Self consistent field theory approximation

Equation (75) expresses the partition function of the many-polymer system in terms of the partition functions of single polymers subjected to external fluctuating fields. The self-consistent field theory approximates this functional integral over the fields by the value of the integrand evaluated at those values of the fields, and Wb, that minimize the functional F[h, 4, vb]. From the definition of F it follows that these functions satisfy the self-consistent equations... 

Figure 11. A schematic representation of the mean-field approximation, a central issue in the self-consistent-field theory. The arrows symbolically represent the lipid molecules. The head of the arrow is the hydrophilic part and the line is the hydrophobic tail. On the left a two-dimensional representation of a disordered bilayer is given. One of the lipids has been selected, as shown by the box around it. The same molecule is depicted on the right. The bilayer is depicted schematically by two horizontal lines. The centre of the bilayer is at z = 0. These lines are to guide the eye the membrane thickness is not preassumed, but is the result of the calculations. Both the potential energy felt by the head groups and that of the tail segments are indicated. We note that in the detailed models the self-consistent potential profiles are of course much more detailed than shown in this graph...

A self-consistent field theory (SCFT) for micelle formation of block copolymers in selective solvents was developed by Yuan el at. (1992). They emphasized the importance of treating the isolated chain at the same level of theoretical approximation at the micelle, in contrast to earlier approaches. This was achieved by modifying the Edwards diffusion equation for the excluded volume of polymers in solution to the case of block copolymers, with one block in a poor solvent. The results of the continuum model were compared to experimental results for PS-PI diblocks in hexadecane, which is a selective solvent for PI and satisfactory agreement was obtained. 

Self-consistent field theory (SCFT, see Sections 2.3.3 and 3.4,2) has recently been applied to the phase behaviour of ordered micellar solutions. Noolandi et al. (1996) compared continuum SCFT to the lattice version of this theory for triblock copolymers such as the Pluronics in aqueous solution. From a different viewpoint, this work represents an extension of the SCFT employed by Hong and Noolandi (1981, 1983) and Matsen and Schick (1994) for the phase behaviour of block copolymer melts to block copolymers in solution. The approximations introduced by the adoption of a lattice model are found to lead to some significant differences in the solution phase behaviour compared with the continuum theory, as illustrated by Fig. 4.44. For example, the continuum theory predicts ordered phases for Pluronic L64 (PE013PP03oPEO 3), whereas the lattice theory (neglecting polydispersity) predicts none. 

The accurate calculation of these molecular properties requires the use of ab initio methods, which have increased enormously in accuracy and efficiency in the last three decades. Ab initio methods have developed in two directions first, the level of approximation has become increasingly sophisticated and, hence, accurate. The earliest ab initio calculations used the Hartree-Fock/self-consistent field (HF/SCF) methodology, which is the simplest to implement. Subsequently, such methods as Mpller-Plesset perturbation theory, multi-configuration self-consistent field theory (MCSCF) and coupled-cluster (CC) theory have been developed and implemented. Relatively recently, density functional theory (DFT) has become the method of choice since it yields an accuracy much greater than that of HF/SCF while requiring relatively little additional computational effort. 

Orbital models are almost invariably the first approximation for the electronic structure of molecules within the Born-Oppenheimer approximation. By considering the motion of each electron in the averaged field of the remaining electrons in the systems in the self-consistent field theory of Hartree and Fock, a model is obtained in which each electron is described by an spin-orbital, a description which is familiar to all chemists. 

As mentioned earlier, studies of simple linear surfactants in a solvent (i.e, those without any third component) allow one to examine the sufficiency of coarse-grained lattice models for predicting the aggregation behavior of micelles and to examine the limits of applicability of analytical lattice approximations such as quasi-chemical theory or self-consistent field theory (in the case of polymers). The results available from the simulations for the structure and shapes of micelles, the polydispersity, and the cmc show that the lattice approach can be used reliably to obtain such information qualitatively as well as quantitatively. The results are generally consistent with what one would expect from mass-action models and other theoretical techniques as well as from experiments. For example. Desplat and Care [31] report micellization results (the cmc and micellar size) for the surfactant h ti (for a temperature of = ksT/tts = /(-ts = 1-18 and... 

Capillary waves do not only broaden the width of the interface but they can also destroy the orientational order in highly swollen lamellar phases (see Fig. 1 for a phase diagram extracted from Monte Carlo Simulations). Those phases occur in mixtures of diblock-copolymers and homopolymers. The addition of homopolymers swells the distance between the lamellae, and the self-consistent field theory predicts that this distance diverges at Lifshitz points. However, general considerations show that mean-field approximations are bound to break down in the vicinity of lifshitz points [61]. (The upper critical dimension is du = 8). This can be quantified by a Ginzburg criterion. Fluctuations are important if... 

The dynamic self-consistent field theory has been widely used in the form of MESODYN [97]. This scheme has been extended to study the effect of shear on phase separation or microstructure formation, and to investigate the morphologies of block copolymers in thin films. In many practical applications, however, rather severe numerical approximations (e.g., very large discretization in space or contour length) have been invoked, that make a quantitative comparison to the the original model of the SCF theory difficult, and only the qualitative behavior could be captured. 

In order to study this effect, Diichs et al. have performed field-theoretic Monte Carlo simulations of the system of Fig. 5 [80,83,104], in two dimensions. (For the reasons explained in Sect. 4.4.2, most of these simulations were carried out in the EP approximation). Some characteristic snapshots were already shown in Fig. 4. Here we show another series of snapshots at = 12.5 for increasing homopolymer concentrations (Fig. 8). For all these points, the self-consistent field theory would predict an ordered lamellar phase. In the... 

Field theoretical simulations [74,75,80] avoid any saddle point approximation and provide a formally exact solution of the standard model of the self-consistent field theory. To this end one has to deal with a complex free energy functional as a fimction of the composition and density. This significantly increases the computational complexity. Moreover, for certain parameter regions, it is very difficult to obtain reliable results due to the sign problem that a complex weight imparts onto thermodynamical averages [80]. We have illustrated that for a dense binary blend the results of the field theoretical simulations and the EP theory agree quantitatively, i.e., density and composi-... 

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