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nbar_Stockmayer_OToole ¤

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

Average number of radicals per particle according to the Stockmayer-O'Toole solution.

\[ \bar{n} = \frac{a}{4} \frac{I_m(a)}{I_{m-1}(a)} \]

where \(a=\sqrt{8 \alpha}\), and \(I\) is the modified Bessel function of the first kind.

References

  • O'Toole JT. Kinetics of emulsion polymerization. J Appl Polym Sci 1965; 9:1291-7.
PARAMETER DESCRIPTION
alpha

Dimensionless entry frequency.

TYPE: float

m

Dimensionless desorption frequency.

TYPE: float

RETURNS DESCRIPTION
float

Average number of radicals per particle.
Source code in src/polykin/emulsion/nbar.py 
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def nbar_Stockmayer_OToole(alpha: float, m: float) -> float:
    r"""Average number of radicals per particle according to the
    Stockmayer-O'Toole solution.

    $$ \bar{n} = \frac{a}{4} \frac{I_m(a)}{I_{m-1}(a)} $$

    where $a=\sqrt{8 \alpha}$, and $I$ is the modified Bessel function of the
    first kind.

    **References**

    *   O'Toole JT. Kinetics of emulsion polymerization. J Appl Polym Sci 1965;
        9:1291-7.

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

    Returns
    -------
    float
        Average number of radicals per particle.
    """
    a = sqrt(8*alpha)
    return (a/4)*iv(m, a)/iv(m-1, a)