Optimization of two-dimensional isocratic gas chromatography: a Pareto approach
Abstract
A mathematical model has been constructed to calculate the two-dimensional peak capacity in GCxGC, as a function of column geometries, mobile phase characteristics and system configuration (pressures and modulation time). The parameters investigated are column length, diameters, film thickness, pressure, and modulation time for both dimensions in isocratic mode. The model is used to relate isocratic peak capacities with total analysis time, and it is used to optimize the parameters above (column geometries and system configuration) to maximize peak capacities and minimize analysis time. The way to approach this (multi-objective) optimization was by the means of the Pareto methodology. The inspection of the Pareto front revealed that the optimal modulation times could be shorter than the usual. Also, in order to optimize time, both columns are operating at sub-optimal conditions relative to the minimum plate height in the plate-height equation.
Contents
1.Introduction 34
2.1. Description of the peak-capacity model for two-dimensional chromatography. 35
2.2. The number of cuts per peak 39
3. Experimental 310
3.1. Parameters used 310
3.2. Equations workflow 311
3.3. Pareto optimization 312
4. Results and discussion 313
Appendix 1: 31717
Appendix 2: 31818
Appendix 3: Error! Bookmark not defined.1921
Appendix 4: 32324
1.Introduction
In separation science, the chromatographer is often confronted with the problem of the optimization of the chromatographic system. This task is relatively straightforward in one-dimensional chromatography. For example, simple rules can be applied to find the optimal column geometries and flow regimes in one-dimensional column chromatography. This task becomes more troublesome in the case of two-dimensional chromatography. The reason is the fact that the different parameters affecting the system (e.g., column geometries, flow rates and modulation times) are no longer independent. Hence, they have to be optimized together, increasing exponentially the number of possible combinations of the parameters.
In the past, a general framework to optimize column geometries, flow rates and modulation times for HPLCxHPLC was developed[1]. In this seminal paper, the problem of the exponential growth of the number of combination of parameters was solved by using the Pareto-optimization methodology. This technique reduces the number of possible combinations to the ones that are actually optimal, in terms of the different objectives that are being maximized (or minimized). In this way, the number of possible combinations to be examined is reduced from several millions to a few hundred. In addition, the changes of the different optimal parameters along the different total analysis times can be inspected, similarly to the work of Poppe [citation Poppe plots].
In that paper, we made use of the Pareto optimization methodology to find those conditions that maximize peak capacity, minimize time, and (eventually) minimize dilution in HPLCxHPLC, in both isocratic and gradient regimes. The parameters optimized concerned column geometries, flow rates and modulation times. In that paper, two effects had demonstrated to be crucial. On one hand, the worsening of the peak capacity of the first dimension was considered. This worsening (sometimes called “Tanaka factor”) is due to the (inevitable) undersampling that occurs when fractions of the first dimension are injected in the second. This effect, commented by many others [citations], was introduced in the Pareto-optimization framework, finding similar results as those found in the bibliography. Another key effect was the worsening of the second dimension peak capacity due to the injection band broadening. Without paying attention to this effect, peak capacity of the second dimension is overestimated.
When optimizing GCxGC systems, several differences exist compared to the liquid chromatography case. On one hand, active modulation is applied between the first and the second dimensions, so the injection band broadening effect in the second dimension can be neglected (similar to the focusing effects that are desirable in the HPLCxHPLC case). On the other hand, no extra mobile phase is injected in the interphase between the first and the second dimension. This means that the chromatographer has less degrees of freedom to manipulate the system, as the complete mass of mobile phase that abandons the first-dimension column (and no more than it) is injected in the second. This does not happen in the HPLCxHPLC case, in which the valves-system in the interphase between the two columns allows to add extra mobile phase in the second dimension, thus changing selectivity and modifying the flow regimes. This restriction in the GCxGC systems forces the diameter of the second dimension column to be smaller than the first, in order to make the chromatographic process of the second-dimension separation fast enough for a true GCxGC separation. As has been demonstrated elsewhere, this in turn forces both (1st and 2nd dimension) columns to work at suboptimal flow velocities according to the plate-height equation.
In this paper, we extend the Pareto optimization concept to the GCxGC case. This allows to examine several effects with a new insight. First, we examine the GCxGC system optimizing the total peak capacity and the analysis time, finding a collection of optimal conditions (opposed to a single optimal condition, as proposed by de Koning et al.). This allows to examine how the column geometries change along the total analysis time (in a similar fashion as the Poppe plots used in HPLC []). Second, the so-called “Tanaka factor” is introduced for the first time in gas chromatography, and its consequences are examined in detail. Third, we want to know the impact of the restrictions mentioned earlier (i.e. working at suboptimal conditions in the plate-height equation) when a collection of optimal conditions is examined. This allows us to inspect whether these restrictions are important at low or at high peak capacities.
2. Theory
2.1. Peak-capacity model for isocratic two-dimensional gas chromatography.
In this section, the equations used to calculate the peak capacity in isocratic two-dimensional gas chromatography are described. As the objective of this work is to optimize GCxGC systems attending to (maximize) peak capacity and (minimize) total analysis time, a model should be constructed that relates both objectives with the factors being optimized (including column lengths, column diameters, film thickness, modulation time, and column pressures).
Schoenmakers et al.[2] defined the conditions to be met for a two-dimensional separation to be called comprehensive, i.e. (i) every bit of the sample is subjected to two different separations and (ii) the resolution of the first-dimension separation is essentially maintained. In other words, this means that the sample is separated following two different retention mechanisms while the separation in both dimensions is maintained. If the two-dimensional chromatography is performed in time, there should be a mechanism (i.e. modulation) that collects small fractions of the first dimension and injects them into the second separation. For the separation to be comprehensive, the first dimension separation should be (normally 100 times) slower than the second dimension.
Therefore, the total analysis time 2Dtw is defined as the sum of the time taken for the last-eluting compound to elute from the first dimension (1tw) and the time taken to run the second dimension(2tw)[3]:
(1)
We will follow through this paper the notation described elsewhere[4] , in which the left-hand superscript indicates the separation dimension (e.g. 1tr stands for the first-dimension retention time, 2L stands for the second-dimension column length, etc.). In Eq. 1, the second term 2tw is equal to the modulation time 1tw. In Eq. (1) this term can be neglected, since (as mentioned) retention times in the first dimension are around 100 times larger than the retention times in the second.
The concept of total peak capacity can be defined as the maximum number of base-line separated peaks that can be separated if the space is occupied with well-ordered, adjacent peaks. The maximum total peak capacity (2Dn) that can be achieved by the system corresponds to a situation in which the two separation mechanisms are completely orthogonal. As we are interested only in maximizing this (theoretical) quantity, the total peak capacity is defined as the product of the first- (1n) and second dimension (2n) peak capacities[5]
(2)
For Gaussian-shaped peaks it is accepted that the peak width at the base is 4σ. Therefore, if base-line separation is required to construct this “well-ordered” peak arrangement, the peak capacity for any dimension becomes
(3)
where t0 is the dead time and tw is defined above.
.
The value of s for the first dimension, 1s, depends on the band broadening contributions of the 1D chromatographic process (1σpeak) and the band broadening contribution due to the relatively low-frequency modulation time (tw):
(4)
Where δdet2 is a constant with values between 12 [[6]] and 4,76 [[7]]. In this model δdet2=12 which is chosen to be sufficiently high in order not to contribute to the total peak width and is a representative value for practical situations. The value δdet2=4,76 has been derived experimentally with statistical overlap theory applied to two dimensional separations, but it will be not used here. The chromatographic band broadening 1speak is a function of the plate height (1H), the first dimension retention time (1tr) and the column length (1L):
(5)
The plate-height 1H can be modeled using the Golay equation[8]:
(6)
Where 1CE is the column efficiency, 1Dm,o is the diffusion coefficient of the analyte in the mobile phase (see eq. 20), 1Ds,o is the diffusion coefficient of the analyte in the stationary phase (see Eq. 21), 1uout is the outlet linear velocity, f(1k) and g(1k) are correction factors (see below), 1dc is column diameter, 1df is the film thickness and 1f1 and 1f2 are pressure correction factors. Superscript 1 indicates that the values are calculated for the first dimension, but an equivalent expression holds for the second dimension. f(1k) is defined as
(7)
Where 1k is the retention factor. Similarly, Function of retention factor g(1k) is
(8)
We will make use of the relationship between the retention factor (1k) and the retention time (1tr) via the column deadtime 1t0
(9)
In Eq. (6), the pressure correction factors are:
(10)
and
(11)
In Eq. (10) and (11), 1p0 is the (dimensionless) ratio between the inlet(1pin) and outlet pressures (1pout).
(12)
The values of 1pin and 1pout are related also through the volumetric flow rate (1F) at the outlet of the column[9]
(13)
Where h is the dynamic viscosity of the carrier gas. We have been using the empirical equation suggested by Etre et al.[10] to calculate this viscosity. For hydrogen (the carrier gas used in all the computations), the viscosity at a certain temperature (Ti) is obtained as follows:
(14)
Similar relationships can be found for other carrier gases.
In Eq. (15), the linear velocity at the outlet of the first-dimension column (1uout) can be related to the volumetric flow rate (1F) at the end of the column and the column diameter (1dc)[11] as described by Poiseuille[12].
(15)
Finally, the average linear velocity in the first dimension (1ū) is a function of the outlet linear velocity (1uout) and the second pressure correction factor in the first dimension (1f2).
(16)
In the following equations, we have to make use of the relationship between the retention time of the un-retained compound and the column length:
(17)
In two-dimensional chromatography, the mass-flow of exiting the first-dimension column is the same as the mass flow that enterings the second dimension column. Applying Poiseuille equation on this equality yields
(18)
Where the superscripts “1” and “2” refer to the first- or second-dimension separations, as described earlier.
To calculate the diffusion coefficient in Eq. (6), we make use of the empirical computation of the mobile phase diffusion coefficient for binary gas mixtures[13], which is calculated from the molar masses of the components (Mm and Mo), the temperature T (K), the pressure p (Pa) and their diffusion volumes ,(m3). In this paper, we have been using hydrogen as carrier gas (m), and we used C12H26 as a model molecule (o).
(19)
From Eq. (19), a relationship between the diffusion coefficient in the first and the second dimensions can be established (considering that the temperature in the first-dimension column is the same as the temperature in the second dimension):
(20)
To calculate the diffusion coefficient in the stationary phase, a good approximation is to consider that Ds is 50.000 times smaller than Dm,o[14]:
(21)
By combining eq. 3 -11, the peak capacity (for any separation dimension) becomes
(22)
When this equation is used to calculate the peak capacity in the first dimension, ti2 is the modulation time and tw is the retention time of the last eluting compound in the first dimension. When the equation is applied to calculate the peak capacity in the second dimension, ti2 becomes the response time of the detector (the time between two adjacent data points) and tw is the modulation time. The solution of this integral can be found in Appendix-1, eq. (23).
2.2. The number of cuts per peak
It is important to get an estimation of the average number of second-dimension injections during the elution of a first-dimension peak (i.e. the number of “cuts per peak”). This is because at very low modulation rates, the number of cuts per peak could be so low that Eq. 4 is no longer applicable. The number of cuts per peak is 41speak/2tw . As 1s is not constant, the average of the number of cuts per peak is calculated by integrating 41speak /2tw over the first-dimension t and dividing the result by the time spanned in the integration limits:
(27)
To simplify the notation, in Eq. 27 all t values correspond to 1t, except the one for the term. Solution of this integral can be found in appendix 2, eq. (28).
3. Experimental
3.1. Parameters used
One should note that the equations described in section 2 are valid within (reasonable) parameter ranges. In this section, some of these parameter ranges are discussed.
As for column diameters, the model is basically valid with column diameters 0.50 mm – 0.03 mm. With diameters below this range the equations are no longer valid due to phenomenon called “slip flow” (in which. the gas interactions with the capillary wall start to become significant[15]). As for the film thickness, the equations describe the processes accurately for thin film columns. Increasing the film thickness at constant temperature affects the capacity ratio[16].