Effective interaction potentials

The effective interaction potential, found by averaging cos d over all angles, is then equal to... 

The second category of methods uses a more general approach, based on fundamental concepts in statistical mechanics of the liquid state. As mentioned above, the Hwang and Freed theory (138) and the work of Ayant et al. (139) allow for the presence of intermolecular forces by including in the formulation the radial distribution function, g(r), of the nuclear spins with respect to the electron spins. The radial distribution function is related to the effective interaction potential, V(r), or the potential of mean force, W(r), between the spin-carrying particles through the relation (138,139) ... 

The second important approximation that enables the resolution of Eq. (1.6) consists in considering that every electron is subject to an effective interaction potential V(ri), which takes into account the full attractive electron-ion interactions as well as somehow a part of the repulsive electron-electron interactions. Ideally we would like to express Hq in the form ... 

The effects of the number of charges Z per particle and of the volume fraction / of the particles on the effective interaction potential between identical charged particles were examined. The number of charges per particle was varied between 150 and 1200 and the particle volume fraction / was varied between 0.005 and 0.12. The electrolyte concentration was assumed to be 10 5 or 10 4 M. 

Fig. 1. Effective interaction potential between identical charged colloidal particles versus the distance between the centers of particles (d is the diameter of the particles) for various charges per particle (Z), particle volume...

When the particle volume fraction / was increased to 0.015, the oscillations of the effective interaction between identical charged particles became larger than those for / = 0.005 at an electrolyte concentration of 10 5 M (Fig. 3). The effective interaction between identical charged particles versus the distance between particles is plotted in Fig. 3 for various values of Z (Z = 300, Z = 600, Z = 1200). As shown in Fig. 4, the colloidal dispersion has a disordered liquidlike structure for Z = 600, but a more ordered structure for Z = 1200. When the electrolyte concentration was increased to 10-4 M, the interaction between identical charged particles became completely screened. As shown in Fig. 5, no oscillations of the effective interaction potential were present for Z = 300,600, and 1200. 

The effective interaction potential is a result of collective behavior. Let us select two neighboring particles immersed in a dispersion of particles of the same kind. Two kinds of... 

As shown in Fig. 17, for Z = 300 and an electrolyte concentration of 10 5 M, the oscillations of the effective interaction potential were weak for low particle-volume frac-... 

Fig. 17. Effective interaction potential between identical charged colloidal particles versus the distance between the centers of particles (d is the diameter of the particles) for various particle volume fractions (/), for charges per particle Z = 300 and electrolyte concentration 10 5 M. (1) / = 0.005 (2) / = 0.015 (3) / = 0.03 (4) / = 0.06 (5) / = 0.12.

Using Monte Carlo simulations and a Debye-Htickel pah-repulsive potential, effective interaction potentials between identical charged particles were calculated and the phase structures were obtained for various charges per particle and various particle volume fractions. The simulation re-... 

Figure 3a shows separately the electrostatic (DLVO-like) and excluded volume contributions to the effective interaction potential between two macroions, W (r). The electrostatic contribution is all the way repulsive and decays mono-... 

Effective interaction potential between two monomers in a solution of other molecules. 

These are chemically homogeneous multiarm star polymers (usually homopolymers) with only excluded volume interactions [21,22], Due to their synthesis procedure (high vacuum anionic polymerization), they are stable and nearly monodisperse [23]. Their softness can be tuned at the synthesis level (number and size of arms) [23,24] and/or by varying the temperature in different solvents [25,26], Moreover, these systems can be functionalized in various ways [27]. What made these systems truly ideal soft colloids were the breakthroughs in both theoretical description and synthesis. The former refers to the ability to describe analytically their internal structure [28] and their softness in terms of an effective interaction potential [24,29]. 

Fig. 2.7. The effective interaction potential between two oxygen atoms joined by a hydrogen hond O-H... O as calculated from the model of Lippincott Schroeder (1955).

Lyubartsev and Laaksonen [1995]. Calculation of effective interaction potentials from radial distribution functions A reverse Monte Carlo approach. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 52,4, pp. 3730-3737. 

It should be pointed out that the above distributions of water over the metal surface as well as the results of the CP simulation of Price and Halley [40] are obtained for the model taking no account of directionality of d-electrons. For such a simple model metal the valence electron density and the attractive electrostatic potential possess maxima at hollow site positions, whereas the full-electron calculations for a copper slab yield the electron density with the maxima at on-top adsorption positions [49]. The explicit account for d-electrons of a transition metal might drastically change the above discussed distribution of water over the surface, making the distribution more biased towards the on-top position. In any case, the effective interaction potential between the metal surface and liquid water differs from that of a single adsorbed water... 

Polymer chains in solution form a loose coil such that two coils can interpenetrate. The resulting effective interaction potential O(lrl) is repulsive and is well approximated by a Gaussian potential of strength Oq and with a range ro which is proportional to the radius of gyration Rg. The excess free energy functional for such soft systems can be approximated by a quadratic form... 

When salts are added into the polyelectrolyte solutions, the effective interaction potential between the ions becomes... 

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