Diffusion constant determination

Studies carried out with the aim of developing models to estimate migration include a study by Aurela and Ketoja (2002). They estimated the diffusion rate of model compounds (butanol, ethanol, butyl acetate and tetrahydrofuran) in air at room temperature. They then measured the diffusion of these substances through papers with different grammages (and hence, porosities) produced from birch Kraft pulp. The model compounds were not in contact with the test papers and hence transferred via the gas phase. They concluded that the diffusion constants determined in air could be used in random walk simulation to predict migration in a fibre network. Random walk simulations are a mathematical means of modelling processes based on probability distribution and are often applied to investigate diffusion processes. 

The concept of two or mote modes of sorption of penetrants in polymers is very familiar to cellulose and protein chemists for the case of water vapor. In fact combined Lan uir and Henry s law sorption was proposed and correctly formulated by Matthes in 1944 for water in cellulose The discovery of dual mode sorption of gases in assy polymers and the aibsequent realization that diffusion constants determined by the time lag method did not have the same ple fundamental significance a ociated whh these parameters for mbbery polymers was of profound importance. Not only were the many carefully determined diffusion coefficients in the literature of questionable value for polymers below their ass transition but a good deal of the careful peculation about solution and diffusion and the effect of... 

The authors wish to thank Dr. B. Lunenfeld, Institute of Endocrinology, Tel-Hashomer Government Hospital (Israel), for the ultracentrifugal studies and for the diffusion constant determination. 

An appropriate value of 7 for a system modeled by the simple Langevin equation can also be determined so as to reproduce observed experimental translation diffusion constants, Dt in the diffusive limit, Dt is related to y hy Dt = kgTmy. See [22, 36], for example. 

An alternative method known as slicing and scaling has been developed (23,24). In this, the rate of diffusion is determined on a thin specimen (6—10 mm thick) and a scaling factor S used to relate the results to a thick specimen. For a material satisfying the requirements of a constant diffusion and constant initial pressure,, the same ratio of time thickness provides the same values of p and %. Thus the thermal resistance of a specimen of thickness at time can be obtained by conditioning a specimen of thickness over a time given by... 

The amplitude of the elastic scattering, Ao(Q), is called the elastic incoherent structure factor (EISF) and is determined experimentally as the ratio of the elastic intensity to the total integrated intensity. The EISF provides information on the geometry of the motions, and the linewidths are related to the time scales (broader lines correspond to shorter times). The Q and ft) dependences of these spectral parameters are commonly fitted to dynamic models for which analytical expressions for Sf (Q, ft)) have been derived, affording diffusion constants, jump lengths, residence times, and so on that characterize the motion described by the models [62]. 

Dynamic information such as reorientational correlation functions and diffusion constants for the ions can readily be obtained. Collective properties such as viscosity can also be calculated in principle, but it is difficult to obtain accurate results in reasonable simulation times. Single-particle properties such as diffusion constants can be determined more easily from simulations. Figure 4.3-4 shows the mean square displacements of cations and anions in dimethylimidazolium chloride at 400 K. The rapid rise at short times is due to rattling of the ions in the cages of neighbors. The amplitude of this motion is about 0.5 A. After a few picoseconds the mean square displacement in all three directions is a linear function of time and the slope of this portion of the curve gives the diffusion constant. These diffusion constants are about a factor of 10 lower than those in normal molecular liquids at room temperature. 

The Debye length of the electrode material can be determined from the constant B, and the sensitivity factor S from C, provided the diffusion length and the diffusion constant for minority carriers are known. 

For an electrode with high interfacial rate constants, for example, relation (28) can be plotted, which yields the flatband potential. It allows determination of the constant C, from which the sensitivity factor S can be calculated when the diffusion constant D, the absorption coefficient a, the diffusion length L, and the incident photon density I0 (corrected for reflection) are known ... 

Example 10.6 A commercial process for the dehydrogenation of ethylbenzene uses 3-mm spherical catalyst particles. The rate constant is 15s , and the diffusivity of ethylbenzene in steam is 4x 10 m /s under reaction conditions. Assume that the pore diameter is large enough that this bulk diffusivity applies. Determine a likely lower bound for the isothermal effectiveness factor. 

The mass of the sedimenting particle could be deduced from its rate of sedimentation at high dilution in a given field, i.e., from its sedimentation constant, if the frictional coefficient / could be determined independently. Hates of diffusion may be utilized to secure this necessary supplementary information, since the diffusion constant D depends also on the frictional coefficient. Thus ... 

One difficulty in Equation 6 lies in the determination of the diffusivity constant for highly concentrated acids over a broad range of temperature. However, available data (13), combined with viscosity values of hydrochloric acid, lead to estimates shown in Tables I and II for the field cases that will be described later on. 

It may be assumed that the accumulation of hydrogen within the pellet is negligible and that it may be treated as being in a quasi-steady-state condition. The finite difference form of Fick s first law may be used to determine the flow rate of hydrogen through the pellet. The diffusion constant appearing in this equation may be considered as an effective Knudsen diffusion coefficient. 

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)... 

The diffusion constants for the gramicidin complex were determined in acetic acid and ethanol solution75. The molecular weight range of gramicidin was calculated as 2800-5000. 

Properties of interest, such as the mean square displacement of an atom or molecule from its starting position, < r2 >, may thus be determined. Since we know this as a function of time, we have a direct route to diffusion coefficients (< r2 > = 6Dt, where D is the diffusion constant). 

Applying this prediction to the cooling rate dependence of a break points in the specific volume curves, one obtains a Vogel-Fulcher temperature of To = 0.35 that agrees well with that determined from the temperature dependence of the diffusion constant in this model, which is T = 0.32. 

An alternative to the common device of determining relative intensities is a study of the fine structure of the scattered beam. This entails resolving the spectrum of scattered light into its three peaks, viz. a central peak and two side ones. The need is thus obviated to refer to I0 or, according to the apparatus, the scattering power of a standard calibration material. The method is used mainly for determining diffusion constants and thermodynamic properties of liquids. 

The diffusion constant D with the underlying microviscosity , and the two order parameters , <(P4> reflecting the degree of orientational constraint have been successfully determined from the fluorescence anisotropy decay in... 

Then, according to the wobble-in-cone model (see Section 5.6.2), the order parameter that is related to the half angle of the cone, and the wobbling diffusion constant (reflecting the chain mobility) can be determined. 

Molecules tend to diffuse randomly, in no particular direction, within any fluid, independently of the flow rate of the mobile phase. Their diffusion rate is determined by the type of molecule, the nature of the mobile phase, and the temperature, and is expressed quantitatively by their diffusion constants. 

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