Time-dependent diffusion coefficient

In particular the use of solution of Eq. (2.37) or Eq. (2.39) allows one to calculate the moments in (2.35) preserving fast vanishing components with / < L. Consequently, the time-dependent diffusion coefficient D(t) makes it possible to recover information about the periphery of the spectrum... 

In this fitting procedure the time-dependent diffusion coefficient as discussed above has been taken explicitly into account. As can be seen, with only two parameters both sets of spectra are well described. This holds also for the other two samples. For the model parameters, the fit reveals Nc = 150 and A (0)/ = 0.18 ns. Nc is very close to Nc = 138 as obtained from the NSE experiments on long-chain PE (see next section). 

If Dh is indeed time dependent as in eq. (5) it is not obvious that C(x, t) will follow an error function expression as in eq. (3) or that >H will be thermally activated as in eq. (4). We now show that eqs. (3) and (4) still apply with a time dependent diffusion coefficient, by making a coordinate transformation (Kakalios and Jackson, 1988). The one-dimensional diffusion equation... 

All the solutions with constant diffusion coefficients can therefore be used for problems with time-dependent diffusion coefficients upon replacement of 3>t/X2 by t. 

Alternatively, we have described the system in terms of a time-dependent diffusion coefficient from the data presented in Fig. 5. This would also be conceivable if the diffusion coefficients described for ternary diffusion in Eqs. (21) and (22) were... 

Lastly, Callaghan and Pinder believed to have detected cooperative diffusion in their experiment48). Their polystyrene specimen of M = 2x 10 displayed a two-component echo attenuation, interpreted in terms of two time-dependent diffusion coefficients. The more rapid component was not observable at long diffusion times, and at short diffusion time (t = 16 ms) scaled with polymer concentration (2, 4, and 8 %) in the manner... 

Concern about fission-product release from coated reactor fuel particles and fission-product sorption by fallout particles has provided stimulus to understand diffusion. In a fallout program mathematics of diffusion with simple boundary conditions have been used as a basis for (1) an experimental method of determining diffusion coefficients of volatile solutes and (2) a calculational method for estimating diffusion profiles with time dependent sources and. time dependent diffusion coefficients. The latter method has been used to estimate the distribution of fission products in fallout. In a fission-product release program, a numerical model which calculates diffusion profiles in multi-coated spherical particles has been programmed, and a parametric study based on coating and kernel properties has provided an understanding of fission product release. 

Interpretation of Drug-Release Kinetics from Hydrogel Matrices in Terms of Time-Dependent Diffusion Coefficients... 

To examine the effect of time-dependent diffusion coefficient on the release behavior from a swellable polymer system containing dissolved or dispersed drug, we consider a polymer sheet with half thickness i, an initial drug loading A, a drug solubility in the polymer matrix C, and a time-dependent drug diffusion coefficient of the following form ... 

Figure 2. Effect of Deborah number (DEB)d on the Characteristic time-dependent diffusion coefficient.

To further illustrate the utility of the present time-dependent diffusion coefficient approach, data from Reference 21 for the thiamine HC1 release from initially dehydrated poly-HEMA sheets with different loading levels are analyzed with equations 3-7. The results are shown in Figure 9 and Table II. It has to be emphasized... 

Figure 9-2 Several sorption (A) and desorption (D) curves, a) normal diffusion with constant D-value b) increasing D-value with increasing permeant concentration c) decreasing D-value with increasing permeant concentration d) sorption with pronounced swelling e) concentration- and time-dependent diffusion coefficient.

P. I. Lee, Interpretation of Drug-Release Kinetics from Hydrogel Matrices in Terms of Time-Dependent Diffusion Coefficients, in Controlled-Release Technology Pharmaceutical Applications, P. I. Lee and W. R. Good (Eds.), ACS Symp. Series No. 348, American Chemical Society, Washington, D.C., 1987. p. 71. 

Schematic diagram of inhomogeneous networks in polymer gels characterised through the observed time-dependent diffusion coefficients... 

Analysis in Terms of the Time-Dependent Diffusion Coefficient... 

In fact, the validity of Eqs. (90) and (91) is not restricted to the simple (i.e., nonretarded) Langevin model as defined by Eq. (73). These formulas can be applied in other classical descriptions of Brownian motion in which a time-dependent diffusion coefficient can be defined. This is for instance, the case in the presence of non-Ohmic dissipation, in which case the motion of the Brownian particle is described by a retarded Langevin equation (see Section V). 

The quantum time-dependent diffusion coefficient D(t) = Cvv(tr) dt1 is an odd function of time given by ... 

The quantum time-dependent diffusion coefficient for t > 0 thus appears as the sum of the classical time-dependent diffusion coefficient (A 7"/t ) (1 — which from now on we shall denote as Dcl(f), and of a supplementary contribution due to quantum effects, which can be shown to be always positive. 

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.

Figure 4. Quantum time-dependent diffusion coefficient D c yj(t) plotted as a function of T/Tc for several different values of yt yt — 0.5 yt = 1 yt — 5 classical limit (y1).

Still assuming that the limit f, —> —oo has been taken from the beginning, one can compute Cxx(t, f to) = J o dt Cvv(fi — 2) < 2- However, the full expression of Cxx(t, t to) is of no use in the analysis of aging effects, which will be seen to rely only on the expression for the time-dependent diffusion coefficient... 

Note that writing formula (121) amounts to making use of the same argument as in the classical case, namely to apply the FDT as expressed by Eq. (46) to the variable x(t) - x(t ). Since one has, in terms of the time-dependent diffusion coefficient,... 

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