Surface density functions

Free surface density functions calculated at step 8 are used as the initial conditions to update the current position of the surface using the following integration... 

The latter free energy can be represented as a surface integral over the solvent accessible surface of the molecule on the basis of a local free energy surface density (FESD) p. This surface density function is represented in terms of a three-dimensional scalar field which is comprised of a sum of atomic increment functions to describe lipophilicity in the molecular environment.The empirical model parameters are obtained by a least squares procedure with experimental log P values as reference data. It is found that the procedure works not only for the prediction of unknown partition coefficients but also for the localization and quantification of the contribution of arbitrary fragments to this quantity. In addition, the... 

In a volume-oriented density function such as that used by ROCS, Gaussian functions are atom-centered. In the surface-oriented formulation, the M. of Equation 2.2.1 are Gaussians with peaks at the atomic surface (set by the atomic radii, denoted fi). By itself, the sum over the M produces internal molecular surfaces in addition to external ones. The E[ of Equation 2.2.2 defines Gaussians on local radial co-ordinates around each observer point from set P, with peaks at the molecular surface (set by the minimum distances from the observers to the molecule, denoted d,). When yis chosen carefully, the integral of the product of two molecules surface-density functions R (defined in Eq. 2.2.3) is very closely approximated by the morphological similarity function used by Surflex-Sim [31]. 

Here, the two surface-density functions are defined with respect to a single set of observations points P. The spheres that pack around each of the two molecules A and B share the same centers, but they have different radii, depending on the minimum distance to each molecular surface. As with the ROCS approach, one can define a similarity metric in terms of the overlap integral of the product of the two surface-density functions. This function is very closely approximated by the function computed by Surflex-Sim, simplified slightly in what follows. 

Figure 1.1 shows typical plots of the orbital radial functions. However, in order to interpret this information as electron density distribution it is often more useful to consider the surface density functions S(r) plotted in Fig. 1.2... 

To draw a chemical picture from a mineral surface using just spectroscopic or electrochemical data is definitively not an easy task. However, the increase of computational power associated with the development of more accurate methodologies and efficient implementations, make computational methods an important tool for investigating mineral surfaces. Density Functional Theory (DFT) is especially important in this scenario, because it has a high accuracy and the computational cost for such calculations is affordable. In fact, many studies have combined spectroscopy data with DFT calculations enhancing the understanding of important phenomena in the surfaces at a molecular level. 

Tokarz-Sobieraj, R., K. Hermann, M. Witko, A. Blume, G. Mestl, and R. Schlogl. 2001. Properties of oxygen sites at the MoOjfOlO) surface Density functional theory cluster studies and photoemission experiments. Surf. Sci. 489 107-125. 

LS. In the LS phase the molecules are oriented normal to the surface in a hexagonal unit cell. It is identified with the hexatic smectic BH phase. Chains can rotate and have axial symmetry due to their lack of tilt. Cai and Rice developed a density functional model for the tilting transition between the L2 and LS phases [202]. Calculations with this model show that amphiphile-surface interactions play an important role in determining the tilt their conclusions support the lack of tilt found in fluorinated amphiphiles [203]. 

S. Chains in the S phase are also oriented normal to the surface, yet the unit cell is rectangular possibly because of restricted rotation. This structure is characterized as the smectic E or herringbone phase. Schofield and Rice [204] applied a lattice density functional theory to describe the second-order rotator (LS)-heiTingbone (S) phase transition. 

Weinert M, Wimmer E and Freeman A J 1982 Total-energy all-electron density functional method for bulk solids and surfaces Phys. Rev. B 26 4571-8... 

Wimmer E, Fu C L and Freeman A J 1985 Catalytic promotion and poisoning all-electron local-density-functional theory of CO on Ni(001) surfaces coadsorbed with K or S Phys. Rev. Lett. 55 2618-21... 

Duarte H A and Salahub D R 1998 Embedded cluster model for chemisorption using density functional calculations oxygen adsorption on the AI(IOO) surface J. Chem. Phys. 108 743... 

Kruger S and Rdsch N 1994 The moderately-large-embedded-cluster method for metal surfaces a density-functional study of atomic adsorption J. Phys. Condens Matters 8149... 

Fig. 17. Adhesion energy G measured as a function of the surface density of the interfacial chains. It may noted that the strength measured in a peel test (a) is about 5 times larger than that measured using the JKR method (b). Further, a maximum exists in the value of G as function of the surface chain density. This is because of swelling effects at larger values of surface chain density. The open symbols represent the data for elastomer molecular weight Mo = 24,000 and the closed symbols represent the data for Mo = 10,000.

Fig. 11. Laboratory OSB board density profile as a function of board thickness when using 10 s and 50 s press closing times. Note the much higher peaks of surface density for the 10 s case, and the more even density profile for the slower press closing time.

D. Henderson, S. Sokolowski, D. Wasan. Structure of a hard-sphere fluid near a rough surface a density-functional approach. Phys Rev E 57 5539-5543, 1998. 

In 1985 Car and Parrinello invented a method [111-113] in which molecular dynamics (MD) methods are combined with first-principles computations such that the interatomic forces due to the electronic degrees of freedom are computed by density functional theory [114-116] and the statistical properties by the MD method. This method and related ab initio simulations have been successfully applied to carbon [117], silicon [118-120], copper [121], surface reconstruction [122-128], atomic clusters [129-133], molecular crystals [134], the epitaxial growth of metals [135-140], and many other systems for a review see Ref. 113. 

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

Sec. Ill is concerned with the description of models with directional associative forces, introduced by Wertheim. Singlet and pair theories for these models are presented. However, the main part of this section describes the density functional methodology and shows its application in the studies of adsorption of associating fluids on partially permeable walls. In addition, the application of the density functional method in investigations of wettability of associating fluids on solid surfaces and of capillary condensation in slit-like pores is presented. 

The problem of adsorption of associating fluids on crystalline surfaces has also been studied by Borowko et al. by using the density functional approach [43]. 

Fig. 10(a) presents a comparison of computer simulation data with the predictions of both density functional theories presented above [144]. The computations have been carried out for e /k T = 7 and for a bulk fluid density equal to pi, = 0.2098. One can see that the contact profiles, p(z = 0), obtained by different methods are quite similar and approximately equal to 0.5. We realize that the surface effects extend over a wide region, despite the very simple and purely repulsive character of the particle-wall potential. However, the theory of Segura et al. [38,39] underestimates slightly the range of the surface zone. On the other hand, the modified Meister-Kroll-Groot theory [145] leads to a more correct picture. 

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