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en:ermittlung_der_materialdaten [2026/05/10 23:27] – deppe2en:ermittlung_der_materialdaten [2026/05/27 23:33] (aktuell) – [Filled polymers] deppe2
Zeile 1: Zeile 1:
 ======Determination of material data====== ======Determination of material data======
  
-======Pure polymers======+=====Pure polymers=====
  
 A thorough knowledge of the material parameters and the behaviour of the material A thorough knowledge of the material parameters and the behaviour of the material
Zeile 52: Zeile 52:
  
 The WLF equation provides a better description than the Arrhenius approach, especially for amorphous polymers whose molten state begins at a temperature slightly above the glass transition temperature. The WLF equation provides a better description than the Arrhenius approach, especially for amorphous polymers whose molten state begins at a temperature slightly above the glass transition temperature.
-This assumes that the segment mobility of polymers near the glass transition temperature is primarily determined by the free volume, which increases approximately linearly with the distance from the glass transition temperature TG. It is assumed that the segment mobility of polymers near the glass transition temperature is predominantly determined by the free volume, which increases approximately linearly with the distance from the glass transition temperature TG [[en:reine_polymere#references|[Fer80]]]. If a temperature approximately 50°C above the glass transition temperature $T_G$ is selected as the standard temperature $T_S$, the parameters $C_1$ and $C_2$ in the equation can be regarded as material-independent [[en:reine_polymere#references|[Mel95]]].+This assumes that the segment mobility of polymers near the glass transition temperature is primarily determined by the free volume, which increases approximately linearly with the distance from the glass transition temperature TG. It is assumed that the segment mobility of polymers near the glass transition temperature is predominantly determined by the free volume, which increases approximately linearly with the distance from the glass transition temperature TG [[en:ermittlung_der_materialdaten#references|[Fer80]]]. If a temperature approximately 50°C above the glass transition temperature $T_G$ is selected as the standard temperature $T_S$, the parameters $C_1$ and $C_2$ in the equation can be regarded as material-independent [[en:ermittlung_der_materialdaten#references|[Mel95]]].
  
 Alternatively, the following approach can also be used: Alternatively, the following approach can also be used:
Zeile 92: Zeile 92:
 The calculation of both the melting behavior and the temperature development in The calculation of both the melting behavior and the temperature development in
 power melts requires a comprehensive knowledge of the thermodynamic material power melts requires a comprehensive knowledge of the thermodynamic material
-behavior [[en:reine_polymere#references|[Hen89]]]. The thermodynamic properties are dependent on both pressure and+behavior [[en:ermittlung_der_materialdaten#references|[Hen89]]]. The thermodynamic properties are dependent on both pressure and
 temperature and show a different material property in the solid state and in the melt temperature and show a different material property in the solid state and in the melt
 state. state.
Zeile 149: Zeile 149:
 In this function, the index $0$ denotes the material property at $0 °C$, while the index $m$ represents the slope of the property function. In this function, the index $0$ denotes the material property at $0 °C$, while the index $m$ represents the slope of the property function.
  
-Since REX / PSI / SIGMA does not account for pressure effects, the determination of the thermodynamic material properties should always be performed at a mean pressure [[en:reine_polymere#references|[Mel95]]].+Since REX / PSI / SIGMA does not account for pressure effects, the determination of the thermodynamic material properties should always be performed at a mean pressure [[en:ermittlung_der_materialdaten#references|[Mel95]]].
  
 The specific enthalpy $∆h$ results from the integration of the specific heat capacity $c$ over the temperature range between $T_1$ and $T_2$, which can be derived from DSC analysis. In this way, the amount of heat per unit mass of the polymer that must be supplied is obtained. The following relation applies: The specific enthalpy $∆h$ results from the integration of the specific heat capacity $c$ over the temperature range between $T_1$ and $T_2$, which can be derived from DSC analysis. In this way, the amount of heat per unit mass of the polymer that must be supplied is obtained. The following relation applies:
Zeile 171: Zeile 171:
 ==== Thermal Conductivity ==== ==== Thermal Conductivity ====
  
-In heat conduction processes, a distinction is made between steady-state and non-steady-state temperature fields. In steady-state temperature fields, the thermal conductivity $λ$ appears as a material property. It is temperature dependent and lower for amorphous materials than for semi-crystalline ones [[en:reine_polymere#references|[Mel95]]].+In heat conduction processes, a distinction is made between steady-state and non-steady-state temperature fields. In steady-state temperature fields, the thermal conductivity $λ$ appears as a material property. It is temperature dependent and lower for amorphous materials than for semi-crystalline ones [[en:ermittlung_der_materialdaten#references|[Mel95]]].
  
 The determination of the thermal conductivity $λ$ can, for example, be carried out using a Thermoflixer device manufactured by SWO Polymertechnik. The measuring principle is illustrated in the following figure. The determination of the thermal conductivity $λ$ can, for example, be carried out using a Thermoflixer device manufactured by SWO Polymertechnik. The measuring principle is illustrated in the following figure.
Zeile 258: Zeile 258:
  
 The pellet diameter $d_{sphere}$ can then be determined using the following equation. The pellet diameter $d_{sphere}$ can then be determined using the following equation.
 +
 +
 +
 +=====Mixture rules for polymer blends=====
 +
 +The rheological as well as the thermodynamic parameters of two-component systems (e.g., solidification and melting enthalpies of polymer blends) generally cannot be determined with sufficient accuracy by a simple linear averaging of the data of the base components. For simulations with REX / PSI / SIGMA, this means that, prior to the actual simulation runs, complete material data for the polymer blends should ideally be determined experimentally. However, this requires a considerable amount of measurements even before the simulation process itself can be carried out. If, on the other hand, only the material data of the individual components are available, the material data of the blends are calculated within REX / PSI / SIGMA from the component data as follows.
 +
 +==== Rheological Parameters of Multicomponent Systems ====
 +
 +For calculating the viscosity of two-phase polymer blends, the literature provides a range of mixing rules that vary in complexity depending on the required level of accuracy—from simpler approaches to more mathematically elaborate ones. In the simulation software packages REX / PSI / SIGMA, the following general mixing rules are applied: 
 +
 +first, the simplest and most widely used logarithmic mixing rule according to Arrhenius:
 +
 +$$log η_{MIX} = w_1 \cdot log η_1 + w_2 \cdot log η_2$$
 +
 +and second, the mixing rule proposed by Mantford:
 +
 +$$(η_{MIX})^{1/3,4} = w_1 \cdot η_1^{1/3,4} + w_2 \cdot η_2^{1/3,4}$$
 +
 +The determination of the blend viscosity from the viscosities of the individual components $η_i$ and their weight fractions $W_i$ using REX / PSI / SIGMA is illustrated in the following figure.
 +
 +{{ :en:rex310_en_103.svg?nolink&700 |}}
 +
 +To calculate the viscosity $η_{MIX}$ at a given shear rate $\dot γ$ for a polymer blend, the blend viscosities $η_{MIX1}$ and $η_{MIX2}$ are first determined for the defined shear rates $\dot γ_1 = 0.9 \cdot \dot γ$ and $\dot γ_2 = 1.1 \cdot \dot γ$. For the Mantford approach, the following applies:
 +
 +$$(η_{MIX1})^{1/3,4} = w_1 \cdot η_1.1^{1/3,4} + w_2 \cdot η_2.1^{1/3,4}$$
 +$$(η_{MIX2})^{1/3,4} = w_1 \cdot η_1.2^{1/3,4} + w_2 \cdot η_2.2^{1/3,4}$$
 +
 +Analogously, the logarithmic mixing rule (Arrhenius) is given by:
 +
 +$$log η_{MIX1} = w_1 \cdot log η_1.1 + w_2 \cdot log η_2.1$$
 +$$log η_{MIX2} = w_1 \cdot log η_1.2 + w_2 \cdot log η_2.2$$
 +
 +If these quantities are known, the flow behavior index $n$ and the consistency $K$ of the approximated blend segment between $\dot γ_1 = \dot γ_{MIX1}$ and $\dot γ_2 = \dot γ_{MIX2}$ can be determined as follows:
 +
 +$$n = 1 + \frac{log(\frac{η_{MIX1}}{η_{MIX2}})}{log(\frac{\dotγ_{MIX1}}{\dotγ_{MIX2}})}$$
 +
 +$$K = \frac{η_{MIX1}}{\dotγ_{MIX1}^{n-1}}$$
 +
 +The desired viscosity $η_{MIX}$ is finally obtained by:
 +
 +$$η_{MIX} = K \cdot \dotγ_{MIX}^{n-1}$$
 +
 +If the dispersed phase is present in solid or highly viscous form, the system can be regarded as a filled polymer. For simulation purposes, in the case of a filled polymer the base polymer and its particle diameter must first be defined. The filler is then specified. This requires a range of material data: the particle diameter $d$, the mass fraction $w$, the solid density $ρ$, the bulk density $ρ_s$, the thermal conductivity of the solid $λ_0$, and the specific heat capacity $c_0$. Similar to polymer blends, two mixing equations are also available for filled polymers. The first is the simple approach according to Einstein:
 +
 +$$η_{MIX} = η_1 \cdot (1 + 2,5 \cdot Φ_2)$$
 +
 +and finally, the approach according to Hashin:
 +
 +$$η_{MIX} = η_1 \cdot [1+2 \cdot \frac{Φ_2}{1-Φ_2}]$$
 +
 +In these equations, $η_1$ denotes the viscosity of the base polymer, and $Φ_2$ the volume fraction of the added component.
 +
 +==== Thermodynamic Parameters of Multicomponent Systems ====
 +
 +Experimentally determined crystalline melting temperatures, both for polymer blends and for filled systems (compounds), are shown in the following figure. For polymer blends, the crystalline melting temperature $T_{K.MIX}$ is calculated in REX / PSI / SIGMA from the weight fractions $w_i$ and the crystalline melting temperatures $T_{Ki}$ as follows:
 +
 +$$T_{K.MIX} = w_1 \cdot T_{K.1} + w_2 \cdot T_{K.2}$$
 +
 +whereas for compounds, the following applies:
 +
 +$$T_{K.MIX} = T_{K.1}$$
 +
 +The crystalline melting temperature $T_{K.MIX}$ of a filled polymer (compound) is therefore assumed to be equal to the crystalline melting temperature $T_{K1}$ of the base polymer.
 +
 +{{ :en:rex310_en_104.svg?nolink&700 |}}
 +
 +To describe the specific heat capacity $c$ of multicomponent systems, a uniform mixing rule is applied for both polymer blends and filled systems. For the specific heat capacity of the mixture $c_{MIX}$, the following applies:
 +
 +$$c_{MIX} = w_1 \cdot c_1 + w_2 \cdot c_2$$
 +
 +where $w_i$ are the weight fractions and $c_i$ the specific heat capacities of the individual or added components.
 +
 +The specific heat capacity $c_{0.MIX}$ of the mixture is therefore given by:
 +
 +$$c_{0.MIX} = w_1 \cdot c_{0.1} + w_2 \cdot c_{0.2}$$
 +
 +The following figure shows the course of the specific heat capacity $c_0$ for polymer blends and filled systems. The slope of the heat capacity curve, $c_{m.MIX}$, for the mixture is determined for both systems analogously according to the following equation:
 +
 +$$c_{m.MIX} = w_1 \cdot c_{m.1} + w_2 \cdot c_{m.2}$$
 +
 +{{ :en:rex310_en_105.svg?nolink&700 |}}
 +
 +The relationship between the specific heat capacity $c_m$ and the weight fractions for polymer blends and filled systems is illustrated as an example:
 +
 +{{ :en:rex310_en_106.svg?nolink&700 |}}
 +
 +The specific enthalpy is given by:
 +
 +$$Δh = w_1 \cdot Δh_1 + w_2 \cdot Δh_2 = w_1 \cdot {(Δh_F + Δh_A)}_1 + w_2 \cdot {(Δh_F + Δh_A)}_2$$
 +
 +Index 1 refers to polymer 1 and index 2 to polymer 2.
 +
 +For describing the specific enthalpy $Δh$ of filled systems (compounds), the following approach is used:
 +
 +$$Δh = w_1 \cdot {(Δh_F + Δh_A)}_1 + w_2 \cdot c_2 \cdot ΔT$$
 +
 +Here, index 1 refers to the polymer, while index 2 denotes the filler.
 +
 +The solid enthalpy $Δh_F$ and melting enthalpy $Δh_A$ for polymer blends can be determined from the enthalpy values of the individual components as follows. For the solid enthalpy $Δh_{F.MIX}$ and melting enthalpy of the mixture $Δh_{A.MIX}$:
 +
 +$$Δh_{F.MIX} = w_1 \cdot Δh_{F.1} + w_2 \cdot Δh_{F.2}$$
 +
 +$$Δh_{A.MIX} = w_1 \cdot Δh_{A.1} + w_2 \cdot Δh_{A.2}$$
 +
 +Analogously, for compounds (filled systems), the solid enthalpy $Δh_{F.MIX}$ and melting enthalpy $Δh_{A.MIX}$ of the mixture are calculated as follows:
 +
 +$$Δh_{F.MIX} = w_1 \cdot Δh_{F.1}$$
 +
 +$$Δh_{A.MIX} = w_1 \cdot Δh_{A.1} + w_2 \cdot c_2 \cdot ΔT$$
 +
 +The following figures show, as examples, results from experimental investigations of the specific solid enthalpy $Δh_F$ and specific melting enthalpy $Δh_A$ of multicomponent systems.
 +
 +{{ :en:rex310_en_107.svg?nolink&700 |}}
 +{{ :en:rex310_en_108.svg?nolink&700 |}}
 +
 +In the simulation programs REX / PSI / SIGMA, the following equations are used to determine the thermal conductivity $λ_{MIX}$ of multicomponent systems from the values of the individual polymers ($w_i$, $λ_i$). For polymer blends:
 +
 +$$λ_{MIX} = w_1 \cdot λ_1 + w_2 \cdot λ_2$$
 +
 +For filled polymers (compounds), the following relationship is assumed for calculating the thermal conductivity:
 +
 +$$λ_{MIX} = λ_2 \cdot \frac{λ_1 + 2 \cdot λ_2 - 2 \cdot Φ_1 \cdot (λ_2 - λ_1)}{λ_1 + 2 \cdot λ_2 + Φ_1 \cdot (λ_2-λ_1)}$$
 +
 +While indices 1 and 2 correspond to the individual polymers 1 and 2 in polymer blends, for filled systems index 1 refers to the base polymer and index 2 to the added filler. For each component, the linear approximation is applied:
 +
 +$$λ_i = λ_{0.i} + λ_{m.i} \cdot T$$
 +
 +Here, $λ_{0.i}$ is the value of the linear approximation for the thermal conductivity of a component at $T = 0°C$, while $λ_{m.i}$ is the slope of the approximated thermal conductivity curve of the component above the melting temperature. For polymer blends, the thermal conductivity of the mixture $λ_{0.MIX}$ is:
 +
 +$$λ_{0.MIX} = w_1 \cdot λ_{0.1} + w_2 \cdot λ_{0.2}$$
 +
 +The slope of the mixture’s thermal conductivity function $λ_{m.MIX}$ is calculated as:
 +
 +$$λ_{m.MIX} = w_1 \cdot λ_{m.1} + w_2 \cdot λ_{m.2}$$
 +
 +For filled polymers (compounds), the following mixing equations are used to determine $λ_{0.MIX}$ and $λ_{m.MIX}$:
 +
 +$$λ_{0.MIX} = λ_{0.2} \cdot \frac{λ_{0.1} + 2 \cdot λ_{0.2} -2 \cdot Φ_1 \cdot (λ_{0.2} - λ_{0.1})}{λ_{0.1} + 2 \cdot λ_{0.2} + Φ_1 \cdot (λ_{0.2} - λ_{0.1})}$$
 +
 +and
 +
 +$$λ_{m.MIX} = λ_{m.2} \cdot \frac{λ_{m.1} + 2 \cdot λ_{m.2} -2 \cdot Φ_1 \cdot (λ_{m.2} - λ_{m.1})}{λ_{m.1} + 2 \cdot λ_{m.2} + Φ_1 \cdot (λ_{m.2} - λ_{m.1})}$$
 +
 +==== Densities of Multicomponent Systems ====
 +
 +The calculation of the solid and melt density $ρ_{MIX}$ of polymer blends and compounds from the densities $ρ_i$ and weight fractions $w_i$ of the individual polymers or of the base polymer and filler follows the same equation:
 +
 +$$\frac {1}{ρ_{MIX}} = \frac{w_1}{ρ_1} + \frac{w_2}{ρ_2}$$
 +
 +The specific volume $v$ of polymer blends and compounds from the specific volumes $v_i$ and weight fractions $w_i$ of the individual polymers or of the base polymer and filler is calculated using:
 +
 +$$v_{MIX} = w_1 \cdot v_1 + w_2 \cdot v_2$$
 +
 +For polymer blends and filled polymers (compounds), the mixture values $v_{0.MIX}$ and $v_{m.MIX}$ are determined using:
 +
 +$$v_{0.MIX} = w_1 \cdot v_{0.1} + w_2 \cdot v_{0.2}$$
 +
 +and
 +
 +$$v_{m.MIX} = w_1 \cdot v_{m.1} + w_2 \cdot v_{m.2}$$
 +
 +To determine the bulk density $ρ_{S.MIX}$ of multicomponent systems such as polymer blends and compounds with granule diameters $d_1$ and $d_2$ (where $d_1 < d_2$) of the individual components or of the base polymer and filler, some considerations must first be made. The following figure illustrates the bulk density $ρ_{S.MIX}$ of a mixture composed of two different particle fractions.
 +
 +{{ :en:rex310_en_109.svg?nolink&700 |}}
 +
 +For both polymer blends and filled systems (compounds), the granule diameter of the smaller fraction is assumed to be much smaller than that of the larger fraction ($d_1 \ll d_2$). If the bulk density $ρ_{S.MIX}$ of a mixture composed of two different particle fractions is plotted against the weight fraction of the smaller fraction $w_1$, each curve exhibits a maximum at the saturation concentration $w_1 = w_{Sät}$, independent of the void fraction $e$. The saturation concentration is calculated as:
 +
 +$$w_{Sät} = \frac {ρ_2 \cdot (1-p_∞) \cdot (1-e)}{ρ_1 + ρ_2 \cdot (1-p_∞) \cdot (1-e)}$$
 +
 +with the maximum packing fraction for a cubic-close-packed arrangement of spheres:
 +
 +$$p_∞ = \frac {V_{sphere}}{V_{total}} = \frac {π}{3 \cdot √2} ≈ 0,74$$
 +
 +Here, $V_{sphere}$ is the volume that the larger material component (Mat. 2) can maximally occupy relative to the total volume $V_{total}$. The void fraction that is too small for the smaller component (Mat. 1) to fit is set to $e = 0.25$. From this, the saturation bulk density (bulk density of the mixed region) is obtained:
 +
 +$$ρ_{Sät} = ρ_1 + (1-ρ_∞) \cdot (1-e) \cdot ρ_2$$
 +
 +To determine the bulk density $ρ_{S.MIX}$ of multicomponent systems in polymer blends and compounds with $d_1 < d_2$, three cases are distinguished. The procedure for the three calculation types is illustrated in the following figure. The subdivision criterion in all calculation types is the ratio of the particle diameters ($d_1 / d_2$). Initially, only two calculation approaches are used ($d_1 < 0.25 \cdot d_2$ or $d_1 > 0.75 \cdot d_2$), while the third calculation approach is obtained by linear averaging of the other two in the transition range ($0.25 \cdot d_2 < d_1 < 0.75 \cdot d_2$).
 +
 +=== Bulk Density of Multicomponent Systems (Polymer Blends and Compounds with $d_1 < d_2$) ===
 +
 +== Calculation Type I (if $d_1 \ll d_2$, i.e., $d_1 < 0.25 \cdot d_2$) ==
 +
 +  * if $w_1 < w_{Sät}$
 +
 +$$\rho_{Mix} = \left(1 - \frac{w_1}{w_{Sät}}\right)\rho_1 + \frac{w_1}{w_{Sät}} \rho_{Sät}$$
 +
 +  * if $w_1 \geq w_{Sät}$
 +
 +$$\rho_{Mix} = \left(1 - \frac{1 - w_1}{1 - w_{Sät}}\right)\rho_2 + \frac{1 - w_1}{1 - w_{Sät}} \rho_{Sät}$$
 +
 +== Calculation Type II (if $d_1 > 0.75 \cdot d_2$) ==
 +
 +$$\frac{1}{\rho_{Mix}} = \frac{w_1}{\rho_1} + \frac{w_2}{\rho_2}$$
 +
 +== Calculation Type III (if $0.25 \cdot d_2 < d_1 < 0.75 \cdot d_2$) ==
 +
 +$$\rho_{Mix} = \frac{(\rho_{Mix I} + \rho_{Mix II})}{2}$$
 +
 +Here, $\rho_{Mix}$ is obtained by linear averaging of **Calculation Types I and II**.
 +
 +=====Polymer blends=====
 +
 +To describe the material behavior of polymer blends, the same material properties must be investigated and characterized as for neat polymers.
 +
 +If, in addition to general calculations, the morphology development of a polymer blend is to be simulated, the interfacial energetic properties of the polymer combination must also be analyzed and described.
 +
 +==== Interfacial Tension of Polymer Blends ====
 +
 +The interfacial tension can be determined experimentally using various methods, including:
 +
 +  * Breaking Thread,
 +  * Pendant Drop, and
 +  * Spinning Drop.
 +
 +=== Breaking Thread method ===
 +The Breaking Thread method is based on the theoretical description of the breakup of a liquid Newtonian filament in a Newtonian matrix. However, this method is limited to multiphase systems in which the melting temperature of the dispersed phase is higher than that of the matrix. Furthermore, the zero-shear viscosity $η_0$ (viscosity $η$ at $\dot γ → 0$) of the matrix should not exceed 40 kPas. A schematic of the experimental setup used for the measurements is shown in the next figure. During the entire test, the heating stage is purged with nitrogen. Before the measurement starts, the system is conditioned in the heating stage at 220 °C for approximately 10 minutes in order to minimize retardation effects during melting. Subsequently, the system is heated up to the desired test temperature. As soon as the filament is completely molten, capillary waves form at its interface with the matrix. The entire process is recorded using a CCD camera.
 +
 +These sinusoidal capillary waves or filament neckings are captured at large time intervals and evaluated by computer analysis.
 +
 +For the filament, the initial diameter $D_0$, the wavelength $λ$, as well as the maximum and minimum filament diameters $D_{max}$ and $D_{min}$ are measured. The following figure shows the characteristic form of such a capillary wave with its relevant dimensions for theoretical consideration.
 +
 +{{ :en:rex310_en_112.svg?nolink&700 |}}
 +
 +The interfacial tension $γ_{12}$ is then a function of the amplitude growth rate $q$, the dimensionless growth rate $Ω$, the matrix viscosity $η_c$, and the initial filament diameter $D_0$, and can be expressed as:
 +
 +$$γ_{12} = \frac {q \cdot η_c \cdot D_0}{Ω(p,X)}$$
 +
 +The amplitude growth rate $q$ can be determined from the slope $S$ of the relative amplitude $log (2 \cdot α_s / D_0)$ plotted against time $t$:
 +
 +$$q = S \cdot ln 10$$
 +
 +{{ :en:rex310_en_113.svg?nolink&700 |}}
 +
 +For the oscillation amplitude, the following relation applies:
 +
 +$$α_s = \frac {D_{max} - D_{min}}{4}$$
 +
 +For calculating the interfacial tension $γ_{12}$ in this work, the dimensionless growth rate $Ω(p,X)$ is used. The viscosity ratio is defined as:
 +
 +$$p = \frac {η_d}{η_c}$$
 +
 +and the wavenumber, determined experimentally, is defined as:
 +
 +$$X= \frac {π \cdot D_0}{λ}$$
 +
 +The next figure shows the behavior of the dimensionless growth rate $Ω$ as a function of the viscosity ratio $p$ and the wavenumber $X$. The solid line indicates the maxima of the dimensionless growth rate $Ω_m$.
 +
 +If the determined $γ_{12}$ values are plotted against different temperatures $T$, the data points above the crystalline melting temperature $T_K$ (for semi-crystalline polymer pairs) or glass transition temperature $T_G$ (for amorphous polymer pairs) can be approximated by a linear function of the form:
 +
 +$$γ_{12} (T) = γ_{12.0} - γ_{12.m} \cdot T$$
 +
 +Here, $γ_{12.0}$ represents the intercept of the approximation function with the ordinate, while $γ_{12.m}$ corresponds to the slope of this function. The following figure illustrates the typical course of interfacial tension $γ_{12}$ as a function of temperature $T$ for a polypropylene (PP) / polyamide (PA6) blend.
 +
 +{{ :en:rex310_en_115.svg?nolink&700 |}}
 +
 +In the literature, the slope of this straight line is typically approximated by $γ_{12.m} = 0.01 , mN/m°C$. In practice, however, this value varies from one polymer pair to another. Therefore, to establish the approximation function, only two measurements of the interfacial tension $γ_{12}$ at two different temperatures $T$ are required. This allows the Breaking Thread Method to provide a rapid determination of interfacial tension $γ_{12}$ above the melting temperature $T_K$ with minimal experimental effort.
 +
 +=== Pendant Drop method ===
 +The //pendant Drop method// is the most versatile and reliable technique for measuring both surface and interfacial tensions of polymers. On the one hand, this is because equilibrium between the polymer phases is usually achieved more quickly compared to other methods, and on the other hand, because measurements can also be performed in an inert atmosphere.
 +
 +The method is based on the optical measurement of the shape of a liquid or molten drop that is in hydrostatic equilibrium with the surrounding phase. The droplet profile is compared with the theoretically predictable droplet shape, which can be calculated based on the Gauss-Laplace equation. The interfacial (or surface) tension is then obtained as:
 +
 +$$γ_{12} = g \cdot Δρ \cdot d_1^2 \cdot \frac{1}{H}$$
 +
 +{{ :en:rex310_en_116b.svg?nolink&250 |}}
 +
 +where $g$ is gravitational acceleration, $Δρ$ is the density difference between the polymer phases, and $\frac {1}{H}$ is a correction factor, whose value depends on the shape factor $S$:
 +
 +$$S= \frac {d_2}{d_1}$$
 +
 +Here, $d_1$ is the maximum droplet diameter and $d_2$ the droplet diameter at a distance $d_1$ from the apex. Values of the correction factor $\frac {1}{H}$ can be obtained numerically from tabulated data using the following equations (piecewise in dependence on $S$):
 +
 +$$\frac {1}{H} = (\frac {0,32720}{S^{2,56651}}) - 0,97553 \cdot S^2 + 0,84059 \cdot S - 0,18069$$
 +
 +for $0.401 ≤ S ≤ 0.46$,
 +
 +$$\frac {1}{H} = (\frac {0,31968}{S^{2,39725}}) - 0,46898\cdot S^2 + 0,50059\cdot S - 0,13261$$
 +
 +for $0.46 ≤ S ≤ 0.59$,
 +
 +$$\frac {1}{H} = (\frac {0,31522}{S^{2,62435}}) - 0,11714\cdot S^2 + 0,15756\cdot S - 0,05285$$
 +
 +for $0.59 ≤ S ≤ 0.68$,
 +
 +$$\frac {1}{H} = (\frac {0,31345}{S^{2,61267}}) - 0,09155\cdot S^2 + 0,14701\cdot S - 0,05877$$
 +
 +for $0.68 ≤ S ≤ 0.90$,
 +
 +$$\frac {1}{H} = (\frac {0,30715}{S^{2,84636}}) - 0,69116 \cdot S^3 + 1,08315\cdot S^2 - 0,18341\cdot S - 0,20970$$
 +
 +for $0.90 ≤ S ≤ 1.00$.
 +
 +Thus, the interfacial or surface tension can be calculated from the two droplet diameters and the melt densities. However, it must be ensured that the molten droplet is in equilibrium with its surrounding phase. For low-viscosity Newtonian fluids, this is usually the case immediately, while for highly viscous or viscoelastic melts, equilibrium can take several hours.
 +
 +Apart from the optical requirements, this method is relatively simple in terms of apparatus, but in practice it requires significant skill to generate droplets suitable for evaluation. Additionally, various experimental conditions must be considered. Since density differences $Δρ$ enter the calculation, reliable melt density data are required. However, for polymers, such data are only sparsely available at arbitrary temperatures and thus must be determined experimentally, which introduces error. Further, practical limitations exist: for interfacial tension measurements, the droplet must be formed in the continuous melt phase of a second polymer. Suitable material combinations must therefore meet the conditions of immiscibility, sufficiently low viscosity of the continuous phase, and transparency to enable optical recording. Moreover, gas release or decomposition during melting may cause bubble formation, which distorts the droplet shape and leads to unrealistic results.
 +
 +=== Spinning Drop Method ===
 +The principle of the spinning drop method is also based on the measurement of droplet shape, but here the contour develops under the influence of centrifugal force. When a cylindrical capillary containing a droplet and a denser liquid is rotated around its longitudinal axis at constant high speeds (2000–8000 rpm), the droplet assumes a cylindrical shape with rounded ends.
 +
 +{{ :en:rex310_en_117.svg?nolink&700 |}}
 +
 +The droplet profile is governed by interfacial (or surface) tension, density difference, and centrifugal force. Since gravitational effects can be neglected, the interfacial tension can be expressed as:
 +
 +$$γ_{12} = \frac {Δρ \cdot ω^2}{4 \cdot Q}$$
 +
 +where $ω$ is the angular velocity of the capillary and $Q$ is a constant defined by:
 +
 +$$L_0 = \frac{(\frac{4}{3}) \cdot (Q \cdot R^3 +1)}{(Q \cdot R^3)^\frac{1}{3}}$$
 +
 +Here, $L_0$ is the equilibrium length of the spinning droplet and $R$ is the droplet radius.
 +
 +This method has proven particularly suitable for systems with extremely low interfacial tensions, with modern instrumentation enabling measurements down to $10^{-5}$–$10^{-6}$ mN/m. Moreover, this principle ensures that the interface is not disturbed by foreign objects, and demixing phenomena as well as interfacial phase formation can be observed. A disadvantage of this method is the slow attainment of equilibrium. For example, in measurements with fluids of intermediate viscosity (300–500 Pas), equilibrium was only reached after more than three hours at 6100 rpm.
 +
 +=====Filled polymers=====
 +
 +==== Tensile Strength ====
 +
 +The tensile strength of agglomerates is defined as the maximum tensile force $F_N$, normalized to the cross-sectional area of an agglomerate, with the force acting perpendicular to the surface [[en:ermittlung_der_materialdaten#references|[Sch75]]].
 +
 +
 +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.
 +
 +{{ :en:rex310_en_120.svg?nolink&700 |}}
 +
 +[[en:ermittlung_der_materialdaten#references|[Sch75]]]
 +
 +The tensile strength is therefore obtained from the sum of the individual adhesive forces, normalized to the cross-sectional area of the agglomerates:
 +
 +$$σ_z = \frac{F_{N,max}}{A} = \frac {1}{A} Σ_{i=1} F_{Np,i} (Δl)$$
 +
 +Since the force–strain behavior of agglomerates is largely unknown, this relationship has little practical relevance. Schubert [[en:ermittlung_der_materialdaten#references|[Sch75]]] compiled various approaches for describing the tensile strength of agglomerates. Among these, an approach by Rumpf [[en:ermittlung_der_materialdaten#references|[Rum61]]] for statistically packed, monodisperse particles is presented:
 +
 +$$σ_z = (1-ε) \cdot k \cdot \frac{F_H}{A_p} = \frac {(1-ε)}{ε} \cdot \frac {F_H}{{d_p}^2}$$
 +
 +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.
 +
 +{{ :en:rex310_en_116.svg?nolink&700 |}}
 +
 +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.
 +
 +==== Measurement of Tensile Strength ====
 +
 +The following figure shows the tensile strength testing device according to Parfitt.
 +
 +{{ :en:rex310_en_120.svg?nolink&700 |}}
 +
 +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:ermittlung_der_materialdaten#references|[Sch75]]], [[en:ermittlung_der_materialdaten#references|[PC97]]], [[en:ermittlung_der_materialdaten#references|[Wic91]]].
 +
 +As an example, the tensile strength of talc is plotted as a function of porosity in the figure. 
 +
 +{{ :en:rex310_en_118.svg?nolink&700 |}}
 +
 +The measurements were performed with an apparatus essentially similar to the one described above, which is described in detail in [[en:ermittlung_der_materialdaten#references|[Wic95]]].
 +
 +==== Porosity ====
 +Porosity is defined as the ratio of void volume to total volume [[en:ermittlung_der_materialdaten#references|[Wic95]]]:
 +
 +$$ψ = \frac{V_H}{V_{ges}} $$
 +
 +with $ψ$ = porosity, $V_H$ = void volume, and $V_{tot}$ = total volume.
 +
 +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.
 +
 +One distinguishes between closed and accessible pores, pores with constant diameter, pores that gradually narrow and are only accessible via narrow capillaries, and permeable pores. Surface roughness must also be considered.
 +
 +Pores in single particles determine the particle porosity $ψ_p$. When such particles are agglomerated, the agglomerate porosity $ψ_a$ results, which is defined as the ratio of void volume between particles to the total agglomerate volume. The definition is problematic with respect to edge zones of the agglomerates.
 +
 +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:
 +
 +$$(1-ψ) = (1-ψ_p)(1-ψ_a)(1-ψ_b)$$
 +
 +{{ :en:rex310_en_119.svg?nolink&700 |}}
 +
 +[[en:ermittlung_der_materialdaten#references|[Wic95]]]
 +
 +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:
 +
 +$$ρ_p = (1-ψ_p) \cdot ρ_f$$
 +
 +The agglomerate density is:
 +
 +$$ρ_a = (1-ψ_p) \cdot (1-ψ_a) \cdot ρ_f$$
 +
 +And the bulk density is:
 +
 +$$ρ_b = (1-ψ_p) \cdot (1-ψ_a) \cdot (1-ψ_b) \cdot ρ_f$$
 +
 +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, and bulk structures. Ideally, multiple maxima (multimodal distribution) appear in the pore size distribution.
 +
 +{{ :en:rex310_en_120b.svg?nolink&700 |}}
 +
 +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.
 +
 +=== 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.
 +
 +== Mercury Intrusion Method ==
 +The mercury intrusion method, proposed by Washburn in 1921 [[en:ermittlung_der_materialdaten#references|[Was21]]], 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.
 +
 +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 **(4.75)**:
 +
 +$$p = \frac {2σcos(Θ)}{r} \tag{4.75}$$
 +
 +with $p$ = applied pressure, $σ$ = surface tension of mercury, $Θ$ = contact angle of mercury, and $r$ = pore radius.
 +
 +Using this equation, the pore size distribution can be calculated [[en:ermittlung_der_materialdaten#references|[PHS79]]]. At higher pressures, the compressibility of mercury must also be taken into account.
  
 ===== References ===== ===== References =====
 +
 [Fer80] [Fer80]
  
Zeile 265: Zeile 675:
  
 [Mel95] [Mel95]
 +
 +[Dul79] Dullien, F.A.L.; Porous Media - Fluid Transport and Pore Structure, Academic Press; 1979 
 +
 +[PC97] Pierrat, P.; Caram, H.S.: Tensile strength of wet granular materials, Powder Technology, 91(1997), 83-93 
 +
 +[PHS79] Polke, R.; Herrmann, W.; Sommer, K.: Charakterisierung von Agglomeraten, Chemie Ingenieur Technik, 51(1979)4, 283-288 
 +
 +[Rum61] Rumpf, H.: Agglomeration, Interm. Symposium Philadelphia, 1961, 379-418 
 +
 +[Rum74] Rumpf, H.: Die Wissenschaft des Agglomerierens, Chemie Ingenieur Technik, 46(1974)1, 1-11
 +
 +[Sch75] Schubert, H.: Tensile Strength of Agglomerates, Powder Technology, 11(1975), 107-119
 +
 +[Was21] Washburn, E.W.:, Proc. Nat. Acad. Sci. U.S., 7(1921), 115 
 +
 +[Wic91] Wicke, R.: Agglomeratkennzeichnung und Prüfmethoden, Technische Akademie Wuppertal, Wuppertal, 1991, 1-32 
 +
 +[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