Theoretical and Experimental Studies of the Effect of Pressure on

Anal. Chem. 1997, 69, 930-943

Theoretical and Experimental Studies of the Effect of Pressure on Solute Retention in Liquid Chromatography Victoria L. McGuffin* and Shu-Hui Chen†

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

Pressure is often assumed to have a negligible influence on solute retention in liquid chromatography because of the small compressibility of the mobile and stationary phases. The range of pressures commonly encountered in reversed-phase separations is considerable, however, and may give rise to significant changes in solute capacity factor. In this study, the retention of model solutes is measured directly along the chromatographic column as a function of the local pressure. The model solutes, a homologous series of derivatized fatty acids, exhibit a significant increase in capacity factor ranging from +9.3% for n-C10 to +24.4% for n-C20 for inlet pressures from 1500 to 5000 psi. These experimental results are compared with a thermodynamic model derived from regular solution theory. This model suggests that state effects alone are not sufficient to describe the measured change in solute retention and that variations in interaction energy with density must also be considered. By using the simple relationship of van der Waals for the interaction energy (E ∝ 1/V), the change in capacity factor with density is slightly underestimated. However, by using an extended relationship that better describes polar fluids (E ∝ 1/V2), good agreement is observed. Finally, the correlation of experimental results with this thermodynamic model reveals that all components in the chromatographic system, including the solute, mobile phase, and stationary phase, must be considered compressible. The results of this study have clear implications for the determination of fundamental physicochemical parameters, as well as for the everyday practice of liquid chromatography. The effect of pressure on fundamental physical properties such as density, viscosity, and diffusion coefficient is well documented.1-6 In addition, pressure has been shown to influence a variety of

equilibrium distribution processes as well.7,8 This fundamental understanding has made possible the theoretical prediction of the influence of pressure on mobile-phase velocity,9 solute retention,10,11 and solute zone dispersion12 in chromatography. In fact, the exploitation and control of pressure-induced changes in equilibrium was first proposed for chromatographic separations more than 20 years ago by Giddings,13,14 leading to the development of supercritical fluid chromatography (SFC). Although variations in equilibrium constants in the liquid phase are not expected to be as large as those in gases or supercritical fluids, the greater pressures commonly applied in liquid chromatography (LC) may be sufficient to induce significant changes. These changes may become particularly important for LC systems where the operating pressure is inherently high, such as multiple column or high-speed separations.15 Unfortunately, only a limited number of experimental investigations have been reported in the literature. Rogers et al.16-18 measured a 300% increase in capacity factor and a concomitant change in selectivity for separations by an adsorption/ion-exchange mechanism under extreme pressure conditions (20 000 psi). Tanaka et al.,19 under more modest pressure conditions (3000 psi), measured a 12% increase in capacity factor for reversed-phase separations performed under ionization control. Under similar pressure conditions, however, Katz et al.20 reported a 5% decrease in retention for a normal-phase adsorption mechanism. These few experimental results indicate that variations in the absolute pressure may, in fact, have a significant effect on solute retention in liquid chromatography. These previous studies have interesting implications for the accurate measurement of solute retention and the associated thermodynamic parameters, as well as their meaningful correlation with theoretical models. If retention is a significant function of pressure, then a retention gradient will arise along the chromatographic column as a direct consequence of the pressure gradient. The retention gradient may not be readily apparent using

† Present address: Department of Chemistry, National Cheng Kung University, Tainan, Taiwan 70101. (1) Bridgman, P. W. Proc. Acad. Arts Sci. 1926, 61, 59. (2) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954. (3) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids; McGraw-Hill: New York, 1977. (4) Stephan, K.; Lucas, K. Viscosity of Dense Fluids; Plenum: New York, 1979. (5) Liley, P. E.; Makita, T.; Tanaka, Y. Properties of Inorganic and Organic Fluids; Hemisphere Publishing: New York, 1988; Vol. V-1. (6) Kestin, J.; Wakeham, W. A. Transport Properties of Fluids: Thermal Conductivity, Viscosity, and Diffusion Coefficient; Hemisphere Publishing: New York, 1988; Vol. I-1.

(7) Hamann, S. D. J. Phys. Chem. 1962, 66, 1359. (8) Macko, T.; Soltes, L.; Berek, D. Chromatographia 1989, 28, 189. (9) Martin, M.; Guiochon, G. Anal. Chem. 1983, 55, 2302. (10) Martire, D. E. J. Liq. Chromatogr. 1987, 10, 1569. (11) Martire, D. E.; Boehm, R. E. J. Phys. Chem. 1987, 91, 2433. (12) Poe, D. P.; Martire, D. E. J. Chromatogr. 1990, 517, 3. (13) Giddings, J. C. Sep. Sci. 1966, 1, 73. (14) Giddings, J. C.; Myers, M. N.; McLaren, L.; Keller, R. A. Science 1968, 162, 67. (15) van der Wal, S. Chromatographia 1986, 22, 81. (16) Bidlingmeyer, B. A.; Hooker, R. P.; Lochmuller, C. H.; Rogers, L. B. Sep. Sci. 1969, 4, 439. (17) Bidlingmeyer, B. A.; Rogers, L. B. Sep. Sci. 1972, 7, 131. (18) Prukop, G.; Rogers, L. B. Sep. Sci. 1978, 13, 59. (19) Tanaka, N.; Yoshimura, T.; Araki, M. J. Chromatogr. 1987, 406, 247. (20) Katz, E.; Ogan, K.; Scott, R. P. W. J. Chromatogr. 1983, 260, 277.

930 Analytical Chemistry, Vol. 69, No. 5, March 1, 1997

S0003-2700(96)00589-6 CCC: $14.00

© 1997 American Chemical Society

conventional postcolumn detectors, because the measured capacity factor is a spatial and temporal average over the entire column length. This inherent averaging process serves to obscure the true magnitude of the pressure effect, making the correlation of experimental measurement with theoretical models nearly impossible. This deleterious process is minimized in the present study by monitoring solute zone migration as a function of spatial position along the chromatographic column. In this manner, the influence of the local pressure on solute retention may be systematically examined. THEORETICAL CONSIDERATIONS A number of theoretical models has been proposed to describe the retention of solutes in liquid chromatography.10,11,21-25 The statistical thermodynamic approach developed by Martire10,11 would clearly be favored in addressing the influence of pressure on solute retention, because of the rigorous derivation and universal applicability of this theory. Unfortunately, many pertinent thermodynamic data, most notably the binary interaction parameters, that are necessary for quantitative evaluation are not available at the present time. Thus, qualitative comparisons between experimental observations and the unified model are possible,26 but quantitative comparisons are not. This distinction is of fundamental importance because correlation of experimental data to an equation with adjustable parameters is insufficient evidence to validate a theoretical model directly. As a result of this limitation, a theoretical model is developed here based on the regular solution theory introduced by Hildebrand and Scott.27-30 In contrast with the more recent approach by Martire,10,11 regular solution theory is somewhat less rigorous and presumes that binary interactions may be approximated as the geometric mean of the individual unitary interactions. However, the unitary interaction parameters that are necessary for implementation of this theory have been well documented in the literature,30 thereby allowing quantitative evaluation of the influence of pressure on solute retention. In regular solution theory, the unitary interaction parameter is defined from the cohesive energy per unit volume of the pure fluid. This parameter, termed the solubility parameter, represents a quantitative measure of the polarity of the fluid:27-30

δ2 ) -E/V ) (∆H - RT)/V

(1)

In this expression, the solubility parameter (δ) is estimated from the molar enthalpy of vaporization (∆H), the molar volume (V), the absolute temperature (T), and the gas constant (R). The regular solution approach has been utilized to describe solute (21) Horvath, C.; Melander, W.; Molnar, I. J. Chromatogr. 1976, 125, 129. (22) Tijssen, R.; Billiet, H. A. H.; Schoenmakers, P. J. J. Chromatogr. 1976, 122, 185. (23) Dill, K. A. J. Phys. Chem. 1987, 91, 1980. (24) Cheong, W. J.; Carr, P. W. J. Chromatogr. 1990, 499, 373. (25) Tijssen, R.; Schoenmakers, P. J.; Bo ¨hmer, M. R.; Koopal, L. K., Billiet, H. A. H. J. Chromatogr. 1993, 656, 135. (26) Evans, C. E.; McGuffin, V. L. J. Microcolumn Sep. 1991, 3, 513. (27) Hildebrand, J. H.; Scott, R. L. Solubility of Nonelectrolytes; Reinhold: New York, 1950. (28) Hansen, C.; Beerbower, A. In Kirk-Othmer Encyclopedia of Chemical Technology, 2nd ed., Suppl. Vol.; Standen, A., Ed.; Wiley: New York, 1971; pp 889-910. (29) Barton, A. F. M. Chem. Rev. 1975, 75, 731. (30) Barton, A. F. M. Handbook of Solubility Parameters and Other Cohesion Parameters, 2nd ed.; CRC Press: Boca Raton, FL, 1991.

retention in liquid chromatographic separations by Martire and Locke31 and by Schoenmakers et al.22,32

ln k ) (Vi/RT)[(δi - δm)2 - (δi - δs)2] - ln β

(2)

where the capacity factor (k) is expressed as a function of the solubility parameters for the solute (i), mobile phase (m), and stationary phase (s), as well as the volume ratio of the mobile and stationary phases (β). This expression provides a clear indication of the careful balance of chemical interactions that is responsible for solute retention. For normal-phase separations, the stationary phase is more polar than the mobile phase (δs . δm), whereas the converse is true for reversed-phase separations (δm . δs). Regardless of separation mode, as the polarity of the solute approaches that of the mobile phase ((δi - δm) f 0), the solute capacity factor decreases logarithmically according to eq 2. Likewise, as the polarity of the solute approaches that of the stationary phase ((δi - δs) f 0), a corresponding increase in solute retention is predicted. Thus, retention can be described both conceptually and mathematically on the basis of the similarity in polarity (cohesive energy density) between the solute and the mobile and stationary phases. As a further refinement on the regular solution approach, the entropy contribution arising from the unequal molar volumes of the solute, mobile phase, and stationary phase may be incorporated into eq 2. This correction was originally proposed by Flory and Huggins to describe the configurational entropy arising upon mixing a solution of large polymer molecules with much smaller solvent species.33 When this correction to the chemical potential is applied to solute retention, the following expression results:

ln k ) (Vi/RT)[(δi - δm)2 - (δi - δs)2] ln β + [Vi/Vs - Vi/Vm + ln (Vs/Vm)] (3) For reversed-phase applications, the most common stationary phase is octadecylsilica and the mobile phases are usually aqueous mixtures of methanol or acetonitrile. Thus, the molar volume of the stationary phase is an order of magnitude greater than that of the mobile phase. The solute molar volume may have a wide range of values but is always expected to be greater than that of the mobile phase. As a result, the Flory-Huggins correction for configurational entropy will generally decrease the predicted capacity factor and will vary with the solute of interest. For the purposes of this discussion, both the original expression and that with Flory-Huggins correction will be considered. It is apparent that pressure may influence nearly all of the variables in eqs 2 and 3. Although pressure is commonly utilized as a state variable, density is chosen for this development as it allows more general applicability to all phases of interest in liquid chromatographic separations. From the Tait equation of state,34,35 the density (F) for a liquid at a specified pressure (P) can be calculated from those same variables in a reference state (F°, P°): (31) Martire, D. E.; Locke, D. C. Anal. Chem. 1971, 43, 68. (32) Schoenmakers, P. J.; Billiet, H. A. H.; de Galan, L. Chromatographia 1982, 15, 205. (33) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953; Chapter 12. (34) Tait, P. G. Scientific Papers; University Press: Cambridge, England, 1900; Vol. 2, pp 343-348. (35) Martin, M.; Blu, G.; Guiochon, G. J. Chromatogr. Sci. 1973, 11, 641.

Analytical Chemistry, Vol. 69, No. 5, March 1, 1997

931

F/F° ) [(P + b)/(P° + b)]c

(4)

where b and c are empirically determined constants. For this study, the reference state is chosen as 25 °C and 14.5 psi (1 bar) because the thermodynamic quantities required to evaluate the solubility parameters via eq 1 are known under these conditions.27-29 On the basis of this relationship, the influence of pressure on each variable in eqs 2 and 3 can be considered: (a) Molar Volume. Because density is inversely proportional to volume, the molar volume at density F is related to that in the reference state (F°, V°) by

(5)

V/V° ) F°/F

(b) Phase Ratio. The number of moles of stationary phase remains constant in the chromatographic system, so that the total volume must decrease with increasing density (eq 5). In contrast, the number of moles of the mobile phase is not limited but is supplied in excess by the pump, so that its total volume changes only slightly with pressure. Consequently, the phase ratio at density F is approximately related to that in the reference state (F°, β°) by

reasonably accurate for nonpolar liquids.27 Because of the strong orientation forces in polar and hydrogen-bonding liquids, however, the radial distribution function may be altered in form and may be a complex function of temperature and pressure.36,37 Although this additional contribution to the total interaction energy is not well understood, Benson proposed that an expression of the general form E ∝ 1/V2 is appropriate for a uniform expansion of the fluid.38 This form shows comparable agreement with experimental data for nonpolar liquids and much better agreement for polar liquids than the classical van der Waals equation.27,38 Because this relationship is confirmed by heat of vaporization data, it is fully consistent with the solubility parameter model (eq 1). For the state effect plus this extended chemical effect, the solubility parameter is given by

(δ/δ°)2 ) (F/F°)3

(10)

By combining all of these effects into eq 2, we obtain a general expression for solute retention as a function of the densities of the solute (i), mobile phase (m), and stationary phase (s):

ln k ) (Vi°(Fi°/Fi)/RT)[{δi°(Fi/Fi°)n - δm°(Fm/Fm°)n}2 {δi°(Fi/Fi°)n - δs°(Fs/Fs°)n}2] - ln β°(Fs/Fs°) (11)

(6)

β/β° ) F/F°

(c) Solubility Parameters. The influence of density on the solubility parameters is considerably more complex because both state and chemical effects are possible,14 as indicated in eq 1. Consider first the situation where the energy of interaction remains constant but the molar volume decreases with density according to eq 5. For this state effect alone, the solubility parameter is given by

(δ/δ°)2 ) F/F°

(7)

where the power n is equal to 1/2 for the state effect alone, 1 for the state effect plus simple chemical effect, and 3/2 for the state effect plus extended chemical effect. A similar expression can be derived from eq 3 that incorporates the Flory-Huggins correction:

ln k ) (Vi°(Fi°/Fi)/RT)[{δi°(Fi/Fi°)n δm°(Fm/Fm°)n}2 - {δi°(Fi/Fi°)n - δs°(Fs/Fs°)n}2] ln β°(Fs/Fs°) + [{Vi°(Fi°/Fi)}/{Vs°(Fs°/Fs)} {Vi°(Fi°/Fi)}/{Vm°(Fm°/Fm)} +

Consider next the influence of chemical effects, where the interaction energy between neighboring molecules also changes with their proximity. The total interaction energy per mole can be expressed as follows:

E ) 2π(N2/V)

∫(r)g(r)r

2

dr

(8)

where (r) is the interaction energy between two molecules separated by distance r and g(r) is the radial distribution function, which represents the probability of finding two molecules separated by distance r relative to the probability expected for a completely random distribution of molecules at the same density. In the simplest approach, the energy is described solely by cohesive interactions ((r) ∝ 1/r6) and the radial distribution function is equal to that for a random distribution (g(r) ) 1). These assumptions are consistent with the classical model of van der Waals,3 where the total interaction energy is of the general form E ∝ 1/V. For the state effect plus this simple chemical effect, the solubility parameter is given by

(δ/δ°)2 ) (F/F°)2

(9)

The simple van der Waals model has been demonstrated to be 932


ln ({Vs°(Fs°/Fs)}/{Vm°(Fm°/Fm)})] (12) Although eqs 11 and 12 appear at first glance to be rather complex, most of the factors are constants that are known or can be reliably estimated. The few factors that are variable may be systematically controlled; for example, δi° may be varied in a continuous, uniform manner by using a homologous series of solutes and F/F° may be controlled by means of the externally applied pressure. Therefore, the validity of these theoretical models can be examined by direct and detailed comparison with experimental measurements. EXPERIMENTAL DESIGN CONSIDERATIONS The success of this investigation is contingent on the ability to isolate the influence of pressure from the many other factors influencing solute retention. In addition, the experimental conditions must be representative of those commonly encountered in practical applications. For these reasons, we have chosen to investigate a reversed-phase separation system with the common octadecylsilica stationary phase. This packing material is con(36) Ben-Naim, A. In Solutions and Solubilities; Dack, M. R. J., Ed.; Wiley: New York, 1975; Part I, pp 29-103. (37) Jorgensen, W. L.; Ibrahim, M. J. Am. Chem. Soc. 1982, 10, 373. (38) Benson, S. W. J. Chem. Phys. 1947, 15, 367.

tained in an optically transparent capillary column in order to facilitate the detection of solutes directly in the high-pressure region of the column. Utilizing this approach, solute retention may be evaluated within isolated regions of the column, wherein the pressure is relatively constant, rather than averaged over the entire column length. In addition, because the extrapolation of retention properties from postcolumn measurements is not necessary, any systematic error in estimating the void time of connecting tubing and unions under varying pressure conditions is avoided. A further advantage of using small-diameter capillary columns is the efficient dissipation of heat that may be generated by viscous flow, thereby minimizing temperature variations within the column. A methanol mobile phase is selected for these studies because its physical properties are well documented as a function of pressure.4-6 In addition, the use of a pure (rather than premixed) mobile phase eliminates any preferential incorporation of mobilephase components into the stationary phase, which may vary with pressure.39 A homologous series of coumarin-labeled fatty acids was chosen as the model solutes for this study. Solute retention is varied in a theoretically predictable manner by altering the chain length in the alkyl portion of the molecule. Interaction of this hydrocarbon moiety with the octadecylsilica stationary phase arises primarily from dispersion (induced-dipole/induced-dipole) forces, the most universal of molecular interactions, making the results of these investigations of general interest and applicability. The coumarin portion of the solute molecule does not appear to contribute to retention due to the low solubility in the nonpolar stationary phase.40 Furthermore, the coumarin allows sensitive fluorescence detection (10-10 M) with a broad linear dynamic range (108).41 Thus, the two portions of these model solutes act in an independent and complementary manner to facilitate predictable retention and sensitive detection within the same molecule. The choice of void marker is also of primary importance, because solute retention must be measured relative to the movement of a nonretained species. This marker must accurately reflect the void time of the column under all pressure conditions. For this study, a fluorescent byproduct of the coumarin derivatization reaction that coelutes with the injection solvent serves as a convenient and reliable choice.42 It has been experimentally confirmed that the change in velocity of this void marker is statistically equal to the change in mobile-phase velocity predicted from the compressibility of methanol.43 Consequently, over the pressure range examined in this study, this void marker remains nonretained (vide infra). As a final consideration in the experimental design, the local pressure within the measurement region must be carefully and independently controlled. To achieve this goal, the absolute pressure at the column inlet is varied in a range consistent with practical operating conditions (1500-5000 psi), while the pressure gradient along the column is maintained at a small, constant value (34 psi/cm). Control of these parameters is accomplished by simultaneously adjusting the length of a splitting capillary at the (39) Berek, D.; Macko, T. Pure Appl. Chem. 1989, 61, 2041. (40) Evans, C. E.; McGuffin, V. L. J. Liq. Chromatogr. 1988, 11, 1907. (41) McGuffin, V. L.; Zare, R. N. Appl. Spectrosc. 1985, 39, 847. (42) Evans, C. E.; McGuffin, V. L. Anal. Chem. 1991, 63, 1393. (43) Evans, C. E. Ph.D. Dissertation, Michigan State University, East Lansing, MI, 1990.

column inlet and a restricting capillary at the column outlet. Thus, the experiment has been designed so that retention may be measured solely as a function of the absolute pressure encountered by the solute zone. EXPERIMENTAL METHODS Reagents. Saturated fatty acids ranging from n-C10 to n-C20 (Sigma Chemical Co., St. Louis, MO) are derivatized with 4-(bromomethyl)-7-methoxycoumarin as described previously.41 Individual fatty acids are isolated and purified using a conventionalscale octadecylsilica column (ODS-224, Applied Biosystems, Inc., San Jose, CA) with a pure methanol mobile phase. The collected fractions are evaporated in a stream of dry nitrogen and redissolved in methanol at a final concentration of 5 × 10-4 M. The polynuclear aromatic hydrocarbons benzo[a]pyrene, tetrabenzonaphthalene, and phenanthro[3,4-c]phenanthrene are obtained from the National Institute of Standards and Technology (Gaithersburg, MD) as Standard Reference Material 869. All organic solvents used in this study are high-purity, distilled-in-glass grade (Burdick and Jackson Division, Baxter Healthcare Corp., Muskegon, MI). Chromatographic System. A schematic representation of the experimental system utilized for this study is shown in Figure 1. A single-piston reciprocating pump (Model 114M, Beckman Instruments, San Ramon, CA) operated in the constant-pressure mode is utilized to deliver the mobile phase. Sample introduction is accomplished by means of a 1.0-µL injection valve (Model ECI4W1, Valco Instruments Co., Houston, TX), and the effluent is subsequently split between the microcolumn and a splitting capillary, resulting in a nominal flow rate of 0.70 µL/min and an injection volume of 11 nL. An open-tubular capillary (50-µm i.d.; 25.7-cm length) is utilized to connect the injector to the microcolumn, which makes possible the placement of detectors in the high-pressure region near the column inlet. The microcolumn is prepared from 0.200-mm-i.d. fused-silica tubing (Hewlett-Packard, Avondale, PA), which is terminated with a quartz wool frit at a length of 43.9 cm. The polyimide coating is carefully removed at 5-cm intervals prior to packing in order to facilitate on-column detection. The packing material is a 3-µm spherical silica with a surface area of 200 m2/g that has been reacted with triethoxyoctadecylsilane at a bonding density of 3.5 ( 0.1 µmol/m2 and capped with chlorotrimethylsilane (MicroPak SP, Varian Associates, Walnut Creek, CA).44 A slurry of this octadecylsilica packing material in methanol is introduced onto the capillary under moderate pressure (5000 psi). The resulting microcolumn has a plate height (H) of 9.5 µm, a total porosity (T) of 0.58, and a flow resistance parameter (φ′) of 550 when evaluated under standard test conditions.45 A 20-µm-i.d. restricting capillary (Polymicro Technologies, Phoenix, AZ) is attached to the column outlet. During the course of the study, the lengths of the restricting and splitting capillaries are simultaneously decreased, so that the mobile-phase linear velocity (0.075 cm/s) and split ratio (1:90) remain constant while the inlet pressure is varied from approximately 5000 to 1500 psi. Detection System. Laser-induced fluorescence is utilized to probe the solute zones at six positions along the microcol-

(44) Simpson, R.; Abbott, S. Varian Instrument Group, Walnut Creek, CA, personal communication, 1988. (45) Gluckman, J. C.; Hirose, A.; McGuffin, V. L.; Novotny, M. Chromatographia 1983, 17, 303.


933

Figure 1. Schematic diagram of on-column fluorescence detection system for capillary liquid chromatography: I, injection valve; T, splitting tee; S, splitting capillary; R, restricting capillary; FOP, fiber-optic positioner; MONO, monochromator; PMT, photomultiplier tube; AMP, amplifier.

umn.42,43,46 The first detector is placed 0.4 cm prior to the packing material to assess the injection profile, while the remaining five detectors are positioned directly on the packed bed at distances of 4.9, 10.4, 15.5, 20.9, and 26.2 cm from the column head. The excitation source is a continuous-wave He-Cd laser (Model 307420M, Omnichrome, Chino, CA) operated at 325 nm with a nominal power of 15 mW. The laser radiation is transmitted to the microcolumn via UV-grade optical fibers (100 µm, Polymicro Technologies), and fluorescence emission is collected in a rightangle geometry using optical fibers of larger diameter (500 µm, Polymicro Technologies). As illustrated in Figure 1, the fluorescence emission at each detector position is transmitted to one of two identical spectrophotometric systems. Within each system, the combined emission is filtered to eliminate the scattered and second-order radiation and is focused onto the entrance slit of a monochromator (Model H1061, Instruments SA, Edison, NJ). The fluorescence emission from the derivatized fatty acids at 420 nm is detected by a photomultiplier tube (Model R1463, Hamamatsu, Middlesex, NJ), and the resulting photocurrent is amplified (100 nA/V, 0.06 s time constant) and converted to the digital domain (Model 2805/5716, Data Translation, Marlborough, MA). An IBM PC-XT computer is used for data acquisition, together with the Forth-based programming language Asyst (Keithley Asyst, Rochester, NY). An acquisition rate of 5 Hz is employed to provide a minimum of 50 points uniformly distributed across the solute zone profile. Data Evaluation and Calculations. To characterize the solute zone, the method of statistical moments is employed because it requires no assumptions concerning the mathematical form of the zone profile. The statistical moments have been calculated by finite summation of the fluorescence intensity as a function of time [I(t)]: (46) Evans, C. E.; McGuffin, V. L. Anal. Chem. 1988, 60, 573.

934


M0 ) M1 )

∑I(t) dt

(13)

∑tI(t) dt/M

0

The zeroth moment (M0) represents the area, while the first moment (M1) represents the centroid of the zone, which is the most accurate measure of the retention time.47 The solute capacity factor (k) within the region between any two detectors is calculated from the difference in the first moment [(∆M1) ) (M1)det2 - (M1)det1], evaluated for both the solute and the void marker:

k)

(∆M1)solute - (∆M1)void (∆M1)void

(14)

It is noteworthy that any change in retention time between the two detectors that arises from variation in the linear velocity due to compression of the mobile phase is exactly proportional for the void marker and the solute. Hence, the on-column measurements of capacity factor are implicitly corrected for the physical manifestations of mobile-phase compressibility. Equations of State. Determination of the relationship between pressure and density is required for the general application of eqs 11 and 12. At the present time, however, no theoretical equation of state accurately describes the complex phases encountered in liquid chromatographic separations. The best choice for liquids appears to be the empirical expression developed by Tait,34,35 given in eq 4. Coefficients for this expression have been determined for the common mobile phases utilized in LC35 but not for the surface-bound stationary phases nor for the many solutes of interest. For this reason, the coefficients for the mobile phase will be considered representative of all components (47) Bidlingmeyer, B. A.; Warren, F. V. Anal. Chem. 1984, 56, 1583A.

Table 1. Estimated Parameters for Mobile Phase, Stationary Phase, and Solutes under Reference State Conditions of 25 °C and 14.5 psi

Mobile-Phase Parameters (Methanol) δm° ) 14.5 cal1/2/cm3/2 Vm° ) 37 cm3/mol F° ) 0.787 g/cm3

Stationary-Phase Parameters (Octadecylsilica) δs° ) 12.5 cal1/2/cm3/2 Vs° ) 403 cm3/mol β° ) 10.3

Solute Parameters (Fatty Acid Derivatives) δi° (cal1/2/cm3/2) carbon no.

Vi° (cm3/mol)

model I

model II

model III

model IV

model V

10 12 14 16 18 20 BaP

439 484 528 573 617 661 265

13.01 12.94 12.88 12.83 12.79 12.76 11.69

12.91 12.84 12.78 12.73 12.69 12.66 11.54

12.91 12.84 12.78 12.73 12.69 12.66 11.54

12.92 12.85 12.79 12.74 12.70 12.67 11.57

12.92 12.85 12.79 12.74 12.70 12.67 11.57

in the chromatographic system. This assumption, although not rigorously correct, is reasonable because the compressibility of these components is expected to be of small and relatively comparable magnitude. For the pure methanol mobile phase utilized in this study, the b and c coefficients are 1210 and 0.148, respectively, when the pressure is expressed in bar and the density in grams per milliliter.35 Thus, if the pressure gradient is assumed to be linear along the column, as verified previously for this mobile phase,43 the local density may be calculated at any distance by means of the Tait equation. Estimation of Parameters. Predictions of the influence of pressure/density on solute retention require knowledge of the many variables in eqs 11 and 12. The specific values for the solute, mobile-phase, and stationary-phase variables utilized in this study are summarized in Table 1. Molar volumes for all components are estimated by the method of LeBas, which is more accurate for condensed phases than that of van der Waals.3 The solubility parameter for the methanol mobile phase is ascertained from wellestablished thermodynamic measurements.29 The parameter for the octadecylsilica stationary phase is obtained from recent chromatographic measurements48 and differs markedly from that for pure octadecane (∼7 cal1/2/cm3/2). Unfortunately, solubility parameters for the model solutes are not available from the literature. Because these solutes comprise a homologous series, however, their parameters are expected to change uniformly with carbon number.49 Once all other constants and variables are known, the solute parameters may be estimated by fitting the retention data at the reference density to eqs 11 and 12, such that relative error is minimized. The magnitude of these parameters, cited in Table 1, is dependent on the form of the theoretical model (vide infra). Although direct confirmation of these values is not possible, the estimated solubility parameters for the coumarin-labeled fatty acids compare favorably with thermodynamic measurements for related molecules such as diphenyl phthalate and dicyclohexyl phthalate (11.33 and 12.24 cal1/2/cm3/2, respectively).22 Finally, the phase ratio (β) is calculated as the ratio of the volume of methanol mobile phase to the volume of octadecyl groups of the stationary phase. This calculation is performed by using the manufacturer’s specifications44 for the (48) Yamamoto, F. M.; Rokushika, S. J. Chromatogr. 1990, 515, 3. (49) Rheineck, A. E.; Lin, K. F. J. Paint Technol. 1968, 40, 611.

Figure 2. Logarithmic dependence of solute capacity factor (k) on carbon number for inlet pressures of 1500 (O), 2400 (0), 3350 (4), 4300 (]), and 4950 psi (3).

density and surface area of the silica packing material, the surface coverage of the monomeric octadecylsilane, and the total porosity of the packed column prepared for this study. RESULTS Experimental evaluation of the pressure/density dependence of the solute capacity factor is accomplished by systematically varying the inlet pressure from 5000 to 1500 psi (337 to 102 bar), while maintaining a small, constant pressure differential (34 psi/ cm; 2.3 bar/cm) along the column. Under these conditions, the variation in pressure along the column length is sufficiently small so as to produce a negligible variation in solute capacity factor with distance.26 Thus, the retention between the first and last on-column detectors (4.9 and 26.2 cm, respectively) is utilized throughout this study, allowing greater precision in the determination of the solute capacity factors ((0.5% rsd). As shown in Figure 2, the model solutes exhibit a logarithmic dependence of capacity factor on carbon number as theoretically expected for a separation mechanism based on dispersion interactions. The ideal retention behavior of this homologous series is observed consistently throughout the pressure regime, with correlation coefficients (R2) for linear regression greater than 0.9995. By extrapolation, the capacity factor for a homologue with Analytical Chemistry, Vol. 69, No. 5, March 1, 1997

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Table 3. Effect of Pressure on Solute Selectivity Factor (r) of Adjacent Solute Pairsa

carbon no.

R 1500 psi

2300 psi

12/10 14/12 16/14 18/16 20/18

1.566 1.543 1.516 1.485 1.472

1.2 0.7 0.0 1.6 2.0

∆R/R (%) 3350 psi 4300 psi 1.4 0.6 0.9 1.0 0.9

1.3 1.4 2.1 2.6 1.1

4950 psi 1.9 2.9 2.4 3.3 2.6

a All experimental data are evaluated between detector 2 (4.9 cm) and detector 6 (26.2 cm).

Figure 3. Capacity factor (k) for n-C14 solute as a function of the average pressure encountered from 4.9 to 26.2 cm along column. Table 2. Effect of Pressure on Solute Capacity Factor (k )a

carbon no.

k 1500 psi

2300 psi

10 12 14 16 18 20

0.52 0.81 1.25 1.90 2.82 4.14

1.4 2.6 3.3 3.3 5.0 7.0

∆k/k (%) 3350 psi 4300 psi 3.3 4.7 5.3 6.2 7.2 8.2

5.2 6.5 8.1 10.3 13.2 14.4

4950 psi 9.3 11.3 14.5 17.3 21.2 24.4

a All experimental data are evaluated between detector 2 (4.9 cm) and detector 6 (26.2 cm).

zero carbon number is 0.065 ( 0.001 under all pressure conditions. This result confirms that the retention of the coumarin moiety itself is small and not significantly influenced by pressure. Further examination of Figure 2 indicates that each of the model solutes exhibits a clear increase in retention with increasing pressure that is significantly greater than the precision of replicate measurements ((0.5% rsd). The systematic, nonlinear nature of this increase in capacity factor with pressure is illustrated more clearly in Figure 3 for n-C14. Quantitative evaluation of the magnitude of the increase is shown in Table 2, where the difference in capacity factor between the lowest pressure (1500 psi) and pressure P is expressed as ∆k/k ) (kP - k1500 psi)/k1500 psi. The increase in retention becomes systematically greater with carbon number, ranging from +9.3% for n-C10 to +24.4% for n-C20 between the lowest and highest pressures. This increase in solute capacity factor leads directly to a corresponding increase in the selectivity factor (R). As shown in Table 3, the increase in the selectivity factor (∆R/R) between the lowest and highest pressures is approximately +2-3% for consecutive solute pairs in the homologous series. The apparent lack of dependence of ∆R/R on carbon number suggests that the observed change in solute retention with pressure arises from an overall increase in the interaction energy and not from a selective change in the separation mechanism of these solutes. DISCUSSION In initial studies, the experimental data were correlated with the unified theoretical approach developed by Martire for gas, supercritical fluid, and liquid chromatography.10,11 In this ap936 Analytical Chemistry, Vol. 69, No. 5, March 1, 1997

proach, Martire employed a statistical thermodynamic approach based on a mean-field lattice model to derive a universal expression for solute retention as a function of the reduced density (F/ Fcrit). Although our experimental data correlated well with the general form of this expression,26 quantitative interpretation was not possible due to the lack of pertinent thermodynamic data, most notably the binary interaction parameters. As a result of this limitation, the theoretical model based on regular solution theory was developed. Although this model is somewhat less rigorous than that of Martire, quantitative interpretation is possible because the unitary interaction parameters are well established in the literature.30 However, the use of unitary interaction parameters imposes a physical picture of the chromatographic system that is greatly simplified and closely circumscribed in its application. The components within the system (solute, mobile phase, stationary phase) must be viewed as discrete, homogeneous liquids with no configurational order. The solubility parameter represents the mean value of the cohesive energy density and is assumed to be invariant throughout each phase. This assumption is clearly an overt simplification for the heterogeneous conditions encountered in liquid chromatographic systems. Although a pure mobile phase has been chosen to minimize the heterogeneity that occurs in mixed solvents,50,51 some short-range order is still likely to occur in methanol due to hydrogen bonding. Similarly, the stationary phase consists of octadecyl groups, silanol and siloxane groups, and any intercalated solvent. To minimize these effects, the model solutes have been chosen to be larger than the dimensional scale of the heterogeneity and to interact solely with the octadecyl groups in the stationary phase.26,42 In addition, because these solutes are similar in structure to the stationary phase, steric exclusion and other entropic effects will also be minimized (vide infra). Consequently, the experimental conditions have been selected for this study to be as consistent as possible with the requirements of the regular solution model, thereby enabling the tractable calculation of solute retention as a function of the density ratio (F/F°). From the regular solution approach, it is apparent that many variables can be considered pressure/density dependent. Because there is no justifiable reason to neglect any of these variables a priori, all combinations have been examined in detail (see Supporting Information). Among the 78 possible models, only four are completely satisfactory in describing the experimental data. All of the successful models utilize the original expression (eq 11), rather than the form that incorporates the Flory-Huggins (50) Karger, B. L.; Snyder, L. R.; Eon, C. Anal. Chem. 1978, 50, 2128. (51) Carr, P. W.; Li, J.; Dallas, A. J.; Eikens, D. I.; Tan, L. C. J. Chromatogr. 1993, 656, 113.

correction (eq 12). Two of these models require that state and simple chemical effects (n ) 1) be considered, whereas the other two require state and extended chemical effects (n ) 3/2). In all models that satisfactorily describe the experimental data, every component in the liquid chromatographic system (solute, mobile phase, stationary phase) must be considered compressible. This result is in clear contrast to gas and supercritical fluid chromatography, where the mobile phase alone is assumed to be compressible.10,11,52,53 For this reason, we will begin our discussion with this commonly assumed model and then compare with the four successful models outlined above.

model I: original expression (eq 11) state and simple chemical effects (eq 9)

Figure 4. Solute capacity factor (k) for all model solutes expressed as a function of the density ratio (F/F°). Solutes: n-C10 (O), n-C12 (0), n-C14 (4), n-C16 (]), n-C18 (3), and n-C20 (+). Experimental regression analysis (- - -) and theoretical analysis (s) for model I.

density-dependent variables δm Evaluation of eq 11 for this case results in the density-dependent expression of the general form:

ln k ) C1 (F/F°)2 + C2 (F/F°) + C3

(15)

where the coefficients are given by

C1 ) (Vi°/RT)(δm°2)

(15-1)

C2 ) (-2Vi°/RT)δm°δi°

(15-2)

C3 ) (Vi°/RT)(2δi° δs° - δs° ) - ln β°

(15-3)

2

In this expression, the C1 coefficient is representative of mobilephase/mobile-phase interactions, which are expected to increase with the square of the density. These interactions effectively serve to decrease the possibility of solute/mobile-phase interactions and, therefore, result in an increase in solute retention (C1 > 0). The C2 coefficient arises from solute/mobile-phase interactions, which increase linearly with density and result in a decrease in solute retention (C2 < 0). The C3 coefficient describes solute/stationaryphase interactions, which are assumed to be independent of density. Thus, this model predicts an overall increase in capacity factor with density in accordance with the familiar quadratic form.10,11,52,53 The experimental retention data, expressed as the logarithm of the capacity factor vs the density ratio (F/F°), are evaluated by means of nonlinear regression to eq 15. The retention data shown in Figure 4 appear to be in good agreement with the general form of this relationship, as indicated by the high correlation coefficients (R2) given in Table 4. However, the coefficients C1-C3 derived from the regression analysis do not agree with the theoretical values determined by direct evaluation of eqs 15-1-15-3, also summarized in Table 4. This poor correlation is clearly evident in Figure 4, where the theoretical curve diverges markedly from the regression line and from the experimental data. The relative error in predicted capacity factor for all solutes is ∼230% across (52) Janssen, H. G.; Snijders, H. M. J.; Rijks, J. A.; Cramers, C. A.; Schoenmakers, P. J. J. High Resolut. Chromatogr. 1991, 14, 438. (53) Janssen, H. G.; Snijders, H.; Cramers, C.; Schoenmakers, P. J. High Resolut. Chromatogr. 1992, 15, 458.

the pressure range examined in this study (Table 9). The predicted increase in capacity factor (∆k/k) between the lowest and highest pressures is +168% for n-C10 to +442% for n-C20, which severely overestimates the experimental measurements (Table 2). The influence of density on solute selectivity (Rij) can, likewise, be predicted based on this model. From eq 15, the selectivity between consecutive homologues i and j is given by

ln Rij ) ln ki - ln kj ) (C1i - C1j)(F/F°)2 + (C2i - C2j)(F/F°) + (C3i - C3j) (16)

where the coefficients are defined in eqs 15-1-15-3. Based on this model and the theoretical values for the coefficients (Table 4), the predicted increase in selectivity (∆R/R) between the lowest and highest pressures is +14-16%, which severely overestimates the measured change of +2-3% for consecutive homologues (Table 3). Thus, the density dependence of both retention and selectivity in liquid chromatography is not accurately described by state and simple chemical effects if the mobile phase alone is assumed to be compressible.

model II: original expression (eq 11) state and simple chemical effects (eq 9) density-dependent variables δi, δm, δs

Evaluation of eq 11 for this case results in the density-dependent expression of the general form

ln k ) C1(F/F°) - ln β°

(17)

where the coefficient is given by Analytical Chemistry, Vol. 69, No. 5, March 1, 1997

937

Table 4. Correlation between Experimental Regression Analysis and Theoretical Analysis for Model I coefficient C1

a

carbon no.

theorya

10 12 14 16 18 20 BaP

154 170 185 201 217 232 93

coefficient C2

coefficient C3

expt

theorya

expt

theorya

expt

R2 b

106 81 129 151 174 175 27

-277 -304 -330 -357 -383 -409 -150

-214 -162 -259 -303 -348 -349 -52

122 133 144 155 166 177 58

107 81 130 153 175 176 27

0.989 0.981 0.979 0.990 0.988 0.972 0.910

Theoretical coefficients calculated from eqs 15-1-15-3. b Correlation coefficient for nonlinear regression of experimental data to eq 15.

Table 5. Correlation between Experimental Regression Analysis and Theoretical Analysis for Model II coefficient C1 carbon no.

theorya

expt

R2 b

10 12 14 16 18 20 BaP

1.732 2.136 2.543 2.951 3.343 3.713 3.474

1.671 2.121 2.556 2.973 3.377 3.768 3.448

0.845 0.872 0.853 0.849 0.837 0.831 0.695

a Theoretical coefficient calculated from eq 17-1. b Correlation coefficient for linear regression of experimental data to eq 17.

Figure 5. Solute capacity factor (k) for all model solutes expressed as a function of the density ratio (F/F°). Solutes: n-C10 (O), n-C12 (0), n-C14 (4), n-C16 (]), n-C18 (3), and n-C20 (+). Experimental regression analysis (- - -) and theoretical analysis (s) for model II.

C1 ) (Vi°/RT)(δm°2 - 2δi°δm° - δs°2 + 2δi° δs°) (17-1)

experimental measurements (Table 3).

model III: original expression (eq 11) state and simple chemical effects (eq 9)

The experimental retention data are evaluated by linear regression analysis of eq 17. As illustrated in Figure 5 and Table 5, the retention data for all solutes appear to be in reasonable agreement with the general form of this relationship. The C1 coefficient derived from the regression analysis is in good agreement with the theoretical value determined from eq 17-1. This agreement results in a relative error in predicted capacity factor for all solutes less than 6.7% across the pressure range examined in this study (Table 9). The predicted increase in capacity factor (∆k/k) between the lowest and highest pressures is +4.6% for n-C10 to +10.1% for n-C20, which underestimates the experimental measurements (Table 2). Further evaluation of this model is accomplished by assessing the influence of density on the solute selectivity. From eq 17, the selectivity between consecutive homologues is given by

ln Rij ) (C1i - C1j)(F/F°)

(18)

where the coefficient C1 is defined in eq 17-1. Based on this model and the theoretical values for the coefficient (Table 5), the predicted increase in selectivity (∆R/R) between the lowest and highest pressures is +1%, which is in good agreement with 938


density-dependent variables δi, δm, δs, β


ln k ) C1(F/F°) - ln (F/F°) - ln β°

(19)

where the coefficient is given by

C1 ) (Vi°/RT)(δm°2 - 2δi° δm° - δs°2 + 2δi° δs°) (19-1)

The experimental retention data are evaluated by nonlinear regression analysis of eq 19. As illustrated in Figure 6 and Table 6, the retention data for all solutes appear to be in reasonable agreement with the general form of this relationship. The C1 coefficients agree well with the theoretically determined values from eq 19-1, resulting in relative error in the predicted capacity factors for all solutes less than 8.7% (Table 9). The predicted increase in capacity factor (∆k/k) between the lowest and highest pressures is +1.9% for n-C10 to +7.3% for n-C20, which underestimates the experimental measurements (Table 2). The dependence of solute selectivity on density is identical to that for model II and is given by eq 18. The predicted increase

Figure 6. Solute capacity factor (k) for all model solutes expressed as a function of the density ratio (F/F°). Solutes: n-C10 (O), n-C12 (0), n-C14 (4), n-C16 (]), n-C18 (3), and n-C20 (+). Experimental regression analysis (- - -) and theoretical analysis (s) for model III. Table 6. Correlation between Experimental Regression Analysis and Theoretical Analysis for Model III coefficient C1 carbon no.

theorya

expt

R2 b

10 12 14 16 18 20 BaP

1.732 2.136 2.543 2.951 3.343 3.713 3.474

1.692 2.142 2.577 2.994 3.398 3.789 3.469

0.605 0.701 0.721 0.737 0.740 0.747 0.890

a Theoretical coefficient calculated from eq 19-1. b Correlation coefficient for nonlinear regression of experimental data to eq 19.

in selectivity for consecutive homologues (+1%) is in good agreement with experimental measurements (Table 3).

model IV:

Figure 7. Solute capacity factor (k) for all model solutes expressed as a function of the density ratio (F/F°). Solutes: n-C10 (O), n-C12 (0), n-C14 (4), n-C16 (]), n-C18 (3), and n-C20 (+). Experimental regression analysis (- - -) and theoretical analysis (s) for model IV.

error of the predicted capacity factor less than 3.1% across the pressure range examined in this study (Table 9). The predicted increase in capacity factor (∆k/k) between the lowest and highest pressures is +9.4% for n-C10 to +21.4% for n-C20, which is in excellent agreement with experimental measurements (Table 2). From eq 20, the selectivity between consecutive homologues is given by

ln Rij ) (C1i - C1j)(F/F°)2

(21)

where the coefficient C1 is defined in eq 20-1. Based on this model, the predicted increase in solute selectivity between the lowest and highest pressures (∆R/R) is +2%, which is in excellent agreement with experimental measurements (Table 3).

model V: original expression (eq 11)

original expression (eq 11)

state and extended chemical effects (eq 10)

state and extended chemical effects (eq 10)

density-dependent variables δi, δm, δs, β

density-dependent variables δi, δm, δs Evaluation of eq 11 for this case results in the density-dependent expression of the general form 2

ln k ) C1(F/F°) - ln β°


ln k ) C1(F/F°)2 - ln (F/F°) - ln β°

(22)

(20) where the coefficient is given by

where the coefficient is given by

C1 ) (Vi°/RT)(δm° - 2δi° δm° - δs° + 2δi° δs°) 2

2

C1 ) (Vi°/RT)(δm°2 - 2δi° δm° - δs°2 + 2δi° δs°) (22-1) (20-1)

The experimental retention data are evaluated by nonlinear regression analysis of eq 20. As shown in Figure 7 and Table 7, all solutes appear to be in very good agreement with the general form of this relationship. In addition, the coefficients derived from regression analysis are in excellent agreement with the values theoretically derived from eq 20-1, resulting in a relative

The experimental retention data are evaluated by nonlinear regression analysis of eq 22. As shown in Figure 8 and Table 8, the retention data are in good agreement with the general form of this relationship. The regression coefficients are in excellent agreement with the values predicted theoretically from eq 22-1, resulting in a relative error in capacity factor less than 4.4% (Table 9). The predicted increase in capacity factor (∆k/k) between the Analytical Chemistry, Vol. 69, No. 5, March 1, 1997

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Table 7. Correlation between Experimental Regression Analysis and Theoretical Analysis for Model IV coefficient C1 carbon no.

theorya

expt

R2 b

10 12 14 16 18 20 BaP

1.703 2.104 2.507 2.912 3.301 3.669 3.421

1.636 2.076 2.502 2.910 3.305 3.688 3.375

0.961 0.962 0.957 0.973 0.972 0.959 1.000


Figure 8. Solute capacity factor (k) for all model solutes expressed as a function of the density ratio (F/F°). Solutes: n-C10 (O), n-C12 (0), n-C14 (4), n-C16 (]), n-C18 (3), and n-C20 (+). Experimental regression analysis (- - -) and theoretical analysis (s) for model V. Table 8. Correlation between Experimental Regression Analysis and Theoretical Analysis for Model V coefficient C1 carbon no. 10 12 14 16 18 20 BaP

theorya

expt

R2 b

1.703 2.104 2.507 2.912 3.301 3.669 3.421

1.656 2.097 2.523 2.931 3.326 3.709 3.395

0.933 0.963 0.954 0.964 0.960 0.952 1.000


lowest and highest pressures is +6.7% for n-C10 to +18.3% for n-C20, which is in good agreement with experimental measurements (Table 2). The dependence of solute selectivity on density is identical to that for model IV and is given by eq 21. The predicted increase in selectivity (+2%) is in excellent agreement with experimental measurements (Table 3). Comparison of Theoretical Models. It is immediately apparent that compression of the mobile phase alone, as commonly assumed for gas and supercritical fluid chromatography,10,11,52,53 cannot account for the observed changes in solute retention for liquid chromatography. This model severely overestimates the effect of density because the polarity of the mobile 940 Analytical Chemistry, Vol. 69, No. 5, March 1, 1997

phase increases proportionately as it is compressed. Because this increase is not compensated by any other density-dependent factors, the quantity (δi - δm)2 in eq 2 increases unrestrained, which leads to a large increase in the predicted capacity factor. The results from models II-V indicate that every component (solute, mobile phase, stationary phase) must be considered compressible in order to represent the experimental data accurately. This conclusion, while clearly novel and important, is not too surprising for liquid chromatography, where the compressibility of all components is likely to be of comparable magnitude. Because solute/mobile-phase (δi - δm)2 and solute/ stationary-phase (δi - δs)2 interactions are carefully balanced in eq 2, any perturbation in this balance caused by density is internally compensated and has a smaller effect on solute retention. The resulting solubility parameter equation that considers enthalpy alone (eq 11) shows very good correlation with experimental data, whereas that with Flory-Huggins correction for configurational entropy (eq 12) does not. By inspection of the models that employ the former equation, it is apparent that the change in enthalpic interactions with density cannot be adequately described by a state effect alone but must also include chemical effects. The simple treatment of chemical effects by means of the van der Waals equation (eq 9) results in a systematic underestimation of the effect of density. The extended treatment (eq 10), which allows for a uniform change in the radial distribution function with density, shows better statistical correlation and lesser error (Tables 7-9). It is also noteworthy that models III and V, in which the phase ratio is considered to be density dependent, show slightly poorer correlation and slightly greater error than models II and IV, respectively (Tables 7-9). On the basis of the experimental data herein, however, models II-V are not statistically distinguishable and must be considered equally viable. Extension of Theoretical Model to Planar and Nonplanar Solutes. The theoretical model describing the influence of pressure on retention has been developed by utilizing solutes with a flexible linear (one-dimensional) structure, much like the stationary phase itself. Further understanding of these effects may be gained by extending the model to rigid planar (two-dimensional) and nonplanar (three-dimensional) structures. For this purpose, a mixture of three polynuclear aromatic hydrocarbons, benzo[a]pyrene (BaP), tetrabenzonaphthalene (TBN), and phenanthro[3,4-c]phenanthrene (PhPh), is employed. Retention measurements for these polynuclear aromatic hydrocarbons are performed concurrently with those for the fatty acids to facilitate direct comparison of the results. The BaP molecule has 20 carbon atoms arranged in a fivering planar structure, whereas TBN and PhPh have 26 carbon atoms in six-ring structures that are twisted and helical, respectively. This combination of solute structures has been used to advantage in the evaluation of stationary-phase selectivity in liquid chromatography.54,55 For silica packing materials that contain low surface coverage of the octadecylsilane, solute retention is controlled primarily by cohesive interactions. Under these conditions, the elution order appears to be based on the accessible surface area of the solute molecule (BaP e PhPh < TBN). For higher surface coverage, however, steric exclusion may prevent the full penetration of nonplanar solutes into the stationary phase (54) Sander, L. C.; Wise, S. A. Anal. Chem. 1987, 59, 2309. (55) Sander, L. C.; Wise, S. A. J. High Resolut. Chromatogr. 1988, 11, 383.

Table 9. Comparison of Relative Error in Predicting Solute Capacity Factor (k) as a Function of Density for Theoretical Models relative error in capacity factor (%) Pinlet (psi)

F/F°

model I

model II

model III

model IV

model V

1500 2300 3350 4300 4950

1.0079 1.0150 1.0222 1.0291 1.0337

2.1 34.2 90.8 169.1 229.7

2.2 2.9 2.5 3.8 6.7

1.9 3.4 3.4 5.1 8.7

2.1 2.4 2.7 3.1 3.0

2.3 2.8 1.7 2.2 4.4

(PhPh < TBN e BaP). An analogous situation is expected in this study, where compression of the stationary phase has already been shown to influence cohesive interactions. Whether or not this compression also influences the selectivity by steric exclusion remains to be determined. The PAH test mixture appears to be ideally suited to probe such changes in the stationary-phase density and structure with local pressure. At all inlet pressures examined in this study (1500-5000 psi), the elution order for the test mixture was PhPh , BaP < TBN. This order suggests that the stationary phase has an intermediate surface coverage,54 which is consistent with the manufacturer’s specifications.44 The logarithm of the capacity factor for the polynuclear aromatic hydrocarbons is shown as a function of the density ratio F/F° in Figure 9. For the planar BaP molecule, the general form of these retention data is similar to that observed for the linear solutes and may be evaluated by using the theoretical models described above. The retention data are analyzed by nonlinear regression to eqs 15, 17, 19, 20, and 22 with reasonable results, as indicated by the correlation coefficients in Tables 4-8. The experimentally and theoretically derived coefficients from these equations are in good agreement for models II-V. The relative error in predicted capacity factor for BaP is comparable to that shown in Table 9 for the linear solutes; however, the models utilizing the simple chemical effect show slightly lesser error (4.1 and 1.8% for models II and III, respectively) than those with the extended chemical effect (10.6 and 7.0% for models IV and V, respectively). Consequently, these theoretical models appear to be successful in describing the influence of pressure on the retention behavior of this rigid, planar molecule. In contrast, the retention of the nonplanar TBN and PhPh solutes is markedly different from that of BaP (Figure 9). Because the capacity factor remains relatively constant with density, cohesive interactions alone are clearly not sufficient to describe the retention behavior for these solutes. The enthalpic effects described by the theoretical models above appear to be compensated by an entropic effect arising from the increased order of the octadecylsilica stationary phase with increasing density. Thus, steric exclusion becomes important for these three-dimensional solute molecules. Further insight can be gained from the selectivity factors (R) for the planar and nonplanar solutes. On the basis of the results obtained for the linear solutes (Table 3), we would expect an increase in selectivity of +2-3% for each additional C2H4 group due solely to the increase in cohesive energy density. Thus, the selectivity factors for TBN/BaP and PhPh/BaP would be predicted to increase by approximately +6-9% between the lowest and highest pressures. As shown in Figure 10, however, the selectivity factors actually decrease by -7 and -5% for TBN/BaP and PhPh/ BaP, respectively. The selectivity factor for TBN/PhPh is

Figure 9. Solute capacity factor (k) for the test mixture of polynuclear aromatic hydrocarbons expressed as a function of the density ratio (F/F°). Solutes: BaP, TBN, and PhPh. The discontinuity at F/F° ) 1.015 is highly reproducible and may be due to a phase transition of the octadecylsilica.58,59

expected to remain relatively constant because these solutes have the same carbon number; however, a small but statistically significant decrease (-1%) is experimentally observed. These preliminary results suggest that the entropic effect is significant for these nonplanar polynuclear aromatic hydrocarbons within the Analytical Chemistry, Vol. 69, No. 5, March 1, 1997

941

Figure 10. Solute selectivity factor (R) for the test mixture of polynuclear aromatic hydrocarbons expressed as a function of the density ratio (F/F°). Solutes: BaP, TBN, and PhPh. The discontinuity at F/F° ) 1.015 is highly reproducible and may be due to a phase transition of the octadecylsilica.58,59

pressure range examined here. These results are directly complementary to the recent experimental studies by Sander and Wise,54,55 Sentell and Dorsey,56 and Buszewski and co-workers57 on the influence of octadecylsilane surface coverage and temperature on solute capacity factor. Additional studies, particularly those where the interrelated variables of temperature, pressure, and surface coverage are examined in a comprehensive and systematic manner, are clearly needed to understand these complex phenomena more completely. IMPLICATIONS The significant increase in solute retention demonstrated in this investigation has clear implications for fundamental investiga(56) Sentell, K. B.; Dorsey, J. G. Anal. Chem. 1989, 61, 930. (57) Buszewski, B.; Suprynowicz, Z.; Staszczuk, P.; Albert, K.; Pfleiderer, B.; Bayer, E. J. Chromatogr. 1990, 499, 305. (58) McGuffin, V. L.; Evans, C. E.; Chen, S. H. J. Microcolumn Sep. 1993, 5, 3. (59) McGuffin, V. L.; Chen, S. H. J. Chromatogr., in press.

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tions as well as for the everyday practice of chromatography. Pressure has previously been considered an unimportant parameter in liquid chromatography, and therefore, many presumptions about solute retention must be reexamined. The first, and perhaps most important, consideration is that solute retention is not constant along the column length. The variation in capacity factor with distance becomes significant even for relatively modest inlet pressures (>2000 psi). A second consideration is that certain variables that are used to control and optimize separations also directly influence the pressure, so that the resulting changes in solute retention may be greater or less than expected from that variable alone. For example, an isocratic change in mobile-phase composition from 100% methanol to 95% methanol/water will cause an increase in pressure of ∼50%. For the ideal solutes considered here, the variation in capacity factor due to this pressure increase is ∼5-10% of the total increase in retention. Similarly, an isothermal change in temperature from 30 to 40 °C will cause a decrease in pressure of ∼10%. For these solutes, the variation in capacity factor due to this pressure decrease is ∼4-7% of the total decrease in retention. It is apparent that the convolution of pressure with all other parameters must be considered for both fundamental and practical studies. A final consideration is that pressure effects, now recognized and understood, may be utilized to advantage for the optimization of difficult separations. Because the most universal of chemical interactions were examined in this study, the results are expected to have general applicability to a wide variety of separations. Although the discussion has been limited to reversed-phase separations, this theoretical model may be applied to normal-phase separations as well. However, the solubility parameter approach presumes that the mobile and stationary phases behave as regular solutions, so that this model is only properly applied to separations that are based on a partition mechanism. In normal-phase separations, the stationary phase is more polar than either the solute or the mobile phase (δs > δi > δm). Because the polarity of each component increases proportionately with pressure, the mobile-phase polarity becomes more similar to that of the solute whereas the stationary-phase polarity becomes more disparate. From a mathematical perspective, (δi - δs)2 increases to a greater extent with pressure than does (δi - δm)2. Consequently, solute retention is expected to decrease with increasing pressure according to eq 2, which is consistent with the experimental results of Katz et al.20 The mobile-phase solvents utilized in normal-phase separations are highly compressible, which may lead to an even greater pressure dependence of solute capacity factor than under reversed-phase conditions. However, the lower viscosity of these solvents and the resulting lower inlet pressures may compensate somewhat for this effect. In any case, the magnitude of the influence of pressure on capacity factor is difficult to predict a priori and requires further investigation for normalphase separations. CONCLUSIONS The capacity factor of model solutes that are retained by dispersion forces, the weakest but most universal of molecular interactions, is shown to increase markedly as a function of the local pressure. These experimental results correlate very well with a comprehensive thermodynamic model based on regular solution theory. From this model, it becomes apparent that all components in the liquid chromatographic system must be considered compressible. The resulting change in enthalpic

interactions is the predominant effect for linear and planar solutes but cannot fully account for the retention behavior of nonplanar solutes. Because the structure of the stationary phase may also be a function of the local pressure, entropic effects may become important for these nonplanar solutes. This hypothesis has interesting implications for the prediction and control of separation processes. Although this study has provided much insight, further systematic investigations will be necessary to elucidate fully the influence of pressure on solute retention in liquid chromatography. ACKNOWLEDGMENT The authors are grateful to Dr. Christine E. Evans (University of Michigan) for assistance with the preliminary experiments, to Dr. George Yefchak (Hewlett-Packard) for assistance with the nonlinear regression analysis, to Dr. Stephen A. Wise (National Institute of Standards and Technology) for providing the poly-

nuclear aromatic hydrocarbon standards, and to Dr. William F. Hug (Omnichrome) for continuing support of the heliumcadmium laser system. This research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences under Contract DE-FG02-89ER14056. SUPPORTING INFORMATION AVAILABLE A table summarizing the 78 possible models arising from eqs 11 and 12, together with their general mathematical expressions and assessment of their viability (3 pages). Ordering information is given on any current masthead page. Received for review June 13, 1996. Accepted December 20, 1996.X AC960589D X

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


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Theoretical and Experimental Studies of the Effect of Pressure on

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