Self-diffusion coefficient calculation

The self-diffusion coefficient calculated for the three body potential is Z) = 1.3 X lO cm /sec. This is to be compared with the experimental value of 2.3 x 10" cmVsec and to the value of 2.25 x 10 cm /sec for the two-body liquid. It could be said that the three-body liquid shows more rigidity in some sense than the two-body liquid. 

To date, D coefficients of carbohydrates established with the PFGSE approactf - " have been undertaken to (1) validate the theoretical self-diffusion coefficients calculated from MD trajectories, (2) demonstrate the complexation of lanthanide cations by sugars,(3) probe the geometry of a molecular capsule formed by electrostatic interactions between oppositely charged P-cyclodextrins, (4) study the influence of concentration and temperature dependence on the hydrodynamic properties of disaccharides, and (5) discriminate between extended and folded conformations of nucleotide-sugars. ... 

Diffusion in zirconia is closely linked to ionic conductivity. Consequently, some diffusion data has already been presented in Sect. 5. This section will include additional results particularly for monoclinic zirconia. Oxygen self-diffusion at a pressure of 300 Torr, as determined by testing zirconia spheres of diameters between 75 and 105 jam, behaves as shown in Fig. 17 [57], where D is the diffusion coefficient, t is time, and a is the sphere radius. At a pressure of 700 Torr, the behavior changes to that shown in Fig. 18 [58]. In this case D is the self-diffusion coefficient and the rest of the terms are as defined before, with a = 100-150 jam. Both of these experiments were performed in an oxygen atmosphere of 180-160. The self-diffusion coefficients calculated from the diffusion data obey Arrhenius expressions as illustrated in Fig. 19 [57, 58]. The linear fits describing the diffusion coefficient at 300 and 700 Torr, are given by ... 

The self-diffusion coefficients calculated for several supercritical thermodynamic states of BJH water are compared with available experimental data and the results of other computer simulations in Table 4. The statistical uncertainty of the calculated values is about 10%, i.e., comparable with the accuracy of experimental data. Thus, the simulated values of D agree very well with experiments. It is also quite surprising that... 

The impedance spectrum of polymer and gel electrolyte appears as a depressed semicircle in the frequency region between 100 kHz and 0.1 Hz, which can be analyzed using the Cole and Cole [1941] approach, as described in Section 2.I.2.3. Typically, polymeric, plasticized, and gel Li-ion conductors show abnormally low conductivity as compared to that expected from self-diffusion coefficients calculated using other methods such as PMFG-NMR (Clericuzio et al. [1995]). In addition to the usual attribution of this effect to ion association, the incomplete removal of the electrode impedance effect during analysis can contribute to an apparent increase in the electrolyte resistance. 

The shear viscosity is a tensor quantity, with components T] y, t],cz, T)yx> Vyz> Vzx> Vzy If property of the whole sample rather than of individual atoms and so cannot be calculat< with the same accuracy as the self-diffusion coefficient. For a homogeneous fluid the cor ponents of the shear viscosity should all be equal and so the statistical error can be reducf by averaging over the six components. An estimate of the precision of the calculation c then be determined by evaluating the standard deviation of these components from tl average. Unfortunately, Equation (7.89) cannot be directly used in periodic systems, evi if the positions have been unfolded, because the unfolded distance between two particl may not correspond to the distance of the minimum image that is used to calculate the fore For this reason alternative approaches are required. 

Following the general trend of looldng for a molecular description of the properties of matter, self-diffusion in liquids has become a key quantity for interpretation and modeling of transport in liquids [5]. Self-diffusion coefficients can be combined with other data, such as viscosities, electrical conductivities, densities, etc., in order to evaluate and improve solvodynamic models such as the Stokes-Einstein type [6-9]. From temperature-dependent measurements, activation energies can be calculated by the Arrhenius or the Vogel-Tamman-Fulcher equation (VTF), in order to evaluate models that treat the diffusion process similarly to diffusion in the solid state with jump or hole models [1, 2, 7]. 

The electrolytic conductivities calculated by this method are given in Table 9 together with the self-diffusion coefficients. The calculated conductivities agree surprisingly well with the experimental ones when a small perturbation on an order of 1 V cm" is applied. 

X 10 cm by measuring molecularly dispersed water in toluene and by correcting for local viscosity differences between toluene and these microemulsions [36]. Values for Dfnic were taken as the observed self-diffusion coefficient for AOT. The apparent mole fraction of water in the continuous toluene pseudophases was then calculated from Eq. (1) and the observed water proton self-diffusion data of Fig. 9. These apparent mole fractions are illustrated in Fig. 10 (top) as a function of... 

Here ZT is calculated from the self-diffusion coefficients of the polymer-penetrant pair, i.e.,... 

Millet determined self-diffusion coefficients for Na and Cs+ ions in hydrated 1200 EW membranes using conductivity measurements and the Einstein equation, D+ = u+kT, where u+ is the absolute mobility of the given cation. u+ can be derived from the equivalent conductivity according to A = 0+IC+ = Fu+, where 0+ is the specific conductivity, C+ is the cation concentration (calculated on the basis of the dry membrane density, EW, and the water content), and F is the Faraday constant. The values of D+ determined via these conductivity measurements... 

The self-diffusion coefficients were calculated, and the effective radii of the dendrimers were estimated from the diffusion coefficients by assuming a spherical geometiy for all dendrimers. The so-obtained radii were 7.8,10.3, and 12.6 A for the first, second, and third generation dendrimer, respectively. 

Figure 10 Size-dependence of the melting point and diffusion coefficient of silica-encapsulated gold particles. The dotted curve is calculated by the equation of Buffat and Borel. The bulk melting temperature of An is indicated by the double arrow as (oo). The solid curve (right-hand side axis) is a calculated An self-diffusion coefficient. (From Ref. 146.)...

After a time r, a 180° RF pulse reverses the spin precession. A second gradient pulse of equal duration 8 and magnitude g follows to tag the spins in the same way. If the spins have not changed their position in the sample, the effects of the two applied gradient pulses compensate each other, and aU spins refocus. If the spins have moved due to self-diffusion, the effects of the gradient pulses do not compensate and the echo-amplitude is reduced. The decrease of the amplitude A with the applied gradient is proportional to the movement of the spins and is used to calculate the self-diffusion coefficient. 

Locate or calculate the binary-component diffusivity or self-diffusion coefficient for a gas, dilute solute, or melt. 

Fig. 17. Translational self-diffusion coefficients for aqueous solutions of five xanthan samples [127], Solid curves, the theoretical values calculated from Eq. (67) along with Eqs. (46) and (58)...

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 ... 

Self-diffusion coefficients were calculated by Carman-Haul equations (16-18). Examples of the percentage attainment of equilibrium with root time plots (Wt/Wa vs. /t) and of the dimensionless time plots (r vs. t) are shown in Figures 1 and 2, respectively. Further calculation (17, 18)... 

Self-diffusion coefficients were calculated from straight-line Wt/Wm vs. /t plots, and Arrhenius plots were constructed (Figures 5, 6, and 7, respectively). Values of D0, 2 a, AS, and AG were estimated again, and are in Table III. A value of d = 500 pm was assumed for the skin studies. 

Figure 2. A pictorial representation of the mode coupling theory scheme for the calculation of the time-dependent friction (f) on a tagged molecule at time t. The rest of the notation is as follows Fs(q,t), self-scattering function F(q,t), intermediate scattering function D, self-diffusion coefficient t]s(t), time-dependnet shear viscosity Cu(q,t), longitudinal current correlation function C q,t), longitudinal current correlation functioa...

Figure 8. The ratio of the self-diffusion coefficient of the solute (Di) to that of the solvent molecules (D ) plotted as a function of the solvent-solute size ratio (

The study is performed at reduced temperature T = 0.75 and reduced density p = 0.844-0.92. This is precisely the system studied in computer simulations [102]. The variation of the self-diffusion coefficient with the solute size is shown in Fig. 8, where the size of the solute molecule has been varied from 1 to 1/20 times that of the solvent molecule. In the same figure the computer-simulated values [102] are also plotted for comparison with the calculated results. The calculated results are in good agreement with the computer simulations. Both the theoretical results and the computer simulation studies show an enhanced diffusion for size ratios TZ TZ = 01/02) between 1.5 and 15. This is due to the sharp decoupling of the solute dynamics from the solvent density mode. 

Spiegler has used the friction model to describe a system consisting of sodium ions (1), chloride ions (2), water (3) and a charged matrix (4). He neglects the interaction of the sodium ions with the chloride ions. Then five independent measurements are needed to calculate the friction coefficients. Spiegler chose to be measured the self-diffusion coefficient... 

The biggest differences are obtained for the lower (p jC pc) density points at each temperature, where the calculated ratios are up to approximately 20 smaller than the molecular dynamics results. This result is not surprising at the lower densities, the effects of attractive forces become important and cause diffusion in a real gas to be slower. Dymond (25) has found that the theory predicted the experimental self-diffusion coefficients for densities down to about 0.7 Pc for T > Tc. 

MD simulation is advantageous for obtaining dynamic properties directly, since the MD technique provides not only particle positions but also particle velocities that enable us to utilize the response theory (e.g., the Kubo formula [175,176]) to calculate the transport coefficients from time-dependent correlation functions. For example, we will examine the self-diffusion process of a tagged PFPE molecular center of mass (Fig. 1.49) from the simulation to gain insight into the excitation of translational motion, specifically, spreading and replenishment. The squared displacement of the center mass of a molecule or a bead is used as a measure of translational movement. The self-diffusion coefficient D can be represented as a velocity autocorrelation function... 

---