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

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

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.

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.

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.

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.

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.

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.

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 the thermograms above:

$$c_p(T) = c_{p,0} + c_{p,m}\cdot T$$

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 $T_E$.

The following figure illustrates, as an example, the determination of the solid enthalpy $∆h_F$ for an amorphous polystyrene.

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.

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.

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.

In the evaluation of DSC analyses, the supplied energy is plotted as a function of temperature. For semi-crystalline thermoplastics, the crystallization temperature $T_K$ can be determined from the peak maximum of the curve. In the case of amorphous thermoplastics, the glass transition temperature $T_G$ can be obtained from the inflection point of the curve.

When plotting the specific volume $v$ as a function of temperature $T$ for a semi-crystalline polymer, it becomes evident that the behavior over the entire temperature range can only be described mathematically with difficulty, if at all.

The following figure additionally shows the specific volume $v$ as a function of temperature $T$ for an amorphous polystyrene.

For the simulation of melt-dominated extruders, however, the primary interest lies in the temperature range above the melting temperature $T_K$ or the glass transition temperature $T_G$. REX / PSI / SIGMA requires a description of the volume or density function in the following form:

$$v(T) = v_0 + v_m \cdot T$$ $$ρ(T) = ρ_0 - ρ_m \cdot T$$

where: $v_0$ = specific volume $v_m$ = slope of the volume function $ρ_0$ = specific density $ρ_m$ = slope of the density function

The procedure for determining the bulk density $ρ_s$ is defined in DIN 53 466.

The bulk density $ρ_s$ is mainly used to determine the maximum mass throughput $\dot m$ of the system and the filling degree $f$ in the solids conveying section. For example, the required screw speed range for a given throughput can be calculated, or conversely, the maximum possible throughput can be determined at a specified screw speed. The evaluation is carried out according to the following equation:

$$ρ_s = \frac{m_1 - m_0}{V_0}$$

with $m_1$ = mass of the container filled with the specimen, $m_0$ = mass of the empty container, and $V_0$ = volume of the container

The solid densities can be determined according to DIN 53 479. This method (buoyancy method) compares the weight of a defined sample mass in air with its apparent weight in a liquid medium (here: distilled water, $ρ_{H_2O}$ = 1.000 g/cm³).

For the solid density of the specimen, the following relation applies:

$$ρ = \frac{m_1 \cdot ρ_{H_2O}}{m_1 - m_2}$$

where $m_1$ = dry mass in air and $m_2$ = specimen mass in the buoyant medium.

For the determination of pellet size, the specimens are first averaged volumetrically from a total of $n$ pellets and subsequently converted into spherical form. The determination of the pellet size is illustrated schematically in the following figure and can be derived from the relation:

An alternative approach is to calculate the pellet diameter $d$ from a total of $n$ specimens using the solid density $ρ$ and the total mass of the specimens $m_{ges}$. For the total of $n$ specimens, the following holds:

$$V_{ges} = \frac{m_{ges}}{ρ}$$

For a single averaged pellet, this results in:

$$V = \frac{V_{ges}}{n}$$

The pellet diameter $d_{sphere}$ can then be determined using the following equation.