Anal. Chem. 1996, 68, 682-689
Informational Orthogonality of Two-Dimensional Chromatographic Separations Patrick J. Slonecker
Henkel Corporation Emery Group, Technical Center, 4900 Este Avenue, Cincinnati, Ohio 45232-1491, and Department of Chemistry, University of Cincinnati, Cincinnati, Ohio 45221-0172 Xiaodong Li and Thomas H. Ridgway
Department of Chemistry, University of Cincinnati, Cincinnati, Ohio 45221-0172 John G. Dorsey*
Department of Chemistry, Florida State University, Tallahassee, Florida 32306-3006
A qualitative informational similarity technique has been used to describe the informational orthogonality of projected two-dimensional (2-D) chromatographic separations of complex mixtures from their one-dimensional 1-D separations. The reversed-phase liquid chromatography (RPLC), supercritical fluid chromatography (SFC), gasliquid chromatography (GLC), and micellar electrokinetic capillary chromatography (MECC) retention behavior of up to 46 solutes of varying molecular properties was studied by 2-D range-scaled retention time plots and information entropy calculations. One hundred five combinations of technique/stationary phase pairs were used to simulate the 2-D chromatographic analyses. The informational entropy of one and two dimensions, the mutual information, the synentropy or “cross information”, and the informational similarity were calculated to describe the informational orthogonality. In addition, pattern descriptors were used to qualitatively describe the 2-D peak distribution. With the solutes tested, informational orthogonality, zero informational similarity, was observed with MECC-SDS/SFC-C1, MECC-SDS/SFC-Carbowax, MECC-TTAB/SFC-Carbowax, HPLC-C18/GLCDB-5, HPLC-PBD/SFC-phenyl, SFC-Carbowax/GLC-DB5, and HPLC-phenyl/SFC-phenyl 2-D chromatographic systems. Conversely, with the solutes tested, informational nonorthogonal behavior described by range-scaled retention time plots of moderate to severe band overlap and data clustering was observed with 2-D chromatographic systems with high informational similarity and moderate to high degrees of synentropy. These results should prove useful for predicting complementary 2-D techniques as well as for choosing a second separation technique for confirmation of separation or peak purity. The optimum informing power of one-dimensional (1-D) chromatographic systems is limited by peak capacity and the sample’s component order and/or disorder or sample dimensionality.1-3 These limitations and the desire to resolve (1) Kaiser, H. Anal. Chem. 1970, 42, 24A-41A. (2) Grushka, E. Anal. Chem. 1970, 42, 1142-1147.
682 Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
increasingly complex mixtures have provided incentive for the development of two-dimensional (2-D) and higher dimensional chromatographic systems for higher total peak capacity or greater separation space. In a 2-D chromatographic system, the maximal theoretical peak capacity is the product of the individual peak capacities. It was once believed that this maximal peak capacity could be obtained if the stationary phase, mobile phase, and solute intermolecular interactions involve completely different separation mechanisms in each 1-D space. Unfortunately, this ideal situation is difficult to attain, and less than perfect separations are more commonplace. To a large extent, the order and disorder of 1-D component peak distributions play significant roles in achieving the optimum resolution.2-4 Much of our knowledge of peak overlap and our awareness of the inability of 1-D separations to separate even relatively simple mixtures is due to the work of Davis et al.5-7 As early as 1983, Davis and Giddings noted, “A chromatogram must be approximately 95% vacant to provide a 90% probability that a given compound of interest will appear as an isolated peak”.8 During the last two decades, improved one-, two-, ..., ndimensional chromatographic systems have been developed to deal with increasingly complex separation tasks. This accomplishment is the result of improved separation techniques, a better understanding of retention mechanisms, and new developments in chromatographic interfaces and computers.9,10 Some early examples include the gel permeation/gel permeation chromatographic system (GPC/GPC) developed by Balke and Patel11 and the liquid chromatography/gas chromatography systems (LC/GC) developed by Raglione and Hartwick.12 Novel research on 2-D analytical techniques, which included LC/GC and (3) Giddings, J. C. J. Chromatogr. A 1995, 703, 3-15. (4) Giddings, J. C. Anal. Chem. 1984, 56, 1258A-1270A. (5) Delinger, S. L.; Davis, J. M. Anal. Chem. 1990, 62, 436-443. (6) Davis, J. M. Anal. Chem. 1991, 63, 2141-2152. (7) Davis, J. M. Anal. Chem. 1994, 66, 735-746. (8) Davis, J. M.; Giddings, J. C. Anal. Chem. 1983, 55, 418-442. (9) Dorsey, J. G.; Cooper, W. T. J. Anal. Chem. 1994, 66, 857A-867A. (10) J. Chromatogr. A 1995, 703, 1-694. (11) Balke, S. T.; Patel, R. D. J. Polym. Sci. B 1980, 18, 453-456. (12) Raglione, T. V.; Hartwick, R. A. Anal. Chem. 1986, 58, 2680-2683. 0003-2700/96/0368-0682$12.00/0
© 1996 American Chemical Society
GPC/GC, was carried out by Cortes et al.13 and Crummett et al.14 Since 1990, there has been great interest in the development of 2-D chromatographic systems using capillary zone electrophoresis (CZE), GC, and SFC. Jorgenson et al. have made major contributions in the development of CZE/CZE and HPLC/CZE systems, and Liu and Phillips have made important contributions in the development of GC/GC systems.10,15-19 Knowledge of retention orthogonality is useful for more than predicting complementary combinations of chromatographies. In many cases, what is needed is a second, independent separation on the original sample, either for confirmation of peak purity or for confirmation of separation. This type of separation process is often used for submissions to regulatory agencies. There have been many papers published on comparative chromatographic stationary phase studies; however, an investigation conducted by Steuer et al.20 piqued our interest. In this work, the strengths and weaknesses of HPLC, SFC, and CZE were compared in a limited drug study using range-scaled retention time plots as a display method for the visualization of informational orthogonality, a term not defined by those authors. Although few papers reference informational orthogonality in terms of chromatography, Liu et al. have reported one approach to measure the “orthogonality” of several 2-D GC/GC systems using factor analysis.21 We were interested in informational orthogonality, a maximal multidimensional informational state, and decided to investigate the concept, as it applies to chromatography, with a wider selection of solutes, stationary phases, and techniques to simulate the analysis of “real-world” or “complex” samples which comprise a range of intermolecular interactions, such as dispersive, dipole-dipole, dipole-induced dipole, and hydrogen bonding effects. A variety of display methods22 have been used historically to map the characterization of chromatographic stationary phases of the same technique, such as the interesting biplot clustering approach used by Massart and Kaufman.23 However, a major goal of many of these methods was to derive theoretical parameters from retention behavior, not to maximize informational orthogonality. Also, these methods required identical chromatographic conditions, a condition which is impossible for multitechnique systems. Therefore, we chose a range-scaled plot approach as the display method because of its applicability to multitechnique comparisons. (13) Cortes, H. J.; Bell, B. M.; Pfeiffer, C. D.; Graham, J. D. J. Microcolumn Sep. 1989, 1, 278-288. (14) Crummett, W. B.; Cortes, H. J.; Fawcett, T. G.; Kallos, G. J.; Martin, S. J.; Putzig, C. L.; Tou, J. C.; Turkelson, V. T.; Yurga, L.; Zakett, D. Talanta 1989, 36, 63-87. (15) Lemmo, A. V.; Jorgenson, J. W. Anal. Chem. 1993, 65, 1576-1581. (16) Larmann, J. P., Jr.; Lemmo, A. V.; Moore, A. W., Jr.; Jorgenson, J. W. Electrophoresis 1993, 14, 439-447. (17) Bushey, M. M.; Jorgenson, J. W. Anal. Chem. 1990, 62, 161-167. (18) Moseley, M. A.; Detering, L. J.; Tomer, K. B.; Jorgenson, J. W. J. Chromatogr. 1989, 480, 197-209. (19) Liu, Z.; Phillips, J. B. J. Chromatogr. Sci. 1991, 29, 227. (20) Steuer, W.; Grant, I.; Erni, F. J. Chromatogr. 1990, 507, 125-140. (21) Liu, Z.; Patterson, D. G., Jr.; Lee, M. L. Anal. Chem. 1995, 67, 3840-3845. (22) Hamoir, T.; Sanchez, F.; Bourguignon, B.; Massart, D. J. Chromatogr. Sci. 1994, 32, 488-489. (23) Massart, D. L.; Kaufman, L. Clustering: Introduction and Preliminary Operations, and Applications. In The Interpretation of Analytical Chemical Data by the Use of Cluster Analysis; Elving, P., Winefordner, J., Kolthoff, I. M., Eds.; Chemical Analysis Series 65; John Wiley & Sons: New York, 1983; Chapter 1, pp 15-29, Chapter 6, pp 162-164.
Table 1. Solutes Used in This Research aniline 4-cyanophenol dimethyl phthalate 4-methyl-2-nitroaniline 1-nitropropane o-nitrophenol benzene toluene benzyl alcohol corticosterone propachlor docosanoic acid fluorobenzene methyl decanoate propyl 4-hydroxybenzoate oleic acid
anthracene dibutyl phthalate ethylbenzene p-nitroaniline 2-nitropropane p-nitrophenol methyl ethyl ketone 4-ethylnitrobenzene cyclohexene
p-chlorophenol N,N-diethylaniline 4-iodophenol 1-nitrohexane o-nitroaniline dl-R-methylbenzylamine naphthalene nitrobenzene n-butyl p-hydroxybenzoate cortisone acetate chloropropham biphenyl diethyl phthalate eicosanol ethyl 4-hydroxybenzoate hexafluorobenzene methyl 4-hydroxybenzoate octadecanophenone phenol triacontane tritolyl phosphate
Information theory has been used in chromatography for component identification and stationary phase comparison and in optimization of chromatographic systems.20,23-30 In the present research, information theory was used to provide a qualitative numerical description of the informational orthogonality in projected 2-D chromatographic separations of a selection of solutes. This was accomplished by calculating the informational entropy, mutual information, and the informational similarity (see Information Theory section below) of the projected 2-D systems from the experimental retention time data from 1-D systems. THEORY Retention Time Transformation and Two-Dimensional Chromatographic Plots. Forty-six solutes of widely varying molecular properties and 15 different stationary phases were used in this study. The solute choice, which was limited to compounds containing molecular functionalities detectable by all of the techniques investigated and intended to simulate real-world samples encountered in everyday separations, was expanded from a list of 26 solutes first used by Ying and Dorsey in 1991 to characterize the retentivity of RPLC columns.31 Table 1 lists these solutes. The number of projected 2-D chromatographic systems resulting from 15 different stationary phases is 105 combinations of 2-D chromatographic systems. Table 2 lists these technique/ stationary phase combinations. Plots of multisource data of very different absolute magnitudes can exhibit skewed data classifications. Therefore, a range-scaling transformation, described by Steuer et al.,20 was applied to the retention time data in this study to compensate for this effect. In this transformation, a solute’s retention time, Rti, and the retention time of the longest-eluting component in common with each (24) Huber, J. F. K.; Kenndler, E.; Reich, G.; Hack, W.; Wolf, J. Anal. Chem. 1993, 65, 2903-2906. (25) Matsuda, R.; Hayashi, Y.; Ishibashi, M.; Takeda, Y. J. Chromatogr. 1989, 462, 13-21. (26) Matsuda, R.; Hayashi, Y.; Ishibashi, M.; Takeda, Y. J. Chromatogr 1989, 462, 23-30. (27) Erni, F.; Frei, R. J. Chromatogr. 1978, 149, 561-569. (28) Dupuis, F.; Dijkstra, A. Anal. Chem. 1975, 47, 379-383. (29) Eskes, A.; Dupuis, F.; Dijkstra, A.; De Clercq, H.; Massart, D. L. Anal. Chem. 1975, 47, 2168-2174. (30) Eckschlager, K.; Danzer, K. Information Theory in Analytical Chemistry; Winefordner, J., Ed.; Chemical Analysis Series 128; John Wiley & Sons: New York, 1994; Chapters 1-13, pp 1-275. (31) Ying, P. T.; Dorsey, J. G. Talanta 1991, 38, 237-243.
Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
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Table 2. Two-Dimensional Chromatography Entropy Data system1/system2
no. of compds
I(1),a bits
I(2),b bits
I(1,2),c bits
syn,d bits
%syn,e
I(1;2),f bits
simg
SFC-biphenyl/MECC-SDS SFC-biphenyl/MECC-TTAB SFC-biphenyl/HPLC-C1 SFC-biphenyl/HPLC-C18 SFC-biphenyl/HPLC-C4 SFC-biphenyl/HPLC-C8 SFC-biphenyl/HPLC-cyano SFC-biphenyl/HPLC-amino SFC-biphenyl/HPLC-PBD SFC-biphenyl/HPLC-phenyl SFC-biphenyl/SFC-C1 SFC-biphenyl/SFC-Carbowax SFC-biphenyl/SFC-phenyl SFC-biphenyl/GC-DB5 MECC-SDS/MECC-TTAB MECC-SDS/HPLC-C1 MECC-SDS/HPLC-C18 MECC-SDS/HPLC-C4 MECC-SDS/HPLC-C8 MECC-SDS/HPLC-cyano MECC-SDS/HPLC-amino MECC-SDS/HPLC-PBD MECC-SDS/HPLC-phenyl MECC-SDS/SFC-C1 MECC-SDS/SFC-Carbowax MECC-SDS/SFC-phenyl MECC-SDS/GC-DB5 MECC-TTAB-HPLC-C1 MECC-TTAB/HPLC-18 MECC-TTAB/HPLC-C4 MECC-TTAB/HPLC-C8 MECC-TTAB/HPLC-cyano MECC-TTAB/HPLC-amino MECC-TTAB/HPLC-PBD MECC-TTAB/HPLC-phenyl MECC-TTAB/SFC-C1 MECC-TTAB/SFC-Carbowax MECC-TTAB/SFC-phenyl MECC-TTAB/GC-DB5 HPLC-C1/HPLC-C18 HPLC-C1/HPLC-C4 HPLC-C1/HPLC-C8 HPLC-C1/HPLC-cyano HPLC-C1/HPLC-amino HPLC-C1/HPLC-PBD HPLC-C1/HPLC-phenyl HPLC-C1/SFC-C1 HPLC-C1/SFC-Carbowax HPLC-C1/SFC-phenyl HPLC-C1/GC-DB5 HPLC-C18/HPLC-C4 HPLC-C18/HPLC-C8 HPLC-C18/HPLC-cyano HPLC-C18/HPLC-amino HPLC-C18/HPLC-PBD HPLC-C18/HPLC-phenyl HPLC-C8/SFC-C1 HPLC-C18/SFC-Carbowax HPLC-C18/SFC-phenyl HPLC-C18/GC-DB5 HPLC-C4/HPLC-C8 HPLC-C4/HPLC-cyano HPLC-C4/HPLC-amino HPLC-C4/HPLC-PBD HPLC-C4/HPLC-phenyl HPLC-C4/SFC-C1 HPLC-C4/SFC-Carbowax HPLC-C4/SFC-phenyl HPLC-C4/GC-DB5 HPLC-C8/HPLC-cyano HPLC-C8/HPLC-amino HPLC-C8/HPLC-PBD HPLC-C8/HPLC-phenyl HPLC-C8/SFC-C1 HPLC-C8/SFC-Carbowax HPLC-C8/SFC-phenyl
12 13 14 15 14 15 14 14 14 14 23 18 23 13 27 28 28 28 28 28 27 28 28 14 9 13 9 30 30 30 30 30 30 30 30 15 9 14 10 34 34 34 34 33 34 34 16 10 15 11 38 39 35 34 37 36 17 11 16 12 39 35 34 37 36 16 11 15 12 35 34 37 36 17 11 16
3.58 3.70 3.81 3.91 3.81 3.91 3.81 3.81 3.81 3.81 4.44 4.17 4.44 3.70 4.61 4.59 4.59 4.59 4.59 4.59 4.61 4.59 4.59 3.66 3.17 3.55 2.95 4.64 4.64 4.64 4.64 4.64 4.64 4.64 4.64 3.77 3.17 3.66 3.32 4.71 4.71 4.71 4.71 4.66 4.71 4.71 3.88 3.12 3.77 3.46 3.98 3.97 4.25 4.18 4.06 4.32 3.85 2.85 3.88 3.25 4.38 4.43 4.37 4.44 4.49 3.62 3.10 3.51 3.25 4.51 4.51 4.16 4.57 3.85 3.10 3.75
3.42 3.70 3.66 3.77 3.38 3.64 3.38 3.81 3.04 3.52 4.26 3.95 4.35 3.39 4.46 4.49 4.08 4.24 4.31 4.11 4.53 3.95 4.31 3.52 3.17 3.70 3.17 4.55 4.06 4.25 4.32 4.19 4.64 3.92 4.35 3.64 3.17 3.81 3.32 4.28 4.37 4.51 4.45 4.80 4.08 4.54 3.75 3.32 3.91 3.46 4.50 4.29 4.51 4.79 4.23 4.59 3.85 3.46 4.00 3.42 4.18 4.51 4.79 4.23 4.59 3.75 3.46 3.91 3.42 4.51 4.79 4.23 4.59 3.85 3.46 4.00
7.00 7.36 7.08 7.65 6.80 7.29 7.04 7.23 6.81 7.29 8.51 7.86 8.37 7.05 8.65 8.86 8.40 8.74 8.81 8.44 8.92 8.38 8.43 7.19 6.34 6.98 6.04 8.99 8.47 8.75 8.60 8.59 9.16 8.53 8.88 7.31 6.34 7.33 6.44 8.82 8.89 8.97 8.78 9.13 8.73 8.69 7.44 6.24 7.58 6.75 7.56 7.11 8.66 8.08 6.63 7.49 7.66 6.25 7.84 6.67 7.03 8.76 8.07 8.01 8.81 6.97 6.38 7.38 6.62 8.49 8.29 8.06 8.09 7.51 6.38 7.72
0.60 0.74 0.99 0.56 0.91 0.80 1.17 0.89 0.26 0.90 1.15 0.91 0.99 1.00 1.04 0.91 0.69 0.91 0.98 1.04 0.89 0.59 0.85 0.60 0.63 0.82 0.73 0.83 0.61 0.76 0.66 0.79 0.74 0.63 0.69 0.83 0.55 0.83 1.06 0.88 1.12 1.38 1.20 1.17 0.51 1.05 0.87 0.58 0.59 0.46 1.75 1.71 0.41 1.49 1.85 1.35 0.94 1.13 0.22 0.56 2.27 1.05 1.36 1.81 1.45 1.04 1.35 0.42 0.56 0.91 1.45 1.98 1.52 1.20 1.49 0.31
8.6 10 14 7.3 13 11 17 12 3.8 12 14 12 12 14 12 10 8.2 10 11 12 10 7.0 10 8.3 9.9 12 12 9.2 7.2 8.7 7.7 9.2 8.1 7.4 7.8 11 8.7 11 16 10 13 15 14 13 5.8 12 12 9.3 7.8 6.8 23 24 4.7 18 28 18 12 18 2.8 8.4 32 12 17 23 16 15 21 5.7 8.5 11 18 25 19 16 23 4.0
0.00 0.04 0.39 0.03 0.39 0.26 0.15 0.39 0.04 0.04 0.19 0.26 0.42 0.04 0.42 0.22 0.27 0.09 0.09 0.26 0.22 0.16 0.47 0.00 0.00 0.27 0.08 0.20 0.23 0.14 0.36 0.24 0.12 0.03 0.11 0.10 0.00 0.14 0.20 0.17 0.19 0.25 0.38 0.33 0.06 0.56 0.19 0.20 0.10 0.17 0.92 1.15 0.10 0.89 1.66 1.42 0.04 0.06 0.04 0.00 1.53 0.18 1.09 0.66 0.27 0.40 0.18 0.04 0.05 0.53 1.01 0.33 1.07 0.19 0.18 0.03
0.00s 0.10ds 0.33cd 0.09a 0.33ad 0.26ad 0.21d 0.32cd 0.11a 0.10ad 0.21cd 0.26cd 0.31cd 0.11d 0.31cd 0.22acd 0.25ac 0.14cd 0.14d 0.25cd 0.22acd 0.19ac 0.33acd 0.00ds 0.00ds 0.28cd 0.16d 0.21cd 0.23acd 0.18cd 0.29cd 0.23cd 0.16acd 0.08ac 0.16acd 0.16acd 0.00d 0.19cd 0.25c 0.20cd 0.21cd 0.23cd 0.29cd 0.27cd 0.12ac 0.35cd 0.22cd 0.25ds 0.16cd 0.22cs 0.48cd 0.55cd 0.15c 0.46cd 0.66cd 0.59cd 0.10acd 0.14ad 0.10as 0.00ads 0.62cd 0.20cd 0.50cd 0.40cd 0.25cd 0.33cd 0.24acd 0.10cs 0.12as 0.35cd 0.48cd 0.28cd 0.50cd 0.22ads 0.24ads 0.09as
684 Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
Table 2 (Continued)
g
system1/system2
no. of compds
I(1),a bits
I(2),b bits
I(1,2),c bits
syn,d bits
%syn,e
I(1;2),f bits
simg
HPLC-C8/GC-DB5 HPLC-cyano/HPLC-amino HPLC-cyano/HPLC-PBD HPLC-cyano/HPLC-phenyl HPLC-cyano/SFC-C1 HPLC-cyano/SFC-Carbowax HPLC-cyano/SFC-phenyl HPLC-cyano/GC-DB5 HPLC-amino/HPLC-PBD HPLC-amino/HPLC-phenyl HPLC-amino/SFC-C1 HPLC-amino/SFC-Carbowax HPLC-amino/SFC-phenyl HPLC-amino/GC-DB5 HPLC-PBD/HPLC-phenyl HPLC-PBD/SFC-C1 HPLC-PBD/SFC-Carbowax HPLC-PBD/SFC-phenyl HPLC-PBD/GC-DB5 HPLC-phenyl/SFC-C1 HPLC-phenyl/SFC-Carbowax HPLC-phenyl/SFC-phenyl HPLC-phenyl/GC-DB5 SFC-C1/SFC-Carbowax SFC-C1/SFC-phenyl SFC-C1/GC-DB5 SFC-Carbowax/SFC-phenyl SFC-Carbowax/GC-DB5 SFC-phenyl/GC-DB5
12 34 35 35 16 10 15 11 34 34 16 10 15 11 36 16 11 15 11 16 10 15 11 18 24 13 18 9 13
3.42 4.45 4.51 4.51 3.50 3.12 3.51 3.28 4.79 4.79 4.00 3.32 3.91 3.46 4.22 3.20 2.85 3.19 3.10 3.75 3.32 3.64 3.28 3.95 4.33 3.55 3.95 3.17 3.70
3.42 4.79 4.15 4.59 3.75 3.32 3.91 3.46 4.14 4.54 3.75 3.32 3.91 3.46 4.59 3.75 3.46 3.91 3.46 3.75 3.32 3.91 3.46 3.95 4.33 3.39 4.06 2.73 3.39
6.79 9.09 8.64 8.20 7.22 6.44 7.38 6.68 8.63 7.98 7.49 6.58 7.61 6.75 8.35 6.92 6.25 7.10 6.38 7.41 6.64 7.55 6.68 7.82 8.57 6.81 7.98 5.90 6.66
0.62 0.83 0.32 0.90 0.73 0.89 0.77 0.40 1.49 1.46 1.12 0.80 0.45 0.63 1.24 0.57 1.12 0.24 0.65 1.08 0.60 0.24 0.67 0.90 0.72 0.96 0.83 0.73 0.76
9.1 9.1 3.7 11 10 14 10 6.0 17 18 15 12 5.9 9.3 15. 8.2 18. 3.4 10 15. 9.0 3.2 10. 12 8.4 14. 10. 12 11.
0.05 0.15 0.02 0.90 0.03 0.00 0.04 0.06 0.30 1.35 0.26 0.06 0.21 0.17 0.46 0.03 0.06 0.00 0.18 0.09 0.00 0.00 0.06 0.08 0.09 0.13 0.03 0.00 0.43
0.12ads 0.18cd 0.07s 0.46cd 0.09ds 0.00ds 0.10ds 0.13s 0.26cd 0.56cd 0.26acd 0.13cd 0.23ac 0.22cd 0.33acd 0.09as 0.14ads 0.00as 0.24ads 0.16cd 0.00ds 0.00s 0.13ad 0.14cds 0.14cd 0.19cd 0.09ds 0.00ds 0.35cd
a First dimension entropy. b Second dimension entropy. c Two-dimensional entropy. d Synentropy. e Percent synentropy. f Mutual information. Informational similarity. Data pattern descriptors: a, data parallel to axis; c, clustered; d, diagonally aligned; s, data scattered.
Figure 2. 32.3%.
HPLC-C4/HPLC-C8: similarity, 0.62; %synentropy,
Figure 1. SFC-biphenyl/HPLC-PBD: similarity, 0.10; %synentropy, 3.8%.
dimension of the 2-D chromatogram, Rtf, are corrected for the retention time of an unretained component, Rto. Corrected solute retention times are transformed to a fractional basis as defined by eq 1. This transformation allows the retention time of each
Xa )
(Rti - Rto) (Rtf - Rto)
(1)
solute in each dimension to be compared on the same scale. The
Xa factors for the solutes previously described and separated were calculated and used to plot 105 2-D chromatograms. Refer to Figures 1-5, where five 2-D chromatographic plots are illustrated. Information Theory. In information theory, event uncertainty is called “information” or “informational entropy,” I, a quantity that is measured in units of bits. In analytical chemistry applications, the informational entropy of a measurement of a sample is redefined as the decrease in the uncertainty of the nature or concentrations of the solutes in a random sample. Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
685
Figure 5. MECC-SDS/MECC-TTAB: similarity, 0.31; %synentropy, 12.0%. Figure 3. HPLC-C1/HPLC-cyano: similarity, 0.29; %synentropy, 13.7%.
I(k), and each 2-D space, I(k,l), where k and l denote the dimensions. In our case, this corresponds to computing the informational entropy from the scaled retention time data of each 1-D chromatographic technique/phase and each 2-D chromatographic technique/phase pair. This is accomplished by summing the informational entropy of each scaled retention time. For example, if there was a 1-D chromatographic separation of nine solutes comprised of three Xa factors ) 0.4, two Xa factors ) 0.6, and four Xa factors ) 0.7, then the total informational entropy for technique/phase 1 would be computed by eq 3:
I(1) )
3 3 2 2 4 4 log2 + log2 + log2 9 9 9 9 9 9
(3)
Theoretically, if no correlation exists between n independent variables, informational entropy is described by eq 4:23 n
I(1, 2, 3, ..., n) )
∑I(j)
(4)
j)1
Figure 4. MECC-SDS/SFC-C1: similarity, 0.00; %synentropy, 8.4%.
The informational entropy of a measurement, I, a probabilistic quantity defined by Shannon32 in communication research, is described by eq 2, where Fk is the probability of the incidence of
I)
∑(-F
k
n
I(1, 2, 3, ..., n)
