Interactions are Long-Ranged

Example 20.2 shows how long-ranged electrostatic interactions hold a salt crystal together. 

EXAMPLE 20.2 Why are sodium chloride crystals stable Let s assume that a crystal of sodium chloride is composed of hard spherical ions that interact solely through electrostatics. To compute the total energy, we sum all the pairwise coulombic interaction energies of one ion with every other ion in the crystal. Crystalline NaCl is arranged in a simple cubic lattice, with interionic spacings a = 2.81 A at 25 °C. 

This series is slow to converge, implying that two charges can have substantial interaction even when they are far apart. The series converges to - hi 2 per row (see Appendix C, Equation (C.4) with a = x = 1), so the energy of interaction of one sodium ion with all other ions in the same row is 

The energy holding each positive charge in the crystal is U12, where the factor of 2 corrects for the double-counting of interactions. To remove a NaCl molecule, both a Na ion and a Cl ion, the energy is U - 206kcalmol , according to this calculation. 

The experimentally determined energy of vaporization (complete ionization) of crystalline NaCl is 183kcal mol . The 23kcal mol discrepancy between the value we calculated and the experimental value is due mainly to the assumption that the ions are hard incompressible particles constrained to a separation of exactly a = 2.81 A. In reality, ions are somewhat compressible. Nevertheless, we can conclude that the coulombic interactions are the dominant attractions that hold ionic crystals together. 

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Ion-ion interactions are long-range interactions decreasing only with r 1 and their mean interaction energy 0M is given by3)... 

The theory of electrolyte solutions developed in this chapter relies heavily on the classical laws of electrostatics within the context of modern statistical mechanical methods. On the basis of Debye-Hiickel theory one understands how ion-ion interactions lead to the non-ideality of electrolyte solutions. Moreover, one is able to account quantitatively for the non-ideality when the solution is sufficiently dilute. This is precisely because ion-ion interactions are long range, and the ions can be treated as classical point charges when they are far apart. As the concentration of ions increases, their finite size becomes important and they are then described as point charges within hard spheres. It is only when ions come into contact that the problems with this picture become apparent. At this point one needs to add quantum-mechanical details to the description of the solution so that phenomena such as ion pairing can be understood in detail. 

Atoms may also undergo other interactions. Charged atoms that have lost or gained one or more electrons are ruled by ionic interaction that we may occasionally encounter. The magnitudes of the enthalpies of these ionic interactions are comparable to those of covalent interactions. Contrary to covalent interactions, however, ionic interactions are long-range interactions when two ionized atoms are separated by a distance R this interaction asymptotically tends towards a Coulomb interaction in 1/R when R increases, which is a relatively slow decrease, much slower than that of a covalent bond with distance. Furthermore, ionic interactions are barely directional, contrary to covalent interactions that are strongly directional. 

Attractive The dispersive interactions are long-range (several A). The strength of the interaction falls off rapidly with distance (1/r6). 

When colliding particles are heavy and their interactions are long range, the impact parameter method conveniently describes the problem (9, 10, 32, 57). This method is based on the concept that the motion of nucleus is described classically and that of electrons is described quantum mechanically. If the angular momentum of colliding system is larger than K, the trajectory of an incident or a scattered particle can be defined. The impact parameter method will be useful where the total scattering is determined mainly by these processes. 

Extension of more advanced methods, in particular, density functional theory, to non-equilibrium phenomena is the principal aim of this survey. We shall consider a simple one-component fluid with van der Waals interactions as a suitable medium for exploration of basic theoretical problems of interfacial dynamics. In the case when intermolecular interactions are long-range, in particular, in the most important case of Lennard-Jones potential, the transformation from the nonlocal (density functional) to local (van der Waals-Landau-Cahn) equations fails due to divergences appearing in the commonly used expansion of the interaction term in the expression for free energy. Setting bound-... 

There are several types of intermolecular interactions, each of which involves electrostatic forces of some kind or other. An example is provided by the ion-ion interactions between cations and anions. Pure electrostatic (Coulombic) interactions are long-range and many, such as hydrogen bonding, are directional. Molecules can, however, be distorted by the electric fields of surrounding molecules, even if the molecules themselves are electrically neutral. 

When 0 is larger than typically a few percent, the above description no longer applies. In the case of hard spheres, the osmotic pressure is well represented by the Carnaham and Starling formula up to large volume fractions [21]. The friction coefficient is more difficult to evaluate because the hydrodynamic interactions are long-range. Microemulsion droplets behave as hard spheres in many circumstances. However, in W/O microemulsions, droplets frequently exhibit supplementary attractive interactions. It has been proposed that the osmotic pressure... 

If interparticle interactions are long range, the interactions between a particle and its own periodic images are substantial, and the symmetry of the lattice, artificially imposed by the periodic boundary conditions, affects properties of the otherwise isotropic system. 

We now turn to consider two charged systems where the membrane consists of a mixture of SDS and pentanol. The first system corresponds to a water dilution (refer to Fig. la) where the only ions in the solvent are the dissociated membrane counter-ions. In this case the electrostatic interactions are long range and significantly larger than all other forces such as hydration, van der Waals, and undulations. 

For this SDS-alcohol series which consists of negatively charged membranes separated by water the electrostatic interactions are long range (Eq. (13)) and significantly larger than all other... 

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