Pair correlation function packing

Fig. 8 Monomer-monomer (gmmir)) pair correlation function for the packing fractions 77=1(T3, 10-4, and 10-5 (left to right) of rod-like (...) and flexible chains (—). The chain length is N=63 and the Bjerrum length ZB=0.5 b...

Fig. 2.8 The pair correlation function g r) for a fluid composed of hard spheres at a packing fraction of = 0.49 calculated as a function of distance, r, using the Ornstein-Zernike equation with the direct correlation function given by equations (2.6.6) and (2.6.10). The data shown as ( ) are from a Monte Carlo calculation.

We now turn to discuss some features of the pair correlation functions that are typical to mixtures of two (or more) components. We have seen in section 2.5 that for spherical particles, the pair correlation has peaks at roughly a, 2a, 3a, etc., where a is the effective diameter of the particles. However, it is not exactly at multiples of a, first because the minimum of the pair potential is at 2s a and not at a, and second because of the randomness of the packing of... 

In most of the illustrations for this book, we calculated the pair correlation functions at volume densities of 0.4 and 0.45. For hard spheres Yau et al. (1999) reported calculations of PYup to t] = 0.52. Even at these, relatively far from the close-packing densities, the convergence of the solutions is very slow and requires up to 1000 iterations. 

Figure 60. Pair correlation function for dense random packings of hard disks. The inset shows an expanded view.

In Fig. 2.26b we show the pair correlation function for the primitive model for water (see Fig. 2.10d). Here, we see a strong peak at R = 2 corresponding to the HBing distance. The height of this peak diminishes as we increase the density while a new peak at R = 1 develops. At p = 0.6, the peak at R = 1 is even higher than the one at R = 2. Clearly, at this density the average intermolecular distance is smaller than R = 2, and hence the particles are more closely packed. 

There is one important conclusion that can be drawn from the study of the pair correlation function for 2-D water-like particles which is relevant to the study of liquid water. If strong directional forces or bonds are operative at some selected directions, then the correlation between the two positions of two particles is propagated mainly through a chain of bonds and less by the filling of space — a characteristic feature of the mode of packing of simple fluids. 

The pair correlation function, introduced in Section 2.4, conveys information on the mode of packing of the molecules in the liquid. This information is often referred to as representing a sort of order, or amount of structure that persists in the liquid. Moreover, the pair correlation function forms an important bridge between molecular properties and thermodynamic quantities. This topic is discussed in Chapter 3. 

In Chapter 2, we outlined the information on the local mode of packing conveyed by the pair correlation function. Similarly, new and complementary information is furnished via the GMDF s, which seem to be of particular value in the study of complex fluids such as water. 

Perhaps some of the most important information on the mode of molecular packing of water in the liquid state is contained in the radial distribution function, which, in principle, can be obtained by processing X-ray or neutron scattering data. There are, however, several difficulties in extracting the proper information from the experimental data. First, it should be kept in mind that the full orientation-dependent pair correlation function cannot be obtained from such an experiment. Instead, only information on the spatial pair correlation function is accessible. We recall the definition of this function,... 

The pair correlation function, introduced in section 5.2.2, conveys information on the mode of packing of the molecules in the liquid. This information is often referred to... 

Packing effects 94 Pair correlation function 60 Pair potential, effective 243 Partial structure factor 220 Partition function 25, 27, 30, 65, 85 Pentamer 78... 

A. Donev, S. Torquato, and F. H. Stillinger. Pair correlation function characteristics of nearly jammed disordered and ordered hard-sphere packings. Phys. Rev. E, 71 011105, 2005. 

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