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--- en:grundlagenhandbuch:materialkenngroessen:mischungsregeln_fuer_polymerblends [2025/05/13 15:54] – angelegt deppe2
+++ en:grundlagenhandbuch:materialkenngroessen:mischungsregeln_fuer_polymerblends [2026/01/27 12:53] (aktuell) – [Densities of Binary Systems] neelest
@@ Zeile 1: / Zeile 1: @@
 ====== Mixing Rules for Binary Systems======
+Both the rheological and the thermodynamical characteristic values of a binary system (e.g. solid and melting enthalpy of polymer blends) are generally deemed as being insufficiently describable over a linear average of the data from the raw components. This means for the simulation with SIGMA that before one starts a simulation, the material data of the binary systems should be determined. This would result in numerous measurements before the simulation process actually starts. If one only has the material data for the single components, the calculations for the binary systems using SIGMA will use the material data of the single components in the following ways:
- --> Mixing Rules for Binary Systems fehlt im Deutschen
-  *[[en:Grundlagenhandbuch:Materialkenngrößen:Mischungsregeln für Polymerblends: Rheologische Kenngrößen von Mehrstoffsystemen]]
+===== Rheological Characteristic Values of Binary Systems =====
-  *[[en:Grundlagenhandbuch:Materialkenngrößen:Mischungsregeln für Polymerblends:Thermodynamische Kenngrößen von Mehrstoffsystemen]]
-  *[[en:Grundlagenhandbuch:Materialkenngrößen:Mischungsregeln für Polymerblends:Dichten von Mehrstoffsystemen]]
+To calculate the viscosity of binary systems one can find both simple and mathematically more complex mixing rules. This is dependent on the level of accuracy required. In the simulation software SIGMA the following general mixing rules are applied. The simplest and prevalent logarithmic mixing rule is from Arrhenius:
+$$\log \eta_{MIX} = w_1 \cdot \log \eta_1 + w_2 \cdot \log \eta_2 \tag{1}$$
+The other one is the mixing rule according to Mantford:
+$$\eta_{MIX}^{1/3.4} = w_1 \cdot \eta_1^{1/3.4} + w_2 \cdot \eta_2^{1/3.4} \tag{2}$$
+The determination of the blend viscosity from the viscosity of the single components $\eta$ and the weight contents $w_i$ with SIGMA is shown in the figure.
+{{ :en:grundlagenhandbuch:materialkenngroessen:mischungsregeln_fuer_polymerblends:en_sigma150_dlg_grundlagenhandbuch_materialkenngroessen_016.svg?700%nolink |}}
+In order to calculate the viscosity $\eta_{MIX}$ at a given shear rate $\dot{\gamma}$ for a polymer blend, the blend viscosities $\eta_{MIX 1}$ and $\eta_{MIX 2}$ have to be determined for the defined shear rates $\dot{\gamma}_1 = 0,9 \cdot \dot{\gamma}$ and $\dot{\gamma}_2 = 1,1 \cdot \dot{\gamma}$.
+The approach Mantford uses is as follows:
+$$\eta_{MIX 1}^{1/3.4} = w_1 \cdot \eta_{1.1}^{1/3.4} + w_2 \cdot \eta_{2.1}^{1/3.4} \tag{3}$$
+$$\eta_{MIX 2}^{1/3.4} = w_1 \cdot \eta_{1.2}^{1/3.4} + w_2 \cdot \eta_{2.2}^{1/3.4} \tag{4}$$
+Analogous to it the logarithm mixing rule (Arrhenius) is:
+$$\log \eta_{MIX 2} = w_1 \cdot \log \eta_{1.1} + w_2 \cdot \log \eta_{2.1} \tag{5}$$
+$$\log \eta_{MIX 2} = w_1 \cdot \log \eta_{1.2} + w_2 \cdot \log \eta_{2.2} \tag{6}$$
+If these values are known the flow index n and the consistence K of the estimated blend segment between $\dot{\gamma}_1 = \dot{\gamma}_{MIX 1}$ and $\dot{\gamma}_2 = \dot{\gamma}_{MIX 2}$ are:
+$$n = 1 + \frac{\log \left(\frac{\eta_{MIX 1}}{\eta_{MIX 2}}\right)}{\log \left(\frac{\dot{\gamma}_{MIX 1}}{\dot{\gamma}_{MIX 2}}\right)} \tag{7}$$
+$$K = \frac{\eta_{MIX 1}}{\dot{\gamma}_{MIX 1}^{n-1}} \tag{8}$$
+The desired viscosity $\eta_{MIX}$ results in:
+$$\eta_{MIX} = K \cdot \dot{\gamma}_{MIX}^{n-1} \tag{9}$$
+If the disperse Phase is available in either solid or high viscous forms one can call it a filled system. For the simulation of a filled polymer one must firstly define its basic polymer content and the size of the granules' diameter. After this the filler must be defined. Therefore data on several materials is necessary: The granule diameter d, the mass content w, the solid density $\rho$, the bulk density $\lambda_0$, the thermal conductivity of the solid $\rho_S$ and the specific heat capacity c0. As with the polymer blend the filled polymers can be defined using two different mixing equations. The former is based on the approach by Einstein:
+$$\eta_{MIX} = \eta_1 \cdot (1 + 2,5 \cdot \phi_2) \tag{10}$$
+and the second is based on the approach by Hashin:
+$$\eta_{MIX} = \eta_1 \cdot \left[1 + 2 \cdot \frac{\phi_2}{1 - \phi_2}\right] \tag{11}$$
+In these eqs. $\eta_1$ represents the viscosity of the basic polymer and $\phi_2$ refers to the volumetric content of the added component.
+===== Characteristic Thermodynamic Values of Binary Systems =====
+The experimentally crystallite melting temperatures measured for the polymer blends as well as for filled systems (compounds) are shown in the figure. For polymer blends SIGMA calculates the crystallite melt temperature $T_{K.MIX}$ from the weight contents wi and the crystalline melt temperature $T_{K i}$ as follows:
+$$T_{K.MIX} = w_1 \cdot T_{K.1} + w_2 \cdot T_{K.2} \tag{1}$$
+for compounds:
+$$T_{K.MIX} = T_{K.1} \tag{2}$$
+So the crystalline melt temperature $T_{K.MIX}$ of a filled polymer (compound) is assumed to be equal to the crystallite melt temperature $T_{K 1}$ of the basic polymer.
+{{ :en:grundlagenhandbuch:materialkenngroessen:mischungsregeln_fuer_polymerblends:en_sigma150_dlg_grundlagenhandbuch_materialkenngroessen_017.svg?700%nolink |}}
+**Figure:** Crystalline melt temperature $T_{K i}$ of binary systems: polymer blends (left) and compounds (right).
+In the description of the specific heat capacity c of binary systems one should use a consistent mixing rule for both the polymer blends and for the filled systems. For the specific heat capacity of the blend $c_{MIX}$ one uses:
+$$c_{MIX} = w_1 \cdot c_1 + w_2 \cdot c_2 \tag{3}$$
+Where the weight content wi and the specific heat capacity ci correspond to the single added component respectively.
+The specific heat capacity $c_{0,MIX}$ of the blend is set to:
+$$c_{0,MIX} = w_1 \cdot c_{0.1} + w_2 \cdot c_{0.2} \tag{4}$$
+In the figure one can see the profile of the specific heat capacity c0 for polymer blends and filled systems.
+Also the slope of the heat capacity profile $c_{m.MIX}$ for the blend is determined for both systems analogous to eqn. (3):
+$$c_{m.MIX} = w_1 \cdot c_{m.1} + w_2 \cdot c_{m.2} \tag{5}$$
+The relationship between the specific heat capacity cm and the weight contents for polymer blends and filled systems is shown in the figure:
+{{ :en:grundlagenhandbuch:materialkenngroessen:mischungsregeln_fuer_polymerblends:en_sigma150_dlg_grundlagenhandbuch_materialkenngroessen_018.svg?700%nolink |}}
+{{ :en:grundlagenhandbuch:materialkenngroessen:mischungsregeln_fuer_polymerblends:en_sigma150_dlg_grundlagenhandbuch_materialkenngroessen_019.svg?700%nolink |}}
+**Figure:** Heat capacity cm of binary systems: polymer blends (left) and filled systems (right).
+The specific enthalpy is set at:
+$$\Delta h = w_1 \cdot \Delta h_1 + w_2 \cdot \Delta h_2 = w_1 \cdot (\Delta h_F + \Delta h_A)_1 + w_2 \cdot (\Delta h_F + \Delta h_A)_2 \tag{6}$$
+Index "1" is related to the polymer 1 and index "2" to the polymer 2. To describe the specific enthalpy δh of filled systems (compounds) one uses the following approach:
+$$\Delta h = w_1 \cdot (\Delta h_F + \Delta h_A)_1 + w_2 \cdot c_2 \cdot \Delta T \tag{7}$$
+Here index "1" is related to the polymer and index "2" to the filler.
+The solid enthalpy $\Delta h_F$ and the melt enthalpy $\Delta h_A$ for polymer blends can be determined using the enthalpy values of the basic components. For the solid enthalpy $\Delta h_{F.MIX}$ melt enthalpy of the blend $\Delta h_{A.Mix}$ one uses:
+$$\Delta h_{F.MIX} = w_1 \cdot \Delta h_{F.1} + w_2 \cdot \Delta h_{F.2} \tag{8}$$
+$$\Delta h_{A.MIX} = w_1 \cdot \Delta h_{A.1} + w_2 \cdot \Delta h_{A.2} \tag{9}$$
+The compounds (filled system) solid enthalpy $\Delta h_{F.MIX}$ and the melt enthalpy $\Delta h_{A.MIX}$ of the blend are calculated as follows:
+$$\Delta h_{F.MIX} = w_1 \cdot \Delta h_{F.1} \tag{10}$$
+$$\Delta h_{A.MIX} = w_1 \cdot \Delta h_{A.1} + w_2 \cdot c_2 \cdot \Delta T \tag{11}$$
+Both figures show examples for results of experimental analysis of the specific solid enthalpy $\Delta h_F$ and the specific melt enthalpy $\Delta h_A$ of binary systems.
+{{ :en:grundlagenhandbuch:materialkenngroessen:mischungsregeln_fuer_polymerblends:en_sigma150_dlg_grundlagenhandbuch_materialkenngroessen_020.svg?700%nolink |}}
+{{ :en:grundlagenhandbuch:materialkenngroessen:mischungsregeln_fuer_polymerblends:en_sigma150_dlg_grundlagenhandbuch_materialkenngroessen_021.svg?700%nolink |}}
+**Figure:** Specific solid enthalpy $\Delta h_A$ of binary systems: polymer blends (left) and filled systems (right).
+In the simulation program SIGMA the following equations are used to determine the thermal conductivity $\lambda_{MIX}$ for binary systems from the values of the single component polymers (wi,λi). For polymer blends the following equation is used:
+$$\lambda_{MIX} = w_1 \cdot \lambda_1 + w_2 \cdot \lambda_2 \tag{12}$$
+For filled polymers (compounds) the following relationship is assumed for the calculation of the thermal conductivity:
+$$\lambda_{MIX} = \lambda_2 \cdot \frac{\lambda_1 + 2 \cdot \lambda_2 - 2 \cdot \phi_1 \cdot (\lambda_2 - \lambda_1)}{\lambda_1 + 2 \cdot \lambda_2 + \phi_1 \cdot (\lambda_2 - \lambda_1)} \tag{13}$$
+The index "1" and "2" correspond to the mixed single components 1 and 2 when using polymer blends. When using the filled system, the index "1" stands for the basic polymer and in the case of the added filler, the index "2". Every component uses a linear estimation like the one mentioned in Chapter Measuring the Porosity:
+$$\lambda_i = \lambda_{0i} + \lambda_{mi} \cdot T \tag{14}$$
+Where λ0i is the value of the linear estimation function for the thermal conductivity of one of the components at the temperature T = 0°C and λm,i is the slope of the approximated thermal conductivity profile of the component for temperatures in the area above the melt temperature. For polymer blends the thermal conductivity of the blend λ0,MIX is:
+$$\lambda_{0,MIX} = w_1 \cdot \lambda_{0,1} + w_2 \cdot \lambda_{0,2} \tag{15}$$
+The slope of the thermal conductivity function λm,MIX of the blend is calculated using:
+$$\lambda_{m,MIX} = w_1 \cdot \lambda_{m,1} + w_2 \cdot \lambda_{m,2} \tag{16}$$
+For filled polymers (compounds) the following mixing rules result if one wants to calculate λ0,MIX and λm,MIX :
+$$\lambda_{0,MIX} = \lambda_{0,2} \cdot \frac{\lambda_{0,1} + 2 \cdot \lambda_{0,2} - 2 \cdot \phi_1 \cdot (\lambda_{0,2} - \lambda_{0,1})}{\lambda_{0,1} + 2 \cdot \lambda_{0,2} + \phi_1 \cdot (\lambda_{0,2} - \lambda_{0,1})} \tag{17}$$
+and
+$$\lambda_{m,MIX} = \lambda_{m,2} \cdot \frac{\lambda_{m,1} + 2 \cdot \lambda_{m,2} - 2 \cdot \phi_1 \cdot (\lambda_{m,2} - \lambda_{m,1})}{\lambda_{m,1} + 2 \cdot \lambda_{m,2} + \phi_1 \cdot (\lambda_{m,2} - \lambda_{m,1})} \tag{18}$$
+===== Densities of Binary Systems ======
+The solid density and the melt density ρMIX of polymer blends and compounds can be calculated by means of the densities ρi and the weight contents wi of the individual components and the following equation:
+$$\frac{1}{\rho_{MIX}} = \frac{w_1}{\rho_1} + \frac{w_2}{\rho_2} \tag{1}$$
+The calculation of the specific volume v of polymer blends and compounds from the specific volume vi and the weight contents wi of the single components resp. of the basic polymer and the filler are described in the following equation:
+$$v_{MIX} = w_1 \cdot v_1 + w_2 \cdot v_2 \tag{2}$$
+For polymer blends and filled polymers (compounds) the following mixing equations result when one wants to calculate v0MIX and vmMIX:
+$$v_{0,MIX} = w_1 \cdot v_{0,1} + w_2 \cdot v_{0,2} \tag{3}$$
+and
+$$v_{m,MIX} = w_1 \cdot v_{m,1} + w_2 \cdot v_{m,2} \tag{4}$$
+To determine the bulk density $\rho_{S.MIX}$ of binary systems like polymer blends and compounds with a granule diameter $d_1$ and $d_2$ (where $d_1 < d_2$) of the single components respective of the viscosity polymer and the filler, one must firstly make the following considerations. In the figure one can see the bulk density $\rho_{S.MIX}$ of a filler of two different particle fractions.
+{{ :en:grundlagenhandbuch:materialkenngroessen:mischungsregeln_fuer_polymerblends:en_sigma150_dlg_grundlagenhandbuch_materialkenngroessen_022.svg?700&nolink |}}
+One must take into account that the diameter of the smaller granule particle fraction is much smaller than the granule diameter of the larger fraction (d1 << d2). If one draws the bulk density ρS MIX of a fill of two different particle fractions as a function of the weight content of the smaller fraction w1 every profile will have a maximum in the saturation concentration w1 = wSät independent of the porosity e. The degree of saturation concentration is defined as follows:
+$$w_{sat} = \frac{\rho_2 \cdot (1 - \rho_{\infty}) \cdot (1 - e)}{\rho_1 + \rho_2 \cdot (1 - \rho_{\infty}) \cdot (1 - e)} \tag{5}$$
+The limiting degree for a face-centred cubic sphere is:
+$$p_{\infty} = \frac{V_{sphere}}{V_{total}} = \frac{\pi}{3 \cdot \sqrt{2}} \approx 0,74 \tag{6}$$
+Thereby, VKugel is the volume occupied by the larger sized material (Mat 2) related to the total volume Vgesamt. The porosity e is set to e = 0,25 and is too small for the smaller material component (mat. 1) to fit in. Hence the following equation results for the saturation bulk density (bulk density of the mixing region):
+$$\rho_{sat} = \rho_1 + (1 - \rho_{\infty}) \cdot (1 - e) \cdot \rho_2 \tag{7}$$
+If one wants to determine the bulk density ρS MIX of binary systems using polymer blends and compounds with $d_1 < d_2$, three cases are distinguished. The strategy of the three cases is shown in the figure. The ratio of the particle fractions ($d_1/d_2$) is set to the sub division criteria for every case. Firstly one assumes two different basic approaches  ($d_1 < 0,25d_2$ resp. $d_1 > 0,75d_2$) while the third basic approach is calculated by the linear average of the first two approaches in the transition zone ($0,25d_2 < d_1 < 0,75d_2$).
+**Bulk density of binary Systems**
+**(Polymerblends and Compounds with $d_1 < d_2$)**
+**Case I**
+**(if $d_1 << d_2$ i. e. $d_1 < 0,25 d_2$)**
+  * if $w < w_{Sät}$
+$$\rho_{Mix} = \left(1 - \frac{w_1}{w_{Sät}}\right)\rho_1 + \frac{w_1}{w_{Sät}} \rho_{Sät}$$
+  * if $w \geq w_{Sät}$
+$$\rho_{Mix} = \left(1 - \frac{1 - w_1}{1 - w_{Sät}}\right)\rho_2 + \frac{1 - w_1}{1 - w_{Sät}} \rho_{Sät}$$
+**Case II**
+**(if $d_1 > 0,75 d_2$)**
+$$\frac{1}{\rho_{Mix}} = \frac{w_1}{\rho_1} + \frac{w_2}{\rho_2}$$
+**Case III**
+**(if $0,25 d_2 < d_1 < 0,75 d_2$)**
+$$\rho_{Mix} = (\rho_{Mix I} + \rho_{Mix II}) / 2$$
+= linear averaging of Case I and II