Potential Yukawa

We will describe integral equation approximations for the two-particle correlation fiinctions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27, 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the hypemetted chain (EfNC) approximation for charged systems, are readily obtained by fast Fourier transfonn methods... 

The solution detennines c(r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to tire PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting witli the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Fluckel limiting law distribution fiinction. 

This r dependence is also known as a Yukawa potential. This type of potential has been used to describe the behaviour of latex suspensions at low ionic strength. 

Charged particles in polar solvents have soft-repulsive interactions (see section C2.6.4). Just as hard spheres, such particles also undergo an ordering transition. Important differences, however, are that tire transition takes place at (much) lower particle volume fractions, and at low ionic strengtli (low k) tire solid phase may be body centred cubic (bee), ratlier tlian tire more compact fee stmcture (see [69, 73, 84]). For tire interactions, a Yukawa potential (equation (C2.6.11)1 is often used. The phase diagram for the Yukawa potential was calculated using computer simulations by Robbins et al [851. 

In the Yukawa potential, A is an inverse range parameter. The value A = 1.8 is appropriate for the inert gases. Each of the above potentials has a hard core. Real molecules are hard but not infinitely so. A slightly softer core is more desirable. The Lennard-Jones potential... 

Since the results for thermodynamics from the Yukawa potential, with A = 1.8, are similar to the results of the LJ potential, it is quite possible that the DHH closure may be applicable to the Yukawa potential. With A = 1.5, the thermodynamics of the square well fluid are also similar. Here, too, the DHH closure may be useful. However, the DHH closure has not been applied to either of these potentials. 

The Yukawa potential is of interest in another connection. According to the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, colloidal... 

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

When applied to spatially extended dynamical systems, the PoUicott-Ruelle resonances give the dispersion relations of the hydrodynamic and kinetic modes of relaxation toward the equilibrium state. This can be illustrated in models of deterministic diffusion such as the multibaker map, the hard-disk Lorentz gas, or the Yukawa-potential Lorentz gas [1, 23]. These systems are spatially periodic. Their time evolution Frobenius-Perron operator... 

This other Lorentz gas is similar to the previous one except that the hard disks are replaced by Yukawa potentials centered here at the vertices of a square lattice. The Hamiltonian of this system is given by... 

Figure 11. Two trajectories of the periodic Yukawa-potential Lorentz gas. They start from the same position but have velocities that differ by one part in a million.

Figure 12. The diffusive modes of the periodic Yukawa-potential Lorentz gas represented by their cumulative function depicted in the complex plane ReFk,hnFk) for two different nonvanishing wavenumbers k. The horizontal straight line is the curve corresponding to the vanishing wavenumber k = 0 at which the mode reduces to the invariant microcanonical equilibrium state.

For large r, G(f, Fo) must vanish, which requires that A = 0. For small distances, where kf < < 1, it should be identical to the Coulomb potential, which requires that 5=1. Finally, we find that the Green s function of the Schrodinger equation in vacuum is the Yukawa potential. 

For many practically important interaction functions, the Fourier coefficients in Eq. (D.9) have finite analytic forms, for example, the Lennard-Jones potential, the Yukawa potential, the Morse potential, and functions that can be derived from those functions. For a power-law interaction... 

Exercise 9.3 (For students of the Fourier transform) Calculate the Fourier transform of the Yukawa potential, e f x, where k > (k Take a limit to show that the Fourier transform of / x is An/ pf. 

Figure 11. The difference between the free-energy densities of fee and bcc phases of particles interacting through a Yukawa potential, as a function of temperature, determined through the FG methods discussed in Section V.C. The error bars reflect the difference between the upper and lower bounds provided by FG switches between the phases (along the Bain path [85]) in the two directions.

Here, ur/p r) describes the interaction between nonbonded units and oa is the effective unit size ([Pg.59]

The introduction of the van der Waals potential in combination with a Yukawa potential produces a curve in which the primary minimum is always deeper than the secondary minimum. This must be so because the primary minimum state is that for which the particles have coalesced and the valency of the nth plate Zn has dropped to zero since Z —> 0 as Xmn —> 2a, —> 0 as —> 2a, and the van der Waals force... 

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