polykin.transport.rheology

aT_WLF

aT_WLF(
    T: float,
    T0: float,
    C1: float | None = None,
    C2: float | None = None,
) -> float

Calculate the temperature shift factor using the Williams-Landel-Ferry equation.

The temperature shift factor \(a_T\) at a given temperature \(T\) is calculated using the following equation:

\[ a_T = 10^{\frac{-C_1 (T - T_0)}{C_2 + (T - T_0)}} \]

where \(T_0\) is the reference temperature, and \(C_1\) and \(C_2\) are the WLF constants.

If \(C_1\) and \(C_2\) are not provided, the universal values (\(C_1 = 17.44\) and \(C_2 = 51.60\) K) are used, in which case \(T_0\) is expected to be the glass-transition temperature \(T_g\).

The application of this equation is tipically limited to the range \(T_g \leq T \leq T_g + 100\) K.

References

• Stephen L. Rosen, "Fundamental principles of polymeric materials", Wiley, 2nd edition, 1993, p. 339.

PARAMETER	DESCRIPTION
T	Temperature (K). TYPE: float
T0	Reference temperature (K), usually taken to be the glass transition temperature. TYPE: float
C1	WLF constant. If None, the universal value is used. TYPE: float | None DEFAULT: None
C2	WLF constant (K). If None, the universal value is used. TYPE: float | None DEFAULT: None

RETURNS	DESCRIPTION
float	Temperature shift factor.

Examples:

Determine the temperature shift factor at 120 °C for polystyrene, given that its glass transition temperature is 100 °C.

>>> from polykin.transport import aT_WLF
>>> T = 120 + 273.15  # K
>>> T0 = 100 + 273.15 # K
>>> aT = aT_WLF(T, T0)
>>> print(f"aT={aT:.2e}")
aT=1.34e-05

Source code in src/polykin/transport/rheology.py

def aT_WLF(T: float,
           T0: float,
           C1: float | None = None,
           C2: float | None = None,
           ) -> float:
    r"""Calculate the temperature shift factor using the Williams-Landel-Ferry
    equation.

    The temperature shift factor $a_T$ at a given temperature $T$ is calculated
    using the following equation:

    $$ a_T = 10^{\frac{-C_1 (T - T_0)}{C_2 + (T - T_0)}} $$

    where $T_0$ is the reference temperature, and $C_1$ and $C_2$ are the WLF
    constants. 

    If $C_1$ and $C_2$ are not provided, the universal values ($C_1 = 17.44$ 
    and $C_2 = 51.60$ K) are used, in which case $T_0$ is expected to be the 
    glass-transition temperature $T_g$.

    The application of this equation is tipically limited to the range
    $T_g \leq T \leq T_g + 100$ K.

    **References**

    * Stephen L. Rosen, "Fundamental principles of polymeric materials", Wiley,
      2nd edition, 1993, p. 339.

    Parameters
    ----------
    T : float
        Temperature (K).
    T0 : float
        Reference temperature (K), usually taken to be the glass transition 
        temperature.
    C1 : float | None
        WLF constant. If None, the universal value is used.
    C2 : float | None
        WLF constant (K). If None, the universal value is used.

    Returns
    -------
    float
        Temperature shift factor.

    Examples
    --------
    Determine the temperature shift factor at 120 °C for polystyrene, given 
    that its glass transition temperature is 100 °C.
    >>> from polykin.transport import aT_WLF
    >>> T = 120 + 273.15  # K
    >>> T0 = 100 + 273.15 # K
    >>> aT = aT_WLF(T, T0)
    >>> print(f"aT={aT:.2e}")
    aT=1.34e-05
    """

    if C1 is None and C2 is None:
        C1 = 17.44
        C2 = 51.60
    elif C1 is not None and C2 is not None:
        pass
    else:
        raise ValueError("Both C1 and C2 must be provided, or neither.")

    return 10 ** (-C1 * (T - T0) / (C2 + T - T0))