Solution phenomenological models

A phenomenological model for redox reactions in solution application to aquocobalt(III) systems. 

Lorentzian line shapes are expected in magnetic resonance spectra whenever the Bloch phenomenological model is applicable, i.e., when the loss of magnetization phase coherence in the xy-plane is a first-order process. As we have seen, a chemical reaction meets this criterion, but so do several other line broadening mechanisms such as averaging of the g- and hyperfine matrix anisotropies through molecular tumbling (rotational diffusion) in solution. 

The broadest class of models, phenomenological models, account explicitly for individual phenomena such as swelling, diffusion, and degradation by incorporation of the requisite transport, continuity, and reaction equations. This class of models is useful only if it can be accurately parameterized. As phenomena are added to the model, the number of parameters increases, hopefully improving the model s accuracy, but also requiring additional experiments to determine the additional parameters. These models are also typically characterized by implicit mean-field approximations in most cases, and model equations are usually formulated such that explicit solutions may be obtained. Examples from the literature are briefly outlined below. 

Analysis of mass transfer in ternary media, until now, has mainly involved experimental studies of model and real food. Phenomenological models could be applied to obtain a more detailed description of the mechanisms involved. However, this would require an understanding of factors such as mass transport properties and transfer dynamics of different active compounds in concentrated solutions, which have yet to be characterized. 

In pulsed vacuum immersion, a phenomenological model would be difficult to develop as solution filtration and solute diffusion mechanisms have to be considered. The main problem to be solved concerning this aspect... 

The multiplicity of solutions at the continuum level can be viewed as arising from a constitutive deficiency in the theory, reflecting the need to specify additional pieces of constitutive information through some kind of phenomenological modeling (see, for instance, Truskinovsky, 1987 Abeyaratne and Knowles, 1991). Here we take a different point of view and interpret the nonuniqueness as an indicator of essential interaction between macro and micro scales. 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

The first phenomenological model for description of polymer dynamics in concentrated solutions and melts was proposed in 1971 by P.de Gennes [50]. In this classical work, it was assumed that due to entanglements, the chain motions in the direction normal to the chain contour are blocked up and only tangential ones are possible. This kind of chain motion in the effective tube was called reptation. In the absence of external fields, the chain can escape from the tube by either of the free tube ends. 

We now turn to the potential dependence of electrosorption of neutral molecules, considering first the model developed by Frumkin. This is a phenomenological model, which depends on considerations of the changes in the electrostatic energy of the interphase caused by adsorption. Assuming that measurements are taken in concentrated solutions of a supporting electrolyte, we can neglect diffuse-double-layer effects and focus our attention on the Helmholtz part of the double layer, considered as a parallel-plate capacitor. In the pure solvent the... 

The equation is a simple case of a mechanistic model. Models such as this may give better predictions but may not always apply because of the complexity of the reactions. Phenomenological models arc expressed by simple rate equations which ignore the details of the reaction. Phenomenological models are typically used to follow cure rates in polymeric systems which are difficult to follow by chemical analysis. This is because reaction products become insoluble during the course of the reaction and, consequently, are not detected in an analysis of the solution. 

In the past 10 years or so, there have been a number of theoretical contributions to the fundamental problem of describing fluids in a mesoscopic context. If one wants to go beyond the usual Debye formulation, it is evident that the simplicity of one-body stochastic models must be abandoned. Stochastic models which are able to describe the dynamical behavior of a complex liquid (for instance, a highly viscous solution), exact their price in terms of a more involved formalism. One must be careful to achieve a balance between complexity in formulation and new information gained from the model. Often one can resort to a phenomenological model, which may or may not be the starting point for a more complete (and complicated) theoretical treatment. 

Cation, anion, and water transport in ion-exchange membranes have been described by several phenomenological solution-diffusion models and electrokinetic pore-flow theories. Phenomenological models based on irreversible thermodynamics have been applied to cation-exchange membranes, including DuPont s Nafion perfluorosulfonic acid membranes [147, 148]. These models view the membrane as a black box and membrane properties such as ionic fluxes, water transport, and electric potential are related to one another without specifying the membrane structure and molecular-level mechanism for ion and solvent permeation. For a four-component system (one mobile cation, one mobile anion, water, and membrane fixed-charge sites), there are three independent flux equations (for cations, anions, and solvent species) of the form... 

The partition and displacement model considers retention to result from a two step process. The first involves formation of a mixed stationary phase by intercalation of solvent molecules from the mobile phase. The composition of the solvents in the stationary phase is established according to thermodynamic equilibrium and is usually different to the bulk mobile phase composition. Competitive sorption of solvents is modeled as a displacement process and is complete before the solute is introduced into the two-phase system. Solute retention is then modeled as a partition process between the solvent modified stationary phase and the mobile phase by taking into account all solute-solvent interactions in both phases. The phenomenological model of solvent effects attempts to model retention as a combination of solute-solvent interactions (the solvation effect) and solvent-solvent interactions (the general medium... 

In what follows, we propose a phenomenological model of the chemisorbed radical-anion standing in the electrochemical double-layer. We shall hence detail the reasons why this chemisorbed radical anion is intrinsically unstable but most probably has a finite lifetime on the polarized metallic surface. We outline the procedure through which we expect that an order of magnitude of the lifetime of the chemisorbed radical-anion may be evaluated numerically via this model. The model potential felt by the radical-anion as it is formed on the polarized electrode is described as the sum of three terms, for which a parametrisation is proposed. One of these terms is meant to include both the surrounding solvent and the repulsion by the polarized electrode, thanks to a mean, locally uniform, electric field. In the present paper, the intensity of this uniform electric field is calibrated on the basis of a conparison between experimental Stark-Tuning shifts for CO chemisorbed on palladium surfaces in solution and Density Functional Theory calculations of field-induced vibrational shifts for CO chemisorbed on palladium clusters. The shape of the resulting model potential is then discussed. 

Numerical simulations of the structure of multidimensional detonations have been carried out by Taki and Fujiwara (19) and by Oran et al. (20). Because of the expense involved in these simulations, both groups used a phenomenological model meant to describe the basic features of energy release, and coupled this to a solution of the conservation equations for the fluid dynamics. 

Whereas the simplest (and older) sorption models do not distinguish between the basic processes contributing to the overall sorption, newer model approaches try to address all relevant processes separately, namely physisorption, chemisorption, coprecipitation, inclusion, diffusion, surface-precipitation, or the formation of solid solutions. Sorption models in a strict sense are usually grouped into two classes, the phenomenological models, and the surface complexation models. 

In the phenomenological model of Kahlweit et al. [46], the behavior of a ternary oil-water-surfactant system can be described in terms of the miscibility gaps of the oil-surfactant and water-surfactant binary subsystems. Their locations are indicated by the upper critical solution temperature (UCST), of the oil-surfactant binary systems and the critical solution temperature of the water-surfactant binary systems. Nonionic surfactants in water normally have a lower critical solution temperature (LCST), Tp, for the temperature ranges encountered in surfactant phase studies. Ionic surfactants, on the other hand, have a UCST, T. Kahlweit and coworkers have shown that techniques for altering surfactant phase behavior can be described in terms of their ability to change the miscibility gaps. One may note an analogy between this analysis and the Winsor analysis in that both involve a comparison of oil - surfactant and water-surfactant interactions. 

The solution ionic radius is arguably one of flie most important microscopic parameters. Although detailed atomic models are needed for a full understanding of solvation, simpler phenomenological models are useful to interpret the results for more complex systems. The... 

This Section presents a consistent nomenclature for the diffusion coefficients and how these diffusion coefficients are measured. A short description is given of literature on diffusion in multimacrocomponent solutions, which provides a basis for interpreting experiments. A sketch of classes of models for polymer dynamics is presented. Proposed classes of phenomenological models are identified. A short sketch is made of alternative theoretical models that treat part or all of polymer dynamics. 

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