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The objective of this work is to establish a means of correcting the theoretical maximum peak capacity of comprehensive two-dimensional (2D) separations to account for the deleterious effect of undersampling first-dimension peaks. Simulations of comprehensive 2D separations of hundreds of randomly distributed sample constituents were carried out, and 2D statistical overlap theory was used to calculate an effective first-dimension peak width based on the number of observed peaks in the simulated separations. The distinguishing feature of this work is the determination of the effective first-dimension peak width using the number of observed peaks in the entire 2D separation as the defining metric of performance. We find that the ratio of the average effective first-dimension peak width after sampling to its width prior to sampling (defined as ) is a simple function of the ratio of the first-dimension sampling time (t(s)) to the first-dimension peak standard deviation prior to sampling (1sigma): = square root1+0.21(t /(s)(1) sigma(2) This is valid for 2D separations of constituents having either randomly distributed or weakly correlated retention times, over the range of 0.2 on t(s)/1 sigma from this expression is in qualitative agreement with previous work based on the effect of undersampling on the effective width of a single first-dimension peak, but predicts up to 35% more broadening of first-dimension peaks than is predicted by previous models. This simple expression and accurate estimation of the effect of undersampling first-dimension peaks should be very useful in making realistic corrections to theoretical 2D peak capacities, and in guiding the optimization of 2D separations.

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The low prediction by statistical-overlap theory of the numbers of singlets and peaks in two-dimensional separations containing zones represented by either circles of small number or eccentric ellipses of any number is shown to result from use of probability expressions for unbound spaces of infinite extent. An exact theory is derived for the probability of singlet formation in a reduced two-dimensional space of unit length, width, and area. The probability is a weighted sum of the probabilities of singlet formation in the interior, edge, and corner regions of the space, which depend only on saturation. The weighting factors are the fractions of area associated with each region and depend on the number of zones, the aspect ratio, the saturation, and the ellipse's spatial orientation. The average numbers of doublets, triplets, and peaks in the space are approximated by combining these results with Roach's equations describing the clustering of circles in an unbound two-dimensional space. Simulations show that theory predicts the number of singlets, doublets, triplets, and peaks, when the number of zones is 25 or more, the aspect ratio is 100 or less, and the saturation is 2 or less. The relationship is derived between the aspect ratios of ellipses in the reduced space and actual separation space. Calculations are presented for comprehensive two-dimensional gas chromatography.

One of the basic tenets of comprehensive two-dimensional chromatography is that the total peak capacity is simply the product of the first and second dimension peak capacities. As formulated the total peak capacity does not depend on the relative values of the individual dimensions but only on the product of the two. This concept is tested here for the experimentally realistic situation wherein the first dimension separation is undersampled. We first propose that a relationship exists between the number of observed peaks in a two-dimensional separation and the effective peak capacity. We then show here for a range of reasonable total peak capacities (500 to 4000) and various contributions of peak capacity in each dimension (10 to 150) that the number of observed peaks is only slightly dependent on the relative contributions over a reasonable and realistic range in sampling times (equal to the first dimension peak standard deviation, multiplied by 0.2 to 16). Most of this work was carried out under the assumption of totally uncorrelated retention times. For uncorrelated separations the small deviations from the product rule are due to the “edge effect” of statistical overlap theory and a recently introduced factor that corrects for the broadening of first dimension peaks by undersampling them. They predict that relatively more peaks will be observed when the ratio of the first to the second dimension peak capacity is much less than unity. Additional complications are observed when first and second dimension retention times show some correlation but again the effects are small. In both cases deviations from the product rule are measured by the relative standard deviations of the number of observed peaks, which are typically 10 or less. Thus although the basic tenet of two-dimensional chromatography is not exact when the first dimension is undersampled, the deviations from the product rule are sufficiently small as to be unimportant in practical work. Our results show that practitioners have a high degree of flexibility in designing and optimizing experimental comprehensive two-dimensional separations.

Equations are derived for the expected numbers of singlet, doublet, and triplet spots In a two-dimensional (2-D) separation bed, In which circular component zones are randomly distributed. The basis of these derivations Is the selective Interpretation of the radial distribution functions governing 2-D Poisson processes. The equations are sufficient to describe overlap In many 2-D separations and are shown to be adequate In describing the overlap of elliptical zones having aspect ratios less than 2. It Is demonstrated that, per unit peak capacity, 2-D separations are considerably worse than their one-dlmenslonal analogues. The equations are verified at low saturations by Interpretation of several hundred computer simulations of spot distributions In rectangular beds. Departures of the equations from the simulations are observed at higher saturations. Caution Is suggested In overinterpreting the quality of 2-D separations.

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