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polykin.copolymerization¤

fit_Finemann_Ross ¤

fit_Finemann_Ross(
    f1: FloatVectorLike, F1: FloatVectorLike
) -> tuple[float, float]

Fit binary copolymer composition data using the Finemann-Ross method.

\[ \left(\frac{x(y-1)}{y}\right) = -r_2 + r_1 \left(\frac{x^2}{y}\right) \]

where \(x = f_1/(1 - f_1)\), \(y = F_1/(1 - F_1)\), \(r_i\) are the reactivity ratios, \(f_1\) is the monomer composition, and \(F_1\) is the instantaneous copolymer composition.

Reference

  • Fineman, M.; Ross, S. D. J. Polymer Sci. 1950, 5, 259.
Note

The Finemann-Ross method relies on a linearization procedure that can lead to significant statistical bias. The method is provided for its historical significance, but should no longer be used for fitting reactivity ratios.

PARAMETER DESCRIPTION
f1

Vector of molar fraction of M1, \(f_1\).

TYPE: FloatVectorLike

F1

Vector of instantaneous copolymer composition of M1, \(F_1\).

TYPE: FloatVectorLike

RETURNS DESCRIPTION
tuple[float, float]

Point estimates of the reactivity ratios \((r_1, r_2)\).
See also

Examples:

>>> from polykin.copolymerization.fitting import fit_Finemann_Ross
>>>
>>> f1 = [0.186, 0.299, 0.527, 0.600, 0.700, 0.798]
>>> F1 = [0.196, 0.279, 0.415, 0.473, 0.542, 0.634]
>>>
>>> r1, r2 = fit_Finemann_Ross(f1, F1)
>>> print(f"r1={r1:.2f}, r2={r2:.2f}")
r1=0.27, r2=0.84
Source code in src/polykin/copolymerization/fitting.py 
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def fit_Finemann_Ross(f1: FloatVectorLike,
                      F1: FloatVectorLike
                      ) -> tuple[float, float]:
    r"""Fit binary copolymer composition data using the Finemann-Ross method.

    $$ \left(\frac{x(y-1)}{y}\right) = -r_2 + r_1 \left(\frac{x^2}{y}\right) $$

    where $x = f_1/(1 - f_1)$, $y = F_1/(1 - F_1)$, $r_i$ are the reactivity
    ratios, $f_1$ is the monomer composition, and $F_1$ is the instantaneous
    copolymer composition.

    **Reference**

    *   Fineman, M.; Ross, S. D. J. Polymer Sci. 1950, 5, 259.

    Note
    ----
    The Finemann-Ross method relies on a linearization procedure that can lead
    to significant statistical bias. The method is provided for its historical
    significance, but should no longer be used for fitting reactivity ratios.


    Parameters
    ----------
    f1 : FloatVectorLike
        Vector of molar fraction of M1, $f_1$.
    F1 : FloatVectorLike
        Vector of instantaneous copolymer composition of M1, $F_1$.

    Returns
    -------
    tuple[float, float]
        Point estimates of the reactivity ratios $(r_1, r_2)$.

    See also
    --------
    * [`fit_copo_data`](fit_copo_data.md): alternative
      (recommended) method.  

    Examples
    --------
    >>> from polykin.copolymerization.fitting import fit_Finemann_Ross
    >>>
    >>> f1 = [0.186, 0.299, 0.527, 0.600, 0.700, 0.798]
    >>> F1 = [0.196, 0.279, 0.415, 0.473, 0.542, 0.634]
    >>>
    >>> r1, r2 = fit_Finemann_Ross(f1, F1)
    >>> print(f"r1={r1:.2f}, r2={r2:.2f}")
    r1=0.27, r2=0.84

    """

    f1 = np.asarray(f1)
    F1 = np.asarray(F1)

    # Variable transformation
    x = f1/(1. - f1)
    y = F1/(1. - F1)
    H = x**2/y
    G = x*(y - 1.)/y

    # Linear regression
    solution = linregress(H, G)
    r1 = solution.slope  # type: ignore
    r2 = - solution.intercept  # type: ignore

    return (r1, r2)