The Lennard-Jones parameters

One of the more difficult decisions to be made is the proper value for the Lennard-Jones parameters. These relate to the interaction between the quantum mechanical atoms and the MM atoms. At the time of writing (1999), there does not appear to be a consensus amongst researchers. Some authors recommend a 10% scaling of the traditional 12-6 parameters. Some authors scale the MM atom charges. 

Table 5.1 Parameters of the united atom force field for polyethylene used as the atomistic input for the coarse-graining procedure. The Lennard-Jones parameters pertain to CH2-group interaction, since chain ends were not considered in the coarse-graining. 

Intermediate states do not have to be physically meaningful, i.e., they do not have to correspond to systems that actually exist. As an example, assume that we want to calculate the difference in hydration free energies of a Lennard-Jones particle and an ion with a positive charge q of le. For simplicity, we further assume that the Lennard-Jones parameters remain unchanged upon charging the particle. Since a direct calculation of the free energy difference is not likely to succeed in this case, we construct intermediate states in which the particle carries fractional charges [Pg.46]

Fig. 9.4. Pa (e) and (e) as a function of the binding energy. The simulations treated 216 water molecules, utilizing the SPC/E water model, and the Lennard-Jones parameters for methane were from [63]. The number density for both the systems is fixed at 0.03333 A 3, and T = 298 K established by velocity rescaling. These calculations used the NAMD program (www.ks.uiuc.edu/namd). After equilibration, the production run comprised 200 ps in the case of the pure water simulation and 500 ps in the case of the methane-water system. Configurations were saved every 0.5 ps for analysis...

In Fig. 1, various elements involved with the development of detailed chemical kinetic mechanisms are illustrated. Generally, the objective of this effort is to predict macroscopic phenomena, e.g., species concentration profiles and heat release in a chemical reactor, from the knowledge of fundamental chemical and physical parameters, together with a mathematical model of the process. Some of the fundamental chemical parameters of interest are the thermochemistry of species, i.e., standard state heats of formation (A//f(To)), and absolute entropies (S(Tq)), and temperature-dependent specific heats (Cp(7)), and the rate parameter constants A, n, and E, for the associated elementary reactions (see Eq. (1)). As noted above, evaluated compilations exist for the determination of these parameters. Fundamental physical parameters of interest may be the Lennard-Jones parameters (e/ic, c), dipole moments (fi), polarizabilities (a), and rotational relaxation numbers (z ,) that are necessary for the calculation of transport parameters such as the viscosity (fx) and the thermal conductivity (k) of the mixture and species diffusion coefficients (Dij). These data, together with their associated uncertainties, are then used in modeling the macroscopic behavior of the chemically reacting system. The model is then subjected to sensitivity analysis to identify its elements that are most important in influencing predictions. 

Estimate the binary diffusion coefficient of triethyl antimony (Sb(C2H5)3, TESb) in H2 at 25°C and 140 Torr pressure. The Lennard-Jones parameters for H2 are a = 2.827 A and e/kg = 59.7 K. Estimate the TESb Lennard-Jones parameters from its density at the boiling point Pb=1.324 g/cm3, and its boiling-point temperature 7], = 159.5°C. 

Figure 2. Experimental and simulated fluorescence Stokes shift function 5(f) for coumarin 343 in water. The curve marked Aq is a classical molecular dynamics simulation result using a charge distribution difference, calculated by semiempirical quantum chemical methods, between ground and excited states. Also shown is a simulation for a neutral atomic solute with the Lennard-Jones parameters of the water oxygen atom (S°). (From Ref. 4.)...

This picture fits very well with the tendency of the sorption isotherm curvature (and hence of the site sorption mode) to disappear at T > Tg. On a more quantitative level, the above characterization of the Henry sorption mode is supported by the smooth temperature dependence of K, found in the PET-C02 system 12), which indicates a roughly unchanged enthalpy of sorption AH, above and below Tg. Additional support is provided by the correlation between K, and the Lennard-Jones parameter s/k characteristic of the gaseous penetrant, in accordance with... 

On the other hand, it is noteworthy that progressively lower s0 values are found for lighter gases. In fact, it appears that s0, like K2, tends to be correlated with the Lennard-Jones parameter of the penetrant gas39), as illustrated in Fig. 3. Such behaviour is expected for K2 (cf. the behaviour of K, above), but not for s0, according to the Langmuir localised sorption concept. 

Fig. 3 a and b. Correlation of the experimental Langmuir capacity (Sq) and affinity (K2) parameters in PC at 35 °C with the Lennard-Jones parameter e/k of the penetrant gas39)... 

Fig. 5. Correlation between experimental D.Di values with the Lennard-Jones parameter of the penetrant gas for four glassy polymers2S) O a copolyester 26i A a polysulphone 23 PC A PPO...

Schwartz et al.26 deduced values of / from the Lennard-Jones parameters e and r0, which have been derived from viscosity measurements, and analysed and tabulated by Hirschfelder et a/.31. A detailed account of the required calculation has been given by Herzfeld and Litovitz32. The problem is to obtain a satisfactory fit of the exponential interaction potential, with the repulsive region of the Lennard-Jones function... 

In order to include curvature-dependence in both the covalent and non-bonding interactions, we used the adaptive intermolecular reactive bond-order (AIREBO) potential,24 with modified van der Waals interactions. This potential uses the same bonding interactions as Brenner s REBO potential,25,26 both of which correctly account for local curvature dependence in the covalent bonding interactions. Chemisorption is thus treated accurately, but there is no explicit or implicit curvature dependence in the Lennard-Jones (L-J) parameters used to describe the non-bonded van der Waals interactions (physisorption). Consequently, we modified the Lennard-Jones parameters to make them explicitly dependent on the curvature of the nanotube. 

The van der Waals interaction was refit for H2 adsorption in SWNT by first fitting the L-J parameters for H-H interactions to the recent high level ab initio results on interactions between H2 molecules reported by Diep and Johnson,27 yielding the center of mass separation of 3.4 A, and the Lennard-Jones parameters nonbonding interaction between carbons is not expected to have a pronounced effect on the H2 adsorption energy and thus we retain the standard AIREBO potential parameters of [Pg.472]

For the interaction potential between hydrogen and carbon, we introduce a new procedure to derive the Lennard-Jones parameters from existing parameters that are appropriate for carbon atoms with sp2 and sp3 hybridizations. These parameters may come from existing force fields, and may have been obtained using either experimental or ab initio results. The L-J parameters a and s are made explicitly dependent on the radius of the nanotube, r, using the following equations ... 

The viscosity may be obtained from the Lennard-Jones parameters by aid... 

The viscosity of a pure monatomic gas of molecular weight M may be expressed in terms of the Lennard-Jones parameters by... 

Read the Lennard-Jones parameters for helium from Table B1 and B2 ... 

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