Brownian movement 267 rotation

For many purposes, it is more convenient to characterize the rotary Brownian movement by another quantity, the relaxation time t. We may imagine the molecules oriented by an external force so that the a axes are all parallel to the x axis (which is fixed in space). If this force is suddenly removed, the Brownian movement leads to their disorientation. The position of any molecule after an interval of time may be characterized by the cosine of the angle between its a axis and the x axis. (The molecule is now considered to be free to turn in any direction in space —its motion is not confined to a single plane, but instead may have components about both the b and c axes.) When the mean value of cosine for the entire system of molecules has fallen to ile(e — 2.718... is the base of natural logarithmus), the elapsed time is defined as the relaxation time r, for motion of the a axis. The relaxation time is greater, the greater the resistance of the medium to rotation of the molecule about this axis, and it is found that a simple reciprocal relation exists between the three relaxation times, Tj, for rotation of each of the axes, and the corresponding rotary diffusion constants defined in equation (i[Pg.138]

We study, following Lewis et al. [77], the rotational Brownian movement of a sphere which is supposed homogeneous, the motion being due... 

Shear-Sensitive Systems. In addition to hydrodynamic effects and simple viscous behavior, the act of pigmentation creates a certain amount of complex behavior (13). If the particles are fine. Brownian movement (14-17) and rotational diffusion (14. 18. 19) are among the phenomena that cause dispersed systems to display complex rheology. The role of van der Waals forces in inducing flocculation (20) and the countervailing role of two electroviscous effects (17. 21. 22) in imparting stability, particularly in aqueous systems, have been noted. Steric repulsions appear to be the responsible factor in nonaqueous systems (23. 24). The adsorbed layer can be quite large (25-28). as detected by diffusion and density measurements of filled systems or by viscometry and normal stress differences (29). 

Because of the symmetry of the sphere, no coupling occurs between the translational and rotational Brownian movements, leading to the further relation... 

Important extensions of kinetic theories to solutions followed the study of the movement of small suspended particles in a liquid, first carefully investigated by R. Brown (1828), and now called the Brownian movement. What seem to have been Brownian movements were noticed by Dutrochet but regarded as optical illusions. Brown found that particles suspended in a liquid in pollen cells are in tremulous motion . He also observed the movement with fine suspended particles of gamboge and a large number of inorganic bodies and noticed the rotation of some of the particles. He seems to have thought that the motion is due to convection currents. 

The kinetic theory of the Brownian movement was developed by Einstein and Smoluchowski, giving equations for the mean square of the displacement in a given direction in a given time, and for the mean square of the angular velocity of rotation. These were verified by experiments. F. M. Exner had previously shown that the square of the velocity of the particles is approximately proportional to the temperature. By counting the number of particles in a given volume microscopically (or ultramicroscopically) it was found that... 

Perrin, F., Brownian movement of an ellipsoid. 11. Free rotation and depolarization of fluorescence. Translation and diffusion of ellipsoidal molecules, /. Phys. Radium, 7,1,1936. 

A large molecule such as a protein, however, will have undergone on the average only a relatively small rotation due to its Brownian movement... 

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. 

Under the action of an applied acoustic field the suggestion was that there would be regions within the polymer where rotation (and vibration) of individual segments were able to take place freely, in phase with the rapid oscillatory movement of the solvent. This segmental movement (termed micro Brovmian motion) was in addition to the movement of the macromolecule as a whole (macro Brownian motion). However, in that segmental motion is a cooperative effect and depends upon the interaction... 

Not only do particles in Brownian agitation move rapidly about in the suspension medium, the magnitude of the movements being capable of exact calculation from the foregoing mathematical considerations, but they are likewise undergoing rotational motion due to an unequal distribution of molecular impacts upon the faces of the parts of a particle on each side of its axis of rotation. 

Normal Brownian motion is a result of solvent molecules impacting on the solute particles, and these give both translational and rotational movement to the solute. An ion on its own will execute this Brownian motion. Since an ion has a charge which can interact with an external electric field, this interaction will perturb the translational Brownian motion, with a cation moving in the direction of the field while an anion will move in the opposite direction. The field will have a minor effect on the rotational Brownian motion, but this will not contribute to the translational mobility. 

Having now discussed hydrated volume, molecular shape and the frictional forces that oppose rotational and translational motion, we are now ready to discuss diffusion, the complete set of processes (including Brownian motion) that together bring about the bulk movement of biological macromolecules from one place to another in aqueous buffer solution. The processes that comprise diffusion are quantified by means of the concept of flux. Flux is defined as... 

P NMR was used to study motion of tri-(2-ethylhexyl) phosphate, TOP, in polycarbonate at different temperatures and concentrations. Brownian rotational motion was observed in TOP but at two different time scales. If TOP was surrounded by other molecules of plasticizer it was capable of rotational diffusion with apparent activation energy of 56 kJ mol". Isolated TOP molecules (surrounded by polymer) showed a temperature dependent movement. These molecules do not diffuse below glass the transition tempera-... 

We have seen that molecules in solution show translational movement caused by the Brownian motion of the solvent. In addition to this translational movement, each solute molecule rotates relative to its centre of mass. This motion is known as rotational diffusion and is described in terms of rotational diflusion coefficient . It has the units of reciprocal seconds and expresses,. 

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