Smoothed-density model

The energy fxmction, W(Rg), was calculated by assuming that the local energy of interaction is proportional to the square of the local subunit density. The subunit density was obtained by assuming that the subchains were distributed relative to the center of mass in a Gaussian distribution. This model is called the smoothed-density model and will be used many times in this chapter. The number density of subunits at a location S relative to the center of mass in a macromolecule of overall radius of gyration Rq is given by ... 

Star polyisopienes 6 greater for methyl i-propylketone than for melhyl-isobulylketone whidt conelates with their different < /1 values. (1/3 —X2) is essentially identical in the two solvente. In dioxane the 6 depression is zero appar tly because the (1 /3 -X2) term is complete n igible. It seems, therefore, that dilute solution properties of star-branched polymers can be at least approximated in terms of three parameters p 1,6 and X2- The smoothed density model is known to predict the variation of a and A2 rather poorly for linear polymers so that quantitative agreement for branched polymers is unlikely. 

Styrene udi the same technique This is slightly below the normal vdue. l/ ] values determined indirectly by the lyrplication of the modified mioothed density model are, however, considerably lower ( 0.03 . This model is known to give low values even for linear polymers but the error is approximately a factor of two. It can be made to yield correct values by changing the constant involved (Cm in Flory s notation) to force a fit with exact perturbation theory valid at low chain expansions Use of the modified smoothed density model on those branched polymers of high molecular weight (0 2 34.5°) shown in Fig. 7 would also lead to abnormally low, values (<0.1), in contrast to the normd values found directly from A2 near 6. The low j values are determined from the equation,... 

Another smoothed density model, based on consideration of the ellipsoidal shape of a segment distribution averaged about any given end-to-end distance, leads to a different power law dependence... 

Abstract. A smooth empirical potential is constructed for use in off-lattice protein folding studies. Our potential is a function of the amino acid labels and of the distances between the Ca atoms of a protein. The potential is a sum of smooth surface potential terms that model solvent interactions and of pair potentials that are functions of a distance, with a smooth cutoff at 12 Angstrom. Techniques include the use of a fully automatic and reliable estimator for smooth densities, of cluster analysis to group together amino acid pairs with similar distance distributions, and of quadratic progrmnming to find appropriate weights with which the various terms enter the total potential. For nine small test proteins, the new potential has local minima within 1.3-4.7A of the PDB geometry, with one exception that has an error of S.SA. 

We therefore use smooth density estimation techniques that are more reliable than the histogram estimates. To improve the reliability for rare amino acid pairs, we use clustering techniques that identify similar pairs that can be modeled by the same density. 

Strutinsky did not address the question of how to calculate microscopically the smooth part E (which necessarily entails specifying the smooth density p). Instead he circumvented this question by substituting for E the empirical energies, ldm of the nuclear liquid-drop model, namely he suggested that... 

When the smoothed or nonlocal density approximation (or NL-DFT model) is used, the weighting function is chosen so that the hard-sphere direct pair-correlation function is well described for the uniform fluid over a wide range of densities. One example of such a weighting function is the model proposed by Tarazona [69], which uses the Percus-Yevick theory for approximating the correlation function over a wide range of density. In this case, the weighting function is expanded as a power series of the smoothed density. The use of a smoothed density in NL-DFT provides an oscillating density profile expected of a fluid adjacent to a sohd surface, the existence of which is corroborated by molecular simulation results [17,18]. 

The form of the function efr ( ) is different in different versions of the smoothed-density approximation proposed by Somo-za and Tarazona [71, 72] and by Poniwier-ski and Sluckin [69, 73]. The density functional model of Somoza and Tarazona is based on the reference system of parallel hard ellipsoids that can be mapped into hard spheres. In the Poniwierski and Sluckin theory the effective weight function is determined by the Maier function for hard sphe-rocylinders and the expression for Ayr (p) is obtained from the Carnahan-Starling ex-... 

The results of several molecular theories that describe the smectic ordering in a system of hard spherocylinders enable us to conclude that the contribution from hardcore repulsion can be described by the smoothed-density approximation. On the other hand, a realistic theory of thermotropic smectics can only be developed if the intermolecular attraction is taken into account, The interplay between hard-core repulsion and attraction in smectic A liquid crystals has been considered by Kloczkow-ski and Stecki [17] using a very simple model of hard spherocylinders with an ad-ditonal attractive r potential. Using the Onsager approximation, the authors have obtained equations for the order parameters that are very similar to the ones found in the McMillan theory but with explicit expressions for the model parameters. The more general analysis has been performed by Me-deros and Sullivan [76] who have treated the anisotropic attraction interaction by the mean-field approximation while the hardcore repulsion has been taken into account using the nonlocal density functional approach proposed by Somoza and Tarazona. 

The complexity of the excess functions in mixtures make an analytical discussion highly desirable. For this purpose the Lennard-Jones and Devonshire model is unfortunately not suitable because of the complicated form of the mean potential model valid in a restricted range of density. For hig densities, as in the solid state, we may use a harmonic potential approximation (cf. Fig. 7.1.2). We shall develop this approximation in more detail in the next paragraph. On the other hand, for the range of densities corresponding to the liquid state we may use the smoothed potential model (Prigogine and Mathot [1952]) (cf. Fig. 7.1.2). This is however an oversimplification and the conclusions have to be used with some caution. 

We test three theories for adsorption and capillary condensation in pores against computer simulation results. They are the Kelvin equation, and two forms of density functional theory, the local density approximation (LDA) and the (nonlocal) smoothed density approximation (SDA) all three theories are of potential use in determining pore size distributions for raesoporous solids, while the LDA and SDA can also be applied to mlcroporous materials and to surface area determination. The SDA is found to be the most accurate theory, and has a much wider range of validity than the other two. The SDA is used to study the adsorption of methane and methane-ethane mixtures on models of porous carbon in which the pores are slit-shaped. We find that an optimum pore size and gas pressure exists that maximizes the excess adsorption for methane. For methane-ethane mixtures we show the variation of selectivity with pore size and temperature. 

The simulated free surface of liquid water is relatively stable for several nanoseconds [68-72] because of the strong hydrogen bonds formed by liquid water. The density decrease near the interface is smooth it is possible to describe it by a hyperbolic tangent function [70]. The width of the interface, measured by the distance between the positions where the density equals 90% and 10% of the bulk density, is about 5 A at room temperature [70,71]. The left side of Fig. 3 shows a typical density profile of the free interface for the TIP4P water model [73]. 

While thin polymer films may be very smooth and homogeneous, the chain conformation may be largely distorted due to the influence of the interfaces. Since the size of the polymer molecules is comparable to the film thickness those effects may play a significant role with ultra-thin polymer films. Several recent theoretical treatments are available [136-144,127,128] based on Monte Carlo [137-141,127, 128], molecular dynamics [142], variable density [143], cooperative motion [144], and bond fluctuation [136] model calculations. The distortion of the chain conformation near the interface, the segment orientation distribution, end distribution etc. are calculated as a function of film thickness and distance from the surface. In the limit of two-dimensional systems chains segregate and specific power laws are predicted [136, 137]. In 2D-blends of polymers a particular microdomain morphology may be expected [139]. Experiments on polymers in this area are presently, however, not available on a molecular level. Indications of order on an... 

At higher pressures only Raman spectroscopy data are available. Because the rotational structure is smoothed, either quantum theory or classical theory may be used. At a mixture pressure above 10 atm the spectra of CO and N2 obtained in [230] were well described classically (Fig. 5.11). For the lowest densities (10-15 amagat) the band contours have a characteristic asymmetric shape. The asymmetry disappears at higher pressures when the contour is sufficiently narrowed. The decrease of width with 1/tj measured in [230] by NMR is closer to the strong collision model in the case of CO and to the weak collision model in the case of N2. This conclusion was confirmed in [215] by presenting the results in universal coordinates of Fig. 5.12. It is also seen that both systems are still far away from the fast modulation (perturbation theory) limit where the upper and lower borders established by alternative models merge into a universal curve independent of collision strength. 

Stelzer et al. [109] have studied the case of a nematic phase in the vicinity of a smooth solid wall. A distance-dependent potential was applied to favour alignment along the surface normal near the interface that is, a homeotropic anchoring force was applied. The liquid crystal was modelled with the GB(3.0, 5.0, 2, 1) potential and the simulations were run at temperatures and densities corresponding to the nematic phase. Away from the walls the molecules behave just as in the bulk. However, as the wall is approached, oscillations appear in the density profile indicating that a layered structure is induced by the interface, as we can see from the snapshot in Fig. 19. These layers are... 

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