Pair potentials calculation

Figure 2.16 The pair potential calculated for rutile particles with a radius of 100 nm, background electrolyte concentration of 10 4moldm 3 and a -potential of —50 mV. Curve a was calculated for an isolated pair of particles and curve b corresponds to the potential for a pair ofparticles in a dispersion at a volume fraction of 0.45. Note how the increased electrolyte content due to the counterions introduced with the particles shorten the range of the repulsion enough for a small secondary minimum to be found at h 70 nm...

Figure 2.17 Pair potentials calculated for 50 nm polystyrene particles in 0.1 moldm 3 electrolyte and with a -potential of —30mV. Curve a is the result for a simple polystyrene surface and curve b was calculated from the model of a 1 nm surfactant layer so that the -potential is taken as occurring at the outer edge of the adsorbed layer. A maximum in the potential of 8kBT is insufficient to provide long-term stability and the curves clearly shows how electrosteric stabilisation can achieve this...

Figure 2.18 The pair potential calculated for polystyrene particles of radius 500 nm, with a measured (.-potential of —12mV in 0.5moldm 3 electrolyte. A steric barrier of S = 3.5 nm was used as the particles had monolayer coverage of a monodisperse non-ionic surfactant. This was C 2E06 which represents a dodecyl hydrophobic moiety linked to hexaethylene glycol via an ether link...

A straightforward explanation for the differences in gel stability is that the attractive particle-particle interactions are not mainly driven by interparticulate H-bond but by van der Waals forces. This idea is supported by pair potential calculations using an approximate expression given by Israelachvili based on the Lifshitz theory of van der Waals forces [7]. 

Fig. 3. Pair potential calculations of SiOa in styrene and toluene using an approximate expression for the Lifohitz theory of van der Waals forces.

In Table 3, the elastic constants obtained from the pair potential calculations are given for two low density forms of crystalline silica a-quartz and... 

Figure 11. The phonon spectrum of a-quartz at ambient pressure. The dotted lines are the results of our pair potential calculations. The symbols represent different sets of inelastic neutron scattering data.

Figure 12. The phonon spectrum for stishovite calculated at ambient pressure from interatomic pair potential calculations. The solid circles are from experiment.

Combining tliis witli the Omstein-Zemike equation, we have two equations and tluee unknowns h(r),c(r) and B(r) for a given pair potential u r). The problem then is to calculate or approximate the bridge fiinctions for which there is no simple general relation, although some progress for particular classes of systems has been made recently. 

To find appropriate empirical pair potentials from the known protein structures in the Brookhaven Protein Data Bank, it is necessary to calculate densities for the distance distribution of Ga-atoms at given bond distance d and given residue assignments ai,a2- Up to a constant factor that is immaterial for subsequent structure determination by global optimization, the potentials then ciiiergo as the negative logarithm of the densities. Since... 

Consider the two-dimensional box in Figure 2.5. If we know the positions of each of the N particles in the square and their pair potential, then we can calculate the total mutual potential energy... 

In calculating a pair potential for dineon, we have to take the separated atom energy as one half of the pair energy for an arbitrarily large distance. 

For each atom type there are two parameters to be determined, the van der Waals radius and the atom softness, Rq and, It should be noted that since the van der Waals energy is calculated between pairs of atoms, but parameterized against experimental data, the derived parameters represent an effective pair potential, which at least partly includes many-body contributions. 

MicroEnv is calculated using standard approximations used in classical molecular mechanics (i.e., pair potentials and analytical expressions for strain energy190) ... 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

As stated above, Afd is related to the contact pair potential Afg(0). In a floe, each particle is in close contact with z other particles. If A/a is small, the z lens-shaped overlap volumes (see Figure A) surrounding each particle do not overlap with each other, and Afd equals zAf (0)/2 where Afg(0) is given by Equation 8. For higher values of A/a, the lenses overlap partly, and Afd < zAfs(0)/2. Above a certain value of A/a (which depends on the packing of the particles in the floe), there is no polymer left within the interstices of the floe and all the solvent in the floe is within a distance A from the surface of at least one particle. Then the volume of solvent which is transferred towards the solution when a particle is added to the floe is readily calculated. 

Any fundamental study of the rheology of concentrated suspensions necessitates the use of simple systems of well-defined geometry and where the surface characteristics of the particles are well established. For that purpose well-characterized polymer particles of narrow size distribution are used in aqueous or non-aqueous systems. For interpretation of the rheological results, the inter-particle pair-potential must be well-defined and theories must be available for its calculation. The simplest system to consider is that where the pair potential may be represented by a hard sphere model. This, for example, is the case for polystyrene latex dispersions in organic solvents such as benzyl alcohol or cresol, whereby electrostatic interactions are well screened (1). Concentrated dispersions in non-polar media in which the particles are stabilized by a "built-in" stabilizer layer, may also be used, since the pair-potential can be represented by a hard-sphere interaction, where the hard sphere radius is given by the particles radius plus the adsorbed layer thickness. Systems of this type have been recently studied by Croucher and coworkers. (10,11) and Strivens (12). 

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