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.

1.Introduction ............................................................................................................................. 4
2. Theory .................................................................................................................................... 6
2.1. Peak-capacity model for isocratic two-dimensional gas chromatography. ..................... 6
2.2. The number of cuts per peak ......................................................................................... 10
3. Experimental ........................................................................................................................ 10
3.1. Parameters used ............................................................................................................ 11
3.2. Equations workflow ...................................................................................................... 11
3.3. Pareto optimization ....................................................................................................... 13
4. Results and discussion ......................................................................................................... 14
4.1. Minimum film thickness and minimum column diameters. ......................................... 14
4.2. Total peak capacity vs. analysis times. ......................................................................... 14
4.3. Plate height equations for a Pareto optimal case .......................................................... 16
4.4. Number of cuts per peak ............................................................................................... 17
5. Conclusions .......................................................................................................................... 19
Appendix 1: .............................................................................................................................. 20
Appendix 2: .............................................................................................................................. 21
Appendix 3: .............................................................................................................................. 23
Appendix 4: .............................................................................................................................. 26

In separation science, the chromatographer is often confronted with the problem of the optimization of the chromatographic system. This task is relatively straightforward in onedimensional 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 Paretooptimization 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 Paretooptimization 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 firstdimension 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
1
G. Vivo -Truyols, Sj. van der Wal, and P. J. Schoenmakers. Anal. Chem. 2010, 82, 8525–8536

separation. As has been demonstrated elsewhere, this in turn forces both (1 st and 2 nd 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.

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 2D t

is defined as the sum of the time taken for the lasteluting compound to elute from the first dimension ( 1 t

) and the time taken to run the second dimension( 2 t

) 3 :
2𝐷 𝑡 𝜔
= 1 𝑡 𝜔
+ 2 𝑡 𝜔
(1)
We will follow through this paper the notation described elsewhere 4 , in which the left-hand superscript indicates the separation dimension (e.g. 1 t r
stands for the first-dimension retention time, 2 L stands for the second-dimension column length, etc.). In Eq. 1, the second term 2 t

is equal to the modulation time 1 t w.
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 ( 2D n) 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- ( 1 n) and second dimension ( 2 n) peak capacities 5
2𝐷 𝑛
(2)
2 P. Schoenmakers et al., LC GC Eur. 16 (2003) 1
3 Vivo-Truyols, G.; van der Wal, Sj.; Schoenmakers, P. J.
Anal. Chem. 2010, 82, 8525–8536
4
Philip J Marriott, Grace Wu Zeying and Peter Schoenmakers. LC-GC Europe , 25 (2012) 266-275.
5
Leonid M. Blumberg, Matthew S. Klee. Journal of Chromatography A, 933 (2001) 1–11

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 𝑛 = 𝑛(𝑡) =
1
4
∫ 𝑡 𝜔 𝑡
0 𝑑𝑡 𝜎(𝑡)
(3) where t
0
is the dead time and t

is defined above.
.
The value of  for the first dimension, 1  , depends on the band broadening contributions of the 1 D chromatographic process ( 1 σ peak
) and the band broadening contribution due to the relatively low-frequency modulation time (t w
):
1 𝜎 = √( 1 𝜎 𝑝𝑒𝑎𝑘
) 2 +
( 𝑡 𝑤
) 2
(𝛿 𝑑𝑒𝑡
) 2
(4)
Where δ det
2 is a constant with values between 12 [6] and 4,76 [7] . In this model δ det
2
=
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 δ det
2 =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 height ( 1 H), the first dimension retention time (
1  peak
is a function of the plate
1 t r
) and the column length ( 1 L):
1 𝜎 𝑝𝑒𝑎𝑘
= √
1
𝐻 ( 1 𝑡 𝑟
) 2
1
𝐿
(5)
The plate-height 1 H can be modeled using the Golay equation 8 :
1
1
𝐻 =
1
1
𝐶 𝐸
((
2 𝐷 𝑚,𝑜
1 𝑢 𝑜
+ 𝑓(
1 𝑘)
1 𝑑
21 𝑐 𝑢
1 𝐷 𝑚,𝑜 𝑜𝑢𝑡
) 𝑓
1
+
2 𝑓
1 𝑢 𝑜𝑢𝑡
1
𝐷 𝑠,𝑜
1 𝑓
2
)
(6)
Where below),
1 CE is the column efficiency, mobile phase (see eq. 20), phase (see Eq. 21),
1
1
1 D m,o
is the diffusion coefficient of the analyte in the
1 D s,o
is the diffusion coefficient of the analyte in the stationary u out is the outlet linear velocity, f( 1 k) and g( 1 d c
is column diameter, 1 k) are correction factors (see d f
is the film thickness and 1 f
1
and 1 f
2
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( 1 k) is defined as 𝑓( 1 𝑘) =
1+6 𝑘 +11 𝑘
96(1+ 𝑘 )
2
(7)
6 Leonid M. Blumberg, FrankDavid, Matthew S. Klee, Pat Sandra. Journal of Chromatography A, 1188 (2008) p.p. 8 -9
7
Henri Snijders, Hans-Gerd Janssen, Carel Cramers. Journal of Chromatography A, 718 (1995) 339-355
8 Jan Beens a, Hans-Gerd Janssen , Mohamed Adahchour , Udo A. Th. Brinkman. Journal of Chromatography A, 1086 (2005) 143.

Where 1 k is the retention factor. Similarly, Function of retention factor g( 1 k) is 𝑔( 1 𝑘) =
3(1+ 𝑘 )
2
(8)
(
We will make use of the relationship between the retention factor ( 1 k) and the retention time
1 t r
) via the column deadtime 1 t
0
1 𝑘 =
1 𝑡 𝑟
− 𝑡
0
1 𝑡
0
(9)
In Eq. (6), the pressure correction factors are:
1 𝑓
1
=
4
0
−1) ( 𝑝
2
0
3
0
−1) 2
−1)
(10) and
1 𝑓
2
=
2
0
3
0
−1)
−1)
(11)
In Eq. (10) and (11), pressures ( 1 p out
).
1 𝑝
0
=
1 𝑝 𝑖𝑛
1 𝑝 𝑜𝑢𝑡
1 p
0
is the (dimensionless) ratio between the inlet( 1 p in
) and outlet
(12)
The values of 1 of the column 9 p in
and 1 p out
are related also through the volumetric flow rate ( 1 F) at the outlet
1
𝐹 = ( 𝜋
1 𝑑
4 𝑐
) (
1 𝑝
2 𝑖𝑛
1 𝑝 𝑜𝑢𝑡
2 𝑜𝑢𝑡 )
(13)
Where  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 (T i
) is obtained as follows: 𝑛 = 83,5 (
𝑇 𝑖
273,15
)
0,680
(14)
Similar relationships can be found for other carrier gases.
In Eq. (15), the linear velocity at the outlet of the first-dimension column ( 1 u out
) can be
( related to the volumetric flow rate ( 1
1 d c
) 11 as described by Poiseuille 12 .
F) at the end of the column and the column diameter
9 Jan Beens a, Hans-Gerd Janssen , Mohamed Adahchour , Udo A. Th. Brinkman. Journal of Chromatography A, 1086 (2005) 141–150.
10 L. S. Ettre. Chromatographia Vol. 18, No. 5. Mai 1984.
11 Jan Beens a, Hans-Gerd Janssen , Mohamed Adahchour , Udo A. Th. Brinkman. Journal of Chromatography A, 1086 (2005) 143.
12 http://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation. Last visit to the website: 13/March/2013.

1 𝑢 𝑜𝑢𝑡
=
1
𝐹
( 𝜋
4
) 𝑑
2 𝑐
(15)
Finally, the average linear velocity in the first dimension ( 1 velocity ( 1 ū) is a function of the outlet linear u out
) and the second pressure correction factor in the first dimension ( 1 f
2
).
1 ū 𝑜𝑢𝑡
1 𝑓
2
(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:
1 𝑡
0
=
1
𝐿
1 𝑢 𝑜𝑢𝑡
(17)
In two-dimensional chromatography, the mass-flow exiting the first-dimension column is the same as the mass flow entering the second dimension column. Applying Poiseuille equation on this equality yields
1 𝑝 𝑜𝑢𝑡
= 𝑝 𝑖𝑛
= √
1 𝑑
42 𝑐
2 𝑑
4 𝑐
2 𝑖𝑛
4 𝑐
1
𝐿 + 1 𝑑
4 𝑐
1
𝐿
2
𝐿
2 𝑝
2 𝑜𝑢𝑡
(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 (M m
and M o
), the temperature T (K), the pressure p (Pa) and their diffusion volumes v m
, v o
(m 3 ). In this paper, we have been using hydrogen as carrier gas
(m), and we used C
12
H
26
as a model molecule (o).
2
𝐷 𝑚,𝑜
=
1,00 𝑥 10
−3
𝑇
1,75
(
1
𝑀𝑚
1 𝑝 [(∑ 𝑣 𝑚
) 3
1
+
𝑀𝑜
1
2
)
1
2
+ + (∑ 𝑣 𝑜
) 3 ]
(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):
1
𝐷 𝑚,𝑜
=
2
𝐷 𝑚,𝑜
2 𝑝 𝑜𝑢𝑡
1 𝑝 𝑜𝑢𝑡
(20)
To calculate the diffusion coefficient in the stationary phase, a good approximation is to consider that D s
is 50.000 times smaller than D m,o
14 :
13 Karaiskakis, G. and D. Gavril.
Journal of Chromatography A, 1037 (2004) 147–189
14 Jan Beens a, Hans-Gerd Janssen , Mohamed Adahchour , Udo A. Th. Brinkman. Journal of Chromatography A, 1086 (2005) 143.

𝐷 𝑠,𝑜
=
𝐷 𝑚,𝑜
50.000
(21)
By combining eq. 3 -11, the peak capacity (for any separation dimension) becomes 𝑛 = 𝑛(𝑡) = 𝜎(𝑡) =
∫ 𝑡 𝜔 𝑡
0
4
√ 𝑡𝑖
2
δ𝑑𝑒𝑡
2
+ 𝑡2
2 𝑑𝑓
2
(
3𝐷𝑠(1+ 𝑡−𝑡0 𝑡0
) 𝑡0
(22)
+𝑓1
2𝐷𝑚,𝑜 𝑢0
+
(
1
𝐶𝐸 𝐿
2
+
6(𝑡−𝑡0) 𝑡0
)𝑢𝑜𝑢𝑡
96 𝐷𝑚,𝑜(1+ 𝑡−𝑡0 𝑡0
)
2
)
)
𝑑𝑡
When this equation is used to calculate the peak capacity in the first dimension, t i
2 is the modulation time and t

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, t i
2 becomes the response time of the detector (the time between two adjacent data points) and t

is the modulation time. The solution of this integral can be found in Appendix-
1, eq. (23).
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 4 1  peak
/ 2 t w
. As 1  is not constant, the average of the number of cuts per peak is calculated by integrating 4 1  peak
/ 2 t w
over the firstdimension t and dividing the result by the time spanned in the integration limits:
1 𝑡 𝜔
−𝑡
0
1
2 𝑡𝜔
∫ 𝑡 𝜔 𝑡
0
4
√ 𝑡 2
2 𝑑𝑓
3𝐷𝑠(1+ 𝑡−𝑡0 𝑡0
)
2 𝑡0
(
+𝑓
1
(
2𝐷𝑚,𝑜 𝑢0
+
𝐶𝐸 𝐿
2 (1+11(𝑡−𝑡0)
2
+
6(𝑡−𝑡0) 𝑡0
)𝑢𝑜𝑢𝑡
96 𝐷𝑚,𝑜(1+ 𝑡−𝑡0 𝑡0
)
2
)
)
(27)
𝑑𝑡
1
1
2 𝑡𝜔

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 .
Table 1 presents the values of the parameters used to optimize the system. Parameters including a range of values are optimized. Parameters with only a single value are fixed..
Table 1: parameters to be varied:
Variable
1 L
Variable range (units)
50- 150 (m)
Constants
1 D m,o
Value (units)
=
2
𝐷 m,0
1 𝑝 𝑜𝑢𝑡
(m 2 /s)
2 L 1- 10 (m)
2 D m,o
3,32631E-05 (m 2 /s)
1 d c
2 d c
1 p in
0,2 (mm)
0,1 (mm)
80- 400 (kPa)
D ɳ s
H2
Temperature
6,65262E-10 (m 2 /s)
0.000103285 (Pa .s)
373,15 (K)
1 t 
2 t 
1 d f
2 d f
1440- 7200 (s)
1- 10 (s)
0,25 (μm)
0,25 (μm)
2 p out
100 (kPa)
Figure 1 represents schematically the way the total time and peak capacity are calculated from a collection of values.
15
James A. McGill. LOW-DENSITY GAS DYNAMIC FACILITY. September 22, 1966.
16
T.H.M. Noij. Trace analysis by capillary gas chromatography. January 1988. p.p. 49.

Figure 1:
Equation scheme for the calculation of isocratic peak capacities
Range of variables: 1 L, 2
Input
L, 1 d c
, 2
Locked variables: 1
Constants: T, 1 CE, 2 d c
, 1 t r
, 2 d f
, 2
CE, 𝛿 d f t r
=t w,
1 p in
, 2 𝑑𝑒𝑡
Calculated constants: ɳ e q. (14), D s,o eq. (21) p out
Calculate for each variable from top to bottom
1 p out
= 2
1 p
0
,
1 f
1
,
1 f
2
,
1 F,
1 u out
,
1 ū,
2
1 t
0
,
2
2
2 f
1
Eq. (10) f p in
2
, (eq. 18)
2 p
0
Eq. (12)
Eq. (11)
F Eq. (13)
2 u out
Eq. (15)
ū Eq. (16)
2 t
0
Eq. (17)
1 D m,o
Eq. (19)
Solving peak capacity Eq. (23) from appendix I
Calculate the total analysis time and the total peak capacity 2D t w
2D
(Eq. 1) ɳ (Eq. 2)
Calculate the average number of cuts per peak eq. (28, appendix
II), and do not consider any particular combination of parameters when the number of cuts per peak is below 1.5.
Search for Pareto front for (maximum) peak capacities and
(minimum) time variables

2
The computation starts by considering the values of
 det
1 L, 2 L, 1 d c
, 2 d c
, 1 t w
, 2 t w
, 1 p in
, 2 p out
, d f
(normally the parameters of the model), along with some of the constants (T, 1
1 d f
and
CE, 2 CE,
). In this step the values of viscosity and diffusion coefficients given at the input were considered constant. These values were calculated using Eqs. (14), (19) and (21) for dodecane /C
12
H
26
on a polysiloxane column.
The second step consists of calculating is used to calculate correction factors dimensions,
1
1 F and
1 p
0
and f
1
and 1
2
1 p out
, via Eq. 18, which is equal to 2 p in
. Next, Eq.(10) p
0
, and Eq. (10) and (11) are applied to calculate the pressure f
2
. Next, Eq. (13) is applied to calculate the flow rate for both
2 F. The value of the flow is used in Eq. (15) to calculate the value of the gas velocity at the outlet, u0. The average linear velocity is then calculated according to Eq.
(16), and the result is used in Eq. 17 to calculate the value of the dead time.
The parameters calculated in the previous equations are used to calculate the peak capacity in the first and in the second dimension via Eq. 27 The total analysis time is simply calculated
 with Eq. (1) and the total peak capacity is calculated with Eq. (2).
To calculate the optimal solution, multiple combinations of parameters (with the ranges specified in Table 1) are used. The protocol explained above is applied for every single combination of parameters to calculate the total peak capacity and the total analysis time.
This results in millions of peak-capacity/time pairs, each one associated to a combination of the parameters described in Table 1. Prior to apply the Pareto optimization, the combination of parameters yielding a number of cuts per peak below 1.5 is discarded. In this paper the value of 1.5 cuts per peak has been selected as a threshold, in accordance with the optimal value observed elsewhere [citation Tanaka, Gabriel].
Pareto optimization is applied then. Pareto optimization has been described elsewhere
[Massart et al., part A], and only a brief definition is going to be given here. In short, Pareto optimization consists basically in the calculation of what is called the “Pareto front”, i.e. those experiments for which it is impossible to improve one objective without worsening one of the others. In other words, the Pareto front in our calculations would be those combination of parameters (column lengths, pressures, etc.) for which it is impossible to obtain higher peak capacities without increasing the total analysis time. The Pareto optimization has been implemented writing home-made routines in Matlab (The Mathworks, Natick, MA, USA).

In the current model, the optimization is performed exclusively in terms of (maximum) peak capacity and (minimum) total analysis time. In other words, the model selects any combination of parameters that yield the maximum peak capacity and yield the minimum time. From this consideration, it is obvious that the model was selecting the minimum film thickness and the minimum column diameters available. This is because both parameters have an impact in the plate height. However, a practical threshold was set up for film thickness and column diameters, in order for the system to be able to hold a reasonable column loadability. As column loadability was not considered in the model, we had to set up this thresholds manually. The values of film thickness and column diameters presented in table 1 are already the minimum logical values attending experimental practice in terms of column loadability.
Fig. (2) depicts the Pareto front for the (maximization of) total peak capacity vs. the
(minimization of) total analysis time. A similar effect described in 4.1. occurred with the column length of the first dimension. The Pareto optimization was selecting all the time the longest column in the first dimension available. In fact there is no theoretical limit for the column length in the first dimension (the longer, the better). We had to set up again a practical threshold, limiting the column length of the first dimension to 150 m. In this case, however, we studied the Pareto fronts when the column length was fixed to different values, as depicted in Fig. (2).
The Pareto fronts depicted in Fig. (2) indicate that for an analysis time of around one hour a range of peak capacities between 18.000 and 25.000 can be achieved with a 50m and a 150m column, respectively. Logically, the peak capacity increases as the column length increases.
Figure 2: Retention time vs. Peak capacity for different column lengths.

Fig. (3) depicts the values of optimal factors along the Pareto front, together with the H/H min ratio (the actual value of the plate height to the optimum (minimum) plate height) for the 100 m column. It can be observed that, for longer analysis times (and more complicated samples), the second dimension column length increases. The increase in the second dimension modulation time as the total peak capacity increases is a logical result from the increase in the second dimension column length. it can be observed also that both and 2 H/ 2
1 H/ 1 H min
H min
become closer to 1 (optimal values) as the peak capacity increases. As time is

available, the chromatographic process is slower, making the flow rate velocity closer to its optimum as, at all times, the system operates at the mass-transfer domain of the plate-height equation (see section 4.3).
Figure 3: Pareto optimized curves, plotted for different parameters, for a 100m column.
Figure 4 depicts the plate height equation for the Pareto point depicted in Fig. 3. Using the column geometries and modulation times found at this optimal point, different inlet pressures were used to generate different values of outlet velocity, to construct the plate-height equation. Table 2 depicts the values obtained at this optimal point. As can be seen, the system

works at the mass-transfer range of the plate-height equation for both dimensions. However, this is more evident for the second dimension, as flow rates are higher to make the twodimensional chromatography possible. This is the reason why the velocity approaches the optimal velocity when the total analysis time is increased.
Table 2: First- and second dimension H-values and velocities for a selected optimal peak capacity.
TPC 1 H 1 H min
1 u
0
1 u
0opt
23435
2 H
0,00024239 0,00020675 0,8911
2 H min
2 u
0
0,4994
2 u
0opt
0,00021314 0,00010733 5,5130 1,3921
Figure 4: van Deemter curves plotted of the first and second dimension.
0,002
0,0018
0,0016
0,0014
0,0012
0,001
0,0008
0,0006
0,0004
0,0002
0
0 0,2 0,4 0,6 0,8 1
Outlet linear velocity (m /s)
1,2 1,4 1,6
0,0005
0,00045
0,0004
0,00035
0,0003
0,00025
0,0002
0,00015
0,0001
0,00005
0
0
1 2 3 4 5 6 7 8
Outlet Linear velocity (m/s)
9 10 11 12
Fig. (xxx) depicts the values of the number of cuts per peak along the Pareto front. One should note that the number of cuts per peak depends on the retention time, since in isocratic

100
80
60
40
160
140
120
20
0
0 elution the band broadening is not constant. As can be seen, the number of cuts per peak is extremely low at times around the dead time. For example, for 100 m columns the number of cuts per peak is around 0.2, which means that a modulation time spans for the elution of 5 peaks. In other words, 5 separated peaks are mixed and analyzed in a single injection.
Obviously, this jeopardizes the separation in the first dimension. However, this is properly accounted in Eq. (4), in which the value of 2 tw=5x(4 1  peak
= 20x 1  peak
. Following Eq. (), this means that 𝜎 = 𝜎 𝑝𝑒𝑎𝑘
√1 + 33.3 = 𝜎 𝑝𝑒𝑎𝑘
5.86
. In other words, this means that the actual band broadening used for the calculation of the peak capacity is 5.86 larger than the chromatographic band broadening. This makes sense, since, as it is mentioned earlier, 5 peaks are embedded into the same second-dimension injection. On the other hand, the number of cuts per peak is at the end of the chromatogram is around 10 (although it varies more along the total peak capacity). At these retention times, 1 𝜎 = 𝜎 𝑝𝑒𝑎𝑘
1.007
, making the contribution to the band broadening due to undersampling almost negligible. As can be seen, the average number of cuts per peak reaches the imposed limit (1.5) in almost all Pareto points. The number of cuts per peak at t
0
decrease with increasing peak capacity, whereas the number of cuts per peak increases at t w
. This is because the difference in band broadening between the beginning and the end of the chromatogram grows as the total analysis time (and column length) increases, which is a logical effect in isocratic chromatography.
4.5. Comparison with other column lengths
When comparing a 100 meter column to a 50 and 150 meter columns, it can be observed that the general optimal pressure increases with increasing column length (Figure 5). This relationship can be related to the increase in flow rate to compensate for the increase in column length. This influence can be observed from eqs.(13) and (5). As can be seen, the
H/H min ratio is almost the same for all column lengths. This is due to the fact that an increase in column length is compensated by an increase of pressure, making the situation respect to the plate-height equations similar for all lengths.
Figure 5: The average overpressure given per column length.
180
50 100
L (m)
150 200
150
125
100
75
50

Typical 2D configurations may consist of a first dimension (60 m length, 0,25 mm i.d., 0,25
μ m film thickness) column together with a second (1,5 m length, 0,25 mm i.d., 0,1 μ m film thickness) column 17 . The total peak capacity can be found for a 50 m column in the first dimension and a 1,5 m column in the second around 10000 peaks. Furthermore it can be observed that this peak capacity can be increased significantly by increasing the second dimension column length. If this is done, the optimal modulation time increases. In practical applications the second dimension column lengths might be smaller than those provided in the literature. Also, modulation times tend to be shorter. This is because we allowed a lower number of cuts per peak in the system (minimally 1.5) opposed to the typical 4 cuts per peak proposed in the literature. The value of 1.5 is the optimal value of number of cuts per peak when the worsening of the first dimension separation due to undersampling is considered properly.
From the van Deemter plots in section 4.3. it can be concluded that both dimensions are operating at higher than optimal flow conditions – the mass transfer region, irrespective to the time needed for the separation. Again, this is a departure from previous studies since the average number of cuts per peak is smaller than the reported cuts per peak in older studies.
As the number of cuts per peak is smaller, the first dimension can be faster, allowing to operate in the mass-transfer range of the Van Deemter equation. Apparently it is beneficial to undersample the first dimension separation to get faster first- and second-dimension analysis.
Finally it can be observed that the pressure (logically) increases with increasing column length and speed, while other parameters remain the same.
17 http://www.restek.com/Technical-Resources/Technical-Library/Petroleum-Petrochemical/petro_PCAN1789-UNV

The solution to the integral in equation 22 is: 𝑛 (𝑡) =
√6𝛂𝐿𝑜𝑔(𝑑𝑒𝑡 (𝛄)) 𝛆
(23)
Where the terms are: 𝛼 = √(96 𝐶𝐸 𝐷𝑚,𝑜
𝐷 𝑠
𝐿 𝑡 2 𝑤 𝑢
0
(24)
+ 𝑑𝑒𝑡 2 (192 𝐷 2 𝑚,𝑜
𝐷 𝑠
𝑓
1
𝑡 2 + 64 𝑑 𝑓
2 𝐷 𝑚,𝑜 𝑓
2
(𝑡 − 𝑡
0
)𝑡
0
𝑢 2 𝑜
+ 𝑑 𝑐
2 𝐷 𝑠
𝑓
1
(11𝑡 2 − 16𝑡 𝑡
0
+ 6𝑡 2
0
)𝑢 2
0
)) 𝛾 = √𝐷 𝑠
√𝑓
1
√192𝐷 2 𝑚,𝑜
+ 11𝑑 𝑐
2 𝑢 2
0 𝛂 + 𝑑𝑒𝑡(192 𝐷 2 𝑚,𝑜
𝐷 𝑠
𝑓
1
(11𝑡 − 8𝑡
0
)𝑢 2
0
+ 32 𝑑 𝑓
2 𝐷 𝑚,𝑜
𝑓
2 𝑡
0 𝑢 2 𝑜
) (25) 𝜀 = 𝑑𝑒𝑡√𝐷 𝑠
√𝑓
1
√192𝐷 2 𝑚,𝑜
+ 11𝑑 𝑐
2 𝑢 2
0
[𝛂/(𝐶𝐸 det 2 𝐷 𝑚,𝑜
𝐷 𝑠
𝐿 𝑢
0
) ] (26)

Solution for Eq. (27) is:
∫ 4
√ 𝑡 2
2 𝑑𝑓
3𝐷𝑠(1+ 𝑡−𝑡0 𝑡0
)
2 𝑡0
(
+𝑓
1
(
2𝐷𝑚,𝑜 𝑢0
+
𝐶𝐸 𝐿
2 (1+11(𝑡−𝑡0)
2
+
6(𝑡−𝑡0) 𝑡0
)𝑢𝑜𝑢𝑡
96 𝐷𝑚,𝑜(1+ 𝑡−𝑡0 𝑡0
)
2
)
) 𝑑𝑡 = 𝜃(𝜄+(𝜅𝜆)) 𝜇
(28)Where the terms are: 𝜂 = 𝑑 2 𝑐
𝐷 𝑠 𝑓
1 𝑢 2
0
(11𝑡 2 − 16𝑡 𝑡
0
+ 6𝑡 2
0
) + 64𝑑 2 𝑓
𝐷 𝑚,𝑜 𝑓
2 𝑡
0 𝑢 2
0
(𝑡 − 𝑡
0
) + 192𝐷 2 𝑚,𝑜
𝐷 𝑠 𝑓
1 𝑡 2 𝛈 𝜃 = √
𝐶𝐸𝐷 𝑚,𝑜
𝐷 𝑠
𝐿𝑢
0 𝜄 = √𝐷 𝑠
√𝑓
1
√11𝑑 2 𝑐 𝑢 2
0
+ 192 𝐷 2 𝑚,𝑜
(𝑑 2 𝑐
𝐷 𝑠 𝑓
1 𝑢 2
0
(11𝑡 − 8𝑡
0
) + 32𝑑 2 𝑓
𝐷 𝑚,𝑜 𝑓
2 𝑡
0 𝑢 2
0
+ 192𝐷 2 𝑚,𝑜
𝐷 𝑠 𝑓
1 𝑡)√𝛈 𝜅 = 2𝑡 2
0 𝑢 2
0
[𝑑 4 𝑐
𝐷 2 𝑠 𝑓
1
2 𝑢 2
0
+ 96𝑑 2 𝑐
𝐷 𝑚,𝑜
𝐷 𝑠 𝑓
1
(6𝐷 𝑚,𝑜
𝐷 𝑠 𝑓
1
− 𝑑
2 𝑓 𝑓
2 𝑢 2 𝑜
) − 512𝑑
2 𝑓
𝐷 2 𝑚,𝑜 𝑓
2
(𝑑
2 𝑓 𝑓
2 𝑢 2
0
+ 12𝐷 𝑚,𝑜
𝐷 𝑠 𝑓
1
)]
√𝐷 𝑠
√𝑓
1
√11𝑑 2 𝑐 𝑢 2
0
+ 192 𝐷 2 𝑚,𝑜
√𝛈 𝜆 = 𝐿𝑜𝑔 [
+11𝑑 2 𝑐
𝐷 𝑠 𝑓
1 𝑡𝑢 2
0 𝜇 = 2√6𝐷 𝑠
3
2 𝑓
1
3
2 𝑡 𝑤
(11𝑑 2 𝑐 𝑢 2
0
− 8𝑑 2 𝑐
𝐷 𝑠 𝑓
1 𝑡
0 𝑢 2
0
+ 192 𝐷 2 𝑚,𝑜
+ 32𝑑
) 3/2
2 𝑓
𝐷 𝑚,𝑜 𝑓
2
√𝛈 𝑡
0 𝑢 2
0
+ 192𝐷 2 𝑚,𝑜
𝐷 𝑠 𝑓
1 𝑡
]
(29)
(30)
(31)
(32)
(33)
(34)