Ellipsoidal Lennard-Jones potential

The ellipsoidal Lennard-Jones potential has recently been applied in molecular dynamics studies of the angular velocity relaxation of hindered rotors, collective reorientation, and the stability of nematic-like orientational ordering. " The model was found to be a convenient and flexible representation of the interactions of polyatomic molecules. 

This finding has significant consequences for molecular dynamics. If the dynamic behavior of sufficiently dense Lennard-Jones fluids is dominated by Vii(r), as was shown for static properties, then one can realize an enormous saving of computer time by using the purely repulsive part of the spherical or ellipsoidal Lennard-Jones potential. Since the computer time used in molecular dynamics depends partially on the number of interactions experienced per molecule, the savings factor is of the order (2 /2.5) = 0.09. 

A recent study examined the effect on molecular dynamics of neglecting the long-range attractive part of the Lennard-Jones potential (spherical and ellipsoidal). This study was motivated by the successful static perturbation theory of simple Lennard-Jones fluids developed by Weeks et a/. The latter theory decomposed the spherical Lennard-Jones potential... 

An important addition compared to previous models was the parameterization of the internucleosomal interaction potential in the form of an anisotropic attractive potential of the Lennard-Jones form, the so-called Gay-Berne potential [90]. Here, the depth and location of the potential minimum can be set independently for radial and axial interactions, effectively allowing the use of an ellipsoid as a good first-order approximation of the shape of the nucleosome. The potential had to be calibrated from independent experimental data, which exists, e.g., from the studies of mononucleosome liquid crystals by the Livolant group [44,46] (see above). The position of the potential minima in axial and radial direction were obtained from the periodicity of the liquid crystal in these directions, and the depth of the potential minimum was estimated from a simulation of liquid crystals using the same potential. 

This algorithm has been applied to calculate the thermal conductivity of a variant of the Gay-Beme fluid where the Lennard-Jones core has been replaced by a purely repulsive 1/r core [20]. Two systems were studied, one consisting of prolate ellipsoids with a length to width ratio of 3 1 and another one consisting of oblate ellipsoids with a length to width ratio of 1 3. The potential parameters are given in Appendix II. They both form nematic phases at high densities. 

The original Gay-Beme potential forms a nematic phase and a Smectic B phase, which is more solid like than liquid like. Ellipsoidal bodies do usually not form smectic A phases because they can easily diffuse from one layer to another layer. However, if one increases the side by side attraction it becomes possible to form smectic A phases [6]. When one calculates transport coefficients very long simulation runs are required. Therefore one sometimes re-places the Lennard-Jones core by a purely repulsive 1/r core in order to decrease the range of the potential. Thereby one decreases the number of interactions, so that the simulations become faster. The Gay-Beme potential can be generalised to biaxial bodies by forming a string of oblate ellipsoids the axes of which are parallel to each other and perpendicular to the line joining their centres of mass [35]. One can also introduce an ellipsoidal core where the three axis are different [38]. 

Although the complete atomistic simulation of ensembles of mesogenic molecules is within reach of present computational facilities, the traditional treatment of liquid crystals in molecular dynamics or Monte Carlo simulations makes use of the Gay-Berne potential, an ingenious computational machine whose aspects deserve to be described here for their epistemological implications. An ordinary Lennard-Jones (LJ) potential, equation 4.38 or 4.40, can be written as a function of the distance between two particles, /Jy, the well depth e and the equilibrium separation a. An ellipsoidal object is identified by the position of its centroid and by an orientation unit vector u, and the Gay-Berne (GB) potential is a modified U that takes into account the anisotropy of the ellipsoid, both in energy and equilibrium separation ... 

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