Diffusion normal

These concepts are tested in Fig. 7(a,b) where one can see that both in the non-adsorbed case (Fig. 7(a)) and in the adsorbed case (Fig. 7(b)) the data is compatible with normal diffusion of the parallel (longitudinal) component, g2/(t) = 4Djyt. For e = —1.5, the chains are freely diffusing in the 3d bulk, far away from the adsorbing wall, g / = For e = —3 most monomers... 

A normal diffusion process, however, runs at a finite concentration of particles different from zero. In this situation it was found [101] that a fractal character (73) of the resulting structure is restricted to an interval a < R < if), where d is the diffusion length (67). Larger clusters have a constant density on a length scale larger than They are no longer fractal there. These observations have various consequences for crystal growth, and will be discussed in the next section. 

FIGURE9.il Phospholipids can be flipped across a bilayer membrane by the action of flippase proteins. Wlien, by normal diffusion through the bilayer, the lipid encounters a flippase, it can be moved quickly to the other face of the bilayer. 

Polarographic maxima. Current-voltage curves obtained with the dropping mercury cathode frequently exhibit pronounced maxima, which are reproducible and which can be usually eliminated by the addition of certain appropriate maximum suppressors . These maxima vary in shape from sharp peaks to rounded humps, which gradually decrease to the normal diffusion-current curve as the applied voltage is increased. A typical example is shown in Fig. 16.3. Curve A is that for copper ions in 0.1 M potassium hydrogencitrate solution, and curve B is the same polarogram in the presence of 0.005 per cent acid fuchsine solution. 

Ay H, Buonanno FS, Rordorf G, Schaefer PW, Schwamm LH, Wu O, Gonzalez RG, Yamada K, Sorensen GA, Koroshetz WJ. Normal diffusion-weighted MRI during strokelike deficits. Neurology 1999 52 1784-1792. 

In many cases, including transport phenomena which differ from normal diffusion, Eq. (3.1.4) turns out to be a good approximation, in particular if only small magnitudes of the gradient intensity ybg (sometimes referred to as the generalized scattering vector of PFG NMR) are considered. Under these circumstances, the genuine diffusivity D is replaced by an effective diffusivity, Deff... 

In contrast to normal diffusion, Ar2n does not grow linearly but with the square root of time. This may be considered the result of superimposing two random walks. The segment executes a random walk on the random walk given by the chain conformation. For the translational diffusion coefficient DR = kBT/ is obtained DR is inversely proportional to the number of friction-performing segments. 

Diffusion regime For times t > xd N3 /4/kBTd2, the dynamics are determined by reptation diffusion. We expect normal diffusive behavior... 

Here we give a brief account of some unimolecular processes other than isomerization. No detailed description of bimolecular processes will be offered, except to remark that (1) the knowledge gained from the unimolecular processes is often useful in interpreting the bimolecular processes and (2) in some cases, the bimolecular processes resemble normal diffusion-influenced reactions in the condensed phase. 

An ingenious variation on the standard fluorescence methods was proposed by Red kin et al. [50]. Water samples were extracted with non-polar solvents, transferred into hexane and the hexane solution frozen at 77 K. At that temperature the normally diffuse luminescence emission bands are present as sharp emission lines, making identification of fluorescing compounds considerably simpler. In the case of a complex mixture, some separation by column or thin layer chromatography might be necessary. 

The connection between anomalous conductivity and anomalous diffusion has been also established(Li and Wang, 2003 Li et al, 2005), which implies in particular that a subdiffusive system is an insulator in the thermodynamic limit and a ballistic system is a perfect thermal conductor, the Fourier law being therefore valid only when phonons undergo a normal diffusive motion. More profoundly, it has been clarified that exponential dynamical instability is a sufRcient(Casati et al, 2005 Alonso et al, 2005) but not a necessary condition for the validity of Fourier law (Li et al, 2005 Alonso et al, 2002 Li et al, 2003 Li et al, 2004). These basic studies not only enrich our knowledge of the fundamental transport laws in statistical mechanics, but also open the way for applications such as designing novel thermal materials and/or... 

Indeed, numerical results in (Li et al, 2003) show that in the irrational case (when the ratio 6/ir and 4>/ir are irrational numbers) the system in Fig 3 exhibits normal diffusion and the heat conduction obeys the Fourier law. In the rational case instead, the system shows a superdiffusive behavior, (a2) = 2Dt1178 (Li et al, 2003)and the heat conductivity diverges with the system size as jy0.25 o.oi ... 

Alternate mass-core hard potential channel In the two billiard gas models just discussed there is no local thermal equilibrium. Even though the internal temperature can be clearly defined at any position(Alonso et al, 2005), the above property may be considered unsatisfactory(Dhars, 1999). In order to overcome this problem, we have recently introduced a similar model which however exhibits local thermal equilibrium, normal diffusion, and zero Lyapunov exponent(Li et al, 2004). 

This relation connects heat conduction and diffusion, quantitatively. As expected, normal diffusion (a =1)corresponds to the size-independent (/ = 0) heat conduction obeying the Fourier law. Moreover, a ballistic motion (a = 2) implies that the thermal conductivity is proportional to... 

Figure 5. Comparison of prediction (4) with numerical data. Normal diffusion ( ). The ballistic motion ( ). Superdiffusion ID Ehrenfest gas channel (Li et al, 2005)(v) the rational triangle channel (Li et al, 2003) (empty box) the polygonal billiard channel with (i = (V > — 1)7t/4), and 2 = 7r/3 (Alonso et al, 2002)(A) the triangle-square channel gas(Li et al, 2005) (<>) / values are obtained from system size L e [192, 384] for all channels except Ehrenfest channel (Li et al, 2005). The FPU lattice model at high temperature regime (Li et al, 2005) ( ), and the single walled nanotubes at room temperature ( ). Subdiffusion model from Ref. (Alonso et al, 2002) (solid left triangle). The solid curve is f3 = 2 — 2/a.

On the other hand, on the bounding hypersurfaces the normal diffusive flux must be null. However, this condition will result naturally from the fact that the conditional joint scalar dissipation rate must be zero-flux in the normal direction on the bounding hypersurfaces in order to satisfy the transport equation for the mixture-fraction PDF.122... 

When one gas diffuses into another, as A into B, even without the quasi-steady-flow component imposed by the burning, the mass transport of a species, say A, is made up of two components—the normal diffusion component and the component related to the bulk movement established by the diffusion process. This mass transport flow has a velocity Aa and the mass of A transported per unit area is pAAa. The bulk velocity established by the diffusive flow is given by Eq. (6.58). The fraction of that flow is Eq. (6.58) multiplied by the mass fraction of A, pA/p. Thus,... 

Normal diffusion rates and the time elapsed during atomic movements are usually such that reaction would be very fast if there were no intermediate free energy barrier. But in most cases rates are not that fast. We conclude that normally there is a free energy barrier. That is. 

In contrast to normal diffusion, in the segmental regime the mean-square displacement does not grow linearly, but with the square route of time. For the... 

Further note that for t=0 Eq. 3.24 does not resemble the Debye function but yields its high Q-limiting behaviour i.e. it is only valid for QR >1. In that regime the form of Dr immediately reveals that the intra-chain relaxation increases in contrast to normal diffusion ocQ, Finally, Fig. 3.2 illustrates the time development of the structure factor. 

Although the shape of the profile of a "spherical diffusion couple" is similar to that of a one-dimensional diffusion couple, one difference is that, whereas the midconcentration position stays mathematically at the initial interface for the normal diffusion couple, the midconcentration position moves with time in the "spherical diffusion couple." Initially, the concentration at the initial interface (r = a) is the mid-concentration Cmid = (Ci + C2)/2. However, as diffusion progresses, the concentration at r = a is no longer the mid-concentration. Rather, the location of the mid-concentration moves to a smaller r. Define the mid-concentration location as Tq. Then Tq x a(l — z /2) for small times. If layer 1 is the solid core (meaning r extends to 0), the concentration at the center begins... 

The case of transport through microporous membranes is different from that of macroporous membranes in that the pore size approaches the size of the diffusing solute. Various theories have been proposed to account for this effect. As reviewed by Peppas and Meadows [141], the earliest treatment of transport in microporous membranes was given by Faxen in 1923. In this analysis, Faxen related a normalized diffusion coefficient to a parameter, X, which was the ratio of the solute radius to the pore radius... 

An unexpected feature of these systems is that whilst an initially rapid but partial redistribution of material appears to occur in the cell, the relaxation of the concentration gradient of absorbing material still remaining at the initial boundary appears follows a normal diffusion process. It should be stressed that the perceived concentration gradient at the boundary is a space-averaged parameter which does not reflect any changes in concentration that occur in the plane at right... 

Finally let us define the normalized diffusivity and the electric current... 

This diffusion constant can be measured by dynamic light scattering. Tanaka and Filmore originally considered the case K P fi, for which swelling occurs from the boundary with the relaxation rate n2D/R2 and the thickness of the diffusion layer is (Df)1/2 after a change of the boundary condition at t = 0. However, this normal diffusion behavior is altered for swelling near the point K = 0. 

Solute-solute Interactions may affect the diffusion rates In the fluid phase, the solid phase, or both. Toor (26) has used the Stefan-Maxwell equations for steady state mass transfer In multicomponent systems to show that, in the extreme, four different types of diffusion may occur (1) diffusion barrier, where the rate of diffusion of a component Is zero even though Its gradient Is not zero (2) osmotic diffusion, where the diffusion rate of a component Is not zero even though the gradient Is zero (3) reverse diffusion, where diffusion occurs against the concentration gradient and, (4) normal diffusion, where diffusion occurs In the direction of the gradient. While such extreme effects are not apparent in this system, it is evident that the adsorption rate of phenol is decreased by dodecyl benzene sulfonate, and that of dodecyl benzene sulfonate increased by phenol. 

As a rule, hydrogen ion is involved not only in the pH-dependency of the reaction term (Thiele modulus) but also as the actively participating species involved in the acid-base equilibrium of all the substrates, reaction intermediates, products, and even the gel matrix. Furthermore, enzymatic reactions are always carried out in the presence of the mobile buffer. By mobile we mean a weak acid or a weak base that can move in and out of the reaction layer, as opposed to the fixed buffer represented by the gel (and by the protein) itself. Thus, we have to include the normalized diffusion-reaction equations for hydrogen ion and for the buffer. 

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