polykin.kinetics.emulsion

compartmentalization_factor

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

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

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

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

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

Compartmentalization factor according to the Stockmayer-O'Toole quasi-steady-state solution.

The compartmentalization factor expresses the reduction of the effective termination rate in a compartmentalized system relative to that in a bulk system with the same overall radical concentration. It is defined as:

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

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

References

• Wulkow, M.; Richards, J. R. Evaluation of the Chain Length Distribution in Free-Radical Emulsion Polymerization—The Compartmentalization Problem. Ind. Eng. Chem. Res. 2014, 53, 7275-7295.

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

RETURNS	DESCRIPTION
float | FloatArray	Compartmentalization factor.

Examples:

Evaluate the compartmentalization factor for α=1e-2 and m=1e-4.

>>> from polykin.kinetics import compartmentalization_factor
>>> Df = compartmentalization_factor(alpha=1e-2, m=1e-4)
>>> print(f"Df = {Df:.2e}")
Df = 1.97e-02

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

def compartmentalization_factor(
    alpha: float | FloatArray,
    m: float | FloatArray,
) -> float | FloatArray:
    r"""Compartmentalization factor according to the Stockmayer-O'Toole quasi-steady-state
    solution.

    The compartmentalization factor expresses the reduction of the effective termination
    rate in a compartmentalized system relative to that in a bulk system with the same
    overall radical concentration. It is defined as:

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

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

    **References**

    *   Wulkow, M.; Richards, J. R. Evaluation of the Chain Length Distribution in
        Free-Radical Emulsion Polymerization—The Compartmentalization Problem.
        Ind. Eng. Chem. Res. 2014, 53, 7275-7295.

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

    Returns
    -------
    float | FloatArray
        Compartmentalization factor.

    Examples
    --------
    Evaluate the compartmentalization factor for α=1e-2 and m=1e-4.
    >>> from polykin.kinetics import compartmentalization_factor
    >>> Df = compartmentalization_factor(alpha=1e-2, m=1e-4)
    >>> print(f"Df = {Df:.2e}")
    Df = 1.97e-02
    """
    nbar = nbar_Stockmayer_OToole(alpha, m)
    return (alpha / nbar - m) / (2 * nbar)

Graphical Illustration

Typical dependence of the compartmentalization factor on the dimensionless entry and desorption frequencies (\(\alpha\) and \(m\)), and the average number of radicals per particle (\(\bar{n}\)).

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

fig, ax = plt.subplots(2, 1)

alpha = np.logspace(-4, 4, 100)
for m in [0, 0.1, 1, 10]:
    Df = compartmentalization_factor(alpha, m)
    nbar = nbar_Stockmayer_OToole(alpha, m)
    ax[0].plot(alpha, Df,  label=rf"$m={m}$")
    ax[1].plot(nbar, Df)

ax[0].set_xlabel(r"$\alpha$")
ax[0].set_ylabel(r"$D_f$")
ax[0].set_xscale("log")
ax[0].set_xlim(1e-4, 1e4)
ax[0].grid(True)
ax[0].legend(loc="best")

ax[1].set_xlabel(r"$\bar{n}$")
ax[1].set_ylabel(r"$D_f$")
ax[1].set_xscale("log")
ax[1].set_xlim(1e-2, 1e2)
ax[1].grid(True)

fig.align_ylabels()
fig.tight_layout()

print(to_html(fig))