polykin.copolymerization

convert_Qe_to_r

convert_Qe_to_r(
    Qe_values: list[tuple[float, float]]
) -> FloatSquareMatrix

Convert Q-e values to reactivity ratios.

According to the Q-e scheme proposed by Alfrey and Price, the reactivity ratios of the terminal model can be estimated using the relationship:

\[ r_{ij} = \frac{Q_i}{Q_j}\exp{\left(-e_i(e_i -e_j)\right)} \]

where \(Q_i\) and \(e_i\) are monomer-specific constants, and \(r_{ij}=k_{ii}/k_{ij}\) is the multicomponent reactivity ratio matrix.

References

• T Alfrey, CC Price. J. Polym. Sci., 1947, 2: 101-106.

PARAMETER	DESCRIPTION
Qe_values	List (N) of Q-e values. TYPE: list[tuple[float, float]]

RETURNS	DESCRIPTION
FloatSquareMatrix(N, N)	Reactivity ratio matrix.

Examples:

Estimate the reactivity ratio matrix for styrene (1), methyl methacrylate (2), and vinyl acetate(3) using Q-e values from the literature.

>>> from polykin.copolymerization import convert_Qe_to_r
>>>
>>> Qe1 = (1.0, -0.80)    # Sty
>>> Qe2 = (0.78, 0.40)    # MMA
>>> Qe3 = (0.026, -0.88)  # VAc
>>>
>>> convert_Qe_to_r([Qe1, Qe2, Qe3])
array([[1.00000000e+00, 4.90888315e-01, 4.10035538e+01],
       [4.82651046e-01, 1.00000000e+00, 1.79788736e+01],
       [2.42325444e-02, 1.08066091e-02, 1.00000000e+00]])

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

def convert_Qe_to_r(Qe_values: list[tuple[float, float]]
                    ) -> FloatSquareMatrix:
    r"""Convert Q-e values to reactivity ratios.

    According to the Q-e scheme proposed by Alfrey and Price, the reactivity
    ratios of the terminal model can be estimated using the relationship:

    $$ r_{ij} = \frac{Q_i}{Q_j}\exp{\left(-e_i(e_i -e_j)\right)} $$

    where $Q_i$ and $e_i$ are monomer-specific constants, and
    $r_{ij}=k_{ii}/k_{ij}$ is the multicomponent reactivity ratio matrix.

    **References**

    *   T Alfrey, CC Price. J. Polym. Sci., 1947, 2: 101-106.

    Parameters
    ----------
    Qe_values : list[tuple[float, float]]
        List (N) of Q-e values.

    Returns
    -------
    FloatSquareMatrix (N, N)
        Reactivity ratio matrix.

    Examples
    --------
    Estimate the reactivity ratio matrix for styrene (1), methyl
    methacrylate (2), and vinyl acetate(3) using Q-e values from the
    literature.

    >>> from polykin.copolymerization import convert_Qe_to_r
    >>>
    >>> Qe1 = (1.0, -0.80)    # Sty
    >>> Qe2 = (0.78, 0.40)    # MMA
    >>> Qe3 = (0.026, -0.88)  # VAc
    >>>
    >>> convert_Qe_to_r([Qe1, Qe2, Qe3])
    array([[1.00000000e+00, 4.90888315e-01, 4.10035538e+01],
           [4.82651046e-01, 1.00000000e+00, 1.79788736e+01],
           [2.42325444e-02, 1.08066091e-02, 1.00000000e+00]])

    """

    Q = [x[0] for x in Qe_values]
    e = [x[1] for x in Qe_values]
    N = len(Q)

    r = np.empty((N, N))
    for i in range(N):
        for j in range(N):
            r[i, j] = Q[i]/Q[j]*exp(-e[i]*(e[i] - e[j]))

    return r