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Falloff

Falloff functions for reactions.

FalloffFunction

Bases: Protocol

Defines a calloff function protocol.

Source code in src/freckll/reactions/falloff.py 
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class FalloffFunction(t.Protocol):
    """Defines a calloff function protocol."""

    def __call__(
        self,
        k0: FreckllArray,
        kinf: FreckllArray,
        m: FreckllArray,
        temperature: FreckllArray,
        *args: float,
    ) -> FreckllArray: ...

no_falloff(k0, kinf, m, temperature, *args)

No falloff function.

Source code in src/freckll/reactions/falloff.py 
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def no_falloff(
    k0: FreckllArray,
    kinf: FreckllArray,
    m: FreckllArray,
    temperature: FreckllArray,
    *args: float,
) -> FreckllArray:
    """No falloff function."""
    return 1.0

sri_falloff(k0, kinf, m, temperature, a_sri, b_sri, c_sri, d_sri, e_sri)

Stanford Research Institute falloff function.

Computes the Stan falloff function, first \(X_{sri}\) is computed as:

\[ X_{SRI} = \frac{1}{1 + [\log_{10}(\frac{k_0 M}{k_\infty})]^2} \]

The center fall-off factor is: $$ F_{c} = d_{SRI}(a_{SRI} \exp(-\frac{b_{SRI}}{T}) + \exp(-\frac{T}{c_{SRI}}))^{X_{SRI}}\times T^{e_{SRI}} $$

Parameters:

Name Type Description Default
k0 FreckllArray

The low-pressure rate constant.

 required
kinf FreckllArray

The high-pressure rate constant.

 required
m FreckllArray

The concentration of the reactants.

 required
temperature FreckllArray

The temperature of the reaction.

 required
a_sri float

The SRI falloff parameter.

 required
b_sri float

The SRI falloff parameter.

 required
c_sri float

The SRI falloff parameter.

 required
d_sri float

The SRI falloff parameter.

 required
e_sri float

The SRI falloff parameter.

 required
Source code in src/freckll/reactions/falloff.py 
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def sri_falloff(
    k0: FreckllArray,
    kinf: FreckllArray,
    m: FreckllArray,
    temperature: FreckllArray,
    a_sri: float,
    b_sri: float,
    c_sri: float,
    d_sri: float,
    e_sri: float,
) -> FreckllArray:
    r"""Stanford Research Institute falloff function.

    Computes the Stan falloff function, first $X_{sri}$ is computed as:

    $$
    X_{SRI} = \frac{1}{1 + [\log_{10}(\frac{k_0 M}{k_\infty})]^2}
    $$

    The center fall-off factor is:
    $$
    F_{c} = d_{SRI}(a_{SRI} \exp(-\frac{b_{SRI}}{T}) + \exp(-\frac{T}{c_{SRI}}))^{X_{SRI}}\times T^{e_{SRI}}
    $$

    Args:
        k0: The low-pressure rate constant.
        kinf: The high-pressure rate constant.
        m: The concentration of the reactants.
        temperature: The temperature of the reaction.
        a_sri: The SRI falloff parameter.
        b_sri: The SRI falloff parameter.
        c_sri: The SRI falloff parameter.
        d_sri: The SRI falloff parameter.
        e_sri: The SRI falloff parameter.
    """

    x_sri = 1 / (1 + np.log10(k0 * m / kinf) ** 2)

    f_cent = (
        d_sri * (a_sri * np.exp(-b_sri / temperature) + np.exp(-temperature / c_sri))
    ) ** x_sri * temperature**e_sri

    return t.cast(FreckllArray, f_cent)

troe_falloff_term(k0, kinf, m, temperature, a_troe, t3, t1, t2)

Troe Falloff function.

Computes the center factor: $$ F_{c} = (1-\alpha)\exp{\frac{-T}{T_3}} + \alpha\exp{\frac{-T}{T_1}} + \exp{\frac{-T_2}{T}} $$ The constants: $$ c = -0.4 - 0.67\log_{10}(F_c) $$ $$ N = 0.75 - 1.27\log_{10}(F_c) $$

The logarithmic falloff factor is: $$ \log_{10}(F) = \frac{\log_{10}(F_c)}{1 + (\log_{10}(\frac{k_0}{k_\infty}) + \frac{c}{N - d\log_{10}(\frac{k_0}{k_\infty}) + c})^2} $$

Finally we return the factor as: $$ F = 10^{\log_{10}(F)} $$

Parameters:

Name Type Description Default
k0 FreckllArray

The low-pressure rate constant.

 required
kinf FreckllArray

The high-pressure rate constant.

 required
m FreckllArray

The concentration of the reactants.

 required
temperature FreckllArray

The temperature of the reaction.

 required
a_troe float

The Troe falloff parameter.

 required
t3 float

The third Troe falloff parameter.

 required
t1 float

The first Troe falloff parameter.

 required
t2 float

The second Troe falloff parameter.

 required

Returns:

Name Type Description
falloff FreckllArray

The falloff term.
Source code in src/freckll/reactions/falloff.py 
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def troe_falloff_term(
    k0: FreckllArray,
    kinf: FreckllArray,
    m: FreckllArray,
    temperature: FreckllArray,
    a_troe: float,
    t3: float,
    t1: float,
    t2: float,
) -> FreckllArray:
    r"""Troe Falloff function.

    Computes the center factor:
    $$
    F_{c} = (1-\alpha)\exp{\frac{-T}{T_3}} + \alpha\exp{\frac{-T}{T_1}} + \exp{\frac{-T_2}{T}}
    $$
    The constants:
    $$
    c = -0.4 - 0.67\log_{10}(F_c)
    $$
    $$
    N = 0.75 - 1.27\log_{10}(F_c)
    $$

    The logarithmic falloff factor is:
    $$
    \log_{10}(F) = \frac{\log_{10}(F_c)}{1 + (\log_{10}(\frac{k_0}{k_\infty}) + \frac{c}{N - d\log_{10}(\frac{k_0}{k_\infty}) + c})^2}
    $$

    Finally we return the factor as:
    $$
    F = 10^{\log_{10}(F)}
    $$


    Args:
        k0: The low-pressure rate constant.
        kinf: The high-pressure rate constant.
        m: The concentration of the reactants.
        temperature: The temperature of the reaction.
        a_troe: The Troe falloff parameter.
        t3: The third Troe falloff parameter.
        t1: The first Troe falloff parameter.
        t2: The second Troe falloff parameter.

    Returns:
        falloff: The falloff term.


    """
    # Falloff terms
    d = 0.14
    if t3 == 0.0 or t1 == 0.0:
        return 1.0

    f_cent = (1 - a_troe) * np.exp(-temperature / t3) + a_troe * np.exp(-temperature / t1) + np.exp(-t2 / temperature)
    log_fcent = np.log10(f_cent)

    c = -0.4 - 0.67 * log_fcent
    nf = 0.75 - 1.27 * log_fcent
    log_k0kinfm = np.log10(k0 * m / kinf)
    log_falloff = log_fcent / (1 + ((log_k0kinfm + c) / (nf - d * (log_k0kinfm + c))) ** 2)

    return t.cast(FreckllArray, 10.0**log_falloff)