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Filled Polymers
Tensile Strength
The tensile strength of agglomerates is defined as the maximum force FN related to the cross section of an agglomerate whereby the force has to operate perpendicular to the surface [1]. 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).
Figure: Force and elongation behavior of agglomerates [1].
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 [1] 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 FH 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.
Figure: Adhesive force for different binding mechanisms according to Rumpf.
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.
Measuring of Tensile Strength
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. These methods and additional test methods are described in detail in [1-3].
In the figure the tensile strength of talcum as a function of the porosity is shown as an example.
Figure: Tensile strength of talcum [4].
The measurements of these values were taken using a device, which is in principle the same as the one shown in the figure. It is described in detail in [4].