Inverse diffusion constant

Figure 15. T-dependence of the inverse diffusion constant, 1/D, from MD simulations. Comparison between MD simulations and NMR data [19] (inset) [51].

The reversible step may be related to the dynamic crossover in protein hydration water at To 345 5K. NMR self-diffusion results [19] indicate that at this temperature a sudden change in hydration water dynamics occurs and the inverse diffusion constant switches from low-temperature super-Arrhenius behavior to high-temperature Arrhenius behavior. Neutron techniques (QENS) have also been used to study protein hydration water at this high-r crossover. Figure 21 shows the atomic MSD of protein hydration water at the low-r crossover measured using MD simulation. These crossovers can also be shown theoretically. Whenever the slope of an Arrhenius plot of the D T) changes, the specific heat has a peak. The well-known Adam-Gibbs equation (AGE) shows this as... 

The proportionality constant in this equation, D, is called the diffusion constant. A typical solute has D 1 X 10 9 m2 s . Values of D do not vary greatly with the solute but are inversely proportional to the viscosity of the solvent. 

Fig. 5.10. Plot of the inverse logarithm of the self-diffusion constant of BPA-PC, for a length N = 20 of the coarse-grained chains, vs. temperature. Straight line indicates the Vogel-Fulcher [187] fit. From [28]...

Diffusion is proportional to the surface area of the blood-gas interface (A) the diffusion constant (D) and the partial pressure gradient of the gas (AP). Diffusion is inversely proportional to the thickness of the blood-gas interface (T). 

The diffusion constant for a gas is proportional to the solubility of the gas and inversely proportional to the square root of the molecular weight of the gas ... 

Figures 8 and 9 show the dependence of the self-diffusion constant and the viscosity of polyethylene melts on molecular weight [47,48]. For small molecular weights the diffusion constant is inversely proportional to the chain length - the number of frictional monomers grows linearly with the molecular weight. This behavior changes into a 1/M2 law with increasing M. The diffusion...

One of the most popular applications of molecular rotors is the quantitative determination of solvent viscosity (for some examples, see references [18, 23-27] and Sect. 5). Viscosity refers to a bulk property, but molecular rotors change their behavior under the influence of the solvent on the molecular scale. Most commonly, the diffusivity of a fluorophore is related to bulk viscosity through the Debye-Stokes-Einstein relationship where the diffusion constant D is inversely proportional to bulk viscosity rj. Established techniques such as fluorescent recovery after photobleaching (FRAP) and fluorescence anisotropy build on the diffusivity of a fluorophore. However, the relationship between diffusivity on a molecular scale and bulk viscosity is always an approximation, because it does not consider molecular-scale effects such as size differences between fluorophore and solvent, electrostatic interactions, hydrogen bond formation, or a possible anisotropy of the environment. Nonetheless, approaches exist to resolve this conflict between bulk viscosity and apparent microviscosity at the molecular scale. Forster and Hoffmann examined some triphenylamine dyes with TICT characteristics. These dyes are characterized by radiationless relaxation from the TICT state. Forster and Hoffmann found a power-law relationship between quantum yield and solvent viscosity both analytically and experimentally [28]. For a quantitative derivation of the power-law relationship, Forster and Hoffmann define the solvent s microfriction k by applying the Debye-Stokes-Einstein diffusion model (2)... 

Cation Size. In their early studies, Hlnsberg (1) and Arcus (3) found that dissolution rates of resists decreased as the size of the cation of the base Increased. Our results support their conclusion. In Figure 5, the dissolution rates of a PMPS(10X)/p-N02-PHMP film 1n different alkali solutions clearly show a decreasing trend with increasing cation size. In fact, the rate 1s Inversely proportional to the cross-sectional area of the unhydrated cation (Figure 6). It is known 1n the diffusion of small molecules 1n polymers, the diffusion constant is Inversely related to the size of the molecule (IS). The observed dependence of dissolution rate on cation size 1s therefore suggestive of cation diffusion as a crucial step. It is... 

The Rouse model12 that yields Eq. [6] also shows that the self-diffusion constant of the chains scales inversely with chain length... 

The diffusion flux J, in mol/cm, is proportional to the concentration gradient and inversely proportional to the diffusion layer s effective thickness 5 (also called the Nemst thickness). The proportionality constant D is the diffusion constant hence,... 

Thus x is proportional to the square of the gel radius and the inverse of the diffusion constant. 

The diffusion equations thus far developed assume that the particles are colloidal and not affected by any motion of the fluid itself which is regarded as stationary. If we limit our discussion to particles in the size range from 0.5 to 5 p, which remain in suspension for rather long periods of time, and if these particles are emitted from a point source and not subject to disturbance by the surrounding fluid, it is obvious that the concentration of particles at any point must be proportional to the diffusion constant and inversely proportional to the square of the distance from the source. Let C be the concentration per unit time, passing a point at any distance R from the source, then... 

For Q<0, this distribution function is peaked around a maximum cluster size (2Q/(2Q-1))< >, where < > is the mean cluster size. 2Q=a+df1 is a parameter describing details of the aggregation mechanism, where a1 is an exponent considering the dependency of the diffusion constant A of the clusters on its particle number, i.e., A NAa. This exponent is in general not very well known. In a simple approach, the particles in the cluster can assumed to diffusion independent from each other, as, e.g., in the Rouse model of linear polymer chains. Then, the diffusion constant varies inversely with the number of particles in the cluster (A Na-1), implying 2Q=-0.44 for CCA-clusters with characteristic fractal dimension d =l.8. 

The diffusion constant is predicted by Eqs. (32) to (35) to be inversely proportional to the total pressure. Experimentally, this is the case to roughly the degree to which the perfect-gas law applies. The equations appear to predict that the diffusion constant will be proportional to the three-halves power of the temperature however, as in the case of viscosity, significant deviations from this behavior occur, as actual molecules are not truly hard spheres and have collision diameters that depend on the relative speeds with which molecules collide with one another. 

We can make a crude estimate of the conditions under which diffusion is likely to be important by comparing the time required for diffusion under given circumstances with the time required for a dependent or competing process. Thus from the kinetic theory (Sec. VI.7) of Brownian motion, the time required for a molecule to diffuse a distance x is given approximately by to = where Z>, the diffusion constant, is inversely pro-... 

The diffusion constant D is regarded as inversely proportional to the solvent viscosity t] as before, while k X) is independent of 77, determined by intramolecular vibrational motions. 

As to the osmotic model, Comper and Preston have questioned its validity as a probable mechanism leading to density inversion. According to their argument, the main diffusion constant of PVP should equal to the cross diffusion constant of dextran if the osmotic model is valid. This expected relation was not found experimentally. 

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