Digital Models of Fluid Structures

Nead I. E., Parvatov G. N.

Prepared by the joint seminar of the sectors
[[unclear: abbreviations]] and [[unclear: abbreviations]]

The results of works \([1, 2, 3]\) make it possible to regard fluids as essentially irregular ensembles of molecules, whose mutual arrangement constantly changes while the average distance between them is preserved. The structures of such ensembles should be considered from the viewpoint of probability geometry. In this case, models prove to be random packings of equal rigid spheres. The basic parameters characterizing the packing of equal spheres are the coordination number, the density, and the radial distribution function \([4]\).

In the present work, two mathematical models of fluid structures are proposed, represented in the form of random sets of equal rigid spheres filling a certain selected volume of three-dimensional space.

The models are realized by means of algorithms for random filling of a bounded space with equal spheres. The nature of the filling is determined by the properties of the fluid: by the distribution of molecules in space and by their mutual connections.

The first model represents a random structure with weak molecular bonds and a uniform distribution of molecules in the volume; the second, one with sufficiently strong bonds.

In the general case, both models describe the processes of packing a bounded space with equal spheres under one or another condition. The coordinates of the centers of the packed spheres are represented by random numbers \(q_1, \ldots, q_s\) from an \(S\)-dimensional space with \(0 < |q_i| < \infty\), where \(i = 1, \ldots, S\).

For the first model, points \(q_1^0, \ldots, q_s^0\) are selected from this space, satisfying the condition \([i]\) \(a < |q_i^0| < b\), in such a way that the distance between the centers of the spheres is not ...

can be defined:

\[ \sum_{i=1}^{S} \left(q_i^{0}-q_i^{1}\right)^{\beta} \ge d \tag{2} \]

This condition physically means that molecules can approach only until they touch one another. Further approach is prevented by the forces of mutual repulsion.

The second model is intended for simple monatomic liquids (of the liquid-gas type, except helium) with sufficiently strong molecular bonds. In this case the molecules have no more than four contacts on one side, thereby forming a stable structure [4]. Here the molecules are represented by rigid solid spheres. Consequently, the second model differs from the first by an additional condition: inequality (2) must be satisfied, with obligatory satisfaction in three-dimensional space \((S = 3)\), by the coordinates of the packed sphere \(x y z\) of the system of equations:

\[ \left\{ \begin{aligned} &\left(x-\frac{x_1+x_2}{2}\right)(x_2-x_1) +\left(y-\frac{y_1+y_2}{2}\right)(y_2-y_1) +\left(t-\frac{t_1+t_2}{2}\right)(t_2-t_1)=0,\\ &\left(x-\frac{x_1+x_3}{2}\right)(x_3-x_1) +\left(y-\frac{y_1+y_3}{2}\right)(y_3-y_1) +\left(t-\frac{t_1+t_3}{2}\right)(t_3-t_1)=0,\\ &(x-x_3)^2+(y-y_3)^2+(t-t_3)^2=\ell . \end{aligned} \right. \tag{3} \]

where \(\ell\) is the diameter of the spheres.

This thereby carries out the random dropping of the next sphere into the hollow between three others until it touches them.

The models were implemented on the M-20 digital computer for a sphere of diameter \(D = 1\). The coordinate centers of the packed spheres of radius 0.05 were represented by triples of \(2^m\) random numbers \(R_i, \theta_i, \Psi_i\), with a uniform distribution over the intervals \([0,5]\), \([0,\pi]\), \([0,2\pi]\), respectively. From these intervals numbers were chosen at random in such a way that \(0 \le |R_i| < 0.5\) for the first model and \(0 \le |R_i| < 0.4\) for the second. The distances between centers had to be no less than \(\ell\), the diameter of the sphere; for the second model, moreover, the mixed system of equations (3) had to be satisfied. The values of the density, the coordination number, and the distribution function with respect to the number of contacts were computed by the method of statistical trials. The results are presented in the form of histograms in Fig. 1.

Fig. 1. Histograms of the number of writings.
\(a\) — first model
\(b\) — second model
\(c\) — histogram from work 4

The obtained density values, 0.32 and 0.61 for the first and second models, respectively, are free from boundary effects. The influence of the boundaries was eliminated by the boundary-transfer method [5].

When calculating the distribution function of spheres by the number of contacts, spheres were considered to be in contact if the distance between them lay in the interval \(d,\ d+\Delta\). Here \(\Delta\) is the algorithm error and is equal to 1.5% of the sphere diameter.

The histograms show that the modes of the contacts lie in the intervals 0–1 for the first model (Fig. 1a) and 8–9 for the second (Fig. 1b). Accordingly, the mean numbers of contacts per sphere are 0.46 and 7.9. The results obtained agree well with analogous data obtained from X-ray diffraction on monatomic liquids [3] and with data obtained using physical models [4] (Fig. 1c). The method of mathematical modeling has certain advantages over physical models. The main one is the speed with which the final result is obtained: an experiment on the M-20 digital computer takes from 2 to 60 min, depending on the complexity of the model. As a result, mathematical models of simple monatomic liquids can apparently be recommended for the study of corresponding structures using a digital computer.

References

1. Stewart G. W. Rev. Mod. Phys. 2, 116 (1941)

2. Prins J. A. Physica III, 3, 147 (1936)

3. Furukawa K. Nature 184, 1208 (1959)

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

5. Naad I. E., Parvatov G. N. “On the determination of the structural properties of random packings.” Collection of TPI, 1969.