7.2. Stochastic Theory#
7.2.1. Brief History of the Stochastic Theory of SEC#
Let us recap the history with the following excerpts from [DCR+02]:
The stochastic theory of chromatography, originally conceived by Giddings and Eyring in 1955 [GE55], was recast by Carmichael to represent SEC processes
Several important contributions to stochastic theory of chromatography appeared after the original Carmichael’s work on SEC.
However this advancement leads to complex mathematics.
With the introduction of the characteristic function (CF) method, the mathematical intractability was completely overcome.
7.2.2. Model Summary#
See the summary of developed models in the following tables.
Model Name |
Charasteristic Function |
PDF formula |
|---|---|---|
GEC Monopore |
\( \phi(\omega)=\exp \Big[ i \omega t_0 + n_1(\frac{1}{1 - i \omega \tau_1} - 1) \Big] \) |
Available |
Dispersive Monopore |
\( \phi(\omega)=\exp \Big[ z + \frac{1}{2 N_0}z^2 \Big]; \quad z = i \omega t_0 + n_1(\frac{1}{1 - i \omega \tau_1} - 1) \) |
NA |
GEC Lognormalpore |
\( \phi(\omega)= \exp \Big[ i \omega t_0 + \int_{R_g}^\infty L_{\mu,\sigma}(r) n_r(\frac{1}{1-i \omega \tau_r} - 1) dr \Big] \) |
NA |
GEC N-pore |
\( \phi(\omega)= \exp \Big[ i \omega t_0 + \sum_{k=1}^N p_{k}n_{k}(\frac{1}{1-i \omega \tau_{k}} - 1) \Big] \) |
NA |
The PDF formula is available only for GEC monopore model while those of other models are only numerically computable.
Model Name |
\(M_1\) |
\(\bar{M_2}\) |
|---|---|---|
GEC Monopore |
\( t_0 + n_1 \tau_1 \) |
\( 2 n_1 \tau_1^2 \) |
Dispersive Monopore |
\( t_0 + n_1 \tau_1 \) |
\( 2 n_1 \tau_1^2 + \frac{ (t_0 + n_1 \tau_1)^2 }{N_0}\) |
GEC Lognormalpore |
\( t_0 + \int_{R_g}^\infty L_{\mu,\sigma}(r) n_r \tau_r dr \) |
\( 2 \int_{R_g}^\infty L_{\mu,\sigma}(r) n_r \tau_r^2 dr \) |
GEC N-pore |
\( t_0 + \sum_{k=1}^N p_k n_k \tau_k \) |
\( 2 \sum_{k=1}^N p_k n_k \tau_k^2 \) |
7.2.3. Stochastic Dispersive Model#
While the above summary outlines the development of the theory, the model we have chosen to use is the stochastic dispersive model [FCRD99] (in its monopore form).
The reasons are as follows:
When accounting for dispersion, other models consider only the stationary phase, not the mobile phase.
For the current use of the model, the monopore form is preferable to avoid computational complexity.