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Simplified Multi-State Steric Mass Action

The simplified multi-state steric mass action is the same as the multi-state SMA model described above (see Section Multi-State Steric Mass Action), but with additional assumptions:

• Molecules are only exchanged between two adjacent states, that is, no transfer from state \(q_{i,1}\) to state \(q_{i,3}\) is allowed.

• Characteristic charge \(\nu_{i,j}\) and shielding factor \(\sigma_{i,j}\) only depend on the index of the state \(j\).

Thus, the exchange parameters \(k^{(i)}_{j\ell}\), the characteristic charge \(\nu_{i,j}\), and the shielding \(\sigma_{i,j}\) can be parameterized with few degrees of freedom. For all \(i = 1,\dots,N_{\text{comp}} - 1\) and \(j,\ell = 0,\dots,M_i - 1\) let

\[\begin{split}\begin{aligned} k^{(i)}_{j\ell} &= \begin{cases} 0, & \text{for } \left\lvert j-\ell\right\rvert \neq 1 \\ K^{(i)}_{ws} + j K^{(i)}_{ws,\text{lin}} - K^{(i)}_{ws,\text{quad}} j(j - M_i+2), & \text{for } \ell = j+1 \\ K^{(i)}_{sw} + \ell K^{(i)}_{sw,\text{lin}} - K^{(i)}_{sw,\text{quad}} \ell(\ell - M_i+2), & \text{for } \ell = j-1, \end{cases}\\ \nu_{i,j} &= \nu_{\text{min},i} + \frac{j}{M_i-1} \left( \nu_{\text{max},i} - \nu_{\text{min},i} \right) - \nu_{\text{quad},i} j (j-M_i+1), \\ \sigma_{i,j} &= \sigma_{\text{min},i} + \frac{j}{M_i-1} \left( \sigma_{\text{max},i} - \sigma_{\text{min},i} \right) - \sigma_{\text{quad},i} j (j-M_i+1). \end{aligned}\end{split}\]

Note that the characteristic charge \(\nu_{i,j}\) has to be monotonically non-decreasing in the second index \(j\) and all other rates and the steric factor \(\sigma_{i,j}\) have to be non-negative.

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