Radial distribution function thermodynamic properties from

Considerable evidence exits of the survival of Zintl ions in the liquid alloy. Neutron diffraction measurements [5], as well as molecular dynamics simulations [6, 7], give structure factors and radial distribution functions in agreement with the existence of a superstructure which has many features in common with a disordered network of tetrahedra. Resistivity plots against Pb concentration [8] show sharp maxima at 50% Pb in K-Pb, Rb-Pb and Cs-Pb. However, for Li-Pb and Na-Pb the maximum occurs at 20% Pb, and an additional shoulder appears at 50% Pb for Na-Pb. This means that Zintl ion formation is a well-established process in the K, Rb and Cs cases, whereas in the Li-Pb liquid alloy only Li4Pb units (octet complex) seem to be formed. The Na-Pb alloy is then a transition case, showing coexistence of Na4Pb clusters and (Pb4)4- ions and the predominance of each one of them near the appropiate stoichiometric composition. Measurements of other physical properties like density, specific heat, and thermodynamic stability show similar features (peaks) as a function of composition, and support also the change of stoichiometry from the octet complex to the Zintl clusters between Li-Pb and K-Pb [8]. 

The starting point is an expression for the intermolecular potential energy, Ul, for two solute particles, i and j, distance r apart in solution. From this expression it is theoretically possible to calculate the thermodynamic properties of the solution. The quantitative link is provided by the radial distribution function, g(r), which provides information concerning the distribution of particles in solution. 

The radial distribution function plays an important role in the study of liquid systems. In the first place, g(r) is a physical quantity that can be determined experimentally by a number of techniques, for instance X-ray and neutron scattering (for atomic and molecular fluids), light scattering and imaging techniques (in the case of colloidal liquids and other complex fluids). Second, g(r) can also be determined from theoretical approximations and from computer simulations if the pair interparticle potential is known. Third, from the knowledge of g(r) and of the interparticle interactions, the thermodynamic properties of the system can be obtained. These three aspects are discussed in more detail in the following sections. In addition, let us mention that the static structure is also important in determining physical quantities such as the dynamic an other transport properties. Some theoretical approaches for those quantities use as an input precisely this structural information of the system [15-17,30,31]. 

We conclude that the proximal radial distribution function (Fig. 1.11) provides an effective deblurring of this interfacial profile (Fig. 1.9), and the deblurred structure is similar to that structure known from small molecule hydration results. The subtle differences of the ( ) for carbon-(water)hydrogen exhibited in Fig. 1.11 suggest how the thermodynamic properties of this interface, fully addressed, can differ from those obtained by simple analogy from a small molecular solute like methane such distinctions should be kept in mind together to form a correct physical understanding of these systems. 

The Kirkwood—Buff (KB) theory of solution (often called fluctuation theory) employs the grand canonical ensemble to relate macroscopic properties, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volnmes, to microscopic properties in the form of spatial integrals involving the radial distribution function. This theory allows one to obtain information regarding some microscopic characteristics of mnlti-component mixtures from measurable macroscopic thermodynamic quantities. However, despite its attractiveness, the KB theory was rarely used in the first three decades after its publication for two main reasons (1) the lack of precise data (in particular regarding the composition dependence of the chemical potentials) and (2) the difficulty to interpret the results obtained. Only after Ben-Naim indicated how to calculate numerically the Kirkwood—Buff integrals (KBIs) for binary systems was this theory used more frequently. 

Another method suggested by the authors for predicting the solubility of gases and large molecules such as the proteins, drugs and other biomolecules in a mixed solvent is based on the Kirkwood-Buff theory of solutions [18]. This theory connects the macroscopic properties of solutions, such as the isothermal compressibility, the derivatives of the chemical potentials with respect to the concentration and the partial molar volumes to their microscopic characteristics in the form of spatial integrals involving the radial distribution function. This theory allowed one to extract some microscopic characteristics of mixtures from measurable thermodynamic quantities. The present authors employed the Kirkwood-Buff theory of solution to obtain expressions for the derivatives of the activity coefficients in ternary [19] and multicomponent [20] mixtures with respect to the mole fractions. These expressions for the derivatives of the activity coefficients were used to predict the solubilities of various solutes in aqueous mixed solvents, namely ... 

SCF-MI (Self Consistent Field for Molecular Interactions) and non orthogonal Cl were used to determine a water-water interaction potential, from which BSSE is excluded in an a priori fashion. The new potential has been employed in molecular dynamics simulation of liquid water at 25°C. The simulations were performed using MOTECC suite of programs. The results were compared with experimental data for water in the liquid phase, and good accordance was found, both in radial distribution functions and thermodynamic properties, as well as in geometric parameters. 

The conformational (non-ionic) free energy, obtained from the radial distribution function for non-ionic chains by Monte Carlo calculations, was used in conjunction with the electrostatic free energy to calculate the actual distribution function of the charged chain segments. The resulting expansion justifies almost quantitatively in many cases the experimental thermodynamic properties (such as pK, H i, etc.) and the dimensional properties (viscosity) of the ionic polysaccharides to which the approach has been applied. 

Two sets of methods for computer simulations of molecular fluids have been developed Monte Carlo (MC) and Molecular Dynamics (MD). In both cases the simulations are performed on a relatively small number of particles (atoms, ions, and/or molecules) of the order of 100simulation supercell. The interparticle interactions are represented by pair potentials, and it is generally assumed that the total potential energy of the system can be described as a sum of these pair interactions. Very large numbers of particle configurations are generated on a computer in both methods, and, with the help of statistical mechanics, many useful thermodynamic and structural properties of the fluid (pressure, temperature, internal energy, heat capacity, radial distribution functions, etc.) can then be directly calculated from this microscopic information about instantaneous atomic positions and velocities. 

Having determined the structure of the polymer liquid, it is in principle possible to compute most thermodynamic properties of interest. Whereas the structure or radial distribution functions at liquid density are primarily controlled by the repulsive part of the intersite potentials, thermodynamic quantities will also be sensitive to the attractive potentials. In the case of a one-component melt, thermodynamic quantities of interest include the pressure P, isothermal compressibility k, and the internal or cohesive energy U. Since in general one theoretically knows g(r) only approximately, the thermodynamic properties derived from structure will be approximate. Moreover, integral equation theory leads to thermodynamically inconsistent results in the sense that the predictions depend on the particular thermodynamic route used to relate the thermodynamic quantity to the structure. ... 

The results of BOSS calculations can be analyzed In ChemEdit. Some plots from a BOSS calculation are shown in Figure 6. Information extracted from a BOSS output includes (a) geometries of the solutes (b) radial distribution functions (c) energy and energy pair distributions (d) average thermodynamic properties (e) components of the solvent-solute energy and (f) AH, AS, and AG for perturbations. 

MD simulations provide the means to solve the equations of motion of the particles and output the desired physical quantities in the term of some microscopic information. In a MD simulation, one often wishes to explore the macroscopic properties of a system through the microscopic information. These conversions are performed on the basis of the statistical mechanics, which provide the rigorous mathematical expressions that relate macroscopic properties to the distribution and motion of the atoms and molecules of the N-body system. With MD simulations, one can study both thermodynamic properties and the time-dependent properties. Some quantities that are routinely calculated from a MD simulation include temperature, pressure, energy, the radial distribution function, the mean square displacement, the time correlation function, and so on (Allen and Tildesley 1989 Rapaport 2004). 

We will demonstrate next that the thermodynamic properties of a fluid can be determined from knowledge of the radial distribution function, g(r), and the pair potential, T(r). To this purpose we will develop expressions for its equation of state and internal energy. 

We conclude that if the assumption of pairwise additivity for the potential energy, Eq. 17.3.8, is valid and the radial distribution function is available, the thermodynamic properties of a fluid can be determined from Eqs 17.4.3 through 17.4.5. provided, of course, that the pair potential be known.)... 

Uncertainties in the prediction of the radial distribution function at high densities from knowledge of the pair potential do not allow for the direct evaluation of thermodynamic properties in such densities through Eqs 17.4.3 and 17.4.5. Three general approaches have been used to circumvent this problem ... 

Since the glass transition is characterized by discontinuous changes in second- (and higher) order thermodynamic properties, it would seem relevant to give some attention to what might be termed second-order structural properties. Just as n r), from which g(r) is derived, is the average number of particles within a sphere of radius r around a reference particle, so we can define moments about this mean distribution. Here we focus on the second moment, termed the radial fluctuation function W(r), and defined as follows ... 

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