Self-consistent mean-field methods

Figure 6.13. The height of a polymer brush in a chemically identical matrix of equal degree of polymerisation 1000 as predicted by the scaling theory and calculated by the numerical self-consistent mean-field method. The SCF calculations were done using a...

A recent survey analyzed the accuracy of tliree different side chain prediction methods [134]. These methods were tested by predicting side chain conformations on nearnative protein backbones with <4 A RMSD to the native structures. The tliree methods included the packing of backbone-dependent rotamers [129], the self-consistent mean-field approach to positioning rotamers based on their van der Waals interactions [145],... 

The determination of the ground state energy and the ground state electron density distribution of a many-electron system in a fixed external potential is a problem of major importance in chemistry and physics. For a given Hamiltonian and for specified boundary conditions, it is possible in principle to obtain directly numerical solutions of the Schrodinger equation. Even with current generations of computers, this is not feasible in practice for systems of large total number of electrons. Of course, a variety of alternative methods, such as self-consistent mean field theories, also exist. However, these are approximate. 

James and Keenan [1959] used a mean-field approximation for potential (7.57). Yamamoto et al. [1977] proposed an expanded JK method (EJK), which takes into account higher-order terms in expansion (7.59). In the self-consistent mean-field approximation we have (see also Kobashi et al. [1984])... 

Many standard search methods have been used in side-chain conformation prediction, including Monte Carlo simulation [176-178], simulated annealing [179], self-consistent mean field calculations [154, 173, 180], and neural networks [170]. Self-consistent mean field calculations represent each side chain as a set of conformations, each with its own probability. Each rotamer of each side chain has a certain probability, p(n). The total energy is a weighted sum of the interactions with the backbone and interactions of side chains with each other ... 

J. Mendes, C. M. Soares, M. A. Carrondo. Improvement of side-chain modeling in proteins with the self-consistent mean field theory method based on an analysis of the factors influencing prediction. Biopolymers. 1999, 50, 111-131. 

The method developed in this book is also used to provide input parameters for composite models which can be used to predict the thermoelastic and transport properties of multiphase materials. The prediction of the morphologies and properties of such materials is a very active area of research at the frontiers of materials modeling. The prediction of morphology will be discussed in Chapter 19, with emphasis on the rapidly improving advanced methods to predict thermodynamic equilibrium phase diagrams (such as self-consistent mean field theory) and to predict the dynamic pathway by which the morphology evolves (such as mesoscale simulation methods). Chapter 20 will focus on both analytical (closed-form) equations and numerical simulation methods to predict the thermoelastic properties, mechanical properties under large deformation, and transport properties of multiphase polymeric systems. 

We have briefly reviewed methods which extend the self-consistent mean-field theory in order to investigate the statics and dynamics of collective composition fluctuations in polymer blends. Within the standard model of the self-consistent field theory, the blend is described as an ensemble of Gaussian threads of extension Rg. There are two types of interactions zero-ranged repulsions between threads of different species with strength /AT and an incompressibility constraint for the local density. 

Since the formulation of the SCFT for block copolymers by Helfand in 1975 [5], great effort has been devoted to the solution of the SCFT equations. To date, the most efficient and accurate method to solve the self-consistent mean-field equations is the reciprocal-space method developed by Matsen and Schick [14], which is based on the expansion in terms of plane wave-like basis functions. Recently, with the availability of increasing computing power and new numerical techniques, real-space methods have been developed to the level that they can be used to explore the possible phases for a given block copolymer architecture [17-19]. 

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