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# Extended Mobile Phase Modulator Langmuir¶

This model is an extension of the mobile phase modulator Langmuir model (see Section Multi Component Langmuir), which allows linear binding of some selected components. A modifier component $$c_{p,\mathrm{mod}}$$ is selected and the remaining components are divided into the index sets $$\mathcal{I}_{\mathrm{lin}}$$ and $$\mathcal{I}_{\mathrm{lang}}$$.

\begin{split}\begin{aligned} \frac{\mathrm{d} q_i}{\mathrm{d} t} &= k_{a,i} e^{\gamma_i c_{p,\mathrm{mod}}} c_{p,i}\: q_{\text{max},i} \left( 1 - \sum_{j=1}^{N_{\text{comp}} - 1} \frac{q_j}{q_{\text{max},j}} \right) - k_{d,i} \: c_{p,\mathrm{mod}}^{\beta_i} \: q_i && i \in \mathcal{I}_{\mathrm{lang}}, \\ \frac{\mathrm{d} q_i}{\mathrm{d} t} &= k_{a,i} c_{p,i} - k_{d,i} \: q_i && i \in \mathcal{I}_{\mathrm{lin}}. \end{aligned}\end{split}

The modifier component is considered to be inert, therefore either

$\frac{\mathrm{d} q_{\mathrm{mod}}}{\mathrm{d} t} = 0$

is used if the modifier component has a bound state, or it can be used without a bound state.

The model can also be used without a modifier component. In this case, the equations are given by

\begin{split}\begin{aligned} \frac{\mathrm{d} q_i}{\mathrm{d} t} &= k_{a,i} c_{p,i}\: q_{\text{max},i} \left( 1 - \sum_{j=1}^{N_{\text{comp}} - 1} \frac{q_j}{q_{\text{max},j}} \right) - k_{d,i} \: q_i && i \in \mathcal{I}_{\mathrm{lang}}, \\ \frac{\mathrm{d} q_i}{\mathrm{d} t} &= k_{a,i} c_{p,i} - k_{d,i} \: q_i && i \in \mathcal{I}_{\mathrm{lin}}. \end{aligned}\end{split}

For more information on model parameters required to define in CADET file format, see Extended Mobile Phase Modulator Langmuir.