Coulombic, generally repulsion

One characteristic feature of surfactants is their amphiphilic nature. These molecules present two moieties the hydrophobic moiety (usually a hydrocarbon chain) interacts with the nanotube sidewalls, while the hydrophilic part, called polar head group, is generally charged or has zwitterionic character. It has the double function of helping solubility in aqueous solvents and of providing additional stabilization towards tubes aggregation by coulombic charge repulsion. 

So far, we have seen that deviation from ideal behavior may affect one or more thermodynamic magnitudes (e.g., enthalpy, entropy, volume). In some cases, we are able to associate macroscopic interactions with real (microscopic) interactions of the various ions in the mixture (for instance, coulombic and repulsive interactions in the quasi-chemical approximation). In practice, it may happen that none of the models discussed above is able to explain, with reasonable approximation, the macroscopic behavior of mixtures, as experimentally observed. In such cases (or whenever the numeric value of the energy term for a given substance is more important than actual comprehension of the mixing process), we adopt general (and more flexible) equations for the excess functions. 

Ionic strength is a useful concept because it allows us to consider some general expressions that depend only on ionic strength and not on the identities of the ions themselves. In 1923, Peter Debye and Erich Hiickel made some simplifying assumptions about all ionic solutions. They assumed that they would be dealing with very dilute solutions, and that the solvent was basically a continuous, structureless medium that has some dielectric constant e. Debye and Hiickel also assumed that any deviations in solution properties from ideality were due to the coulombic interactions (repulsions and attractions) between the ions. 

DFT methods compute electron correlation via general functionals of the electron density (see Appendix A for details). DFT functionals partition the electronic energy into several components which are computed separately the kinetic energy, the electron-nuclear interaction, the Coulomb repulsion, and an exchange-correlation term accounting for the remainder of the electron-electron interaction (which is itself... 

The general idea of using different orbitals for different spins" seems thus to render an important extension of the entire framework of the independent-particle model. There seem to be essential physical reasons for a comparatively large orbital splitting depending on correlation, since electrons with opposite spins try to avoid each other because of their mutual Coulomb repulsion, and, in systems with unbalanced spins, there may further exist an extra exchange polarization of the type emphasized by Slater. 

Ionic compounds such as halides, carboxylates or polyoxoanions, dissolved in (generally aqueous) solution can generate electrostatic stabilization. The adsorption of these compounds and their related counter ions on the metallic surface will generate an electrical double-layer around the particles (Fig. 1). The result is a coulombic repulsion between the particles. If the electric potential associated with the double layer is high enough, then the electrostatic repulsion will prevent particle aggregation [27,30]. 

However, billiard balls are a pretty bad model for electrons. First of all, as discussed above, electrons are fermions and therefore have an antisymmetric wave function. Second, they are charged particles and interact through the Coulomb repulsion they try to stay away from each other as much as possible. Both of these properties heavily influence the pair density and we will now enter an in-depth discussion of these effects. Let us begin with an exposition of the consequences of the antisymmetry of the wave function. This is most easily done if we introduce the concept of the reduced density matrix for two electrons, which we call y2. This is a simple generalization of p2(x1 x2) given above according to... 

The first two tenns in Eq. (2) represent the kinetic energy of the nuclei and the electrons, respectively. The remaining three terms specify the potential energy as a function of the interaction between the particles. Equation (3) expresses the potential function for the interaction of each pair of nuclei. In general, this sum is composed of terms that are given by Coulomb s law for the repulsion between particles of like charge. Similarly, Eq. (4) corresponds to the electron-electron repulsion. Finally, Eq. (5) is the potential function for the attraction between a given electron (<) and a nucleus (j). 

The question now arises of what simplification is possible in the treatment of orientationally structured adsorbates and what general model can be involved to rationalize, within a single framework, a diversity of their properties. Intermolecular interactions should include Coulomb, dispersion, and repulsive contributions, and the adsorption potential should depend on the substrate constitution and the nature of adsorbed molecules. However difficult it may seem, all these factors can be taken into account if we follow the description pattern put forward in this book. Its fundamentals are briefly sketched below. 

The density functional theory (DFT) [32] represents the major alternative to methods based on the Hartree-Fock formalism. In DFT, the focus is not in the wavefunction, but in the electron density. The total energy of an n-electron system can in all generality be expressed as a summation of four terms (equation 4). The first three terms, making reference to the noninteracting kinetic energy, the electron-nucleus Coulomb attraction and the electron-electron Coulomb repulsion, can be computed in a straightforward way. The practical problem of this method is the calculation of the fourth term Exc, the exchange-correlation term, for which the exact expression is not known. 

In general, when a charged solid surface faces an ion of similar charge in an aqueous suspension, the ion is repelled from the surface by Coulomb forces. The Coulomb repulsion produces a region in the aqueous solution that is relatively... 

The enthalpy change related to associative process (A//ab) is due essentially to coulombic interactions and, subordinately, to polarization, repulsion, covalent bonding, elastic interactions, and vibrational effects. The latter two causes are generally negligible and may have some effects only at low T. 

Like the Coulombic forces, the van der Waals interactions decrease less rapidly with increasing distance than the repulsive forces. They include interactions that arise from the dipole moments induced by nearby charges and permanent dipoles, as well as interactions between instantaneous dipole moments, referred to as dispersion forces (Israelachvili 1992). Instantaneous dipole moments can be thought of as arising from the motions of the electrons. Even though the electron probability distribution of a spherical atom has its center of gravity at the nuclear position, at any very short instance the electron positions will generally not be centered on the nucleus. 

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