Frequency self-diffusion coefficient

Nowadays, self-diffusion coefficients are almost exclusively measured by NMR methods, through the use of methods such as the 90-8-180-8-echo technique (Stejs-kal and Tanner sequence) [10-12]. The pulse-echo sequence, illustrated in Figure 4.4-2, can be divided into two periods of time r. After a 90° radio-frequency (RF) pulse the macroscopic magnetization is rotated from the z-axis into the x-y-plane. A gradient pulse of duration 8 and magnitude g is appHed, so that the spins dephase. 

Experiments snch as the one illnstrated in Fignre 4.38 not only give us self-diffusion coefficients for certain snbstances, bnt as the temperatnre of the experiment is varied, they give us the temperature dependence of the process and a measurement of the activation energy barrier to diffnsion. Diffusion in solid systems, then, can be modeled as an activated process that is, an Arrhenius-type relationship can be written in which an activation energy, Ea, and temperatnre dependence are incorporated, along with a preexponential factor. Do, sometimes called ht frequency factor ... 

Various factors govern autohesive tack, such as relaxation times (x) and monomer friction coefficient (Co) and have been estimated from the different crossover frequencies in the DMA frequency sweep master curves (as shown in Fig. 22a, b). The self-diffusion coefficient (D) of the samples has been calculated from the terminal relaxation time, xte, which is also called as the reptation time, xrep The D value has been calculated using the following equation ... 

The self-diffusion coefficients in supercritical ethylene were measured using the pulsed NMR spectrometer described elsewhere (9,10), automated for the measurement of diffusion coefficients by the Hahn spin echo method (11). The measurements were made at the proton resonance frequency of 60 MHz using a 1 1.2 kG electromagnet. 

The Space Fractional Dimension the Cutoff Time of the Scaling in the Time Domain t0, the Characteristic Frequency co5, and Estimated Self-Diffusion Coefficient for the Polymer-Water Mixtures ... 

Fig. 18. Self-diffusion coefficients of benzene in NaX at 458 K PFG NMR, O (97) and (92) (JENS, A (13) deduced from NMR lineshape analysis, (10). Comparison with nonequilibrium measurements T, sorption uptake with piezometric control (93) , zero-length column method (96) o, frequency-response and single-step frequency-response technique (98). The region of the results of gravimetric measurements with different specimens (92) is indicated by the hatched areas. Asterisked symbols represent data obtained by extrapolation from lower temperatures with an activation energy confirmed by NMR measurements.

The results described in this report compare well with data of Van-Den-Begin et al. [15] obtained on silicalite samples with an equivalent radius of 31pm by means of Single-Step Frequency-Response. The authors report a self diffusion coefficient for n-hexane of about 2 10 cmVs at a temperature of444 K. However, it has to be considered that, due to the shape of the sorption isotherm, the self-diffusion coefficient will be somewhat smaller than the transport diffusion coefficient. Caro et al [16] report a transport diffusion coefficient of 1.8 10 cmVs for the system n-hexane/HZSM-5 at 298 K, determined gravimetrically. The crystals used in that study were of prismatic shape, the dimensions being 330 pm (z-axis), 110 pm... 

Physical Mechanisms. The simplest interpretation of these results is that the transport coefficients, other than the thermal conductivity, of the water are decreased by the hydration interaction. The changes in these transport properties are correlated the microemulsion with compositional phase volume 0.4 (i.e. 60% water) exhibits a mean dielectric relaxation frequency one-half that of the pure liquid water, and ionic conductivity and water selfdiffusion coefficient one half that of the bulk liquid. In bulk solutions, the dielectric relaxation frequency, ionic conductivity, and self-diffusion coefficient are all inversely proportional to the viscosity there is no such relation for the thermal conductivity. The transport properties of the microemulsions thus vary as expected from simple changes in "viscosity" of the aqueous phase. (This is quite different from the bulk viscosity of the microemulsion.)... 

Fig. 14.9 The isomerization rate on Zraw.v-stilbene displayed as a function of the inverse self-diffusion coefficient of the solvent at T = 298 K. The measurements are done at different pressures in supercritical and hquid alkane solvents Ethane (circles), propane (triangles), and -butane (squares). The solid lines represent fit to the Markovian Kramers theory that use solvent modified barrier height (A n) and barrier frequency ( >b ) From Jorg Schroeder, Ber. Bunsenges. Phys. Chem. 95, 233 (1991).

By way of introduction let us note that the depolarized spectrum Ivh(co) calculated in Section 7.5 for independent rotors consists of a superposition of Lorentzian bands all centered at zero frequency. In the simplest case of symmetric top rotors the spectrum consists of a single band with a width [q2D + 6<9] which depends only on the translational self-diffusion coefficient D and on the rotational diffusion coefficient 0. This should be compared and contrasted with the depolarized spectrum Ivh(co) of certain pure liquids (e.g., aniline, nitrobenzene, quinoline, hexafluorobenzene) shown schematically in Fig. 12.1.1. The spectrum appears to be split. This entirely novel fea-... 

Figure 9 Concentration dependence of the self-diffusion coefficient of /j-hexane in zeolite NaX with mean crystallite diameters of 55 p-m (O), 20 xm([ 1), 15 xm ( 0 ), and 4 p.m (A) at 293 K. The proton resonance frequencies were 60 MHz (open symbols) and 16.6 MHz (full symbols), corresponding to external magnetic fields of 1.41 and 0.39 T, respectively. The error bars indicate the uncertainty in the diffusivities and concentrations. (From Ref. 108.)...

Note literature values for self-diffusion coefficients vary widely, indicating the difficulty of making reliable measurements. The values in this table are intended to be representative only. The values of diffusion coefficients in the literature are mostly given in cm s to convert the values given here to cm s multiply by 10. Do, pre-experimental, or frequency, factor E, activation... 

AcAc, acetylacetonate EPR, electron paramagnetic resonance DPM, dipivaloylmethane Tc, Correlation time for molecular tumbling A/x, concentration of spins X (per unit volume) D, mutual translational self-diffusion coefficient of the molecules containing A and X a, distance of closest approach of A and X ye, magnetogyric ratio for the electron C, spin-rotation interaction constant (assumed to be isotropic) Ashielding anisotropy <7 <7j ) coo, Debye frequency 0d, the corresponding Debye temperature Fa, spin-phonon coupling constant. 

For collision frequency v, dynamic dispersed phase viscosity p, pulsation energy transport coefficient Ti, of the pseudo-gas (analog of the thermal conductivity coefficient in a molecular gas), and particle self-diffusion coefficient D, we have [16-18]... 

Here, to and k are the fluctuation frequency and wave-number, respectively, H(x) designates the Heaviside step function, D is the particle self-diffusion coefficient identified in Equation 4.5 and Equation 4.7, and is understood as the maximal wave number possible in a dispersion containing spherical particles of radius a, particle volume concentration being equal to ([). 

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