Numerical self-consistent field

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. 

Garza, J., Vargas, R. and Vela, A. 1988. Numerical self-consistent-field method to solve the Kohn-Sham equations in confined many-electron atoms. Phys. Rev. E. 58 3949-54. 

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). 

The result that the decay length , of the order parameter oscillations remains finite at yc is also suggested by Shull [20] using a numerical self-consistent field treatment. On the other hand, for not too strong Hj and %<%c his treatment is in good numerical agreement with the result of Fredrickson [12], and hence one can conclude that the extension to thin films [61] described above is sensible. 

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). 

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 same group used analytical arguments and numerical self-consistent field calculations to compare the interfacial tensions for a system of A and B homopolymers, immiscible with each other, having AB diblocks and comb copolymers localized at the interface [330,331]. In this case the overall molecular weight of the compatibilizer molecule was kept constant and the architecture was varied. At fixed N and > as the number of teeth increased the comb becomes less efficient at reducing the interfacial tension as compared to the diblock. At high values of N and x> the difference between the architectures becomes more pronounced. The diblock produces the lowest interfacial tension at the lowest concentrations. On the other hand, long combs with multiple teeth were found to be more efficient compatibilizers than short diblocks. 

D. Andrae, Numerical self-consistent field method for polyatomic molecules, Mol. Phys. 99 (2001) 327-334. 

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. 

The numerical self-consistent field method can also be used to construct adsorption isotherms, from which estimates of sticking energies may be... 

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

The question has been considered more carefiilly by Israels et al. (1995), who have used numerical self-consistent field calculations both in one and in two dimensions to evaluate the relative stabilities of a microemulsion phase compared with an interface with non-vanishing tension coexisting either with micelles or with multi-lamellar phases. Their calculations suggested that vanishing interfacial tension was possible for symmetrical copolymers with a much larger degree of polymerisation than the homopolymers and they also... 

In this article, we will focus in particular on the so-called coupled monolayer models [18,20-26,28,29], where membranes are described as stacks of two sheets (monolayers), each with their own elastic parameters. Mraiolayers are bound to each other by a local harmonic potential that accounts for the areal compressibility of lipids within the membrane and their constant volume [22, 28]. Li et al. have recently compared the elastic properties of amphiphilic bilayers with those of the corresponding monolayers within a numerical self-consistent field study of... 

Scheutjens and Heer [106] developed a numerical self-consistent field (SCF) method that enables the calculation of equilibrium SCF concentration profiles near interfaces. This SCF method was applied to depletion effects in Ref. [107], showing that the depletion layer thickness is close to Rg at low polymer concentrations, but... 

Such a repulsive contribution to the depletion interaction originates from excluded volume interactions between the depletants in case of ideal polymers and penetrable hard spheres it is absent One might expect accumulation effects also in the case of interacting polymers. From Monte Carlo simulation studies [51] and numerical self-consistent field computations [52, 53] it follows that interacting polymers do contribute to repulsive depletion interactions but with a strength of the repulsion that is nearly imperceptible. 

For a -solvent with y = 1 (4.21) is in quantitative with numerical self-consistent field results. For a good solvent with y = 0.77 (4.21) compares favourably with computer simulation results, see [41]. 

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. 

Our numerical self-consistent field (SCF) calculations are based on the model developed by Scheutjens and Fleer. In this treatment, the phase behavior of polymer systems is modeled by combining Markov chain statistics with a mean field approximation for the free energy. The equations in this lattice model are solved numerically and self-consistendy. The self-consistent potential is a function of the polymer segment density distribution and the Flory-Huggins interaction parameters, or... 

In the meantime, Scheutjens and Fleer [35] had formulated their now classical numerical self-consistent field (lattice) (SF) fheory for equilibrium adsorption of polymers, which provides a very detailed partition function from which strucfural and fhermodynamic information can be derived. The key idea of a self-consisfenf field approach is that, on the one hand, the interactions between molecules are treated as if they constitute an external field on an individual molecule (fhereby defining potential energies), and that each molecule finds its optimum distribution of conformations and locations as a Boltzmann distribution in that field. On the other hand, the distribution found has to be such that it will indeed produce the field, hence the term self-consistent. An iterative scheme is employed to reach consistency, and as soon as the solution is found, energy and entropy of fhe system are known as well. 

FIG. 8 Loop and tail profile according to the numerical self-consistent field model (solid curves) and according to the analytical approximation (dotted). (From Ref. 23.)... 

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