Mean or molecular field

Mean or Molecular Field Approximation in Any Dimension and Any Value of 5... 

The most general treatment of intermolecular interactions gives rise to the Curie-Weiss law described above (Equation 3.5), allowing intermolecular interactions to be modelled by the parameter, B. This is based on the mean or molecular field approximation, which can be generalised to... 

The significance of 6 is most easily demonstrated by application of the mean or molecular field theory (MFT). The central idea here is that magnetic moments on neighboring atoms give rise to an internal or exchange or molecular field, H , in addition to any applied field as in (8)... 

It is obvious that Fig. 2 represents a drastic simplification of reality for complicated molecules The merit of this theory is that it gives a good qualitative picture of real gelation, the first indication for the universality principle that complicated molecular details are not very relevant for the main results. Physicists call Fig. 2 a Bethe lattice, and this gelation process is then called percolation on a Bethe lattice . Hie macromolecules are also designated as Cayley trees since, like trees in a forest, they have no cyclic links between their branches. Many other problems of theoretical physics, besides percolation, have been studied on Bethe lattices. When critical exponents were found they usually agreed with those obtained by using simple approximations for real lattices like mean field (or molecular field) approximation, Landau ansatz for phase transitions, van der Waals equation, etc. We will thus also denote them as mean field approximations. 

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. 

Volatile or volatilizable compounds may be introduced into the spectrometer via a pinhole aperture or molecular leak which allows a steady stream of sample molecules into the ionization area. Non-volatile or thermally labile samples are introduced directly by means of an electrically heated probe inserted through a vacuum lock. Numerous methods of sample ionization are available of which the most important are electron impact (El), chemical ionization (CY), field ionization (FI), field desorption (FD), fast atom bombardment (FAB), and radio-frequency spark discharge. 

CNTs can be easily doped by noncovalent means via molecular adsorption, an aspect that has been considerably exploited to develop ultrasensitive field effect transistor sensors [88-91]. However, substitutional doping with B and N to confer p and n character to the CNTs has also been carried out [92]. Such doped systems can be more susceptible to react with donors or acceptors molecules (depending on the doping) allowing the chemically reactivity to increase. 

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] ... 

The quantum theory of molecular collisions in external fields described in this chapter is based on the solutions of the time-independent Schrodinger equation. The scattering formalism considered here can be used to calculate the collision properties of molecules in the presence of static electric or magnetic fields as well as in nonresonant AC fields. In the latter case, the time-dependent problem can be reduced to the time-independent one by means of the Floquet theory, discussed in the previous section. We will consider elastic or inelastic but chemically nonreac-tive collisions of molecules in an external field. The extension of the formalism to reactive scattering problems for molecules in external fields has been described in Ref. [12]. 

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. 

The VSEPR approach is largely restricted to Main Group species (as is Lewis theory). It can be applied to compounds of the transition elements where the nd subshell is either empty or filled, but a partly-filled nd subshell exerts an influence on stereochemistry which can often be interpreted satisfactorily by means of crystal field theory. Even in Main Group chemistry, VSEPR is by no means infallible. It remains, however, the simplest means of rationalising molecular shapes. In the absence of experimental data, it makes a reasonably reliable prediction of molecular geometry, an essential preliminary to a detailed description of bonding within a more elaborate, quantum-mechanical model such as valence bond or molecular orbital theory. 

In the preceding chapter we have been considering the equilibrium of two phases of the same substance. Some of the most important cases of equilibrium come, however, in binary systems, systems of two components, and we shall take them up in this chapter. Wo can best understand what is meant by this by some examples. The two components mean simply two substances, which may be atomic or molecular and which may mix with each other. For instance, they may be substances like sugar and wrater, one of which is soluble in the other. Then the study of phase equilibrium becomes the study of solubility, the limits of solubility, the effect of the solute on the vapor pressure, boiling point, melting point, etc., of the solvent. Or the components may be metals, like copper and zinc, for instance. Then we meet the study of alloys and the whole field of metallurgy. Of course, in metallurgy one often has to deal with alloys with more than two components—ternary alloys, for instance, with three components—but they arc considerably more complicated, and we shall not deal with them. 

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