Lennard-Jones constants

Lennard-Jones Constants Determined from Viscosity Data 

Source Reid, R. C., J. M. Praunitz, and B. E. Poling, The Properties of Gases and Liquids, 4th ed., McGraw-Hill, New York (1987). 

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In the absence of specific penetrant/polymer interactions, solubility of the penetrant is determined mainly by its chemical namre and depends on condensability, which is represented by boiling temperamre (Tb), critical temperature (Ter), or Lennard-Jones constant (s/fe) [7,8]. It is known that in the hydrocarbon series the increase in condensability is accompanied by a parallel increase in the size of molecules (Table 9.1 [9-17]). It is therefore not surprising that in both glassy and rubbery polymers correlations of hydrocarbon solubility in the polymers with condensability and sizes of hydrocarbon molecules are observed (Figures 9.1 through 9.3). 

Since the Lennard-Jones potential-energy function is used, the equation is strictly valid only for nonpolar gases. The Lennard-Jones constants for the unlike molecular pair AB can be estimated from the constants for like pairs A A and BB ... 

Table ll-l Lennard-Jones constants and critical properties... 

Modelling Gas Separation in Porous Membranes 99 Table 5.2 Lennard-jones constants, molecular masses and average velocities at room... 

In this expression, 82 is the second coefficient calcnlated as we saw above and b is the co-volume, which is involved in the Van der Waals eqnation (equation [7.113]). As we initially did to calculate the second coefficient, the authors calculated the following coefficients using the expression of two Lennard-Jones constants for the interaction between molecules ... 

Using the Chapman-Enskog equation Svehia [18] gives the following Lennard-Jones constants for the gases ... 

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. 

An approximate value for dc in the equation for tire Lennard-Jones potential, quoted above, may be obtained from the van der Waals constant, b, since... 

Here Tq are coordinates in a reference volume Vq and r = potential energy of Ar crystals has been computed [288] as well as lattice constants, thermal expansion coefficients, and isotope effects in other Lennard-Jones solids. In Fig. 4 we show the kinetic and potential energy of an Ar crystal in the canonical ensemble versus temperature for different values of P we note that in the classical hmit (P = 1) the low temperature specific heat does not decrease to zero however, with increasing P values the quantum limit is approached. In Fig. 5 the isotope effect on the lattice constant (at / = 0) in a Lennard-Jones system with parameters suitable for Ne atoms is presented, and a comparison with experimental data is made. Please note that in a classical system no isotope effect can be observed, x "" and the deviations between simulations and experiments are mainly caused by non-optimized potential parameters. 

An intrinsic surface is built up between both phases in coexistence at a first-order phase transition. For the hard sphere crystal-melt interface [51] density, pressure and stress profiles were calculated, showing that the transition from crystal to fluid occurs over a narrow range of only two to three crystal layers. Crystal growth rate constants of a Lennard-Jones (100) surface [52] were calculated from the fluctuations of interfaces. There is evidence for bcc ordering at the surface of a critical fee nucleus [53]. 

Here C and C2 are suitable constants. The Lennard-Jones potential can also be written as... 

Figure 2.10. Part of the better description of the Morse and Exp.-6 potentials may be due to the fact that they have three parameters, while the Lennard-Jones potential only employs two. Since the equilibrium distance and the well depth fix two constants, there is no additional flexibility in the Lennard-Jones function to fit the form of the repulsive interaction.

In some force fields, especially those using the Lennard-Jones form in eq. (2.12), the /fg parameter is defined as the geometrical mean of atomic radii, implicitly via the geometrical mean rale used for the C and C2 constants. 

The quantities e and a are the force constants of the Lennard-Jones interaction solute molecule K with an element of the wall of the hydroquinone cage (cf. Eq. 30). It is assumed that for this interaction the frequently-used combining rules for the interaction between two unlike particles hold,... 

When calculating the rate constants, two potentials were used the anisotropic 6-12 Lennard-Jones from [209] and the anisotropic Morse [216] for comparison. The results appeared to be very similar, thus indicating low sensitivity of the line widths to the potential surface details. The agreement with experimental data shown in Fig. 5.6(h) is fairly good. Moreover, the SCS approximation gives a qualitatively better approach to the problem than the purely non-adiabatic IOS approximation. As is seen from Fig. 5.6 the significant decrease of the experimental line widths with j is reproduced as soon as adiabatic corrections are made [215]. 

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