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Multi-State Steric Mass Action

The multi-state steric mass action model adds \(M_i-1\) additional bound states \(q_{i,j}\) (\(j = 0, \dots, M_i - 1\)) for each component \(i\) to the steric mass action model (see Section Steric Mass Action) and allows the exchange between the different bound states \(q_{i,0}, \dots, q_{i,M-1}\) of each component. In the multi-state SMA model a variable number of states of the bound molecule (e.g., different orientations on the surface, binding strength of tentacle adsorbers) is added which are more and more strongly bound, i.e.,

\[\begin{aligned} \nu_{i,j} \leq \nu_{i,j+1} \qquad i = 1, \dots, N_{\text{comp}} - 1, \quad j = 0,\dots, M_i - 1. \end{aligned}\]

The exchange between the different states of each component is allowed and, since the molecules can potentially bind in all states at the same binding site, competitive effects are present.

\[\begin{split}\begin{aligned} \frac{\mathrm{d} q_{i,j}}{\mathrm{d} t} =& \phantom{+} k_{a,i}^{(j)} c_{p,i} \left(\frac{\bar{q}_0}{q_{\text{ref}}}\right)^{\nu_{i,j}} - k_{d,i}^{(j)}\: q_{i,j}\: \left(\frac{c_{p,0}}{c_{\text{ref}}}\right)^{\nu_{i,j}} \\ &- \sum_{\ell = 0}^{j-1} \underbrace{k^{(i)}_{j\ell}\: q_{i,j}\: \left(\frac{c_{p,0}}{c_{\text{ref}}}\right)^{\left(\nu_{i,j} - \nu_{i,\ell}\right)}}_{\text{to weak state}} - \sum_{\ell = j+1}^{M_i - 1} \underbrace{k^{(i)}_{j\ell}\: q_{i,j}\: \left(\frac{\bar{q}_0}{q_{\text{ref}}}\right)^{\left(\nu_{i,\ell} - \nu_{i,j}\right)}}_{\text{to strong state}} \\ &+ \sum_{\ell = 0}^{j-1} \underbrace{k^{(i)}_{\ell j}\: q_{i,\ell}\: \left(\frac{\bar{q}_0}{q_{\text{ref}}}\right)^{\left(\nu_{i,j} - \nu_{i,\ell}\right)}}_{\text{from weak state}} + \sum_{\ell = j+1}^{M_i - 1} \underbrace{k^{(i)}_{\ell j}\: q_{i,\ell}\: \left(\frac{c_{p,0}}{c_{\text{ref}}}\right)^{\left(\nu_{i,\ell} - \nu_{i,j}\right)}}_{\text{from strong state}} & \begin{aligned} i &= 1, \dots, N_{\text{comp}} - 1, \\ j &= 0, \dots, M_i - 1, \end{aligned} \end{aligned}\end{split}\]

where \(c_{p,0}\) and \(q_0\) denote the salt concentrations in the liquid and solid phase of the beads respectively. The number of available salt ions \(\bar{q}_0\) is given by

\[\begin{aligned} \bar{q}_0 = \Lambda - \sum_{j=1}^{N_{\text{comp}} - 1} \sum_{\ell=0}^{M_j - 1} \left( \nu_{j,\ell} + \sigma_{j,\ell} \right) q_{j,\ell}. \end{aligned}\]

A neutrality condition compensating for the missing equation for \(\frac{\mathrm{d} q_0}{\mathrm{d}t}\) is required:

\[\begin{aligned} q_0 = \Lambda - \sum_{j=1}^{N_{\text{comp}} - 1} \sum_{\ell=0}^{M_j - 1} \nu_{j,\ell} q_{j,\ell}. \end{aligned}\]

The concept of reference concentrations (\(c_{\text{ref}}\) and \(q_{\text{ref}}\)) is explained in the respective paragraph in Section Reference concentrations.

For more information on model parameters required to define in CADET file format, see Multi-State Steric Mass Action.