Lennard-Jones interaction, diffusion

Figure 2.5. Molecular D)oiainics simulation of self-diffusion in a dense fluid of "soft" spherical particles near a "hard" solid wail. The "wall" exerts no force on the particles but reverses the z-component of the velocity if a molecule attempts to cross it a is the length parameter in the repulsive part of the Lennard-Jones interaction. (Redrawm from J.N. Cape. J. Chem. Soc., Faraday Trans. II 78 (1982) 317.)...

Figure 2.5 shows a Molecular Dynamics simulation, which is the counterpart of the previous example. Particle trajectories are shown for fluid particles, interacting via an law (as in the Lennard-Jones interaction), with a structureless "hard" wall. A striking feature is that the diffusion between the... 

The transport coefficients like viscosity, thermal conductivity and self-diffusivity for a pure mono-atomic gas and the diffusivity for binary mixtures obtained from the rigorous Chapman-Enskog kinetic theory with the Lennard-Jones interaction model yield (e.g., [39], sect 8.2 [5], sects 1-4, 9-3 and 17-3) ... 

Modern kinetic theory is able to predict the transport coefficients of the Lennard-Jones liquid (1-center Lennard-Jones interaction between particles) to a fairly good approximation (Karkheck 1986 Hoheisel 1993). The results of these theories have been compared in detail with the exact MD computation results (Borgelt et al. 1990). Comparisons for self-diffusion, shear viscosity and thermal conductivity are presented in Figures 9.2-9.4. 

We now describe a relatively simple MD model of a low-index crystal surface, which was conceived for the purpose of studying the rate of mass transport (8). The effect of temperature on surface transport involves several competing processes. A rough surface structure complicates the trajectories somewhat, and the diffusion of clusters of atoms must be considered. In order to simplify the model as much as possible, but retain the essential dynamics of the mobile atoms, we will consider a model in which the atoms move on a "substrate" represented by an analytic potential energy function that is adjusted to match that of a surface of a (100) face-centered cubic crystal composed of atoms interacting with a Lennard-Jones... 

Xenon has been considered as the diffusing species in simulations of microporous frameworks other than faujasite (10-12, 21). Pickett et al. (10) considered the silicalite framework, the all-silica polymorph of ZSM-5. Once again, the framework was assumed to be rigid and a 6-12 Lennard-Jones potential was used to describe the interactions between Xe and zeolite oxygen atoms and interactions between Xe atoms. The potential parameters were slightly different from those used by Yashonath for migration of Xe in NaY zeolite (13). In total, 32 Xe atoms were distributed randomly over 8 unit cells of silicalite at the beginning of the simulations and calculations were made for a run time of 300 ps at temperatures from 77 to 450 K. At 298 K, the diffusion coefficient was calculated to be 1.86 X 10 9 m2/s. This... 

June et al. (12) used TST as an alternative method to investigate Xe diffusion in silicalite. Interactions between the zeolite oxygen atoms and the Xe atoms were modeled with a 6-12 Lennard-Jones function, with potential parameters similar to those used in previous MD simulations (11). Simulations were performed with both a rigid and a flexible zeolite lattice, and those that included flexibility of the zeolite framework employed a harmonic term to describe the motion of the zeolite atoms, with a force constant and bond length data taken from previous simulations (26). 

June et al. (85) presented united-atom calculations for butane and for hexane in silicalite, whereby the bond and dihedral angles of the alkanes were allowed to vary. In addition, the calculation of hexane took account of an additional intramolecular Lennard-Jones potential for nonbonded atoms more than three bonds apart (which prevents the alkane crossing over itself). The interaction parameters for the alkane molecules were taken from Ryckaert and Bellmans (3), and those governing the interaction of the alkanes with the zeolite from a previous study of the low-occupancy sorption of alkanes in silicalite (87). Variable loadings of alkanes were considered from 1 to 8 molecules per unit cell were considered, and calculations were allowed to run for 500 ps for diffusion at 300 K. 

Benzene-benzene interactions were modeled with a Buckingham potential that was shown to yield reasonable predictions of the properties of liquid and solid benzene. Benzene-zeolite interactions were modeled by a short-range Lennard-Jones term and a long-range electrostatic term. In total, 16 benzene molecules were simulated in a unit cell of zeolite Y, corresponding to a concentration of 2 molecules per supercage. Calculations ran for 24 ps (after an initial 24-ps equilibration time) for diffusion at 300 K. 

Auerbach et al. (101) used a variant of the TST model of diffusion to characterize the motion of benzene in NaY zeolite. The computational efficiency of this method, as already discussed for the diffusion of Xe in NaY zeolite (72), means that long-time-scale motions such as intercage jumps can be investigated. Auerbach et al. used a zeolite-hydrocarbon potential energy surface that they recently developed themselves. A Si/Al ratio of 3.0 was assumed and the potential parameters were fitted to reproduce crystallographic and thermodynamic data for the benzene-NaY zeolite system. The functional form of the potential was similar to all others, including a Lennard-Jones function to describe the short-range interactions and a Coulombic repulsion term calculated by Ewald summation. 

There have been several studies of the iodine-atom recombination reaction which have used numerical techniques, normally based on the Langevin equation. Bunker and Jacobson [534] made a Monte Carlo trajectory study to two iodine atoms in a cubical box of dimension 1.6 nm containing 26 carbon tetrachloride molecules (approximated as spheres). The iodine atom and carbon tetrachloride molecules interact with a Lennard—Jones potential and the iodine atoms can recombine on a Morse potential energy surface. The trajectives were followed for several picoseconds. When the atoms were formed about 0.5—0.7 nm apart initially, they took only a few picoseconds to migrate together and react. They noted that the motion of both iodine atoms never had time to develop a characteristic diffusive form before reaction occurred. The dominance of the cage effect over such short times was considerable. 

The influence of size and shape on the diffusion of hydrophobic solutes was estimated by simulations involving artificial Lennard-Jones particles those intermolecu-lar interaction parameters were based on those for ammonia or oxygen, respectively. The results on the size dependence of diffusion confirmed that the membrane interior differs strongly from a bulk hydrocarbon. In the center of the bilayer, the excess free energy for hydrophobic Lennard-Jones particles remained low irrespective of the size of the particles. This can be explained by the large fraction of accessible volume in that region. 

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]. 

Koddermann et al. calculated the heats of vaporization for imidazolium-based ILs [Cnmim][NTf2] with n = 1, 2, 4, 6, 8 by means of MD simulations [81], The authors applied a force field which they had developed recently. Within this force field the authors reduced the Lennard-Jones parameters in order to reproduce experimental diffusion coefficients [81], The refined force field also led to absolute values of heats of vaporization as well as their increase with the chain length of the imidazolium cation such that this quantities were described correctly. The overall heats of vaporization were split in several contributions and discussed in detail. The authors observed that with increasing alkyl chain length, the Coulomb contribution to the heat of vaporization remained constant at around 80 kJ mol 1, whereas the van der Waals interaction increased continuously. The calculated grow of about 4.7 kJ mol"1 per CH2-group of the van der Waals contribution in the IL exactly matched the increase in the heats of vaporization for n-alcohols and n-alkanes, respectively. The results support the importance of van der Waals interactions even in systems completely composed of ions [81],... 

The complex rotational behavior of interacting molecules in the liquid state has been studied by a number of authors using MD methods. In particular we consider here the work of Lynden-Bell and co-workers [60-62] on the reorientational relaxation of tetrahedral molecules [60] and cylindrical top molecules [61]. In [60], both rotational and angular velocity correlation functions were computed for a system of 32 molecules of CX (i.e., tetrahedral objects resembling substituted methanes, like CBt4 or C(CH3)4) subjected to periodic boundary conditions and interacting via a simple Lennard-Jones potential, at different temperatures. They observe substantial departures of both Gj 2O) and Gj(() from predictions based on simple theoretical models, such as small-step diffusion or 7-diffusion [58]. Although we have not attempted to quantitatively reproduce their results with our mesoscopic models, we have found a close resemblance to our 2BK-SRLS calculations. Compare for instance our Fig. 13 with their Fig. 1 in [60]. 

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