Continuum models mean-field theories

There have been few attempts to generalize mean-field theories to the unrestricted case. Netz and Orland [227] applied their field-theoretical model to the UPM. Because such lattice theories yield quite different critical properties from those of continuum theories, comparison of their results with other data is difficult. Outhwaite and coworkers [204-206] considered a modification of their PB approach to treat the UPM. Their theory was applied to a few conditions of moderate charge and size asymmetry. 

The older mean-field theories are often based on the lattice model of ill-defined a priori size and shape, neglecting variability of monomer structures in PO copolymers and blends. Several newer approaches have been proposed, viz., the polymer reference interaction site model (PRISM) (Schweizer and Curro 1989, 1997), the Monte Carlo (MC) simulations (Sariban and Binder 1987 Muller and Binder 1995 Weinhold et al. 1995 Escobedo and de Pablo 1999), the continuum field theory (CFT) (Fredrickson et al. 1994), or an analytical lattice models (Dudowicz and Freed 1991). The latter model leads to relatively simple mathematical expressions, which offer an insight into basic thermodynamics, but again do not predict how monomer structure affects blend miscibility. 

Theoretical treatments of liquid crystals such as nematics have proved a great challenge since the early models by Onsager and the influential theory of Maier and Saupe [34] mentioned before. Many people have worked on the problems involved and on the development of the continuum theory, the statistical mechanical approaches of the mean field theory and the role of repulsive, as well as attractive forces. The contributions of many theoreticians, physical scientists, and mathematicians over the years has been great - notably of de Gennes (for example, the Landau-de Gennes theory of phase transitions), McMillan (the nematic-smectic A transition), Leslie (viscosity coefficients, flow, and elasticity). Cotter (hard rod models), Luckhurst (extensions of the Maier-Saupe theory and the role of flexibility in real molecules), and Chandrasekhar, Madhusudana, and Shashidhar (pre-transitional effects and near-neighbor correlations), to mention but some. The devel-... 

Certain difficulties remain, however, with this approach. First, such an important feature as a secondary structure did not find its place in this theory. Second, the techniques of sequence design ensuring exact reproduction of the given conformation are well developed only for lattice models of polymers. The existing techniques for continuum models are complex, intricate, and inefficient. Yet another aspect of the problem is the necessity of reaching in some cases beyond the mean field approximation. The first steps in this direction were made in paper [84], where an analog of the Ginzburg number for the theory of heteropolymers was established. 

Theoretical bases of continuum models including their mathematical formulation and numerical implementation have already been discussed in the previous chapter of this book. We have therefore restricted our review to the environment effects on the NMR observables, without going into the theory of continuum models. This contribution is divided into five sections. After the Introduction, the definitions of the NMR parameters are recalled in the second section. The third section is focused on methodological aspects of the calculation of the NMR parameters in continuum models. The fourth section reviews calculations of the solvent effects on the nuclear magnetic shielding constants and spin-spin coupling constants by means of continuum models, and the final section presents a survey on the perspectives of this field. 

Another kind of self-consistent theory used a continuum diffusion representation to describe the distribution of segments.21 23 The segments were assumed to be subjected to an external potential of a mean field. An analytical approximation of the latter self-consistent theory was suggested by Milner et al. (the MWC model)24 on the basis of the observation that at high stretching the partition function of the brush is dominated by the classical path as the most probable distribution. Under this assumption, it was found that the self-consistent field is parabolic and leads to a parabolic distribution of the monomer density. Similar theories for polyelectrolyte brushes25 27 also adopted the parabolic distribution approximation. 

Exactly as In lattice models, the walks are assumed to take place In a (self-consistent) field Ulz), which depends on the concentration profile

relation between U[z) and (p[z) one may use the Floiy-Hugglns theory usually in an expanded form, but other models, such as a generalized Van der Waals equation of state ) can also be taken. The most general expression for the self-consistent mean field U z) has been given by Hong and Noolandl K It has been shown ) that this expression is the continuum analogue of the lattice version of Scheutjens and Fleer, to be discussed in sec. 5.5. 

A basic approach for the description of polymer chains in the continuum is the Gaussian thread model [26, 31]. Treating interactions among monomers in a mean-field-like fashion, one obtains the self-consistent field theory (SCFT) [11, 32-36] which can also be viewed as an extension of the Hory-Huggins theory to spatially inhomogeneous systems (like polymer interfaces in blends, nucrophase separation in block copolymer systems [11, 13], polymer bmshes [37, 38], etc.). However, with respect to the description of the equation of state of polymer solutions and blends in the bulk, it is stiU on a simple mean-field level, and going beyond mean field to include fluctuations is very difficult [11, 39-42] and outside the scope of this article. 

For the simulation of more complex flows, one needs a constitutive equation or a rheological equation of state. Nearly all of the many equations that have been proposed over the past fifty years are basically empirical in nature, and only in the last twenty-five years have such models been developed on the basis of mean field molecular theories, e.g., tube models. Although the early models were often developed with a molecular viewpoint in mind, it is best to think of them as continuum models or semi-empirical models. The relaxation mechanisms invoked were crude, involving concepts such as network rupture or anisotropic friction without the molecular detail required to predict a priori the dependence of viscoelastic behavior on molecular structure. While these lack a firm molecular basis and thus do not have universal validity or predictive capability, they have been useful in the interpretation of experimental data. In more recent times, constitutive equations have been derived from mean field models of molecular behavior, and these are described in Chapter 11. We describe in this section a few constitutive equations that have proven useful in one or another way. More complete treatments of this subject are given by Larson [7] and by Bird et al. [8]. 

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