Weighted density functional

A mass (weight) density function, given the symbol m and having dimensions mass/(volume)(length), can be defined analogously to population density letting AM be the mass of crystals per unit system volume in the size range E to L + AL,... 

B. Laird, D. Kroll. Freezing of soft spheres a critical test for weighted density functional techniques. Phys Rev A 42 4810, 1990 D. Kroll, B. Laird. Comparison of weighted density functional theories for inhomogeneous liquids. Phys Rev A 42 4806, 1990. 

Also, in addition to the zero-ranged repulsion between imlike species, other interactions can be included into the theory. In order to capture some fluid-like correlations and describe the details of packing in the vicinity of surfaces weighted density functionals have been used successfully [21]. 

Fig. 4.28 Left Transformed number weighted density function of the Stokes diameter, the translational hydrodynamic diameter, the maximum Feret diameter (xstokes. Xh,t. xperekmax). and the diameter of g)Tation Xg, calculated for DLCA aggregates with primary particles of 20 nm diameter and a lognormal distribution of N (Nso,o = 200 and crin = 0.4) right corresponding density functions of the Stokes, the translational, and the effective hydrodynamic diameter (xstokes. Xh,t. Xh.eff) weighted by hght extinction (t/ext) and scattering intensity at 90° and 173° ( int.90 /i73°) and number (qo)...

In this section, we discuss one such calculation on Si and Ge using a generalized weighted density functional scheme. 

Yu Y-X A novel weighted density functional theory for adsorption, fluid-solid interfacial tension, and disjoining properties of simple hquid films on planar solid surfaces, / Chem Phys 131(2) 024704, 2009. 

Choudhury, N., and Ghosh, S. K. 2003. J. Phys. Chem. B. New weighted density functional theory based on perturbative approach. 107 7155. 

Curtin, W. A., and Ashcroft, N.W. Weighted-density-functional theory of inhomogeneous liquids and the freezing transition. 1985. Phys. Rev. A Atom. Mol. Opt. Phys. 32 2909. 

These quantities are thus analogous to those we have already defined for a size distributed particle population. Instead of particle size, we have a distribution variable r, which is intrinsic to a given chemical species. For example, for a flash vaporizer single-entry separator (Figure 2.4.7), the material balance for 1 mole of feed having a molecular weight density function off/(M) is... 

The two molecular weight density functions frequentiy used in such types of calculations are ... 

Consider a continuous chemical mixture present in the vapor phase. Its molecular weight density function is known to be/ (M). Determine the expression of (M) for the first liquid drop formed as the vapor is cooled to temperature T. Make appropriate simplifications for low-pressure gas and ideal liquid solution. 

Approximate solution formalisms have been developed by Gotterman and Prausnitz (1985) and Kehlen et al. (1985) based on the assumption that the same type of density function may describe the molecular weight based distribution in all three streams. Other approximate calculation procedures have also been developed. One such procedure employs the first and second moments of each molecular weight density function and develops appropriate relations from (6.3.68) between the parameters of each density function. Suppose a T distribution (see (4.1.331)) describes the behavior of all three streams. The mean and the standard deviation for each stream are given, respectively, by... 

In general, most classical DFTs fall into two classes (a) methods based on functional Taylor series expansions about some reference state - usually the homogeneous phase and (b) non perturbative methods based on the concept of weighted density functionals. 

Weak coupling 128, 129 Weighted density functional 32, 95 Wetting, interfacial 91, 94 Woodcock 120, 126, 136, 141... 

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