polykin.thermo.acm

FloryHuggins_activity

FloryHuggins_activity(
    phi: FloatVector, m: FloatVector, chi: FloatSquareMatrix
) -> FloatVector

Calculate the activities of a multicomponent mixture according to the Flory-Huggins model.

\[ \ln{a_i} = \ln{\phi_i} + 1 - m_i \left(\sum_j \frac{\phi_j}{m_j} - \sum_j \phi_j \chi_{ij} + \sum_j \sum_{k>j} \phi_j \phi_k \chi_{jk} \right) \]

where \(\phi_i\) are the volume, mass or segment fractions of the components, \(\chi_{ij}\) are the interaction parameters, and \(m_i\) is the characteristic size of the components.

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 \(\phi\) is taken to be the same variable used to estimate \(\chi\). In this particular implementation, the matrix of interaction parameters is independent of molecular size and, thus, symmetric.

References

• P.J. Flory, Principles of polymer chemistry, 1953.
• Lindvig, Thomas, et al. "Modeling of multicomponent vapor-liquid equilibria for polymer-solvent systems." Fluid phase equilibria 220.1 (2004): 11-20.

PARAMETER	DESCRIPTION
phi	Volume, mass or segment fractions of all components, \(\phi_i\). The composition variable must match the one used in the determination of \(\chi_{ij}\). TYPE: FloatVector
m	Characteristic size of all components, typically equal to 1 for small molecules and equal to the average degree of polymerization for polymers. TYPE: FloatVector
chi	Matrix (N,N) of interaction parameters, \(\chi_{ij}\). It is expected (but not checked) that \(\chi_{ij}=\chi_{ji}\) and \(\chi_{ii}=0\). TYPE: FloatSquareMatrix

RETURNS	DESCRIPTION
FloatVector	Activities of all components.

See also

• FloryHuggins2_activity: equivalent method for binary solvent-polymer systems.

Source code in src/polykin/thermo/acm/floryhuggins.py

def FloryHuggins_activity(phi: FloatVector,
                          m: FloatVector,
                          chi: FloatSquareMatrix
                          ) -> FloatVector:
    r"""Calculate the activities of a multicomponent mixture according to the
    Flory-Huggins model.

    $$ 
    \ln{a_i} = \ln{\phi_i} + 1 - m_i \left(\sum_j \frac{\phi_j}{m_j} - 
    \sum_j \phi_j \chi_{ij} + \sum_j \sum_{k>j} \phi_j \phi_k \chi_{jk} \right)
    $$

    where $\phi_i$ are the volume, mass or segment fractions of the
    components, $\chi_{ij}$ are the interaction parameters, and $m_i$ is the
    characteristic size of the components. 

    !!! 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 $\phi$ is taken to be the same
        variable used to estimate $\chi$.
        In this particular implementation, the matrix of interaction parameters
        is independent of molecular size and, thus, symmetric.

    **References**

    *   P.J. Flory, Principles of polymer chemistry, 1953.
    *   Lindvig, Thomas, et al. "Modeling of multicomponent vapor-liquid
        equilibria for polymer-solvent systems." Fluid phase equilibria 220.1
        (2004): 11-20.

    Parameters
    ----------
    phi : FloatVector
        Volume, mass or segment fractions of all components, $\phi_i$. The
        composition variable must match the one used in the determination of
        $\chi_{ij}$.
    m : FloatVector
        Characteristic size of all components, typically equal to 1 for small
        molecules and equal to the average degree of polymerization for
        polymers.
    chi : FloatSquareMatrix
        Matrix (N,N) of interaction parameters, $\chi_{ij}$. It is expected
        (but not checked) that $\chi_{ij}=\chi_{ji}$ and $\chi_{ii}=0$.

    Returns
    -------
    FloatVector
        Activities of all components.

    See also
    --------
    * [`FloryHuggins2_activity`](FloryHuggins2_activity.md): equivalent
      method for binary solvent-polymer systems.

    """
    A = dot(phi, 1/m)
    B = dot(chi, phi)
    C = 0.5*dot(phi, dot(phi, chi))
    return phi*exp(1 - m*(A - B + C))