Mixing semiempirical

The Prandtl mixing length concept is useful for shear flows parallel to walls, but is inadequate for more general three-dimensional flows. A more complicated semiempirical model commonly used in numerical computations, and found in most commercial software for computational fluid dynamics (CFD see the following subsection), is the A — model described by Launder and Spaulding (Lectures in Mathematical Models of Turbulence, Academic, London, 1972). In this model the eddy viscosity is assumed proportional to the ratio /cVe. 

Flory proposed a semiempirical expression to predict the concentration dependence of the melting curve of long-chain polymers mixed with small solvent molecules [75] ... 

Equation (6-37) represents the friction factor for Newtonian fluids in smooth tubes quite well over a range of Reynolds numbers from about 5000 to 105. The Prandtl mixing length theory and the von Karman and Blasius equations are referred to as semiempirical models. That is, even though these models result from a process of logical reasoning, the results cannot be deduced solely from first principles, because they require the introduction of certain parameters that can be evaluated only experimentally. 

The earliest equations for Gibbs excess energy, like Margules and van Laar, were largely empirical. More recent equations and NRTL and UNIQUAC are based on a semiempirical physical model, called the two-liquid theory, based on local composition. The molecules do not mix in a random way, but because of different bonding effects, the molecules prefer a certain surrounding. This results in a composition at the molecular level, the local composition, which differs from the macroscopic composition. 

These semiempirical models postulate that local strain associated with different atomic sizes of the elements is the major contribution to the solid-solution enthalpy of mixing. An estimate of lattice strain energy has been compared to fitted values of the enthalpy of mixing for several group III-V systems (156). The results led to a calculated enthalpy of mixing that was a... 

This restriction rules out all discrete models exclusively based on semiempirical force fields, leaving among the discrete models the MC/QM and the MD/QM procedures, in which the second part of the acronyms indicates that the solute is described at the quantum mechanical (QM) level, as well as the full ab initio MD description, and some mixed procedures that derive the position of some solvent molecules from semiclassical simulations, replace the semiclassical description with the QM one, and repeat the calculation on these small supermolecular clusters. The final stage is to perform an average on the results obtained with these clusters. These methods can be used also to describe electronic excitation processes, but at present, their use is limited to simple cases, such as vertical excitations of organic molecules of small or moderate size. This limitation is due to the cost of computations, and there is a progressive trend toward calculations for larger systems. 

Modem quantum-mechanical Valence-Bond (VB) theory has firm roots back to classical ideas even of a century and a half ago. These connections are of special interest, especially if greater general insight and extension of the classical concepts can be made. The interconnecting simpler semiempirical approaches, such as are of the prime focus here, are historically inextricably mixed with that of the ab initio theory, and the development has been via a peculiarly torturous road toward quantitative relevance. Thence here some brief historical commentary which also sets some nomenclature and ideas is first made. 

The topic of interactions between Lewis acids and bases could benefit from systematic ab initio quantum chemical calculations of gas phase (two molecule) studies, for which there is a substantial body of experimental data available for comparison. Similar computations could be carried out in the presence of a dielectric medium. In addition, assemblages of molecules, for example a test acid in the presence of many solvent molecules, could be carried out with semiempirical quantum mechanics using, for example, a commercial package. This type of neutral molecule interaction study could then be enlarged in scope to determine the effects of ion-molecule interactions by way of quantum mechanical computations in a dielectric medium in solutions of low ionic strength. This approach could bring considerable order and a more convincing picture of Lewis acid base theory than the mixed spectroscopic (molecular) parameters in interactive media and the purely macroscopic (thermodynamic and kinetic) parameters in different and varied media or perturbation theory applied to the semiempirical molecular orbital or valence bond approach [11 and references therein]. 

Taking the solvation into account in such models is both very difficult and necessary for the best possible understanding of these exceedingly complex phenomena. The disolvation of the amides dimer (one solvent per lithium in THF, THF + HMPA or THF + DMPU)50 seems to be indicated, while trisolvated dimers appear relatively unstable. However, a very extensive semiempirical theoretical (MNDO) study on the various cyclic and open mixed aggregates formed by LDA and LiTMP with LiCl or three different enolates, solvated by discrete molecules of THF or HMPA, showed that general conclusions are almost impossible to draw48. A complex interplay of steric effects, induced by the partners of the aggregate and the solvent, seems to be the dominant influence on the relative stabilities of the species characterized. 

The mixed (heterogeneous) complexes of a lithium amide (LDA or LiTMP) and a ketone lithium enolate (acetone, cyclohexanone or diisopropyl ketone) have been examined by semiempirical methods (MNDO) by Romesberg and Collum48. If the stabilization associated with these mixed complexes was not determined, the solvation (by THF and HMPA) of the mixed cyclic dimers and trimers was calculated to be generally exothermic (but decreasingly with the steric demand of the enolate) and led to disolvated entities. A set of solvated dimers, trimers and tetramers, cyclic or not, has thus been identified... 

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