The Radial Distribution Function g r

We notice that if is the radius of molecule j, then because of the strong repulsive forces it is unlikely that there will be an other molecule, whose center lies in a sphere of radius equal to 2R.  

If we now move just beyond 2R, we will find more molecules (per unit volume) than the bulk average, because of the effect of attractive forces. A little farther out, however, this effect diminishes and the number of molecules present becomes equal to the bulk average. 

If we divide the local densities at various distances from molecule j by the bulk average p = NIV, we obtain the radial, or pair, distribution function, g r). In addition to r, g is a. strong function of density p, and a weak function of T  

Typically, however, we use the notation g(r) with an implied dependency on temperature and density. 

They can also be estimated theoretically if the intermolecular potential is known, but the accuracy declines with increasing density. 

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Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

Figure A2.3.7 The radial distribution function g r) of a Lemiard-Jones fluid representing argon at T = 0.72 and p = 0.844 detennined by computer simulations using the Lemiard-Jones potential.

Microscopic theory yields an exact relation between the integral of the radial distribution function g(r) and the compressibility... 

Distribution functions measure the (average) value of a property as a function of an independent variable. A typical example is the radial distribution function g r) which measmes the probability of finding a particle as a function of distance from a typical ... 

The second category of methods uses a more general approach, based on fundamental concepts in statistical mechanics of the liquid state. As mentioned above, the Hwang and Freed theory (138) and the work of Ayant et al. (139) allow for the presence of intermolecular forces by including in the formulation the radial distribution function, g(r), of the nuclear spins with respect to the electron spins. The radial distribution function is related to the effective interaction potential, V(r), or the potential of mean force, W(r), between the spin-carrying particles through the relation (138,139) ... 

A number of approximate integral equations for the radial distribution function g(r) of fluids have been proposed in recent years. Two particularly useful approximations are the Percus-Yevick (PY)1,2 and the Convolution Hypernetted Chain (CHNC)3-4 equations. In this paper an efficient numerical method of solving these equations is described and the results obtained bv applying the method to the PY equation are discussed. A later paper will describe the behavior of the... 

Modem developments are centered around the calculations of the radial distribution function g (r), which is the ratio of the densiiy of molecules al a distance r from a given molecule, to the average density in the gas. The compressibility can be expressed straightforwardly in terms of g (r) as follows. 

The MD simulations show that second shell water molecules exist and are distinct from freely diffusing bulk water. Freed s analytical force-free model can only be applied to water molecules without interacting force relative to the Gd-complex, it should therefore be restricted to water molecules without hydrogen bonds formed. Freed s general model [91,92] allows the calculation of NMRD profiles if the radial distribution function g(r) is known and if the fluctuation of the water-proton - Gd vector can be described by a translational motion. The potential of mean force in Eq. 24 is obtained from U(r) = -kBT In [g(r)] and the spectral density functions have to be calculated numerically [91,97]. 

The total nonbonded contribution to the stress ty is then < " = ]C/ (T y>(P) which is the sum of er"fc(/j) over all atoms /i that engage in nonbonded interactions. This sum may be written in an integral form by use of the radial distribution function g(r), where... 

The effective potential U was calculated from the radial distribution function (g(r)) using the expression... 

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. 

For colloidal liquids, Eqs. (19-21) refer to the excess energy [second term of the right-hand side of Eq. (19)], the osmotic pressure and osmotic compressibility, respectively. They show one of the important features of the radial distribution function g(r), namely, that this quantity bridges the (structural) properties of the system at the mesoscopic scale with its macroscopic (thermodynamic) properties. 

Having defined the different Interactions occurlng In [3.6.1], we now need to specify the probability of finding an Ion a at some position r. The one-particle (singlet) density p fr jls defined In sec. I.3.9d as the number of particles per volume at position r. Now we apply the definition to Ions. The radial distribution function g (r)and the ion-wall total correlation function h (r) follow from (1.3.9.22 and 23] as... 

For atomic liquids with only spherically symmetric interactions, the pair distribution function will contain no angular dependence and hence the structure in the system (at the pairwise level) is completely given by the radial distribution function, g(r), where r=lrl is simply the magnitude of the separation vector. For a molecular system, the radial distribution function is obtained from the full angle average of the pair distribution function. 

The radial distribution function, g(R), and the mean-square fluctuation, MSF, were obtained using equations (4) and (5) [39] ... 

A snapshot of an equilibrated conformation is shown in Fig. 3. The radial distribution functions (g(r)) between the various types of oxygen atoms in the PGA chain are shovra in Fig. 4. 

Step 3 The PS particles are removed by an O2 plasma treatment (2 min at 250 W, and SOOmTorr). In Chemical Properties we present XPS results, which describe the effect of O2 plasma treatment regarding removal of the PS particles by O2 plasma and oxidation of Pt. It should be noted that it is not possible to dissolve the PS particles in acetone after the Ar+ etching process, which is believed to be due to ion-induced crosslinking of the polymer chains during ion etching (111), making them resistant to normal solvents for PS. The radial distribution function, g r), from the initial colloidal adsorption step is preserved throughout the nanofabrication procedure. 

Thus, integration of equation (2.8.9) with respect to /T yields the Helmholtz energy A. This in turn requires that the radial distribution function g r) be known as a function of temperature. Since g r) is generally not available as a function of temperature, equation (2.8.15) is not a convenient route to obtain A. On the basis of equation (1.3.19), one obtains the relationship... 

This definition simplifes considerably in the case of an isotropic liquid for which the single particle density is a constant (i.e. p(r) = N/V), resulting in the radial distribution function g(r). As said above, these definitions will form a part of our later thinking. 

Here, S(Q) is the so-called structure factor which takes into account the particle-particle interaction. 5(0 is related to the radial distribution function g(r) (which gives the number of particles in shells surrounding a central particle) [12],... 

The theoretical analysis is similar to the case of an isolated polyion as outlined in the previous section. Here the three work components are added, and the derivative of the sum with respect to 9 r) at fixed r is set to zero. The In cs and constant terms are isolated and independently set to zero, producing formulas for both 0(r) and Q(r). Substituted back into the expression for the overall work (with reference state at infinity), these optimized functions generate the pair potential of mean force w(r). The radial distribution function g(r) is given by cxp w(r) (the pair potential is in units of kBT). 

Plots of the pair potentials w(r) exhibiting inverted forces may be seen in our other publications. The counterion-polyion potential is attractive in the near and far regions but inverted (repulsive) in the intermediate region. For like-charged polyions, the polyion-polyion potential is repulsive in the near and far regions but attractive for intermediate distances. Here we show graphs of the radial distribution functions g(r) = exp[—w(r)]. In Figure 5... 

The radial distribution function, g(r), can be determined experimentally from X-ray diffraction patterns. Liquids scatter X-rays so that the scattered X-ray intensity is a function of angle, which shows broad maximum peaks, in contrast to the sharp maximum peaks obtained from solids. Then, g(r) can be extracted from these diffuse diffraction patterns. In Equation (273) there is an enhanced probability due to g(r) > 1 for the first shell around the specified molecule at r = o, and a minimum probability, g(r) < 1 between the first and the second shells at r = 1.5cr. Other maximum probabilities are seen at r = 2(7, r = 3 o, and so on. Since there is a lack of long-range order in liquids, g(r) approaches 1, as r approaches infinity. For a liquid that obeys the Lennard-Jones attraction-repulsion equation (Equation (97) in Section 2.7.3), a maximum value of g(r) = 3 is found for a distance of r = <7. If r < cr, then g(r) rapidly goes to zero, as a result of intermolecular Pauli repulsion. 

Kassel and Muskat1 were the first to attempt to calculate the surface energy after the quantum-mechanical theory of intermole-cular potential was completed in 1930. They assumed the distribution function of liquid in two alternative ways. First, they assumed that the radial distribution function g(R) is given by... 

Other melting rules are formulated in terms of properties of the saturated liquid phase, focusing in particular on some feature of the structure as quantified by the radial distribution function g r). Ideally, these rules could be applied to predict the freezing transition of a molecular model using a correlation function given by (say) an integral equation theory for g r). 

The radial distribution function g (r) calculated from a (finite size) RMC model is a section of the complete radial distribution function gc(r), multiplied by a step function... 

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