Diffusion bimodal behavior

The movement of chemicals undergoing any number of reactions with the soil and/or in the soil system (e.g., precipitation-dissolution or adsorption-desorption) can be described by considering that the system is in either the equilibrium or nonequilibrium state. Most often, however, nonequilibrium is assumed to control transport behavior of chemical species in soil. This nonequilibrium state is thought to be represented by two different adsorption or sorption sites. The first site probably reacts instantaneously, whereas the second may be time dependent. A possible explanation for these time-dependent reactions is high activation energy or, more likely, diffusion-controlled reaction. In essence, it is assumed that the pore-water velocity distribution is bimodal,... 

The process of analyte retention in high-performance liquid chromatography (HPLC) involves many different aspects of molecular behavior and interactions in condensed media in a dynamic interfacial system. Molecular diffusion in the eluent flow with complex flow dynamics in a bimodal porous space is only one of many complex processes responsible for broadening of the chromatographic zone. Dynamic transfer of the analyte molecules between mobile phase and adsorbent surface in the presence of secondary equilibria effects is also only part of the processes responsible for the analyte retention on the column. These processes just outline a complex picture that chromatographic theory should be able to describe. 

Simulations—isoergic and isothermal, by molecular dynamics and Monte Carlo—as well as analytic theory have been used to study this process. The diagnostics that have been used include study of mean nearest interparticle distances, kinetic energy distributions, pair distribution functions, angular distribution functions, mean square displacements and diffusion coefficients, velocity autocorrelation functions and their Fourier transforms, caloric curves, and snapshots. From the simulations it seems that some clusters, such as Ar, 3 and Ar, 9, exhibit the double-valued equation of state and bimodal kinetic energy distributions characteristic of the phase change just described, but others do not. Another kind of behavior seems to occur with Arss, which exhibits a heterogeneous equilibrium, with part of the cluster liquid and part solid. 

The dynamic behavior of linear charged polyelectrolytes in aqueous solution is not yet understood. The interpretation of dynamic light scattering (DLS) of aqueous solutions of sodium poly(styrene sulfonate) (NaPSS) is particularly complicated. The intensity correlation function shows a bimodal shape with two characteristic decay rates, differing sometimes by two or three orders of magnitude, termed fast and slow modes. The hrst observations in low salt concentration or salt free solution were reported by Lin et al. [31] for aqueous solutions of poly(L-lysine). Their results are described in terms of an extraordinary-ordinary phase transition. An identical behavior was hrst observed by M. Drifford et al. in NaPSS [32], Extensive studies on this bimodal decay on NaPSS in salt-free solution, or solutions where the salt concentration is increased slowly, have been reported [33-36]. The fast mode has been attributed to different origins such as the coupled diffusion of polyions and counterions [34,37,38] or to cooperative fluctuations of polyelectrolyte network [33,39] in the semidilute solutions. 

The most surprising feature of the behavior of PS-PVP-PEO micelles with water-soluble PVP (protonized) and PEO blocks in acidic media is their aggregation in the region of low pH. Because it is a rather unexpected phenomenon, we studied it in more detail. The distributions of relaxation times obtained by DLS are bimodal (Eig. 8). Angular dependences (not shown) prove that both fast and slow relaxation modes correspond to diffusive processes. The intensity of the slow mode decreases with increasing pH and decreasing copolymer concentration. At very low copolymer and HCl concentrations, the slow mode disappears completely. The DLS measurements thus show that PS-PVP-PEO solutions contain two types... 

In all these investigations, very small input amplitudes were used to justify the use of a linear technique for investigation of generally nonlinear adsorption systems. One of the main drawbacks of the classical, linear FR method for investigation of adsorption kinetics is that, in a number of cases, same shapes of linear FR characteristic functions are obtained for different kinetic mechanisms, and the method is reduced to estimation of kinetic parameters of an assumed model. One example of such behavior is the same shape of characteristic curves obtained for adsorption governed by micropore, pore, or pore-siuface diffusion mechanisms [28]. Another very characteristic example is the case of bimodal characteristic functions (with two maxima of the out-of-phase and two inflection points of the in-phase characteristic functions [31,33,35]). These results were shown to fit equally well to three different kinetic models [19,33,35]. As it will be shown in Section III, the nonlinear FR method can overcome this problem. 

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