Self-diffusion coefficients temperature dependence

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]. 

Figure 2 The temperature dependence of the self-diffusion coefficient of 2,3-dimethyl-butane predicted by Cohen and Turnbull s free volume model. (From Ref. 25.)...

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. ... 

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.)...

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 ... 

Wool and O Connor [33] stated that the self-diffusion coefficient should follow a Williams-Landel-Ferry (WLF) temperature dependence providing that the mode of failure remains the same between samples healed at different temperatures between Tg and Tg + 100°C. Using a reference temperature of 196°C (469 K), the WLF relationship for PI700 polysulfone can be written as follows [38] ... 

Figure 4 Temperature dependence of the experimental self-diffusion coefficient of propane molecules in NaX.

Self diffusion coefficients can be obtained from the rate of diffusion of isotopically labeled solvent molecules as well as from nuclear magnetic resonance band widths. The self-diffusion coefficient of water at 25°C is D= 2.27 x 10-5 cm2 s 1, and that of heavy water, D20, is 1.87 x 10-5 cm2 s 1. Values for many solvents at 25 °C, in 10-5 cm2 s 1, are shown in Table 3.9. The diffusion coefficient for all solvents depends strongly on the temperature, similarly to the viscosity, following an Arrhenius-type expression D=Ad exp( AEq/RT). In fact, for solvents that can be described as being globular (see above), the Stokes-Einstein expression holds ... 

The variation in the self-diffusion coefficient is primarily determined by the change in density varied over a wide range. There is no significant temperature dependence within the error of the measurements. The data cover a range of densities 0.34 <. p/pc < 2.5, and temperature 1.14 T/Tc <. 1.41. 

The experimental temperature dependence is much more closely reproduced by the empirical correlations of Dawson et al. (43) and of Mathur and Thodos (44). The exact experimental values are not reproduced by any of the methods. However, considering the difference in molecular weight between toluene and tolu-ene-d0,the approximations involved,and the error in the experimental values (which gets higher as the density decreases), the correlation of Mathur and Thodos gives a very good estimation of the self diffusion coefficient in supercritical toluene. 

The practical advantage of these relations is that, in MD simulations, single molecule properties like the self-diffusion coefficient and rotational relaxation times converge much faster than system properties due to additional averaging over the number of molecules in the ensemble. We applied eqs. 10 and 11 to our MD results using data at 800 K as a reference point in order to predict the viscosity over the entire temperature interval. In Fig. 7 we compare the predicted values with those obtained from simulation. It appears that in the temperature interval 600 K to 800 K predictions of Eq. (10) are more consistent with MD results than are the predictions of Eq. (11). This leads us to conclude that the viscosity temperature dependence in liquid HMX is more correlated... 

Here, Ds and Dd are the coefficients representing the Soret and Dufour effects, respectively, Du is the self-diffusion coefficient, and Dik is the diffusion coefficient between components / and k. Equations (7.149) and (7.150) may be nonlinear because of, for example, reference frame differences, an anisotropic medium for heat and mass transfer, and temperature- and concentration-dependent thermal conductivity and diffusion coefficients. 

Figure 5.3 depicts the Arrhenius plots of the apparent self-diffusion coefficient of the cation (Dcation) and anion (Oanion) for EMIBF4 and EMITFSI (Figure 5.3a) and for BPBF4 and BPTFSI (Figure 5.3b). The Arrhenius plots of the summation (Dcation + f anion) of the cationic and anionic diffusion coefficients are also shown in Figure 5.4. The fact that the temperature dependency of each set of the self-diffusion coefficients shows convex curved profiles implies that the ionic liquids of interest to us deviate from ideal Arrhenius behavior. Each result of the self-diffusion coefficient has therefore been fitted with VFT equation [6].

The temperature dependencies of the viscosity (Figure 5.6) and the summation of the self-diffusion coefficient (Dcation + Oanion) (Figure 5.4) interestingly show the contrasted profiles with the indication of inverse relationship between viscosity and self-diffiision coefficient. This can be explained in terms of the Stokes-Einstein equation, which correlates the self-diffusion coefficient (Dcation Danion) with viscosity (q) by the following relationship ... 

Self-diffusivity, cooperatively with ionic conductivity, provides a coherent account of ionicity of ionic liquids. The PGSE-NMR method has been found to be a convenient means to independently measure the self-diffusion coefficients of the anions and the cations in the ionic liquids. Temperature dependencies of the self-diffusion coefficient, viscosity and ionic conductivity for the ionic liquids, cannot be explained simply by Arrhenius equation rather, they follow the VFT equation. There is a simple correlation of the summation of the cationic and the anionic diffusion coefficients for each ionic liquid with the inverse of the viscosity. The apparent cationic transference number in ionic liquids has also been found to have dependence on the... 

We present here the results of such a systematic investigation on the dependence of the self-diffusion coefficient of flexible polymer chains as a function of P and N, conducted on polydimethylsiloxane (PDMS). This model polymer is well above its glass temperature at room temperature (Ta = - 120°C), so that one can expect that spurious effects associated with the variation of the free volume and of the local monomer-monomer friction coefficient with the molecular weights of the chains are minimised. 

As is seen, when the temperature rises, a considerable decoupling of the Li ions in liquid LiCl seems to take place. Decoupling of the cations is likely to help cationic self-diffusion more than cationic conductance, because in self-diffusion cations must interchange positions while in conductance they may move in parallel. In fact, corresponding to the temperature dependence of n in molten LiCl, an increase of D+/Ta (D+ = self-diffusion coefficient of the cations, A = equivalent conductance) with absolute temperature T has been observed for all molten alkali halides checked (30). (Also work is in progress at Mainz to check LiCl.)... 

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