Brownian motion, mass transport

The friction coefficient is one of the essential elements in the Langevin description of Brownian motion. The derivation of the Langevin equation from the microscopic equations of motion provides a Green-Kubo expression for this transport coefficient. Its computation entails a number of subtle features. Consider a Brownian (B) particle with mass M in a bath of N solvent molecules with mass m. The generalized Langevin equation for the momentum P of the B... 

Diffusion of small molecular penetrants in polymers often assumes Fickian characteristics at temperatures above Tg of the system. As such, classical diffusion theory is sufficient for describing the mass transport, and a mutual diffusion coefficient can be determined unambiguously by sorption and permeation methods. For a penetrant molecule of a size comparable to that of the monomeric unit of a polymer, diffusion requires cooperative movement of several monomeric units. The mobility of the polymer chains thus controls the rate of diffusion, and factors affecting the chain mobility will also influence the diffusion coefficient. The key factors here are temperature and concentration. Increasing temperature enhances the Brownian motion of the polymer segments the effect is to weaken the interaction between chains and thus increase the interchain distance. A similar effect can be expected upon the addition of a small molecular penetrant. 

Atoms taking part in diffusive transport perform more or less random thermal motions superposed on a drift resulting from field forces (V//,-, Vrj VT, etc.). Since these forces are small on the atomic length scale, kinetic parameters established under equilibrium conditions (i.e., vanishing forces) can be used to describe the atomic drift and transport, The movements of atomic particles under equilibrium conditions are Brownian motions. We can measure them by mean square displacements of tagged atoms (often radioactive isotopes) which are chemically identical but different in mass. If this difference is relatively small, the kinetic behavior is... 

The inhalation airflow comes to a rest in the alveolar region. In still air, the collision of gas molecules with each other results in Brownian motion. The same happens with sufficiently small particles (which can be seen when the dust particles in a nonventilated room are hit by a sunbeam). For very small or ultrafine particles (when the particle size is similar to the mean free path length of the air molecules), the motion is not determined by the flow alone but also by the random walk called diffusion. The diffusion process is always associated with a net mass transport of particles from a region of high particle concentration to regions of lower concentration in accordance with the laws of statistical... 

In the foregoing discussion of the Brownian motion method, the ensemble averages are all constructed from an ensemble of replica systems of the subset of h molecules, the behavior of each replica having been time-smoothed over the interval r. However, in a steady-state transport process dfiN)/dt = 0 at every point in phase space, where f N) is the instantaneous phase density of the N molecules. In principle, at least, it should thus be possible to express the steady-state pressure tensor and the mass and heat currents in terms of ensemble averages constructed without preliminary time-averaging in the replica systems. Thus it is desirable to examine the possibility of obtaining solutions to the reduced Liouville equation, Eq. 8, without preliminary timeaveraging. 

Two things must occur before a particle in suspension in a fluid deposits on a surface to become part of the foulant layer. First the particle has to be transported to the surface by one or a combination of mechanisms including Brownian motion, turbulent diffusion, as described in Chapter 5, or by virtue of the momentum possessed by the particle. The size of the particle will have a large influence on the dominant mechanism. For instance very small particles would be expected to be subject to Brownian diffusion and eddy diffusion whereas the larger particles because of their mass would move under momentum forces. Having arrived at the surface the particle must stick if it is to be regarded as part of the foulant layer residing on the surface. 

Particles suspended in the fluid are carried by the fluid as it flows across the surface. If the fluid is flowing under laminar conditions the transport of the particles across the fluid layers to the surface will be by Brownian motion. Under turbulent conditions particles will be brought to the laminar sub-layer by eddy diffusion, but the remainder of the journey to the surface, across the laminar sublayer is generally ascribed to Brownian motion. Under these conditions for the small particles involved, they may be treated as molecules. In other words mass... 

About the turn of the century cuid shortly thereafter, certain developments in mathematical physics and in physical chemistry were realized which were to prove important in the theory of mass and charge transport in solids, later. Einsteinand Smoluchowski( ) initiated the modern theory of Brownian motion by idealizing it as a problem in random flights. Then some seventeen years or so later, Joffee A proposed that interstitial defects could form inside the lattice of ionic crystals and play a role in electrical conductivity. The first tenable model for ionic conductivity was proposed by Frenkel, who recognized that vaccin-cies and interstitials could form internally to account for ion movement. 

Evanescent wave microscopy has already yielded a number of contributions to the fields of micro-and nanoscale fluid and mass transport, including investigation of the no-slip boundary condition, applications to electrokinetic flows, and verification of hindered Brownian motion. With more experimental data and improvements to TIRE techniques, the accuracy and resolution of these techniques are certain to improve. Areas of potential improvements include development of rmiform-sized and bright tracer particles, creation of high-NA imaging optics and high-sensitivity camera systems, and further development of variable index materials for better control of the penetration-depth characteristics. 

When multiple scattering is discarded from the measured signal, DLS can be used to study the dynamics of concentrated suspensions, in which the Brownian motion of individual particles (self-diffusion) differs from the diffusive mass transport (gradient or collective diffusion), which causes local density fluctuations, and where the diffusion on very short time-scales (r < c lD) deviates from those on large time scales (r c D lones and Pusey 1991 Banchio et al. 2000). These different diffusion coefficients depend on the microstructure of the suspension, i.e. on the particle concentration and on the interparticle forces. For an unknown suspension it is not possible to state a priori which of them is probed by a DLS experiment. For this reason, a further concentration limit must be obeyed when DLS is used for basic characterisation tasks such as particle sizing. As a rule of thumb, such concentration effects vanish below concentrations of 0.01-0.1 vol%, but certainty can only be gained by experiment. 

Brownian motion is a crucial factor in partide assembly. The mass transport from a source, the bulk colloid, to a drain, the pattern, can be accomplished by diffusion. After assembly, the particles have to be trapped to prevent further random motion. 

It has to be remembered that even under turbulent flow conditions there is a laminar sub-layer of slow moving fluid adjacent to the solid surface (see Section 5.2). The transport of material across this "boundary layer" will in general, only be possible by Brownian or molecular motion. The viscous sub-layer represents a resistance not only to heat transfer, but also the transfer of mass. 

In the mid-1800s. Pick [3,4] introduced two differential equations that provide a mathematical framework to describe the otherwise random phenomenon of molecular diffusion. The flow of mass by diffusion across a plane was proportional to the concentration gradient of the diffnsant across it. The components in a mixture are transported by a driving force dnring diffusion. The molecnlar motion is Brownian. The ability of... 

The term hydrodynamic interactions describes the dynamic correlations between the particles, induced by diffusive momentum transport through the solvent. The physical picture is the same, whether the particle motion is Brownian (i.e., driven by thermal noise) or the result of an external force (e.g., sedimentation or electrophoresis). The motion of particle i perturbs the surrounding solvent, and generates a flow. This signal spreads out diffusively, at a rate governed by the kinematic viscosity of the fluid J]kin = tl/p (t] is the solvent shear viscosity and p is its mass density). On interesting (long) time scales, only the transverse hydrodynamic modes [14] remain, and the fluid may be considered as incompressible. The viscous momentum field around a particle diffuses much faster than the particle itself, so that the Schmidt number... 

---