polykin.copolymerization

radical_fractions_multi

radical_fractions_multi(
    f: FloatVectorLike, k: FloatSquareMatrix
) -> FloatVector

Calculate the radical fractions for a multicomponent system.

In a multicomponent system, the radical fractions \(p_i\) can be determined by solving the following set of linear algebraic equations:

\[ \sum_{j\ne i}^N k_{ij} p_i f_j = \sum_{j\ne i}^N k_{ji} p_j f_i \]

where \(k_{ij}\) are the cross-propagation rate coefficients and \(f_i\) are the monomer compositions. 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.

PARAMETER	DESCRIPTION
f	Vector of instantaneous monomer compositions, \(f_i\). TYPE: FloatVectorLike(N)
k	Matrix of cross-propagation rate coefficients. The diagonal elements \(k_{ii}\) are not used. TYPE: FloatSquareMatrix(N, N)

RETURNS	DESCRIPTION
FloatVector(N)	Vector of radical fractions, \(p_i\).

See also

• radical_fractions_ternary: specific method for terpolymer systems.

Examples:

>>> from polykin.copolymerization import radical_fractions_multi
>>> import numpy as np

Define the cross-propagation coefficient matrix.

>>> k = np.zeros((3, 3))
>>> k[0, 1] = 500.
>>> k[1, 0] = 50.
>>> k[0, 2] = 30.
>>> k[2, 0] = 200.
>>> k[1, 2] = 300.
>>> k[2, 1] = 40.

Evaluate the radical fractions at f1=0.5, f2=0.3, f3=0.2.

>>> f = [0.5, 0.3, 0.2]
>>> p = radical_fractions_multi(f, k)
>>> p
array([0.25012791, 0.47956341, 0.27030868])

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

def radical_fractions_multi(f: FloatVectorLike,
                            k: FloatSquareMatrix,
                            ) -> FloatVector:
    r"""Calculate the radical fractions for a multicomponent system.

    In a multicomponent system, the radical fractions $p_i$ can be determined
    by solving the following set of linear algebraic equations:

    $$ \sum_{j\ne i}^N k_{ij} p_i f_j = \sum_{j\ne i}^N k_{ji} p_j f_i $$

    where $k_{ij}$ are the cross-propagation rate coefficients and $f_i$ are
    the monomer compositions. 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.

    Parameters
    ----------
    f : FloatVectorLike (N)
        Vector of instantaneous monomer compositions, $f_i$.
    k : FloatSquareMatrix (N, N)
        Matrix of cross-propagation rate coefficients. The diagonal elements
        $k_{ii}$ are not used.

    Returns
    -------
    FloatVector (N)
        Vector of radical fractions, $p_i$.

    See also
    --------
    * [`radical_fractions_ternary`](radical_fractions_ternary.md):
      specific method for terpolymer systems.

    Examples
    --------
    >>> from polykin.copolymerization import radical_fractions_multi
    >>> import numpy as np

    Define the cross-propagation coefficient matrix.
    >>> k = np.zeros((3, 3))
    >>> k[0, 1] = 500.
    >>> k[1, 0] = 50.
    >>> k[0, 2] = 30.
    >>> k[2, 0] = 200.
    >>> k[1, 2] = 300.
    >>> k[2, 1] = 40.

    Evaluate the radical fractions at f1=0.5, f2=0.3, f3=0.2.
    >>> f = [0.5, 0.3, 0.2]
    >>> p = radical_fractions_multi(f, k)
    >>> p
    array([0.25012791, 0.47956341, 0.27030868])
    """
    f = np.asarray(f)

    A = k.T * f[:, np.newaxis]
    x = A.sum(axis=0)
    np.fill_diagonal(A, A.diagonal() - x)
    A[-1, :] = 1.
    b = np.zeros(A.shape[0])
    b[-1] = 1.
    p = np.linalg.solve(A, b)

    return p