Method for Calculating Random Structures of Composite Materials

B. A. Vorob’ev, V. I. Ses’

Presented by the joint seminar of the sectors
[[unclear: DSI]] and [[unclear: IRD]] of the [[unclear: NII EI]]

At various stages of work on the study of composite materials, the need arises to know the statistical characteristics of their internal structure. The dispersed phase—these may be molecules, atoms, grains of sand or gravel—has particles of random shape and size, and their arrangement is irregular and random. The principal difficulty in the theoretical investigation of such structures arises from the fact that the particles are impermeable closed bodies; hence it is impossible to expect independence in their positions relative to one another. This obvious fact is so substantial that, up to now, no satisfactory definition has been found for random dense packings that would withstand strict logical criticism and at the same time agree with our practical notions. Quite often such structures are illustrated by spheres, but the characteristics of the structures are determined by semiempirical formulas.

In the present work it is proposed, in order to obtain numerical characteristics of the internal structure, to use a probabilistic-geometrical model of the spatial arrangement of structure-forming elements, with subsequent computation of this model on a digital computer.

A convenient model for many structures may be a random packing of unequal spheres, with a distribution of spheres by size analogous to the distribution of particle sizes. To facilitate the computation of the model, we assume that a limiting dense packing can be obtained. For an explanation, let us turn to Fig. 1.

Fig. 1

Fig. 2

Figure 1 schematically shows a fragment of the packing around the sphere designated 1; all spheres 2–9 touch sphere 1. There are several groups in which each sphere touches three others (1, 2, 3, 4; 1, 3, 4, 9; 1, 4, 9, 8; ...). Between spheres 4–5 and 6–8 there are no contacts; their approach is prevented by previously packed spheres. In an extremely dense packing, such gaps between nearest neighbors are absent; then, by connecting with segments the centers of tangent spheres, the entire space of the packing will be filled by a system of tetrahedra whose vertices lie at the centers of four spheres. Each sphere belonging to a given tetrahedron will touch the other three.

In this case, to obtain the geometric characteristics of the packing, one can use similar characteristics obtained for systems of tetrahedra. Figure 2 shows an example of a tetrahedron formed by spheres 1, 2, 3, 4. In a tetrahedron the edges are known, being equal to the sum of the contacting spheres; therefore it is not difficult to determine its volume. By studying the triangles that are the faces of the tetrahedra, using the formulas [1] one can find all the solid angles that measure the part of the area of the sphere enclosed in the tetrahedron, or part of the volume, since solid angles are central. The ratio of the sum of the volumes cut out by the tetrahedron from the spheres to the volume of the tetrahedron determines its geometric density. The ratio of the sum of the areas of the parts of the spheres enclosed in the tetrahedron to the volume of the tetrahedron gives the specific surface.

The smallest cross section of the pore, an important parameter in filtration theory, is determined by the difference between the areas of the face and the sectors, and the largest cross section can be determined from the formula [2]

\[ \frac{1}{\chi_0} = \frac{1}{2} \left\{ \left( \frac{1}{\chi_1} + \frac{1}{\chi_2} + \frac{1}{\chi_3} + \frac{1}{\chi_4} \right) + (3P)^{1/3} \right\}, \tag{1} \]

where \(\chi_0\) is the radius of the fifth sphere in contact with the given \(\chi_1, \chi_2, \chi_3, \chi_4\), which form the tetrahedron.

If the solid angle cut out by the \(j\)-th tetrahedron from the \(i\)-th sphere is denoted by \(A_{ij}\), then the number of the \(j\)-th tetrahedra containing the \(i\)-th sphere will be equal to

\[ N_{ij}=\frac{1}{A_{ij}}. \tag{2} \]

The number of neighbors for a sphere will be equal to [2]

\[ T=\frac{1}{2}N+2. \tag{3} \]

In addition to the foregoing, a number of other characteristics can be estimated, depending on the purpose of the model.

Thus, all possible combinations of spheres forming different tetrahedra are enumerated; the characteristics of interest are calculated for each type and averaged with allowance for the frequency of occurrence of the given type in the system. In [2] a method is described for determining the relative frequency of a given type of tetrahedron in a system, which consists in the following. Let the packing consist of spheres of three sizes \(r_1\), \(r_2\), and \(r_3\) with relative frequencies \(f_1\), \(f_2\), and \(f_3\), respectively. Then the frequency of tetrahedra containing the \(i\)-th sphere will be proportional to the number of \(i\)-th spheres and to the average number of tetrahedra containing one \(i\)-th sphere [2]:

\[ F_1 : F_2 : F_3 = f_1 \overline{N}_1 : f_2 \overline{N}_2 : f_3 \overline{N}_3 . \tag{4} \]

The total number of tetrahedra is equal to:

\[ n = (F_1 + F_2 + F_3)^4 . \tag{5} \]

Expanding equation 5, we obtain the sum; each term contains four factors \(F\) with the corresponding indices \((F_1, F_2, F_3)\), denoting the radii of the spheres forming a tetrahedron whose frequency is equal to the given product.

The calculation is carried out by an iterative method. First, the frequencies expressed by 5 are assigned an arbitrary value, for example unity, and the tetrahedral system is calculated; the average solid angle \(A_i\) is determined for each sphere size. Then, from the known values of the sphere frequencies \(f_i\) (the initial data) and from the calculated values \(A_i\), with allowance for 2, refined values of \(F_i\) are determined, formula 4. The computations are repeated until the newly obtained value of \(F_i\) differs from the preceding one by the prescribed amount. The process converges rapidly, and it is usually possible to limit oneself to three cycles.

In order to assess the efficiency of the described method of calculating packings, the authors carried out a factorial experiment on steel balls, and then the same granulometric compositions were calculated on an M-20. From the experimental data and the calculated data, regression equations were constructed for the packing density

\[ \rho_M = 0.8046 + 0.0044X_1 - 0.0089X_2 - 0.0005X_3 , \]

\[ \rho_{\mathrm{sh}} = 0.5875 + 0.01X_1 - 0.01X_2 - 0.0035X_3 , \]

where \(\rho_m\), \(\rho_{\text{sh}}\) are the density of the model and of the packing of spheres; \(X_1\), \(X_2\), \(X_3\) are the relative frequencies of spheres with radii equal to 1, 2, and 3, respectively, in pieces.

The overestimated value of the density calculated is quite understandable; moreover, the density on the spheres was determined without compaction, since under vibration the small and large spheres should become arranged.

It is more advantageous to use the method described for cases in which what matters is not the absolute value of the characteristics, but the direction of their variation, that is, for solving extremal problems, such as finding the greatest density, the specific surface area, etc.

The merit of this calculation method is that the computations are carried out on a digital computer, and with three sphere sizes about 7 seconds of machine time are required. Comparison with experimental data shows that such a model is applicable for obtaining the characteristics of somewhat mixed, real packings.

Literature

1. Bronshtein I. N. and Semendyaev K. A. Handbook of Mathematics. Publishing House “Nauka,” Moscow, 1964.

2. M. J. Hogendijk, “Random dense packing of spheres with a discrete distribution of the radii,” Philips Res. Repts. 18, 119–126, 1963.