Anal. Chem. 1997, 69, 145-151
Vector Model of Multiphase Separations Heather Smith and Richard Sacks*
Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109
A vector model is developed for separations using combinations of two or more stationary phase chemistries. The model is useful for the development of separation optimization strategies for methods involving series-coupled columns, parallel columns, or mixed stationary phases. In the model, each mixture component occupies a single point in a multidimensional vector space, with each orthogonal axis representing the retention on a particular stationary phase. For every pair of points, a separation vector is defined. The length and direction of the separation vector give all the necessary information on the separation of this binary mixture. The ensemble of separation vectors for the entire mixture is used to describe the multiphase separation. For the case of a separation using series-coupled columns or mixed stationary phases, the multidimensional space collapses by orthogonal projections onto a linear separation axis. The orientation of this axis defines the fractional contributions of the various phases to the overall separation selectivity. An example of the use of the vector model in establishing the optimal separation axis and thus phase fraction in a two-phase separation is presented. The combined use of two or more stationary phase chemistries is widespread in chemical separation science. Methods using two or more stationary phases have been developed for increasing peak capacity,1 enhancing selectivity,2-4 and increasing qualitative information content5,6 in gas chromatography. The analysis of a given mixture on two parallel columns may nearly double the peak capacity of the separation and greatly increases qualitative information content. Most GC instruments have two or more detectors and can accommodate two or more parallel columns. Series-coupled columns with intermediate trapping7,8 are frequently used for two-phase separations involving heart cutting. Often, critical portions of the eluent from a preliminary chromatogram are transferred to a second column after refocusing for subsequent analysis. This can greatly facilitate the analysis of critical components in complex mixtures. Liu and Phillips9,10 used series-coupled columns with a desorption thermal modulator to achieve comprehensive two-dimensional analysis of very complex (1) Liu, Z.; Sirimanne, S.; Patterson, D.; Needham, L.; Phillips, J. Anal. Chem. 1994, 66, 3086. (2) Sandra, P.; Van Roelenbosch, M. Chromatographia 1981, 14, 345. (3) Laub, R.; Purnell, J. J. Chromatogr. 1975, 112, 71. (4) Akard, M.; Sacks, R. Anal. Chem. 1994, 66, 3036. (5) Krupcik, J.; Guiochon, G.; Schmitter, J. M. J. Chromatogr. 1981, 213, 189. (6) Maurer, T.; Engewald, W.; Steinborn, A. J. Chromatogr. 1990, 517, 77. (7) Deans, D. Chromatographia 1968, 1, 18. (8) Schomburg, G.; Hysmann, H.; Weeke, F. J. Chromatogr. 1975, 112, 205. (9) Liu, Z.; Phillips, J. J. Microcolumn Sep. 1989, 1, 249. (10) Liu, Z.; Phillips, J. J. Chromatogr. Sci. 1991, 29, 227. S0003-2700(96)00652-X CCC: $14.00
© 1997 American Chemical Society
mixtures. Extraordinarily large peak capacities are obtained in these two-dimensional separations. Mixed stationary phases2,11 and series-coupled columns12-19 have been used to tune the selectivity of a separation for a specified set of target compounds. Usually, two stationary phases are employed. Window diagrams, in which relative retention for the most difficult to separate pair of components (critical pair) is plotted versus the fraction of either of the phases, often are used for optimization. For the case of mixed stationary phases, the phase fraction and thus the selectivity are adjusted by changing the volume or weight ratio of the two phases.3,20 For the case of tandem columns, the phase fraction is adjusted by changing the lengths of the columns14,21-25 or the pressure at the midpoint between the columns.26-28 Akard and Sacks29,30 used series-coupled columns with adjustable intermediate pressure for tuning the selectivity of high-speed GC separations. High speed is achieved by the use of a cryofocusing inlet system with relatively short capillary columns operated at unusually high carrier gas flow rates. They introduced a new resolution function which results in more accurate window diagrams than are obtained with relative retention as the dependent variable.29 Akard also extended the use of pressure tunable selectivity by developing an optimization procedure for the case of three series-coupled columns.30 Despite significant progress in the use of multiphase separations, there is no unified approach to the optimization of these potentially powerful techniques. For more complex mixtures, selectivity optimization by window diagrams is accomplished with iterative algorithms, which can be very computationally intensive. In addition, these approaches provide little insight for the selection (11) Hildebrand, G. P.; Reilley, C. M. Anal. Chem. 1964, 36, 47. (12) Purnell, J.; Wattan, M. Anal. Chem. 1991, 63, 1261. (13) Jones, J.; Purnell, J. Anal. Chem. 1990, 62, 2300. (14) Sandra, P.; David, F.; Proot, M.; Diricks, G.; Verstappe, M.; Verzele, M. J. High Resolut. Chromatogr. 1985, 8, 782. (15) Kaiser, R. E.; Reider, R. I. J. High Resolut. Chromatogr. 1979, 2, 416. (16) Matisova, E.; Kovacicova, E.; Garaj, J.; Kraus, G. Chromatographia 1989, 27, 494. (17) Benicka, E.; Krupcik, J.; Kuljovsky, P.; Repka, D.; Garaj, J. Mikrochim. Acta 1990, 3, 1. (18) Purnell, J.; Williams, P. J. High Resolut. Chromatogr. 1983, 6, 569. (19) Purnell, J.; Williams, P. J. Chromatogr. 1984, 292, 197. (20) Laub, R.; Purnell, J. Anal. Chem. 1976, 48, 799. (21) Vililobos, R.; Pearson, R. IST Trans. 1986, 25, 55. (22) Mehran, M. F.; Cooper, W. J.; Jennings, W. J. High Resolut. Chromatogr. 1984, 7, 215. (23) Krupcik, J.; Guiochon, G.; Schmitter, J. J. Chromatogr. 1981, 213, 189. (24) Krupcik, J.; Mocak, J.; Simova, A.; Garaj, J.; Guiochon, G. J. Chromatogr. 1982, 238, 1. (25) Ingraham, D.; Schoemaker, C.; Jennings, W. J. Chromatogr. 1982, 239, 39. (26) Deans, D.; Scott, I. Anal. Chem. 1973, 45, 1137. (27) Maurer, T.; Engewald, W.; Steinborn, A. J. Chromatogr. 1990, 517, 77. (28) Kaiser, R.; Lenning, L.; Blomberg, L.; Rieder, R. J. High Resolut. Chromatogr. 1985, 8, 92. (29) Akard, M.; Sacks, R. Anal. Chem. 1995, 67, 2733. (30) Akard, M. Ph.D. Dissertation, University of Michigan, 1994.
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Figure 1. Pressure tunable tandem column ensemble. Pi, inlet pressure; Po, outlet pressure; Pt, tuning pressure.
of stationary-phase chemistry combinations for a specific analysis problem. In this report, a vector model is presented for multiphase separations. This model provides significant insight into the separation of complex mixtures with multiphase techniques. The model successfully predicts the retention patterns and relative resolutions of all component pairs. In addition, window diagrams which point to the optimal phase fraction for the complete separation of a mixture are obtained from the model with fewer iterative calculations. EXPERIMENTAL SECTION Experimental data were obtained with a Varian 3700 GC, which was modified for high-speed operation. The Varian flame ionization detector (FID) and the split inlet system were used without change. A cryofocusing inlet system (Chromatofast, Inc., Ann Arbor, MI) was mounted on the side of the Varian instrument and connected between the split inlet and the column ensemble. This inlet system, which has been described in detail,31 delivers to the column a highly concentrated vapor plug about 10 ms in width. The FID current was amplified with an electrometer having a time constant of about 20 ms. A 12-bit A/D board operated with a sampling rate of 100 Hz was used for data acquisition. A Gateway 200 4DX2-66V computer was used for data collection and instrument control. The software for A/D board control was LabTech Notebook for Windows. Figure 1 shows the pressure-controlled tandem column ensemble used for selectivity tuning. Carrier gas is supplied at points CG. The inlet pressure Pi is the highest pressure in the system and is held constant during these experiments. The outlet pressure Po also is constant at 1 atm. If the tuning pressure Pt at the junction point between the columns is increased, the pressure drop along the first column C1 is decreased, and the pressure drop along C2 is increased. This results in a decrease in carrier gas velocity in C1 and an increase in that in C2. The resulting increase in sample component residence times in C1 and decrease in residence times in C2 increase the influence of C1 on the overall separation selectivity. The tuning pressure Pt was adjusted manually by means of a two-stage pressure regulator. Connections between the columns and the pressure control point were made with an all-glass splitter. A 10-cm length of 0.1-mm-i.d. fused silica tubing was used as a pneumatic restriction between the column junction and the tuning pressure control point. The columns were both 0.25-mm i.d. and used 0.25-µm-thick bonded stationary phases. Column C1 was a 5.0-m length of (31) Klemp, M.; Akard, M.; Sacks, R. Anal. Chem. 1993, 65, 2516.
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Figure 2. Vector representation of a two-phase separation. kp, capacity factor on polar column; knp, capacity factor on nonpolar column; k, retention pattern vector; R, retention vector; S, separation vector; F, phase fraction axis.
nonpolar 5% phenyl-95% dimethyl polysiloxane phase (DB-5). Column C2 was a 4.7-m length of very polar poly(ethylene glycol) (wax) phase. Hydrogen carrier gas was used after purification with filters for water vapor, oxygen, and hydrocarbons. Samples were obtained from Aldrich and were of 99% or greater purity. Capacity factors on the individual columns were obtained by injecting 0.01 µL of each compound as a neat liquid using a split ratio of 20:1. Mixtures were prepared with about equal volumes of the pure components, and 0.1 µL of the mixture was injected to obtain chromatograms with the tandem column ensemble. All calculations were made with the use of Microsoft Excel spread sheet run on a 486 PC. Vector calculations were performed using Excel. Chromatograms were plotted using Grams 386 software. VECTOR MODEL The vector model is illustrated by a two-phase system using a polar column p and a nonpolar column np. For every component, the retention on the two columns occupies a single point in a plot of retention on the polar column versus retention on the nonpolar column. Capacity factor k, as defined in eqs 1 and 2, is used to express retention since it is easy to measure and is directly related to thermodynamic properties. Here, tR is the retention time, and
k ) (tR - tm)/tm
(1)
tm ) L/u
(2)
tm is the holdup time. The holdup time is just the ratio of the column length L to the average carrier gas velocity u. Note that the capacity factor is defined as the product of phase volume ratio of the separation column and partition ratio for the compound. Figure 2 shows a plot of kp versus knp for a four-component mixture. The location of the points for the four compounds (labeled 1-4) is based on their retention on the two phases and is determined by their polarity with respect to the chemistries of the two phases. Retention pattern vectors (k) can be defined from the origin of the coordinate system to the coordinates of each
point. Two such vectors are labeled as k1 and k2. The orthogonal components of these vectors are the capacity factors for the corresponding mixture components on the individual phases. If the slope of a retention pattern vector relative to the knp axis (angles β1 and β2 in Figure 2) is less than 1.0, the compound shows greater retention on the nonpolar phase and is considered nonpolar with respect to the pair of phases considered. For every pair of points x and y in this two-dimensional retention space, a separation vector Sx,y can be defined. Two examples, S1,2 and S3,4, are shown in Figure 2. The length of a separation vector describes the degree of retention differences between the two components on the two phases. In general, a longer separation vector implies easier separation of the component pair. The angle of the separation vector gives information regarding elution order and the sensitivity of elution order to changes in the relative contributions of the phases. A positive slope occurs when one of the components shows greater retention on both phases. This precludes elution order reversals with changes in the relative contributions of the two phase chemistries. Both examples in Figure 2 have negative slopes, and coelutions can occur. The location of the separation vector is a measure of overall retention and selectivity, with increasing distance from the origin implying greater overall retention. Also shown in Figure 2 is a phase fraction axis F. The angle θ of the phase fraction axis with the polar retention axis is defined by eq 3. Here, fnp and fp are the fractional contributions of the
θ ) tan-1(fnp/fp)
(3)
two phases to the overall selectivity. For mixed stationary phases, these values can be the weight or volume fractions of the two phases. For tandem column systems, they can represent the holdup time fractions for the two column segments. Orthogonal projection of the retention pattern vectors onto the phase fraction axis gives a set of retention vectors R along the phase fraction axis. These vectors have the general form given in eq 4, where Fi is the unit vector along the F axis. The scalar
Rx ) [projFikx]Fi
(4)
length of Rx is found as the dot product of kx and Fi, which in two dimensions is given by ||kx|| cos[π/2 - (θ + βx)]. Here, ||kx|| is the length of vector kx, and π/2 - (θ + βx) is the included angle. Using the identity cos(A - B) ) cos(A) cos(B) + sin(A) sin(B) and the facts that cos(π/2) ) 0 and sin(π/2) ) 1, eq 4 can be recast as eq 5. These retention vectors provide information on
Rx ) ||kx|| sin(θ + βx)Fi
(5)
elution order and on the relative location of chromatographic peaks using the specified phase fractions fnp and fp. The orthogonal projection of a separation vector onto the phase fraction axis gives the separation of the two components using the indicated phase fractions. Note that, if the separation vector is orthogonal to the phase fraction axis, as is the case for S3,4 in Figure 2, its projection has zero length, and the two components coelute. If the separation vector has a positive slope with respect to the knp axis, then it can never be orthogonal to the separation axis, and coelution is impossible. If the separation vector has a
Figure 3. Complete set of separation vectors for a five-component mixture. Vectors shown as broken lines are for component pairs which are unconditionally separated.
negative slope, as for both cases in Figure 2, then for some characteristic phase fraction coelution will occur. If the separation vector has a negative slope and makes a minor angle R with respect to the knp axis, as is shown for S3,4 in Figure 2, then the phase fraction resulting in coelution is given by eq 6.
fnp/fp ) tan(R)
(6)
For a complex mixture of n components, the number of unique component pair separation vectors N is given by eq 7. Figure 3
N ) (n2 - n)/2
(7)
shows the 10 separation vectors for a five-component mixture. The six vectors shown as broken lines have positive slopes and are sufficiently long for a relatively easy separation. Thus, the corresponding compound pairs are unconditionally separated and need not be considered further. One short vector with a positive slope is shown as a solid line since, under certain conditions, it may represent the critical pair, even though coelution cannot occur. The three vectors with negative slopes will result in coelutions for different phase fractions. RESULTS AND DISCUSSION Scalar Quantities. All important scalar quantities including overall capacity factors, component pair separations, and component pair relative resolutions can be derived directly from the vector model. For mixed stationary phases or tandem column ensembles, an overall capacity factor kox can be defined for each component x as described in eq 8.32,33 For a specified value of θ,
kox ) fnpknpx + fpkpx
(8)
kox for each component is proportional to the length of the (32) Mayer, H.; Karpathy, O. J. Chromatogr. 1962, 8, 308. (33) Purnell, J.; Jones, J.; Wattan, M. J. Chromatogr. 1987, 399, 99.
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corresponding retention vector as described in eqs 9 and 10. In
kox ) [1/C(θ)]||Rx||
(9)
kox ) [1/C(θ)]||kx|| sin(θ + βx)
(10)
these equations, double vertical bars indicate the length of the corresponding vector. For a specified value of θ, C(θ) is constant for all components and is given by eq 11. The value of C(θ) ranges
C(θ) ) sin(θ) + cos(θ)
(11)
from 1.0 for θ values of 0 and π/2 to a value of 1.4 (21/2) for a θ value of π/4. The separation ∆kox,y of two components can be found as the difference in the overall capacity factor values for the components or as the scalar length of the projection of the corresponding separation vector onto the phase fraction axis. This is given by eq 12. The scalar length of this projection is found as the dot
∆kox,y ) [1/C(θ)] projFiSx,y
(12)
product of Sx,y and Fi. If R is defined as the minor angle of the separation vector with respect to the nonpolar (horizontal) axis, then the range of R is from 0 to π/2. Further, expressing the included angle in terms of π/2, θ, and R, and using the identities cos(A - B) ) cos(A) cos(B) + sin(A) sin(B) and sin(-A) ) -sin(A), eq 12 can be recast as eq 13. In eq 13, the plus sign is used
∆kox,y ) [1/C(θ)]||Sx,y|| |sin(θ ( R)|
(13)
if Sx,y has a positive slope (coelution not possible), and the minus sign is used if the slope is negative (coelution inevitable for some phase fraction). Also note that the absolute value of sin(θ ( R) is used in eq 13. This is necessary to account for cases where the separation vector is either above or below the phase fraction axis. In previous work,29 the relative resolution Rrel, as defined by eq 14, was shown to be a more reliable measure of separation quality than relative retention (kox/koy) for all capacity factor values.
Rrel ) ∆kox,y/(koa + 1)
(14)
Here, koa is the average overall capacity factor for the two components of interest. By the use of eqs 4 and 12, eq 14 can be put into a vector format. This is given in eq 15, where ka is the
Rrel )
projFiSx,y projFika + C(θ)
(15)
average retention pattern vector having length ||ka|| and angle βa equal to the average corresponding values for kx and ky. An alternative and very useful presentation of eq 15 is obtained by substitution from eqs 5 and 13:
Rrel ) 148
||Sx,y|| |sin(θ ( R)| ||ka|| sin(θ + βa) + C(θ)
Analytical Chemistry, Vol. 69, No. 2, January 15, 1997
(16)
Figure 4. Relative resolution as a function of phase fraction axis angle θ and separation vector angle R. Darker regions correspond to smaller relative resolution. The separation vector length is 1.0, and the coordinates of its center are (2.0,2.0).
Figure 4 shows the magnitude of Rrel, indicated by the shading in the figure, as function of angles θ and R. For the horizontal axis, the phase fraction axis is rotated through its range from 0 to π/2. Note that this is equivalent to changing the phase fraction from 0 (only the polar phase used for the separation) to 1.0 (only the nonpolar phase used). For the vertical axis, the separation vector of fixed length was rotated about its geometric center, thus generating a continuously changing angle R. The rotation axis was located at coordinate (2,2), and the vector length was 1.0. Note that changing R implies changing the specific component pair of interest. The value of R is fixed for any specified component pair using a specified pair of stationary phases. The lower portion of the figure corresponds to a separation vector with a negative slope, and the top portion is for the case of a positive slope. Dark regions of the figure correspond to low relative resolution, and the dark trough extending from the lower left corner of the figure to the center of the right edge is the locus of θ and R combinations which results in coelution. For the lower left corner of the figure, the phase fraction axis is vertical (θ ) 0), corresponding to using only the polar phase for the separation. Since the separation vector is horizontal (R ) 0), the corresponding components have the same capacity factor values on the polar column, and thus coelution occurs. As the separation vector is rotated in a clockwise direction about its center, the angle R required for coelution increases, and the dark trough in the figure tracks toward the right center region of the figure. When R has increased to π/2 (vertical separation vector), the two components have the same capacity factors on the nonpolar column, and coelution will occur for a phase fraction axis angle of π/2. For the upper portion of the figure, where the + sign is used in the (θ ( R) term, the minor angle of the separation vector is in the first quadrant, and the slope is positive. Coelution of this
Table 1. Capacity Factor Data for Components in the Test Mixture label A B C D E F G H I J K L M N
compound benzene cis-1,2-dichloroethylene trichloroethylene nonane toluene decane ethylbenzene p-xylene m-xylene o-xylene 3-ethyltoluene 2-ethyltoluene 1,3,5-trimethylbenzene 1,2,3-trimethylbenzene
bp (°C) k(DB5), 60 °C k(wax), 60 °C 80.1 60.3 87 150.8 110.6 174.1 136.2 138.3 139.1 144.4 165.2 161.3 164.7 176.1
0.41 0.24 0.56 2.87 0.98 6.47 2.11 2.26 2.26 2.72 4.95 4.68 5.40 7.48
0.69 0.97 0.98 0.45 1.37 0.94 2.50 2.64 2.75 3.70 5.48 4.85 6.15 9.92
component pair cannot occur, and the relative resolution is always greater than zero. In all cases, the relative resolution is largest on one of the individual phases and decreases monotonically with changing phase fraction. Note that, in the upper half of the figure, for an R value of π/4, the relative resolution is constant for all phase fractions. This is because the center of the separation vector is located such that kp ) knp. For other cases, the upper half of the figure shows less symmetry, and one of the phases may show significantly larger relative resolution values. Optimization Procedures. The vector model was used to perform a critical pair window diagram analysis for the separation of a 14-component mixture containing 10 aromatic compounds. Capacity factor values were obtained separately on the DB-5 and the poly(ethylene glycol) columns. The compounds, their boiling points, and their capacity factor values at 60 °C are given in Table 1. Figure 5 shows the chromatograms using the DB-5 column (a) and the poly(ethylene glycol) column (b). Note the time scale of the separation. Using only the nonpolar column, components H (p-xylene) and I (m-xylene) coelute. Also note that component D (nonane) elutes after component J, and F (decane) elutes late in the chromatogram after M. Using only the polar column, some separation of H and I is achieved, but B, C, and F all coelute. Also, components G (ethylbenzene) and H are less well separated. Note the very large retention shifts for the nonpolar compounds D and F relative to chromatogram a. For a mixture having n components and (n2 - n)/2 unique separation vectors, a plot of relative resolution as a function of θ can be obtained for every separation vector. These plots will be similar to horizontal slices through Figure 4 at the appropriate values of R. The scaling of these plots will vary with the length and location of each separation vector as described in eq 16. For this mixture, there are 91 unique separation vectors. However, for many of them, coelutions do not occur, and separation is relatively easy. Figure 6 shows the plot of k(DB-5) versus k(poly(ethylene glycol)) for the 14-component mixture. The letters identifying the points are the same as in Table 1. Also shown are the 21 separation vectors which must be considered because the corresponding component pairs can coelute or because the separation vector is very short, and thus the component pair may be the critical pair for some phase fraction values. Component pair B-C is an example of this latter case. From the angles that these separation vectors make with the nonpolar axis, all coelution pairs and the corresponding values of θ and phase fraction fnp/fp were determined. Table 2 summarizes
Figure 5. Chromatograms of a 14-component mixture using a 5.0m-long DB-5 (nonpolar) column (a) and a 4.7-m-long poly(ethylene glycol) (polar) column (b). See Table 1 for component identification and capacity factor values.
Figure 6. Two-dimensional retention plots for the 14-component mixture described in Figure 5 and Table 1. The 21 separation vectors correspond to all component pairs which will be the critical pairs for certain phase fraction axis angles. The coelution phase fraction angles and the corresponding phase fractions are given in Table 2.
this information. Note that, while there are 91 total separation vectors, only 20 coelutions are observed, and only 21 separation vectors need be considered. Figure 7a shows plots of relative resolution as functions of θ for the 21 critical pairs. The largest relative resolution considered in the figure (0.03) is just adequate for an actual resolution of about 1.0 for 0.25-mm-i.d. capillary columns with stationary phase film thickness of 0.25 µm or less and total column length of about 10 m. The zero points in the figure correspond to coelutions, Analytical Chemistry, Vol. 69, No. 2, January 15, 1997
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Table 2. Separation Vector Data for all Critical Pairs in the Test Mixture critical pair
θ (rad)
θ (deg)
fraction
||S||
H-I F-B F-C B-C F-E D-A D-B D-C F-G F-H F-I D-E F-J A-B F-L D-G F-K D-H D-I F-M D-J
0 0.00474 0.00630 0.0245 0.0778 0.0976 0.194 0.223 0.344 0.384 0.405 0.452 0.635 1.03 1.14 1.22 1.25 1.30 1.31 1.37 1.52
0 0.272 0.361 1.40 4.55 5.59 11.1 12.8 19.7 22.0 23.2 25.9 36.4 59.2 65.5 69.7 71.5 74.3 75.0 78.4 87.3
1 0.995 0.994 0.976 0.928 0.911 0.835 0.815 0.736 0.712 0.700 0.673 0.576 0.373 0.313 0.270 0.251 0.219 0.211 0.171 0.0439
0.106 6.22 5.91 0.315 5.50 2.48 2.68 2.37 4.62 4.54 4.58 2.10 4.65 0.322 4.29 2.18 4.78 2.27 2.37 5.32 3.25
Figure 8. (a) Plot of kp vs knp for the mixture described in Table 1, showing the orthogonal projections onto a phase fraction axis which results in coelution of nonane (D) and toluene (E). (b) Chromatogram obtained with the corresponding phase fraction.
Figure 7. (a) Plots of relative resolution vs phase fraction axis angle for all component pairs which could be critical pairs in the two-phase separation of the mixture described in Table 1. (b) Critical pair window diagram for the mixture described in Table 1.
and the θ values are from Table 2. For most component pairs, a nearly symmetric, V-shaped plot is obtained centered at the phase fraction corresponding to coelution. The flanks of each plot are steeper for the longer separation vectors. The plot for p-xylene and m-xylene is within the range of Rrel values covered in the figure for the entire range of phase fractions with coelution occurring for a θ value of π/2 (only the nonpolar phase used). The separation of these two components is best using only the poly(ethylene glycol) column, but other coelutions preclude the use of this column only. 150 Analytical Chemistry, Vol. 69, No. 2, January 15, 1997
By considering only the minimum values of Rrel as a function of θ, the window diagram of Figure 5b is obtained. This is similar to a conventional window diagram, except that the independent variable is θ rather than phase fraction, and the dependent variable is relative resolution rather than relative retention. Note that this window diagram was obtained with a large reduction in the number of calculations relative to other procedures and is reliable even for very fast separations involving low-capacity-factor components. The best separation of the critical pair occurs at a θ value of about 8.6°, corresponding to a polar phase fraction of 0.87. Figure 8a shows a plot of knp vs kp for the test mixture. The identifier letters labeling the points in the figure correspond to the letters used in Table 1. The figure also shows a phase fraction axis with θ ) 27°. This value is indicated by a vertical line in Figure 7b. For each of the 14 components, an orthogonal projection onto the phase fraction axis is shown. The corresponding points on the phase fraction axis indicate the relative peak positions in a chromatogram using a phase fraction ratio defined by eq 3 as 0.34 nonpolar and 0.66 polar. The critical pair is n-nonane and toluene (D and E), with a relative resolution of 0.0135 indicated on the window diagram. At this phase fraction, m-xylene and p-xylene have a relative resolution of 0.0197. Figure 8b shows a chromatogram of the mixture using this phase fraction ratio. The phase fraction ratio was measured by the use of eq 8 and the constraint that fnp + fp ) 1. Values were computed for all 14 components in the mixture, and the average value was adjusted by changing the pressure at the junction between the columns. Note that the chromatogram is complete
pair with a relative resolution of 0.020. This is confirmed by the chromatogram shown in Figure 9b. Again, the agreement with the peak positions predicted from Figure 9a is excellent. For example, in Figure 8a, component F (decane) is predicted to elute after components G, H, and I, while in Figure 9a, component F is predicted to elute before G, H, and I. This is confirmed in the corresponding chromatograms.
Figure 9. (a) Plot of kp vs knp for the mixture described in Table 1, showing the orthogonal projections onto a phase fraction axis which results in separation of nonane (D) and toluene (E) and improved separation of p-xylene (H) and m-xylene (I). (b) Chromatogram obtained with the corresponding phase fraction.
in about 70 s, which is about twice as long as in Figure 5. This is because the column ensemble is nearly 10 m in length, while the individual columns used for Figure 5 were about half this length. The peak ordering and the relative positions of peaks in the chromatogram of Figure 8b are in excellent agreement with the projections along the phase fraction axis. The coelution of components D and E, which is indicated by the projections onto the phase fraction axis, is observed in the chromatogram. Figure 9a is similar to Figure 8a except that θ has been changed to 17°. This is also indicated by a vertical line in Figure 7b. Note that this phase fraction angle should result in complete separation of components D and E and should also result in a better separation of m-xylene and p-xylene, which is the critical
CONCLUSIONS The vector model for multiphase separations described in this report obtains the same results as previous scalar models, but the additional organization of numerical capacity factor data in the vector model provides more insight into the multiphase separation of complex mixtures and allows for a significant reduction in calculations. The model, developed here in detail for two GC stationary phase chemistries, should be useful for any separation where total component migration can be represented as a linear combination of the migrations obtained using two different phase chemistries. The window diagram optimization strategy described here using the vector model results in the identification of the best phase fraction ratio for the complete separation of the mixture. The vector model also should be useful in targeting the separation strategy for a specified subset of the components present in the mixture. In these cases, a significant reduction in the number of relevant separation vectors can be achieved. This results in simplified calculations and, in some cases, considerably faster analyses. While the window diagram approach to the optimization of multiphase separations is very useful in some applications, other optimization techniques may be suggested by the use of vector representation. With the vector model, iterative calculations for window diagram optimization have been reduced, thus significantly reducing the required computing time. The use of vector notation should facilitate the development of optimization strategies and associated algorithms for systems involving more than two stationary phase chemistries. The vector model should simplify both the mathematics and the visualization of more complex multiphase separations. Received for review July 2, 1996. Accepted October 30, 1996.X AC960652Z X
Abstract published in Advance ACS Abstracts, December 15, 1996.
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