Diffusion coefficient small solutes

Diffusion of small solute particles (atoms, molecules) in a dense liquid of larger particles is an important but ill-understood problem of condensed matter physics and chemistry. In this case one does not expect the Stokes-Einstein (SE) relation between the diffusion coefficient D of the tagged particle of radius R and the viscosity r/s of the medium to be valid. Indeed, experiments [83, 112-115] have repeatedly shown that in this limit SE relation (with slip boundary condition) significantly underestimates the diffusion coefficient. The conventional SE relation is D = C keT/Rr]s, where k T is the Boltzmann constant times the absolute temperature and C is a numerical constant determined by the hydrodynamic boundary condition. To explain the enhanced diffusion, sometimes an empirical modification of the SE relation of the form... 

This brief analysis shows that mixing only by diffusion is viable when small mixing paths are used. However, if the channel geometry is very small, the large fluid molecules must collide often with the channel wall and not with other molecules. Moreover, diffusion coefficients of solutions containing large molecules (e.g., enzymes and some proteins) are two orders of magnitude... 

When the concentration of macromolecules such as proteins increases, the diffusion coefficient would be expected to decrease, since the diffusivity of small solute molecules decreases with increasing concentration. However, experimental data (G4, C7) show that the diffusivity of macromolecules such as proteins decreases in some cases and increases in other cases as protein concentration increases. Surface charges on the molecules appear to play a role in these phenomena. 

Coefficient a = 1.4 x 10 cm /sec for water at room temperature (two orders of magnitude faster than diffusion of small solutes), making At about 6 minutes. 

The Turing mechanism requires that the diffusion coefficients of the activator and inlribitor be sufficiently different but the diffusion coefficients of small molecules in solution differ very little. The chemical Turing patterns seen in the CIMA reaction used starch as an indicator for iodine. The starch indicator complexes with iodide which is the activator species in the reaction. As a result, the complexing reaction with the immobilized starch molecules must be accounted for in the mechanism and leads to the possibility of Turing pattern fonnation even if the diffusion coefficients of the activator and inlribitor species are the same 62. 

In ulttafUttation, the flux,/ through the membrane is large and the diffusion coefficient, D, is small, so the ratio cjcan teach a value of 10—100 or mote. The concentration of retained solute at the membrane surface, may then exceed the solubility limit of the solute, and a precipitated semisohd gel forms on the surface of the membrane. This gel layer is an additional battier to flow through the membrane. 

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

Electrically assisted transdermal dmg deflvery, ie, electrotransport or iontophoresis, involves the three key transport processes of passive diffusion, electromigration, and electro osmosis. In passive diffusion, which plays a relatively small role in the transport of ionic compounds, the permeation rate of a compound is deterrnined by its diffusion coefficient and the concentration gradient. Electromigration is the transport of electrically charged ions in an electrical field, that is, the movement of anions and cations toward the anode and cathode, respectively. Electro osmosis is the volume flow of solvent through an electrically charged membrane or tissue in the presence of an appHed electrical field. As the solvent moves, it carries dissolved solutes. 

First, when a large excess of inert elec trolyte is present, the electric field will be small and migration can be neglected for minor ionic components Eq. (22-19) then applies to these minor components, where D is the ionic-diffusion coefficient. Second, Eq. (22-19) apphes when the solution contains only one cationic and one anionic species. 

Interdiffusion of bilayered thin films also can be measured with XRD. The diffraction pattern initially consists of two peaks from the pure layers and after annealing, the diffracted intensity between these peaks grows because of interdiffusion of the layers. An analysis of this intensity yields the concentration profile, which enables a calculation of diffusion coefficients, and diffusion coefficients cm /s are readily measured. With the use of multilayered specimens, extremely small diffusion coefficients (-10 cm /s) can be measured with XRD. Alternative methods of measuring concentration profiles and diffusion coefficients include depth profiling (which suffers from artifacts), RBS (which can not resolve adjacent elements in the periodic table), and radiotracer methods (which are difficult). For XRD (except for multilayered specimens), there must be a unique relationship between composition and the d-spacings in the initial films and any solid solutions or compounds that form this permits calculation of the compo-... 

The behavior of ionic liquids as electrolytes is strongly influenced by the transport properties of their ionic constituents. These transport properties relate to the rate of ion movement and to the manner in which the ions move (as individual ions, ion-pairs, or ion aggregates). Conductivity, for example, depends on the number and mobility of charge carriers. If an ionic liquid is dominated by highly mobile but neutral ion-pairs it will have a small number of available charge carriers and thus a low conductivity. The two quantities often used to evaluate the transport properties of electrolytes are the ion-diffusion coefficients and the ion-transport numbers. The diffusion coefficient is a measure of the rate of movement of an ion in a solution, and the transport number is a measure of the fraction of charge carried by that ion in the presence of an electric field. 

Beuche (j[ ) gives the following expression for the diffusion coefficient of a small molecule in a polymer solution. This equation also known as the Dolittle equation is... 

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

The values of hj for different ions are between 0 and 15 (see Table 7.2). As a rule it is found that the solvation number will be larger the smaller the true (crystal) radius of the ion. Hence, the overall (effective) sizes of different hydrated ions tend to become similar. This is why different ions in solution have similar values of mobilities or diffusion coefficients. The solvation numbers of cations (which are relatively small) are usually higher than those of anions. Yet for large cations, of the type of N(C4H9)4, the hydration number is zero. 

We have applied FCS to the measurement of local temperature in a small area in solution under laser trapping conditions. The translational diffusion coefficient of a solute molecule is dependent on the temperature of the solution. The diffusion coefficient determined by FCS can provide the temperature in the small area. This method needs no contact of the solution and the extremely dilute concentration of dye does not disturb the sample. In addition, the FCS optical set-up allows spatial resolution less than 400 nm in a plane orthogonal to the optical axis. In the following, we will present the experimental set-up, principle of the measurement, and one of the applications of this method to the quantitative evaluation of temperature elevation accompanying optical tweezers. 

The theory has been verified by voltammetric measurements using different hole diameters and by electrochemical simulations [13,15]. The plot of the half-wave potential versus log[(4d/7rr)-I-1] yielded a straight line with a slope of 60 mV (Fig. 3), but the experimental points deviated from the theory for small radii. Equations (3) to (5) show that the half-wave potential depends on the hole radius, the film thickness, the interface position within the hole, and the diffusion coefficient values. When d is rather large or the diffusion coefficient in the organic phase is very low, steady-state diffusion in the organic phase cannot be achieved because of the linear diffusion field within the microcylinder [Fig. 2(c)]. Although no analytical solution has been reported for non-steady-state IT across the microhole, the simulations reported in Ref. 13 showed that the diffusion field is asymmetrical, and concentration profiles are similar to those in micropipettes (see... 

The mobilities of alkylpyridines were modeled and predicted in capillary zone electrophoresis.35 The model predicted that compounds adopt a preferred orientation, and additionally predicted mobilities of structural isomers to within 4%, a higher degree of accuracy than can be obtained from simple considerations of van der Waal s radius. Quantitative prediction of the mobilities of some pyridines, such as alkenylpyridines, was not possible. Mobilities of small solutes in capillaries filled with oligomers of ethylene glycol were related to solution viscosity and the diffusion coefficient.36... 

Shimizu, T. and Kenndler, E., Capillary electrophoresis of small solutes in linear polymer solutions Relation between ionic mobility, diffusion coefficient and viscosity, Electrophoresis, 20, 3364, 1999. 

---