Lennard-Jones potential diameters

Figure 7-12. Plot of the van der Waals interaction energy according to the Lennard-Jones potential given in Eq. (27) (Sj, = 2.0 kcal mol , / (, = 1.5 A). The calculated collision diameter tr is 1.34 A.

We have studied, by MD, pure water [22] and electrolyte solutions [23] in cylindrical model pores with pore diameters ranging from 0.8 to more than 4nm. In the nonpolar model pores the surface is a smooth cylinder, which interacts only weakly with water molecules and ions by a Lennard-Jones potential the polar pore surface contains additional point charges, which model the polar groups in functionalized polymer membranes. 

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... 

A1P04-31 (structure type ATO) has unidimensional channels with nominal diameter 5.4 A. To model Xe/AlP04-31 atomistically, we assume that A1P04-31 is rigid and defect free with the experimentally determined crystal structure [7]. Xe atoms are represented as spheres, and Xe-Xe and Xe-0 interactions are taken to be Lennard-Jones potentials using previously derived parameters [5,8]. 

The molecular size of a solvent can be characterized in several ways. One of them is to assign the solvent a molecular diameter, as if its molecules were spherical. From a different aspect, this diameter characterizes the cavity occupied by a solvent molecule in the liquid solvent. From a still further aspect, this is the mean distance between the centers of mass of two adjacent molecules in the liquid. The diameter plays a role in many theories pertaining to the liquid state, not least to those treating solvent molecules as hard spheres, such as the scaled particle theory (SPT, see below). Similar quantities are the collision diameters a of gaseous molecules of the solvent, or the distance characterizing the minimum in the potential energy curve for the interaction of two solvent molecules. The latter quantity may be described, e.g., according to the Lennard-Jones potential (Marcus 1977)... 

Here /is the ionization potential of the quenching molecule and = [c/(Q) + (/(I )] /2 is the distance of closest approach of the collision pair, where the d values are taken as Lennard-Jones collision diameters deduced from viscosity measurements. Thus a plot of In (cr ) versus In aQfj j HI ) would be predicted to be linear. This model also predicts some variability for different v vibrational levels due to Franck-Condon effects, but this can be ignored in the present experiment where mainly the v = 32 level is excited by the 532-nm source. 

Notes (Tlj = molecular collision diameter calculated from the Lennard-Jones potential ... 

FIGURE 9.1 Dependence of solubility coefficient of various hydrocarbons in natural rubber on hydrocarbon boiling temperature, 7], (a) and on diameter of their molecules calculated from Lennard-Jones potential, ctu (b). (From Semenova, S.I., Membranes (in Russian),... 

For the adsorbate-adsorbate interaction, Ugg, the usual Lennard-Jones potential for two gas molecules separated a distance r is assumed with a cutoff at 3 molecular diameters in order to speed up calculations. 

Fig. 7. Typical system configurations for the same parameters as in Fig. 6. The density is Ap = 0,0.184,0.286,0.498. Only a thin slice of the simulation box is shown in the right-most image. Each monomer is represented by a sphere of diameter 1.12 <7, which corresponds to the minimum of the Lennard-Jones potential...

Fig. 9.20. The relative potential U = L7(Z)/LfA of molecule A as a function of the distance Z from the pore centre for several values of the relative (dimensionless) pore radius r/OA for three different regions (a-c). Ua is the absolute value of the (Lennard Jones) potential on a free surface, oa the molecular diameter. After de Lange et al. [59,63].

A plot of the Lennard-Jones potential against distance between two molecules is shown in fig. 2.2. It is clear that the repulsive component dominates for values of r less than the molecular diameter, where it rises to values over 5eLj for r = 0.9a. The stable minimum occurs at r = 1.12a. By the time the separation between the two molecules is equal to three times their diameter, the intermolecular... 

Fig. 2.2 Plot of the Lennard-Jones potential u r) in units of the attractive potential energy Clj against intermolecular distance, r, in units of the molecular diameter a. The vertical and horizontal straight lines at r = a show the potential energy for a hard-sphere representation of the system.

This section discusses simulations in which the parameters are explicitly mapped to experimental systems. In particular, this mapping affects ion diameter cr, rod radius r0, Bjerrum length B, and line charge density A. In order to have a rod radius different from cr this requires the introduction of a new potential for ion-rod interactions, for which a modified truncated and shifted Lennard-Jones potential has been used ... 

The kinetic diameter is defined as the intermolecular distance of closest approach for two molecules colliding with zero initial kinetic energy (see Ref. 1(b) and is calculated from a Lennard - Jones potential. The difference between the apparent pore diameter (from crystallography) and the kinetic diameter is explained on the basis of activated diffusion of adsorbates into the zeolite pores. 

Let us plot the logarithmic values of the shear viscosities calculated by the PNM model, divided by the dilute gas values (which vary as the square root of temperature 7 as a function of ejkT, e being the depth of the Lennard-Jones potential well. The results are given in Fig. 1 for a reduced density of na = 0.818, a being the hard core diameter. The experimental points are well fitted by two approximately straight lines. We observe that Eyring s formula for shear viscosity at constant density... 

Here Op is the quenching cross-section of molecules P by molecule Q, p is the collision-reduced mass, p the quencher polariabilizty. Ip and 7p are the ionization potentials of P and Q, respectively, and Rq the Lennard-Jones collision diameter. C is a constant, and A is given by... 

Following standard practice the realistic potential elastic collision diameters (Sch3CF3 m) have been estimated using Lennard-Jones collision diameters (o ch3CF3-m) and the Lennard-Jones universal... 

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