Lennard-Jones potential dynamics simulations

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. 

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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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... 

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. 

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

In the second place, we shall study rotational dynamics. Rotational processes are of fundamental importance for dielectric relaxation. To shed light on some controversial issues in dielectric relaxation, Brot and co-workers did a computer simulation of a system of disks interacting via both Lennard-Jones potentials and electric dipole-dipole couplings. This is pre-... 

In such molecular dynamic simulations, one starts with an array of atoms or molecules , initially on a lattice, interacting with one another via an interatomic potential. These interacting potentials were taken by Paskin et al (1980, 1981) to be the Lennard-Jones potential (l> rij) = e[(l/rij) — 2(l/rij) ], where e denotes the depth of the potential energy and rij denotes the interatomic separation of the atoms. This potential is assumed to have a sharp cut-off at an arbitrarily chosen value 1.6 (lattice constant) of the interatomic separation. The external stress or force is applied only at the boundary surface atoms. In order to investigate the Griffith fracture phenomena, one can consider for example a two-dimensional lattice of linear size L, remove a few I L) consecutive bonds along a horizontal row in the middle of the network, and apply tensile force on the upper and lower surface atoms in the vertical direction. 

Fig. 3.17. Molecular dynamic simulation results for the onset of fracture growth instablity in a triangular lattice network with Lennard-Jones potential, having an initial crack at the left-side boundary, (a) Initial stages of growth, and (b) late stage unstable growth with large propagation velocities (Abraham et al 1994).

Fig. 3.19. Molecular dynamic simulation results for the fracture propagation in amorphous structures (with Lennard-Jones potential) show that the average fracture velocity crosses over to a higher value (ufinai from Uinitiaij indicated by the dotted lines) at the late stages of growth, as the crack size exceeds the typical size (correlation length) of the voids in the network. The inset shows that a corresponding crossover in the fractured surface roughness exponent also occurs along with the crossover in the fracture velocity (from Nakano et al 1995).

In classical molecular dynamics simulations, atoms are generally considered to be points which interact with other atoms by some predehned potential form. The forms of the potential can be, for example, Lennard-Jones potentials or Coulomb potentials. The atoms are given velocities in random directions with magnitudes selected from a Maxwell-Boltzman distribution, and then they are allowed to propagate via Newton s equations of motion according to a finite-difference approximation. See the following references for much more detailed discussions Allen and Tildesley (1987) and Frenkel... 

The plain Lennard-Jones potential is shifted up, so that its minimum located at 21/6ct has value 0, and set to zero beyond that point. The advantage of including the r 6 contribution instead of merely using the purely repulsive r 12 is that Eq. 4 is exactly zero beyond rcut and merges smoothly to this value at rCM. The use of a smooth hard core in molecular dynamics simulations is advantageous since the force is the derivative of the potential therefore the latter should be differentiable. In fact, the derivative must also be bounded to ensure numerical stability of the discrete integrator. 

The parameter A tunes the stiffness of the potential. It is chosen such that the repulsive part of the Lennard-Jones potential makes a crossing of bonds highly improbable (e.g., k=30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Lennard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly form liquid crystalline phases. 

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