Physical 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 classical kinetic theoty of gases treats a system of non-interacting particles, but in real gases there is a short-range interaction which has an effect on the physical properties of gases. The most simple description of this interaction uses the Lennard-Jones potential which postulates a central force between molecules, giving an energy of interaction as a function of the inter-nuclear distance, r. 

Extension to many dimensions provides insight into more sophisticated aspects of the method and into the nature of molecular interactions. In the second stage of this unit, the students perform molecular dynamics simulations of 3-D van der Waals clusters of 125 atoms (or molecules). The interactions between atoms are modeled using the Lennard-Jones potentials with tabulated parameters. Only pairwise interactions are included in the force field. This potential is physically realistic and permits straightforward programming in the Mathcad environment. The entire program is approximately 50 lines of code, with about half simply setting the initial parameters. Thus the method of calculation is transparent to the student. 

The Lennard-Jones potential continues to be used in many force fields, particularly those targeted for use in large systems, e.g., biomolecular force fields. In more general force fields targeted at molecules of small to medium size, slightly more complicated functional forms, arguably having more physical justification, tend to be used (computational times for small molecules are so short dial the efficiency of the Lennard-Jones potential is of little consequence). Such forms include the Morse potential [Eq. (2.5)] and the Hill potential... 

The Lennard-Jones potential is the one most commonly used at present since the Buckingham form has an additional parameter, exhibits a physically unrealistic maximum at very short intemuclear distances, and approaches —00 as the intemuclear distance approaches zero. While these artifacts can he eliminated by proper computer programming, they increase the time of computation. Nearly the same curves result in the region of r of interest if eqs. 5 and 6 are made to coincide at their minima. 

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. 

Consider the simple case where the radial distribution function in the fluid is zero for radii less than a cut-off value determined by the size of the hard core of the solute, and one beyond that value. Calculate the value of the parameter a appearing in the equation of state Eq. (4.1) for a potential of the form cr , where c is a constant and n is an integer. An example is the Lennard-Jones potential where = 6 for the long-ranged attractive interaction. What happens if n <37 Explain what happens physically to resolve this problem. See Widom (1963) for a discussion of the issue of thermodynamic consistency when constructing van der Waals and related approximations. 

Figure 4. Reversible work of formation of a physically consistent cluster as a function of the number of molecules in the cluster, /, and the cluster volume, v. Monte Carlo simulation for supercooled Ar vapor (Lennard-Jones potential) at 70 K and various pressures [67, 5].

Wolf GH, Bukowinski MST (1987) Theoretical study of the stmctural properties and equations of state of MgSiOs and CaSiOs perovskites implications for lower mantle composition. In High Pressure Research in Mineral Physics. Manghnani MH, Syono Y (eds), Terrapub, Tokyo p 313-331 Wood WW, Parker RF (1957) Monte Carlo equation of state of molecules interacting with the Lennard-Jones potential. 1. Supercritical isotherm at about twice the critical temperature. J Chem Phys 27 720-733... 

The object of any statistical mechanical theory of polymer systems is ultimately to relate the measurable physical properties of the system to the properties of the constituent monomers and their mutual interactions. It is imperative that the initial statistical mechanical theories of these physical properties of polymer systems not depend on the exact details of a particular polymer. Instead, these theories should reflect those generic properties of polymer systems that are a result of the chainlike structures of polymer molecules. Once the properties of simple, yet general, models of polymers are well understood, it is natural to focus attention upon the particular aspects of a polymer of interest. The initial use of simple models of polymers is not solely dictated by an attempt to obtain those general features of polymer systems. The mathematical simplicity of the model is required so that we avoid the use of uncontrollable mathematical approximations which necessarily arise with the use of more complicated models. When the model is sufficiently simple, yet physically nontrivial, we are able to test different approximation schemes to find those that are useful. Presumably these methods of approximation would also be useful for more complicated models. This emphasis upon mathematical simplicity has its analog in the theory of fluids. First hard-core interactions can be used to test the physical principles associated with various methods of approximation. Once physically sound approximation schemes have been obtained with this model, they may be applied with more realistic potentials, e.g., the Lennard-Jones potential, which require subsequent numerical approximations. Thus we wish to separate approximations of a physical origin from those of purely a numerical nature. This separation... 

The FENE model with short chains of ten beads interacting with a truncated Lennard-Jones potential has been used for an MD investigation (381,382) of mode coupling theory just above the predicted critical temperatiue, which in turn is above the Tg. It is too early to say whether this theory will prove to be of practical use for polymer science, but simulations of this kind are probing deeply into the natiue of the glass transition, which can only help to illiuninate the physics of this important transition. 

It comes, of course, as no surprise that introducing a molecular cut-off and applying a kinetic slip condition to the first molecular layer resolves the notorious singularities of hydrodynamic description. The hydrodynamic singularities are eliminated, however, only at molecular distances, and are still felt in sharp interface curvatures at microscopic distances identified here as the intermediate asymptotic region. The computations are eased considerably when non-physical divergence of both viscous stress and attractive Lennard-Jones potential beyond the cut-off limit... 

The fact that dipoles interact with a disteince dependence gives a physical basis for the form of the attractive interaction in a widely used energy function called the Lennard-Jones potential,... 

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 ... 

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