polykin.copolymerization

sequence_multi

sequence_multi(
    Pself: FloatVectorLike,
    k: Optional[Union[int, IntArrayLike]] = None,
) -> FloatArray

Calculate the instantaneous sequence length probability or the number-average sequence length.

For a multicomponent system, the probability of finding \(k\) consecutive units of monomer \(i\) in a chain is:

\[ S_{i,k} = (1 - P_{ii})P_{ii}^{k-1} \]

and the corresponding number-average sequence length is:

\[ \bar{S}_i = \sum_{k=1}^{\infty} k S_{i,k} = \frac{1}{1 - P_{ii}} \]

where \(P_{ii}\) is the self-transition probability \(i \rightarrow i\), which is a function of the monomer composition and the reactivity ratios.

References

• NA Dotson, R Galván, RL Laurence, and M Tirrel. Polymerization process modeling, Wiley, 1996, p. 177.

PARAMETER	DESCRIPTION
Pself	Vector of self-transition probabilities, \(P_{ii}\), corresponding to the diagonal of the matrix of transition probabilities. TYPE: FloatVectorLike(N)
k	Sequence length, i.e., number of consecutive units in a chain. If None, the number-average sequence length will be computed. TYPE: int | IntArrayLike(M) | None DEFAULT: None

RETURNS	DESCRIPTION
FloatArray(N, M)	If k is None, the number-average sequence lengths, \(\bar{S}_i\). Otherwise, the sequence probabilities, \(S_{i,k}\).

See also

• transitions_multi: instantaneous transition probabilities.
• tuples_multi: instantaneous tuple fractions.

Examples:

>>> from polykin.copolymerization import sequence_multi
>>> from polykin.copolymerization import transitions_multi
>>> import numpy as np

Define reactivity ratio matrix.

>>> r = np.ones((3, 3))
>>> r[0, 1] = 0.2
>>> r[1, 0] = 2.3
>>> r[0, 2] = 3.0
>>> r[2, 0] = 0.9
>>> r[1, 2] = 0.4
>>> r[2, 1] = 1.5

Evaluate self-transition probabilities.

>>> f = [0.5, 0.3, 0.2]
>>> Pself = transitions_multi(f, r).diagonal()
>>> Pself
array([0.24193548, 0.29487179, 0.20930233])

Evaluate number-average sequence lengths for all monomers.

>>> S = sequence_multi(Pself)
>>> S
array([1.31914894, 1.41818182, 1.26470588])

Evaluate probabilities for certain sequence lengths.

>>> S = sequence_multi(Pself, k=[1, 5])
>>> S
array([[0.75806452, 0.00259719],
       [0.70512821, 0.00533091],
       [0.79069767, 0.00151742]])

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

def sequence_multi(Pself: FloatVectorLike,
                   k: Optional[Union[int, IntArrayLike]] = None,
                   ) -> FloatArray:
    r"""Calculate the instantaneous sequence length probability or the
    number-average sequence length.

    For a multicomponent system, the probability of finding $k$ consecutive
    units of monomer $i$ in a chain is:

    $$ S_{i,k} = (1 - P_{ii})P_{ii}^{k-1} $$

    and the corresponding number-average sequence length is:

    $$ \bar{S}_i = \sum_{k=1}^{\infty} k S_{i,k} = \frac{1}{1 - P_{ii}} $$

    where $P_{ii}$ is the self-transition probability $i \rightarrow i$, which
    is a function of the monomer composition and the reactivity ratios.

    **References**

    * NA Dotson, R Galván, RL Laurence, and M Tirrel. Polymerization
    process modeling, Wiley, 1996, p. 177.

    Parameters
    ----------
    Pself : FloatVectorLike (N)
        Vector of self-transition probabilities, $P_{ii}$, corresponding to
        the diagonal of the matrix of transition probabilities.
    k : int | IntArrayLike (M) | None
        Sequence length, i.e., number of consecutive units in a chain.
        If `None`, the number-average sequence length will be computed.

    Returns
    -------
    FloatArray (N, M)
        If `k is None`, the number-average sequence lengths, $\bar{S}_i$.
        Otherwise, the sequence probabilities, $S_{i,k}$.

    See also
    --------
    * [`transitions_multi`](transitions_multi.md):
      instantaneous transition probabilities.
    * [`tuples_multi`](tuples_multi.md):
      instantaneous tuple fractions.

    Examples
    --------
    >>> from polykin.copolymerization import sequence_multi
    >>> from polykin.copolymerization import transitions_multi
    >>> import numpy as np

    Define reactivity ratio matrix.
    >>> r = np.ones((3, 3))
    >>> r[0, 1] = 0.2
    >>> r[1, 0] = 2.3
    >>> r[0, 2] = 3.0
    >>> r[2, 0] = 0.9
    >>> r[1, 2] = 0.4
    >>> r[2, 1] = 1.5

    Evaluate self-transition probabilities.
    >>> f = [0.5, 0.3, 0.2]
    >>> Pself = transitions_multi(f, r).diagonal()
    >>> Pself
    array([0.24193548, 0.29487179, 0.20930233])

    Evaluate number-average sequence lengths for all monomers.
    >>> S = sequence_multi(Pself)
    >>> S
    array([1.31914894, 1.41818182, 1.26470588])

    Evaluate probabilities for certain sequence lengths.
    >>> S = sequence_multi(Pself, k=[1, 5])
    >>> S
    array([[0.75806452, 0.00259719],
           [0.70512821, 0.00533091],
           [0.79069767, 0.00151742]])

    """

    Pself = np.asarray(Pself)

    if k is None:
        S = 1/(1. - Pself + eps)
    else:
        if isinstance(k, (list, tuple)):
            k = np.array(k, dtype=np.int32)
        if isinstance(k, np.ndarray):
            Pself = Pself.reshape(-1, 1)
        S = (1. - Pself)*Pself**(k - 1)

    return S