# Progressive Cellular Architecture in Microscale Gas Chromatography for Broad Chemical Analyses
## 2 Theoretical Rationale  
In gas chromatography, two important metrics are the retention times of the analytes and the separation resolutions of target analyte pairs. The retention time (\(t_{Ri}\)) of analyte \(i\) is the time taken to elute from a given column [31]:  
\[t_{Ri}=(1+k_{i})t_{0} \tag{1}\]  
where \(k_{i}\) is the retention factor for analyte \(i\), and \(t_{0}\) is the elution time of an unretained analyte. The value of \(k_{i}\) depends on the stationary phase material, its polarity, and its thickness; the temperature of the separation column; and the polarity and volatility of the analyte. In many applications, a large value of \(k_{i}\) is undesirable because it increases \(t_{R}\), thereby forcing the carrier gas flow and column heating to be prolonged, consuming excessive energy. The separation resolution (\(R_{s}\)) of two analytes is also affected by the properties of the column and the analytes [31]:  
\[R_{s}=\frac{1}{4}(N)^{1/2}\bigg{(}\frac{\alpha-1}{\alpha}\bigg{)}\bigg{(}\frac {k_{2}}{1+k_{2}}\bigg{)};\ \alpha=\frac{k_{2}}{k_{1}} \tag{2}\]  
where \(\alpha\) is the separation factor, and \(N\) is the efficiency of the separation column, which is approximated as a constant for the given column geometry. The \(\alpha\) is only dependent on the analyte properties, i.e., volatility and polarity. As evident from Equation (2), if \(k_{2}\) is <1, \(R_{s}\) is highly sensitive to \(k_{2}\), decreasing rapidly with decreasing \(k_{2}\). If \(k_{2}\) is >10, an increase in \(k_{2}\) only contributes to a modest increase in \(R_{s}\). Considering Equations (1) and (2), to achieve large \(R_{s}\) while maintaining small \(t_{R}\), \(k_{i}\) in the range of 3-10 is favored for the target analytes. Note that Equations (1) and (2) are only applicable to isothermal separations; temperature programmed separations require a more complex analytical model. The \(R_{s}\) can be also determined experimentally from chromatograms [31]:  
\[R_{s}=\frac{1.18\Delta t_{R}}{PWHH_{1}+PWHH_{2}} \tag{3}\]  
where \(\Delta t_{R}\) is the difference in retention time between two peaks; and the \(PWHH_{1}\) and \(PWHH_{2}\) are the peak widths at half height of the two peaks. Equation (3) assumes a perfectly Gaussian peak, whereas the actual peak shape may have peak distortion depending on the polarity of the analyte. To evaluate the benefit of using a stationary phase thickness that is tailored to the target analytes, six hypothetical cases were analyzed; specifically, the retention time and separation resolution were estimated based on the carrier gas and flow, column type and length, and column temperature. In all cases, columns were assumed to have polydimethylsiloxane (PDMS) stationary phase, 3 m length and 250 \(\upmu\)m inner diameter (ID). The carrier gas was assumed to be nitrogen with 0.6 sccm flow rate. The analytes were assumed to be alkanes C3-C11. (This type of analysis can be performed using any type of chromatogram modeler using well-established rules of solubility, including commercial modelers, e.g., Pro EZGC(r) [32]). Cases 1-3 were used for comparing isothermal separations at 25 \({}^{\circ}\)C, which was assumed to be the ambient temperature. Case 1 incorporated a single column with 0.1 \(\upmu\)m stationary phase thickness (Column B), and Case 2 incorporated a single column with 5.0 \(\upmu\)m stationary phase thickness (Column A), whereas Case 3 mimicked a 2-cell PCA that incorporated Column A for alkanes C\({}_{3}\)-C\({}_{7}\) and Column B for alkanes C\({}_{8}\)-C\({}_{11}\). As evident from the computed results (Table 2), Case 1 required only 636 s analysis time for eluting C\({}_{11}\), but provided \(R_{s}\)\(\leq\) 2.1 for the high-volatility analytes, C\({}_{3}\)-C\({}_{5}\). Case 2 provided much higher \(R_{s}\)\(\geq\) 6 for C\({}_{3}\)-C\({}_{5}\), but required 28,704 s analysis time, which was 45-fold longer than Case 1. Case 3 maintained \(R_{s}\)\(\geq\) 6 across alkanes C\({}_{3}\)-C\({}_{11}\), and required 1309 s analysis time, which was only twice that of Case 1.