cm^-1 and an attempt frequency of (2.9 ± 4.2) × 10¹³ s^-1 for the methyl rotation process. Comparison with the harmonic frequency of a methyl group in a 3-fold cosine potential suggests that the hindered rotation occurs via uncorrelated jumps of single methyl groups rather than concerted gear-like rotation.

Restricted diffusion of methyl groups in proteins revealed by deuteron NMR: manifestation of intra-well dynamics

The three-site hops of methyl groups are usually used as an approximation of the mechanistic description of motions responsible for the longitudinal NMR relaxation. Distinguishing between three-site hops and a more realistic mechanism of diffusion in a potential requires extended experimental and computational analysis. In order to achieve this goal, in this work the restricted diffusion is decomposed into two independent modes, namely, the jumps between potential wells and intra-well fluctuations, assuming time scale separation between these modes. This approach allows us to explain the rise in the theoretical value of T1 minimum for the restricted diffusion mechanism compared with the three-site hops mechanism via rescaling the three-site hops correlation function by the order parameter of intra-well motions. The main result of the paper is that, in general, intra-well dynamics can be visible in NMR even in the limit of large barrier heights in contrast to the common view that this limit converges to the three-site hops mechanism. Based on a previously collected detailed set of deuteron NMR relaxation and spectral data in the villin headpiece subdomain protein over a wide temperature range of 300-31 K, we are then able to conclude that the mechanism of diffusion in the threefold potential is likely to be the main source of the dynamics in this system.

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.

Rotational diffusion of colloidal particles near confining walls

We study the rotational diffusion of a spherical colloid confined in a narrow channel between parallel plane hard walls. The walls damp translational diffusion much more than rotational diffusion so that there is expected to be little translation-rotation coupling. Using a recent calculation of the nonisotropic rotational mobilities arising from the hydrodynamic interactions with the walls, we set up the rotational Smoluchowski equation for either a particle with a permanent dipole moment or a polarizable particle with axisymmetric polarizabilities subject to an external electric field. Using the Smoluchowski equation dynamics we calculate the time-correlation functions of orientation that are measured in depolarized light scattering for the cases of no external field, external field normal to the walls, and external field parallel to the walls. The decay of correlations is shown to be given by a weighted sum of decaying exponentials and can be characterized by an initial and a mean characteristic decay time. The weights and decay rates of each component and the characteristic decay times are studied numerically for a range of field strengths. The nonisotropic rotational mobilities make these decay times highly sensitive to the distance of the particle from the confining walls. This position dependence can be used as a method of measuring the rotational mobilities or, conversely, the rate of decay of correlations can be used as a probe of particle position between the confining walls.

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.