Temperature-dependent diffusivity coefficients

The second approach employs a detailed reaction model as well as the diffusion of EG in solid PET [98, 121-123], Commonly, a Fick diffusion concept is used, equivalent to the description of diffusion in the melt-phase polycondensation. Constant diffusion coefficients lying in the order of Deg, pet (220 °C) = 2-4 x 10 10 m2/s are used, as well as temperature-dependent diffusion coefficients, with an activation energy for the diffusion of approximately 124kJ/mol. 

Fig. 4.23 also indicates a slight decrease of the signal plateau which, at a first glance, was unexpected. In the following, a reactive dispersion model given in ref. [37] is applied to deduce rate constants for different reaction temperatures. A trapezoidal response function will be used. The temperature-dependent diffusion coefficient was calculated according to a prescription by Hirschfelder (e.g., [80], p. 68 or [79], p. 104] derived from the Chapman-Enskog theory. For the dimensionless formulation, the equation is divided by M/A (with M the injected mass and A the cross-section area). This analytical function is compared in Fig. 4.24 with the experimental values for three different temperatures. The qualitative behavior of the measured pulses is well met especially the observed decrease of the plateau is reproduced. The overall fit is less accurate than for the non-reactive case but is sufficient to now evaluate the rate constant. 

The pressure and temperature dependent diffusion coefficient of the adsorbate in the carrier gas is approximated according to Gilliland [18] ... 

With the value of e obtained from the analysis of isotherm data, it was possible to obtain temperature dependent diffusion coefficients in the Henry s Law region for CH4 in silica gel that are in fair agreement with experiment [17]. 

Figure 7.14 Arrhenius plots of temperature-dependent diffusivity coefficients of C6 alkanes within silicalite indicate that diffusion is increasingly hindered in the medium-pore channels as the degree of branching increases. Cyclohexane also diffuses very slowly. [Reproduced from reference 133 with permission. Copyright 1995 American Chemical Society.]...

Figure 10.16 shows the Arrhenius plots with the temperature-dependent diffusion coefficients of several solids. The differences can be considerable. For some applications good ionic conductivity is required while other materials are chosen because they are good diffusion barriers. 

The electrical and optical characteristics of Pt diffusion in n-type GaN film were investigated. The diffusion extent was characterized by secondary ion mass spectrometry. The temperature-dependent diffusion coefficients of Pt in n-GaN were 4.158 X 10-4, 1 572 x 10 3 and 3.216 x 10-3cm2/s at 650, 750 and 850C, respectively. The data could be described by ... 

The development of a scientific understanding of diffusion in liquid-phase polymeric systems has been largely due to Duda et al. (1982), Ju et al. (1981), and Vrentas and Duda (1977a,b, 1979) whose work in this area has been signal. In their most recent work, Duda et al. (1982) have developed a theory which successfiilly predicts the strong dependence of the diffusion coefficient on temperature and concentration in polymeric solutions. The parameters in this theory are relatively easy to obtain, and in view of its predictive capability this theory would seem to be most appropriate for incorporating concentration-dependent diffusion coefficients in the diffusion equation. 

Another distinctive feature of strong tunnelling recombination could be seen after a step-like (sudden) increase (decrease) of temperature (or diffusion coefficient - see equation (4.2.20)) when the steady-state profile has already been reached. Such mobility stimulation leads to the prolonged transient stage from one steady-state y(r,T ) to another y(r,T2), corresponding to the diffusion coefficients D(T ) and >(72) respectively. This process is shown schematicaly in Fig. 4.2 by a broken curve. It should be stressed that if tunnelling recombination is not involved, there is no transient stage at all since the relevant steady state profile y(r) — 1 - R/r, equation (4.1.62), doesn t depend on >( ). 

Results are shown in Figs. 12 and 13. All blend specimens were set iso-thermally above LCST and kept there for a maximum of 5 min. As will be seen, this corresponds only in some cases to an early stage of spinodal decomposition depending on temperature. The diffusion coefficients governing the dynamics of phase dissolution below LCST are in the order of 10"14 cm2 s"1. Figure 12 reflects the influence of the mobility coefficient on the phase dissolution. As can be seen, the apparent diffusion coefficient increases with increasing temperature of phase dissolution which expresses primarily the temperature dependence of the mobility coefficient. Furthermore, it becomes evident that the mobility obeys an Arrhenius-type equation. Similar results have been reported for phase dis-... 

Figure 3. In the short-memory limit (rac —> oo), the quantum time-dependent diffusion coefficient DP - 00 piotte(j as a function of yt for several different bath temperatures on both sides of Tc (full lines) yrth = 0.25(T — 2TC, classical regime) yt — 0.5 (T = Tc, crossover) yrth = 1 (T = Tc/2, quantum regime) yt = +oo (T = 0). The corresponding curves for the classical diffusion coefficient Dcl( ) are plotted in dotted lines in the same figure.

When compared with the right-hand side of Eq. (126), the right-hand side of Eq. (129) appears to be formally identical to the time-dependent diffusion coefficient Drcff(t — f) at the effective temperature 7 err = l//cpeff. One can thus rewrite Eq. (129) in the following more compact form ... 

Equation (130), in turn, allows the determination of the effective temperature as a function of t — t and t — to- Since the time-dependent diffusion coefficient is, at any positive time, a monotonic increasing function of the temperature, Eq. (130) yields for Teff t t. t — to) a uniquely defined value. 

In the classical limit in which the time-dependent diffusion coefficient is proportional to the temperature, the inverse effective temperature (S f deduced in this limit from Eq. (130) satisfies the equation... 

In contrast with its classical counterpart, the inverse effective temperature is not given by a simple ratio of time-dependent diffusion coefficients like Eq. (131), but has to be deduced from the implicit equation (130), which in general can only... 

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