polykin.copolymerization

inst_copolymer_multi

inst_copolymer_multi(
    f: Optional[FloatVectorLike],
    r: Optional[FloatSquareMatrix],
    P: Optional[FloatSquareMatrix] = None,
) -> FloatVector

Calculate the instantaneous copolymer composition for a multicomponent system.

In a multicomponent system, the instantaneous copolymer composition \(F_i\) can be determined by solving the following set of linear algebraic equations:

\[ \begin{bmatrix} P_{11}-1 & P_{21} & \cdots & P_{N1} \\ P_{12} & P_{22}-1 & \cdots & P_{N2} \\ \vdots & \vdots & \vdots & \vdots \\ 1 & 1 & \cdots & 1 \end{bmatrix} \begin{bmatrix} F_1 \\ F_2 \\ \vdots \\ F_N \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix} \]

where \(P_{ij}\) are the transition probabilitites, which can be computed from the instantaneous monomer composition and the reactivity matrix.

References

• H. K. Frensdorff, R. Pariser; Copolymerization as a Markov Chain. J. Chem. Phys. 1 November 1963; 39 (9): 2303-2309.

PARAMETER	DESCRIPTION
f	Vector of instantaneous monomer compositions, \(f_i\). TYPE: FloatVectorLike(N) | None
r	Matrix of reactivity ratios, \(r_{ij}=k_{ii}/k_{ij}\). TYPE: FloatSquareMatrix(N, N) | None
P	Matrix of transition probabilities, \(P_{ij}\). If None, it will be computed internally. When calculating other quantities (e.g., sequence lengths, tuples) that also depend on \(P\), it is more efficient to precompute \(P\) once and use it in all cases. TYPE: FloatSquareMatrix(N, N) | None DEFAULT: None

RETURNS	DESCRIPTION
FloatVector(N)	Vector of instantaneous copolymer compositions, \(F_i\).

See also

• inst_copolymer_binary: specific method for binary systems.
• inst_copolymer_ternary: specific method for terpolymer systems.
• monomer_drift_multi: monomer composition drift.
• transitions_multi: instantaneous transition probabilities.

Examples:

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

Define the 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 the instantaneous copolymer composition at f1=0.5, f2=0.3, f3=0.2.

>>> f = [0.5, 0.3, 0.2]
>>> F = inst_copolymer_multi(f, r)
>>> F
array([0.32138111, 0.41041608, 0.26820282])

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

def inst_copolymer_multi(f: Optional[FloatVectorLike],
                         r: Optional[FloatSquareMatrix],
                         P: Optional[FloatSquareMatrix] = None
                         ) -> FloatVector:
    r"""Calculate the instantaneous copolymer composition for a multicomponent
    system.

    In a multicomponent system, the instantaneous copolymer composition $F_i$
    can be determined by solving the following set of linear algebraic
    equations:

    $$ \begin{bmatrix}
    P_{11}-1  & P_{21}   & \cdots  & P_{N1} \\
    P_{12}    & P_{22}-1 & \cdots  & P_{N2} \\
    \vdots    & \vdots   & \vdots  & \vdots \\
    1         & 1        & \cdots  & 1
    \end{bmatrix}
    \begin{bmatrix}
    F_1    \\
    F_2    \\
    \vdots \\
    F_N
    \end{bmatrix} =
    \begin{bmatrix}
    0      \\
    0      \\
    \vdots \\
    1
    \end{bmatrix} $$

    where $P_{ij}$ are the transition probabilitites, which can be computed
    from the instantaneous monomer composition and the reactivity matrix.

    **References**

    *   H. K. Frensdorff, R. Pariser; Copolymerization as a Markov Chain.
        J. Chem. Phys. 1 November 1963; 39 (9): 2303-2309.

    Parameters
    ----------
    f : FloatVectorLike (N) | None
        Vector of instantaneous monomer compositions, $f_i$.
    r : FloatSquareMatrix (N, N) | None
        Matrix of reactivity ratios, $r_{ij}=k_{ii}/k_{ij}$.
    P : FloatSquareMatrix (N, N) | None
        Matrix of transition probabilities, $P_{ij}$. If `None`, it will
        be computed internally. When calculating other quantities (e.g.,
        sequence lengths, tuples) that also depend on $P$, it is more efficient
        to precompute $P$ once and use it in all cases.

    Returns
    -------
    FloatVector (N)
        Vector of instantaneous copolymer compositions, $F_i$.

    See also
    --------
    * [`inst_copolymer_binary`](inst_copolymer_binary.md):
      specific method for binary systems.
    * [`inst_copolymer_ternary`](inst_copolymer_ternary.md):
      specific method for terpolymer systems.
    * [`monomer_drift_multi`](monomer_drift_multi.md):
      monomer composition drift.
    * [`transitions_multi`](transitions_multi.md):
      instantaneous transition probabilities.

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

    Define the 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 the instantaneous copolymer composition at f1=0.5, f2=0.3, f3=0.2.
    >>> f = [0.5, 0.3, 0.2]
    >>> F = inst_copolymer_multi(f, r)
    >>> F
    array([0.32138111, 0.41041608, 0.26820282])
    """

    if P is not None and (f is None and r is None):
        pass
    elif P is None and (f is not None and r is not None):
        P = transitions_multi(f, r)
    else:
        raise ValueError("Invalid combination of inputs.")

    N = P.shape[0]
    A = P.T - np.eye(N)
    A[-1, :] = 1.
    b = np.zeros(N)
    b[-1] = 1.
    F = np.linalg.solve(A, b)

    return F