Self-consistent mean field theory model

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...

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. 

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. 

Figure 10.1 Schematic representation of length and time scales involved in various types of physical models of polymeric and biological systems. CFD = computational fluid dynamics CG-MD = coarse-grained molecular dynamics DPD = dissipative particle dynamics FEA = finite element analysis SCMFT = self-consistent mean field theory ...

The theoretical approach based on the HNC integral equation is described in the context of ionic specificity. Two levels of description of the water medium are considered. Within the Primitive Model (continuous solvent), ionic specificity is introduced via effective, solvent-averaged, dispersion forces. The agreement with experimental data in bulk or at air-water interfaces is only partial and illustrates the limits of that approach. Within the Born-Oppenheimer model, the molecular HNC equation is solved with an explicit description of the solvent molecules (SPC water). Ionic and solvent profiles in bulk and at interfaces are enriched by short-range osdUated structures. The ionic polaris-ability is introduced via the self-consistent mean-field theory, the polarisable ions carrying an effective, fixed dipole moment. The study of the air-water interface reveals the limits of the conventional HNC approach and the needs for improved integral equations. 

Despite the simple and universal structure of the nonrelativistic Hamiltonian for N interacting electrons, it produces a broad spectrum of physical and chemical phenomena that are difficult to conceptualize within the full -electron theory. Starting with the work of Hartree [162] in the early years of quantum mechanics, it was found to be very rewarding to develop a model of electrons that interact only indirectly with each other, through a self-consistent mean field. A deeper motivation lies in the fact that the relativistic quantum field theory of electrons is... 

Most of the theories were based on the self-consistent mean field approximation.12-32 The other ones include the scaling analysis, - molecular dynamics simulations and Monte Carlo simulations.40,41 The self-consistent mean field theories12-32 were developed along the following three lines (1) on the basis of a lattice model,12-22 (2) on the basis of a diffusion type equation,23-28 and (3) analytical approaches.29-32... 

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. 

This name covers all polymer chains (diblocks and others) attached by one end (or end-block) at ( external ) solid/liquid, liquid/air or ( internal ) liquid/liq-uid interfaces [226-228]. Usually this is achieved by the modified chain end, which adsorbs to the surface or is chemically bound to it. Double brushes may be also formed, e.g., by the copolymers A-N, when the joints of two blocks are located at a liquid/liquid interface and each of the blocks is immersed in different liquid. A number of theoretical models have dealt specifically with the case of brush layers immersed in polymer melts (and in solutions of homopolymers). These models include scaling approaches [229, 230], simple Flory-type mean field models [230-233], theories solving self-consistent mean field (SCMF) equations analytically [234,235] or numerically [236-238]. Also first computer simulations have recently been reported for brushes immersed in a melt [239]. 

Hurter, RN., Scheutjens, J.M.H.M., and Hatton, T.A., Molecular modeling of micelle formation and solubilization in block copolymer micelles. 1. A self-consistent mean-field lattice theory. Macromolecules, 26, 5592, 1993. 

This theory, when applied to the SPC water geometry but with a fixed dipole equal to that of an isolated H2O molecule, p = 1.85D, leads to the result = 2.62D. This is in good agreement with the well-known extended simple point charged model SPC/E which allows for the polarisation correction. This validates the self-consistent mean-field procedure. As an illustration of the anion polarisation effect. Fig. 7... 

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