The tensile strength of agglomerates is defined as the maximum force $F_N$ related to the cross section of an agglomerate whereby the force has to operate perpendicular to the surface. Problems arise when calculating the tensile strength, as the agglomerates are not continuums but rather a fill of primary particles that are irregularly formed and generally arranged inordinate in the agglomerate. If one assumes that the forces in agglomerates are only transited at the contact points of the individual primary particles and the inordinate arrangement of the particles it can be said that the maximum traction of the individual primary particles $F_{Np}$ results as a function of the elongation of the agglomerates (see figure).

The tensile strength results from the sum of the individual adhesive force located on the cross section of the agglomerates:

$$\sigma_z = \frac{F_{N,max}}{A} = \frac{1}{A} \sum_{i=1}^n F_{Np,i}(\Delta l) \tag{1}$$

As the behavior of the force and elongation of the agglomerates is not known in detail, this equation is not of much importance. Schubert identified different approaches to describe the tensile strength of agglomerates. For example the approach of Rumpf for statistically packed monodisperse particles:

$$\sigma_z = (1 - \varepsilon) \cdot k \cdot \frac{F_H}{A_p} = \frac{(1 - \varepsilon)}{\varepsilon} \cdot \frac{F_H}{d_p^2} \tag{2}$$

This approach excels because it does not need any adaptation factors and also has an excellent correlation with experimental results for special cases. Only the adhesive forces $F_H$ are indefinite. The formation of adhesive forces that are important is dependent on the agglomerate size, the degree of fluid saturation and the electrical potential. One can see the results comparing the adhesive forces in the calculation of the different adhesive forces of a sphere–sphere model in the figure.

One recognizes that the fluid bridges and the van-der-Waals forces have the most significant influence. Electrostatic binding forces have a larger range making them important for clustering processes. Additionally one can see that the gravity influence is off balance at large particle diameters (1,6 mm). One must bear in mind that this value is calculated for a sphere. For a real system this value is much lower.

In order to measure the tensile strength, the agglomerates are positioned in the circular sample mount. After which they are compressed and twisted. After the sample mount is filled, the two halves of the device are drawn apart and the forces needed are measured. The tensile strength can be calculated by relating the force needed to rupture the sample to the cross section of the apparatus.

In the figure the tensile strength of talcum as a function of the porosity is shown as an example.

The measurements of these values were taken using a device, which is in principle the same as the one shown in the figure.

Porosity is defined using the ratio of void volume to total volume:

$$\psi = \frac{V_H}{V_{ges}} \tag{1}$$

with:

• $\psi$ = porosity,
• $V_H$ = void volume,
• $V_{ges}$ = total volume.

The porosity can differ by the look of the agglomerate without being recognized. In the figure different types of pores are shown in the model of a single particle.

There are open and closed pores, pores with a fixed diameter, pores, which continually taper, and pores, which are only accessible over narrow capillaries, as well as through flow pores. Surface roughness must be considered.

The porosity of single particles $\psi_p$ is due to pores. From a cluster of particles one gets the agglomerate porosity $\psi_a$. This is the relationship of the void volume between the particles to the agglomerate volume. The problem with this definition is the specification of the agglomerate volume with respect to the outer zones.

In the loose fill of agglomerates one can find the bulk porosity $\psi_b$, which is the relationship between the agglomerates and the total volume of the loose fill within the mold cavity. The total porosity $\psi$ is composed of individual porosities. It considers:

$$(1 - \psi) = (1 - \psi_p)(1 - \psi_a)(1 - \psi_b) \tag{2}$$

Figure: Porosity of a) single particles, b) bulk

From measurements one cannot not distinguish between the porosity of the primary particles $\psi_p$, the porosity of the agglomerates $\psi_a$ and the bulk porosity $\psi_b$. Densities are also usually measured. A porous material has a smaller density than the solid material $\rho_f$. The density of the individual particle $\rho_p$ is related to the porosity as follows:

$$\rho_p = (1 - \psi_p) \cdot \rho_f \tag{3}$$

The agglomerate density is:

$$\rho_a = (1 - \psi_p) \cdot (1 - \psi_a) \cdot \rho_f \tag{4}$$

Appropriately considered for the bulk density $\rho_b$:

$$\rho_b = (1 - \psi_p) \cdot (1 - \psi_a) \cdot (1 - \psi_b) \cdot \rho_f \tag{5}$$

Besides the different pore types there are also various pore sizes to be considered. In the figure, the distribution density curve of the pore radius is represented for a fill of agglomerates. Generally speaking, pore sizes can be distinguished into: single particle pores, agglomerate pores and loose material pores. In the ideal case they yield various maximums of the distribution density (multi-modal).

In the determination of the porosity of the agglomerates or the individual particles, there are a range of measuring techniques at ones disposal. Mentioned here are the figure analysis and the mercury porosity techniques.

For the image analysis one or more cuts are made from an agglomerate and the ratio of the void areas to the solid material fraction are determined. With the help of sequential cuts, the difference between open and closed pores can be seen. In addition, the distribution of the values and the forming area can be determined. From a single individual sectional drawing only the value distribution for the spherical pores can be calculated. The intermediate porosity areas agree well however with the total porosity.

The method suggested by Washburn 1921 can be used to determine the volumes of the pores and of the distribution of the pore radius. Mercury wets very badly and wraps itself around the agglomerates. The volume of the mercury, which is displaced from the agglomerate, is measured with a mercury porosity meter.

Through known solid densities, the agglomerate density yields corresponding to DIN 53193 or DIN 51057 from the displaced volume and the difference in mass. In dependence of the pressure needed, the mercury penetrates the small pores, corresponding to the Gauß-Laplace equation:

$$p = \frac{2\sigma \cos(\Theta)}{r} \tag{1}$$

with:

• $p$ = pressure,
• $\sigma$ = surface tension of the mercury,
• $\Theta$ = angle of contact of the mercury,
• $r$ = radius of pore

With this equation the distribution of the pore radius lets itself be calculated. At higher pressures, the compressibility of the mercury must be considered.