Energy/distance profiles

PVA molecules show a typical energy-distance profile of adsorbed amphiphilic macromolecules (Figure 9.7) with a train, loop, tail arrangement. 

FIG. 3 Energy-distance profiles for polystyrene latex particles bearing carboxylic acid surface groups (lyophobic colloid) suspended in different aqueous media (a) 0.15 mol/L NaCl at pH 3.0 (b) 0.00015 mol/L NaCl at pH 3.0, and (c) 0.00015 mol/L NaCl at pH 7.0. 

Figure 3.2 Schematic representation of the energy distance profile for two molecules interacting AE represents the stabilising interaction energy...

Figure 3.3 Energy-distance profile when the electrode is irreversible...

Figure 3.5 Energy-distance profile for a reduction reaction (r = reduction, f = forward reaction)...

Fig. 3 —Schematic energy versus distance profiles of DLVO interaction.

The difference between the energy profiles in the solvent and in the gas phase provides the solvation free energy, Gr.wx- (profile c in Fig. 13). The solvent reorganization energy, Ao, may then be obtained from equation (60) as a function of the R—X distance (profile d in Fig. 13). 

This chapter comprises two sections. The first describes the most usual techniques to directly measure force versus distance profiles between solid or liquid surfaces. We then describe different long-range forces (range >5 nm) accessible to evaluation via these techniques for different types of surface active species. The second section is devoted to attractive interactions whose strong amplitude and short range are difficult to determine. In the presence of such interactions, emulsion droplets exhibit flat facets at each contact. The free energy of interaction can be evaluated from droplet deformation and reveals interesting issues. 

Detailed Kinetic Modeling. Recent advances in computation techniques (11) have made it much easier to compute concentration-distance profiles for flame species. The one-dimensional isobaric flame equations are solved via a steady state solution using finite difference expressions. An added simplification is that the energy equation can be replaced with the measured temperature profile. In the adaptive mesh algorithm, the equations are first solved on a relatively coarse grid. Then additional grid points could be included if necessary, and the previous solution interpolated onto the new mesh where it served as the initial solution estimate. This process was continued until several termination criteria were satisfied. 

Figure 8.7 Schematic free-energy profile for proton-transfer step between intermediates having a parabolic free-energy/distance relationship. See text.

Fig. 14 Computed free energy reaction profile (kcal/mol) for catalytic azidocarbonylation at (Xantphos)Pd(CO)2. Inset shows computed geometries of the oxidative addition and C-N coupling transition states, with key distances in A...

Within this contimiiim approach Calm and Flilliard [48] have studied the universal properties of interfaces. While their elegant scheme is applicable to arbitrary free-energy fiinctionals with a square gradient fomi we illustrate it here for the important special case of the Ginzburg-Landau fomi. For an ideally planar mterface the profile depends only on the distance z from the interfacial plane. In mean field approximation, the profile m(z) minimizes the free-energy fiinctional (B3.6.11). This yields the Euler-Lagrange equation... 

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