Repulsive Interactions, Total Interaction Pair Potentials

7 Repulsive Interactions, Total Interaction Pair Potentials 

---

The atomic interactions of the system are derived from a many-body empirical potential, the attractive part of which is expressed within the SMA of the TB theory ", while the repulsive term is a pair-potential of Bom-Mayer type. Accordingly, the total energy of the system is written as ... 

It is possible to obtain the total intermolecular pair potential by simply summing the attractive and repulsive potentials. In 1903, Mie proposed a semi-empirical interaction pair potential in the form... 

A force sensor in an AFM can only work if the probe interacts with the force field associated with a surface. The interaction force between the probe and the surface in ambient air is illustrated in Figure 9.2. The total intermolecular pair potential is obtained by assuming one attractive (-Ci/z ) and another repulsive potential (C2/z )-Superimposing the two gives an expression for the well-known Lennard-Jones potential, where Ci and C2 are the corresponding coefficients for the attractive and repulsive interactions, respectively, and z is the distance between the sample surface and rest position of the cantilever. 

The pair potential of colloidal particles, i.e. the potential energy of interaction between a pair of colloidal particles as a function of separation distance, is calculated from the linear superposition of the individual energy curves. When this was done using the attractive potential calculated from London dispersion forces, Fa, and electrostatic repulsion, Ve, the theory was called the DLVO Theory (from Derjaguin, Landau, Verwey and Overbeek). Here we will use the term to include other potentials, such as those arising from depletion interactions, Kd, and steric repulsion, Vs, and so we may write the total potential energy of interaction as... 

The expressions for any will derive from the quantity 0, the total potential energy of aU the interactions for the electrolyte solution. Contributions to 0 can be as simple or as complex as is wished, e.g. hard core terms or other short range repulsions, ion—dipole interactions, Gurney terms, cavity terms, ion—solvent interactions, solvent—solvent interactions and others where the molecular structure of the solvent is considered. All these can be included, or selections can be made, and all the resultant g compared. In these cases, the total potential energy, 0, is then taken to be a sum of pair-wise terms for interactions between the ions concerned ... 

The basic assumptions underlying the use of most atom-atom potential calculations are that only central forces operate between pairs of atoms and that the total interaction energy is the sum of the interactions between all pairs of atoms—the additivity assumption. The individual atom-atom interaction energies include a repulsive term with a steep rise in the energy at small interatomic distances, an attractive term designed to allow for London-type dispersion attractions and, sometimes, an additional coulombic interaction as well. With an exponential function as the repulsive term, the interaction energy between a pair of atoms can be written as... 

Figure 3.12 The potential energy for an NaCl ion pair versns ion separation in nnits of kJ moP. The total potential energy curve is the sum of the attractive Coulomb interaction between the charges and the repulsive interaction. The equilibrium bond length (rg = 236 pm) corresponds to the minimum of the total potential energy curve. The potential energy at the minimum is approximately —560 kJ mol .

Hamaker. The contribution of the surface extension energy and/or bending elasticity to the pair interaction potential is also included. The extension of the drop surface upon the deformation corresponds to a soft interdroplet repulsion. All the remaining possible interactions (electrostatic, steric, depletion, etc.) can usually be treated in the framework of Deqaguin s approximation, which allows one to account for the two contributions of the total interaction energy (i) across the flat film and (ii) between the spherical surfaces surrounding the fllm. " Combined with relevant expressions for the hydrodynamic interactions, this approach could be used for studying the coalescence of Brownian emulsion and microemulsion droplets. ... 

Exact calculations of AE have been carried out by Kelley and Wolfsberg [19] for colinear collisions between an atom and a diatomic molecule. The oscillator potential was considered to be both harmonic and Morse-type, and the interaction between the colliding pair was taken both as an exponential repulsion and as a Lennard-Jones 6 12 potential. Two important conclusions were reached First, when the initial energy of the oscillator increases, the total energy transferred from translation to vibration, AE, decreases. Second, the effect of using a Morse-oscillator potential in place of the harmonic oscillator was generally to decrease AE, often by more than a factor of 10. 

The forces acting on a colloidal system include gravitational, diffusion, viscous, inertial, attractive Van der Waals, and electrical repulsive forces. Because most of these forces are functions of the particle size, it is important to know both particles size and size distribution. The classical Derjaguin-Landau-Verwey-Overbeek (DLVO) theory describes colloid stability on the basis of pair interaction, considering only attractive van der Waals forces and repulsive electrostatic forces (Molina-Bolfvar and Ortega-Vinuesa, 1999). The total potential energy of interaction, Ujc, between two particles is defined as ... 

---