Translational Brownian diffusion coefficient

To determine the value of the diffusivity that connects the two approaches, we follow Einstein s thermodynamic arguments given in Section 5.2 for evaluating the translational Brownian diffusion coefficient. The basis for this is the random Brownian motion of the monomer units in the gel, which translates into the gel osmotic pressure. If, as above, the flow through the gel is assumed to follow Darcy s law (Eq. 4.7.7), then we may write the applied hydrodynamic force per mole of solution flowing through the gel as... 

For example, in the case of PS and applying the Smoluchowski equation [333], it is possible to estimate the precipitation time, fpr, of globules of radius R and translation diffusion coefficient D in solutions of polymer concentration cp (the number of chains per unit volume) [334]. Assuming a standard diffusion-limited aggregation process, two globules merge every time they collide in the course of Brownian motion. Thus, one can write Eq. 2 ... 

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

For a single fluorescent species undergoing Brownian motion with a translational diffusion coefficient Dt (see Chapter 8, Section 8.1), the autocorrelation function, in the case of Gaussian intensity distribution in the x, y plane and infinite dimension in the z-direction, is given by... 

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. 

The r-average translational diffusion coefficient l> is calculated from the equation Dj = V/q2. For a collection of identical spheres undergoing ordinary Brownian motion in solution. 

Because the assumption of simple Brownian diffusion breaks down, the diffusion in biomembranes cannot be described by a single diffusion coefficient. For instance, FRAP experiments in the plasma membrane showed that the observed translational diffusion rates depend on the size of the initial photobleached spot, which is also inconsistent with a simple Singer-Nicolson model. 

Generally, mean size and size distribution of nanoparticles are evaluated by quasi-elastic light scattering also named photocorrelation spectroscopy. This method is based on the evaluation of the translation diffusion coefficient, D, characterizing the Brownian motion of the nanoparticles. The nanoparticle hydro-dynamic diameter, is then deduced from this parameter from the Stokes Einstein law. 

The difference between elastic and "quasielastic" measurements is that in the latter, small changes in the frequency due to the translational ("Brownian") movement of the scattering particles are also measured. The broadness of the intensity distribution of the emitted light for frequencies around the primary monocluomatic beam frequency is directly related to the diffusion coefficient of the particles, which can then be related to the hydrodynamic radius if a model for the particle shape is available Dynamic light scattering can thus be used to follow the kinetics of particle coagulation by following the decrease in diffusion coefficient as the particle size increases. ... 

As seen from Ref 9, data for the rotational diffusion coefficient also shows a far milder dependence wit temperature than its translational counterpart. However the observed decoupling from the SE and SED behaviors is there seen to follow a far less drastic behaviour than that here found. On such a basis we deem that the presence of a strong directional interaction cannot account for the higher mobility observed by experiment if compared to the Brownian dynamics estimates of SE and SED. Finally it is worth to emphasize that the observed breakdown of both SE and SED approximations only appear for the miscible phase below Tl but not after the re-entrance above Tu into the high temperature, miscible phase. Such fact is thus suggestive of the existence of phenomena additional to those responsible of the re-entrant behavior being... 

Rotational Brownian motion results in the disordering of anisometric particles previously oriented in some particular way owing to the flow of the dispersion medium (see Chapter IX) or the application of an electric field. From the time of the disordering one can determine the rotational diffusion coefficient, and, for known particle size and shape, also Avogadro s number. If the particles are able to undergo co-orientation, they usually are of substantially anisometric shape, and their translational and rotational diffusion coefficients differ from those obtained for spherical particles. For example, for prolate ellipsoids of revolution with a ratio of their main diameters d] d2-= 1 10, the diffusion coefficient, D, is about 2/3 of the value obtained for spherical particles of the same volume. 

One of the important applications of Stokes law occurs in the theory of Brownian motion. According to Einstein (El a) the translational and rotary diffusion coefficients for a spherical particle of radius a diffusing in a medium of viscosity n are, respectively,... 

Thus the coefficient of Brownian diffusion of particles with small volume concentration W, suspended in a liquid that is at rest or undergoing translational motion with a constant velocity, has a constant value and is identical in all directions. 

Bulk polymer properties such as viscosity and elasticity are concerned with averaged responses of an assembly of polymer chains to external stimuli. On the other hand, the self-diffusion coefficient has something to do with the average speed of translation of the centers-of-mass of individual chains. Thus its study should give us a clue to the clarification of the modes of Brownian motion of a single chain on long timescales. This expectation must have been in the mind of polymer workers for many years, but, except in dilute solutions, few measurements of Ds were undertaken until recently, probably on the one hand because of experimental difficulties and on the other because of the lack of an adequate guiding theory. 

In quiescent, dilute suspensions, the light fluctuations result essentially from the Brownian displacement of the single particles. Thus, the decay rate F can be traced back to the particles translational diffusion coefficient D, ... 

Diffusion. The translational diffusion coefficient D is the most commonly measured transport property of polymer solutions, but as there are several distinct types of diffusion, care must be taken to interpret D properly. For c < c, Brownian motion of isolated chains in a homogeneous solvent defines the dilute solution diffusion coefficient Dq. As c increases toward c and above, chain-chain interactions modify the friction felt during chain motion. Under these conditions, the tracer- or self-diffusion coefficient Dtr is measured by tracking the path of a single chain in a macroscopically imiform mixture of chains and solvent. To distinguish the test chain from neighbors so that its path can be identified, the chain... 

Brownian motion of particles in solution gives rise to a spectral distribution in the scattered light. From measurements on the line width of scattered laser light by photon correlation spectroscopy (PCS) the translational diffusion coefficient may be determined. Unlike the traditional method of investigating translational diffusion, PCS does not require a macroscopic concentration gradient and therefore can more readily be applied to investigate association processes. The method has been used to determine the translational diffusion coefficients, of micelles formed by a number of block copolymers in selectively bad solvents. ... 

The entire description of the translational diffusion of a polymer molecule in solution requires the determination of the mutual or cooperative diffusion coefficient, D, which characterizes the relaxation of a concentration gradient and the self-diffusion coefficient, Ds, which describes the Brownian or random motions of the polymer molecule (860-863). 

A comparison of rotational and translational diffusion results obtained in l-octyl-3-imidazolium tetrafluoroborate, [omim][BF4], and in 1-propanol and isopropyl benzene has been given for TEMPONE. Measurements at different temperatures and concentrations indicate that rotational motion can be described by isotropic Brownian diffusion only for the classical organic solvents used, but not for the IL. Simulation of the EPR spectra fit with the assumption of different rotational motion around the different molecular axes. Rotational diffusion coefficients >rot follow the Debye-Stokes-Einstein law in all three solvents, whereas the translational diffusion coefficients do not follow the linear Stokes-Einstein relation D ot versus Tlr ). The activation energy for rotational motions Ea,rot in [omim][BF4] is higher than the corresponding activation energies in the organic solvents. 

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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