Mean molecular field

Our quantum chemical approach has been applied to explain the thermodynamics of the TKHS-like materials on the level of simple mean (molecular) field approximation [1,4] which, however, is not well adapted to provide for quantitative estimations of the critical temperature of phase transitions. To examine the... 

A is a constant which depends on the nature of the liquid crystal, V is the molar volume of the liquid crystal and is the order of the nematic phase. 6 is the angle between the long axis of the molecule C and the optic axis of the liquid crystal. In the approximation of the so-called mean (molecular) field approach [117, 118] the average orientation of molecule in the liquid crystal is completely determined by the above anisotropic potential. Applying simple Boltzmann statistics, the average orientation of the long axis of a molecule can then be expressed as... 

There is, however, one further approxhnation that has been used above, and that is the one we now call a mean-field appmxinuitwn, or the assumption of the existence of a mean molecular field. It is an approximation which is sometimes difficult to avoid even in modern statistical mechanics, and is worth full discussion. 

In 1.6 we saw how important was the role played in the historical development of ideas about the constitution of fluids generally, and that of the liquid-gas interface particularly, by the mean-molecular-field (or mean-field) approximation. We asserted in 3.5 (anticipating the development in the present chapter) that this approximation contains the assumption that within the range of the attractive forces of every molecule there is always found the same number of neighbouring molecules. This means that the potential energy of attraction felt by the molecules is assumed to be a constant, not varying from molecule to molecule at any one time or with time for any one molecule. 

In Section 7.1.2 a method for the calculation of mean molecular polarizability was presented. Mean molecular polarizability can be calculated from additive contributions of the atoms in their various hybridization states in a molecule (see Eq. (6)). Mean molecular polarizability, a, expresses the magnitude of the dipole moment, fi, induced into a molecule imder the influence of an external field, E (Eq. (15))... 

Once the mechanisms of dynamic processes are understood, it becomes possible to think about controlling them so that we can make desirable processes to occur more efficiently. Especially when we use a laser field, nonadiabatic transitions are induced among the so-called dressed states and we can control the transitions among them by appropriately designing the laser parameters [33 1]. The dressed states mean molecular potential energy curves shifted up or down by the amount of photon energy. Even the ordinary type of photoexcitation can be... 

Partial Least Squares (PLS) regression (Section 35.7) is one of the more recent advances in QSAR which has led to the now widely accepted method of Comparative Molecular Field Analysis (CoMFA). This method makes use of local physicochemical properties such as charge, potential and steric fields that can be determined on a three-dimensional grid that is laid over the chemical stmctures. The determination of steric conformation, by means of X-ray crystallography or NMR spectroscopy, and the quantum mechanical calculation of charge and potential fields are now performed routinely on medium-sized molecules [10]. Modem optimization and prediction techniques such as neural networks (Chapter 44) also have found their way into QSAR. 

At high Reynolds numbers, it is usually possible to assume that the mean scalar fields (e.g., (cc are independent of molecular-scale quantities such as the molecular-diffusion coefficients. In this case, it is usually safe to assume that all scalars have the same molecular diffusivity T. The conserved-scalar transport equation then simplifies to37... 

In addition to the above effects, the intermolecular interaction may affect polymer dynamics through the thermodynamic force. This force makes chains align parallel with each other, and retards the chain rotational diffusion. This slowing down in the isotropic solution is referred to as the pretransition effect. The thermodynamic force also governs the unique rheological behavior of liquid-crystalline solutions as will be explained in Sect. 9. For rodlike polymer solutions, Doi [100] treated the thermodynamic force effects by adding a self-consistent mean field or a molecular field Vscf (a) to the external field potential h in Eq. (40b). Using the second virial approximation (cf. Sect. 2), he formulated Vscf(a), as follows [4] ... 

An important step in developing the mean-field concept was done by Landau [8, 10]. Without discussing the relation between such fundamental quantities as disorder-order transitions and symmetry lowering, we just want to note here that his theory is based on thermodynamics and the derivation of the temperature dependence of the order parameter via the thermodynamic potential minimization (e.g., the free energy A(r),T)) which is a function of the order parameter. It is assumed that the function A(rj,T) is analytical in the parameter 77 and thus near the phase transition point could be expanded into the series in 77 usually it is a polynomial expansion with temperature-dependent coefficients. Despite the fact that such a thermodynamical approach differs from the original molecular field theory, they are quite similar conceptually. In particular, the r.h.s. of the equation of state for the pressure of gases or liquids and the external field in ferromagnetics, respectively, have the same polynomial form. 

It is known that the classical molecular field theory discussed above is not suited for describing a close vicinity of the critical point. Experimentally obtained values of the parameter (3 (called the critical exponent) are essentially less than (3q = 1/2 predicted by the mean-field theory. On the other hand, the experimental values of (3 = 0.33-0.34 turn out to be universal for many different systems (except for quantum liquid-helium where (3... 

No attempts were made, in the above analysis, to locate the overcrowded hydrogen atoms at the 4- and 5-positions, but they almost certainly deviate from the mean molecular plane (see Section III, A, 2). When the 4- and 5-positions are substituted by bulkier groups, the overcrowding becomes more severe and is sufficient to distort the molecule from overall planarity. Optical activity may be induced by this distortion if the molecule is suitably substituted. The work of Newman and co-workers in this field has been discussed briefly in... 

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