Brownian diffusion translational

Translational motion is the change in location of the entire molecule in three-dimensional space. Figure 11 illustrates the translational motion of a few water molecules. Translational motion is also referred to as self-diffusion or Brownian motion. Translational diffusion of a molecule can be described by a random walk, in which x is the net distance traveled by the molecule in time At (Figure 12). The mean-square displacement (x2) covered by a molecule in a given direction follows the Einstein-derived relationship (Eisenberg and Crothers, 1979) ... 

Fig. 6.12. A Typical CARS signal trajectory revealing the particle number fluctuations of 110-nm polystyrene spheres undergoing free Brownian diffusion in water. The epi-detected CARS contrast arises from the breathing vibration of the benzene rings at 1003cm 1. B Measured CARS intensity autocorrelation function for an aqueous suspension of 200-nm polystyrene spheres at a Raman shift of 3050 cm-1 where aromatic C-H stretch vibrations reside. The corresponding translational diffusion time, td, of 20 ms is indicated. (Panel B courtesy of Andreas Zumbusch, adapted from [162])...

Brownian diffusion and enter another tube. If it repeats this action of small angular motions followed by translation into new tubes, it can work its way around a large circle of radius R. Through simple geometric arguments, this radius is R = 2 (L/ac) L. Since the motion... 

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. 

At one extreme, a suspension may be considered dilute if the thermal motion (Brownian diffusion) of the particles predominate over the imposed interparticle interaction [30-32]. In this case, the particle translational motion is large and only occasional contacts will occur between the particles that is, the particles do not see each other until a collision occurs, giving a random arrangement of particles. 

Hint Rotational Brownian diffusivity is the manifestation of random walks of the orientation of the rod. By analogy with translational diffusion, the rotational diffusivity D,- = kTMaG, where M eis the mobility tensor relating angular velocity and torque. 

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... 

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. 

When the reciprocal of the scattering vector, S, is large in comparison to the size of the polymer coil, the DLS fluctuation spectrum is due to macromolecular coil center of mass motions. These motions are generated by translation (Brownian) diffusion and/or any bulk fluid flow of the solution. Unwanted scattering can be produced by slight thermal differences or vibration in the solution which can cause bulk fluid flow. 

The DLS technique involves measurement of the Doppler broadening of the Rayleigh-scattered light as a result of Brownian motion (translational diffusion) of the particles. This thermal motion causes time fluctuations in the scattering intensity and a broadening of the Rayleigh line. The Rayleigh line has a Lorentzian line shape. In macromolecular solutions, concentration... 

The standard model for diffusive motion in polymers is Brownian diffusion, which occurs as a series of infinitesimal reorientational steps. This model is most appropriate for intermediate-to-large sized spin probes and spin-labeled macromolecules, where the macromolecule is much larger than any solvent molecules. Because of this broad applicability, the Brownian diffusion model is the most widely used. This type of rotational diffusion is completely analogous to the one-dimensional random walk used to describe translational diffusion in standard physical chanistry texts, with the difference that the steps are described in terms of a small rotational step 59 that can occur in either the positive or negative direction. In three dimensions, rotations about each of three principal axes of the nitroxide must be taken into account. A diffusion constant may be defined for each of these rotations motions, in a way that is completely analogous to the definition of translational diffusion constant for the one-dimensional random walk. 

In magnetic suspensions, in addition to the thermal fluctuations of the particle moment, the particle itself moves with respect to the liquid carrier (translational and rotational Brownian diffusion). This leads to the change of the orientational distribution of the particle anisotropy axes under the influence of the applied field. 

The continuous and erratic motion of individual particles (i.e., pollen grains) as the result of random collisions with the adjoining molecules of fluid (water) was observed by the botanist Robert Brown in 1827. The term Brownian diffusion is used for colloidal particles to distinguish it from solute molecular diffusion. However, both are end members of a continuum of particle sizes and a fundamental consequence of kinetic theory is that all particles have the same average translational kinetic energy. The average particle velocity increases with decreasing mass (Shaw, 1978). The Stokes-Einstein s equation for particle diffusivity is based on this concept. It is... 

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. 

Further support for this approach is provided by modern computer studies of molecular dynamics, which show that much smaller translations than the average inter-nuclear distance play an important role in liquid state atom movement. These observations have conhrmed Swalin s approach to liquid state diffusion as being very similar to the calculation of the Brownian motion of suspended particles in a liquid. The classical analysis for this phenomenon was based on the assumption that the resistance to movement of suspended particles in a liquid could be calculated by using the viscosity as the frictional force in the Stokes equation... 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

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 ... 

FIG. 12 Translational diffusion (also called Brownian motion) of a water molecule can be described by a random walk starting at t = 0 and ending at t = At, where x is the net distance traveled during At and t is time. 

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... 

The last point to be made is the famous Stokes-Einstein relationship that was found by Einstein by comparing the Brownian motion with common diffusion processes [66,67]. Accordingly the translational diffusion was found to depend... 

Diffusion is defined as the random translational motion of molecules or ions that is driven by internal thermal energy - the so-called Brownian motion. The mean movement of a water molecule due to diffusion amounts to several tenth of micrometres during 100 ms. Magnetic resonance is capable of monitoring the diffusion processes of molecules and therefore reveals information about microscopic tissue compartments and structural anisotropy. Especially in stroke patients diffusion sensitive imaging has been reported to be a powerful tool for an improved characterization of ischemic tissue. 

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>. 

However, in reality, all macromolecules in solution are undergoing constant Brownian motion, and this fact leads to fluctuation in I(t). The fluctuation rate can be related to the translational diffusion of the macromolecules. The faster the diffusion, the faster the fluctuation will be. 

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. 

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