Self-diffusion water model

FIG. 24 Water self-diffusion in polyacrylamide gels with various Bis content with comparison to various models from the literature. (Reprinted from Ref. 290, Copyright 1998, Academic Press.)... 

A simple two-state model for the observed water proton self-diffusion may be put forward in the form... 

FIG. 4 Apparent mole fraction (x) water in continuous phase of brine, decane, and AOT microemulsion system derived from the water self-diffusion data of Fig. 3 using the two-state model of Eq. (1). 

Independent self-diffusion measurements [38] of molecularly dispersed water in decane over the 8-50°C interval were used, in conjunction with the self-diffusion data of Fig. 6, to calculate the apparent mole fraction of water in the pseudocontinuous phase from the two-state model of Eq. (1). In these calculations, the micellar diffusion coefficient, D ic, was approximated by the measured self-dilfusion coefficient for AOT below 28°C, and by the linear extrapolation of these AOT data above 28°C. This apparent mole fraction x was then used to graphically derive the anomalous mole fraction x of water in the pseudocontinuous phase. These mole fractions were then used to calculate values for... 

One of the most convincing tests of the AG relationship appeared in the work of Scala et al.92 for the SPC/E model of water,57 which is known to reproduce many of water s distinctive properties in its super-cooled liquid state qualitatively. In this study, the dynamical quantity used to correlate with the configurational entropy was the self-diffusivity D. Scala et al. computed D via molecular dynamics simulations. The authors calculated the various contributions to the liquid entropy using the methods described above for a wide range of temperature and density [shown in Figure 12(a-c)]. 

Figure 13 Self-diffusivity D versus configurational entropy, Sc = Scon, for the SPC/E water model at various density p values. The lines are fits to the AG form given by Eq. [10] with tr tx 1/D. Reprinted with permission from Ref. 92.

Results in Table I illustrate some of the strengths and weaknesses of the ST2, MCY and CF models. All models, except the MCY model, accurately predict the internal energy, -U. Constant volume heat capacity, Cv, is accurately predicted by each model for which data is available. The ST2 and MCY models overpredict the dipole moment, u, while the CF model prediction is identical with the value for bulk water. The ratio PV/NkT at a liquid density of unity is tremendously in error for the MCY model, while both the ST2 and CF models predictions are reasonable. This large error using the MCY model suggests that it will not, in general, simulate thermodynamic properties of water accurately (29). Values of the self-diffusion coefficient, D, for each of the water models except the CF model agree fairly well with the value for bulk water. 

The simple water charmel models can explain the ionomer peak and the small-angle upturn in the scattering data of fhe unoriented samples as well as of the oriented films. Interestingly, the helical structure of backbone segments is responsible for fhe sfabilify of fhe long cylindrical charmels. The self-diffusion behavior of wafer and protons in Nation is well described by the water channel model. The existence of parallel wide channels af high wafer uptake favors large hydrodynamic confributions to electro-osmotic water transport and hydraulic permeation. 

Pulsed field gradient (PFG)-NMR experiments have been employed in the groups of Zawodzinski and Kreuer to measure the self-diffusivity of water in the membrane as a function of the water content. From QENS, the typical time and length scales of the molecular motions can be evaluated. It was observed that water mobility increases with water content up to almost bulk-like values above T 10, where the water content A = nn o/ nsojH is defined as the ratio of the number of moles of water molecules per moles of acid head groups (-SO3H). In Perrin et al., QENS data for hydrated Nation were analyzed with a Gaussian model for localized translational diffusion. Typical sizes of confining domains and diffusion coefficients, as well as characteristic times for the elementary jump processes, were obtained as functions of A the results were discussed with respect to membrane structure and sorption characteristics. ... 

In that case the self diffusion coefficient - concentration curve shows a behaviour distinctly different from the cosurfactant microemulsions. has a quite low value throughout the extension of the isotropic solution phase up to the highest water content. This implies that a model with closed droplets surrounded by surfactant emions in a hydrocarbon medium gives an adequate description of these solutions, found to be significantly higher them D, the conclusion that a non-negligible eimount of water must exist between the emulsion droplets. 

Packer and Rees [3] extended the work of Tanner and Stejskal by the development of a theoretical model using a log-normal size distribution function. Measurements made on two water-in-oil emulsions are used to obtain the self-diffusion coefficient, D, of the water in the droplets as well as the parameters a and D0 0. Since then, NMR has been widely used for studying the conformation and dynamics of molecules in a variety of systems, but NMR studies on emulsions are sparse. In first instance pulsed field gradient NMR was used to measure sdf-diffusion coefficients of water in plant cells (e.g. ref. [10]). In 1983 Callaghan... 

Fleisher et al. [12] studied the self-diffusion of oil and water in rape seeds. The selfdiffusion of oil was found to be completely restricted. The experiments could be explained in toms of the model of diffusion within spherical droplets and a Gaussian mass distribution of the droplet radii. At the same time Van den Enden et al. [9] introduced the technique described above. It is a rapid method for the determination of water droplet size distributions in spreads by using low resolution pulsed field gradient NMR. Their method was based on the recognition that a set of echo attenuation values (R) as a function of the field gradient pulsed width, obtained under conditions where R is independent of the time allowed for diffusion, contains all the necessary information on the water droplet size distribution (see above). A log-normal distribution of water droplet sizes was assumed. 

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... 

Abstract We use Nuclear Magnetic Resonance relaxometry (i.e. the frequency variation of the NMR relaxation rates) of quadrupolar nucleus ( Na) and H Pulsed Gradient Spin Echo NMR to determine the mobility of the counterions and the water molecules within aqueous dispersions of clays. The local ordering of isotropic dilute clay dispersions is investigated by NMR relaxometry. In contrast, the NMR spectra of the quadrupolar nucleus and the anisotropy of the water self-diffusion tensor clearly exhibit the occurrence of nematic ordering in dense aqueous dispersions. Multi-scale numerical models exploiting molecular orbital quantum calculations, Grand Canonical Monte Carlo simulations, Molecular and Brownian Dynamics are used to interpret the measured water mobility and the ionic quadrupolar relaxation measurements. 

Our previous study (J 6) of self diffusion in compressed supercritical water compared the experimental results to the predictions of the dilute polar gas model of Monchick and Mason (39). The model, using a Stockmayer potential for the evaluation of the collision integrals and a temperature dependent hard sphere diameters, gave a good description of the temperature and pressure dependence of the diffusion. Unfortunately, a similar detailed analysis of the self diffusion of supercritical toluene is prevented by the lack of density data at supercritical conditions. Viscosities of toluene from 320°C to 470°C at constant volumes corresponding to densities from p/pQ - 0.5 to 1.8 have been reported ( 4 ). However, without PVT data, we cannot calculate the corresponding values of the pressure. 

Another descriptor of the mobility of water molecules in contact with the clay layers is the water self-diffusion coefficient. A fine recent review summarizes the theoretical and practical aspects of measurement by spin-echo nmr methods of this parameter (36) The plot of the decrease in the water self-diffusion coefficient as a function of C, the amount of suspended clay, for the same samples, is again a straight line going through the origin. By resorting once more to a similar analysis in terms of a two-state model (bound and "free water), one comes up (25) with a self-diffusion coefficient, for those water molecules pinched in-between counterions and the clay surface, of 1.6 10 15 m2.s 1,... 

While the clathrate model is attractive, it is not correct to assume that the water is organized in some long-lived structure the observation that the self-diffusion coefficient for co-sphere water is larger than that for the solute rules this out. However, the rotational correlation time is shorter for ethanol and t-butyl alcohol in water (in the clathrate cage ) than in the pure liquid (Goldammer and Hertz, 1970 Goldammer and Zeidler, 1969). Nmr experiments show that in water the solvent dipole moments point away from the apolar groups (Hertz and Radle, 1973). 

Anderko and Lencka find. Eng. Chem. Res. 37, 2878 (1998)] These authors present an analysis of self-diffusion in multicomponent aqueous electrolyte systems. Their model includes contributions of long-range (Coulombic) and short-range (hard-sphere) interactions. Their mixing rule was based on equations of nonequilibrium thermodynamics. The model accurately predicts self-diffusivities of ions and gases in aqueous solutions from dilute to about 30 mol/kg water. It makes it possible to take single-solute data and extend them to multicomponent mixtures. 

Experiments were carried out with Ionac MC 3470 to determine the self-diffusion coefficient values for H+ and Al + in the coupled transport. Data points were used from the experiment involving 2N acid sweep solution in Figure 34.24b, presented later. These values formed the basis for aluminum transport rate or flux (7ai) calculation at different time intervals. The equilibrium data generated in Figure 34.20b were used in conjunction with Equation 34.25 to determine the interdiffusion coefficient values. Local equilibrium was assumed at the membrane-water interface. Eigure 34.24a shows computed Dai,h values for this membrane. When compared with Dai,h values for Nafion 117, it was noticed that the drop in interdiffusion coefficient values was not so steep, indicative of slow kinetics. The model discussed earlier was applied to determine the self-diffusion coefficient values of aluminum and hydrogen ions in Ionac MC 3470 membrane. A notable point was that the osmosis effect was not taken into account in this case, as no significant osmosis was observed in a separate experiment. 

The spin-lattice relaxation time 7] as a function of temperature T in liquid water has been studied by many researchers [387-393], and in all the experiments the dependence T (T) showed a distinct non-Arrhenius character. Other dynamic parameters also have a non-Arrhenius temperature dependence, and such a behavior can be explained by both discrete and continuous models of the water structure [394]. In the framework of these models the dynamics of separate water molecules is described by hopping and drift mechanisms of the molecule movement and by rotations of water molecules [360]. However, the cooperative effects during the self-diffusion and the dynamics of hydrogen bonds formation have not been practically considered. 

Many physical properties undergo dramatic changes in value as water is heated and pressurized from sub- to supercritical conditions, particularly in the region of the critical point where some properties such as heat capacity reach a singularity. This change in behavior means that more familiar correlations of properties measured at subcritical conditions are likely to be inaccurate when applied at supercritical conditions. There have been some experimental studies performed to measure, tabulate, and in some cases correlate values of key properties of supercritical water, such as the self-diffusion coefficient, viscosity,thermal conductivity," heat capacity at constant volume," dielectric constant," and selfdissociation constant." " Far more work has been devoted to calculation of property values from models fitted empirically to data or developed more rigorously through molecular simulation. For PVT data and its derivatives, several attempts... 

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