CALCULATION OF THE PARTICLE-SIZE DISTRIBUTION OF PARTICLES TO ENSURE THEIR DENSEST PACKING IN SINTERED PRODUCTS

V. M. BELOUSOV

(Presented by the Scientific Seminar on Dielectrics)

In powder metallurgy, in the production of ceramic materials and refractories, the densest possible packing of particles before sintering is of great importance. One of the criteria of sintering, according to the theories of Frenkel [1, 2] and Pines with co-workers [3—7], is the degree to which the pores between particles are closed. The presence of gas in the pores may serve as an obstacle to the complete closure of pores. When pores of different sizes are present, large pores may grow at the expense of the reduction of small ones [5]. One way to reduce the porosity of a sintered article to a minimum is the limiting reduction of the gaps between particles before sintering, since in this case, along with the reduction of the gaps, the amount of gas in these gaps is also reduced.

In this connection, much attention has been devoted to the question of the particle-size distribution of particles. Poluboyarinov and Gracheva [8] gave a broad review of work in this direction and presented the data of their own investigations. In their work (as well as in all the works cited by them), mixtures of particles of different sizes were selected empirically; the mass was compacted by repeated shaking (10,000 times), after which the bulk density was determined. We have not found in the literature any attempt at a computational solution of the problem of the particle-size distribution of particles. In the present work an attempt at such a solution is undertaken.

1. Basic calculation

Let us have at our disposal a cubical box with edge \(L\) and balls of identical diameter \(d\), where \(d = \dfrac{L}{n}\), and \(n\) is an integer. Let us fill the box so that the lines of contact of the balls are parallel to the edges of the cube, as shown in Fig. 1. It is assumed that all the balls have the same mass (density). In this case it may be asserted that the packing density (the bulk density, or simply the weight of the box with the balls) does not depend on the diameter of equal balls.

This assertion is easy to prove in the following way. Let us describe, about each sphere, a cube with edge \(l=d\). The volume of the large cubical box is

\[ V_1=L^3=n^3l^3, \tag{1} \]

and the volume of all the spheres is

\[ V_2=n^3\cdot \frac{1}{6}\pi d^3. \tag{2} \]

The ratio of the volume of the spheres to the volume of the cube is

\[ \frac{V_2}{V_1}=\frac{\pi}{6}=\mathrm{const}\quad(\text{does not depend on } n). \]

Since small cubes in a large one (under the condition \(L=nl\)) can be packed absolutely densely, then for any sphere diameter \(d=\dfrac{L}{n}\) (i.e., for any integer \(n\)) the packing density will not change.

Fig. 1.

This circumstance makes it possible to reduce the calculation to the filling of a small cube of edge \(l=d\) with spheres of decreasing radii. The filling scheme is shown in Fig. 2.

Let us denote the radius of the largest sphere, the central one, equal to half the edge of the cube, by \(r_0\). The eight corners of the cube are then to be filled. We denote the radius of the next smaller sphere, tangent to the central sphere and to three faces of the cube, by \(r'_1\). This radius is easily related to \(r_0\). The semidiagonal of the cube is \(AO=r_0\sqrt{3}\). By analogy, \(AO'=r'_1\sqrt{3}\). From Fig. 2 it is evident that

\[ r'_1=AO-AO'-r_0=r_0\sqrt{3}-r'_1\sqrt{3}-r_0, \]

whence

\[ r'_1=r_0\frac{\sqrt{3}-1}{\sqrt{3}+1}. \tag{3} \]

Fig. 2.

The next corner sphere of smaller size, tangent to the sphere of radius \(r'_1\) and to three faces of the cube, will be

\[ r'_2=r'_1\cdot\frac{\sqrt{3}-1}{\sqrt{3}+1} = r_0\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)^2. \]

The corner of the cube will be filled by a series of spheres with radii

\[ r'_n=r_0\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)^n \tag{4} \]

for \(n\) arbitrarily large.

The next substantial cavity to be filled may be taken as the gap opposite the middle of an edge of the cube. The radius of a sphere tangent to the central sphere and to the faces of the dihedral angle of the cube,

denote by \(r_1''\). Analogously to the derivation of formula (3), for this radius we obtain

\[ r_1''=r_0\frac{\sqrt{2}-1}{\sqrt{2}+1}. \tag{5} \]

The general formula for a successive series of spheres filling the dihedral angle opposite the middle of an edge will be

\[ r_n''=r_0\left(\frac{\sqrt{2}-1}{\sqrt{2}+1}\right)^n . \tag{6} \]

An approximate relative estimate of the radii of the spheres filling the cube leads to the following values:

\[ \frac{r_1'}{r_0}=0.268, \]

\[ \frac{r_1''}{r_0}=0.171. \]

For selecting finer fractions, formulas (4) and (6) may be used. Using formula (4), one can obtain, for example:

\[ \frac{r_2'}{r_0}=0.072, \]

\[ \frac{r_3'}{r_0}=0.019 \]

and so on.

By counting the number of spheres of the corresponding size, one can calculate the relative mass of each fraction. For example, the first four fractions may be estimated from the following considerations: to fill a cube with edge \(l\), one sphere of radius \(r_0\), eight spheres of radius \(r_1'\), and twelve spheres of radius \(r_1''\) will be required; the remaining voids can be filled with finer fractions (dust). Then:

1) the fraction of spheres of radius \(r_0\) should amount to 52.3%,
2) the fraction of spheres of radius \(r_1'\) should amount to 8.1%,
3) the fraction of spheres of radius \(r_1''\) should amount to 3.2%,
4) lighter fractions (including dust) should amount to 36.4%.

2. Some clarifications concerning intermediate fractions

Spheres of radius \(r_1'\) fill the largest voids between the walls of the cube and a sphere of radius \(r_0\). When large spheres are arranged in the order shown in Fig. 1, it is necessary to fill voids between the spheres that are considerably larger than in the case considered above. With less generality in the calculation, in this case as well it is possible to calculate the radii of spheres of two intermediate fractions.

Let us place eight spheres of radius \(r_0\) inside a cube with edge \(4r_0\). The largest void will be at the center of the cube. The radius of the sphere \(r_1\) touching all eight large spheres is easily calculated:

\[ \frac{r_1}{r_0}=\sqrt{3}-1 \simeq 0.732 . \tag{7} \]

The sphere of the next-size fraction must touch two adjacent spheres of radius \(r_1\), located at the centers of two neighboring cubes. Let us denote this radius by \(r_2\). It is easy to verify that

\[ \frac{r_2}{r_0}=2-\sqrt{3} \simeq 0.268 . \tag{8} \]

Thus, this refinement has made it possible to introduce an additional intermediate fraction of spheres of radius \(r_1\).

The relative mass of the four fractions in this case may be calculated approximately from the following considerations. Construct a cube whose vertices will be located at the centers of the large spheres. Inside the cube there will be one eighth of each sphere of radius \(r_0\), one sphere of radius \(r_1\), and six halves of spheres of radius \(r_2\). The remaining voids are to be filled with finer fractions. Calculation of the relative share of each fraction gives:

Thus, the introduction of an additional intermediate fraction, having a significant relative mass, changed the ratio of the fine fractions. The fraction of spheres of radius \(r_1\), not taken into account in the calculation in § 1, enters in a considerable amount—20.5%.

3. Estimation of fractions for the densest packing of the first fraction of spheres

It is natural to expect that, under conditions of frictionless sliding, after shaking the spheres of the first fraction of radius \(r_0\) will arrange themselves relative to one another in the densest possible manner. The known methods of densest packing of spheres of identical size are mutual arrangements corresponding to hexagonal and cubic face-centered crystallographic lattices.

It is known from the literature that in these cases the large spheres occupy 74% of the space, with one large void and two small ones corresponding to each large sphere [9]. The dimensions of the particles and the number of fractions can be calculated most simply for the cubic face-centered lattice.

The large voids are formed among six neighboring spheres of radius \(r_0\), the centers of which lie at the vertices of an octahedron. Each edge of the octahedron is equal to \(2r_0\). The diagonal of the octahedron \(AB\) in Fig. 3 is equal to \(2r_0+2r_1\), where \(r_1\) is the radius of the sphere filling the large void.

and tangent to all six large spheres. The diagonal of the octahedron, on the other hand, is equal to \(AB = 2r_0\sqrt{2}\), therefore

\[ \frac{r_1}{r_0}=(\sqrt{2}-1)\simeq 0.414. \tag{9} \]

The small voids are enclosed between four spheres of radius \(r_0\), whose centers lie at the vertices of a tetrahedron. A sphere of radius \(r_2\), the center of which coincides with the center of the tetrahedron, must be tangent to the four spheres of radius \(r_0\). The distance from a vertex of the tetrahedron to the center is equal to

Fig. 3

\[ r_0\sqrt{\frac{3}{2}}. \]

This distance, according to Fig. 4, is equal to \(r_0 + r_2\). Consequently,

\[ \frac{r_2}{r_0}=\sqrt{\frac{3}{2}}-1\simeq 0.225. \]

Fig. 4.

From the ratio of the numbers of spheres, the fractions for filling the space may be estimated approximately:

\[ \begin{array}{rll} \text{fraction of spheres of radius } r_0 & 74\%,\\ \text{”} \quad \text{”} \quad \text{”} \quad r_1 & 5\%,\\ \text{”} \quad \text{”} \quad \text{”} \quad r_2 & 2\%,\\ \text{finer fractions (including dust)} & 19\%. \end{array} \]

4. Results of the calculations and their agreement with known data on the selection of granulometric composition

The calculation of particle sizes for filling the gap between the surface of a sphere and the faces of the circumscribed cube has the greatest generality. Formulas (4) and (6) make it possible to calculate the radii of successively decreasing spheres for any number of fractions. Two circumstances give rise to doubt: 1) in the system of packing of spheres shown in Fig. 1, there are voids more significant than the largest gap filled by a sphere of radius \(r'_1\) (see Fig. 2); 2) when spheres of radius \(r_0\) are packed without friction, their arrangement corresponding to the greatest density is most probable. The first question is considered in § 2, the second in § 3. The refinement of the calculations in § 2 led to an additional intermediate fraction of spheres, calculated by formula (7), the proportion of which in filling the volume is considerable—20.5%, whereas the intermediate fractions in the calculation of § 1 enter in insignificant amounts. The calculation of § 3, carried out on the basis of the densest arrangement of spheres of radius \(r_0\), leads to a predominant proportion of large spheres and small proportions of intermediate fractions. But the calculations of §§ 2 and 3, unlike the first variant of the calculation, do not possess generality; the calculation of the radius of the fourth fraction of spheres by size already encounters great difficulties.

The answer to the question of which of the calculations should be preferred must be given by experimental data on the selection of a granulometric composition ensuring the densest bulk packing. The data presented in the work of Polyuboyarinov and Gracheva [8] cannot give a direct answer, because, first, each of the poured fractions contained particles of different sizes within certain limits and, second, in the empirical selection the size composition of the fractions corresponding to the densest packing might not have been found. But we have no other data at our disposal; therefore, until a special experimental check is carried out, only an approximate comparison with the empirical data of that work can be made.

Polyuboyarinov and Gracheva selected, in different proportions, three size fractions of particles for the synthetic mixture, the third fraction consisting of a mixture of particles from a certain largest size down to the finest dust. The greatest density was obtained for three variants of the granulometric composition of coarse chamotte in the proportions given in Table 1.

Table 1

For fine chamotte, the greatest density was obtained with the fraction proportions given in Table 2.

If the grain size of each fraction is taken to be the mean value, i.e., in the first table to assume \(r_0 = 3.5\) mm, \(r_1 = 1.25\) mm and

\(r_2=0.5\) mm, then \(\dfrac{r_1}{r_0}=0.36\) and \(\dfrac{r_2}{r_0}=0.14\). Taking, respectively, for the fine grog in the second table \(r_0=1.25\) mm, \(r_1=0.3\) mm, \(r_2=0.1\) mm, we obtain \(\dfrac{r_1}{r_0}=0.24\) and \(\dfrac{r_2}{r_0}=0.08\). In the calculation of the fractions of § 1 the following ratios were obtained:

\[ \frac{r'_1}{r_0}=0.268;\qquad \frac{r''_1}{r_0}=0.171;\qquad \frac{r'_2}{r_0}=0.072. \]

There is no reason to expect complete agreement, since each fraction includes particles differing in size by a factor of 4–5. The mean value of the sizes is a conventional characteristic. Moreover, the particles are not spherical in shape. Nevertheless, some sizes are in good agreement with the calculated data, especially the fine-grog fractions. The shares of the fractions in the mixtures agree satisfactorily with the calculation given in § 1.

Table 2

Poluboyarinov and Gracheva note in their conclusions that the maximum packing density is obtained with a sufficient quantity of fine fractions and a sufficiently wide interval between the grain sizes of the finest and the coarsest fractions, with an insignificant content of intermediate fractions; the intermediate fractions of particles may be discarded without detriment.

This is in good agreement with the calculated data of § 1: the coarse fraction should amount to 52.3%; the intermediate fractions should be present in insignificant amounts (8.1% and 3.2%). The refinement of the calculation carried out in § 2 gave an additional intermediate fraction \(\dfrac{r_1}{r_0}=0.732\), which should amount to 20.5%. In Tables 1 and 2 there is no corresponding fraction, but in the work of Poluboyarinov and Gracheva three more variants of the densest packings are given, of which two contain eight fractions each, and one contains six fractions (two intermediate ones excluded), while the second fraction in order of coarseness has relative particle sizes \(\dfrac{r_1}{r_0}=0.78\) and is present in amounts of 17% (in two cases) and 13% (in one case).

There is no set of fractions among the most densely packed particles corresponding to the calculation of § 3, either in the experiments of Poluboyarinov and Gracheva or in the works by Fuller, Scola, Lintsov, and Rikke cited by them. Apparently, this method of packing is not practically realized.

This can be explained by the fact that the particles slide during mutual displacement with friction; moreover, with repeated shaking the fine particles are displaced more intensively and earlier than the coarse pa-

particles manage to arrange themselves in the order of their densest packing (a face-centered cube or a hexagonal system), while the fine particles fill all the voids and immobilize the large ones.

Thus, the calculation given in § 1, with the refinement carried out in § 2, may serve as a basis for calculating the granulometric composition of fine particles for shaping (molding, pressing, casting of a diluted mass, etc.) before sintering.

REFERENCES

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3. Geguzin, Ya. E., Morkon, L. O., Pines, B. Ya. Self-diffusion and viscous flow (sintering and creep) in pressed metal powders. DAN SSSR, 27, 577, 1952.

4. Pines, B. Ya. On sintering (in the solid phase). ZhTF, 16, 747, 1946.

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