Attractive potentials

Chemisorption occurs when the attractive potential well is large so that upon adsorption a strong chemical bond to a surface is fonued. Chemisorption involves changes to both the molecule and surface electronic states. For example, when oxygen adsorbs onto a metal surface, a partially ionic bond is created as charge transfers from the substrate to the oxygen atom. Other chemisorbed species interact in a more covalent maimer by sharing electrons, but this still involves perturbations to the electronic system. 

For purely repulsive potentials (u(r) > 0 ),f(r) is negative is positive. For purely attractive potentials,... 

The assumption of Gaussian fluctuations gives the PY approximation for hard sphere fluids and tire MS approximation on addition of an attractive potential. The RISM theory for molecular fluids can also be derived from the same model. 

In order for the primary photoelectron, which is bound to the surface atom with binding energy to be detected ia xps, the electron must have sufficient kinetic energy to overcome, ia addition to E the overall attractive potential of the spectrometer described by its work function, Thus, the measured kinetic energy of this photoelectron, Ej is given by... 

Sorption of nonionic, nonpolar hydrophobic compounds occurs by weak attractive interactions such as van der Waals forces. Net attraction is the result of dispersion forces the strength of these weak forces is about 4 to 8 kj/mol ( 1 2 kcal/mol). Electrostatic interactions can also be important, especially when a molecule is polar in nature. Attraction potential can develop between polar molecules and the heterogeneous sod surface that has ionic and polar sites, resulting in stronger sorption. 

Fig. 3. Attraction—repulsion potentials as a function of distance between particle centers. Curve 1 represents the attractive potential caused by van der Waals forces, curve 2 is the repulsive potential caused by double-layer forces, and curve 3 is the resultant force experienced by the two particles.

In a force-displacement curve, the tip and sample surfaces are brought close to one another, and interact via an attractive potential. This potential is governed by intermolecular and surface forces [18] and contains both attractive and repulsive terms. How well the shape of the measured force-displacement curve reproduces the true potential depends largely on the cantilever spring constant and tip radius. If the spring constant is very low (typical), the tip will experience a mechanical instability when the interaction force gradient (dF/dD) exceeds the... 

As an adsorption geometry one considers a semi-infinite system with an impenetrable wall at z = 0, such that monomer positions are restricted to the positive half-space z > 0. At the wall acts a short-range attractive potential, either as a square well... 

We use the off-lattice MC model described in Sec. IIB 2 with a square-well attractive potential at the wall, Eq. (10), and try to clarify the dynamic properties of the chains in this regime as a function of chain length and the strength of wall-monomer interaction. 

The simplest atomistic model for the formation of a crystal in continuous space requires the definition of some effective attractive potential between any two atoms, which is defined independently of the other atoms in the cluster or crystal. The most frequently studied potential is the Lennard-Jones potential... 

Another problem arises from the presence of higher terms in the multipole expansion of the electrostatic interaction. While theoretical formulas exist for these also, they are even more approximate than those for the dipole-dipole term. Also, there is the uncertainty about the exact form of the repulsive interaction. Quite arbitrarily we shall group the higher multipole terms with the true repulsive interaction and assume that the empirical repulsive term accounts for both. The principal merit of this assumption is simplicity the theoretical and experimental coefficients of the R Q term are compared without adjustment. Since the higher multipole terms are known to be attractive and have been estimated to amount to about 20 per cent of the total attractive potential at the minimum, a rough correction for their possible effect can be made if it is believed that this is a preferable assumption. 

Fig. 2. Sketch of the interaction potential between segments m and n. The potential can be decomposed into a hard core repulsive potential Unm (hard) and a weak attractive potential Unn, (attr)...

Thus the net result of all interactions is an attractive potential behaving as... 

The attractive potential due to the positively charged cores is not strong enough to keep the valence electrons inside the metal. As a result, the electrons spill out into the vacuum, i.e. the electron density just outside the surface is not zero. Because the charge of these electrons is not compensated by positive ions, a dipole layer exists at the surface, with the negative end to the outside. An electron traveling from the solid to the vacuum must overcome this barrier of height [Pg.228]

Technically, these simulations are very similar to those of Huber et al. [12] the major difference is the introduction of the attractive potential between the pairing ions. In addition, these authors introduce different dielectric constants ej and for the two solutions. Near the interface the dielectric constant varies locally according to the simple prescription ... 

In porous media, liquid-gas phase equilibrium depends upon the nature of the adsorbate and adsorbent, gas pressure and temperature [24]. Overlapping attractive potentials of the pore walls readily overcome the translational energy of the adsorbate, leading to enhanced adsorption of gas molecules at low pressures. In addition, condensation of gas in very small pores may occur at a lower pressure than that normally required on a plane surface, as expressed by the Kelvin equation, which relates the radius of a curved surface to the equilibrium vapor pressure [25],... 

The first possibility is that the attractive potential associated with the solid surface leads to an increased gaseous molecular number density and molecular velocity. The resulting increase in both gas-gas and gas-wall collision frequencies increases the T1. The second possibility is that although the measurements were obtained at a temperature significantly above the critical temperature of the bulk CF4 gas, it is possible that gas molecules are adsorbed onto the surface of the silica. The surface relaxation is expected to be very slow compared with spin-rotation interactions in the gas phase. We can therefore account for the effect of adsorption by assuming that relaxation effectively stops while the gas molecules adhere to the wall, which will then act to increase the relaxation time by the fraction of molecules on the surface. Both models are in accord with a measurable increase in density above that of the bulk gas. 

The attractive potential exerted on the electrons due to the nuclei - the expectation value of the second operator VNe in equation (1-4) - is also often termed the external potential, Vext, in density functional theory, even though the external potential is not necessarily limited to the nuclear field but may include external magnetic or electric fields etc. From now on we will only consider the electronic problem of equations (1 -4) - (1 -6) and the subscript elec will be dropped. 

If this is combined with the classical expression for the nuclear-electron attractive potential and the electron-electron repulsive potential we have the famous Thomas-Fermi expression for the energy of an atom,... 

The van der Waals attractive potential [27] increases with an increase in nanopartide radius or with a decrease in center-to-center separation distance between nanoparticles ... 

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