Temperature radial distribution

Figure 28. Low-temperature radial distribution functions for model C melt with A = 500 and/=0.5.

Flere g(r) = G(r) + 1 is called a radial distribution function, since n g(r) is the conditional probability that a particle will be found at fif there is another at tire origin. For strongly interacting systems, one can also introduce the potential of the mean force w(r) tln-ough the relation g(r) = exp(-pm(r)). Both g(r) and w(r) are also functions of temperature T and density n... 

Fig. 6.2 Radial distribution function determined from a lOOps molecular dynamics simulation of liquid argon at a temperature of 100K and a density of 1.396gcm. ...

One of the primary features of the Gay-Berne potential is the presence of anisotropic attractive forces which should allow the observation of thermally driven phase transitions and this has proved to be the case. Thus using the parametrisation proposed by Gay and Berne, Adams et al. [9] showed that GB(3.0, 5.0, 2, 1) exhibits both nematic and isotropic phases on varying the temperature at constant density. This was chosen to be close to the transitional density for hard ellipsoids with the same ellipticity indeed it is generally the case that to observe a nematic-isotropic transition for Gay-Berne mesogens the density should be set in this way. The long range orientational order of the phase was established from the non-zero values of the orientational correlation coefficient, G2(r), at large separations and the translational disorder was apparent from the radial distribution function. 

Fig. 5.2 Radial distribution curves, Pv

The radial distribution functions in ab initio simulations agree with experiment if the temperature is raised by roughly 50 K, consistent with results from the... 

After this computer experiment, a great number of papers followed. Some of them attempted to simulate with the ab-initio data the properties of the ion in solution at room temperature [76,77], others [78] attempted to determine, via Monte Carlo simulations, the free energy, enthalpy and entropy for the reaction (24). The discrepancy between experimental and simulated data was rationalized in terms of the inadequacy of a two-body potential to represent correctly the n-body system. In addition, the radial distribution function for the Li+(H20)6 cluster showed [78] only one maximum, pointing out that the six water molecules are in the first hydration shell of the ion. The Monte Carlo simulation [77] for the system Li+(H20)2oo predicted five water molecules in the first hydration shell. A subsequent MD simulation [79] of a system composed of one Li+ ion and 343 water molecules at T=298 K, with periodic boundary conditions, yielded... 

Ludwig s (2001) review discusses water clusters and water cluster models. One of the water clusters discussed by Ludwig is the icosahedral cluster developed by Chaplin (1999). A fluctuating network of water molecules, with local icosahedral symmetry, was proposed by Chaplin (1999) it contains, when complete, 280 fully hydrogen-bonded water molecules. This structure allows explanation of a number of the anomalous properties of water, including its temperature-density and pressure-viscosity behaviors, the radial distribution pattern, the change in water properties on supercooling, and the solvation properties of ions, hydrophobic molecules, carbohydrates, and macromolecules (Chaplin, 1999, 2001, 2004). 

It should be noted, however, that some of the tendencies described above may become invalid for very small droplets (for example, smaller than 10 pm under conditions in Ref. 156). Such small droplets may require a longer flight time to a given axial distance far from the atomizer due to the high deceleration, and their cooling rates may decrease as a result of the reduced relative velocity and temperature. In addition, the two-way coupling 576] may affect the momentum and heat transfer between atomization gas and droplets so that the droplet behavior may be different from that discussed above, particularly the radial distributions of droplet sizes and velocities. 

The radial distribution of oxygen atoms in H20(as) prepared at 10 K is qualitatively different from that in H20(as) prepared at 77 K. The high-temperature... 

Unlike premixed flames, which have a very narrow reaction zone, diffusion flames have a wider region over which the composition changes and chemical reactions can take place. Obviously, these changes are principally due to some interdiffusion of reactants and products. Hottel and Hawthorne [5] were the first to make detailed measurements of species distributions in a concentric laminar H2-air diffusion flame. Fig. 6.5 shows the type of results they obtained for a radial distribution at a height corresponding to a cross-section of the overventilated flame depicted in Fig. 6.2. Smyth et al. [2] made very detailed and accurate measurements of temperature and species variation across a Wolfhard-Parker burner in which methane was the fuel. Their results are shown in Figs. 6.6 and 6.7. 

The ordering of the anions in bmimX ionic liquids has also been suggested by our recent large-angle x-ray scattering experiment on liquid bmimi [23]. Figure 13 shows a differential radial distribution function obtained for liquid bmimi at room temperature. Clear peaks in the radial distribution curve are... 

The vapor sample under investigation may not eontain only one kind of speeies. It is desirable to learn as mueh as possible about the vapor composition from independent sources, but here the different experimental conditions need to be taken into account. For this reason, the vapor composition is yet another unknown to be determined in the electron diffraction analysis. Impurities may hinder the analysis in varying degrees depending on their own ability to scatter electrons and on the distribution of their own intemuclear distances. In case of a conformational equilibrium of, say, two conformers of the same molecule may make the analysis more difficult but the results more rewarding at the same time. The analysis of ethane-1,2-dithiol data collected at the temperature of 343 kelvin revealed the presence of 62% of the anti form and 38% of the gauche form as far as the S-C-C-S framework was concerned. The radial distributions calculated for a set of models and the experimental distribution in Figure 6 serve as illustration. 

Figure 6. Ethane-1,2-dithiol has a mixture of the anti (a) and gauche (g) forms at the experimental temperature of 343 K, with respect to the S-C-C-S framework. The top three curves were calculated for models, the bottom curve is the experimental radial distribution.

The radiation nozzle system has been used for studying a series of transition metal dihalide molecules. Typical molecular intensity distributions are shown in Fig. 4 for manganese(II) chloride. The quickly damping character of the intensity distribution relates to the large-amplitude motion in the molecule due to the high temperature ( 750 °C) conditions of the experiment. Fig. 5 shows the radial distribution from the same experiment which also well demonstrates the straightforward manner of structure determination of such simple molecules. 

Mass spectrometric studies on iron(II) chloride and iron(II) bromide showed that monomers are the major species in the vapor phase, while concentration of the dimer, Fe2X4, increases with temperature in its certain interval. The electron diffraction data on iron(II) chloride could be well approximated by monomers only, while the data on the bromide indicated the presence of a detectable amount of dimeric species. This can be seen on the radial distribution of Fig. 12. It was found that there was about 10% dimeric qjecies present under the experimental conditions (nozzle temperature around 625 °C). As regards the relative scattering power, this constituted about 20%, and allowed only the determination of a limited amount of structural information. The electron diffraction date were consistent with a bridge stmcture characterized by the same Fe-Br, bond length as that of the monomeric... 

One has to be able to image the exact position of each adsorbate to obtain the radial distribution function. This requires a low adsorbate mobility, since a full scan typically takes one minute to do. This method is therefore mainly applied to atoms on low-temperature surfaces. The consequence of keeping the temperature low to suppress surface mobility is that the absolute value of the repulsive interactions that can be determined is also small, typically less than 10 kJ/mol at room temperature. 

The calculations of g(r) and C(t) are performed for a variety of temperatures ranging from the very low temperatures where the atoms oscillate around the ground state minimum to temperatures where the average energy is above the dissociation limit and the cluster fragments. In the course of these calculations the students explore both the distinctions between solid-like and liquid-like behavior. Typical radial distribution functions and velocity autocorrelation functions are plotted in Figure 6 for a van der Waals cluster at two different temperatures. Evaluation of the structure in the radial distribution functions allows for discussion of the transition from solid-like to liquid-like behavior. The velocity autocorrelation function leads to insight into diffusion processes and into atomic motion in different systems as a function of temperature. 

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