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FloryHuggins2_activity ¤

FloryHuggins2_activity(
    phi1: Union[float, FloatArray],
    m: Union[float, FloatArray],
    chi: Union[float, FloatArray],
) -> Union[float, FloatArray]

Calculate the solvent activity of a binary polymer solution according to the Flory-Huggins model.

\[ \ln{a_1} = \ln \phi_1 + \left(\ 1 - \frac{1}{m} \right)(1-\phi_1) + \chi_{12}(1-\phi_1)^2 \]

where \(\phi_1\) is the volume, mass or segment fraction of the solvent in the solution, \(\chi\) is the solvent-polymer interaction parameter, and \(m\) is the ratio of the molar volume of the polymer to the solvent, often approximated as the degree of polymerization.

Note

The Flory-Huggins model was originally formulated using the volume fraction as primary composition variable, but many authors prefer mass fractions in order to skip issues with densities, etc. This equation can be used in all cases, as long as \(\phi_1\) is taken to be the same variable used to estimate \(\chi\). For example, if a given publication reports values of \(\chi\) estimated with the mass fraction version of the Flory-Huggins model, then \(\phi_1\) should be the mass fraction of solvent.

References

  • P.J. Flory, Principles of polymer chemistry, 1953.
PARAMETER DESCRIPTION
phi1

Volume, mass, or segment fraction of solvent in the solution, \(\phi_1\). The composition variable must match the one used in the determination of \(\chi\).

TYPE: float | FloatArray

m

Ratio of the molar volume of the polymer chain to the solvent, often approximated as the degree of polymerization.

TYPE: float | FloatArray

chi

Solvent-polymer interaction parameter, \(\chi\).

TYPE: float | FloatArray

RETURNS DESCRIPTION
FloatVector

Activity of the solvent.
See also

Examples:

Calculate the activity of ethylbenzene in a ethylbenzene-polybutadiene solution with 25 wt% solvent content. Assume \(\chi=0.29\).

>>> from polykin.thermo.acm import FloryHuggins2_gamma
>>> gamma = FloryHuggins2_gamma(phi1=0.25, m=1e6, chi=0.29)
>>> print(f"{gamma:.2f}")
0.62
Source code in src/polykin/thermo/acm/floryhuggins.py 
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def FloryHuggins2_activity(phi1: Union[float, FloatArray],
                           m: Union[float, FloatArray],
                           chi: Union[float, FloatArray]
                           ) -> Union[float, FloatArray]:
    r"""Calculate the solvent activity of a binary polymer solution according
    to the Flory-Huggins model.

    $$ \ln{a_1} = \ln \phi_1 + \left(\ 1 - \frac{1}{m} \right)(1-\phi_1)
                  + \chi_{12}(1-\phi_1)^2 $$

    where $\phi_1$ is the volume, mass or segment fraction of the solvent in
    the solution, $\chi$ is the solvent-polymer interaction parameter, and $m$
    is the ratio of the molar volume of the polymer to the solvent, often
    approximated as the degree of polymerization. 

    !!! note

        The Flory-Huggins model was originally formulated using the volume
        fraction as primary composition variable, but many authors prefer mass
        fractions in order to skip issues with densities, etc. This equation
        can be used in all cases, as long as $\phi_1$ is taken to be the same
        variable used to estimate $\chi$. For example, if a given publication
        reports values of $\chi$ estimated with the mass fraction version of
        the Flory-Huggins model, then $\phi_1$ should be the mass fraction of
        solvent.

    **References**

    *   P.J. Flory, Principles of polymer chemistry, 1953.

    Parameters
    ----------
    phi1 : float | FloatArray
        Volume, mass, or segment fraction of solvent in the solution, $\phi_1$.
        The composition variable must match the one used in the determination
        of $\chi$.
    m : float | FloatArray
        Ratio of the molar volume of the polymer chain to the solvent, often
        approximated as the degree of polymerization.
    chi : float | FloatArray
        Solvent-polymer interaction parameter, $\chi$.

    Returns
    -------
    FloatVector
        Activity of the solvent.

    See also
    --------
    * [`FloryHuggins_activity`](FloryHuggins_activity.md): equivalent method
      for multicomponent systems.

    Examples
    --------
    Calculate the activity of ethylbenzene in a ethylbenzene-polybutadiene
    solution with 25 wt% solvent content. Assume $\chi=0.29$.
    >>> from polykin.thermo.acm import FloryHuggins2_gamma
    >>> gamma = FloryHuggins2_gamma(phi1=0.25, m=1e6, chi=0.29)
    >>> print(f"{gamma:.2f}")
    0.62

    """
    phi2 = 1 - phi1
    return phi1*exp((1 - 1/m)*phi2 + chi*phi2**2)