Intermolecular forces Lennard-Jones potential

One fascinating feature of the physical chemistry of surfaces is the direct influence of intermolecular forces on interfacial phenomena. The calculation of surface tension in section III-2B, for example, is based on the Lennard-Jones potential function illustrated in Fig. III-6. The wide use of this model potential is based in physical analysis of intermolecular forces that we summarize in this chapter. In this chapter, we briefly discuss the fundamental electromagnetic forces. The electrostatic forces between charged species are covered in Chapter V. 

The fundamental importance of bonding energies between bodies are traditionally divided into two broad classes chemical bond (short-range force), and physical or intermolecular bond (long-range force). The energies are largely dependent on the distance at which one body feels the presence of the other. Usually, they are called a Lennard-Jones potential [34] which has a minimum value at a certain distance. 

Because of their importance to nucleation kinetics, there have been a number of attempts to calculate free energies of formation of clusters theoretically. The most important approaches for the current discussion are harmonic models, " Monte Carlo studies, and molecular dynamics calcula-tions. In the harmonic model the cluster is assumed to be composed of constituent atoms with harmonic intermolecular forces. The most recent calculations, which use the harmonic model, have taken the geometries of the clusters to be those determined by the minimum in the two-body additive Lennard-Jones potential surface. The oscillator frequencies have been obtained by diagonalizing the Lennard-Jones force constant matrix. In the harmonic model the translational and rotational modes of the clusters are treated classically, and the vibrational modes are treated quantum mechanically. The harmonic models work best at low temjjeratures where anharmonic-ity effects are least important and the system is dominated by a single structure. 

To provide a more quantitative explanation of the magnitudes of the properties of different materials, we must consider several types of intermolecular forces in greater detail than we gave to the Lennard-Jones model potential in Chapter 9. The Lennard-Jones potential describes net repulsive and attractive forces between molecules, but it does not show the origins of these forces. We discuss other intermolecular forces in the following paragraphs and show how they arise from molecular structure. Intermolecular forces are distinguished from intramolecular forces, which lead to the covalent chemical bonds discussed in Chapters 3 and 6. Intramolecular forces between atoms in the covalent bond establish and maintain... 

The forces of attraction and repulsion between molecules must be considered for a more accurate and rigorous representation of the gas flow. Chapman and Enskog proposed a well-known theory in which they use a distribution function, the Boltzmann equation, instead of the mean free path. Using this approach, for a pair of non-polar molecules, an intermolecular potential, V (r), is given in the potential function proposed by the Lennard-Jones potential ... 

The theory describing diffusion in binary gas mixtures at low to moderate pressures has been well developed. Modem versions of the kinetic theory of gases have attempted to account for the forces of attraction and repulsion between molecules. Hirschfelder et al. (1949), using the Lennard-Jones potential to evaluate the influence of intermolecular forces, presented an equation for the diffusion coefficient for gas pairs of nonpolar, nonreacting molecules ... 

The intramolecular and intermolecular solvent forces can be obtained in several ways. For rare gas solvents, it is most commonly the case that Lennard— Jones potentials are used to represent the intermolecular interactions (although more sophisticated potential energy functions are available from atom—atom scattering data). Molecular solvents, such as water, require more complicated models. The construction of these models is an active field and well beyond the scope of this review. Many of these models aim to reproduce bulk properties of the solvent and are quite successful in doing that (although some properties. 

A force sensor in an AFM can only work if the probe interacts with the force field associated with a surface. The interaction force between the probe and the surface in ambient air is illustrated in Figure 9.2. The total intermolecular pair potential is obtained by assuming one attractive (-Ci/z ) and another repulsive potential (C2/z )-Superimposing the two gives an expression for the well-known Lennard-Jones potential, where Ci and C2 are the corresponding coefficients for the attractive and repulsive interactions, respectively, and z is the distance between the sample surface and rest position of the cantilever. 

S. F. Boys and 1. Shavitt, Intermolecular Forces and Properties of Fluids. 1. The Automatic Calculation of Higher Virial Coefficients and Some Values of the Fourth Coefficient for the Lennard-Jones Potential, Proc. Roy. Soc. London, A254, 487 98 (1960). 

From the definitions of Wa and Wq, it can be seen that is a function of surface energies. If the structure of a material and the molecular potential energy-separation relationships are known, the surface energy can be calculated by evaluating the work required to separate to infinity the material either side of a chosen plane. For a material in which the dominant intermolecular forces are dispersion force interactions, the Lennard-Jones potential (see Dispersion forces and Polar forces) will apply, and the calculation is relatively simple. It gives work of cohesion on phase 1... 

In the original version of this model an attractive term based on Lennard-Jones potential constants was included in order to account for intermolecular attraction between adsorbed molecules. However, it was found that the fit of the experimental isotherm was not significantly improved by the inclusion of these factors so they were omitted in later studies. This observation suggests that, at least for nonpolar sorbates, the adsorption equilibrium behavior is governed by the attractive potential of the framework, which determines the Henry constant, and the repulsive interaction between molecules due to their finite size. Attractive forces between adsorbed molecules appear to be of only secondary importance. 

Eigure 4.33 shows a plot of the Lennard-Jones potential between two atoms or molecules. The quantity s in Equation 4.12 is equal to the depth of the potential well and is a measure of the strength of the intermolecular force. The quantity a- is a measure of the size of the molecules and is defined as the separation at which 7=0. Table 4.7 lists the Lennard-Jones parameters, s and a, for a few atoms and molecules. The Lennard-Jones potential is a good representation of generic intermolecular interactions in nonionic systems and is used extensively in the modeling and simulation of chemical systems. 

When we come to calculate the second virial coefficients of pure and mixed gases we come to one of the principal difficulties of all methods of prediction based on statistical mechanics - what do we know of intermolecular forces Here caution is needed, for twenty years ago we thought that we knew more about them than we did. The Lennard-Jones potential is... 

We have seen above that the 6-12 Lennard-Jones potential closely approximates intermolecular forces for many molecules. Equation (12) can be made dimensionless by dividing F by e. This results in a universal function in which the dimensionless poten-ial is a function of the dimensionless distance of separation between the molecules, r/a. The energy parameter e. and the distance parameter a. are characteristic values for a given molecule. This is a microscopic theory of corresponding states. It is related to the macroscopic theory through the critical properties of a fluid. Because the critical temperature is a measure of the kinetic energy of fluids in a common physical state, there should be a simple proportionality between the energy parameter e. and the critical temperature Tc. Because the critical volume reflects molecular size, there should also be a simple proportionality between a. and the cube root of Vc. For simple non-polar molecules which can be described by the 6-12 Lennard-Jones potential, the proportionalities have been found to be ... 

Intermolecular potential functions can be used to represent intermolecular forces and are often used directly in thermodynamic models. The Lennard-Jones potential includes both repulsive and attractive forces. The distance dependency of the attractive part of the Lennard-Jones potential is the same as for the vdW forces. 

The second simulation technique is molecular dynamics. In this technique, which was pioneered by Alder, initial positions of theparticles of a system of several hundred particles are assigned in some way. Displacements of the particles are determined by numerically simulating the classical equations of motion. Periodic boundary conditions are applied as in the Monte Carlo method. The first molecular dynamics calculations were done on systems of hard spheres, but the method has been applied to monatomic systems having intermolecular forces represented by the square-well and Lennard-Jones potential energy functions, as well as on model systems representing molecular substances. Commercial software is now available to carry out molecular dynamics simulations on desktop computers. ... 

The equilibrium distance between two molecules is achieved when the intermolecular force F(r) is 0 from (1.13) it is seen that the constant ro denotes the equilibrium distance between two molecules. Further it is seen that the Lennard-Jones potential (1.12) has minimum (r) = —for r = ro, i.e. when the molecules are at their mutual equilibrium distance. For r —> oo, (P r) and F r) — 0. 

---