Path theory

Since they are just as elementary as the "mean free path" theories. 

One must be very careful in reviewing the older, and some more recent, literature in consideration of the tortuosity and constriction factors some work has attempted to separate these two factors however, more modem developments show that they cannot be strictly decoupled. This aspect will be particularly important when reviewing the barrier and tortuous-path theories of electrophoresis, as discussed later. 

The tortuous-path and barrier theories consider the effects of the media on the electrophoretic mobility in a way similar to the effect of media on diffusion coefficients discussed in a previous section of this chapter. The tortuons-path theory seeks to determine the effect of increased path length on electrophoretic mobility. The barrier theory considers the effects of the barrier or media conductivity on the electrophoretic mobility. 

The early theories for the transport coefficients were based on the concept of the mean free path. Excellent summaries of these older theories and their later modifications are to be found in standard text books on kinetic theory (J2, K2). The mean-free-path theories, while still very useful from a pedagogical standpoint, have to a large extent been supplanted by the rigorous mathematical theory of nonuniform gases, which is based on the solution of the Boltzmann equation. This theory is... 

G.D. Billing, Quantum corrections to the classical path theory, J. Chem. Phys., 99 (1993) 8849. 

Diffusion path theory. predicts a linear relationship between interface positions and t1 when there is no convection. Based on the correlation coefficients in Table I, this relationship appears to hold for the systems at these conditions. The low coefficients for the upper interfaces resulted from the measurement uncertainty ( 0.05 mm) being the same order of magnitude as the total movement of the interfaces, which is also why these plots are pot included in Figure 7. Division of the best-fit slopes by 2t gives an estimate of the interface velocities at any elapsed time t. 

In contrast, the brine phase tended to form large pockets in the TRS system, causing extensive movement of the liquid crystal to the brine-microemulsion interface. This behavior is schematically illustrated in Figure 9. Not surprisingly, interface movements were inconsistent with diffusion path analysis. Figure 10 shows such a plot for the TRS/C12 system at 1.5 gm/dl salinity. The nonlinearity results from convection, which speeds up equilibration. Experimentally, the inconsistency with diffusion path theory was evident from the time-dependent appearance of the upper microemulsion interface, an indication of variable interface compositions. 

A comparison between experimental and theoretical results shows that diffusion path analysis can qualitatively predict what is observed when an anionic surfactant solution contacts oil. Experimentally, one or two intermediate phases formed at all salinities. The growth of these phases was easily observed through the use of a vertical-orientation microscope. Except when convection occurred due to an intermediate phase being denser than the phase below it, interface positions varied as the square root of time. As a result, diffusion path theory could generally he used to correctly predict the direction of movement and relative speeds of the interfaces. 

Although the contacting experiments were performed with surfactant systems typical of those used in enhanced oil recovery, application of the results to detergency processes may be possible. For example, the growth of oil-rich intermediate phases is sometimes a means for removing oily soils from fabrics. Diffusion path theory predicts that oil is consumed fastest in the oil-soluble end of the three-phase regime where an oil-rich intermediate microemulsion phase forms. 

E. Vanden-Eijnden Transition Path Theory, Lect. Notes Phys. 703, 453-493 (2006)... 

In this chapter, we have shown why the recent transition path theory (TPT) offers the correct probabilistic framework to understand the mechanism by which rare events occur by analyzing the statistical properties of the reactive trajectories involved in these events. The main results of TPT are the probability density of reactive trajectories and the probability current (and associated streamlines) of reactive trajectories, which also allows one to compute the probability flux of these trajectories and the rate of the reaction. It was also shown that TPT is a constructive theory under the assumption that the reaction channels are local, TPT naturally leads to algorithms that allow to identify these channels in practice and compute the various quantities that TPT offers. 

Maxwell [65] was able to obtain these fairly accurate expressions for the transport coefficients which describe their primary dependence of upon temperature, pressure, mass and size of the molecules in the gas based on rather crude arguments. Historically, the mean free path theory given by Maxwell [65] predates the more accurate theory based on the Boltzmaim equation by about half a century. 

Mixtures containing 1 wt% of the pure nonionic surfactant C,2E5 in water were contacted with pure n-hexadecane and n-tetradecane at various temperatures between 25 and 60°C using the vertical cell technique. Similar experiments were performed with C,2E4 and n-hexadecane between about 15 and 40°C. In both cases the temperature ranged from well below to well above the phase inversion temperature (PIT) of the system, i.e., the temperature where hydrophilic and lipophilic properties of the surfactant are balanced and a middle phase microemulsion forms (analogous to the optimal salinity for ionic surfactants mentioned above). The different intermediate phases that were seen at different temperatures and the occurrence of spontaneous emulsification in some but not all of the experiments could be understood in terms of known aspects of the phase behavior, e.g., published phase diagrams for the C12E 5-water-n-tetradecane system, and diffusion path theory. That is, plausible diffusion paths could be found that showed the observed intermediate phases and/or spontaneous emulsification for each temperature. 

It is possible to define a classical path theory by introducing a single configuration type of approximation. Consider, e.g., a system with two degrees of freedom r and R where the first is the vibrational and the second the translational degree of freedom. Thus we... 

In the string method [37-Al] and the associated transition path theory (TPT)... 

Metzner, P, Schutte, C., Vanden-Eijnden, E. Illustration of transition path theory on a collection of simple examples, J. Chem. Phys. 2006,125,084110. 

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