M-diffusion model

The proposed specification of the kernel for m- and J-diffusion models is mathematically closed, physically clear and of quite general character. In particular, it takes into consideration that any collisions may be of arbitrary strength. The conventional m-diffusion model considers only strong collisions (0(a) = 1 /(27c)), while J-diffusion considers either strong (y = 0) or weak (y = 1) collisions. Of course, the particular type of kernel used in (1.6) restricts the problem somewhat, but it does allow us to consider kernels with arbitrary y < 1. 

It should be noted in conclusion that, by virtue of the impact condition (t,/ > tc) and adiabaticity of the m-diffusion model (cotc > 1), the latter relates to the limit alternative to the perturbation theory ... 

Consequently, the m-diffusion model does not extend to the domain where the Hubbard relation holds. Therefore, the J-diffusion model is the only realistic description of rotational diffusion within the framework of impact theory. 

This is the result for the 7-diffusion model. In the M-diffusion model the molecules have a distribution of molecular speeds, p(coo), but each molecule retains the same speed through all collisions. Letting Cj<°>(/o wo), Ci(n,t a>o) denote the free particle and n-free step-correlation... 

In the limit of small-step diffusion where Xj J-diffusion reduces to Hubbard s relation (2.12). Thus in this limit the extended J-diffusion model agrees with the classical theory. This is not true for the M-diffusion model. In the other limit when Xj becomes large, as in a dilute gas, where many rotations occur between... 

In both cases, the time-resolved anisotropy decay acquires a complex form of an infinite series. Gordon [77] has shown that the decay may, in special cases, exhibit damped oscUlatiOTis, which has been observed experimentally, but the original RD model was unable to offer an explanation of this rather exotic behavior. Numerical calculations by McClung [79] validated the ED prediction for spherical molecules. For asymmetric molecules, the calculations confirmed the formulae based on the J-diffusion model and indicated that the assumptions used in the M-diffusion model are less realistic [81]. 

Braun, M., and Renz, U., Investigation of Multicomponent Diffusion Models in Turbulent Flow, Procc. Engineering Foundation Conf. on Condensation and Condenser Design, pp81- 92, 1993. 

In the Keilson-Storer model of J-diffusion, non-adiabatic relaxation is assumed to extend to the whole energy spectrum of a rotator. Actually, for large J the relaxation becomes adiabatic. The considerable difference between the times appears in the adiabatic limit since xe = 00, while xj is defined by m-diffusion according to Eq. (1.12). As is seen from Eq. (1.5) and Eq. (1.6), both J- and m-diffusion are just approximations which hold for low- and high-excited rotational levels, respectively. In general 0 < xj/xE < 1 + y. 

It is clear that J-diffusion is a good approximation for rotational relaxation as a whole, if the centre of equilibrium distribution over J is within the limits of non-adiabatic theory. In the opposite case m-diffusion is preferable. Consequently, the J-diffusion model is applicable, if the following inequality holds ... 

FIG. 23 Comparison of various one-parameter diffusion models. (Reproduced with permission from Ref. 448, Analysis of Diffusion and Structure in Polyacrylamide Gels by Nuclear Magnetic Resonance, M.S. Thesis, Florida State University, Copyright 1997, Brigita Penke.)... 

Mahlman and Sworski (1967) found that curves for scavenging of H2 by NOj- have nearly the same shape for neutral and 0.1 M acid solutions, over the concentration range 1 mM to 0.4 M of the solute, despite the fact that nitrate reacts much faster with eh than with H. Schwarz points out, however, that when two solutes (H+ and N03-) compete for the same intermediate (eh), the extrapolation to zero scavenger concentration is not a valid procedure. The calculation of the diffusion model agrees with experiment over the entire N03- concentration range. Further, the model predicts a lower H2 yield at very low scavenger concentrations. 

S. J. Madsen, B. C. Wilson, M. S. Patterson, Y. D. Park, S. L. Jacques, and Y. Hefetz. Experimental tests of a simple diffusion model for the estimation of scattering and absorption coefficients of turbid media from time-resolved diffuse reflectance measurements. Applied Optics, 31 3509-3517, 1992. 

A small step rotational diffusion model has been used to describe molecular rotations (MR) of rigid molecules in the presence of a potential of mean torque.118 120,151 t0 calculate the orientation correlation functions, the rotational diffusion equation must be solved to give the conditional probability for the molecule in a certain orientation at time t given that it has a different orientation at t = 0, and the equilibrium probability for finding the molecule with a certain orientation. These orientation correlation functions were found as a sum of decaying exponentials.120 In the notation of Tarroni and Zannoni,123 the spectral denisities (m = 0, 1, 2) for a deuteron fixed on a reorienting symmetric top molecule are ... 

Dissolution time, tdi (for tablet) Tablet mass, m Diffusivity, D Grain particle size, dp Tablet size, 77 Porosity, e Order-of-magnitude model derived from Fick s and Darcy s laws [6] 2m2 td x2d2pH4Ds(l-s)2... 

Bashford D Weaver D. L. and Karplus M. Diffusion-collision model for the folding kinetics of the lambda-repressor operatorbinding domain. J. Biomol. Str. Dyn. (1984) 1 1243-1255. 

In Fig.la a lower slope ((j)=2.25) and higher Intercept (g=1.32) Is obtained for the low pressure region as compared to the slope (((>=2.60) and Intercept (g=1.09) for high pressure region. Since from Eq(13), a value of g close to 1 implies adherence to the solution-diffusion model. It appears that at low pressures where g = 1.32>1.0, a combined model of viscous and diffusive flow Is operative. This correlates with previous SEM studies In our laboratory (unpublished), where mlcro-pln holes were postulated to exist In the skin. The presence of such m-LcAO-p-Ln hoZ 6 In the surface can be used to explain the high g-value. Above 10 atm, the DDS-990 membrane Is compressed or compacted and the mlcro-pln holes filled. Thus g = l.O l.O implies adherence to the solution-diffusion model. 

Nowak M., Schreen D., and Spickenbom K. (2004) Argon and CO2 on the race track in silicate melts a tool for the development of a CO2 speciation and diffusion model. Geochim. Cosmochim. Acta 68, 5127-5138. 

Figure 3 Escape probability as a function of the initial electron-cation distance. The lower broken curve is calculated from the Onsager equation [Eq. (15)]. The numerical results for different mean free times x were taken from Ref. [22]. The unit of x is rJ(ksTlmf, where m is the electron mass. The upper broken curve was calculated using the energy diffusion model. (From Ref. 23.)...

Strommen, M. R., and R. M. Kamens, Development and Application of a Dual-Impedance Radial Diffusion Model to Simulate the Partitioning of Semi-Volatile Organic Compounds in Combustion Aerosols, Environ. Sci. Technol., 31, 2983-2990 (1997). 

Fig. 13. Concentration profiles within an ATRllow-tiougncen convection-diffusion model (dj). The models are descri m...

The radial diffusion model and the linear sorption model are compared in Fig. 18.76. Since according to Eq. 19-76 the total mass of the chemical associated with the particle aggregate, (M(r), and the macroscopic solid-water distribution ratio, Kd(t), are linearly related ... 

As pointed out by Peeters et al. (1996), based on their own experiments and on the reinterpretation of published field data, the adequate model to describe horizontal diffusion in lakes and oceans is the shear diffusion model by Carter and Okubo (1965). The model is described in Box 22.4. The most important consequence of this model is that the 4/3 law and the equivalent t3-power law for c2(t) expressed by Eq. 22-42 are replaced by an equation which corresponds to a continuous increase of the exponent m from 1 to 2 (Box 22.4, Eq.l) ... 

The nearly constant peroxynitrite concentration observed in neutral solution changes dramatically when [Ft] of the solution is increased. Fig. 2 compares the transient absorption of aqueous nitrate at [Ft ] = 10-7 M and [Ft] = 0.140 M. The peroxynitrite concentration drops rapidly as protonation leads to the formation of peronitrous acid (peroxynitrous acids absorbs relatively weakly around 240 nm and is not observable in Fig. 2.). In Fig. 3 the peroxynitrite concentration is represented by the transient absorption at 310 nm as a function of [it]. As expected the formation of peroxynitrous acid increases with the concentration of protons. The protonation of peroxynitrite can be viewed as a prototypical diffusion limited bimolecular reaction and thus constitutes an excellent test bed for diffusion models. 

Kuppermann, A., Actions Chimiques et Biologiques des Radiations, Ve S rie, Haissinsky, M., ed., Academic, New York, 1961. (In this reference the diffusion model is discussed at length.)... 

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