904 Anal. Chem. 1988, 58, 904-906

Prediction of Retention for Coupled Column Gradient Elution High-Performance Liquid Chromatography

Sterling A. Tomellini* Department of Chemistry, University of New Hampshire, Durham, New Hampshire 03824

Shih-Hsien Hsu and Richard A. Hartwick Department of Chemistry, Rutgers University, Busch Campus, Piscataway, New Jersey 08854

An expansion of previously developed numerical integration techniques is described for the calculation of retention in multiple-column, gradient elution HPLC. An algorithm is de- scribed whereby up to six coupled columns, using up to a quaternary mobile phase, can be readily calculated. Data are presented for two-column, linear binary gradient experiments.

A powerful approach to improving selectivities among various compounds in HPLC is the coupling of columns containing different stationary phases (1-6). While coupled columns offer interesting advantages when used under isoc- ratic elution conditions, the inherent resolving power of a multiphasic system can best be realized when solvent pro- gramming is employed. In the isocratic case the retention times of the compounds are not affected by the order of the columns, as opposed to the coupled column gradient exper- iment, where the order of the columns can be significant. Thus, the scientist may be able to achieve a desired separation by varying the order of a set of columns and/or by changing the solvent gradient curve. Attempting to develop and op- timize a separation empirically with such a large number of variables becomes quite complex, with a diminished proba- bility of randomly finding a solution. In such situations, the ability to calculate predicted retention times given minimal experimental data becomes a more efficient approach.

Exact mathematical solutions of the integrals involved in predicting retention under single-column gradient elution conditions can quickly become intractable. It can be appre- ciated that exact solutions of the appropriate equations under coupled column, multiple solvent gradient elution can become even more complex and inflexible. The absence of the nec- essary mathematical tools has hindered the study of gradient elution, multiple-column approaches in optimization schemes.

Computer-based numerical integration algorithms offer a logical approach to solving such complex problems. This paper will describe the application of a numerical integration-based approach allowing for calculation of retention under gradient elution, serial column conditions.

EXPERIMENTAL SECTION Apparatus and Reagents. Six test solutes were used uracil

(unretained), guaifenesin, p-cresol, p-ethylphenol, nitrobenzene, and methyl benzoate. All solutes were reagent grade or better. HPLC grade methanol ( J. T. Baker, Phillipsburg, NJ) was used as an organic modifier. Mobile phases were prepared volumet- rically using double-distilled, deionized water and were sparged with helium prior to use.

Stationary phases were synthesized by refluxing the appropriate chlorosilyl derivative with Whatman Partisil-10 (Whatman, Inc., Clifton, NJ), a support material having a nominal 10-pm particle diameter (batch no. 100591).

The phenyl phase was slurry packed into a 150 X 1 mm i.d. glass-lined column using a previously described method (7). The

0003-2700/86/0358-0904$0 1.50/0

C18 phase was packed into a 100 X 1 mm i.d. glass-lined column in a similar manner.

The chromatographic system consisted of a Brownlee MPLC microbore gradient HPLC (Applied Biosystems, Foster City, CA) a Kratos 753 variable-wavelength detector (Spectros, Inc., Ramsey, NJ) fitted with 0.5-pL flow cells, and a Kipp and Zonin strip chart recorder. A Rheodyne 7413 microinjector (Rhecdyne, Cotati, CA) was used with a 0.5-pL injection volume.

A value for the instrumental gradient delay of 220 pL was used throughout this study. Although the delay time can be measured by means of an acetone gradient, in practice the delay volume is most accurately treated as simply a fitting coefficient for the overall equations. Once fit to a given system however, its value will not change with that instrument and column. Solvent pro- portioaing to obtain the isocratic retention data was made by the MPLC pump. The flow rate was held constant at 50 pL min-' for all experiments.

Peak standard deviations were measured by using the method of Foley and Dorsey (8). A measured extra-column variance of 4 mma was used, accounted for in the reported data. Computer programs were written in FORTRAN and run on a Digital Equipment Corp. (Maynard, MA) VAX 11/780 computer. Source code listings are available from the authors.

THEORY The application of numerical integration techniques for the

calculation of retention under gradient elution conditions on a single column has been previously described (9). The basic computational steps are-similar for the coupled column ex- periment, with the exception of the delay time before the gradient front overtakes the solute band. This isocratic migration distance depends not only on the mobile-phase composition during the gradient delay but also upon the ca- pacity factor vs. mobile phase relationship for the solute over the length of each column encountered.

The key to understanding the effect of multiple columns on the isocratic travel distance is to treat each column as a separate delay volume. If the gradient overtakes the band within the first column, then the previously derived rela- tionship (9) between instrumental delay time, tD, and the isocratic travel distance, Lis,, can be used,

where Lcol(l) is the length of column 1, tD is the external in- strumental delay time, t,(l) is the retention time of an un- retained solute for column 1, and k'(,, is the capacity factor for the band of interest under the initial solvent composition on column 1. If, however, the band travels completely through the first column isocratically, then the distance traveled in the second column before being overtaken by the gradient must be determined.

Considering such a situation just as the solute band reaches the head of the second column in a two-column experiment, then it can be seen that the first column has acted simply as an additional delay volume, and an equation similar to eq 1 can be derived. The delay time used for column 2 (tD@)),

0 1986 American Chemical Society

ANALYTICAL CHEMISTRY, VOL. 58, NO. 4, APRIL 1986 905

however, is equal to the external instrumend delay, tD, plus the retention time of an unretained solute for column 1, tm(l), minus the elution time of the solute band from column 1, t ~ ( 1 ) . If the elution time of the band in column 1 is derived in terms of the solute capacity factor in column 1, then the delay time to be used for column 2 becomes

tD(2) = (tD + (1 + ~ ’ ( l ) ) ( L c o l ( l ) / ~ m ( l ) ) + tm(l)) (2)

where is the mobile-phase linear velocity of a solute in column 1. The same logic can be extended to any number of columns and forms the basis of the developed programs.

Once the position at which the gradient overtakes the solute band is known, the final calculations are essentially identical with that used for single-column gradient elution. The integral for the gradient is evaluated by incrementing the time over some small step (0.01 min in this study), calculating the corresponding h’ for the resulting time, and then calculating the length traveled during this time interval. The sum of the time steps will equal t$ when the s u m of the lengths traveled is equal to L,. A continuous correction must be made, since each solute band is traveling a t a changing rate, relative to both the column and to the gradient velocity. As previously shown (9) one way to eliminate length from the problem is to calculate a “gradient” time as well as an actual time. While the actual time interval is specified, the “gradient” time in- terval, tg, can be calculated by using the relationship

t g = t a - ( L a / u n ) (3)

where La is the length traveled by the solute band in the actual time interval and u, is the linear velocity of the gradient front in column n, which is assumed to be Lcol/tm (i.e., adsorption of the solvent components is neglected). Thus, the sum of t , intervals is equal to the “gradient” time seen by the solute band, and it is this time that must be used to calculate the instanteous capacity factors for the stepwise integration. Integration is completed when the retention time for the solute equals the sum of the time spent traveling both isocratically and under gradient conditions.

Bandwidth Calculation under Serial Column Condi- tions. The width (in time or volume units) of a band eluting from a set of serial columns is determined by the instantaneous velocity (k’inst) of the band as it exits the final column and by the physical width of the zone at this point. The physical zone width is fixed by the total variance, in distance units, accumulated during transit through the system. Thus,

(4)

where s2t is the total band variance, s2,c is the extra column variance due to injection volume, connecting tube dispersion, etc., and s2n is the variance contribution of each of n columns. The total column variance (in distance units) is then given by

S2t = SZec + S21 + S22 + ... + S2n

(5)

where L i and N , are the length and efficiency, respectively, of each column in the system.

The width of the eluting peak in time units resulting from dispersion in the multiple columns can be calculated directly from the above;

where u, is the mobile phase linear velocity of the nth column and k tt is the instantaneous velocity of the zone at its first statistical moment (retention time) as it exits the final (nth)

l I ( ” ” ’ 0 I O 2 0 30

I I , I l ’ ” - - )

0 10 2 0 ‘3 0 4 0

I , , ! I I I I ,

5 I O 20 3 3 4 3

min

Figure 1. Series of chromatograms showing the separation of test solutes under several solvent gradients with different column orders: (A) C,,/C,Ph (10 cm/15 cm), uslng a gradient of 50-70% methanol over 30 min (linear; (B) same column order as (A), but with a 45-70% methanol gradient over 30 min; (C) reversal of column order (C,Ph/C,,) using a 30-70% methanol gradient (linear) over 40 min. Identity of solutes corresponds to the solute order in Tables 1-111.

column. The k$,st and position of each zone are readily available throughout the separation process due to the iterative nature of the computer program. The above treatment as- sumes that all portions of the solute zone are traveling at the same velocity. Under conditions of either rapid solvent composition change or very steep solute retention/solvent composition curves, eq 6 may overestimate the bandwidth, due to physical compression of the solute zone. While em- pirical corrections can be employed to correct for this (9, IO), a full theoretical derivation quantifying such effects has yet to be satisfactorily derived.

RESULTS AND DISCUSSION Retention Time Accuracy. A FORTRAN program based

upon the described algorithm was developed and tested against experimental results. A variety of gradient conditions were examined to determine the accuracy of these programs. The results of one of these experiments are presented in Tables 1-111.

Four isocratic data points were acquired for the five test solutes for both an octadecyl (C18) and a propylphenyl (C,Ph) column. These isocratic data are presented in Table I. The data were fitted to a second-order polynomial as previously described (8). Two gradient coupled column experiments were

900 ANALYTICAL CHEMISTRY, VOL. 58, NO. 4, APRIL 1986

Table I. Isocratic Retention Data”

vol % methanol solute 65 55 45 35

Column: 10 cm CI8

uracil 1.4 1.4 1.4 1.6 guaifenesin 2.0 2.4 3.3 6.4

nitrobenzene 3.3 5.0 8.2 15.2 p-ethylphenol 3.3 6.0 11.2 25.0 methyl benzoate 4.1 7.2 14.0 31.6

Column: 15 cm Phenyl Phase uracil 2.0 2.0 2.0 2.2 guaifenesin 3.5 5.0 7.4 15.5 p-cresol 4.1 6.4 10.0 13.5 p-ethylphenol 5.5 10.0 18.2 42.6 nitrobenzene 6.6 11.0 18.2 36.0 methyl benzoate 6.6 12.6 24.2 58.2

p-cresol 2.5 3.8 5.9 11.2

a All retention data are in minutes.

Table 11. Observed (obsd) and Calculated (calcd) Retention Times and Peak Widths for Coupled C18 and Phenyl Columns

t R % ST compd obsd calcd difference obsd calcd

A. Gradient Conditions, Linear, 50-70% Methanol over 30 min; Column Order, C18/C3Ph

uracil 3.5 guaifenesin 9.0 8.7 3.3 0.22 0.20 p-cresol 12.2 12.1 0.8 0.27 0.28 p-ethylphenol 17.9 18.0 0.5 na 0.31 nitrobenzene 17.9 17.8 0.6 na 0.35 methyl benzoate 21.0 21.0 0.0 0.32 0.32

B. Gradient Conditions, Linear, 45-70% Methanol over 30 min; Column Order, C18/C3Ph

uracil 3.6 guaifenesin 11.2 10.8 3.6 0.25 0.24 p-cresol 15.0 14.8 1.3 0.27 0.31 nitrobenzene 21.0 20.9 0.5 na 0.32 p-ethylphenol 21.5 21.6 0.4 na 0.35 methyl benzoate 24.4 24.5 0.4 0.27 0.32

performed with a 10-cm CI8 column first, connected to a 15-cm C3Ph column.

In the f i t experiment, a 30-min linear binary gradient from 50 to 70% methanol was run, with a delay volume of 220 pL. The actual and calculated retention times under these con- ditions for the five test solutes are given in Table IIA.

A second gradient was run from 45% to 70% methanol in 30 min with both the actual and calculated retention times for the test solutes being given in Table IIB. Excellent agreement (better than 3.3% relative standard deviation in all instances) between the experimentally determined reten- tion times and those calculated by the program was observed.

Iteration of possible elution conditions to achieve the sep- aration of nitrobenzene from ethylphenol was performed on the computer. Calculations indicated that a reversal of the column order, with a slight change of gradient, should be effective. The order of the columns was thus reversed so that the solute entered the 15-cm C3Ph column and eluted from the 10-cm C18 column. A 40-min gradient from 30 to 70%

Table 111. Observed (obsd) and Calculated (calcd) Retention and Peak Widths for Coupled Phenyl and CIS Columnsn

compd t R % ST

obsd calcd difference obsd calcd

uracil 3.7 guaifenesin 20.4 20.5 0.5 0.31 0.31 p-cresol 25.4 24.9 2.0 0.33 0.37 nitrobenzene 32.2 31.9 0.9 0.30 0.35 p-ethylphenol 33.8 33.5 0.9 0.30 0.36 methyl benzoate 37.0 36.8 0.5 0.27 0.33

“Gradient conditions, 30-70% methanol over 40 min; column order, CqPh/Cla.

methanol was run. The resulting data are presented in Table 111. Again, excellent agreement between the actual and calculated retention times was found for all five compounds.

The predicted and observed peak time-standard deviations are also listed in Tables 1-111. The peak time-standard de- viations at elution depend upon the instaneous velocity (k’mt) at elution for each particular solute. This value, which is difficult to obtain in such a complex system using exact mathematical solutions, is readily available using numerical techniques.

The developed approach and corresponding computer programs can predict retention and peak widths under com- plicated experimental conditions with accuracy commensurate to those levels commonly observed in liquid chromatography. The program as written can accomodate up to six coupled columns in any order, each with their own particular k’vs. mobile-phase composition function, from isocratic elution to complex gradients with multiple columns. Given this math- ematical tool, the problem of determining which chromato- graphic conditions can achieve, or as importantly cannot achieve, the desired separation within any chosen constraints becomes the next problem to be approached.

The solution to such an “optimization” problem is of course complex. However, this report represents the necessary universal mathematical tools available for researchers in this area to develop approaches to use for optimization strategies (12).

LITERATURE CITED

!

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Sci. 1973, 7 7 , 96. (4) Dolphin, R. J.; Hoogeveen, C. P. J.; Wllimott, F. W. J . Chromatogr.

1976, 722, 259. (5) Huber, J. F. K.; Linden, R. V. D. J . Chromafogr. 1973, 83 , 267. (6) Rocca, J. L.; Black, 8. D.; Higgins, J. W. “Prediction of Gradient Elution

Behavior for HPLC Columns Packed with Mixed Stationary Phases”, Presented at 8th International Symposium on Column Liquid Chroma- tography, New York, 1964.

(7) Meyer, R. M.; Hartwick, R. A. Anal. Chem. 1984, 56, 2211. (8) Foley. J. P.; Dorsey, J. G. Anal. Chem. 1981, 55, 730. (9) Tomellini, S. A.; Woodruff, H. 8.; Hartwlck, R. A. Anal. Chem. 1985,

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971.

RECEIVED for review September 23,1985. Accepted December 2,1985. The support of the Petroleum Research Fund, Grant 16161-AC3, the Rutgers Research Council, and a Biomedical Research Support grant are gratefully acknowledged.