Numerical self-consistent field theory

The effect of AB diblock size relative to the homopolymers on the compati-bilization of A/B homopolymer blends was examined using numerical self-consistent field theory (in two dimensions) by Israels et al. (1995). They found that the interfacial tension between homopolymers can only be reduced to zero if the blocks in the diblock are longer than the corresponding homopolymer. Short diblocks were observed to form multilamellar structures in the blend, whereas a microemulsion was formed when relatively long copolymers were added to the homopolymer mixture. These observations were compared to experiments on blends of PS/PMMA and symmetric PS-PMMA diblocks reported in the same paper. AFM was used to measure the contact angle of dewetted PS droplets on PMMA, and the reduction in the interfacial tension caused by addition of PS-PMMA diblocks was thereby determined. The experiments revealed that the interfacial tension was reduced to a very small value by addition of long diblocks, due to emulsification of the homopolymer by the diblock, in agreement with the theoretical expectation (Israels et al. 1995). 

Such adsorption isotherms may be predicted using numerical self-consistent field theory. This involves a relatively straightforward extension of the ap-... 

A MC study (262) of two different types of relatively short chains on a lattice, with solvent molecules represented as vacancies, was used to explore phase separation of the UCST type the conclusion was that the Freed-Dudowicz lattice theory (263,264) gives good agreement with simulations. A similar lattice model was used to investigate the behavior of chains tethered to a surface, with the MC results being compared with numerical self-consistent field theory (135). The essential identity of density profiles obtained by the two methods was taken to validate the self-consistent field theory. 

The simplest model of polymers comprises random and self-avoiding walks on lattices [11,45,46]. These models are used in analytical studies [2,4], in particular in the numerical implementation of the self-consistent field theory [4] and in studies of adsorption of polymers [35,47-50] and melts confined between walls [24,51,52]. 

A selection of the predictions of the equilibrium structure of DPPC bilayers as found by numerical self-consistent-field calculations is given in the following figures. In a series of articles, the SCF predictions for such membranes were published, starting in the late 1980s. As discussed above, we will update these early predictions for the theory outlined above with updated parameter sets. The calculations are very inexpensive with respect to the CPU time, and thus variations of the parameter-set will also provide deeper insight into the various subtle balances that eventually determine the bilayer structure - the mechanical properties as well as the thermodynamic properties. 

The scientific interests of Huzinaga are numerous. He initially worked in the area of solid-state theory. Soon, however, he became interested in the electronic structure of molecules. He studied the one-center expansion of the molecular wavefunction, developed a formalism for the evaluation of atomic and molecular electron repulsion integrals, expanded Roothaan s self-consistent field theory for open-shell systems, and, building on his own work on the separability of many-electron systems, designed a valence electron method for computational studies on large molecules. 

Numerical self-consistent fields have been established (see, for example [9,30,31]) and we shall record below a few of the consequences of the calculations, in particular for the chemical potential p, when we have considered the relativistic generalization of the TF theory (Sect. 6.4 below). 

An almost identical conclusion was obtained analytically by way of the self-consistent field theory. Edward [23] showed Pcy j 9/5 for d = 3 in good accord with the numerical calculation mentioned above, together with the mean end-to-end distance that scales as... 

The conformations of the molecules in the polymer layer and the resulting steric interaction energy can be calculated by means of a numerical self-consistent field model. The free energy of the polymer layers then is minimized by considering all possible conformations (including adsorbed segments) of the chains. We will not discuss the theory because it can rarely... 

The relativistic theory and computation of atomic structures and processes has therefore attained some sort of maturity and the various codes now available are widely used. Those mentioned so far were based on ideas originating from Hartree and his students [28], and have been developed in much the same way as the non-relativistic self-consistent field theory recorded in [28-30]. All these methods rely on the numerical solution, using finite differences, of the coupled differential equations for radial orbital wave-functions of the self-consistent field. This makes them unsuitable for the study of molecules, for which it is preferable to expand the radial amplitudes in a suitably chosen set of analytic functions. This nonrelativistic matrix Hartree-Fock method, as it is often termed, was pioneered by Hall and Lennard-Jones [31], Hall [32,33] and Roothaan [34,35], and it was Roothaan s students, Synek [36] and Kim [37] who were the first to attempt to solve the corresponding matrix Dirac-Hartree-Fock equations. Kim was able to obtain solutions for the ground state of neon in 1967, but at the expense of some numerical instability, and it seemed at the time that the matrix Dirac-Hartree-Fock scheme would not be a serious competitor to the finite difference codes. 

The statistical mechanical treatment is carried out on the basis of Barker s self-consistent field theory. This treatment accounts for local order and yields a set of non linear equations, which are solved numerically by iterations. The most interesting point of this model is that it does not use adjustable parameters. However, the predictions are rather poor. 

Abstract We present an overview of statistical thermodynamic theories that describe the self-assembly of amphiphilic ionic/hydrophobic diblock copolymers in dilute solution. Block copolymers with both strongly and weakly dissociating (pH-sensitive) ionic blocks are considered. We focus mostly on structural and morphological transitions that occur in self-assembled aggregates as a response to varied environmental conditions (ionic strength and pH in the solution). Analytical theory is complemented by a numerical self-consistent field approach. Theoretical predictions are compared to selected experimental data on micellization of ionic/hydrophobic diblock copolymers in aqueous solutions. 

Asakura and Oosawa (5) first identified depletion as a mechanism for generating an attractive interparticle potential. Numerous elaborations of their simple model followed, including sophisticated lattice and self-consistent field theories. Recently, Evans (27) resolved some inconsistencies in the evaluation of the effective pair potentid between the original niave model and the subsequent detailed analyses and achieved quantitative consistency between predictions and the detailed experiments employing bilayer membranes in a micropipette device. 

We have now found all necessary equations to numerically calculate the time evolution of the densities in a binary polymer mixture. This leads us to the following procedure which we refer to as the dynamic self consistent field theory (DSCFT) method ... 

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. 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

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