polykin.kinetics.emulsion

nbar_Li_Brooks

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

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

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

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

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

Average number of radicals per particle according to the Li-Brooks approximate quasi-steady-state solution.

\[ \bar{n} = \frac{2 \alpha}{m + \sqrt{m^2 + \frac{8 \alpha \left( 2 \alpha + m \right)}{2 \alpha + m + 1}}} \]

This formula agrees well with the exact Stockmayer-O'Toole solution, with a maximum deviation of about 4%.

References

• Li B-G, Brooks BW. Prediction of the average number of radicals per particle for emulsion polymerization. J Polym Sci, Part A: Polym Chem 1993;31:2397-402.

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

RETURNS	DESCRIPTION
float | FloatArray	Average number of radicals per particle.

See Also

• nbar_Stockmayer_OToole: Preferred exact solution; simpler and numerically robust.
• nbar_Ugelstad: Alternative exact solution based on continued fractions.

Examples:

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

>>> from polykin.kinetics import nbar_Li_Brooks
>>> nbar = nbar_Li_Brooks(alpha=1e-2, m=1e-4)
>>> print(f"nbar = {nbar:.2e}")
nbar = 5.02e-01

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

def nbar_Li_Brooks(
    alpha: float | FloatArray,
    m: float | FloatArray,
) -> float | FloatArray:
    r"""Average number of radicals per particle according to the Li-Brooks approximate
    quasi-steady-state solution.

    $$ \bar{n} = \frac{2 \alpha}{m + \sqrt{m^2 +
        \frac{8 \alpha \left( 2 \alpha + m \right)}{2 \alpha + m + 1}}} $$

    This formula agrees well with the exact Stockmayer-O'Toole solution, with a maximum
    deviation of about 4%.

    **References**

    *   Li B-G, Brooks BW. Prediction of the average number of radicals per particle for
        emulsion polymerization. J Polym Sci, Part A: Polym Chem 1993;31:2397-402.

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

    Returns
    -------
    float | FloatArray
        Average number of radicals per particle.

    See Also
    --------
    * [`nbar_Stockmayer_OToole`](nbar_Stockmayer_OToole.md):
      Preferred exact solution; simpler and numerically robust.
    * [`nbar_Ugelstad`](nbar_Ugelstad.md):
      Alternative exact solution based on continued fractions.

    Examples
    --------
    Evaluate the average number of radicals per particle for α=1e-2 and m=1e-4.
    >>> from polykin.kinetics import nbar_Li_Brooks
    >>> nbar = nbar_Li_Brooks(alpha=1e-2, m=1e-4)
    >>> print(f"nbar = {nbar:.2e}")
    nbar = 5.02e-01
    """
    return (
        2 * alpha / (m + sqrt(m**2 + 8 * alpha * (2 * alpha + m) / (2 * alpha + m + 1)))
    )

Graphical Illustration

Illustration of the typical error obtained with the Li-Brooks approximation.

import matplotlib.pyplot as plt
import numpy as np
from polykin.kinetics.emulsion import nbar_Li_Brooks, nbar_Stockmayer_OToole
from polykin.utils.docs import to_html

fig, ax = plt.subplots()
alpha = np.logspace(-4, 4, 100)
error = np.zeros_like(alpha)
for m in [0, 0.1, 1, 10]:
    for i in range(len(alpha)):
        nbar_SO = nbar_Stockmayer_OToole(alpha[i], m)
        nbar_LB = nbar_Li_Brooks(alpha[i], m)
        error[i] = 1e2*(nbar_LB - nbar_SO)/nbar_SO
    ax.plot(alpha, error,  label=rf"$m={m}$")
ax.set_xscale("log")
ax.grid(True)
ax.legend(loc="best")
ax.set_xlabel(r"$\alpha$")
ax.set_ylabel("Relative Error (%)")

print(to_html(fig))