Diffusion coefficients classical calculations

The classical MD simulations performed in task I provide self-diffusion coefficients for water and also for hydronium ions, which is strictly the vehicular component of the proton diffusivity. These diffusion coefficients are calculated from the mean square displacement of H2O and HsO using the Einstein relation. The numerical values for Nation and SSC membranes at the four hydration levels are hsted in Table 5 along with the experimental values. ... 

Transport properties of films on a basis chitosan and medicinal substance are investigated. Sorption and diffusive properties of films are studied. Diffusion coefficients are calculated. Kinetic curves of release of the amikacin, having abnormal character is shown. The analysis of the obtained data showed that a reason for rejection of regularities of process of transport of medicinal substance from chitosan films from the classical fikovsky mechanism are stractural changes in a polymer matrix, including owing to its chemical modification at interaction with medicinal substance. 

The classical equation for 7 sis provided in Section VII.A of Chapter 2. It depends only on the spin quantum number S, on the molar concentration of paramagnetic metal ions, on the distance d, and on a diffusion coefficient D, which is the sum of the diffusion coefficients of both the solvent molecule (Dj) and the paramagnetic complex (Dm), usually much smaller. The outer-sphere relaxivity calculated with this equation at room temperature and in pure water solution, by assuming d equal to 3 A, is shown in Pig. 25. It appears that the dispersions do not have the usual Lorentzian form. 

The computational efficiency of a FF approach also enables simulations of dynamical behavior—molecular dynamics (MD). In MD, the classical equations of motion for a system of N atoms are solved to generate a search in phase space, or trajectory, under specified thermodynamic conditions (e.g., constant temperature or constant pressure). The trajectory provides configurational and momentum information for each atom from which thermodynamic properties such as the free energy, or time-dependent properties such as diffusion coefficients, can be calculated. 

For the light molecules He and H2 at low temperatures (below about 50°C.) the classical theory of transport phenomena cannot be applied because of the importance of quantum effects. The Chapman-Enskog theory has been extended to take into account quantum effects independently by Uehling and Uhlenbeck (Ul, U2) and by Massey and Mohr (M7). The theory for mixtures was developed by Hellund and Uehling (H3). It is possible to distinguish between two kinds of quantum effects— diffraction effects and statistics effects the latter are not important until one reaches temperatures below about 1°K. Recently Cohen, Offerhaus, and de Boer (C4) made calculations of the self-diffusion, binary-diffusion, and thermal-diffusion coefficients of the isotopes of helium. As yet no experimental measurements of these properties are available. 

These calculated intracrystalline diffusion coefficients are particularly appropriate for comparison with those determined from pulsed field gradient (PFG) NMR experiments. Time-independent equilibrium properties such as adsorbate conformations are also readily accessible. The classical nature of the simulations allows a particle s trajectory to be followed, and from this it is possible to determine all kinds of information, such as how often a particle diffuses through a certain region. 

Using the reaction free energy we are now in the position of calculating the reaction (classical) kinetics by means of the equations outlined in Section 5.2. In particular, for solving DE using Eq. 8-49, it is first necessary to evaluate the related diffusion coefficient D of the reaction coordinate. 

The diffusion coefficients of water calculated from the MD simulations exhibited good agreement with experiment both in terms of the trend with respect to increasing water content as well as the trend with respect to length of the side chain. The diffusivity of the hydronium ions calculated from classical MD simulation agreed with experiment in terms of the trend with respect to increasing water content, but were consistently too low and did not reflect the experimental dependence on length of the side chain,... 

A fundamental study was performed to demonstrate that flow FFF is a good alternative technique for the rapid measurement of protein diffusion coefficients [10]. The results obtained for 15 proteins were in good agreement (within 4%) with the literature data based on classical methods and a group of modern methods such as photon correlation spectrometry (PCS), laminar flow analysis, a chromatographic relaxation method, and analytical split-flow thin-cell (SPLITT) fractionation. The advantages of flow FFF are the high-speed separations and the calculation of D values directly from retention data. 

Going beyond an atomistic description of the aqueous phase and the membrane, Paddison and coworkers [79-88] employed statistical mechanical models, incorporating solvent friction and spatially dependent dielectric properties, to the calculation of the proton diffusion coefficient in Nation and PEEKK membrane pores. They concluded from their studies that, in accordance with NMR based evidence [50], the mechanism of proton transport is more vehicular (classical ion transport) in the vicinity of the pore surface and more Grotthus-like in the center. 

The absolute magnitude of the calculated value of Qa depends on the way in which the relation between diffusion coefficient and mobility for ion states differs from the classical Einstein form. If we introduce parameters 0+ and 6 and write... 

Calculation of diffusion coefficient from size and shape of solute molecules and viscosity of solvent, by classical mechanics... 

These are noteworthy relations. They express neatly the way in which the diffusion coefficient (to which diffusion rates are proportional) depends on a balance between thermal energy, represented by kT, and frictional resistance to motion, represented by /. They allow the calculation of diffusion coefficients by classical hydrodynamic methods. Such values of D are important when, as often, experimental values are not available and when they are available, comparison with experiment permits tests of the assumptions of the theoretical models. 

Diffusion coefficients, both translational and rotational, can be calculated from the equations D = kT/f and =kT/f ((3.38) and (3.39)) if the frictional coefficient / or f is known. For particles of simple shape, in ideal dilute solution in a continuous fluid, / can be expressed in terms of the size and shape and the solvent viscosity, by the methods of classical hydrodynamics [7]. 

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