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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.