Statistical Study of Random Structures on Digital Models

V. A. Vorob’ev, I. E. Naats, G. N. Papratov

Presented by the joint seminar of the sectors
of DSM and MED of the Scientific Research Institute of Electronic Technology

Most problems of radiation chemistry and defectoscopy deal with materials in whose structure there is no long-range order. To determine the interrelation of the structural features with the physical properties of such materials, it is necessary to know the parameters characterizing the structure. The principal ones are the radial function, the pair distribution function, and the coordination number [1].

In determining these structural parameters, physical models of various types prove useful [2, 3].

Physical models of random structures are bulky engineering constructions [3]; therefore their possibilities in statistical investigation are limited.

More promising in this respect are mathematical models [4, 5], which allow modern digital computers to be used for their statistical investigation.

In the present work, a statistical study of random structures is carried out, using as an example single-atom fluids on the M-20 digital computer, with the aid of their mathematical models.

In [5], two mathematical models of fluids are described, representing molecules in the form of random collections of nonintersecting spheres in a bounded space with filling density \(0.32 \ldots 0.61\). For these models, we compute the pair distribution function \(g_i(R_j)\) of the centers of spheres located at a certain distance from one another, and the radial function \(\rho_i(R_j)\).

To determine the radial function, consider a sphere with center at point “\(O\)” and a volume element \(dV\) at distance \(R\) from

“0”, then the product

\[ \rho_i(R_j)\,dV \]

is the probability that the center of another sphere lies in \(dV\).

Analogously, for an ensemble of \(N\) spheres the pair function \(g_i(R_j)\) determines the probability that two elements \(dV_1\) and \(dV_2\), containing the centers of spheres, are separated by a distance \(R\); more precisely, the product

\[ N g_i(R_j)\,dV_1\,dV_2 \]

is the mutual probability of finding a pair of spheres separated by a distance \(R\). Here \(N\) is the number of spheres per unit volume.

The functions were computed by the method of statistical tests on the M-20 digital computer according to the following algorithms for both models. To compute the radial functions in the central region of the packing (in order to reduce the influence of the boundaries [6]), the center of an arbitrarily chosen sphere was taken as the origin, and the spheres located from it at a distance \(R_i + 0.005\) were counted. The computations were repeated for ten arbitrary centers, and for each interval the average number of spheres \(\overline{N_i}\) was found. The mean value of the radial function for each interval was computed by the formula

\[ \rho(R_i)=\overline{N_i}\,V_{\mathrm{sh}}/V_i . \]

Here

\[ \begin{aligned} V_i &\text{ is the volume of the } i\text{-th sphere,}\\ V_{\mathrm{sh}} &\text{ is the volume of a ball,}\\ \overline{N_i} &\text{ is the average number of centers in the } i\text{-th sphere.} \end{aligned} \]

The pair distribution functions were obtained for the central regions of the packings by direct enumeration; that is, pairs with identical distances between centers were counted. Thus, the total number of enumerations for a fixed distance was equal to

\[ 2N_{\mathrm{pairs}} = n(n-1), \]

where \(n\) is the number of spheres participating in the enumeration.

The function \(g_iR_j\) was found in the interval between \(R\) and \(R_i + 0.005\). The smallest distance between spheres is equal to the sphere diameter \(0.1\), and the largest is the diameter of the inner region, equal to \(0.6\). The results of the computations are averages over ten realizations with a standard deviation of 3%. The computation time was 20 min.

From analysis of the pair distribution functions (Fig. 3) it is seen that pairs with distances between centers in the intervals \(0.2\text{–}0.3\) for the dense-packing model and \(0.25\text{–}0.45\) for the sparse-packing model occur most often.

Fig. 1. Dependence
\(a\) — for the first model
\(b\) — taken from work 4
\(R\) — measured in sphere diameters

Fig. 2. Dependence
Second model
\(R\) — in sphere diameters

Fig. 3. Distribution function of pairs.
\(a\) — first model
\(b\) — second model
\(R\) — distance in sphere diameters
\(N_n\) — average number of pairs.

The radial functions (Figs. 1, 2) agree with the results obtained experimentally on physical models and from diffraction patterns [3, 7].

Investigation of the second model (density 0.61) yielded a new radial function with a sharper peak in the interval 0.1–0.11 (Fig. 2), which is closer to the radial function of liquids than in the first model (density 0.32). From the first interval 0.1–0.11, the number of nearest neighbors was calculated to be 8.0, which corresponds to the value obtained by direct counting of the number of contacts [5]. This result agrees well with the coordination numbers obtained in [3, 7] for liquid helium, neon, and argon. They are, respectively, 8.5; 8.8; 8.0.

Thus, the algorithm presented above for the statistical study of random structures makes it possible, on a computer, to obtain comparatively quickly (20 min.), with an accuracy of not less than 3%, the radial function and the distribution function of pairs of particles.

References

1. Bernal J. D., Nature 183, 141 (1959).

2. Scott G. D., Nature 188, 908 (1960).

3. Bernal J. D., Mason J., Nature 188, 910 (1960).

4. Vorob’eva V. A., Kindrar V. K., Naats I. E. “Digital modeling of random packings of spheres of different diameters with low packing density.” Collection of TPI, 1969.

5. Naats I. E., Parvaten G. N. “Digital models of fluid structures.” Present collection.

6. Naats I. E., Parvaten G. N. “On estimating density in digital modeling.” Collection of TPI, 1969.

7. Furukawa H., Nature 184, 1208 (1959).