Calculation of the Coordination Number of Random Structures

V. I. Ses, A. B. Chigorko

(Presented by the scientific seminar of the Scientific Research Institute of Electron Introscopy)

In works [1, 2] a connection is shown between the physicomechanical properties of composite materials and the structural characteristics of the filler, and the use of a model for finding estimates of these characteristics is substantiated. As the model, a packing of rigid unequal spheres with a prescribed size distribution is proposed.

In the present work a method is proposed for finding the coordination number of the packing described above, in which the size distribution of the spheres is specified by the frequency \((f_i)\) of spheres with radius \(\rho_i\).

The method is based on the assumption that the coordination number for a sphere of radius \(\rho_i\) depends essentially on the function \(\psi(\rho_j \rho_i^{-1})\), which is equal to the possible number of neighbors of a sphere of radius \(\rho_i\) when it is covered by spheres of radius \(\rho_j\). The function \(\psi(\rho_j \rho_i^{-1})\) is discrete and depends only on the ratio of the radii \(\rho_j\) to \(\rho_i\). The coordination number for a sphere of radius \(\rho_i\) is determined by random combinations of mutually contacting spheres.

The expected frequency of contact of the \(i\)-th sphere with the \(j\)-th is proportional to the relative frequency of spheres \(f_j f_i^{-1}\) in the packing and to the quantity \((\rho_i+\rho_j)^2\), and the expression for finding \(\langle \psi_i\rangle\) as the weighted mean value of the function \(\psi(\rho_j \rho_i^{-1})\) is written in the following form:

\[ \langle \psi_i\rangle = \frac{\sum f_j f_i^{-1}(\rho_i+\rho_j)^2 \psi(\rho_j \rho_i^{-1})} {\sum f_j f_i^{-1}(\rho_i+\rho_j)^2}. \tag{1} \]

Expression (1) contains the function \(\psi(\rho_j \rho_i^{-1})\), which has been found exactly for six values of \((\rho_j \rho_i^{-1})\).

The given values of \(\psi(\rho_j \rho_i^{-1})\) are limiting, and for real packings they are somewhat smaller; moreover, the function is unknown for intermediate values of the argument, and therefore we propose to find the function by the method of digital simulation. On a digital computer one obtains a random packing of spheres of radius \(\rho_j\) on the surface of the \(i\)-th sphere. The function \(\psi(\rho_j \rho_i^{-1})\) is numerically equal to the number of packed spheres. From the center of the \(i\)-th sphere the touching spheres are seen under a certain solid angle, which cuts a circle out of the surface of the \(i\)-th sphere. Thus, the problem of finding the function \(\psi(\rho_j \rho_i^{-1})\) reduces to obtaining a random packing of circles on a sphere.

The packing-generation algorithm provides for a random-number generator to draw the coordinates of the center of the next circle to be placed, and a procedure for checking this circle for intersections with those already placed. The packing is considered complete if the number of consecutive failures reaches a prescribed value. An intersection of the trial circle with those already placed is considered a failure.

The appendix gives a program for finding the function $\psi$. Below are the values of $\psi$ computed on a digital computer for $\rho=1, 2, 3$.

The values of $\psi$ were found from several realizations and therefore turned out to be fractional. Table 1 gives the results of a $2^3$-factor experiment for the packing of steel balls embedded in paraffin; the granulometric composition in the experiments was selected in accordance with the planning matrix; the coordination number was determined by direct counting for randomly selected balls. The radii of the steel balls in the three-fraction packing are equal to 6.5; 9.9; 14.4. In the second column of the table, the fourth row indicates the coordination number, and below it the number of balls in the sample with $\rho_1=6.5$ and with the corresponding coordination number. In the third column the coordination number of the sample is given, and in the fourth column the calculated value $\langle\psi_1\rangle$, obtained from formula (1) taking into account the tabulated values of the function $\psi(\rho_j\rho_i^{-1})$.

The calculated values $\langle\psi_{6.5}\rangle$ deviate from the experimental ones by no more than 11%. The greatest deviation is observed in the first experiment for a sample of 25 balls with a standard deviation of 1.1. The confidence estimate in accordance with the expression $|a-\bar{x}|<t(P(k))\dfrac{s}{\sqrt{n}}$ and with probability 0.99 will be

\[ \varepsilon = 2{,}787\,\frac{1{,}1}{\sqrt{25}}\simeq 0{,}61 . \]

The greatest deviation, equal to 0.5, does not exceed the confidence estimate.

Table 1

The statistically insignificant deviation of the experimental results from the calculated ones confirms the validity of the estimate

coordination number by the proposed method. And although the estimates are somewhat biased, the bias does not exceed the confidence interval. Thus, the proposed method can be used to find absolute values of coordination numbers as a function of the granulometric composition of the packing.

REFERENCES

1. V. I. Soroker, V. I. Galaktionov. Selection of optimal mixtures of fractionated aggregates for concrete at reinforced-concrete product plants. Izvestiya vuzov, “Construction and Architecture,” Novosibirsk, 1966, No. 5.

2. V. A. Vorob’ev, I. E. Naats, V. I. Ses’. Calculation of the granulometric composition of concrete aggregates. Izvestiya vuzov, “Construction and Architecture,” Novosibirsk, 1970, No. 10.