Parametric Modulation in Liquid Chromatography: Multivariate

Anal. Chem. 1997, 69, 2963-2971

Parametric Modulation in Liquid Chromatography: Multivariate Optimization of Mobile Phase Composition and Temperature Patrick H. Lukulay† and Victoria L. McGuffin*

Department of Chemistry, Michigan State University, East Lansing, Michigan 48824-1322

The concept and theory of parametric modulation are presented, and the strategy is demonstrated for multivariate optimization of mobile phase composition and temperature in liquid chromatography. Because each parameter to be optimized is maintained in separate and distinct zones along the column, the solutes are able to interact independently within each environment. Under these conditions, solute retention is a simple and rigorously predictable summation of the retention in each environment. Hence, parametric modulation is more accurate and requires fewer preliminary experiments than traditional optimization methods. This approach is demonstrated by application to the separation of isomeric polynuclear aromatic hydrocarbons using a polymeric octadecylsilica stationary phase, with methanol and acetonitrile mobile phases at temperatures from 23 to 45 °C. A variety of systematic methods has been developed in order to optimize the parameters of interest in chromatography.1-4 These methods are of four general types: simultaneous, sequential, regression, and theoretical. In the simultaneous methods, experimental measurements are performed concurrently for all desired levels of the parameters. The conditions that yield the best separation, as judged by empirical means or by a suitable quality criterion, are then identified as the optimum conditions. In the sequential methods, a few initial experiments are performed at selected levels of the parameters. Based on the results of these experiments, a search routine such as the sequential simplex method is used to direct and guide subsequent experiments toward the optimum conditions. In the regression methods, a few experiments are performed at selected levels of the parameters, and the data are fit to a predefined mathematical function. Once the regression coefficients have been determined, this function may be used to estimate the quality of the separation at all intermediate values of the parameters, from which the optimum is identified. Finally, in the theoretical methods, a mathematical equation that is wholly derived from theoretical principles and requires no experimental measurements is used to estimate the optimum conditions. Each of these methods has inherent † Current address: Wyeth-Ayerst Research, 401 N. Middletown Rd., Pearl River, NY 10965. (1) Berridge, J. C. Techniques for the Automated Optimization of HPLC Separations; Wiley: New York, 1985. (2) Schoenmakers, P. J. Optimization of Chromatographic Selectivity; Elsevier: Amsterdam, 1986. (3) Glajch, J. L.; Snyder, L. R. Computer-Assisted Method Development for HighPerformance Liquid Chromatography; Elsevier: Amsterdam, 1990. (4) Lukulay, P. H.; McGuffin, V. L. J. Microcolumn Sep. 1996, 8, 211.

S0003-2700(96)01228-0 CCC: $14.00

© 1997 American Chemical Society

advantages and limitations for optimization of the relevant parameters in chromatographic separations.3,4 Among these methods, regression has proven to be the most widely used and successful approach for both univariate and multivariate optimization. The regression approach is attractive because, by using a simple mathematical model, only a few initial experiments are necessary in order to characterize the complete response surface. While this approach can lead to rapid identification of the most promising experimental conditions, it requires that the model used to express the relationship between the parameter of interest and the chromatographic property such as capacity factor, plate number, etc. be well defined. Unfortunately, deviations from ideal thermodynamic behavior can lead to inexact or ill-defined models for the parameters of interest in liquid chromatography, including mobile phase composition,5,6 stationary phase composition,7 and temperature.8-10 Such deviations cannot be predicted a priori and, hence, serve to limit the accuracy with which the optimum conditions can be identified. This problem, which is detrimental in univariate optimization, can become prohibitive when two or more parameters are to be optimized simultaneously. To overcome this problem, a new approach has been developed for univariate and multivariate optimization called parametric modulation.4,11-15 The fundamental strategy of this approach is that chromatographic retention may be accurately predicted if the solute is constrained to undergo interactions independently within each environment of mobile phase, stationary phase, temperature, etc. This strategy is implemented by maintaining each of these parameters in discrete and separate zones along the chromatographic column, as illustrated schematically for modulation of mobile phase and temperature in Figure 1. Under these conditions, the overall solute retention is a simple summation of the retention in each of the individual environments. This approach should allow accurate prediction of solute retention and, hence, allow univariate and multivariate optimization with a minimum number of preliminary experiments. (5) Colin, H.; Guiochon, G.; Jandera, P. Anal. Chem. 1983, 55, 442. (6) Drouen, A. C. J. H.; Billiet, H. A. H.; de Galan, L. J. Chromatogr. 1986, 352, 127. (7) Issaq, H. J.; Mellini, D. W.; Beesley, T. E. J. Liq. Chromatogr. 1988, 11, 333. (8) Nahum, A.; Horvath, C. J. Chromatogr. 1981, 203, 53. (9) Hammers, W. E.; Verschoor, P. B. A. J. Chromatogr. 1983, 282, 41. (10) Cole, L. A.; Dorsey, J. G.; Dill, K. A. Anal. Chem. 1992, 64, 1324. (11) Wahl, J. H.; Enke, C. G.; McGuffin, V. L. Anal. Chem. 1990, 62, 1416. (12) Wahl, J. H.; Enke, C. G.; McGuffin, V. L. Anal. Chem. 1991, 63, 1118. (13) Wahl, J. H.; McGuffin, V. L. J. Chromatogr. 1989, 485, 541. (14) Lukulay, P. H.; McGuffin, V. L. J. Chromatogr. 1995, 691, 171. (15) Lukulay, P. H.; McGuffin, V. L. J. Liq. Chromatogr. 1995, 18, 4039.

Analytical Chemistry, Vol. 69, No. 15, August 1, 1997 2963

up )

F πrp2p

(3)

where F is the volumetric flow rate and rp and p are the radius and the total porosity, respectively, of the column within temperature zone p. The total retention time (ti) of the solute is the summation of residence times in each solvent and each temperature zone, q

ti )

n

∑∑

F

p)1 j)0

Figure 1. Schematic illustration of the parametric modulation concept for the simultaneous optimization of n solvents and q temperatures.

tijp kijp ) -1 tojp

(1)

where tijp and tojp are the elution time of a retained and a nonretained solute, respectively, in solvent j and at temperature p. It may be readily shown11,16 that the residence time of a solute in a solvent zone of length xj is given by

(

)

xj 1 + kijp tijp ) up kijp

xj

n

∑k

(16) Wahl, J. H. Ph.D. Dissertation, Michigan State University, 1991.

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Analytical Chemistry, Vol. 69, No. 15, August 1, 1997

(σijp)l2 ) hpdplp

(6)

where hp is the reduced plate height, dp is the mean particle diameter, and lp is the length of the column within temperature zone p. Since variances are independent and additive only in the time domain,17 the length variance must be converted to the corresponding temporal variance (σijp)t2. The total temporal variance is then expressed as the summation of the contributions from each solvent and each temperature zone, q

2

n

∑∑(σ p)1 j)0

In this equation, up is the mobile phase linear velocity, which may be expressed as

(5)

) lp

ijp

If the limit has a noninteger value for the first temperature zone (p ) 1), then the summation in eq 4 is performed in the normal manner for the integer solvent zones 0-n, and the fraction is treated as a multiplier for the last solvent zone. This solvent zone subsequently enters the second temperature zone, so that the remainder is used as a multiplier for the first solvent zone (n ) 0) in the second temperature zone (p ) 2). The computation is performed in an analogous manner for all subsequent temperature zones up to the summation index (p ) q). Theory of Solute Band Broadening. The theoretical basis of solute band broadening under the conditions of parametric modulation has been established previously on a single column11,12 and on serially coupled columns14 and is extended here to include temperature modulation. If the broadening arises from the column and is independent of extracolumn effects, the variance in length units (σijp)l2 can be expressed as

(σi)t ) (2)

(4)

kijp

The limit of the summation index (n), which represents the total number of solvent zones required to elute the solute from a temperature zone of length lp, is determined by evaluating the expression

j)0

THEORY Theory of Solute Retention. The theoretical basis of solute retention under the conditions of parametric modulation has been established previously for univariate modulation of the mobile phase11,12,15 as well as for multivariate modulation of the mobile and stationary phases.14 A similar approach is adopted here for multivariate modulation of the mobile phase and temperature. The inherent assumption underlying this theory is that solute retention is controlled independently within each solvent and temperature zone. This requires that the individual solvent and temperature zones remain distinct with minimal mixing at the boundaries (vide infra). In addition, the solute must be able to achieve steadystate conditions rapidly after each change in solvent or temperature. This latter requirement may limit the selection of mobile and stationary phases to those exhibiting rapid kinetics and linear isotherms for the solutes of interest. The validity of these assumptions has already been established for solvent modulation on a single column11,12 and on serially coupled columns14 and is extended here to include temperature modulation. The capacity factor (kijp), which is the thermodynamic measure of solute retention, may be expressed as

( )

πrp2pxj 1 + kijp

ijp)t

2

)

(∑ ( ))

q

hpdp

n

p)1

lp

j)0

∑

πrp2pxj 1 + kijp F

kijp

2

(7)

Finally, the equivalent number of theoretical plates (N) for the chromatographic system under the conditions of parametric modulation is given by (17) Giddings, J. C. Dynamics of Chromatography; Marcel Dekker: New York, 1965.

ti2

N)

(8)

(σi)t2

where the total retention time and temporal variance are evaluated from eqs 4 and 7, respectively. Although the plate number is not a theoretically meaningful concept, it provides a useful and practical basis for the comparison of different modulation sequences. Theory of Parametric Zone Broadening. To ensure that solute retention is controlled independently within each solvent and temperature zone, it is necessary that the zones exhibit minimal mixing at the boundaries. Such mixing will adversely influence the accuracy of predicting solute retention from the theoretical models. Moreover, distortion and even splitting of solute peaks may occur if elution occurs at or near a solvent zone boundary. The zone purity can be conveniently defined as the ratio of the extant zone length that is not mixed to the theoretical or desired zone length.14 When expressed in this manner, the zone purity approaches unity when no mixing occurs and approaches zero when complete mixing occurs between adjacent zones. For example, the solvent zone purity is

xj - ξxj )1-ξ xj

(9)

If the solvent zone is broadened predominantly by column processes, such as diffusion and mass transport, the mixed fraction ξ is given by

ξ)

4(σo)l xj

(10)

where the total variance for a nonretained solvent zone (σo)l2 in length units is q

∑l ) (∑h d l r

(σo)l2 )

ξ)

p

p p p p

4

p)1

p)1

∑l r

(14)

lp ) 20lp,mix

(15)

The zone lengths given by eqs 12 and 15 represent the minimum values that may be used with confidence under the conditions of parametric modulation in order to ensure adherence to the assumptions of the theoretical models. Optimization Strategy. The strategy for optimizing separations by parametric modulation requires preliminary measurement of the solute capacity factors at each desired value of each parameter. For a system comprised of n solvents and q temperatures, a total of nq measurements (where measurement is understood to be the mean and standard deviation of several trials) is required for each solute. From these measurements, the retention time and variance can then be predicted by using eqs 4 and 7, respectively, for any given sequence and length of the solvent and temperature zones. After the retention time and variance are calculated for each solute, the extent of separation between adjacent solutes is given by the resolution (Ri,i+1):

Ri,i+1 )

2

p )

(ti+1 - ti)

(16)

2[(σi+1)t + (σi)t]

(11)

q

(

lp,mix lp

If the minimum acceptable value for the zone purity is arbitrarily defined to be 95%, the temperature zone length required to maintain this value may be determined from eq 14 as

q

2

(

temperature difference between adjacent zones and the effective thermal conductivity of the packed column and solvent.18,19 Alternatively, if the temperature zone is broadened by convection, the mixed length is dependent on the temperature difference and the linear velocity, as well as the thermal conductivity, heat capacity, and density of the solvent.19 In either of these cases, the mixed region will be of constant length (lp,mix) for a given chromatographic system under constant experimental conditions, so that the mixed fraction ξ is given by

p)2

2

p p

The quality of the overall separation is then assessed by using a modified and improved form of the multivariate function known as the chromatographic resolution statistic (CRS):14,20

p)1

If the minimum acceptable value for the zone purity is arbitrarily defined to be 95%, the solvent zone length required to maintain this value may be determined from eq 10 as

xj ) 80(σo)l

CRS )

∑ i)1

Ri,i+1 - Ropt

)

2

m-1

1

Ri,i+1 - Rmin Ri,i+1

+

∑

(Ri,i+1)2

i)1 (m

)

tf

- 1)Rav2 m

(17)

(12)

Similarly, the temperature zone purity is

lp - ξlp )1-ξ lp

(( m-1

(13)

If the temperature zone is broadened predominantly by static heat transfer (Fourier’s law), the length that is mixed depends on the

where m is the total number of solutes, tf is the elution time of the final solute, Ropt is the optimum or desired resolution, Rmin is the minimum acceptable resolution, and Rav is the average resolution, which is given by (18) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. (19) Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems; Cambridge University Press: Cambridge, England, 1984. (20) Schlabach, T. D.; Excoffier, J. L. J. Chromatogr. 1988, 439, 173.


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Rav )

1 m

m

∑

Ri,i+1

(18)

i)1

The first term of the CRS function is a measure of the extent of separation between each pair of adjacent solutes in the chromatogram. This term approaches zero when each individual resolution element approaches the optimum value and approaches infinity when any resolution element approaches the minimum value. The second term of the CRS function reflects the uniformity of spacing between solute peaks and approaches a minimum value of unity when the sum of the individual resolution elements is equal to the average value. The final term of the CRS function is intended to minimize the analysis time and may be neglected if this is not a primary goal of the optimization. To optimize the separation, the experimental conditions that yield the minimum value for the CRS function must be determined. This is achieved by systematically varying the sequence and length for each parameter to be optimized in order to generate a complete CRS response surface. The optimum conditions may be identified in two ways: by visual inspection of the response surface or by using an iterative search routine such as the simplex method.13-15,21 EXPERIMENTAL METHODS Reagents. Reagent-grade chrysene, benz[a]anthracene, benzo[c]phenanthrene, pyrene, perylene, benzo[a]pyrene, and benzo[e]pyrene are obtained from Sigma Chemical Co. (St. Louis, MO). Tetrabenzonaphthalene and phenanthro[3,4-c]phenanthrene are obtained from the National Institute of Standards and Technology (Gaithersburg, MD). Standard solutions of each polynuclear aromatic hydrocarbon are prepared by dissolution in methanol at 10-4 M concentration. Organic solvents are high-purity, distilled-in-glass grade (Baxter Healthcare, Burdick and Jackson Division, Muskegon, MI). Experimental System. An experimental system has been developed to achieve parametric modulation of both solvent and temperature. In this system, a dual-syringe pump (Model 140, Applied Biosystems, Foster City, CA) is programmed to deliver the individual solvent zones for a specified time period. If more than two solvents are to be modulated, the solvent delivery system may be constructed by using an automated multiple-port switching valve,12 as shown in Figure 1. The sample is then introduced by means of a 1.0-µL injection valve (Model ECI4W1, Valco Instruments, Houston, TX), after which the effluent stream is split (50: 1) to yield a nominal injection volume of 20 nL and a nominal flow rate of 1.2 µL/min. The splitter is constructed from a stainless-steel tube, with volume approximately equal to the split ratio multiplied by the column volume, which is connected to a fused-silica capillary restrictor (200-µm o.d., 50-µm i.d., 91-cm length, Polymicro Technologies, Phoenix, AZ). This design ensures that the split ratio does not vary significantly during the changes in solvent zone composition. To ensure rapid thermal transport and equilibration,22-24 the chromatographic column is prepared from fused-silica tubing of capillary dimensions (350-µm o.d., 200-µm i.d., Polymicro Technologies), which is terminated at the desired length with a quartz (21) Deming, S. N.; Morgan, S. L. Anal. Chem. 1973, 45, 278A. (22) Yoo, J. S.; Watson, J. T.; McGuffin, V. L. J. Microcolumn Sep. 1992, 4, 349. (23) Poppe, H.; Kraak, J. C.; Huber, J. F. K.; van den Berg, J. H. M. Chromatographia 1981, 14, 515. (24) Bowermaster, J.; McNair, H. M. J. Chromatogr. Sci. 1984, 22, 165.

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wool frit. The column is packed according to the procedure described previously25 with an irregular silica material having a mean particle size of 5.5 µm, pore size of 190 Å, and surface area of 240 m2/g (IMPAQ 200, PQ Corp., Conshohocken, PA), which has been chemically bonded with a polymeric octadecyl stationary phase (5.4 µmol/m2). The resulting column has a total porosity (p) of 0.90, separation impedance (φp) of 450, and reduced plate height (hp) of 4.7 at 23 °C, 4.5 at 35 °C, 4.3 at 40 °C, and 3.7 at 45 °C using pyrene as a model solute under standard test conditions.25 For independent control of each temperature zone, the column is enclosed in water jackets constructed of nylon tubing (0.32-cm o.d., 0.20-cm i.d., Cole-Parmer, Niles, IL) and fittings (Swagelok NY-200-3, Crawford Fitting Co., Solon, OH). Water circulation is provided by thermostatically controlled baths (Model RTE-9B, Neslab Instruments, Portsmouth, NH), with an operating range of -20 to 100 °C ((0.1 °C). The polynuclear aromatic hydrocarbon solutes are detected by using both laser-induced fluorescence and UV absorbance in a fused-silica capillary flow cell (350-µm o.d., 75-µm i.d., 10- and 35-cm length, respectively, Polymicro Technologies). In the fluorescence detection system,26 a helium-cadmium laser (325 nm, 25 mW, Model 3074-40M, Omnichrome, Chino, CA) is reflected by a dielectric mirror and is focused on the capillary with a quartz lens. The fluorescence emission is collected perpendicular and coplanar to the excitation beam with another quartz lens, spectrally isolated with a bandpass interference filter (420 nm, S10-420-F, Corion, Holliston, MA), and focused onto a photomultiplier tube (Model Centronic Q4249B, Bailey Instruments, Saddle Brook, NJ). The photocurrent is amplified and converted to voltage by a picoammeter (Model 480, Keithley Instruments, Cleveland, OH). A variable-wavelength UV absorbance detector (Model Uvidec 100-V, Japan Spectroscopic Co., Tokyo, Japan), operated at 254 nm, is placed in series after the fluorescence detector. Data from both detectors are displayed simultaneously by using a chart recorder (Model 585, Linear Instruments Corp., Reno, NV). Computer Optimization Program. The computer program developed to optimize chromatographic separations by the parametric modulation approach is written in the FORTRAN 77 language for execution on a VAXstation 3200 computer (Digital Equipment, Maynard, MA).14 This program utilizes a systematic mapping procedure in which the lengths for each parameter to be optimized are varied independently within prescribed limits. For the present study, the temperature zone length is varied from 0 to 90 cm, and the solvent zone length is varied from 5 cm to the maximum length required to elute all solutes. For each combination of temperature and solvent zone lengths, the total retention time and variance of each solute are calculated by means of eqs 4 and 7, respectively. The resolution between each pair of adjacent solutes is calculated by using eq 16. Finally, the overall quality of the separation is assessed by means of the modified CRS function in eq 17, using selected values for the optimum and minimum acceptable resolution of 1.5 and 1.0, respectively. By graphing the CRS value as a function of the temperature and solvent zone lengths, a complete multivariate response surface may be constructed. The minimum CRS value is then identified by visual inspection or computer search of the response surface. (25) Gluckman, J. C.; Hirose, A.; McGuffin, V. L.; Novotny, M. Chromatographia 1983, 17, 303. (26) McGuffin, V. L.; Zare, R. N. Appl. Spectrosc. 1985, 39, 847.

RESULTS AND DISCUSSION The goals of this preliminary study are to demonstrate the conceptual and theoretical basis of parametric modulation and to apply this strategy for the multivariate optimization of the mobile phase and temperature. In preparation to meet these goals, it is worthwhile to elucidate clearly the conditions under which a spatial gradient in temperature will be most beneficial. In a spatial gradient, unlike in a temporal gradient, all solutes will encounter all temperature zones in an analogous manner according to eq 4. The capacity factor can be expressed by using the classical thermodynamic relationship of van’t Hoff

ln k )

∆H ∆S + - ln β RT R

(19)

where ∆H and ∆S are the molar enthalpy and entropy for solute transfer between the mobile and stationary phases, β is the volume ratio of the mobile and stationary phases, R is the gas constant, and T is the absolute temperature.27-29 For an ideal solute, ∆H and ∆S are invariant with temperature, and a linear relationship is observed for ln k versus 1/T. For these ideal solutes, there exists an isothermal temperature that will yield capacity factors that are exactly equivalent to the spatial temperature gradient, whether that gradient is discontinuous (as described herein) or continuous in nature.30 Consequently, a spatial temperature gradient can be used to optimize separations for all solutes but will be superior to isothermal separations only for solutes that deviate from ideal thermodynamic behavior. Based on this consideration, the chromatographic system and solutes chosen for this preliminary study should exhibit a distinct change in the retention mechanism with temperature. Although there are a variety of chromatographic systems that meet this criterion, octadecylsilica has been selected as the stationary phase because of its widespread use and general applicability in liquid chromatography. This type of stationary phase has been shown to undergo a phase transition, where the transition temperature is dependent on the bonding density of the alkyl group, the alkyl chain length, the mobile phase composition, and other variables.31-35 Whereas the exact nature of this phase transition remains the subject of continued study and controversy, it is nevertheless apparent that a significant change occurs in the molecular structure and morphology. These changes have been examined by a variety of analytical techniques, including nuclear magnetic resonance spectroscopy,34-37 Fourier transform infrared spectroscopy,34,38,39 and thermal analysis,40 as well as chromatographic (27) Melander, W. R.; Campbell, D. E.; Horvath, C. J. Chromatogr. 1978, 158, 215. (28) Melander, W. R.; Chen, B. K.; Horvath, C. J. Chromatogr. 1979, 185, 99. (29) Tchapla, A.; Heron, S.; Colin, H.; Guiochon, G. Anal. Chem. 1988, 60, 1443. (30) Moore, L. K.; Synovec, R. E. Anal. Chem. 1993, 65, 2663. (31) Wheeler, J. F.; Beck, T. L.; Klatte, S. J.; Cole, L. A. Dorsey, J. G. J. Chromatogr. 1993, 656, 317. (32) Tchapla, A.; Heron, S.; Lesellier, E.; Colin, H. J. Chromatogr. 1993, 656, 81. (33) Sander, L. C.; Wise, S. A. J. Chromatogr. 1993, 656, 335. (34) Gilpin, R. K. J. Chromatogr. 1993, 656, 217. (35) Sentell, K. B. J. Chromatogr. 1993, 656, 231. (36) Sindorf, D. W.; Maciel, G. E. J. Am. Chem. Soc. 1983, 105, 1848. (37) Sindorf, D. W.; Maciel, G. E. J. Am. Chem. Soc. 1983, 105, 3767. (38) Sander, L. C.; Callis, J. B.; Field, L. R. Anal. Chem. 1983, 55, 1068. (39) Leyden D. E.; Kendall, D. S.; Burggraf, L. W.; Pern, F. J.; DeBello, M. Anal. Chem. 1982, 54, 101. (40) Morel, D.; Tabar, K.; Serpinet, J.; Claudy, P.; Letoffe, J. M. J. Chromatogr. 1987, 395, 73.

methods.31,32 At temperatures above the transition point, the stationary phase appears to be flexible and randomly oriented, so that solute retention is dominated by enthalpic interactions. As temperature is decreased below the transition point, the stationary phase becomes progressively more rigid and ordered, so that solute retention acquires a correspondingly greater entropic contribution. Under “entropy-dominated” conditions, there is a significant increase in the selectivity of octadecylsilica with respect to the size and shape of solute molecules.41-44 Throughout the phase transition, both ∆H and ∆S are temperature dependent for many solutes, leading to substantial deviations from ideal behavior. For this study, a polymeric octadecylsilica material with a high bonding density of 5.4 µmol/m2 is utilized as the stationary phase. From previous studies,45 the transition temperature for this material is known to be approximately 45 °C. Because of the progressive nature of the phase transition, it is desirable to examine a broad range of temperatures in the vicinity of the transition temperature. In this manner, the temperature that yields the most optimal variations in selectivity for the model solutes of interest can be identified. The model solutes consist of isomeric four-, five-, and six-ring polynuclear aromatic hydrocarbons (PAHs). Representative chromatograms for a mixture of these solutes are shown in Figure 2 in the range between room temperature and the transition temperature. From these chromatograms, it is apparent that the solutes exhibit significant changes in retention and selectivity with temperature in this range. The capacity factors of each PAH were measured at temperatures of 23, 35, 40, and 45 °C using pure methanol (Table 1) and acetonitrile (Table 2) as mobile phases. In general, the capacity factors increase with the carbon number or number of aromatic rings at all temperatures and in both mobile phases. The notable exceptions to this behavior are tetrabenzonaphthalene and phenanthro[3,4-c]phenanthrene, both of which have 26 carbon atoms arranged in six aromatic rings with a distinctly nonplanar structure. These solutes, together with benzo[a]pyrene, have been recommended by Sander and Wise46-48 to characterize the selectivity of octadecylsilica stationary phases. The retention order phenanthro[3,4-c]phenanthrene < tetrabenzonaphthalene , benzo[a]pyrene, which is observed here for all temperatures and mobile phases, is fully consistent with the high bonding density of the polymeric octadecylsilica stationary phase.46 To examine the thermodynamic influence of temperature on solute retention, a van’t Hoff plot has been prepared for the methanol (Figure 3) and acetonitrile (Figure 4) mobile phases. It is evident that the logarithm of the capacity factor increases nonlinearly with decreasing temperature for many of the PAH solutes. This is most clearly demonstrated by tetrabenzonaphthalene and chrysene in the methanol mobile phase (Figure 3). This deviation from ideal behavior is accompanied by a change in the elution order. At temperatures well below the transition point, where the octadecylsilica stationary phase is rigidly ordered and entropic forces predominate, the planar chrysene is more (41) Snyder, L. R. J. Chromatogr. 1979, 179, 167. (42) Chmielowiec, J.; Sawatzky, H. J. Chromatogr. Sci. 1979, 17, 245. (43) Sentell, K. B.; Henderson, A. N. Anal. Chim. Acta 1991, 246, 139. (44) Sander, L. C.; Wise, S. A. Anal. Chem. 1989, 61, 1749. (45) Chen, S. H.; McGuffin, V. L. J. Chromatogr., manuscript to be submitted. (46) Sander, L. C.; Wise, S. A. Anal. Chem. 1984, 56, 504. (47) Sander, L. C.; Wise, S. A. J. Chromatogr. 1984, 316, 163. (48) Sander, L. C.; Wise, S. A. LC-GC 1990, 8, 378.


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Table 2. Capacity Factors (kijp) for Polynuclear Aromatic Hydrocarbons at Various Temperatures on Polymeric Octadecylsilica Stationary Phase Using Acetonitrile as Mobile Phase

Figure 2. Representative chromatograms of polynuclear aromatic hydrocarbons as a function of temperature. Column: 200-µm-i.d. × 90-cm fused-silica capillary, packed with 5.5-µm polymeric octadecylsilica. Mobile phases: methanol at 1.2 µL/min. Detector: laserinduced fluorescence with excitation at 325 nm and emission at 420 nm. Solutes: (1) benzo[c]phenanthrene, (2) pyrene, (3) phenanthro[3,4-c]phenanthrene, (4) benz[a]anthracene, (5) tetrabenzonaphthalene, (6) chrysene, (7) benzo[e]pyrene, (8) perylene, and (9) benzo[a]pyrene. Table 1. Capacity Factors (kijp) for Polynuclear Aromatic Hydrocarbons at Various Temperatures on Polymeric Octadecylsilica Stationary Phase Using Methanol as Mobile Phase

capacity factor (kijp)

polynuclear aromatic hydrocarbon

23 °C

35 °C

40 °C

45 °C

benzo[c]phenanthrene pyrene phenanthro[3,4-c]phenanthrene benz[a]anthracene tetrabenzonaphthalene chrysene benzo[e]pyrene perylene benzo[a]pyrene

0.51 0.58 0.77 0.83 1.06 1.04 1.48 1.68 2.47

0.39 0.41 0.56 0.55 0.84 0.64 0.96 1.00 1.41

0.34 0.38 0.48 0.48 0.71 0.54 0.79 0.85 1.14

0.31 0.34 0.40 0.41 0.61 0.45 0.67 0.71 0.92

Figure 3. Effect of temperature on logarithm of capacity factor for polynuclear aromatic hydrocarbons using methanol as mobile phase. Experimental conditions as given in Figure 3. Solutes: (4) benzo[c]phenanthrene, (b) pyrene, (0) phenanthro[3,4-c]phenanthrene, (2) benz[a]anthracene, (]) tetrabenzonaphthalene, (9) chrysene, (O) benzo[e]pyrene, ([) perylene, and (3) benzo[a]pyrene.

capacity factor (kijp)


23 °C

35 °C

40 °C

45 °C


0.77 0.82 0.95 1.53 1.60 2.04 2.57 3.14 4.79

0.63 0.65 0.74 1.05 1.35 1.32 1.72 1.98 2.85

0.58 0.61 0.67 0.94 1.24 1.15 1.53 1.73 2.46

0.52 0.53 0.58 0.80 1.10 0.97 1.28 1.43 2.00

retained than the nonplanar tetrabenzonaphthalene. However, as the temperature approaches the transition point where the stationary phase becomes more flexible, enthalpic forces favor the retention of tetrabenzonaphthalene because of the greater carbon number or number of aromatic rings. Similar behavior is observed in the acetonitrile mobile phase (Figure 4); tetrabenzonaphthalene and chrysene coelute at low temperature, but tetrabenzonaphthalene becomes more highly retained near the transition temperature. It is apparent from the results summarized in Tables 1 and 2 that one or more critical solute pairs exist at each of the examined conditions of mobile phase and temperature. Because the identity 2968 Analytical Chemistry, Vol. 69, No. 15, August 1, 1997

Figure 4. Effect of temperature on logarithm of capacity factor for polynuclear aromatic hydrocarbons using acetonitrile as mobile phase. Experimental conditions as given in Figure 3; solutes as given in Figure 4.

of the critical pair differs, none of these conditions alone will be sufficient to separate all of the isomeric PAH solutes. Moreover, because there are significant deviations from ideal behavior (Figures 3 and 4), a regression approach cannot be used to predict accurately the most optimal conditions for the separation. Nev-

Table 3. Evaluation of Selected Permutations of a One-Solvent/Two-Temperature and Two-Solvent/ Two-Temperature Chromatographic System for the Separation of Polynuclear Aromatic Hydrocarbonsa temp 1 (°C)

temp 2 (°C)

solvent 1

23 23 23 23 23 23 23 40 23 40

35 40 45 35 40 45 40 23 40 23

100% CH3OH 100% CH3OH 100% CH3OH 100% CH3CN 100% CH3CN 100% CH3CN 100% CH3OH 100% CH3OH 100% CH3CN 100% CH3CN

solvent 2

tf (min)

(Ri,i+1)min

CRSmin

100% CH3CN 100% CH3CN 100% CH3OH 100% CH3OH

120.5 120.0 119.5 67.0 67.0 64.0 102.0 80.0 87.0 83.0

1.29 1.31 1.28 1.29 1.29 1.32 1.33 1.34 1.35 1.34

2.6 2.4 2.6 3.1 3.1 2.6 2.3 2.3 2.2 2.2

a The analysis time (t ), minimum resolution ((R f i,i+1)min), and minimum chromatographic resolution statistic (CRSmin) corresponding to the optimum conditions for each permutation are given.

ertheless, it seems reasonable to expect that the solutes can be resolved by parametric modulation of the mobile phase and temperature, because each solute pair is well resolved under at least one of the examined conditions. The parametric modulation approach provides a high degree of power and versatility in the design of a chromatographic system to achieve the desired separation. Among the solvents and temperatures examined in this study, there are 64 possible permutations: 8 one-solvent/one-temperature systems, 24 onesolvent/two-temperature systems, 8 two-solvent/one-temperature systems, and 24 two-solvent/two-temperature systems. It would clearly be impractical to examine all of these possibilities by timeconsuming, trial-and-error experimental measurements. However, the computer program described previously provides a rapid and effective means to identify the most promising of these permutations. The individual capacity factors (kijp) for the PAH solutes shown in Tables 1 and 2 are used to calculate the retention time and variance according to eqs 4 and 7, respectively. The resolution for each pair of adjacent solutes is calculated from eq 16, and the overall quality of the separation is assessed by the modified CRS function in eq 17. For each permutation, the sequence and length of each solvent and temperature are systematically varied over the prescribed range. During this preliminary search, the minimum CRS values are stored in a file and are used to identify the most promising permutation for further study. The results for selected permutations of a one-solvent/twotemperature system and a two-solvent/two-temperature system are summarized in Table 3. The minimum CRS value will be used as the primary criterion for comparison, although the minimum resolution of the critical solute pair and the analysis time are also cited. For the one-solvent/two-temperature systems, the best separation is predicted to occur using methanol as the mobile phase with a temperature sequence of 23 °C followed by 40 °C. These conditions should provide a minimum resolution of 1.31, with a total analysis time of 120 min. For the two-solvent/twotemperature systems, the best separation is predicted for a solvent sequence of acetonitrile followed by methanol, with a temperature sequence of 23 °C followed by 40 °C. Under these conditions, the minimum resolution is 1.35, with a total analysis time of 87 min. It is apparent that the overall quality of the predicted separation using modulation of both mobile phase and temperature is not significantly better than that using temperature alone. Furthermore, the analysis time will not be substantially different,

Figure 5. Topographic (A) and contour (B) maps of the CRS response surface as a function of the column length (cm) maintained at temperatures of 23 °C (temp 1) and 40 °C (temp 2).

because the two-solvent modulation scheme will require a reequilibration period between separations (at least one column Analytical Chemistry, Vol. 69, No. 15, August 1, 1997

2969

Table 4. Comparison of Experimental and Theoretical Retention Time, Peak Width, and Resolution under the Predicted Optimum Conditions Illustrated in Figures 6 and 7 retention time (min)

width (min)

resolution


expt

theorya

% errorb

expt

theoryc

% errorb

expt

theoryd

% errorb


37.7 38.6 41.9 53.3 55.9 63.2 74.4 85.7 119.2

37.3 38.3 41.0 53.0 54.7 63.4 74.5 86.1 120.0

1.1 0.8 2.2 0.6 2.2 -0.3 -0.1 -0.5 -0.7

0.68 0.82 0.95 1.02 1.22 1.36 1.50 2.04 2.58

0.78 0.80 0.86 1.11 1.15 1.33 1.57 1.81 2.53

-12.8 2.5 10.5 -8.1 6.1 2.3 -4.5 12.7 2.0

1.20 3.72 11.6 2.32 5.65 7.84 6.40 14.5

1.31 3.21 12.1 1.55 7.02 7.64 6.90 15.6

-8.4 15.9 -4.1 49.7 -19.5 2.6 -7.2 -7.1

average

(0.9

(6.8

(14.3

a Calculated from eq 4. b % error ) 100(expt - theory)/theory. c Calculated from eq 7, assuming a peak width of 4(σ ) . d Calculated from eqs i t 4, 7, and 16, assuming a peak width of 4(σi)t.

Figure 6. Predicted chromatogram for the polynuclear aromatic hydrocarbons at the optimum conditions identified from Figure 6. Solutes as given in Figure 3.

volume, e.g., 21 min). Because the overall performances of the systems are nearly equivalent, the optimal one-solvent/twotemperature system was selected for further study because of the greater ease and simplicity of experimental implementation. To identify the most favorable conditions, the computer program is used next to generate topographic and contour maps of the CRS response surface as a function of the length of each temperature zone. As shown in Figure 5, the overall quality of the separation improves continuously with the temperature zone length at 23 °C. In contrast, the temperature zone at 40 °C has a lesser effect, improving the separation notably for lengths up to 5 cm but only slightly thereafter. Thus, the optimum conditions for parametric modulation consist of an 85-cm length of the octadecylsilica column maintained at 23 °C, followed by a 5-cm length at 40 °C, using pure methanol as the mobile phase. The isomeric PAHs are predicted to be well separated under these conditions, as shown in Figure 6, with a CRS value of 2.4 and an analysis time of 120 min. The critical solute pair is benzo[c]phenanthrene and pyrene, with a minimum resolution of 1.31. All other solute pairs have resolution greater than the optimum value (Ropt ) 1.5), which is desirable for accurate qualitative and quantitative analysis. Because the CRS response surface shown in Figure 5 is relatively flat in the vicinity of the optimum, small discrepancies in temperature zone length should have little effect 2970 Analytical Chemistry, Vol. 69, No. 15, August 1, 1997

Figure 7. Experimental chromatogram for the polynuclear aromatic hydrocarbons obtained under the predicted optimum conditions. Column: 200-µm-i.d. × 90-cm fused-silica capillary, packed with 5.5µm polymeric octadecylsilica, 85-cm length maintained at 23 °C and 5-cm length maintained at 40 °C. Mobile phase: methanol at 1.2 µL/ min. Other experimental conditions as given in Figure 3; solutes as given in Figure 3.

on the quality of the separation. Thus, the analytical method should be relatively reproducible and rugged. To demonstrate the practical application of this method, the separation of the isomeric four-, five-, and six-ring PAHs was performed under the predicted optimum conditions. The experimental chromatogram (Figure 7) shows excellent separation of all PAHs with resolution comparable to that in the predicted chromatogram (Figure 6). The experimental retention time and peak width for each PAH agree well with the theoretically predicted values from eqs 4 and 7, respectively, as summarized in Table 4. The average relative error is (0.9% for retention time and (6.9% for peak width. Thus, the theoretical models developed herein can accurately predict the experimental results for parametric modulation of both mobile phase and temperature.4 CONCLUSIONS The multivariate optimization of liquid chromatographic separations by means of mobile phase and temperature is a challenging task due, in large part, to deviations from ideal behavior. Parametric modulation appears to be a promising approach to this problem. Because the individual solvents and temperatures are spatially separated in discrete zones, solute retention is a simple, time-weighted average in each environment to which the solute is exposed. For a system composed of n solvents and q temperatures, only nq retention measurements are necessary. From these measurements, solute retention may be accurately

predicted for any combination, sequence, and length of the solvent and temperature zones. Other physical parameters of interest in eqs 4 and 7, such as particle size, column dimensions, and flow rate, can be optimized simultaneously with no additional experimental measurements. This approach has been demonstrated for the separation of isomeric polynuclear aromatic hydrocarbons using polymeric octadecylsilica as the stationary phase and methanol and acetonitrile as mobile phases. Based on these results, the parametric modulation approach appears to be a powerful and versatile strategy for the multivariate optimization of mobile phase and temperature.

synthesis of the polymeric octadecylsilica stationary phase, and to Dr. Stephen A. Wise (National Institute of Standards and Technology) for providing the polynuclear aromatic hydrocarbon standards. This research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-FG02-89ER14056.

ACKNOWLEDGMENT The authors are grateful to Dr. Lane C. Sander (National Institute of Standards and Technology) for performing the custom

AC9612281

Received for review December 3, 1996. Accepted May 9, 1997.X

X

Abstract published in Advance ACS Abstracts, July 1, 1997.


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Parametric Modulation in Liquid Chromatography: Multivariate

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