.. _multi_component_sips_model:

Multi Component Sips
~~~~~~~~~~~~~~~~~~~~~~~~

The Sips binding model is a combination of the :ref:`Freundlich<freundlich_ldf_model>` and the :ref:`Langmuir adsorption model<multi_component_langmuir_model>`.

.. math::

    \begin{aligned}
        \frac{\mathrm{d} q_i}{\mathrm{d} t} = k_{a,i}\: \left( \frac{c_{p,i}}{ c_{\text{ref}} }\right)^{1 / n_i}\: q_{\text{max},i} \left( 1 - \sum_{j=0}^{N_{\text{comp}} - 1} \frac{q_j}{q_{\text{max},j}} \right) - k_{d,i} \left( \frac{q_i}{q_{\text{ref}}} \right) && i = 0, \dots, N_{\text{comp}} - 1.
    \end{aligned}


Here, :math:`c_{\text{ref}}` is a :ref:`reference concentration <reference_concentrations>`, :math:`n_i` is the Freundlich exponent, :math:`k_{a,i}, k_{d,i}` are the adsorption and desorption rates, and :math:`q_{\text{max},j}` is the adsorption capacity.

As for the :ref:`Freundlich<freundlich_ldf_model>` isotherm, the first order Jacobian :math:`\left(\frac{dq^*}{dc_p}\right)` tends to infinity as :math:`c_{p} \rightarrow 0` for :math:`n>1`.
Additionally, the isotherm is undefined for :math:`c_{p} < 0` if :math:`\frac{1}{n_i}` can be expressed as :math:`\frac{p}{q}` with :math:`p,q \in \mathbb{N}` where :math:`q` is an even number.
Negative concentrations can arise during simulations due to numerical fluctuations.
To address these issues an approximation of the isotherm is considered below a threshold concentration :math:`c_p < \varepsilon`.
This approximation matches the isotherm in such a way that :math:`q=0` at :math:`c_p=0` and also matches the value and the first derivative of the istotherm at :math:`c_p = \varepsilon`, where :math:`\varepsilon` is a very small number, for example :math:`1e-10`.
The form of approximation and its derivative is given below for :math:`c_p < \varepsilon`:

.. math::

    \begin{aligned}
        c_{p,i,lin}  &= \left(\frac{\varepsilon}{c_{p,\text{ref}}}\right)^{\frac{1}{n_i} - 2} \frac{c_{p,i}}{{c_{p,\text{ref}}}^2} \left( \left(2-\frac{1}{n_i}\right)\varepsilon + c_{p,i}\left(\frac{1}{n}-1\right) \right)  \\
        \frac{\mathrm{d}q_i}{\mathrm{d}t} &= k_{a,i} c_{p,i,lin} q_{\text{max},i} \left( 1 - \sum_j \frac{q_j}{q_{\text{max},j}} \right) - k_{d,i} \frac{q_i}{q_{i,\text{ref}}}
    \end{aligned}

For more information on model parameters required to define in CADET file format, see :ref:`multi_component_sips_config`.
For more information on the model and its origin, please refer to :cite:`sips1948`.