Diffusion critical mutual

The territory of an expanded single coil is much larger than the volume the monomers actually occupy. Therefore, in comparison to the collisions of small molecules, the probability of mutual collisions between two coils is significantly enhanced. When such fluffy coils diffuse, the mutual friction yields a solution with a high viscosity. In the early history of polymer science, the high viscous polymer solution was misimderstood as a colloidal gel. However, polymer solutions are actually the molecular dispersions of long chains in the solvent molecules. With the increase of the polymer concentration, the coils start to interpenetrate into each other. We can define an illusive critical overlap concentration C, as illustrated in Fig. 4.2. Then, polymer solutions with the concentrations beyraid C are called concentrated solutions. 

Consider the bimolecular reaction of A and B. The concentration of B is depleted near the still-unreacted A by virtue of the very rapid reaction. This creates a concentration gradient. We shall assume that the reaction occurs at a critical distance tab- At distances r tab. [B] = 0. Beyond this distance, at r > rAB, [B] = [B]°, the bulk concentration of B at r = °°. We shall examine a simplified, two-dimensional derivation the solution in three dimensions must incorporate the mutual diffusion of A and B, requiring vector calculus, and is not presented here. 

In accordance with theoretical predictions of the dynamic properties of networks, the critical concentration of dextran appears to be independent of the molecular weight of the flexible polymeric diffusant although some differences are noted when the behaviour of the flexible polymers used is compared e.g. the critical dextran concentrations are lower for PEG than for PVP and PVA. For ternary polymer systems, as studied here, the requirement of a critical concentration that corresponds to the molecular dimensions of the dextran matrix is an experimental feature which appears to be critical for the onset of rapid polymer transport. It is noteworthy that an unambiguous experimental identification of a critical concentration associated with the transport of these types of polymers in solution in relation to the onset of polymer network formation has not been reported so far. Indeed, our studies on the diffusion of dextran in binary (polymer/solvent) systems demonstrated that both its mutual and intradiffusion coefficients vary continuously with increasing concentration 2. ... 

The large fluctuations in temperature and composition likely to be encountered in turbulence (B6) opens the possibility that the influence of these coupling effects may be even more pronounced than under the steady conditions rather close to equilibrium where Eq. (56) is strictly applicable. For this reason there exists the possibility that outside the laminar boundary layer the mutual interaction of material and thermal transfer upon the over-all transport behavior may be somewhat different from that indicated in Eq. (56). The foregoing thoughts are primarily suppositions but appear to be supported by some as yet unpublished experimental work on thermal diffusion in turbulent flow. Jeener and Thomaes (J3) have recently made some measurements on thermal diffusion in liquids. Drickamer and co-workers (G2, R4, R5, T2) studied such behavior in gases and in the critical region. 

In these experiments the number of initiating sites is a set value and cannot be increased. The rate of mutual destruction, the rate of diffusion of the monomer into the starch polymer, and the propagation rate are the controlling factors in the degree of grafting. The diffusion rate would be the critical factor governing the number of starch to acrylonitrile bonds achieved. These factors determine whether a block or a graft copolymer will be obtained. 

Because of the similarity between the mechanisms of viscous flow, diffusion and electrical conductivity, which are all activated processes, a relationship between these phenomena was sought. It has been established empirically that the temperature dependence of viscosity and resistivity of glass melts are often mutually dependent according to the relationship log t/ 3 log 2, or log = a log — b (cf. Morey, 1954). However, it should be borne in mind that mobility of cations is critical for transfer of electric charges while mobility of anionic structural units (network formers) is involved in the case of vi.scous flow. This is why the relation between the two quantities is difficult to interpret. 

A final note concerns what effect the precipitous decline of mutual diffusion coefficients might have as the critical point is approached. We saw no effect in our experimental results which might be attributable to this decline. Its effect must be either, masked by other mechanisms, or, as Cussler has suggested (University of Minnesota, personal communication, 1988), it may simply result in a steep concentration gradient over a very small distance in most engineering experiments where, as is here the case, a relatively large temperature spread is used and the resulting increase in mass transfer resistance is small. 

Let us find the collision frequency of conducting uncharged spherical drops in a turbulent fiow of a dielectric liquid in the presence of a uniform external electric field. Just as before, we assume a developed fiow, with drop sizes smaller than the inner scale of turbulence. We assume the drops to be undeformed, which is possible if the external electric field strength Eo does not exceed the critical value and the size of drops is sufficiently small. Under these conditions, the factor of mutual diffusion of drops of two types 1 and 2 with regard to hydrodynamic interaction is given by (13.86), while h and are given by the expressions (13.85) that apply to drops with a completely retarded surface. We must also take into account molecular and electric interaction forces acting on the drops. 

Dioxane is a cyclic diether forming a six-membered ring [20]. Thus it is a nearly nonpolar symmetric molecule. 1,4-Dioxane is an extraordinary solvent, capable of solubilizing most organic compounds, and water in all proportions, and many inorganic compounds. The self-diffusion coefficient of dioxane is 1.1 x 10 cm /s, about half that of a water molecule. The effective diameter of dioxane is 5.5 A - about twice that of a water molecule. One should not forget that a water-dioxane mixture narrowly avoids a lower critical consolute point. However, the effects of criticality are reflected in the values of ffie mutual diffusion coefficient and viscosity. Note that binary mixtures are often chosen so that they are mixable (do not phase separate). Thus, the two components interact attractively and strongly. 

Equation 10.26 would be valid if colloidal diffusion processes were exactly analogous to those for individual molecules. However, the interactions between particles in colloidal systems tend to extend over distances much greater than those involved in the formation of atomic or molecular activated complexes (say, 10-100 run vs. O.l-l.O nm). As a result, the effects of those interactions will begin to be felt by the particles well before they approach to the critical distance r. Their mutual diffusion rate will therefore be reduced and the collision frequency will drop accordingly. The collision frequency will also be reduced by the hydrostatic effect mentioned above for rapid coagulation. 

This approximation is almost always made. Lo and Kawasaki have investigated the effect of the first vertex correction, shown in Eq. (124), on the critical behavior of the mutual diffusion coefficient the effect is seen to be tiny. Bedeaux and Mazur, who do not use Kawasaki s method, have shown that corrections to Eq. (126) give rise to a tail on which is dominated at... 

The coefficient of sell-diffusion does not appear to have an anomaly near the critical point. For the engineer, however, the mutual dift usion coefficient is the more important property. The binary dilfusion coefficient approaches zero at the mixture critical point ("critical slowing-down"). In dilute mixtures, however, the decrease of the binary dilfusion coefficient is not seen until the critical line is approached very closely. For many practical purposes, such as supercritical extraction and chromatography, the mixture is dilute, and it can be assumed that the coefficient of binary diffusion is intermediate between that in the vapor and that in the liquid. Since the diffusion coefficient decreases roughly inversely proportional to the density, dilfusion in supercritical solvents is much more rapid than in liquid solvents, thus increasing the speed of diffusion-controlled chemical processes. 

The concept of universality classes, mentioned in the previous section, can be extended so as to be applicable to the characterization of the asymptotic critical behavior of dynamic properties (Hohenberg Halperin 1977). Two systems belong to the same dynamic universality class when they have the same number and types of relevant hydro-dynamic modes. Thus the asymptotical critical behavior of the mutual mass diffusivity D 2 and of the viscosity rj of liquid mixtures near a consolute point will be the same as that of the thermal diffusivity a and the viscosity j of one-component fluids near the vapor-liquid critical point (Sengers 1985). Hence, in analogy with equation (6.16) for liquid mixtures near a consolute point it can be written... 

As an example, Figure 6.1 shows the mutual diffusivity of a mixmre of hexane and nitrobenzene at the critical composition as a function of T — Tc as measured by Taylor dispersion (Matos Lopes et al. 1992) and by light scattering (Wu et al. 1988) the solid curve represents the Stokes-Einstein diffusion law (6.21) with = 104 0.06. In Figure 6.2 a log-log plot of the viscosity ratio of a mixture of 3-methylpentane and nitroethane at the critical composition is shown as a function of the correlation length. The solid curve represents the power law (Q y with Q = 1.4nm and z = 0.063 (Burstyn et al. 1983). Experiments in fluids near the vapor-liquid critical point are consistent with these results (Guttinger Cannell 1980 Berg Moldover 1990). 

Fig. 6.1. Mutual diffusivity T> of a mixture of hexane and nitrobenzene at the critical composition as a function of T — Tc. The solid curve represents equation (6.21) (Matos Lopes et al. 1992).

Hence, the behavior of the mass-diffusion coefficient of a mixture in the critical region is controlled by the behavior of (dn/dx)T,p = x As pointed out in Section 6.1, in binary mixtures the osmotic susceptibility x diverges near a consolute point and leads to the critical slowing down of the mass diffusion. Near a vapor-liquid critical point, on the other hand, where the thermodynamic properties undeigo a crossover from pure-fluid-like behavior before they display their asymptotic mixture behavior (Jin et al. 1993), the osmotic susceptibility does not exhibit a critical behavior except at temperatures very close to the plait-point temperature (for the system mentioned above, the reduced temperature has to be smaller than 5 x 10 ). Therefore, not too close to the critical point, the mutual diffiisivity is dominated by its background value d/(px)> and the critical slowing down that follows the Stokes-Einstein diffusion law is not seen in the mass diffusion coefficient. 

DIPPR Project 882 is organized to develop, maintain and make available to its sponsors a computer databank of selected and evaluated physical, thamodynamic and transport properties for binary mixtures. The properties include viscosity, thermal conductivity, mutual-diffusion coefficient, excess volume and density, surface tension, critical temperature, pressure and density and the solubility of sparingly soluble materials. The data from the original literature have been compiled in their original units. Computer routines have been developed to provide the data in SI units for final dissemination. Assessments of the imprecisions and inaccuracies for each of the variables (temperature, pressure, composition and property) are made, and the results have been screened and adjusted, where applicable, to be consistent with the pure component data calculated from a variety of reliable sources. The data may be drawn from electronic database as tables and plots of the experimental data in the original or SI units. 

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