polykin.copolymerization

radical_fractions_ternary

radical_fractions_ternary(
    f1: Union[float, FloatArrayLike],
    f2: Union[float, FloatArrayLike],
    k12: float,
    k21: float,
    k13: float,
    k31: float,
    k23: float,
    k32: float,
) -> tuple[
    Union[float, FloatArray],
    Union[float, FloatArray],
    Union[float, FloatArray],
]

Calculate the radical fractions for a ternary system.

In a ternary system, the radical fractions \(p_i\) are related to the monomer composition \(f_i\) by:

\[\begin{aligned} a &= k_{21}k_{31}f_1^2+k_{21}k_{32}f_1f_2+k_{23}k_{31}f_1f_3 \\ b &= k_{12}k_{31}f_1f_2+k_{12}k_{32}f_2^2+k_{13}k_{32}f_2f_3 \\ c &= k_{12}k_{23}f_2f_3+k_{13}k_{21}f_1f_3+k_{13}k_{23}f_3^2 \\ p_1 &= \frac{a}{a+b+c} \\ p_2 &= \frac{b}{a+b+c} \\ p_3 &= \frac{c}{a+b+c} \end{aligned}\]

where \(k_{ij}\) are the cross-propagation rate coefficients. Note that the homo-propagation rate coefficients \(k_{ii}\) do not appear in the equations. For this reason, radical fractions cannot be evaluated from reactivity ratios alone.

References

• Hamielec, A.E., MacGregor, J.F. and Penlidis, A. (1987), Multicomponent free-radical polymerization in batch, semi- batch and continuous reactors. Makromolekulare Chemie. Macromolecular Symposia, 10-11: 521-570.

PARAMETER	DESCRIPTION
f1	Molar fraction of M1. TYPE: float | FloatArrayLike
f2	Molar fraction of M2. TYPE: float | FloatArrayLike
k12	Propagation rate coefficient. TYPE: float
k21	Propagation rate coefficient. TYPE: float
k13	Propagation rate coefficient. TYPE: float
k31	Propagation rate coefficient. TYPE: float
k23	Propagation rate coefficient. TYPE: float
k32	Propagation rate coefficient. TYPE: float

RETURNS	DESCRIPTION
tuple[float | FloatArray, ...]	Radical fractions, \((p_1, p_2, p_3)\).

See also

• radical_fractions_multi: generic method for multicomponent systems.

Examples:

>>> from polykin.copolymerization import radical_fractions_ternary
>>> p1, p2, p3 = radical_fractions_ternary(
...              f1=0.5, f2=0.3, k12=500., k21=50.,
...              k13=30., k31=200., k23=300., k32=40.)
>>> print(f"p1 = {p1:.2f}; p2 = {p2:.2f}; p3 = {p3:.2f}")
p1 = 0.25; p2 = 0.48; p3 = 0.27

Source code in src/polykin/copolymerization/multicomponent.py

def radical_fractions_ternary(f1: Union[float, FloatArrayLike],
                              f2: Union[float, FloatArrayLike],
                              k12: float,
                              k21: float,
                              k13: float,
                              k31: float,
                              k23: float,
                              k32: float,
                              ) -> tuple[Union[float, FloatArray],
                                         Union[float, FloatArray],
                                         Union[float, FloatArray]]:
    r"""Calculate the radical fractions for a ternary system.

    In a ternary system, the radical fractions $p_i$ are related to the
    monomer composition $f_i$ by:

    \begin{aligned}
    a &= k_{21}k_{31}f_1^2+k_{21}k_{32}f_1f_2+k_{23}k_{31}f_1f_3 \\
    b &= k_{12}k_{31}f_1f_2+k_{12}k_{32}f_2^2+k_{13}k_{32}f_2f_3 \\
    c &= k_{12}k_{23}f_2f_3+k_{13}k_{21}f_1f_3+k_{13}k_{23}f_3^2 \\
    p_1 &= \frac{a}{a+b+c} \\
    p_2 &= \frac{b}{a+b+c} \\
    p_3 &= \frac{c}{a+b+c}
    \end{aligned}

    where $k_{ij}$ are the cross-propagation rate coefficients. Note that the
    homo-propagation rate coefficients $k_{ii}$ do not appear in the equations.
    For this reason, radical fractions cannot be evaluated from reactivity
    ratios alone.

    **References**

    *   Hamielec, A.E., MacGregor, J.F. and Penlidis, A. (1987), Multicomponent
    free-radical polymerization in batch, semi- batch and continuous reactors.
    Makromolekulare Chemie. Macromolecular Symposia, 10-11: 521-570.

    Parameters
    ----------
    f1 : float | FloatArrayLike
        Molar fraction of M1.
    f2 : float | FloatArrayLike
        Molar fraction of M2.
    k12 : float
        Propagation rate coefficient.
    k21 : float
        Propagation rate coefficient.
    k13 : float
        Propagation rate coefficient.
    k31 : float
        Propagation rate coefficient.
    k23 : float
        Propagation rate coefficient.
    k32 : float
        Propagation rate coefficient.

    Returns
    -------
    tuple[float | FloatArray, ...]
        Radical fractions, $(p_1, p_2, p_3)$.

    See also
    --------
    * [`radical_fractions_multi`](radical_fractions_multi.md):
      generic method for multicomponent systems.

    Examples
    --------
    >>> from polykin.copolymerization import radical_fractions_ternary
    >>> p1, p2, p3 = radical_fractions_ternary(
    ...              f1=0.5, f2=0.3, k12=500., k21=50.,
    ...              k13=30., k31=200., k23=300., k32=40.)
    >>> print(f"p1 = {p1:.2f}; p2 = {p2:.2f}; p3 = {p3:.2f}")
    p1 = 0.25; p2 = 0.48; p3 = 0.27
    """

    f1 = np.asarray(f1)
    f2 = np.asarray(f2)
    f3 = 1. - (f1 + f2)

    p1 = k21*k31*f1**2 + k21*k32*f1*f2 + k23*k31*f1*f3
    p2 = k12*k31*f1*f2 + k12*k32*f2**2 + k13*k32*f2*f3
    p3 = k12*k23*f2*f3 + k13*k21*f1*f3 + k13*k23*f3**2

    denominator = p1 + p2 + p3

    p1 /= denominator
    p2 /= denominator
    p3 /= denominator

    return (p1, p2, p3)