Diffusion constant, calculation

The steady-state methods involve theoretical analysis of magnetic resonance spectra observed under steady-state conditions. This typically involves assumptions regarding the adequacy of magnetic resonance line shape theory, some model for molecular motions and distances of closest approach on collision, and a comparison of calculated spectra for various assumed diffusion constants, and observed spectra. In general, the agreement between diffusion constants calculated using the transient and steady-state methods has been excellent. 

Figure 6.13 Squares and solid curve probability of Ag + cross-linking phosphate chains in (AgI)x(AgP03)i x glasses as a function of x. Circles and dotted curve effective Ag+ diffusion constant, calculated as described in the text. There is a transition between local and macroscopic diffusion between x = 0.2 and 0.3.

Diffusion constants. The diffusion constants calculated from Eq. (5.203) using VACF described above are depicted in Fig. 5.19 against the ion radius, which is taken as half of the Lennard-Jones a parameter. The behavior is striking in a sense that it entirely breaks the Einstein-Stokes law, which predicts monotonic decrease of the diffusion constant as the ion size increases. (Though we have not shown it here, our theory, in fact, predicts the monotonic decrease, when the electrostatic interaction between the ion and solvent molecules is turned off . See [91].) Instead, our results exhibit just the opposite behavior to the Einstein-Stokes law when the ion size is small. We will discuss the phys-... 

Diffusion constant or any other related properties such as viscosity always differ from original value because of the fast dynamic of the CG beads. Therefore, absolute number for dynamical quantities from CG MD is not comparable with the experimental or atomistic simulation results. However, relative numbers are meaningful for different system of interest. It is a normal practice to scale the dynamical quantities by a scaling factor obtained from known values from atomistic details calculations or experimental observation. In case of PS the diffusion constant calculated from the Einstein s relation (ref equation 14) and CG MD for = 9 chain was 4.8 x 10 enrols and for N = 350 chain the value was 4.6 x 10 cm /s at 500 K. 

The symbol E+ denotes the total energy of the crystal with the positron in its lowest state (Bloch state at — 0), and v is the crystal volume. In metals, Ed is typically on the order of - p (Ep is the Fermi energy) [73]. Positron diffusion constants calculated using Equation 4.105 are listed in Table 4.19. Some experimental values of D+ that can be found in the literature are also included for comparison. We can see that there is relatively good agreement with the experimental values. 

Ifihe Bath relaxation con start t, t, is greater than 0.1 ps. yon should be able Lo calculate dyriani ic p roperlies, like time correlation fun c-tioris and diffusion constants, from data in the SNP and/or C.SV files (sec "Collecting Averages from Simulations"... 

Tlere, y Is the friction coefficien t of the solven t. In units of ps, and Rj is th e random force im parted to th e solute atom s by the solvent. The friction coefficien t is related to the diffusion constant D oflh e solven l by Em stem T relation y = k jT/m D. Th e ran doin force is calculated as a ratulom number, taken from a Gaussian distribn-... 

Here, y is the friction coefficient of the solvent, in units of ps and Rj is the random force imparted to the solute atoms by the solvent. The friction coefficient is related to the diffusion constant D of the solvent by Einstein s relation y = kgT/mD. The random force is calculated as a random number, taken from a Gaussian distribu-... 

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

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. 

RCS (.sec Rate-controlling step) Racemization, 49, 95 Rate constant calculation of. 17 composite, 161-164 diffusion-controlled, 200-201 Rate-controlling step, 9, 82-86 Rate-determining step (see Ratecontrolling step)... 

The diffusivity of the vapour of a volatile liquid in air can be conveniently determined by Winkdmann s method in which liquid is contained in a narrow diameter vertical tube, maintained at a constant temperature, and an air stream is passed over the top of the tube sufficiently rapidly to ensure that the partial pressure of the vapour there remains approximately zero. On the assumption that the vapour is transferred from the surface of the liquid to tile air stream by molecular diffusion alone, calculate the diffusivity of carbon tetrachloride vapour in air at 321 K and atmospheric pressure from the experimental data given in Table 10.3. 

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 ... 

The diffusion constant of a primary radical must be of the order of 10 cm.2 sec.- the radius r is about 5X10 cm., and as we have seen 1 10 " per second. Hence ]ag l0 radicals per cc. But the radicals are being generated at a rate of 10 cc. sec. hence the average lifetime of a radical from generation to capture by a polymer particle will be only 10 sec. " The rate of termination by reaction between two radicals in the aqueous phase at the calculated equilibrium concentration, 10 radicals per cc., will be given by... 

The diffusion constant Dj of neutral particle j is calculated in two steps. First, the binary diffusion coefficient Dij in each of the background gas species (SiH4, Si2H6, H2) is calculated, following Perrin et al. [192]. Then Dj is approximated using Blanc s law [219] ... 

Within the Rouse model for polymer dynamics the viscosity of a melt can be calculated from the diffusion constant of the chains using the relation [22,29,30] ... 

Figure 7-3. Active site properties of CAII from SCC-DFTB/MM-GSBP simulations [91]. (a) The root mean square differences between the RMSFs calculated from GSBP simulations (WT-20 and WT-25 have an inner radius of 20 and 25 A respectively) and those from Ewald simulation, for atoms within a certain distance from the zinc, plotted as functions of distance from the zinc ion that die center of die sphere in GSBP simulations is the position of the zinc ion in the starting (crystal) structure, (b) The diffusion constant for TIP3P water molecules as a function of the distance from the zinc ion in different simulations...

Table I shows the results of calculating a soil diffusion coefficient and soil diffusion half-lives for the pesticides. The 10% moisture level specified means that the soil is relatively dry and that 40% of the soil volume is air available for diffusion. Complete calculations were not made for methoxychlor, lindane, and malathion because, based on Goring s criteria for the Henry s law constant, they are not volatile enough to diffuse significantly in the gas phase. This lack of volatility is reflected in their low values of X. These materials would move upward in the soil only if carried "by water that was moving upward to replace the water lost through evapotranspiration at the surface. Mirex has a very high Henry s law constant. On the basis of Goring s criteria, Mirex should diffuse in the soil air but, because of its strong adsorption, it has a very large a and consequently a very small soil air diffusion coefficient. The behavior of Mirex shows that Goring s criteria must be applied carefully.

The solution to this equation was obtained for each airway in the cast for each set or experimental conditions using the measured values for length of the airway and measured fraction of the total flow wnich passes through the airway. The diffusion constants are shewn in Table I. The mean deposition for each generation was obtained from the deposition calculated for all cast airways of the given generation. 

The short delays between Ca2+ influx and exocytosis have important implications for the mechanism of fusion of synaptic vesicles (see Ch. 9). In this short time, a synaptic vesicle cannot move significant distances and must be already at the release site. From the diffusion constant of Ca2+ in squid axoplasm, one can calculate that Ca2+ could diffuse a distance of only 850 A, somewhat greater than the diameter of a synaptic vesicle. Therefore, in fast synapses, vesicle exocytosis sites must be close to the triggering Ca2+ channels.. Vesicles are exposed to [Ca2+] of a few hundred micromoles near the cytoplasmic mouth of the channels. 

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. 

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