Self consistent mean field theory

P Koehl, M Delarue. Application of a self-consistent mean field theory to predict protein side-chains conformation and estimate their conformational entropy. J Mol Biol 239 249-275, 1994. 

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. 

From the theoretical viewpoint, much of the phase behaviour of blends containing block copolymers has been anticipated or accounted for. The primary approaches consist of theories based on polymer brushes (in this case block copolymer chains segregated to an interface), Flory-Huggins or random phase approximation mean field theories and the self-consistent mean field theory. The latter has an unsurpassed predictive capability but requires intensive numerical computations, and does not lead itself to intuitive relationships such as scaling laws. 

Fig. 2.44 Phase diagram for a conformationally-symmetric diblock copolymer, calculated using self-consistent mean field theory. Regions of stability of disordered, lamellar, gyroid, hexagonal, BCC and close-packed spherical (CPS), phases are indicated (Matsen and Schick 1994 ). All phase transitions are first order, except for the critical point which is marked by a dot.

The theory of Jones and Richmond based on the self-consistent mean field theory predicts Eq. (B-122), which may be rewritten... 

Fig. 1 Phase diagram of self-assembled structures in AB diblock copolymer melt, predicted by self-consistent mean field theory [31] and confirmed experimentally [33]. The MesoDyn simulations [34, 35] demonstrate morphologies that are predicted theoretically and observed experimentally in thin films of cylinder-forming block copolymers under surface fields or thickness constraints dis disordered phase with no distinct morphology, C perpendicular-oriented and Cy parallel-oriented cylinders, L lamella, PS polystyrene, PL hexagonally perforated lamella phase. Dots with related labels within the areal of the cylinder phase indicate the bulk parameters of the model AB and ABA block copolymers discussed in this work (Table 1). Reprinted from [36], with permission. Copyright 2008 American Chemical Society...

Analytical self-consistent mean field theories were developed independently by Zhulina el al.29 30 and Milner et al.31,32 They are based on the assumption that for large stretchings of the grafted chains with respect to their Gaussian dimension, one can approximate the set of conformations of a stretched grafted chain by a set of most likely trajectories, and predict for such cases a parabolic density profile. In the calculations of the interactions between the two brushes, the interdigitation between the chains was ignored. 

Noolandi and coworkers [82-84] investigated the equilibrium interfacial properties for the multi-component mixtures containing a block copolymer by the numerical calculations based on the self-consistent mean field theory. For both A/B/A-b-B and A/B/C-h-D systems, they found that the interfacial tension decreases and the concentration of block copolymer at the interface increases with increasing the chain length of block copolymer. Israels et al. [85] examined the interfacial behavior of symmetric A-b-B diblock copolymers in a blend of... 

A more comprehensive theory for the thermodynamics of semi crystalline diblocks has been developed using self-consistent mean field theory applied to diblocks with one amorphous block and one crystallizable block [20].The amorphous regions were modelled as flexible chains, and the crystalline regions as folded chains. Both monolayers and bilayers of once-folded chains were considered. Expressions were derived for the thickness of the amorphous and crystalline region and the number of folds. The central result is the domain spacing scaling [Eq. (1)]. 

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. 

For the static (thermodynamic equilibrium) behavior, self-consistent mean field theory, which will be discussed in greater detail in Chapter 19, can be utilized to calculate y]2. This... 

The book by Hamley [65] is a good general resource for self-consistent mean field theory. This formalism is based on the assumptions that (a) every chain in the system obeys Gaussian statistics, (b) the fluid is incompressible, and (c) the interactions between different structural units are local so that they depend only on the chemical nature but not on the positions of the units along their respective chains. As a result, the equations describe an ensemble of ideal chains in an external field which, in turn, is determined self-consistently from the structural unit probability distributions. As illustrated by Matsen and Schick [66], solving the exact equations requires a significant amount of computational effort to determine the equilibrium... 

As mentioned above, Edwards has used a self-consistent mean field theory to derive similar (though not identical) results to those obtained from scaling theory (Edwards, 1966 Edwards and Jeffers, 1979). One virtue of this approach is that the relationships were obtained from a single extrapolation formula which encompasses all concentrations from an infinitely dilute solution to bulk polymer. This means that a somewhat more intuitively acceptable picture of polymer solutions emerges, one where there is a gradual change from one concentration regime to another. 

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