Temperature Variation of the Diffusion Coefficient

Diffusion coefficients vary considerably with temperature. This variation is generally expressed in terms of the Arrhenius equation  

The activation energy can be determined from the gradient of a plot of In D versus 1 IT (Fig. 5.19). Such graphs are known as Arrhenius plots. Diffusion coefficients found in the literature are usually expressed in terms of the Arrhenius equation D0 and Ea values. Some representative values for self-diffusion coefficients are given in Table 5.2. 

The probability that an atom will successfully move can be estimated by using Maxwell-Boltzmann statistics. The probability p that an atom will move from one position of minimum energy to an adjacent position is 

TABLE 5.2 Representative Values for Self-diffusion Coefficients  

The atoms in a crystal are vibrating continually with frequency v, which is usually taken to have a value of about 1013 Hz at room temperature. It is reasonable to suppose that the number of attempts at a jump, sometimes called the attempt frequency, will be equal to the frequency with which the atom is vibrating. The number of successful jumps that an atom will make per second, the jump frequency T, will be equal to the attempt frequency v, multiplied by the probability of a successful move, that is, 

Note literature values for self-diffusion coefficients vary widely, indicating the difficulty of making reliable measurements. The values in this table are intended to be representative only. The values of diffusion coefficients in the literature are mostly given in cm s to convert the values given here to cm s multiply by 10. Do, pre-experimental, or frequency, factor E, activation 

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There is experimental evidence to suggest that anion and cation diffusion can have different mechanisms [70]. The temperature variation of the diffusion coefficients of 1,1 P and 7 Li in aPEO-LiPF6 shows quite different trends. The 31P diffusion coefficients follow a VTF-type de-... 

The diffusion coefficients, as expected, increase with increasing temperature. Variation of the diffusion coefficient as a function of temperature can be expressed in terms of the Arrhenius equation, which, in logarithmic form, is... 

In Figure 5, the accumulation real point, is related to the temperature variation of the diffusion coefficient. According to the Pick s law and tacking account Eq. 2, the following equation may be obtained... 

The mobility or diffusion of die atoms over the surface of die substrate, and over the film during its formation, will occur more rapidly as the temperature increases since epitaxy can be achieved, under condition of ctystallographic similarity between die film and the subsuate, when the substrate temperamre is increased. It was found experimentally that surface diffusion has a closer relationship to an activation-dependent process than to the movement of atoms in gases, and the temperamre dependence of the diffusion of gases. For surface diffusion the variation of the diffusion coefficient widr temperature is expressed by the Anhenius equation... 

Whatever the technique used, it is important to note that (i) only an equivalent viscosity can be determined, (ii) the response of a probe may be different in solvents of the same viscosity but of different chemical nature and structure, (iii) the measured equivalent viscosity often depends on the probe and on the fluorescence technique. Nevertheless, the relative variations of the diffusion coefficient resulting from an external perturbation are generally much less dependent on the technique and on the nature of the probe. Therefore, the fluorescence techniques are very valuable in monitoring changes in fluidity upon an external perturbation such as temperature, pressure and addition of compounds (e.g. cholesterol added to lipid vesicles alcohols and oil added to micellar systems). 

The diffusion coefficients in EFLs with alcohol/H20 mixtures were also studied [23,24]. Figure 9.3 shows the variation of the diffusion coefficient of benzene as a function of temperature (299-393 K) for EFL mixtures where the mole ratio of methanol/H20 was maintained at 0.70/0.30 and the amount of CO2 was increased from 0 to 0.40 mole fraction [23]. At 313 K, the addition of 40 mol% CO2 caused a 100% increase in the diffusion coefficient of benzene. However, increasing the temperature and the proportion of CO2 caused the largest increase in the diffusion coefficient. Over a 500% increase in the diffusion coefficient of benzene is observed when the temperature is increased to 363 K and 0.30 mole fraction CO2 is combined with the 0.70/0.30 mole ratio methanol/H20 mixture. 

Souvignet et al. [24] studied the variation of the diffusion coefficients of nonpolar compounds, such as benzene and anthracene, and polar compounds, such as m-cresol and nitrophenol, in ethanol/H20/C02 mixtures. Figure 9.4 shows the variation of benzene s diffusion coefficient in 0.61/0.39 mole ratio ethanol/H20 mixtures as a function of added CO2 (0 0 mol%) and temperature (299-333 K). For the ethanol/H20 mixture increasing the temperature from 298 to 333 K caused a 95% increase in the diffusion coefficient of benzene while adding 40 mol% CO2 to the ethanol/H20 mixtures increased the diffusion coefficient by 213%. However, the combination of both the addition of 40 mol% CO2 and increasing the temperature to 333 K provided a... 

Figure 2.10 Arrhenius plot showing temperature variation of He diffusion coefficient in carbonado (diamond), a indicates the size (in micrometers) of the pulverized powder. Note that the diffusion is characterized by two distinct activation energies. Reproduced from Zashu and Hiyagon (1995).

The experimental a versus x dependence for these samples, together with the fitting curves, are shown in Fig. 53. Note that in contrast to the previous example, these data are obtained at a constant sample composition. Now, Variations of the parameters a and x are induced by temperature variation. As mentioned above, the exponents a as well as the relaxation time x are functions of different experimentally controlled parameters. The same parameters can affect the structure or the diffusion simultaneously. In particular, both a and x are functions of temperature. Thus, the temperature dependence of the diffusion coefficient in (144) should be considered. Let us consider the temperature dependence of the diffusion coefficient D ... 

Figure 6. Variation of the diffusion coefficient of polyacrylamide gels on a double logarithmic scale as a function of their concentration at swelling equilibrium. The temperature was 22 C. The slope was 0.74 and agreed with the exponent predicted by equation 21.

Katsanos, N.A. Karaiskakis, G. Temperature variation of gas diffusion coefficients measured hy the reversed-flow sampling technique. J. Chromatogr. 1983, 254,15 25. 

For the most part, the separation factor decreases as the temperature increases. The transport model shows that the diffusion coefficient for ethylene is more sensitive to temperature variation than the diffusion coefficient of the Ag -ethylene complex. It seems reasonable that the diffusion coefficient of ethane is increased by a larger number than that of the complex. The decreasing separation factor may simply be explained by the fact that temperature favors the transport of ethane over the Ag" - ylene complex. When the ethylene partial pressure becomes small, the transport mechanism is controlled by gas solubility and the concentration of gas at gas-membrane interface. The amount of Ag -ethylene complex is decreased by the snuill amount of ethylene in the membrane and the effect of temperature becomes pronounced. 

The variation of the diffusion coefficient with concentration, temperature, Tg and permeant size can be readily explained in terms of the free volume of the system. In particular, a model proposed by Fujita " has been found to describe satisfactorily the diffusion of a number of organic vapours and liquids in polymers above their Tg. 

Values for G(unknown) were experimentally determined by using the previously calibrated cells, and these data were used to calculate values for D(unknown) using the cell constants. The overall average value of D(unknown) was 1.11 x 1(T5, which compares well with a reported value of 1.1 X 10 5. The coefficient of variation associated with the diffusion coefficient was 2.7% for one cell and 1.7% for a second cell. This calibration procedure thus provided information about the accuracy and precision of the method as well as the effect of temperature and concentration on the determination of the diffusion coefficient. 

Base Polymer. The water uptake properties of polymers can be assessed by immersing films in water and recording increases in weight. The diffusion coefficient can be obtained from such data.49 Values of the diffusion coefficient were given in Table 15.13. There is a wide variation in the maximum amount of water absorbed by polymeric resins. Certain systems have a very low absorption at the lower temperatures, but this breaks down at higher temperatures. 

The contact point = c is a critical consolute point. The calculated critical values of the virial coefficient and of the droplet volume fraction (B =-21 and <(ic JO. 13) for a hard-sphere model with an attractive potential are in qualitative agreement with the experimental observations (Figure 2). Around those critical values, a very large turbidity is observed. If the temperature is varied, the microemulsion separates into two turbid microemulsions. Angular variations of the scattered intensity and of the diffusion coefficient are observed (16) but the correlation function remains exponential. All these features are characteristic of the vicinity of a critical consolute point. The data can be fitted with theoretical predictions (17) ... 

Fig. 14.2. Variation of the hydrodynamic swelling of a polystyrene chain in cyclohexane, with respect to temperature. The results come from measurements of the diffusion coefficient of a polymer, by light scattering (according to Perzynski, Adam, and De-Isanti3) and were extrapolated to zero concentration. The figure shows how the ratio...

Fig. 3. Variation with temperature of the diffusion coefficients for various simulated fluids and actual laboratory fluids. Sources of data are, from left to right LJ argon, simulated Refs. 7 (DC) and 12 (C) laboratory. Ref. 41 bard spheres (for which temperature axis corresponds to pV/NkT X.50), Ref. 82 soft spheres. Ref. 20 xenon. Ref. 41 toluene. Ref. 42 methyl cyclohexane. Ref. 43 carbon tetrachloride. Ref. 44 o-terphenyl. Ref. 45 molten KQ, simulated using Tosi-Fumi (TF) potential parameters. Ref. S repellent Gaussian core particles. Ref. 21 (F. H. Stillinger kindly deduced the values his simulation results would infer for argonlike particles in familiar units) Na ions diffusing in molten 6KN03-4Ca(N0j)2 solvent medium. Ref. 46. The dashed extension of lower temperature in the case of xenon is based on the Arrhenius parameters quoted for the data. ...

The variation of a diffusion coefficient with temperature is given by ... 

The membrane conductivity data used in the present calculation are those for a carboxylic acid membrane [97-99], as the overall conductivity of a composite membrane is dictated by the carboxylic layer. Diffusion coefficient data for chloride and chlorate ions across the commercial composite membranes are not available. Hence, an average of the diffusion coefficient data for the chloride ion for carboxylic acid membranes [100,101] was used, assuming the same temperature dependence as that of membrane conductivity. The diffusion coefficient of the chlorate ion was assumed to be the same as that of the chloride ion. Variations in Dq- and k with anolyte concentration, under commercial operating conditions, were reported [102] to be weak, and hence, these dependencies were not considered here. 

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