Diffusion simulation

Figure 3. Comparison of the simulated diffusion coefficient (0) with the theoretical value Do (solid line). The simulation parameters are a = n/2, L/a = 100, i. m 1. and kBT =...

H. -C. Shin et al., A study on the simulated diffusion-limited current transient of a self-affine fractal electrode based upon the scaling property, J. Electroanal. Chem., 531 p. 101, Copyright 2002, with permission from Elsevier Science. 

Figure 8. Simulated current transients obtained from the self-affine fractal profiles h(x) of various morphological amplitudes rj of (a) 0.1, 0.3, and 0.5 (b) 1.0, 2.0, and 4.0 in h(x) = 7]fws(x). Reprinted from H.-C. Shin et al., A study on the simulated diffusion-limited current transient of a self-affine fractal electrode based upon the scaling property, J. Electroanal. Chem., 531, p. 101, Copyright 2002, with permission from Elsevier Science.

Although detailed microscopic calculations of the problem mentioned above are not available, there exist several computer simulation studies [102, 117], which also find the anomalous enhanced diffusion, even for simple model potentials such as the Lennard-Jones. The physical origin of the enhanced diffusion is not clear from the simulations. The enhancement can be as large as 50% over the hydrodynamic value. What is even more surprising is that the simulated diffusion constant becomes smaller than the hydro-dynamic prediction for very small solutes, with sizes less than one-fifteenth of the solvent molecules. These results have defied a microscopic explanation. 

Transport of Silica along a Temperature Gradient. This example simulates diffusion of silicic acid into a domain where temperature decreases with increasing distance from the inlet at X = 0. The temperature field is steady in time. Quartz precipitates according to the reaction... 

Much interest remains, however, in the question of the long time limiting density because Fig. 4 indicated that the deviation from the Batchinski equation seen in all laboratory studies at low D becomes greater than the uncertainty in the simulation diffusion coefficients (only) at Z)<5x 10" cm sec. At this diffusivity, relations discussed in Section II. B imply that the time needed for full structural equilibration would exceed 10 psec. The dependence of D on the degree of equilibration is not known at this time, but it seems probable that a major computing effort will be needed to determine whether the volume dependence of the dense atomic liquid diffusivity is as simple and significant as (13) implies or is more complex, as for the known behavior of low-diffusivity laboratory liquids (Fig. 4). 

Figure 9.11. Simulated diffusion attenuation profiles illusttating die influence of convection. Trace (a) illusttates signal decay for a diffusion coefficient of 10 x in...

Figure 9.7 Difference from experiment of simulated diffusion constant, AD [solid symbois, ieft axis], and of simulated temperature of maximum density, A md [open symbols, right axis], as a function of S. The linear regressions of D [dashed line] and Tmd [dotted line] given in Eqs. 9.5 and 9.6 are also shown. Symbols Indicate three- [triangles], four- [diamonds], five-[squares], and multipole [circles] models, with TIP3P, TlP4P-Ew, T1P5P-E, and SSDQOl indicated by larger symbols.

Qi - Q2 refer to the difference in concentration between the surface and the bulk, and HR C is analogous to D X. We thus simulate diffusion within the Wagner approximation by choosing two capacities whose values are the same. 

The analysis of fast polymerisation reactions has shown that the effects, revealed during the mathematical simulation (diffusion model), are identical to the experimental effects of the cationic polymerisation of isobutylene (as an example). The important consequence of process nonisothermicity is its adverse effect on polymer quality, while the external thermostating is not effective enough in this case [52],... 

The models mentioned so far are limited in their application as they represent only first order reaction kinetics with Fickian diffusion, therefore do not allow for multicomponent diffusion, surface diffusion or convection. Wood et al. [16] applied the algorithms developed by Rieckmann and Keil [12,44] to simulate diffusion using the dusty gas model, reaction with any general types of reaction rate expression such as Langmuir-Hinshelwood kinetics and simultaneous capillary condensation. The model describes the pore structure as a cubic network of cylindrical pores with a random distribution of pore radii. Transport in the single pores of the network was expressed according to the dusty gas model as... 

The advantage of computer simulations is the possibility of obtaining transport data that cannot or can only barely be measured. It is possible in this way to simulate diffusion coefficients of solvent molecules in the ionic solvation shells and to compare them with those of the bulk solvent molecules and with those of the ions, or to study transport coefficients at different time scales. 

Figure 4.40. Simulated diffusion constants in faujasite as a function of hydrocarbon chain length .

Figure la. Fractal aggregate, constructed by computer simulated diffusion limited aggregation. Fractal dimension D = 1.44... 

Diffusion of 0.075 g of three Hair Relaxers through a semi-permeable membrane toward a NaCI 9 g/L solution, simulating diffusion through skin... 

The most straightforward approach for simulating diffusion through a glassy polymer is the frozen polymer method. The essential idea of the frozen polymer method is that polymer chains remain fixed in place and provide a static external field, through which a small-molecule penetrant can diffuse. 

We introduce here the way in which we solve analogue (i.e. continuous) problems digitally, for diffusion processes in the bulk of the solution. That is to say, we digitally simulate diffusion in solution, given a boundary concentration at the electrode and one at some large distance from it. How large this must be will be discussed in this chapter also. The way in which we get Cq, the concentration at the electrode, is not a diffusion problem and will be dealt with in Chapt. 4. 

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