Translational Brownian motion

M. Tokuyama and I. Oppenheim, Statistical-mechanical theory of Brownian motion— translational motion in an equilibrium fluid, Physica A 94, 501 (1978). 

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) ... 

Rodlike polymers do two kinds of Brownian motion, translation and rotation. The translational Brownian motion is the random motion of the position vector R of the centre of mass, and the rotational Brownian motion is the random motion of the unit vector u which is parallel to the polymer. 

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... 

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 relationships follow from considerations of Brownian motions and thermal fiuctuations which also infiuence the internal motions in flexible objects. is the translation coefficient of the particle s center of mass where the subscript indicates the z-average over the molar mass distribution. The first bracket in Eq. (12) describes the concentration dependence which often is well represented by a linear dependence... 

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. 

Particles subject to Brownian motion tend to adopt random orientations, and hence do not follow these rules. A particle without these symmetry properties may follow a spiral trajectory, and may also rotate or wobble. In general, the drag and torque on an arbitrary particle translating and rotating in an unbounded quiescent fluid are determined by three second-order tensors which depend on the shape of the body ... 

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>. 

During the first diffusional step in which the molecule executes a kind of translational and rotational Brownian motion in the soft-fluctuating force field of its neighbors, its direction cosines are represented as... 

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. 

In solution things are more complex. The reaction partners are no longer free in their translational motion as they are in the gas phase they have to move in a condensed medium, and their motion is governed by other physical phenomena which for economy of exposition we shall not consider in detail. It is sufficient to recall that the physical models for the most important terms, Brownian motions, diffusion forces, are expressed in their basic form using a continuum description of the medium. 

The incorporation of Brownian motion can be effected in a manner similar to that of Batchelor (1976b), who found the translational diffusion flux due to Brownian motion to be equivalent to one produced by steady forces acting on the particles. In this context, the force exerted on particle i in an ensemble of N particles is taken to be... 

Small particles suspended in a gas undergo random translational motion because they are being buffeted by collisions with swiftly moving gas molecules. This motion appears almost as a vibration of the ensemble of particles, although there is a net displacement with time of any given particle. Observation of this motion in a liquid was first made in 1828 by the British naturalist Robert Brown (1828), and the phenomenon thus has been called brownian motion (also known as brownian movement). Bodaszewski (1883) studied the brownian motion of smoke particles and other suspensions in air and likened these movements to the movements of gas molecules as postulated by the kinetic theory. The principles governing brownian motion are the same, whether the particles are suspended in a gas or in a liquid. 

One consequence of kinetic theory is that particles will have the same average translational energy as molecules when the gas is in equilibrium. Thus it is possible to compute the average velocity of a particle as it moves in brownian motion. Denoting this velocity as v0,... 

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 random fluctuations in the positions of particles in space are often translational, the kinetics of these processes can be considered comparable to the decay of a concentration gradient by translational Brownian motion. Likewise, as the orientation of any molecule undergoes similar random fluctuations in space,... 

---