2D separations

The peaks are revealed in the 2D separation space as oval-shaped peaks in a contour plot format, reproduced as a schematic diagram in Figure 4.9. The contour-peaks are now completely separated in the 2D space, whereas they were severely overlapping on the first column. 

Figure 4.13 GC X GC analysis of vetiver essential oil column 1, BPX5 column 2, BPX50 (0.8 m in length). The lower trace presents the pulsed peaks obtained from the modulation process, and shows such peaks in a manner that represents the normal cliromatograpliic result presentation. Tliis nace is many times more sensitive than a normal GC trace. In the upper plot, the 2D separation space shows that the BPX50 column is not very effective in separating components of the oils based on polarity, since all the components are bunched up along the same region of 2D time.

Multidimensional methods thus involve a combination of single mechanisms and systems. In any multidimensional (usually 2D) approach, it is desirable that each dimension be as pure as possible in terms of selectivity of the separation mechanism. In comprehensive 2D separations, the precision (or chromatographic resolution) becomes a limiting factor and is ultimately determined by the quality of the separation in both dimensions. 

This chapter examines another measure of the space used by 2D separations subject to correlation. Some researchers use the words, peak capacity, to express the maximum number of zones separable under specific experimental conditions, regardless of what fraction of the space is used. By definition, however, the peak capacity is the maximum number of separable zones in the entire space. No substantive reason exists to change this definition. The ability to use the space, however, depends on correlation. In deference to previous researchers (Liu et al., 1995 Gilar et al., 2005b), the author adopts the term, practical peak capacity, to describe the used space. The practical peak capacity is the peak capacity, when the separation mechanisms are orthogonal, but is less than the peak capacity when they are not. The subsequent discussion is based on practical peak capacity. 

Equation 3.2 was proposed by Roach (1968) almost 40 years ago to model the overlap of coal particulates sampled from air onto a flat surface. The equation was verified by studying the clustering of randomly distributed circles in a square representing the reduced space of a 2D separation (Oros and Davis, 1992). It then was modified (Rowe and Davis, 1995) to study the clustering of inhomogeneous random distributions of circles (Rowe et al., 1995 Davis, 2004), in which more circles are found in parts of the reduced square than in others, and to address the clustering of ellipses and reduction of clustering that occurs near the reduced-square boundaries (Davis, 2005). For simplicity, only Equation 3.2 is used in this chapter. 

FIGURE 3.2 Graph of average minimum resolution R versus saturation a for 2D separations of randomly distributed bi-Gaussian single-component peaks. Reprinted from Liu and Davis, copyright (2006), with permission from Elsevier. 

The goal of this chapter is to review the recent significant advances achieved in the study of 2D maps (Marchetti et al., 2004 Pietrogrande et al., 2002,2003,2005,2006 a Campostrini et al., 2005). Fundamental aspects concerning the intimate structure of multicomponent mixtures and separations will be discussed in Section 4.2. Description of the methods recently developed by the present authors for characterizing the separation pattern complexity of a 2D multicomponent map will be presented in Sections 4.3 and 4.4. These methods allow one to describe complex 2D separations in terms of SC number (m), the detection of hidden homologous series, spot shape features and separation performance. For these reasons they are named as decoding methods. In Section 4.5, the most recent achievements derived from the application... 

FIGURE 4.4 Example of 2D separations of five homologous series having (a) constant phase (red vector in inset) and uncorrelated frequencies and (b) constant frequency (blue vector in inset) and uncorrelated phases. (See color plate.)... 

Obviously, when going from 2D to ID, we observe an inverse behavior, that is, a randomness is created. However, such a feature does not have only virtual interest but is of surprisingly practical value, as will be discussed in the subsequent paragraph. In fact the SMO approach, based on a ID random pattern, was successfully applied to multicomponent 2D separations (Pietrogrande et al., 2002, 2003 Campostrini et al., 2005). 

Consequently, several hidden quantities can be estimated on the basis of the SMO approach. The procedure based on Equation 4.13 can be simply extended even to 2D separations as described in Fig. 4.7. In practice, the 2D pattern, in terms of spot positions and abundances, is divided into several strips. Each strip is transformed into a ID line chromatogram and the procedure described in Fig. 4.7 is then applied. Equation 4.13 is employed to calculate the m value of each strip from which the total m value is obtained. Applications to this procedure will be reported in Section 4.5. At this point, the reader s attention is drawn to the fact that the procedure of transforming 2D strips into ID chromatograms (see Fig. 4.7) once more corresponds to the overlapping mechanisms described in Fig. 4.2 and has been evocated in comparing Fig. 4.4 with Fig. 4.3. In this way, if random structures (e.g., such as those marked in Fig. 4.1b) are present, their memory is lost and the 2D pattern is reduced to a Poissonian ID one. Therefore, the number of SCs can be correctly estimated, even if the 2D pattern was not Poissonian. 

FIGURE 4.7 Procedure developed to reduce a 2D separation into several 1D chromatograms suitable for applying SMO method on 2D maps. As an example, only the first two strips are reported. 

Equations 4.22 1.24 are the 2D equivalents of Equations 4.17 1.20. The comparison of the two sets of equations shows a surprising consequence. If the peak capacities of the 1D and 2D separation systems were identical, the 2D separation would lead to more severe overlap. In order to have the same number of components isolated as singlets with a 1D and a 2D separation system, the peak capacity of the 2D system ( 2d) should be double of that of the ID system (nw). Ideally, in an orthogonal system 112D = n j D, but part of the gain in peak capacity is lost due to the increased probability of peak overlap provided the 2D chromatogram is disordered. 

An adaptation of Fourier analysis to 2D separations can be established by calculating the autocovariance function (Marchetti et al., 2004). The theoretical background of that approach is that the power spectrum and the autocovariance function of a signal constitute a Fourier pair, that is, the power spectmm is obtained as the Fourier transform of the autocovariance function. 

A nonlinear curve fitting procedure of the experimental (Eq. 4.28) to the theoretical (Eq. 4.27) 2D autocovariance function can serve to perform some fundamental characterization of the 2D separation. The total volume (Vy) and the peak height dispersion (/a() can be readily measured in the chromatogram, thus the number of components (m) and the peak widths (a, and ay) can be estimated (Marchetti et al., 2004). 

FIGURE 4.9 The autocovariance function close to the origin of a 2D separation. The maximum amplitude and the characteristic half-widths at half-heights are used in the simplified analysis. Reproduced from Marchetti et al., (2004) with permission from the American Chemical Society. 

Most often, real 2D chromatograms exhibit a composite ordered and disordered characteristic, that is, a series of disordered spots are superimposed over ordered spot sequences. When the chromatogram is derived from a mixture of several chemical families, a superficial look at the 2D separation map may give the impression of randomness. In that case, the autocovariance function, however, can resolve and help identify the hidden structured nature of the map. 

As an X-ray diffraction image map helps identify the lattice structure of a crystal, the autocovariance function of a 2D separation map may help recognize the chemical structure of complex mixtures. 

This section reviews the most recent results obtained by applying the above-described methods to complex 2D separations. In particular, the case of 2D-PAGE... 

Estimation of the Separation Parameters The extension of the SDO procedure to 2D separations implies that the 2D map is divided into many strips considered as ID separations on which computations are performed. Different... 

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