Brownian motion rotational diffusivity

The techniques described under Photon Correlation exploit the correlation between the photons emitted by single molecules and by a small number of molecules. Picosecond photon correlation techniques investigate effects driven by the absorption of a single photon of the excitation light. The effects investigated by FCS are driven by Brownian motion, rotation, diffusion effects, intersystem crossing, or conformational changes. Because of these random and essentially sample-internal stimulation mechanisms, correlation techniques do not necessarily depend on a pulsed laser. 

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

To work out the time-dependence requires a specific model for the movement of the paramagnet, for example, Brownian motion, or lateral diffusion in a membrane, or axial rotation on a protein, or jumping between two conformers, etc. That theory is beyond the scope of this book the math can become quite hairy and can easily fill another book or two. We limit the treatment here to a few simple approximations that are frequently used in practice. 

Molecular motions in low molecular weight molecules are rather complex, involving different types of motion such as rotational diffusion (isotropic or anisotropic torsional oscillations or reorientations), translational diffusion and random Brownian motion. The basic NMR theory concerning relaxation phenomena (spin-spin and spin-lattice relaxation times) and molecular dynamics, was derived assuming Brownian motion by Bloembergen, Purcell and Pound (BPP theory) 46). This theory was later modified by Solomon 46) and Kubo and Tomita48 an additional theory for spin-lattice relaxation times in the rotating frame was also developed 49>. 

In addition to translational Brownian motion, suspended molecules or particles undergo random rotational motion about their axes, so that, in the absence of aligning forces, they are in a state of random orientation. Rotary diffusion coefficients can be defined (ellipsoids of revolution have two such coefficients representing rotation about each principal axis) which depend on the size and shape of the molecules or particles in question28. 

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

How rapidly diffusion occurs is characterized by the diffusion coefficient D, a parameter that provides a measure of the mean of the squared displacement x of a molecule per unit time f. For diffusion in two dimensions such as a membrane, this is given by = 4Ht. The Saffman-Delbrtlck model of Brownian motion in biologic membranes describes the relationship between membrane viscosity, solvent viscosity, the radius R and height of the diffusing species, and D for both lateral and rotational diffusion of proteins in membranes (3, 4). This model predicts for example that for lateral diffusion, D should be relatively insensitive to the radius of the diffusing species, scaling with log (1/R). 

For solutions of nonspherical particles the situation is more complicated and the physical picture can be described qualitatively as follows for a system of particles in a fluid one can define a distribution function, F (Peterlin, 1938), which specifies the relative number of particles with their axes pointed in a particular direction. Under the influence of an applied shearing stress a gradient of the distribution function, dFfdt, is set up and the particles tend to rotate at rates which depend upon their orientation, so that they remain longer with their major axes in position parallel to the flow than perpendicular to it. This preferred orientation is however opposed by the rotary Brownian motion of the particles which tends to level out the distribution or orientations and lead the particles back toward a more random distribution. The intensity of the Brownian motion can be characterized by a rotary diffusion coefficient 0. Mathematically one can write for a laminar, steady-state flow ... 

Four different models for the molecular dynamics have been tested to simulate the experimental spectra. Brownian rotational diffusion and jump type diffusion [134, 135] have been used for this analysis, both in their pure forms and in two mixed models. Brownian rotational diffusion is characterized by the rotational diffusion constant D and jump type motion by a residence time t. The motions have been assumed to be isotropic. In the moderate jump model [135], both Brownian and jump type contributions to the motion are eou-pled via the condition Dx=. ... 

Above 100 K, motional effects on spectrum become pronounced with increasing temperature and, above 230 K, the spectra consist of essentially an isotropic and equally spaced hyperfine triplet, but with different relative intensities. The line shape simulations were carried out by adopting a Brownian rotational diffusion model in order to evaluate the associated (average) rotational correlation time, and its degree of anisotropy, = zpy, /... 

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker-Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

In order to describe the fractional rotational diffusion, we use the FKKE for the evolution of the probability density function W in configuration angular-velocity space for linear molecules in the same form as for fixed-axis rotators—that is, the form of the FKKE suggested by Barkai and Silbey [30] for one-dimensional translational Brownian motion. For rotators in space, the FKKE becomes... 

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