Boltzmann analysis

This method of extracting concentration-dependent D is usually referred to as Boltzmann analysis. 

In summary, Boltzmann analysis for a diffusion-couple profile involves the following steps ... 

If D depends on concentration, the first indication would come from the asymmetry of the diffusion-couple concentration profile. That is, there is no center symmetry with respect to x = 0, which means that one side approaches the end concentration more rapidly than the other side (Figure 3-28a). In such cases, D as a function of C can be obtained by Boltzmann analysis (Equation 3-58e) ... 

Figure 3-37 Compositional dependence of diffusivities. (a) Fe-Mg interdiffusivity along the c-axis in olivine as a hinction of fayalite content at P — O.l MPa and log /b2 =-6.9 0.1. Diffusion data are extracted using Boltzmann analysis. Some of the nonsmoothness is likely due to uncertainty in extracting interdiffusivity using the Boltzmann method. Data are from Chakraborty (1997). (b) Ar and CO2 diffusivity in melt as a function of H2O content. Data are from Watson (1991b) and Behrens and Zhang (2001).

The following diffusion data are adapted from experimental diffusion data for water diffusion in a basaltic melt (Zhang and Stolper, 1991). The experiment was carried out at 1 300°C and the duration of the experiment is 10 minutes. Using Boltzmann analysis to obtain diffusion coefficients or water as a function of water concentration. Hint You will probably need to use a spreadsheet program to do simple integration and differentiation. You may also try to write a simple program. You may fix the concentration at one end to be 0.410 and the other end to be 0.100.)... 

FIGURE 2.3-10 Matano-Boltzmann analysis of tree-diffusion experiment. Matano plane (2 = 0) Located by area ABC — area CDE. 

Sun C. and Munn L.L. Particulate nature of blood determines macroscopic rheology a 2-D lattice Boltzmann analysis. Biophys. J. 88 1635-1645, 2005. 

Coimterion condensation has detractors (28-34), who point to flaws in the concept s derivation, such as artificial subdivision of the counterions into two populations, inappropriate extrapolation of the Debye-Hiickel approximation to regions of high electrostatic potential, and inconsistent treatment of counterions. The full nonlinear Poisson-Boltzmann equation offers a more rigorous way to interpret electrostatic phenomena in electrolyte solutions, but the physical picture obtained through this equation is different in some ways from the one suggested by condensation (21,34,35). In particular, a Poisson-Boltzmann analysis does not readily identify distinct populations of condensed and free counterions but rather a smoothly varying Gouy-Chapman layer. Nevertheless, Poisson-Boltzmann-based... 

According to a nonlinear Poisson-Boltzmann analysis, the initial radial decay of f from a cylindrical chain segment is much steeper than predicted by a Debye-Hiickel analysis see Figure 3 (28). The steep decay lessens at large distance, and eventually adopts an asymptotic functional form compatible with a Debye-Hiickel approximation. However, to superimpose across this distant region the predictions of the Poisson-Boltzmann analysis onto those formidated... 

Fig. 3. Schematic plots of the decay of the electrostatic potential if(r) near a polyelectrolyte chain of unspecified chain radius. The potential decay predicted by a nonlinear Poisson-Boltzmann analysis can be superimposed onto one predicted by a linearized Debye-Hiickel approximation in the far field region if the surface potential of the Debye-Hiickel is appropriately adjusted. However, in this case there remains strong deviation between the two approaches in the region nearer the chain.

M. D. Paulsen, C. F. Anderson, and M. T. Record Jr., Biopolymers, 27, 1249 (1988). Counterion Exchange Reactions on DNA Monte Carlo and Poisson-Boltzmann Analysis. 

Dissociative excitation of molecules by electrons is a key process in many industrially important plasmas because it is the mechanism that provides the activated radicals that initiate the surface chemistries of interest. For example, many of the gases used in the etching of silicon do not display any reactivity in the absence of plasma. The construction of detailed models of these plasmas relies on a reliable data base of cross sections. Unfortunately, electron-impact dissociation cross sections are extremely difficult to measure and there are only a handful of cases where good data exist. Chlorine gas, which is widely used in the plasma etching of semiconductors, is one such example. Cross sections for ionization and dissociative electron attachment were measured during the 1970s and there has been one experimental study of electron impact dissociation. Cross sections for other dominant electron collision processes have been derived from Boltzmann analysis and early swarm measurements. ... 

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