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--- en:reine_polymere [2025/09/11 10:44] – neelest
+++ en:reine_polymere [2026/05/10 23:37] (aktuell) – gelöscht deppe2
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-======Pure polymers======
-A thorough knowledge of the material parameters and the behaviour of the material
-are vitally important to be able to obtain results.
-In order to describe the behavior of the polymers, the following properties must be
-examined:
-  * Rheological properties (shear viscosity)
-  * Thermodynamic properties (crystalline melting / glass transition temperature, heat capacity, specific enthalpy, thermal conductivity)
-  * Density (specific volumes, bulk density, solid material density)
-  * Size of the pellet granule
-===== Rheological Material Parameters =====
-The behavior of the flow of the fluids is described using the law:
-$$τ=η\cdot\dotγ$$
-with the shear stress $τ$, viscosity $η$ and shear rate $\dotγ$.
-Constant viscosities are usually found only for very small and very high shear rates
-for polymer melts. More often polymer melts show a pseudo-plastic behavior, which
-can be described using the power law according to Ostwald and de Waale.
-$$τ=K\cdot\dotγ^n$$
-Here, $n$ is the exponent of the flow law (n‹1) and $K$ is the consistency factor.
-In double logarithmic scale, the viscosity over the shear rate yields a linear pro-file
-with gradient ($n-1$).The gradient of this straight line is dependent on the shear rate,
-therefore for the value of $n$ shear rate ranges must always be given. In addition to this the independent shear rate with zero viscosity cannot be described using the
-power flow law. This difficulty can be solved using the Carreau equation.
-The simulation programmes REX / PSI / SIGMA offer two equations to describe the
-rheological behavior, firstly the Carreau–WLF (Williams, Landel-Ferry) equation and
-secondly the Carreau-Arrhenius equation. The equations differ in that they offer
-different descriptions for the dependence of the temperature on the viscosity. This
-difference was introduced so that data from different origins (CAMPUS; BAYMAT;
-VISCOSITY) are able to be entered without conversions. The evaluation of the
-viscosity function with the Carreau-estimation program offers not only the zero
-viscosity $a$, but the reciprocal transitional shear rate $b$, the gradient $c$, the reference
-temperature $T_b$ and the standard temperature $T_s$ which are necessary for the
-following equation:
-WLF approach:
-$$log(a_T) = \frac {C_1\cdot(T_B-T_S)} {C_2+(T_B-T_S)} - \frac {C_1\cdot(T-T_S)} {C_2+(T-T_S)}$$
-with: $C_1$ = 8.86, $C_2$ = 101.6, $T_B$ = reference temperature, $T_S$ = standard temperature ($T_S ≈ T_G + 50°C$), $T$ = current 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 **[Ferr80]**. 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.
-Alternatively, the following approach can also be used:
-$$ln(a_T) = - \frac {C_1 \cdot (T-T_B)} {C_2+(T-T_B)}$$
-with: $T_B$ = reference temperature, $T$ = current temperature, $C_1$ = adjustment constant, $C_2$ = adjustment constant
-can also be used.
-The Arrhenius approach, which is also available, is:
-$$ln(a_T)=\frac{E}{R} (\frac{1}{T+273.15}-\frac{1}{T_B+273.15}) $$
-where: $T_B$ = reference temperature, $T$ = current temperature, $E$ = activation energy, $R$ = universal gas constant
-{{ :en:rex310_en_088.svg?nolink&700 |}}
-===== Measuring Series with the High Pressure Capillary Rheometer =====
-In order to be able to define the necessary Carreau parameters, measurements have
-to be taken, for instance, with the high-pressure capillary rheometer. Usually the
-experimental series are performed at 3 different temperatures.
-In a high-pressure capillary rheometer, pre-heated material flows through a capillary
-with a circular cross-section. During this process, the total area of the important viscosities is measured. For low viscosity fluids, long thin capillaries are used and for
-high viscosity fluids appropriately high pressures are used. By using this
-discontinuous method, the necessary pressure is made available through carrier gas,
-gravity or by means of pistons. The volume flow rate is constant at a constant piston
-speed. The pressure gradient at the inlet and at the outlet is not constant due to
-vortex formation as a result of viscoelastic effects (Bagley-Correction), due to the
-change of the flow speed (Hagenbach-Correction) and due to the variation of the
-friction at the wall (Couette-Correction). To calculate the exact viscosity of the
-polymer melt, the pressure gradient is measured using two pressure sensors on a
-designated length in the capillary, because here are simple rheological flow
-relationships.
-===== Thermodynamic Material Parameters =====
-The calculation of both the melting behavior and the temperature development in
-power melts requires a comprehensive knowledge of the thermodynamic material
-behavior. The thermodynamic properties are dependent on both pressure and
-temperature and show a different material property in the solid state and in the melt
-state.
-The following data is required for REX / PSI / SIGMA: the crystalline melting
-temperature, glass transition temperature, specific heat capacity and specific
-enthalpy. These can be detected using the DSC (Difference Scanning Calorimetric)-
-analysis.
-===== General Principle of the DSC-Analysis =====
-The principle is based on the measurement of the heat flow between one specimen
-and a comparable substance in a twin calorimeter. The sample and the reference
-substances are arranged symmetrically to each other so that the temperature
-difference is zero. Many chemical and physical transitions like melting, crystallisation,
-oxidation or decomposition of a substance are related to a heat flow i.e. changes in
-enthalpy appear. These enthalpy changes are detected by the DSC-analysis with
-regard to their position in the temperature range and their calorimetric magnitude.
-The specific heat and the enthalpy as a function of temperature can be quantified
-very quickly and very easily. The measurement principle is shown in the following figure.
-{{ :en:rex310_en_089.svg?nolink&700 |}}
-Through a temperature sensor designed as a thermal resistor, a heat flow $\dot Q$ is conducted from the electrically heated furnace body to both the sample crucible and the reference crucible, which are usually identical (same dimensions, same material).
-The heat flow to the reference crucible $\dot Q_R$ is determined by the heat capacity of the crucible material and the inherent thermal losses.
-This also applies to the sample crucible in the case of identical crucible materials and a symmetrical measuring cell.
-However, the specimen enclosed in the sample crucible ($\dot Q_S = \dot Q_R$) generates an additional heat flow $\dot H$ ($dH/dt$), which can now be determined by difference calculation:
-$$\dot H = \dot Q_S - \dot Q_R = \frac{T_P-T_S}{R_t} - \frac{T_P-T_R}{R_t} = \frac{T_S-T_R}{R_t} = - \frac{ΔT}{R_t}$$
-Here, $\dot H$ = heat flow of the specimen, $\dot Q_S$ and $\dot Q_R$ = heat flow to the sample and reference crucible, $R_T$ = thermal resistance of the sensor, $T_P$ = furnace temperature (temperature program), and $T_S$ and $T_R$ = sample and reference temperature.
-Semi-crystalline thermoplastics do not exhibit a sharp melting point like metals but rather a melting range, due to the presence of crystallites of different lamellar thicknesses. Smaller, less perfectly ordered crystallites melt at lower temperatures than larger crystallites.
-A characteristic feature of every semi-crystalline polymer material is the melting or crystallite peak temperature $T_K$. The position of this peak on the temperature axis is defined by the onset temperature ($T_A$), the peak temperature ($T_K$), and the end temperature ($T_E$), which are also of importance when determining other thermodynamic properties, such as the specific enthalpy.
-When plotting the heat flow $\dot H (T)$ as a function of temperature $T$, semi-crystalline thermoplastics exhibit the following characteristic profile.
-{{ :en:rex310_en_090.svg?nolink&700 |}}
-===== Description of Thermodynamic Material Properties =====
-The specific heat capacity $c$ of polymers typically lies in the range of 0.1 to 5 kJ/kg K.
-The following figure illustrates, as an example, the temperature dependence of the specific heat capacity $c$ for a semi-crystalline polyamide.
-{{ :en:rex310_en_091.svg?nolink&700 |}}
-Amorphous polymers exhibit a different heat capacity profile $c(T)$ compared to semi-crystalline polymers. At the glass transition temperature $T_G$, a distinct change in the level of the specific heat capacity $c$ can be observed.
-The next figure shows the specific heat capacity $c$ as a function of temperature for an amorphous polystyrene.
-{{ :en:rex310_en_092.svg?nolink&700 |}}
-For simulation purposes with REX / PSI / SIGMA, the material data obtained from DSC analysis are approximated above the melting end temperature or the end temperature of the softening range $T_E$ by a linear function, as indicated by the dashed lines in Figures 88 and 89 FIXME :
-$$c_p(T) = c_{p,0} + c_{p,m}\cdot T$$
-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 [Meli95].
-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:
-$$Δh = \int \limits_ {T_1}^{T_2} c_p(T)dT$$
-For simulations with REX / PSI / SIGMA, the functional relationship of specific enthalpy $∆h$ over temperature $T$ is not of primary interest. Instead, the characteristic values for semi-crystalline polymers or blends, such as the solid enthalpy $∆h_F$ and the melting enthalpy $∆h_A$, are required.
-The determination of these parameters is exemplified below using a semi-crystalline polyamide. The melting process is initiated by the supply of heat (latent heat of fusion) into the system and is completed upon reaching the end temperature $T_E$. The total enthalpy difference observed at this temperature corresponds to the energy required to completely melt the polymer.
-By constructing tangents in the lower and upper temperature ranges, this total enthalpy $∆h$ can be separated into the partial enthalpies $∆h_F$ and $∆h_A$. In the lower temperature range, a suitable approximation can be achieved with statistical software by comparing the coefficients of determination for different interval widths [$T_F$ ; $T$]. In the upper temperature range, the interval [$T_E$ ; $T_{MAX}$] serves as the basis for tangent determination.
-Amorphous systems, by contrast, show a fundamentally different enthalpy behavior due to the absence of a melting transition. For each amorphous polymer, only the solid enthalpy $∆h_F$ is used as a characteristic value. This corresponds to the enthalpy level at the end of the softening range, at the end temperature $T_E$.
-{{ :en:rex310_en_093.svg?nolink&700 |}}
-The following figure illustrates, as an example, the determination of the solid enthalpy $∆h_F$ for an amorphous polystyrene.
-{{ :en:rex310_en_094.svg?nolink&700 |}}
-===== 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 [Meli95].
-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 measurement is performed in a heated test chamber containing a defined specimen mass $m$ and a sensor in direct contact with the sample. Over a defined measuring period $t$, an energy amount $\dot Q$ is supplied to the specimen.
-{{ :en:rex310_en_092b.svg?nolink&700 |}}
-The temperature change $∆T$ experienced by the specimen during this measuring period is detected by the sensor. From the supplied heat $Q$ and the measured temperature change $∆T$, the thermal conductivity $λ = f(\dot Q , ∆T)$ can be determined. The advantage of this measuring principle compared to the plate method according to DIN 52 612 lies in its ability to measure thermal conductivities $λ$ even at very high temperatures, which are of particular importance for polymeric materials.
-The next figure shows a typical course of thermal conductivity $λ$ as a function of temperature $T$ for semi-crystalline polymers, using polypropylene as an example.
-{{ :en:rex310_en_093b.svg?nolink&700 |}}
-For simulation purposes, a linear equation of the form
-$$λ(T) = λ_0 + λ_m \cdot T$$
-is required, according to the dashed line shown in the figure. The parameters to be determined are the specific thermal conductivity at $0 °C$ ($λ_0$) and the slope of the thermal conductivity function ($λ_m$).
-The temperature dependence of the thermal conductivity $λ$ for an amorphous polystyrene is also shown for comparison.
-{{ :en:rex310_en_094b.svg?nolink&700 |}}