Unterschiede
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| en:gefuellte_polymere [2026/04/20 17:23] – [Porosity] neelest | en:gefuellte_polymere [2026/05/10 23:37] (aktuell) – gelöscht deppe2 | ||
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| - | ======Filled polymers====== | ||
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| - | ===== Tensile Strength ===== | ||
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| - | The tensile strength of agglomerates is defined as the maximum tensile force $F_N$, normalized to the cross-sectional area of an agglomerate, | ||
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| - | Difficulties in calculating the tensile strength arise from the fact that agglomerates are not continuous solids but rather a packing of primary particles, which are irregularly shaped and generally arranged in a random manner within the agglomerate. Assuming, in a simplified way, that forces in agglomerates are transmitted only at the contact points between individual primary particles, it follows from this and from the random arrangement of the particles that the maximum tensile force of the individual primary particles $F_{Np}$ is a function of the agglomerate strain. | ||
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| - | [[en: | ||
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| - | The tensile strength is therefore obtained from the sum of the individual adhesive forces, normalized to the cross-sectional area of the agglomerates: | ||
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| - | $$σ_z = \frac{F_{N, | ||
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| - | Since the force–strain behavior of agglomerates is largely unknown, this relationship has little practical relevance. Schubert [[en: | ||
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| - | $$σ_z = (1-ε) \cdot k \cdot \frac{F_H}{A_p} = \frac {(1-ε)}{ε} \cdot \frac {F_H}{{d_p}^2}$$ | ||
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| - | This approach is particularly notable because it does not require adjustment factors and, in specific cases, shows excellent agreement with experimental data. Only the adhesive forces $F_H$ remain undetermined. The type of adhesive forces that are decisive for tensile strength depend significantly on the agglomerate size, the degree of liquid saturation, and the electrical potential or surface charge density. To compare adhesive forces, calculations based on a sphere–sphere contact model are shown. | ||
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| - | It can be observed that liquid bridges and van der Waals forces exert the strongest influence on adhesion forces. Electrostatic binding forces have a longer range and are therefore primarily relevant for particle deposition processes. It can also be seen that gravitational influence only becomes dominant for large particle diameters (approx. 1.6 mm). It should be noted that this value was calculated for ideally smooth spheres; for real systems, the value is expected to be significantly lower. | ||
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| - | ===== Measurement of Tensile Strength ===== | ||
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| - | The following figure shows the tensile strength testing device according to Parfitt. | ||
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| - | In this setup, the agglomerates under investigation are placed into the circular specimen holder, compressed, and then twisted. Twisting refers to the rotation of the compression piston in the specimen holder, as known from the Jenike shear cell, allowing rearrangement and reorientation processes to take place. After the specimen holder is filled, its two halves are pulled apart, and the force required to separate them is measured. The tensile strength can then be calculated from the ratio of the breaking force to the cross-sectional area of the device. This and other testing methods are described in detail in [[en: | ||
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| - | As an example, the tensile strength of talc is plotted as a function of porosity in the figure. | ||
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| - | The measurements were performed with an apparatus essentially similar to the one described above, which is described in detail in [[en: | ||
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| - | ===== Porosity ===== | ||
| - | Porosity is defined as the ratio of void volume to total volume [[en: | ||
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| - | $$ψ = \frac{V_H}{V_{ges}} $$ | ||
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| - | with $ψ$ = porosity, $V_H$ = void volume, and $V_{tot}$ = total volume. | ||
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| - | Porosity can occur in different forms, which are not necessarily visible from the outside of an agglomerate. The figure illustrates different types of pores in the model of a single particle. | ||
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| - | One distinguishes between closed and accessible pores, pores with constant diameter, pores that gradually narrow and are only accessible via narrow capillaries, | ||
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| - | Pores in single particles determine the particle porosity $ψ_p$. When such particles are agglomerated, | ||
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| - | When a bulk bed of agglomerates is formed, the bulk porosity $ψ_b$ arises, defined as the ratio of void volume between agglomerates to the total bulk volume. The total porosity $ψ$ is composed of the different contributions and can be expressed as: | ||
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| - | $$(1-ψ) = (1-ψ_p)(1-ψ_a)(1-ψ_b)$$ | ||
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| - | [[en: | ||
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| - | In general, porosity measurements cannot distinguish between contributions from particle porosity $ψ_p$, agglomerate porosity $ψ_a$, and bulk porosity $ψ_b$. Instead, densities are usually measured. A porous material exhibits a lower density than the true solid density $ρ_f$. The particle density $ρ_p$ is related to porosity as follows: | ||
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| - | $$ρ_p = (1-ψ_p) \cdot ρ_f$$ | ||
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| - | The agglomerate density is: | ||
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| - | $$ρ_a = (1-ψ_p) \cdot (1-ψ_a) \cdot ρ_f$$ | ||
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| - | And the bulk density is: | ||
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| - | $$ρ_b = (1-ψ_p) \cdot (1-ψ_a) \cdot (1-ψ_b) \cdot ρ_f$$ | ||
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| - | In addition to the various pore types, different pore sizes must also be considered. The following figure shows the pore radius distribution curve of a bulk solid composed of agglomerates. In general, pore size distributions differ significantly between single particles, agglomerates, | ||
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| - | For determining the porosity of agglomerates or individual particles, a number of measurement methods are available. Among them, only image analysis and mercury intrusion porosimetry will be mentioned here. | ||
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| - | ==== Measurement of Porosity ==== | ||
| - | === Image Analysis === | ||
| - | In image analysis, one or more cross-sections of an agglomerate are prepared, and the area fractions of voids and primary particle solids are determined. By means of serial sections, a distinction can be made between open and closed pores. In addition, pore size distribution and shape factor can be evaluated. From a single cross-sectional image, pore size distribution can only be calculated for spherical pores. However, the mean area porosity generally agrees well with the total porosity, regardless of pore shape. | ||
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| - | === Mercury Intrusion Method === | ||
| - | The mercury intrusion method, proposed by Washburn [**8**] in 1921, is suitable for determining pore volume and pore size distribution. Mercury exhibits very poor wetting behavior and envelopes the agglomerates. The volume of mercury displaced by the agglomerate is measured with the mercury porosimeter illustrated in the figure. | ||
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| - | FIXME PLATZHALTER ABBILDUNG 121: Quecksilber-Porosimeter | ||
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| - | If the true solid density is known, the agglomerate density can be calculated in accordance with DIN 53193 or DIN 51057 from the displaced volume and the mass difference. Depending on the applied pressure, mercury penetrates into smaller pores in accordance with the Gauss–Laplace equation FIXME**(5.75)**: | ||
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| - | $$p = \frac {2σcos(Θ)}{r}$$ | ||
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| - | with $p$ = applied pressure, $σ$ = surface tension of mercury, $Θ$ = contact angle of mercury, and $r$ = pore radius. | ||
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| - | Using this equation, the pore size distribution can be calculated [**7**]. At higher pressures, the compressibility of mercury must also be taken into account. | ||
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| - | ===== References ===== | ||
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| - | [Dul79] Dullien, F.A.L.; Porous Media - Fluid Transport and Pore Structure, Academic Press; 1979 | ||
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| - | [PC97] Pierrat, P.; Caram, H.S.: Tensile strength of wet granular materials, Powder Technology, 91(1997), 83-93 | ||
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| - | [PHS79] Polke, R.; Herrmann, W.; Sommer, K.: Charakterisierung von Agglomeraten, | ||
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| - | [Rum61] Rumpf, H.: Agglomeration, | ||
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| - | [Rum74] Rumpf, H.: Die Wissenschaft des Agglomerierens, | ||
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| - | [Sch75] Schubert, H.: Tensile Strength of Agglomerates, | ||
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| - | [Was21] Washburn, E.W.:, Proc. Nat. Acad. Sci. U.S., 7(1921), 115 | ||
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| - | [Wic91] Wicke, R.: Agglomeratkennzeichnung und Prüfmethoden, | ||
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| - | [YFY82] Yokoyama, T.; Fujii, K.; Yokoyama, T.: Measurement of the tensile Strength of a Powder Bed by a Swing Method Measuring Instrument, Powder Technology, 32(1982), 44 - 52 | ||