Molecular mean field theory

Molecular mean-field theory and the Bean-Rodbell model... 

Figure 1.5 In a molecular mean field theory, one molecule (here dark) is assumed to interact with all the others through an average field (symbolized here by grey contours)...

In this section we consider a general model that has broad applicability to phase transitions in soft materials the Landau theory, which is based on an expansion of the free energy in a power series of an order parameter. The Landau theory describes the ordering at the mesoscopic, not molecular, level. Molecular mean field theories include the Maier-Saupe model, discussed in detail in Section 5.5.2. This describes the orientation of an arbitrary molecule surrounded by all others (Fig. 1.5), which set up an average anisotropic interaction potential, which is the mean field in this case. In polymer physics, the Flory-Huggins theory is a powerful mean field model for a polymer-solvent or polymer-polymer mixture. It is outlined in Section 2.5.6. 

It has been assumed that molecular properties contribute additively to the macroscopic tensor components, which are consequently proportional to the number density. If intermolecular interactions contribute to the physical property, then deviations from a linear dependence of the property on density are expected. Also the contribution of orientational order will be more complex, since the properties will depend on the degree of order of interacting molecules. Effects of molecular interactions contribute to the dielectric properties of polar mesogens, and are particularly important for elastic and visoelastic properties. Molecular mean field theories of elastic properties predict that elastic constants should be proportional to the square of the order parameter this result highlights the significance of pairwise interactions. 

Analytical approaches to understanding the effect of molecular flexibility on orientational order have concentrated on both the isotropic-nematic and the nematic-smectic transition [61, 62] and mean field theory has shown that cholesteric pitch appears not to depend on the flexibility of the molecule [63]. 

In what follows, we use simple mean-field theories to predict polymer phase diagrams and then use numerical simulations to study the kinetics of polymer crystallization behaviors and the morphologies of the resulting polymer crystals. More specifically, in the molecular driving forces for the crystallization of statistical copolymers, the distinction of comonomer sequences from monomer sequences can be represented by the absence (presence) of parallel attractions. We also devote considerable attention to the study of the free-energy landscape of single-chain homopolymer crystallites. For readers interested in the computational techniques that we used, we provide a detailed description in the Appendix. ... 

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... 

A major ingredient for an RG treatment is a simple and transparent characterization of the molecular forces driving phase separation. This situation calls for mean-field theories of the ionic phase transition. The past decade has indeed seen the development of several approximate mean-field theories that seem to provide a reasonable, albeit not quantitative, picture of the properties of the RPM. Thus, the major forces driving phase separation seem now to be identified. Moreover, the development of a proper description of fluctuations by GDH theory has gone some way to establish a suitable starting point for RG analysis. Needless to say, these developments are also of prime importance in the more general context of electrolyte theory. 

Recently, the stiff-chain polyelectrolytes termed PPP-1 (Schemel) and PPP-2 (Scheme2) have been the subject of a number of investigations that are reviewed in this chapter. The central question to be discussed here is the correlation of the counterions with the highly charged macroion. These correlations can be detected directly by experiments that probe the activity of the counterions and their spatial distribution around the macroion. Due to the cylindrical symmetry and the well-defined conformation these polyelectrolytes present the most simple system for which the correlation of the counterions to the macroion can be treated by analytical approaches. As a consequence, a comparison of theoretical predictions with experimental results obtained in solution will provide a stringent test of our current model of polyelectrolytes. Moreover, the results obtained on PPP-1 and PPP-2 allow a refined discussion of the concept of counterion condensation introduced more than thirty years ago by Manning and Oosawa [22, 23]. In particular, we can compare the predictions of the Poisson-Boltzmann mean-field theory applied to the cylindrical cell model and the results of Molecular dynamics (MD) simulations of the cell model obtained within the restricted primitive model (RPM) of electrolytes very accurately with experimental data. This allows an estimate when and in which frame this simple theory is applicable, and in which directions the theory needs to be improved. 

In the intermediate domain of values for the parameters, an exact solution requires the specific inspection of each configuration of the system. It is obvious that such an exact theoretical analysis is impossible, and that it is necessary to dispose of credible procedures for numerical simulation as probes to test the validity of the various inevitable approximations. We summarize, in Section IV.B.l below, the mean-field theories currently used for random binary alloys, and we establish the formalism for them in order to discuss better approximations to the experimental observations. In Section IV.B.2, we apply these theories to the physical systems of our interest 2D excitons in layered crystals, with examples of triplet excitons in the well-known binary system of an isotopically mixed crystal of naphthalene, currently denoted as Nds-Nha. After discussing the drawbacks of treating short-range coulombic excitons in the mean-field scheme at all concentrations (in contrast with the retarded interactions discussed in Section IV.A, which are perfectly adapted to the mean-field treatment), we propose a theory for treating all concentrations, in the scheme of the molecular CPA (MCPA) method using a cell... 

It has been the merit of Picken (1989, 1990) having modified the Maier-Saupe mean field theory successfully for application to LCPs. He derived the stability of the nematic mesophase from an anisotropic potential, thereby making use of a coupling constant that determines the strength of the orientation potential. He also incorporated influences of concentration and molecular weight in the Maier-Saupe model. Moreover, he used Ciferri s equation to take into account the temperature dependence of the persistence length. In this way he found a relationship between clearing temperature (i.e. the temperature of transition from the nematic to the isotropic phase) and concentration ... 

The formation and equilibrium structure of polymer layered silicate nanocomposites, in particular with organically modified layered silicates, has been shown to be a strong function of the nature of the polymer (polar or apolar), the charge carrying capacity of the layered silicate, as well as the chain length and structure of the cationic surfactant. However, both the polymer/silicate compatibility and hybrid equilibrium structure for these nanocomposites are observed to be independent of polymer molecular weight. The experimental results have been summarized by Vaia et al. and a lattice based mean field theory has been developed to explain these results [26]. 

Like all mean-field theories, SCF theories replace the detailed, configuration-dependent interaction potentials with a mean potential averaged over the distribution of molecular configurations. Unlike other mean-field theories, SCF theory explicitly calculates the mean field by accounting for the polymer chain statistics. This field, in turn, controls the distribution of polymer configurations Hence the term self-consistent. ... 

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