Self-consistent field calculations models

Fig. 4.2. Charge distribution and surface potential in a jellium model, (a) Distribution of the positive charge (a uniform background abruptly drops to zero at the boundary) and the negative charge density, determined by a self-consistent field calculation. (b) Potential energy as seen by an electron. By including all the many-body effects, including the exchange potential and the correlation potential, the classical image potential provides an adequate approximation. (After Bardeen, 1936 see Herring, 1992.)...

In addition to these experimental methods, there is also a role for computer simulation and theoretical modelling in providing understanding of structural and mechanical properties of mixed interfacial layers. The techniques of Brownian dynamics simulation and self-consistent-field calculations have, for example, been used to some advantage in this field (Wijmans and Dickinson, 1999 Pugnaloni et al., 2003a,b, 2004, 2005 Parkinson et al., 2005 Ettelaie et al., 2008). 

C. Amovilli and B. Mennucci, Self-consistent-field calculation of Pauli repulsion and dispersion contributions to the solvation free energy in the polarizable continuum model, J. Phys. Chem. B, 101 (1997) 1051. 

Adiabatic electron affinity, energy difference between the ground state of the anion and the most stable state of the neutral molecule. A particular semi-empirical self-consistent field calculation. It stands for Austin Model-1. 

Figure 2 A simplified flow diagram for self-consistent-field calculations. This diagram is modeled after one given by Wilson. ...

This situation is quite analogous to that encountered in electronic structure applications of dimensional scaling. Here, too, the dimensional approach could be used on its own as a conceptual tool or semi-quantitative model, or it could be used quantitatively as a corrective for more familiar approximation methods. In particular, dimensional scaling could be used to treat the many- body effects omitted from self-consistent field calculations, in much the same way that it has been used here to treat the cluster integrals omitted from the HNC and PY calculations. 

Note that the sharpness of the transition in the change on going from the depletion zone to the parabolic zone is due to limitations in the analytical function and does not reflect real transition behaviour. The more gentle transitions indicated in the theoretical SCF profile are more realistic. In the self-consistent field calculation a lattice model is not presumed the volume fraction of the tethered chains is calculated from a diffusion equation that involves polymer propagators and a (z-dependent) potential function that includes enthalpic interactions between the two copolymer blocks and between each block and the solvent. Initially the potential function is set to zero, the pol)nner propagators are calculated and then the volume fraction variation of the tethered block. A new potential is calculated from this volume fraction profile and the process reiterated imtil the difference in volume fraction profiles calculated by sequential iterations is smaller than some defined tolerance. The approach bears similarities to the SCF approach of Shull (1991) but makes no allowance for the dry brush case, i.e. that in which the relative molecular mass of the solvent approaches that of the tethered polymer molecule. 

To summarize, the example of homopolymer/copolymer mixtmes demonstrates nicely how field-theoretic simulations can be used to study non-trivial fluctuation effects in polymer blends within the Gaussian chain model The main advantage of these simulations is that they can be combined in a natural way with standard self-consistent field calculations. As mentioned earlier, the self-consistent field theory is one of the most powerful methods for the theoretical description of polymer blends, and it is often accurate on a quantitative level hi many regions of the parameter space, fluctuations are irrelevant for large chain lengths (large Jf) and simulations are not necessary. Field-theoretic simulations are well suited to complement self-consistent field theories in those parameter regions where fluctuation effects become important. 

The self-consistent field approach models the microstructure of the composite in order to deal more realistically with the internal distribution of stress and strain. The various treatments in this category have been brought together by Wu and McCullough [30]. The treatment relates to an arbitrary reference material and encompasses a bounding approach. The lower bound of calculated modulus refers to a dispersion of filler particles in a continuous matrix of polymer and the upper bound refers to inclusions of polymer in a continuous phase that has the properties of the filler. The approach is further modified in reference [29] and the derived equations which relate modulus to Vf are shown next ... 

Our study was extended to the binary system silica-calcia. The parameters of the Born-Mayer potential (Equation 3.4.2) were taken from Ref. [15]. The interatomic potential was calculated from the potential used for pure silica performed by TTAM [3] and derived from the ab initio Hartree-Fock self-consistent field calculations for model clusters of silica. The parameters used in Equation 3.4.2 are summarized in Table 3.4.2. 

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