Diffusion coefficient hydrodynamic effect

Moreover, the influence of the motions of the particles on each other (i.e., when the motion of a particle affects those of the others because of communication of stress through the suspending fluid) can also influence the measured diffusion coefficients. Such effects are called hydrodynamic interactions and must be accounted for in dispersions deviating from the dilute limit. Corrections need to be applied to the above expressions for D and Dm when particles interact hydrodynamically. These are beyond the scope of this book, but are discussed in Pecora (1985), Schmitz (1990), and Brown (1993). 

Figure 4.13 Estimation of reduced diffusion coefficient by effective medium approximation. Combination of steric and hydrodynamic effects on reduced diffusion coefficient. The solid lines represent hydrodynamic effect for probe radii of 3.4 and 10 A calculated using Brinkman s equation (see Figure 4.9). The dashed lines represent the combined steric and hydrodynamic effect using Equation 4-40 for the steric effect.

The interactions between a lipophilic or hydrophilic drug and micellar phases are caused by weak physicochemical forces such as hydrophobic (unspecific) and electrostatic effects (specific dipole-dipole, dipole-mdacoA dipole) and steric effects, whereas the hydrophobic binding to the micellar systems is dominant. An indirect indication for the presence of interactions between the micellar phase and drugs is given by molecular and dynamic parameters of the drug and the micelles (ionic mobiUty, diffusion coefficient, hydrodynamic radius, apparent molecular mass), which are altered by the solubilization of lipophilic substances in a characteristic manner. 

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). 

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 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... 

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. 

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 accounting for diffusion in these models, in fact, is in many cases a formality. This is because, as can be seen from Equations 20.19 and 20.21, the contribution of the diffusion coefficient D to the coefficient of hydrodynamic dispersion D is likely to be small, compared to the effect of dispersion. If we assume a dispersivity a of 100 cm, for example, then the product av representing dispersion will be larger than a diffusion coefficient of 10-7-10-6 cm2 s-1 wherever groundwater velocity v exceeds 10 9-10-8 cm s 1, or just 0.03-0.3 cm yr-1. 

If instead of semi-infinite diffusion, some distance (5m acts as an effective diffusion layer thickness (Nernst layer approximation), then a modified expression of equation (63) applies where ro is substituted by 1 / (1 /Vo + 1 /<5m ) (see equation (38) above). For some hydrodynamic regimes, which for simplicity, are not dealt with here, the diffusion coefficient might need to be powered to some exponent [57,58],... 

A reciprocal proportionality exists between the square root of the characteristic flow rate, t/A, and the thickness of the effective hydrodynamic boundary layer, <5Hl- Moreover, f)HL depends on the diffusion coefficient D, characteristic length L, and kinematic viscosity v of the fluid. Based on Levich s convective diffusion theory the combination model ( Kombi-nations-Modell ) was derived to describe the dissolution of particles and solid formulations exposed to agitated systems [(10), Chapter 5.2]. In contrast to the rotating disc method, the combination model is intended to serve as an approximation describing the dissolution in hydrodynamic systems where the solid solvendum is not necessarily fixed but is likely to move within the dissolution medium. Introducing the term... 

Thus, the thickness of the effective hydrodynamic boundary layer hl obviously depends on the diffusion coefficient. The diffusion coefficient D further correlates to the diameter of the particle or molecule as demonstrated by the relation... 

At low Q the experiments measure the collective diffusion coefficient D. of concentration fluctuations. Due to the repulsive interaction the effective diffusion increases 1/S(Q). Well beyond the interaction peak at high Q, where S(Q)=1, the measured diffusion tends to become equal to the self-diffusion D. A hydrodynamics factor H(Q) describes the additional effects on D ff=DaH(Q)/S Q) due to hydrodynamics interactions (see e.g. [342]). Variations of D(Q)S(Q) with Q (Fig. 6.28) may be attributed to the modulation with H(Q) displaying a peak, where S(Q) also has its maximum. For the transport in a crowded solution inside a cell the self-diffusion coefficient is the relevant parameter. It is strongly... 

The z-averag translational diffusion coefficient aj infinite dilution, D, could be determined by extrapolating r/K to zero scattering angle and zero concentration as shown typically in Figs. 4 and 5. D is related to the effective hydrodynamic radius, by the Stokes-Einstein relation ... 

When the solution is dilute, the three diffusion coefficients in Eq. (40a, b) may be calculated only by taking the intramolecular hydrodynamic interaction into account. In what follows, the diffusion coefficients at infinite dilution are signified by the subscript 0 (i.e, D, 0, D10> and Dr0). As the polymer concentration increases, the intermolecular interaction starts to become important to polymer dynamics. The chain incrossability or topological interaction hinders the translational and rotational motions of chains, and slows down the three diffusion processes. These are usually called the entanglement effect on the rotational and transverse diffusions and the jamming effect on the longitudinal diffusion. In solving Eq. (39), these effects are taken into account by use of effective diffusion coefficients as will be discussed in Sect. 6.3. 

At finite polymer concentrations, the intermolecular hydrodynamic interaction may also alter polymer dynamics. Except for spherical particles, the hydrodynamic calculations of the effective diffusion coefficients including this... 

Another consequence of the solvent s presence on the rate of reactant diffusion towards (and away from) each other is that solvent has to be squeezed out of ( sucked into ) the intervening space between the reactants. Because this takes time, the approach (or separation) of reactants is slowed. Effectively, the solvent diffusion coefficient is reduced at distances of separation between reactants from one to several solvent diameters. Figure 38 (p. 216) shows the diffusion coefficient as a function of reactant separation distances. This effect is known as hydrodynamic repulsion and it more than cancels the net increase of reaction rate due to the potential of mean force. It is discussed further in Chap. 8 Sect. 2.5 and Chap. 9 Sect. 3. Both the steady-state and transient terms in the rate coefficient depend on these effects. 

Up to now, only hydrodynamic repulsion effects (Chap. 8, Sect. 2.5) have caused the diffusion coefficient to be position-dependent. Of course, the diffusion coefficient is dependent on viscosity and temperature [Stokes—Einstein relationship, eqn. (38)] but viscosity and temperature are constant during the duration of most experiments. There have been several studies which have shown that the drift mobility of solvated electrons in alkanes is not constant. On the contrary, as the electric field increases, the solvated electron drift velocity either increases super-linearly (for cases where the mobility is small, < 10 4 m2 V-1 s-1) or sub-linearly (for cases where the mobility is larger than 10 3 m2 V 1 s 1) as shown in Fig. 28. Consequently, the mobility of the solvated electron either increases or decreases, respectively, as the electric field is increased [341— 348]. 

The other source of an effective electric field dependence of the diffusion coefficient is due to hydrodynamic repulsion. As the ions approach (or recede from) one another, the intervening solvent has to be squeezed out of (or flow into) the intervening space. The faster the ions move, the more rapidly does the solvent have to move. A Coulomb interaction will markedly increase the rate of approach of ions of opposite charge and so the hydrodynamic repulsion is correspondingly larger. It is necessary to include such an effect in an analysis of escape probabilities. Again, the force is directed parallel to the electric field and so the hydro-dynamic repulsion is also directed parallel to the electric field. Perpendicular to the electric field, there is no hydrodynamic repulsion. Hence, like the complication of the electric field-dependent drift mobility, hydro-dynamic repulsion leads to a tensorial diffusion coefficient, D, which is similarly diagonal, with components... 

Consequently, while the effect of an electric field dependence of both drift mobility and diffusion coefficient and also hydrodynamic repulsion decreases, the recombination probability, dielectric saturation and relaxation effects increase the recombination probability. 

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