Waals Attractive Interaction Energy

Hamaker [1] in 1917 showed the potential energy of attractian between two spherical interacting partides d the same material separated by a distance, h, is given by 

Two flat plates sizes 6 x g x Semi-infinite parallel plates 6 x 6 Two spheres of radii Oj, 

Two spheres of equal radius, o Sphere of radius a with an infinite slab 

The Hamaker constant for two particles of materials 1 and 2, respectively, interacting across a vacuum is given by [3]  

Alternatively, this result can be expressed in terms of the dielectric constants of the various materials through the relationship [4], bj = 

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AVhen the particles are coated with a pol3mier of thickness S, the van der Waals attractive interaction energy is calculated by [13—16]... 

The Three Contributions to the Total van der Waals Attractive Interaction Energy"... 

Influence of Addition of Electrolyte and Increase of Temperature Addition of electrolyte or increase of temperature at a given electrolyte concentration to a sterically stabilized dispersion may result in its flocculation at a critical concentration or temperature, which in many cases coincides with the theta point for the stabilizing chain. At the theta point the mixing term in the steric interaction is zero and any yield value measured should correspond to the residual van der Waals attraction. The energy arising from van der Waals attraction may be calculated from the following approximate relationship,... 

Fig. 1.4 Resultant interaction energy between two particles with van der Waals attractive interactions and electron overlap repulsion interactions. 

The Boyle temperature T is defined as the temperature for which the second virial coefficient is zero, i.e. TR = 1. The ratio T /Tc = 27/8 is well known for a van der Waals fluid. In this transition, two opposing contributions to the free energy, i.e. translational entropy of the fluid molecules and the van der Waals attractive interaction, are balanced. 

When the van der Waal s attraction brings a particle (molecule) closer to the surface, a repulsive force develops between the core electrons of the particle (molecule) surface and those of the atoms in the surface. The equilibrium separation is determined by a balance between repulsive and the attractive forces and decreases with increasing radius of the particle. Note also that the dielectric constant of the medium separating the particle (molecule) and the surface also affects the magnitude of E in Equations (9.1)-(9.3), as the constant C is inversely proportional to the dielectric constant. The van der Waal s attractive interaction energy is small, and thus the physisorption bands are weak and can be easily broken, especially when cleaning in high dielectric constant liquids. 

As the two particles approach, there will be two (at least) types of interaction the repulsive interaction just described and the relentless van der Waals attractive interactions, which make most colloids inherently unstable. The total interaction energy for the system under consideration will be the sum of the two energies... 

The van der Waals attraction between two water molecules is due to the induced dipole moment in the molecules and permanent dipole moments of the water molecules. The attractive potential energy between two molecules varies as the inverse sixth power of the distance between the molecules. The total van der Waals attractive potential energy of a soap film of thickness, t, due to the sum of the interactions between all the molecules in the film is given approximately by... 

Hard-sphere models lack a characteristic energy scale and, hence, only entropic packing effects can be investigated. A more realistic modelling has to take hard-core-like repulsion at small distances and an attractive interaction at intennediate distances into account. In non-polar liquids the attraction is of the van der Waals type and decays with the sixth power of the interparticle distance r. It can be modelled in the fonn of a Leimard-Jones potential Fj j(r) between segments... 

In equation (2) Rq is the equivalent capillary radius calculated from the bed hydraulic radius (l7), Rp is the particle radius, and the exponential, fxinction contains, in addition the Boltzman constant and temperature, the total energy of interaction between the particle and capillary wall force fields. The particle streamline velocity Vp(r) contains a correction for the wall effect (l8). A similar expression for results with the exception that for the marker the van der Waals attraction and Born repulsion terms as well as the wall effect are considered to be negligible (3 ). 

Two repulsive contributions, osmotic and elastic contributions [31, 32], oppose the van der Waals attractive contribution where the osmotic potential depends on the free energy of the solvent-ligand interactions (due to the solvation of the ligand tails by the solvent) and the elastic potential results from the entropic loss due to the compression of ligand tails between two metal cores. These repulsive contributions depend largely on the ligand length, solvent parameters, nanopartide radius, and center-to-center distance ... 

Here we present and discuss an example calculation to make some of the concepts discussed above more definite. We treat a model for methane (CH4) solute at infinite dilution in liquid under conventional conditions. This model would be of interest to conceptual issues of hydrophobic effects, and general hydration effects in molecular biosciences [1,9], but the specific calculation here serves only as an illustration of these methods. An important element of this method is that nothing depends restric-tively on the representation of the mechanical potential energy function. In contrast, the problem of methane dissolved in liquid water would typically be treated from the perspective of the van der Waals model of liquids, adopting a reference system characterized by the pairwise-additive repulsive forces between the methane and water molecules, and then correcting for methane-water molecule attractive interactions. In the present circumstance this should be satisfactory in fact. Nevertheless, the question frequently arises whether the attractive interactions substantially affect the statistical problems [60-62], and the present methods avoid such a limitation. 

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