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FloryHuggins ¤

Flory-Huggings multicomponent activity coefficient model.

This model is based on the following Gibbs energy of mixing per mole of sites:

\[ \frac{\Delta g_{mix}}{R T}= \sum_i \frac{\phi_i}{m_i}\ln{\phi_i} + \sum_i \sum_{j>i} \phi_i \phi_j \chi_{ij} \]

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.

In this particular implementation, the interaction parameters are allowed to depend on temperature according to the following empirical relationship (as used in Aspen Plus):

\[ \chi_{ij} = a_{ij} + b_{ij}/T + c_{ij} \ln{T} + d_{ij} T + e_{ij} T^2 \]

Moreover, \(\chi_{ij}=\chi_{ji}\) and \(\chi_{ii}=0\).

References

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

Number of components.

TYPE: int

a

Matrix (N,N) of parameters, by default 0. Only the upper triangle must be supplied.

TYPE: FloatSquareMatrix | None DEFAULT: None

b

Matrix (N,N) of parameters, by default 0. Only the upper triangle must be supplied.

TYPE: FloatSquareMatrix | None DEFAULT: None

c

Matrix (N,N) of parameters, by default 0. Only the upper triangle must be supplied.

TYPE: FloatSquareMatrix | None DEFAULT: None

d

Matrix (N,N) of parameters, by default 0. Only the upper triangle must be supplied.

TYPE: FloatSquareMatrix | None DEFAULT: None

e

Matrix (N,N) of parameters, by default 0. Only the upper triangle must be supplied.

TYPE: FloatSquareMatrix | None DEFAULT: None

Source code in src/polykin/thermo/acm/floryhuggins.py 
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class FloryHuggins():
    r"""[Flory-Huggings](https://en.wikipedia.org/wiki/Flory–Huggins_solution_theory)
    multicomponent activity coefficient model.

    This model is based on the following Gibbs energy of mixing per mole of
    sites:

    $$ \frac{\Delta g_{mix}}{R T}= \sum_i \frac{\phi_i}{m_i}\ln{\phi_i}
    + \sum_i \sum_{j>i} \phi_i \phi_j \chi_{ij} $$

    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. 

    In this particular implementation, the interaction parameters are allowed
    to depend on temperature according to the following empirical relationship
    (as used in Aspen Plus):

    $$ \chi_{ij} = a_{ij} + b_{ij}/T + c_{ij} \ln{T} + d_{ij} T + e_{ij} T^2 $$

    Moreover, $\chi_{ij}=\chi_{ji}$ and $\chi_{ii}=0$.

    **References**

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

    Parameters
    ----------
    N : int
        Number of components.
    a : FloatSquareMatrix | None
        Matrix (N,N) of parameters, by default 0. Only the upper triangle must
        be supplied.
    b : FloatSquareMatrix | None
        Matrix (N,N) of parameters, by default 0. Only the upper triangle must
        be supplied.
    c : FloatSquareMatrix | None
        Matrix (N,N) of parameters, by default 0. Only the upper triangle must
        be supplied.
    d : FloatSquareMatrix | None
        Matrix (N,N) of parameters, by default 0. Only the upper triangle must
        be supplied.
    e : FloatSquareMatrix | None
        Matrix (N,N) of parameters, by default 0. Only the upper triangle must
        be supplied.

    """

    _N: int
    _a: FloatSquareMatrix
    _b: FloatSquareMatrix
    _c: FloatSquareMatrix
    _d: FloatSquareMatrix
    _e: FloatSquareMatrix

    def __init__(self,
                 N: int,
                 a: Optional[FloatSquareMatrix] = None,
                 b: Optional[FloatSquareMatrix] = None,
                 c: Optional[FloatSquareMatrix] = None,
                 d: Optional[FloatSquareMatrix] = None,
                 e: Optional[FloatSquareMatrix] = None
                 ) -> None:
        """Construct `FloryHuggins` with the given parameters."""

        # Set default values
        if a is None:
            a = np.zeros((N, N))
        if b is None:
            b = np.zeros((N, N))
        if c is None:
            c = np.zeros((N, N))
        if d is None:
            d = np.zeros((N, N))
        if e is None:
            e = np.zeros((N, N))

        # Check shapes
        for array in [a, b, c, d, e]:
            if array.shape != (N, N):
                raise ShapeError(
                    f"The shape of matrix {array} is invalid: {array.shape}.")

        # Check bounds (same as Aspen Plus)
        check_bounds(a, -1e2, 1e2, 'a')
        check_bounds(b, -1e6, 1e6, 'b')
        check_bounds(c, -1e6, 1e6, 'c')
        check_bounds(d, -1e6, 1e6, 'd')
        check_bounds(e, -1e6, 1e6, 'e')

        # Ensure chi_ii=0 and chi_ij=chi_ji
        for array in [a, b, c, d, e]:
            np.fill_diagonal(array, 0.)
            enforce_symmetry(array)

        self._N = N
        self._a = a
        self._b = b
        self._c = c
        self._d = d
        self._e = e

    @functools.cache
    def chi(self, T: float) -> FloatSquareMatrix:
        r"""Compute the matrix of interaction parameters.

        $$
        \chi_{ij} = a_{ij} + b_{ij}/T + c_{ij} \ln{T} + d_{ij} T + e_{ij} T^2
        $$

        Parameters
        ----------
        T : float
            Temperature. Unit = K.

        Returns
        -------
        FloatSquareMatrix
            Matrix of interaction parameters.
        """
        return self._a + self._b/T + self._c*log(T) + self._d*T + self._e*T**2

    def Dgmix(self,
              T: Number,
              phi: FloatVector,
              m: FloatVector) -> Number:
        r"""Gibbs energy of mixing per mole of sites, $\Delta g_{mix}$.

        Parameters
        ----------
        T : float
            Temperature. Unit = K.
        phi : FloatVector
            Volume, mass or segment fractions of all components.
        m : FloatVector
            Characteristic size of all components, typically equal to 1 for
            small molecules and equal to the average degree of polymerization
            for polymers.

        Returns
        -------
        float
            Gibbs energy of mixing per mole of sites. Unit = J/mol.
        """
        p = phi > 0.
        gC = dot(phi[p]/m[p], log(phi[p]))
        gR = 0.5*dot(phi, dot(phi, self.chi(T)))
        return R*T*(gC + gR)

    def Dhmix(self,
              T: float,
              phi: FloatVector,
              m: FloatVector) -> float:
        r"""Enthalpy of mixing per mole of sites, $\Delta h_{mix}$.

        $$ \Delta h_{mix} = \Delta g_{mix} + T \Delta s_{mix} $$

        Parameters
        ----------
        T : float
            Temperature. Unit = K.
        phi : FloatVector
            Volume, mass or segment fractions of all components.
        m : FloatVector
            Characteristic size of all components, typically equal to 1 for
            small molecules and equal to the average degree of polymerization
            for polymers.

        Returns
        -------
        float
            Enthalpy of mixing per mole of sites. Unit = J/mol.
        """
        return self.Dgmix(T, phi, m) + T*self.Dsmix(T, phi, m)

    def Dsmix(self,
              T: float,
              phi: FloatVector,
              m: FloatVector) -> float:
        r"""Entropy of mixing per mole of sites, $\Delta s_{mix}$.

        $$ \Delta s_{mix} = -\left(\frac{\partial \Delta g_{mix}}
           {\partial T}\right)_{P,\phi_i,m_i} $$

        Parameters
        ----------
        T : float
            Temperature. Unit = K.
        phi : FloatVector
            Volume, mass or segment fractions of all components.
        m : FloatVector
            Characteristic size of all components, typically equal to 1 for
            small molecules and equal to the average degree of polymerization
            for polymers.

        Returns
        -------
        float
            Entropy of mixing per mole of sites. Unit = J/(mol·K).
        """
        return -derivative_complex(lambda t: self.Dgmix(t, phi, m), T)[0]

    def a(self,
          T: float,
          phi: FloatVector,
          m: FloatVector
          ) -> FloatVector:
        r"""Activities, $a_i$.

        Parameters
        ----------
        T : float
            Temperature. Unit = K.
        phi : FloatVector
            Volume, mass or segment fractions of all components.
        m : FloatVector
            Characteristic size of all components, typically equal to 1 for
            small molecules and equal to the average degree of polymerization
            for polymers.

        Returns
        -------
        FloatVector
            Activities of all components.
        """
        return FloryHuggins_activity(phi, m, self.chi(T))

Dgmix ¤

Dgmix(
    T: Number, phi: FloatVector, m: FloatVector
) -> Number

Gibbs energy of mixing per mole of sites, \(\Delta g_{mix}\).

PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

phi

Volume, mass or segment fractions of all components.

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

RETURNS DESCRIPTION
float

Gibbs energy of mixing per mole of sites. Unit = J/mol.
Source code in src/polykin/thermo/acm/floryhuggins.py 
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def Dgmix(self,
          T: Number,
          phi: FloatVector,
          m: FloatVector) -> Number:
    r"""Gibbs energy of mixing per mole of sites, $\Delta g_{mix}$.

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    phi : FloatVector
        Volume, mass or segment fractions of all components.
    m : FloatVector
        Characteristic size of all components, typically equal to 1 for
        small molecules and equal to the average degree of polymerization
        for polymers.

    Returns
    -------
    float
        Gibbs energy of mixing per mole of sites. Unit = J/mol.
    """
    p = phi > 0.
    gC = dot(phi[p]/m[p], log(phi[p]))
    gR = 0.5*dot(phi, dot(phi, self.chi(T)))
    return R*T*(gC + gR)

Dhmix ¤

Dhmix(T: float, phi: FloatVector, m: FloatVector) -> float

Enthalpy of mixing per mole of sites, \(\Delta h_{mix}\).

\[ \Delta h_{mix} = \Delta g_{mix} + T \Delta s_{mix} \]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

phi

Volume, mass or segment fractions of all components.

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

RETURNS DESCRIPTION
float

Enthalpy of mixing per mole of sites. Unit = J/mol.
Source code in src/polykin/thermo/acm/floryhuggins.py 
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def Dhmix(self,
          T: float,
          phi: FloatVector,
          m: FloatVector) -> float:
    r"""Enthalpy of mixing per mole of sites, $\Delta h_{mix}$.

    $$ \Delta h_{mix} = \Delta g_{mix} + T \Delta s_{mix} $$

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    phi : FloatVector
        Volume, mass or segment fractions of all components.
    m : FloatVector
        Characteristic size of all components, typically equal to 1 for
        small molecules and equal to the average degree of polymerization
        for polymers.

    Returns
    -------
    float
        Enthalpy of mixing per mole of sites. Unit = J/mol.
    """
    return self.Dgmix(T, phi, m) + T*self.Dsmix(T, phi, m)

Dsmix ¤

Dsmix(T: float, phi: FloatVector, m: FloatVector) -> float

Entropy of mixing per mole of sites, \(\Delta s_{mix}\).

\[ \Delta s_{mix} = -\left(\frac{\partial \Delta g_{mix}} {\partial T}\right)_{P,\phi_i,m_i} \]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

phi

Volume, mass or segment fractions of all components.

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

RETURNS DESCRIPTION
float

Entropy of mixing per mole of sites. Unit = J/(mol·K).
Source code in src/polykin/thermo/acm/floryhuggins.py 
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def Dsmix(self,
          T: float,
          phi: FloatVector,
          m: FloatVector) -> float:
    r"""Entropy of mixing per mole of sites, $\Delta s_{mix}$.

    $$ \Delta s_{mix} = -\left(\frac{\partial \Delta g_{mix}}
       {\partial T}\right)_{P,\phi_i,m_i} $$

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    phi : FloatVector
        Volume, mass or segment fractions of all components.
    m : FloatVector
        Characteristic size of all components, typically equal to 1 for
        small molecules and equal to the average degree of polymerization
        for polymers.

    Returns
    -------
    float
        Entropy of mixing per mole of sites. Unit = J/(mol·K).
    """
    return -derivative_complex(lambda t: self.Dgmix(t, phi, m), T)[0]

__init__ ¤

__init__(
    N: int,
    a: Optional[FloatSquareMatrix] = None,
    b: Optional[FloatSquareMatrix] = None,
    c: Optional[FloatSquareMatrix] = None,
    d: Optional[FloatSquareMatrix] = None,
    e: Optional[FloatSquareMatrix] = None,
) -> None

Construct FloryHuggins with the given parameters.

Source code in src/polykin/thermo/acm/floryhuggins.py 
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def __init__(self,
             N: int,
             a: Optional[FloatSquareMatrix] = None,
             b: Optional[FloatSquareMatrix] = None,
             c: Optional[FloatSquareMatrix] = None,
             d: Optional[FloatSquareMatrix] = None,
             e: Optional[FloatSquareMatrix] = None
             ) -> None:
    """Construct `FloryHuggins` with the given parameters."""

    # Set default values
    if a is None:
        a = np.zeros((N, N))
    if b is None:
        b = np.zeros((N, N))
    if c is None:
        c = np.zeros((N, N))
    if d is None:
        d = np.zeros((N, N))
    if e is None:
        e = np.zeros((N, N))

    # Check shapes
    for array in [a, b, c, d, e]:
        if array.shape != (N, N):
            raise ShapeError(
                f"The shape of matrix {array} is invalid: {array.shape}.")

    # Check bounds (same as Aspen Plus)
    check_bounds(a, -1e2, 1e2, 'a')
    check_bounds(b, -1e6, 1e6, 'b')
    check_bounds(c, -1e6, 1e6, 'c')
    check_bounds(d, -1e6, 1e6, 'd')
    check_bounds(e, -1e6, 1e6, 'e')

    # Ensure chi_ii=0 and chi_ij=chi_ji
    for array in [a, b, c, d, e]:
        np.fill_diagonal(array, 0.)
        enforce_symmetry(array)

    self._N = N
    self._a = a
    self._b = b
    self._c = c
    self._d = d
    self._e = e

a ¤

a(
    T: float, phi: FloatVector, m: FloatVector
) -> FloatVector

Activities, \(a_i\).

PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

phi

Volume, mass or segment fractions of all components.

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

RETURNS DESCRIPTION
FloatVector

Activities of all components.
Source code in src/polykin/thermo/acm/floryhuggins.py 
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def a(self,
      T: float,
      phi: FloatVector,
      m: FloatVector
      ) -> FloatVector:
    r"""Activities, $a_i$.

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    phi : FloatVector
        Volume, mass or segment fractions of all components.
    m : FloatVector
        Characteristic size of all components, typically equal to 1 for
        small molecules and equal to the average degree of polymerization
        for polymers.

    Returns
    -------
    FloatVector
        Activities of all components.
    """
    return FloryHuggins_activity(phi, m, self.chi(T))

chi cached ¤

chi(T: float) -> FloatSquareMatrix

Compute the matrix of interaction parameters.

\[ \chi_{ij} = a_{ij} + b_{ij}/T + c_{ij} \ln{T} + d_{ij} T + e_{ij} T^2 \]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

RETURNS DESCRIPTION
FloatSquareMatrix

Matrix of interaction parameters.
Source code in src/polykin/thermo/acm/floryhuggins.py 
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@functools.cache
def chi(self, T: float) -> FloatSquareMatrix:
    r"""Compute the matrix of interaction parameters.

    $$
    \chi_{ij} = a_{ij} + b_{ij}/T + c_{ij} \ln{T} + d_{ij} T + e_{ij} T^2
    $$

    Parameters
    ----------
    T : float
        Temperature. Unit = K.

    Returns
    -------
    FloatSquareMatrix
        Matrix of interaction parameters.
    """
    return self._a + self._b/T + self._c*log(T) + self._d*T + self._e*T**2