polykin.properties.diffusion

DL_Stokes_Einstein

DL_Stokes_Einstein(T: float, R: float, mu: float) -> float

Estimate the diffusion coefficient of spherical particles using the Stokes-Einstein equation.

The diffusion coefficient of a dilute solution of spherical particles in a homogeneous fluid is given by:

\[ D = \frac{k_B T}{6 \pi \mu R} \]

where \(k_B\) is the Boltzmann constant, \(T\) is the temperature, \(\mu\) is the fluid viscosity, and \(R\) is the particle radius.

Note

The Stokes-Einstein equation is only valid for systems where the particle radius \(R\) is significantly larger than the radius of the fluid molecules. When the particle sizes are comparable, the actual diffusion coefficient is typically larger than predicted.

PARAMETER	DESCRIPTION
T	Temperature [K]. TYPE: float
R	Particle radius [m]. TYPE: float
mu	Viscosity [Pa.s]. TYPE: float

RETURNS	DESCRIPTION
float	Diffusion coefficient [m²/s].

Examples:

Estimate the diffusion coefficient of latex particles with a 100 nm radius in water at 298 K.

>>> from polykin.properties.diffusion import DL_Stokes_Einstein
>>> D = DL_Stokes_Einstein(T=298.0, R=100e-9, mu=1e-3)
>>> print(f"{D:.1e} m²/s")
2.2e-12 m²/s

Source code in src/polykin/properties/diffusion/liquid.py

def DL_Stokes_Einstein(
    T: float,
    R: float,
    mu: float,
) -> float:
    r"""Estimate the diffusion coefficient of spherical particles using the
    Stokes-Einstein equation.

    The diffusion coefficient of a dilute solution of spherical particles in a
    homogeneous fluid is given by:

    $$ D = \frac{k_B T}{6 \pi \mu R} $$

    where $k_B$ is the Boltzmann constant, $T$ is the temperature, $\mu$ is the
    fluid viscosity, and $R$ is the particle radius.

    !!! note

        The Stokes-Einstein equation is only valid for systems where the 
        particle radius $R$ is significantly larger than the radius of the
        fluid molecules. When the particle sizes are comparable, the actual
        diffusion coefficient is typically larger than predicted.

    Parameters
    ----------
    T : float
        Temperature [K].
    R : float
        Particle radius [m].
    mu : float
        Viscosity [Pa.s].

    Returns
    -------
    float
        Diffusion coefficient [m²/s].

    Examples
    --------
    Estimate the diffusion coefficient of latex particles with a 100 nm radius
    in water at 298 K.
    >>> from polykin.properties.diffusion import DL_Stokes_Einstein
    >>> D = DL_Stokes_Einstein(T=298.0, R=100e-9, mu=1e-3)
    >>> print(f"{D:.1e} m²/s")
    2.2e-12 m²/s
    """
    return kB*T/(6*pi*mu*R)