Hydrodynamic diffusion coefficients

Once the mechanical dispersion associated with water is accounted for, the dispersion coefficient, D, is a function of two components—molecular ( ) ) and mechanical (Dj) dispersion. D can be represented as follows  

Equation (3.4.34) can be altered by making assumptions as to the influence of molecular diffusion and rate of water flow. For example, if the molecular diffusion is negligible and the dispersion carries with the pore-water velocity, 

The values for w, n, and all depend on the soils and their water contents (Scott 2000). 

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The simple form in Eq. (7) can be maintained by replacing the Brownian diffusion coefficient in the expression kc = /-An /5 by the shear-induced hydrodynamic diffusion coefficient for the particles, Ds. Shear-induced hydrodynamic diffusion of particles is driven by random displacements from the streamlines in a shear flow as the particles interact with each other. For particle volume fractions between 20 and 45%, Ds has been related to... 

The fast mode slowly decreases when the salt concentration increases. Since the added salt screens the electrostatic interaction, Dfeff tends in excess of salt to the hydrodynamic diffusion coefficient (D-point in Figure 15). 

The Stokes law or hydrodynamic diffusion coefficient should then be observed. Thus at high salt concentration, the effects of ionic shielding discussed in Section 9.2 are unimportant, and the diffusion coefficient reduces to that of the isolated macroion. Physically this occurs because now a fluctuation in the counterions can be neutralized by a concentration fluctuation in ions of opposite charge, so that large electrical forces need not be exerted on the macroion. 

In dilute solutions, tire dependence of tire diffusion coefficient on tire molecular weight is different from tliat found in melts, eitlier entangled or not. This difference is due to tire presence of hydrodynamic interactions among tire solvent molecules. Such interactions arise from tire necessity to transfer solvent molecules from tire front to tire back of a moving particle. The motion of tire solvent gives rise to a flow field which couples all molecules over a... 

Thus one must rely on macroscopic theories and empirical adjustments for the determination of potentials of mean force. Such empirical adjustments use free energy data as solubilities, partition coefficients, virial coefficients, phase diagrams, etc., while the frictional terms are derived from diffusion coefficients and macroscopic theories for hydrodynamic interactions. In this whole field of enquiry progress is slow and much work (and thought ) will be needed in the future. 

The scaling dependence of the diffusion coefficient on N and Cobs Iso poses a number of questions. While the original scaling predictions, based on reptation dynamics [26,38], oc N, have been verified by some measurements [91,98], significant discrepancies have been reported too [95,96]. Attempts to interpret existing data in terms of alternative models, e.g., by the so-called hydrodynamic scaling model [96], fail to describe observations [100,101]. 

From various studies" " it is becoming clear that in spite of a heat flux, the overriding parameter is the temperature at the interface between the metal electrode and the solution, which has an effect on diffusion coefficients and viscosity. If the variations of these parameters with temperature are known, then / l (and ) can be calculated from the hydrodynamic equations. 

The hydrodynamic radius reflects the effect of coil size on polymer transport properties and can be determined from the sedimentation or diffusion coefficients at infinite dilution from the relation Rh = kBT/6itri5D (D = translational diffusion coefficient extrapolated to zero concentration, kB = Boltzmann constant, T = absolute temperature and r s = solvent viscosity). 

Dynamic light scattering (DLS) Translational diffusion coefficient, hydrodynamic or Stokes radius branching information (when Rh used with Rg) Fixed 90° angle instruments not suitable for polysaccharides. Multi-angle instrument necessary. [3]... 

Theoretically, the diffusion coefficient can be described as a function of the disjoining pressure 77, the effective viscosity of lubricant, 77, and the friction between lubricant and solid surfaces. In relatively thick films, an expression derived from hydrodynamics applies to the diffusion coefficients. 

The values for the lipid molecules compare well (althoughJgiey are still somewhat larger) with the experimental value of 1.5x10 cm /s as measured with the use of a nitroxide spin label. We note that the discrepancy of one order of magnitude, as found in the previous simulation with simplified head groups, is no longer observed. Hence we may safely conclude that the diffusion coefficient of the lipid molecules is determined by hydrodynamic interactions of the head groups with the aqueous layer rather than by the interactions within the lipid layer. The diffusion coefficient of water is about three times smaller than the value of the pure model water thus the water in the bilayer diffuses about three times slower than in the bulk. 

The hydrodynamic drag experienced by the diffusing molecule is caused by interactions with the surrounding fluid and the surfaces of the gel fibers. This effect is expected to be significant for large and medium-size molecules. Einstein [108] used arguments from the random Brownian motion of particles to find that the diffusion coefficient for a single molecule in a fluid is proportional to the temperature and inversely proportional to the frictional coefficient by... 

Altenberger and Tirrell [11] utilized the Langevin equation for particle motion coupled with hydrodynamics described by the Navier-S takes equation to determine particle diffusion coefficients in porous media given by... 

Trinh et al. [399] derived a number of similar expressions for mobility and diffusion coefficients in a similar unit cell. The cases considered by Trinh et al. were (1) electrophoretic transport with the same uniform electric field in the large pore and in the constriction, (2) hindered electrophoretic transport in the pore with uniform electric fields, (3) hydrodynamic flow in the pore, where the velocity in the second pore was related to the velocity in the first pore by the overall mass continuity equation, and (4) hindered hydrodynamic flow. All of these four cases were investigated with two different boundary condi-... 

For displacements shorter than the mean pore dimension, (z2) < a, where flow velocities tend to be spatially constant and homogeneously distributed, Brownian diffusion is the only incoherent transport phenomenon that contributes to the hydrodynamic dispersion coefficient. As a direct consequence, the dispersion coefficient approaches the ordinary Brownian diffusion coefficient,... 

In this section, we consider flow-induced aggregation without diffusion, i.e., when the Peclet number, Pe = VLID, where V and L are the characteristic velocity and length and D is the Brownian diffusion coefficient, is much greater than unity. For simplicity, we neglect the hydrodynamic interactions of the clusters and highlight the effects of advection on the evolution of the cluster size distribution and the formation of fractal structures. 

In addition to the fact that MPC dynamics is both simple and efficient to simulate, one of its main advantages is that the transport properties that characterize the behavior of the macroscopic laws may be computed. Furthermore, the macroscopic evolution equations can be derived from the full phase space Markov chain formulation. Such derivations have been carried out to obtain the full set of hydrodynamic equations for a one-component fluid [15, 18] and the reaction-diffusion equation for a reacting mixture [17]. In order to simplify the presentation and yet illustrate the methods that are used to carry out such derivations, we restrict our considerations to the simpler case of the derivation of the diffusion equation for a test particle in the fluid. The methods used to derive this equation and obtain the autocorrelation function expression for the diffusion coefficient are easily generalized to the full set of hydrodynamic equations. 

The dynamical properties of polymer molecules in solution have been investigated using MPC dynamics [75-77]. Polymer transport properties are strongly influenced by hydrodynamic interactions. These effects manifest themselves in both the center-of-mass diffusion coefficients and the dynamic structure factors of polymer molecules in solution. For example, if hydrodynamic interactions are neglected, the diffusion coefficient scales with the number of monomers as D Dq /Nb, where Do is the diffusion coefficient of a polymer bead and N), is the number of beads in the polymer. If hydrodynamic interactions are included, the diffusion coefficient adopts a Stokes-Einstein formD kltT/cnr NlJ2, where c is a factor that depends on the polymer chain model. This scaling has been confirmed in MPC simulations of the polymer dynamics [75]. 

The method of hydrodynamic stability involves bringing two solutions of different concentrations into contact in a capillary tube, forming a stable and measurable concentration gradient. The diffusion coefficient is a function of the... 

Equations (18) and (19) show that the diffusion coefficient is inversely proportional to both solvent viscosity and the molar volume of the hydrodynamic... 

Hydrodynamic stability (free boundary method) Diffusion coefficient determination 5... 

M Southard, L Dias, K Himmelstein, V Stella. Experimental determinations of diffusion coefficients in dilute aqueous solution using the method of hydrodynamic stability. Pharm Res 8 1489-1491, 1991. 

Second, hydration of drug molecules may affect their hydrodynamic radii. The diffusion coefficient D is related to the frictional resistance,/, that the diffusing particle experiences in moving through a medium by the equation [51]... 

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