Brownian motion rotational dynamics

We discuss the rotational dynamics of water molecules in terms of the time correlation functions, Ciit) = (P [cos 0 (it)]) (/ = 1, 2), where Pi is the /th Legendre polynomial, cos 0 (it) = U (0) U (it), u [, Is a unit vector along the water dipole (HOH bisector), and U2 is a unit vector along an OH bond. Infrared spectroscopy probes Ci(it), and deuterium NMR probes According to the Debye model (Brownian rotational motion), both... 

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>. 

Applications of optical methods to study dilute colloidal dispersions subject to flow were pioneered by Mason and coworkers. These authors used simple turbidity measurements to follow the orientation dynamics of ellipsoidal particles during transient shear flow experiments [175,176], In addition, the superposition of shear and electric fields were studied. The goal of this work was to verify the predictions of theories predicting the orientation distributions of prolate and oblate particles, such as that discussed in section 7.2.I.2. This simple technique clearly demonstrated the phenomena of particle rotations within Jeffery orbits, as well as the effects of Brownian motion and particle size distributions. The method employed a parallel plate flow cell with the light sent down the velocity gradient axis. 

Here (Oe and co are delivered by the corresponding Langevin equations of the theory of the rotational Brownian motion. In order to obtain these equations, one must include in the dynamic equations (4.308) and (4.310) the random thermal torques. We do that in the following way ... 

Brownian motion effects are weak, then we may transit from the statistical description to the dynamical one. In such a situation the particle rotation is determined simply by the balance of viscous and field-induced torques and thus is governed by the equation... 

Classic Brownian motion has been widely applied in the past to the interpretation of experiments sensitive to rotational dynamics. ESR and NMR measurements of T and Tj for small paramagnetic probes have been interpreted on the basis of a simple Debye model, in which the rotating solute is considered a rigid Brownian rotator, sueh that the time scale of the rotational motion is much slower than that of the angular momentum relaxation and of any other degree of freedom in the liquid system. It is usually accepted that a fairly accurate description of the molecular dynamics is given by a Smoluchowski equation (or the equivalent Langevin equation), that can be solved analytically in the absence of external mean potentials. 

One can hope that these will not greatly affect comparisons of macroscopic relaxation functions rather than microscopic functions and that better treatments as from Fulton s methods for example will clarify these questions. Even so, It seems fair to claim that a better basis now exists for extracting useful information from Kerr effect measurements and to explore questions of whether rotational reorientations in time are diffusion or Brownian motion like at one extreme infrequently by large jumps at the other or something in between. With developments in instrumentation of the sort suggested above there appear to be real possibilities for studies of dynamics of simpler molecules to complement those by other methods. 

Dynamic light scattering (also known as photon correlation spectroscopy) is a measurement technique sensitive to the motions of particles in solution. As we learned previously in this chapter, all suspended particles in a colloidal solution are constantly subject to Brownian motion. Bombardment by solvent molecules leads to rotational, translational, and even more complicated conformational motions (in structured molecules such as polymer chains or proteins, for example). If we probe the solution with visible light, this constant particle motion will result in a time-varying fluctuation of the scattered intensity 7(0). Dynamic... 

The function / is called the orientation distribution function. Changes in this function are governed by a dynamic conservation equation. It takes into account contributions to the rotational flux from Brownian motion (jj) and from the hydrodynamic convection (j/,) (Brenner 1972 Hinch and Leal 1972, Schowalter 1978 Bird et al., 1987)... 

Before discussing theoretical models for the rheology of fiber suspensions and its connection to fiber orientation, there are three topics that must be discussed Brownian motion, concentration regimes, and fiber flexibility. Brownian motion refers to the random movement of any sufficiently small particle as a result of the momentum transfer from suspending medium molecules. The relative effect that Brownian motion may have on orientation of anisotropic particles in a dynamic system can be estimated using the rotary Peclet number, Pe s y Dm, where y is the shear rate and Ao is the rotary diffusivity, which defines the ratio of the thermal energy in the system to the resistance to rotation. Doi and Edwards (1988) estimated the rotary diffusivity, Ao, to be... 

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=. ... 

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