Analytical SCF theory

Inherent in the analytical SCF theory are (1) that the free ends of the chains can sample any position within the brush rather than be constrained to reside at the peripheral surface of the layer and, (2) that there is strong stretching, that is... 

As described in Sect. 2.4, SCF calculations are useful in determining local details of density profiles. A more local examination of profiles is indeed necessary to study the question of interpenetration in more detail. The analytical SCF theory [56, 57] shares with the adapted Alexander model embodied in Eq. 35 the characteristic of impenetrability. The full numerical SCF theory is necessary to... 

No informative experimental data have been obtained on the precise shape of segment profiles of tethered chains. The only independent tests have come from computer simulations [26], which agree very well with the predictions of SCF theory. Analytical SCF theory has proven difficult to apply to non-flat geometries [141], and full SCF theory in non-Cartesian geometry has been applied only to relatively short chains [142], so that more detailed profile information on these important, nonplanar situations awaits further developments. 

Since the addition of functionalized polymers (or diblocks) has yielded a robust method of exfoliating the sheets, we developed an analytical SCF theory that allows us to gain further insight into the behavior of this system and test the results from the numerical SCF calculations. Below, we describe the analytical model, discuss the results from this theory and compare the findings from the two SCF studies. 

In the RISM-SCF theory, the ab initio MO calculation and the RISM equation must be solved in a self-consistent manner. In other words, solvent distribution around the solute molecule is determined by the electronic structure of the solute, and the electronic structure of the solute is affected by the surrounding solvent distribution. To achieve this, the iteration cycle is repeated until mutual convergence between the ab initio MO calculation, which provides the partial charge on the solute, and the RISM equation, which provides PCFs, is attained. As mentioned above, it is noteworthy that the computational cost is drastically reduced by the analytic expression of the theory, compared with the QM/MM method. This means that the combination with more sophisticated (and computationally more expensive) methods such as CASPT2 and MCQDPT can be realized with moderate computational costs. Moreover, solvent distribution at the molecular level can be obtained simultaneously. These are remarkable merits compared with other hybrid-type methods. 

Evaluation of the integrals that arise in the calculations was for some time a primary problem in the field. A most important development in this context was the introduction of Gaussian-type basis functions by Boys [32], who showed that all of the integrals in SCF theory could be evaluated analytically if the radial parts of the basis functions were of the form P(x,y, z) exp(-r2). The first ten functions are listed by Hehre, Radom, Schleyer and Pople [33] and we repeat them here ... 

This computer laboratory presents the simplest analytical (Roothaan-like) SCF calculation imaginable. The basis set is a linear combination of Slater-type orbitals truncated after the first term.i s The calculation was developed in response to a well-known series of related problems suggested by Slater. S jhe simplicity of the calculation ensures that the fundamentals of SCF theory will not be obscured by complicated mathematics and computer code. This calculation has been done using BASIC, Mathematica, and Mathcad. 

We start with a brief reminder of the theory of self-assembly in a selective solvent of non-ionic amphiphilic diblock copolymers. Here, the focus is on polymorphism of the emerging copolymer nanoaggregates as a function of the intramolecular hy-drophilic/hydrophobic balance. We then proceed with a discussion of the structure of micelles formed by block copolymers with strongly dissociating PE blocks in salt-free and salt-added solutions. Subsequently, we analyze the responsive behavior of nanoaggregates formed by copolymers with pH-sensitive PE blocks. The predictions of the analytical models are systematically complemented by the results of a molecularly detailed self-consistent field (SCF) theory. Finally, the theoretical predictions are compared to the experimental data that exist to date. 

Numerical SCF theory can be used to probe the self-assembly of both long and short copolymers [72, 73, 77], for non-ionic [78-82] as well as ionic systems [83-86]. For long polymers, we can use the numerical SCF theory to check the scaling predictions and to test the validity of the analytical approximations. For short chains, the numerical theory is still expected to give reasonable predictions and results can be used to analyze experiments on the one hand, and to complement computer simulation results on the other hand. 

Below, we begin with a brief description of the numerical SCF model. In prior studies, we used this method to determine the free energies as a function of surface separation for polymer-coated surfaces in solution. Here, we describe our findings for die interactions between solid surfaces immersed in (1) a singlecomponent melt and (2) a melt that contains polymers with surface-active end-groups We also introduce an analytical SCF model for the melt containing end-functionalized chains and present the results from this theory. Comparisons are made between the numerical and analytical SCF results and die implications of these findings are discussed furdier in the Conclusions section. 

In the remainder of this section we describe the behavior of the coarse-grained model in the framework of the SCF theory. To allow for an analytical treatment, the SCF calculations use an even simpler model, which only reproduces the properties of the polymer model on long-length scales. 

The SCF theory describes the vanishing of the nucleation barrier at the spinodal [163, 183, 164] and is free of the thermodynamic inconsistency of the classical nucleation theory. In the vicinity of the spinodal, the density difference between the inside and the outside of the bubble becomes small, the bubble size large, and its interface to the mother phase very broad. These effects make the behavior in the vicinity of the spinodal amenable to an analytic description, the Cahn-Hilliard theory [146], While the theory was originally formulated for systems with a single order... 

High level HF/SCF calculations on ethylene using analytic derivative methods have been reported by Frisch et al. [341]. Their results are presented in Table 10.4. Vibrational frequencies and infrared intensities are also given. Interestingly, even at this high level of SCF theory the frequencies for vio(B]u) and vi](B2g) are predicted with reversed order. There are, however, no difficulties in assigning these bands for the correct normal mode on the basis of symmetry properties of die respective normal modes and the fact that B2g mode is Raman active wdiile Biu -infrared active. Both infrared and Raman intensities are relatively well predicted at the SCF large basis set calculations. 

Various models of SFE have been published, which aim at understanding the kinetics of the processes. For many dynamic extractions of compounds from solid matrices, e.g. for additives in polymers, the analytes are present in small amounts in the matrix and during extraction their concentration in the SCF is well below the solubility limit. The rate of extraction is then not determined principally by solubility, but by the rate of mass transfer out of the matrix. Supercritical gas extraction usually falls very clearly into the class of purely diffusional operations. Gere et al. [285] have reported the physico-chemical principles that are the foundation of theory and practice of SCF analytical techniques. The authors stress in particular the use of intrinsic solubility parameters (such as the Hildebrand solubility parameter 5), in relation to the solubility of analytes in SCFs and optimisation of SFE conditions. 

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. 

The previous result is an important one. It indicates that there can be yet another fruitful route to describe lipid bilayers. The idea is to consider the conformational properties of a probe molecule, and then replace all the other molecules by an external potential field (see Figure 11). This external potential may be called the mean-field or self-consistent potential, as it represents the mean behaviour of all molecules self-consistently. There are mean-field theories in many branches of science, for example (quantum) physics, physical chemistry, etc. Very often mean-field theories simplify the system to such an extent that structural as well as thermodynamic properties can be found analytically. This means that there is no need to use a computer. However, the lipid membrane problem is so complicated that the help of the computer is still needed. The method has been refined over the years to a detailed and complex framework, whose results correspond closely with those of MD simulations. The computer time needed for these calculations is however an order of 105 times less (this estimate is certainly too small when SCF calculations are compared with massive MD simulations in which up to 1000 lipids are considered). Indeed, the calculations can be done on a desktop PC with typical... 

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