Lennard-Jones diameter

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... 

Gas molecule Kinetic diameter (A) Lennard-Jones diameter (A)... 

Figure 4.11 illustrates a carbon membrane with pores in the range suitable for molecular sieving [78]. As expected, there is a clear and indisputable correlation between flux and molecular size. In Figure 4.12, the carbon membrane is more open (pore size in the range 6-10 A). The gas pair reported is CO2 and CH4, and as can be seen, the selectivity is clearly in favor of CO2 indicating selective surface flow. The critical temperatures, 7)., and Lennard-Jones diameters, for the two gases are... 

Fig. 7. A random configuration of atoms (black) surrounded by exclusion spheres (gray). The disconnected pockets of space that lie outside of the generally overlapping exclusion spheres are termed cavities (cross-hatched). A natural choice for the effective exclusion radius for the Lennard-Jones fluid is r.v = ct, the Lennard-Jones diameter.

Figure 9.5 Molecular dynamics calculations of the nondimcnsional surface pressure (difference between pressure inside a drop and the gas) for a Lennard-Jones intermolecular potential. Classical liquid drop theory begins to break down fordroplei radii smallcrihan about 10 times the Lennard-Jones diameter CTij. Calculations for Ar/su = 0.71 and = 0,58. (After Thompson et al, 1984.)...

Because zeolites have highly uniform pore sizes with dimensions comparable to those of small to medium-sized molecules, they exhibit a phenomenon known as shape selectivity. To help put this idea in more concrete terms Table 10.3 lists some Lennard-Jones diameters of various molecules. Since zeolites possess uniform pore sizes small changes in the size of a molecule can mean it is too large to either adsorb into the pores of the material or diffuse through the zeolite pores. 

Here r, e, and (T,y are the internuclear distance, the dispersion well depth, and the Lennard—Jones diameter, respectively. The 12th power term describes the repulsive interaction, whereas the 6th power term represents the attractive term. Nonbonded interactions are calculated between atoms that are three or more atoms apart. 

At petrochemical plants there are numerous gas streams that contain valuable components which need to be recovered and reused. These are typically non-reacted monomers, by-products from reactors, inerts, solvents and carrier gas. There is a nice potential for using CMS membranes for many of these applications, and thereby also save money if complicated systems with columns, refrigeration and compressors can be avoided. A study on separation of alkanes-alkenes was performed by Hagg et alP Their systems were the separation of propane-propene and propan-ethene. As the alkanes-alkenes are chemically and physically quite similar compounds with almost identical critical properties, they must be separated on the basis of their molecular size. The Lennard-Jones diameter is 4.7 A and 5.1 A for propene and propane, respectively hence, a carefully tailored CMS membrane would be able to separate these two components according to the molecular sieving mechanism. A selectivity of 23 for this gas pair was documented at 30 °C, and even much higher selectivity at 50 °C this is believed to be a result of a transition of separation mechanisms for propane at lower temperature propane will... 

Lennard-Jones diameter of component i characteristic time of diffusion out of phase j characteristic time of propagation in phase j characteristic volume of component i ratio between characteristic times of termination and interphase mass transport of a chain of length x in phase J... 

Fig. 2.36 A sample of water-like particles in two dimensions. The circles indicate the Lennard-Jones diameter of the particles. The arrows attached to each particle are unit vectors along which a hydrogen bond may be formed. See Sec. 2,6.2.

Lennard-Jones diameter of methane, au = 3.82 A. If m solutes of diameter [Pg.491]

It is known that glassy polymer membranes can have a considerable size-sieving character, reflected mainly in the diffusive term of the transport equation. Many studies have therefore attempted to correlate the diffusion coefficient and the membrane permeability with the size of the penetrant molecules, for instance expressed in terms of the kinetic diameter, Lennard-Jones diameter or critical volume [40]. Since the transport takes place through the available free volume in the material, a correlation between the free volume fraction and transport properties should also exist. Through the years, authors have proposed different equations to correlate transport and FFV, starting with the historical model of Cohen and Turnbull for self diffusion [41], later adapted by Fujita for polymer systans [42]. Park and Paul adopted a somewhat simpler form of this equation to correlate the permeability coefficient with fractional free volume [43] ... 

The correlation function equals -1 as long as r is smaller than the diameter of the hard core, or if it is well within the Lennard-Jones diameter a. It falls off to 0 for large r. Fig. 6a shows the behavior of the coirelation function in supercritical Ixnnard-Jonesium, at f = kTIz = 2, about 30% above the critical point s is the Lennard-Jones well-depth and k Boltzmann s constant. It was calculated by De Boer in 1949, and is reported in [9]. 

Fig. 35 Snapshot pictures of two spherical polymer brushes with 92 chains, with number of effective subunits N = 40 grafted to a nanoparticle with radius = 1.9 Lennard-Jones diameters...

Of course, if the box is very small, e. g. L 3 coex [4], then no phase separation within a box is possible at all, and then the state of the system is homogeneous irrespective of density. Thus, the mean field free energy density/ (c) discussed above can be given a well-defined meaning when we re-interpret it as a coarse-grained free energy density flic) of a coarse-graining cell which has a linear dimension L of the order of coex or smaller. Note, however, that coex is typically only of the order of a Lennard-Jones diameter, except in the immediate vicinity of the critical point. 

The solvation of an HS solvaton in a real fluid is important in the study of the solvation thermodynamic of any real solvaton. The solvation process of a real particle may be performed in two steps first, the creation of a suitable cavity, then turning on the other parts of the interactions between the solvaton and the (real) solvent. In order to define the size of the cavity we need to assign an effective hard-core diameter to the solvent molecules. For simple spherical molecules, such as the noble gases, a natural choice of the effective diameter might be the van der Waals or Lennard-Jones diameter of the molecules. For more complex molecules there is no universal way of defining an effective HS diameter to be assigned to the solvent molecules. 

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