Lennard-Jones potential, molecular

Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. 

We have two interaction potential energies between uncharged molecules that vary with distance to the minus sixth power as found in the Lennard-Jones potential. Thus far, none of these interactions accounts for the general attraction between atoms and molecules that are neither charged nor possess a dipole moment. After all, CO and Nj are similarly sized, and have roughly comparable heats of vaporization and hence molecular attraction, although only the former has a dipole moment. 

Molecular dynamics calculations have been made on atomic crystals using a Lennard-Jones potential. These have to be done near the melting point in order for the iterations not to be too lengthy and have yielded density functioi). as one passes through the solid-vapor interface (see Ref. 45). The calculations showed considerable mobility in the surface region, amounting to the presence of a... 

The MYD analysis assumes that the atoms do not move as a result of the interaetion potential. The eonsequenees of this assumption have recently been examined by Quesnel and coworkers [50-55], who used molecular dynamic modeling techniques to simulate the adhesion and release of 2-dimensional particles from 2-D substrates. Specifically, both the Quesnel and MYD models assume that the atoms in the different materials interact via a Lennard-Jones potential

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Kihara20 used a core model in which the Lennard-Jones potential is assumed to hold for the shortest distance between the molecular cores instead of molecular centers. By use of linear, tetrahedral, and other shapes of cores, various molecules can be approximated. Thomaes,41 Rowlinson,35 Hamann, McManamey, and Pearse,14 Atoji and Lipscomb,1 Pitzer,30 and Balescu,4 have used other models of attracting centers and other mathemtical methods, but obtain similar conclusions. The primary effect is to steepen the potential curve so that in terms of inverse powers of the inter-... 

Physisorption is very similar to the molecular van der Waals interaction, which makes gases condense in multilayers. The Van der Waals interaction between molecules is often described by the Lennard-Jones potential, which has the form... 

The interaction of hydrogen (deuterium) molecules with a transition metal surface c an be conveniently described in terms of a Lennard--Jones potential energy diagram (Pig. 1). It cxxislsts of a shallcw molecular precursor well followed by a deep atomic chemisorption potential. Depending on their relative depths and positions the wells m or may not be separated by an activation energy barrier E as schematically Indicated by the dotted cur e in Fig. 1. 

Figure 4.4 shows a section of a perfect single crystal surface [such as Pt(lll)] which is approached by a diatomic molecule (say 02) undergoing dissociative chemisorption. The progress of this process is illustrated by a contour plot of the energetics as a function of the distance x of the molecule from the surface and of the separation y between the two atoms, together with the well-known one-dimensional Lennard-Jones potential diagram. (The molecular axis is assumed to be parallel to the surface... 

Fig. 5.1 A schematic projection of the 3n dimensional (per molecule) potential energy surface for intermolecular interaction. Lennard-Jones potential energy is plotted against molecule-molecule separation in one plane, the shifts in the position of the minimum and the curvature of an internal molecular vibration in the other. The heavy upper curve, a, represents the gas-gas pair interaction, the lower heavy curve, p, measures condensation. The lighter parabolic curves show the internal vibration in the dilute gas, the gas dimer, and the condensed phase. For the CH symmetric stretch of methane (3143.7 cm-1) at 300 K, RT corresponds to 8% of the oscillator zpe, and 210% of the LJ well depth for the gas-gas dimer (Van Hook, W. A., Rebelo, L. P. N. and Wolfsberg, M. /. Phys. Chem. A 105, 9284 (2001))...

Molecular dynamics has been used extensively to explore the solid-liquid interface. In one such study, a modified Lennard-Jones potential has been used to model this interaction in the spreading of a droplet [4], of the form... 

Figure 4.30. Molecular model for dialkyl succinic acid. The C -C distance is 1.54 A the C-C-C angles are tetrahedral, Qj- = 109.47 °. The center of the negative charge was placed at a distance of 1.25/2 = 0.625 A from Ae carboxyl carbon atom, and the positive charge at a distance of 0.7 A from the negative charge. The effective radii for the various alkyl groups were computed from the van der Waals volumes given by Bondi (1968). These are methyl, 1.75 A ethyl, 1.78 A isopropyl, 2.0 A tert-butyl, 2.2 A. These radii were used to construct the Lennard-Jones potentials between the various groups [see Eq. (4.8.46)].

We now present results from molecular dynamics simulations in which all the chain monomers are coupled to a heat bath. The chains interact via the repiflsive portion of a shifted Lennard-Jones potential with a Lennard-Jones diameter a, which corresponds to a good solvent situation. For the bond potential between adjacent polymer segments we take a FENE (nonhnear bond) potential which gives an average nearest-neighbor monomer-monomer separation of typically a 0.97cr. In the simulation box with a volume LxL kLz there are 50 (if not stated otherwise) chains each of which consists of N -i-1... 

Here t0 is an average vibrational period in a surface potential well having a value of the order of subpicoseconds for a simple molecule, and U() is a depth of the potential, which strongly depends on the type of interaction. For strong interaction, t exceeds 1 s or more, while in the case of no interaction, such as elastic collision, it is less than T(>. If we assume the Lennard-Jones potential for the interaction potential and expand it at the equilibrium position r0, t() is given by relevant molecular parameters as... 

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. 

Equations (1) and (4) or other variations of the 12-6 power law are often called the Lennard-Jones potential. The numerical values of the constants in the Lennard-Jones potential may be obtained from studies of the compressibility of condensed phases, the virial coefficients of gases, and by other methods. A summary of these methods and other expressions for the molecular interaction energy can be found in the book by Moelwyn-Hughes (1964). 

The values of C and D are evaluated at the critical point and normal boiling point. U. is the vertical molecule-cation interaction energy and U isJthe corresponding molecule-anion term. U and w are calculated as the sums of all the appropriate dielectric and Lennard-Jones potentials. The actual calculation of an x/m isotherm is the superposition of several solution models. The principal one corresponds to the partial filling by molecules on the cation sites. The value of x/m is a constant times Xg, summed over all sites, where the constant is the molecular weight ratio. 

We studied the distribution of a supercritical solvent around an infinitely dilute solute molecule via molecular dynamics simulations. The Lennard-Jones potential... 

The molecular size of a solvent can be characterized in several ways. One of them is to assign the solvent a molecular diameter, as if its molecules were spherical. From a different aspect, this diameter characterizes the cavity occupied by a solvent molecule in the liquid solvent. From a still further aspect, this is the mean distance between the centers of mass of two adjacent molecules in the liquid. The diameter plays a role in many theories pertaining to the liquid state, not least to those treating solvent molecules as hard spheres, such as the scaled particle theory (SPT, see below). Similar quantities are the collision diameters a of gaseous molecules of the solvent, or the distance characterizing the minimum in the potential energy curve for the interaction of two solvent molecules. The latter quantity may be described, e.g., according to the Lennard-Jones potential (Marcus 1977)... 

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