polykin.kinetics.emulsion

n2bar

n2bar(alpha: float, m: float) -> float

n2bar(alpha: FloatArray, m: FloatArray) -> FloatArray

n2bar(alpha: FloatArray, m: float) -> FloatArray

n2bar(alpha: float, m: FloatArray) -> FloatArray

n2bar(
    alpha: float | FloatArray, m: float | FloatArray
) -> float | FloatArray

Second moment of the normalized radical number distribution according to the Stockmayer-O'Toole quasi-steady-state solution.

\[ \overline{n^2} = \bar{n} \left(1 + \frac{m}{2} \right) + \frac{\alpha}{2} \]

where \(\bar{n}(\alpha, m)\) is the average number of radicals per particle.

References

• Stockmayer, W. H. Note on the Kinetics of Emulsion Polymerization. J. Polym. Sci. 1957, 24, 314-317.
• O'Toole, J. T. Kinetics of Emulsion Polymerization. J. Appl. Polym. Sci. 1965, 9, 1291-1297.

PARAMETER	DESCRIPTION
alpha	Dimensionless entry frequency. TYPE: float | FloatArray
m	Dimensionless desorption frequency. TYPE: float | FloatArray

RETURNS	DESCRIPTION
float | FloatArray	Second moment of the normalized radical number distribution.

Examples:

Evaluate the average square number of radicals per particle for α=1e-2 and m=1e-4.

>>> from polykin.kinetics import n2bar
>>> n2bar_ = n2bar(alpha=1e-2, m=1e-4)
>>> print(f"n2bar = {n2bar_:.2e}")
n2bar = 5.07e-01

Source code in src/polykin/kinetics/emulsion/smithewart.py

def n2bar(
    alpha: float | FloatArray,
    m: float | FloatArray,
) -> float | FloatArray:
    r"""Second moment of the normalized radical number distribution according to the
    Stockmayer-O'Toole quasi-steady-state solution.

    $$ \overline{n^2} = \bar{n} \left(1 + \frac{m}{2} \right) + \frac{\alpha}{2} $$

    where $\bar{n}(\alpha, m)$ is the average number of radicals per particle.

    **References**

    *   Stockmayer, W. H. Note on the Kinetics of Emulsion Polymerization. J. Polym. Sci.
        1957, 24, 314-317.
    *   O'Toole, J. T. Kinetics of Emulsion Polymerization. J. Appl. Polym. Sci.
        1965, 9, 1291-1297.

    Parameters
    ----------
    alpha : float | FloatArray
        Dimensionless entry frequency.
    m : float | FloatArray
        Dimensionless desorption frequency.

    Returns
    -------
    float | FloatArray
        Second moment of the normalized radical number distribution.

    Examples
    --------
    Evaluate the average square number of radicals per particle for α=1e-2 and m=1e-4.
    >>> from polykin.kinetics import n2bar
    >>> n2bar_ = n2bar(alpha=1e-2, m=1e-4)
    >>> print(f"n2bar = {n2bar_:.2e}")
    n2bar = 5.07e-01
    """
    nbar = nbar_Stockmayer_OToole(alpha, m)
    return nbar * (1 - m / 2) + alpha / 2