polykin.kinetics.emulsion

kentry_diffusion_reversible

kentry_diffusion_reversible(
    r: float, Dw: float, Dp: float, q: float, k: float
) -> float

Radical entry coefficient assuming reversible diffusion-controlled entry.

The entry coefficient is given by:

\[ k_e = 4 \pi r D_w N_A \frac{q D_p (X \coth{X} - 1)}{D_w + q D_p (X \coth{X} - 1)} \]

where

\[ X = r \sqrt{\frac{k}{D_p}} \]

Here, \(r\) is the particle radius, \(D_w\) and \(D_p\) are the diffusion coefficients of the radical in the aqueous and particle phases, respectively, \(q\) is the partition coefficient (particle/aqueous), and \(N_A\) is Avogadro's number.

The parameter \(k\) is an effective first-order rate coefficient describing radical disappearance within the particle:

\[ k = k_p^p [M^p] + \frac{n k_t^p}{N_A v_p} \]

where \(k_p^p\) and \(k_t^p\) are the propagation and termination rate coefficients in the particle, \([M^p]\) is the monomer concentration in the particle, \(n\) is the number of radicals per particle, and \(v_p\) is the particle volume.

References

• Hansen, F. K.; Ugelstad, J. Particle Nucleation in Emulsion Polymerization. I. A Theory for Homogeneous Nucleation. J. Polym. Sci., Polym. Chem. Ed. 1978, 16, 1953-1979.

PARAMETER	DESCRIPTION
r	Particle radius [m]. TYPE: float
Dw	Diffusion coefficient in the aqueous phase [m²/s]. TYPE: float
Dp	Diffusion coefficient in the particle phase [m²/s]. TYPE: float
q	Partition coefficient of the radical (particle/aqueous). TYPE: float
k	First-order rate coefficient for radical disappearance inside the particle [s⁻¹]. TYPE: float

RETURNS	DESCRIPTION
float	Radical entry coefficient [m³/(mol·s)].

See Also

• kentry_diffusion: Related method assuming irreversible diffusion-controlled entry.

Examples:

Evaluate the radical entry coefficient for a particle of radius 100 nm, with diffusion coefficients of 1e-9 m²/s in the aqueous phase and 1e-8 m²/s in the particle phase, a partition coefficient of 100, and a first-order rate coefficient of 1e5 s⁻¹.

>>> from polykin.kinetics import kentry_diffusion_reversible
>>> ke = kentry_diffusion_reversible(r=100e-9, Dw=1e-9, Dp=1e-8, q=100, k=1e5)
>>> print(f"ke = {ke:.2e} m³/(mol·s)")
ke = 7.35e+08 m³/(mol·s)

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

def kentry_diffusion_reversible(
    r: float,
    Dw: float,
    Dp: float,
    q: float,
    k: float,
) -> float:
    r"""Radical entry coefficient assuming reversible diffusion-controlled entry.

    The entry coefficient is given by:

    $$ k_e = 4 \pi r D_w N_A
       \frac{q D_p (X \coth{X} - 1)}{D_w + q D_p (X \coth{X} - 1)} $$

    where

    $$ X = r \sqrt{\frac{k}{D_p}} $$

    Here, $r$ is the particle radius, $D_w$ and $D_p$ are the diffusion coefficients of
    the radical in the aqueous and particle phases, respectively, $q$ is the partition
    coefficient (particle/aqueous), and $N_A$ is Avogadro's number.

    The parameter $k$ is an effective first-order rate coefficient describing radical
    disappearance within the particle:

    $$ k = k_p^p [M^p] + \frac{n k_t^p}{N_A v_p} $$

    where $k_p^p$ and $k_t^p$ are the propagation and termination rate coefficients in
    the particle, $[M^p]$ is the monomer concentration in the particle, $n$ is the number
    of radicals per particle, and $v_p$ is the particle volume.

    **References**

    *   Hansen, F. K.; Ugelstad, J. Particle Nucleation in Emulsion Polymerization. I.
        A Theory for Homogeneous Nucleation. J. Polym. Sci., Polym. Chem. Ed. 1978, 16,
        1953-1979.

    Parameters
    ----------
    r : float
        Particle radius [m].
    Dw : float
        Diffusion coefficient in the aqueous phase [m²/s].
    Dp : float
        Diffusion coefficient in the particle phase [m²/s].
    q : float
        Partition coefficient of the radical (particle/aqueous).
    k : float
        First-order rate coefficient for radical disappearance inside the particle [s⁻¹].

    Returns
    -------
    float
        Radical entry coefficient [m³/(mol·s)].

    See Also
    --------
    * [`kentry_diffusion`](kentry_diffusion.md):
      Related method assuming irreversible diffusion-controlled entry.

    Examples
    --------
    Evaluate the radical entry coefficient for a particle of radius 100 nm, with diffusion
    coefficients of 1e-9 m²/s in the aqueous phase and 1e-8 m²/s in the particle phase, a
    partition coefficient of 100, and a first-order rate coefficient of 1e5 s⁻¹.
    >>> from polykin.kinetics import kentry_diffusion_reversible
    >>> ke = kentry_diffusion_reversible(r=100e-9, Dw=1e-9, Dp=1e-8, q=100, k=1e5)
    >>> print(f"ke = {ke:.2e} m³/(mol·s)")
    ke = 7.35e+08 m³/(mol·s)
    """
    X = r * sqrt(k / Dp)
    Y = X / tanh(X) - 1.0
    f = (q * Dp * Y) / (Dw + q * Dp * Y)
    return kentry_diffusion(r, Dw, f)