Lennard-Iones potential

We have studied, by MD, pure water [22] and electrolyte solutions [23] in cylindrical model pores with pore diameters ranging from 0.8 to more than 4nm. In the nonpolar model pores the surface is a smooth cylinder, which interacts only weakly with water molecules and ions by a Lennard-Jones potential the polar pore surface contains additional point charges, which model the polar groups in functionalized polymer membranes. 

In crystal NaCl, each Na+ or Cl- ion is surrounded by 6 nearest neighbors of opposite charge and 12 nearest neighbors of the same charge. Two sets of forces oppose each other the coulombic attraction and the hard-core repulsion. The potential energy u(r) of the crystal is given by the Lennard-Jones potential expression,... 

A considerable amount of work has been done on the development of water-ion potential energy functions." " Most of these functions are of the standard Lennard-Jones plus Coulomb form, with parameters selected to give the experimental free energy or enthalpy of solvation. ... 

One other aspect of nonprimitive electric double layer theories which is particularly relevant to the inner Stern region are the models for the water molecule and the ions. The simplest models for a water molecule and an ion are a hard-sphere point dipole and point charge, respectively. A more realistic model of the hard-sphere water molecule would include quadrupoles and octupoles and also polarizability. However the hard-sphere property is best avoided and replaced, for example, by a Lennard-Jones potential. An alternative to a multipolar water model are three point charge sites associated with the atoms within the water molecule. 

We next consider the interactions at the second level, in terms of Lennard-Jones potentials. We considered the electronic structure of a system like KCI by beginning with the inert gas Ar and transferring protons between alternate nuclei to produce K and Cl. Certainly, the extra proton will reduce the size of the K ion in comparison to Ar, but the Cl" ion will be correspondingly expanded, and we may hope that the Cl" overlap interaction will not be changed too much. 

One of the simplest and therefore computationally less expensive potential functions for ion-water consists of the sum of long-range Coulorabic electrostatic interactions plus short-range dispersion interactions usually represented by the Lennard-Jones potential. This last term is a combination of 6 and 12 powers of the inverse separation between a pair of sites. Two parameters characterize the interaction an energetic parameter e, given by the minimum of the potential energy well, and a size parameter a, that corresponds to the value of the pair separation where the potential energy vanishes. The 6-th power provides the contribution of the attractive forces, while repulsive forces decay with the 12-th power of the inverse separation between atoms or sites. 

The MC perturbation method was used to obtain the pmf. The central carbon-Cl distance was defined as r and the free energy changes were computed at 0.125 or 0.25 A intervals with the Cl on the C3 axis. At this point it is clear that there are contact and solvent-separated minima near 3 and 5.75 A with the transition state at about 4 A, as shown in Fig. 5. XRISM calculations reveal a similar shape, although again the XRISM pmf is much flatter than the MC or MD result. Figure 5 also contains the primitive model prediction which is the sum of the Lennard-Jones potentials between the ions and the Coulombic interaction divided by the experimental dielectric constant, 78.4. The importance of the solvent structure is clear. 

Q.19.10 Consider Figs. 20.8 and 20.10, comment on the behavior of the superposition of the interactional force and the Lennard-Jones potential when the interactional force between the double layers i/fg = (a) 50, (b) 10, (c) 2.5, and (d) 0.5. Sketching graphs may help. (These numbers are on the same arbitrary magnitude scale as the one used in Fig. 20.10.) Propose a mechanism that could change the interactional force between double layers other the infusion of counter ions (which is mentioned in the text.)... 

This section discusses simulations in which the parameters are explicitly mapped to experimental systems. In particular, this mapping affects ion diameter cr, rod radius r0, Bjerrum length B, and line charge density A. In order to have a rod radius different from cr this requires the introduction of a new potential for ion-rod interactions, for which a modified truncated and shifted Lennard-Jones potential has been used ... 

Here, r is the radial distance from the rod axis to the center of the ion, rs is a parameter that shifts the Lennard-Jones potential towards larger r, and rcut = rs + 21,6cr. In Figure 15 the relation between rs and the rod radius is... 

FIG. 15 Connection between several important length scales. The distance of closest approach, closest approach, in the sense of the thermal distance already employed for the Lennard-Jones potential, is found to be = rs I connection between the translation parameter rs and the rod radius is given by r0 = rs + cr/2. 

For ionic solids interacting with Coulomb pair potentials, similar calculations can be carried out. However, this is a rather complex matter because Coulomb, van der Waals attraction and Pauli repulsion should all be taken into account. In addition, there are uncertainties in the choice of suitable pair-potential equation (many inter-atomic potential equations, including Lennard-Iones were tried), and the calculated Gf results are highly dependent on the particular choice of pair-potential model. As an example, Gf = 212m) m 2 was calculated theoretically for the NaCl (100) crystal, which is near to the experimental value of Gf = 190 m) m 2 from extrapolation of the molten salt surface tension values, but far away from Gf = 300 mj m 2, which was found from crystal cleavage experiments. 

Alkali and halide ion adsorption near metal surfaces has been investigated by a variety of groups. A simple (9-3) Lennard-Jones potential, first used by Lee et al. 

It is possible to infer these structures, that of lowest energy and others of higher energy as well, from simple models for some kinds of clusters. This is particularly so for atomic clusters whose binding forces are well represented by pairwise interactions such as the Lennard-Jones potentials that approximate van der Waals interactions, or the Born-Mayer interactions that describe the forces between the ions in alkali halide clusters. Many molecular clusters also behave much like their simple models, models... 

The bond between atoms is described by the curve that traces the way the energy changes as the atomic coordinates change. The potential of the interatomic attraction is the energy as a function of the normal vibration coordinate strain, which is usually taken to be the distance between the bound atoms (Figure 2.1). Many variants exist. A Morse curve is representative for the potential between covalently bonded atoms. If the bond is between ions the interatomic potential is well described by a Lennard-Jones potential with a Coulomb term. 

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