Local density distribution

An alternative, already implicit in van der Waals theory of the surface tension [237], exploits an interconnection between the coefficient Co and the pair correlation function g(ra, rp) of the inhomogeneous system. As the local density distribution determining g(ra, rp) is unknown, one has to resort to a... 

The subject of main interest in the present study is the layering phenomenon in films that are formed from like-charged particles confined between two uncharged surfaces. In the case when confining surfaces are parallel, the particle layering is characterized by a local density distribution p(z) across the slit. Two kinds of films in a plane-parallel slit can be distingushed. The first one is that formed from the macroion suspension adsorbed into a slit of the fixed thickness. The other kind of film can be formed in the case when slit surfaces are movable. 

Figure 7. MC data for the normalized local density distribution of the macroions in a slit formed by two uncharged surfaces fixed on a distance H/D = 10. The bulk volume fraction of macroions is / 0.05 (high surfactant concentration) and macroion charge number is fixed...

Figure 8. MC data for the normalized local density distribution of the macroions in a film formed at macroion bulk volume fraction rj = 0.05. The macroion charge is fixed at Z = 30. (a) The film has thickness H/D = 7.5 and contains four particle layers two surface monolayers and two middle-film layers. The dashed line shows MC data obtained from the simulation without excluded volume forces, (b) The film has thickness H/D = 5 and contains three particle layers two surface monolayers and one middle-film layer, (c) The film has thickness H/D = 3.5 and contains two surface monolayers only.

In the case of higher charge, Z 30 on Fig. 8a, both models result that the like-charged particles being confined to a film that has a thickness around H/D = 7.5 tend to be organized into four particle layers. For the middle-film layers formed with and without excluded volume forces, only some quantitative differences in the particle local density distribution are observed. The main difference introduced by excluded volume forces is found in the surface layers. Taking into account the discrete nature of the solvent results that the surface layers themselves show a structuring with respect to the film surfaces. 

In the following, this qualitative characterization of the macroion layering in a wedge slit is supported and detailed by analyzing the profiles p(x) and px(z) of the macroion local density distribution along and across the wedge slit, respectively. 

The fact that or + is not a discrete level but the maximum local density distribution of the surface group orbital reduces the coupling between adsorbate orbital ao and surface orbital a + n. In order to compute the bond strength we have to return to Eq.(2.229b) ... 

The diffusion and reaction of the various species in the groups for initial spatial distributions representative for the different LET values have been treated theoretically by solving numerically the differential equations describing the development in time of the average local density distribution of the species in the groups ... 

Here the fluid density p and velocity u can be obtained from local density distributions through... 

Figure 3 Illustrations of ID and 2D density profiles (A) typical one-dimensional local density distribution p r) for spherical solvent molecules around a spherical solute (B) three-dimensional local density distribution (r) on X-Y plane.

The excess bulk chemical potential for HS system can be accurately evaluated by using Camahan-StarUng (CS) EOS (Carnahan and Starling, 1969). The local excess chemical potential p r) can be derived from the functional derivative of excess Helmholtz free energy with respect to the local density distribution p r), which follows... 

Here p,(r) is the local density distribution of HS component /. The weighting function takes the same mathematical expression as in Eq. (16) with the involved HS diameter being replaced by The extension to nonadditive HS mixture is a little bit tricky, and a few methods (Ayadim and Amokrane, 2010 Matthias, 2004, 2011) have been developed, and the accuracies of these extensions are rather striking. Recendy, the FMT has been extended to sticky HS fluids (Hansen-Goos and Wetdaufer, 2011) and to nonspherical hard-convex systems (Hansen-Goos and Mecke, 2009 Hendrik and Klaus, 2010). 

The second line in Eq. (37) is derived straightforwardly by referring to Eq. (30). The reference HS system has identical local density distribution p r) as the LJ system and the diameter of the HS can be estimated by using BH method (Barker and Henderson, 1967)... 

The microscopic configuration of methane on graphiti2ed carbon black obtained with the 5-Site model is shown in Figure 2 where we plot the local density distribution versus the distance from the surface and the angle formed between the normal of the graphite surface and the vector pointing from the carbon atom to one of the four hydrogen atoms. 

Figure 2. Local density distribution of methane. The conditions are 113 K and 1000 Pa.

Fig. 9 Localization of 2D wavefunctions due to the fluctuation of the hopping integral illustrated by a numerical example [61]. The electronic Hamiltonian of a supercell containing 3,200 molecules was built using the known disorder in the hopping integral at 300 K. The localized density distribution p (x, y) = I yr (x, y) is shown for two eigenfunctions of such a disordered Hamiltonian...

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