Supported Material Models

The Specific Volume (pvT) supported material models for Material Modeler are described here. These models describe the behavior of a single cooling rate.

Dual Domain Tait

Thermodynamic Parameters: Equations of state are often written using parameters for pressure $(\tilde{P})$ , volume $(\tilde{v})$ , and temperature $(\tilde{T})$ , which relate to universal constants.

van der Waals Equation: The vander Waals equation improves the ideal gas law. However, it cannot adequately describe the complex interactions present in real fluids, such as non-Newtonian polymer melts.

Tait Empirical PVT Relation: Developed in 1888, the Tait empirical PVT (Pressure-Volume-Temperature) relation can accurately reproduce experimental liquid densities for polymers in both melt and glassy states. The equation, as it is used for injection molding simulation software, is as follows:

$V (T, p) = V_{0} (T) [1 - C 1 n (1 + \frac{p}{B (T)})] + V_{t r} (T, p)$

Where:

$V (T, p)$ is the specific volume at temperature $T$ and pressure $p$ .
$V_{0} (T)$ is the specific volume at a reference state.
$C$ is a constant.
$B (T)$ is a temperature-dependent function.
$V_{t r} (T, p)$ accounts for additional volume changes due to temperature and pressure.

Tait Equation - Solid and Liquid Domains

Figure 1. Specific Volume vs Temperature

Note: Plot of specific volume versus temperature and pressure for a semicrystalline polymer.

Solid Domain: $V_{0} = b_{1 s} + b_{2 s} (T - b_{5})$ $B (T) = b_{3 s} e^{[- b_{4 s} (T - b_{5})]}$ $V_{t r} (T, p) = b_{7} e^{[(b_{8} (T - b_{5}) - (b_{9} p))]}$ The data fit coefficients are: $b_{1 s'} b_{2 s'} b_{3 s'} b_{4 s'} b_{5'} b_{6'} b_{7'} b_{8'} b_{9'}$

Molten Domain: $V_{0} = b_{1 m} + b_{2 m} (T - b_{5})$ $B (T) = b_{3 m} e [- b_{4 m} (T - b_{5})]$ $V_{t r} (T, p) = 0$ The data fit coefficients are: $b_{1 m'} b_{2 m'} b_{3 m'} b_{4 m'} b_{5} a n d b_{6}$

$T_{t r} (p) = b_{5} + b_{6} p$ $b_{5} = T_{t r} (0)$

Understanding the following distinctions and the role of the B7 parameter is important for accurately fitting specific volume data.

The Tait model describes the behavior of specific volume in different phases of materials, specifically for semi-crystalline and amorphous polymers. When applying this model, we consider different domains and parameters to capture the material's characteristics accurately.

Morphological Types

Semi-crystalline Materials observe distinct behaviors in both solid and liquid domains. These materials transition between phases, and this transition can be described using the Tait equation parameters.

Amorphous Materials exhibit a different behavior compared to semi-crystalline ones. The key difference lies in the B7 parameter, which helps to describe the specific volume behavior in the amorphous state. For purely amorphous materials, the B7 parameter is zero, resulting in two linear segments that meet at a transition point.

Hybrid Morphology Some materials exhibit hybrid morphology, which combines characteristics of both amorphous and semi-crystalline materials. These materials present a non-trivial curvature in their specific volume data, which cannot be described by a zero B7 parameter. Instead, the B7 parameter takes on a real value to accurately represent this curvature.

Other Morphologies Some materials exhibit volumetric behavior different than the three described above. Polyetherimide (PEI) and several polymer alloys exhibit a volume reduction as temperature increases when the pressure is well above one atmosphere. Their curves would seem to change their morphological behavior as the pressure increases. Because the molecules are large, and especially in the case of polymer alloys, when the temperature increases, the ability of the molecules to mix increases (i.e. they become more miscible). This results in a volume reduction similar to, but more exaggerated than the often used example of mixing ethanol and water in equal volume amounts (50ml of water + 50 ml of ethanol → 96-97ml of mixture, not 100ml. These morphologies are not yet supported by Material Modeler's automated Specific Volume Fitter.

Dual Domain IKV

Equations for Specific Volume:

Molten: $V_{s p} = \frac{P_{1}}{P_{4} + P} + \frac{P_{2}}{P_{3} + p} \cdot T$
Solid: $V_{s p} = \frac{P_{1}}{P_{4} + P} + \frac{P_{2}}{P_{3} + p} \cdot T + P_{5} \cdot e^{P_{6} T - P_{7} p}$

Parameters:

$P T_{1'} P T_{2}$
$P_{1 m'} P_{2 m'} P_{3 m'} P_{4 m}$ $P_{1 s'} P_{2 s'} P_{3 s'} P_{4 s'} P_{5'} P_{6'} P_{7}$

Note: Operates directly on Temperature, not

∆ T

. Temperature units are fixed to those from fitting.

Schmidt and IKV

Differ only by arrangement of variables
Equations operate directly on temperature

Material Modeler assumes cgs units for IKV, and SI for Schmidt.

Modified Schmidt

Uses the Schmidt arrangement of variables.
Operates on (T - T₀), not directly on the temperature value.
Units can be converted for this model.

Renner: Continuous Model

Renner with Crystallization:

Continuous model with crystallization fraction as an input

Wang Polynomial:

Includes cooling rate
18 Parameters.

Dual Domain Schmidt

Equations of State:

Molten: $V_{s p} = \frac{A_{1}}{A_{2} + P} + \frac{B_{1}}{B_{2} + P} \cdot T$
Solid: $V_{s p} = \frac{A_{1}}{A_{2} + P} + \frac{B_{1}}{B_{2} + P} \cdot T + C_{1} \cdot e^{C_{2} T - C_{3} P}$

Parameters:

$P T_{1'} P T_{2}$
$A_{1 m'} A_{2 m'} B_{1 m'} B_{2 m}$
$A_{1 s'} A_{2 s'} B_{1 s'} B_{2 s'} C_{1'} C_{2'} C_{3}$

Important Note:

Operates directly on Temperature, not $∆ T$ .
Parameters for pressure dependence are fixed to those from fitting because of the way temperature is managed.

Dual Domain Modified Schmidt

$∆ T = T - (P T_{1} + P T_{2} \cdot P)$

$V_{s p} = \frac{A_{1}}{A_{2} + P} + \frac{B_{1}}{B_{2} + P} \cdot ∆ T + C_{1} e^{C_{2} \cdot ∆ T - C_{3} \cdot P}$

Because this model operates on the difference of the temperature to its transition temperature, the parameters in this model can be converted to any consistent unit system.