Invisible Analyte Peak Deformations in Single-Component Liquid

peak consists of displaced plateau molecules; the injected molecules (mass ... in such systems,1 the prerequisite being that at least one eluent addit...
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Anal. Chem. 2006, 78, 2765-2771

Invisible Analyte Peak Deformations in Single-Component Liquid Chromatography Jo 1 rgen Samuelsson, Robert Arnell, and Torgny Fornstedt*

Department of Physical and Analytical Chemistry, Uppsala University, BMC Box 577, SE-751 23, Uppsala, Sweden

It is well known that if a small excess of solute is injected into a chromatographic system equilibrated with an eluent containing the same solute, a single so-called perturbation peak will appear in the chromatogram. It was recently shown (Samuelsson, J.; Forsse´ n, P.; Stefansson, M.; Fornstedt, T. Anal. Chem. 2004, 76, 953-958) that this peak consists of displaced plateau molecules; the injected molecules (mass peak) elute later, together with a deficiency of plateau molecules and are therefore not detected. In this article, we investigated what happens if a large rather than a small excess of solute molecules is injected. To study this systematically, the experimental method involved an enantiomer pair in an achiral separation system. It was found that the invisible mass peak was extremely deformed and that its shape depended on the amount of excess injected, the eluent concentration, and the column length. Depending on these operational conditions, the mass peak changed from a classical Langmuirian (tailing) to an anti-Langmuirian (leading) shape, with deformed shapes observable in the transition. The visible, overloaded perturbation peak was always Langmuirian, regardless of the mass peak shape. Chromatography is often performed with more or less complex eluents to achieve suitable retention times and resolution. It has been demonstrated that extreme peak deformations might happen in such systems,1 the prerequisite being that at least one eluent additive should adsorb strongly to the stationary phase.2,3 Injection of the analyte dissolved in the eluent generally perturbs the equilibrium of the additive; the peak generated is called the “system peak” by the analytical community4-6 and the “perturbation peak” by the engineering community.7-10 The retention times * To whom correspondence should be addressed. E-mail: torgny.fornstedt@ ytbioteknik.uu.se. (1) Golshan-Shirazi, S.; Guiochon, G. Anal. Chem. 1986, 61, 2380-2388. (2) Fornstedt, T.; Guiochon, G. Anal. Chem. 1994, 66, 2116-2128. (3) Fornstedt, T.; Guiochon, G. Anal. Chem. 1994, 66, 2686-2693. (4) Arvidsson, E.; Crommen, J.; Schill, G.; Westerlund, D. J. Chromatogr. 1989, 461, 429-441. (5) Sokolowski, A.; Fornstedt, T.; Westerlund, D. J. Liq. Chromatogr. 1987, 10, 1629-1662. (6) Golshan-Shirazi, S.; Guiochon, G. Anal. Chem. 1990, 62, 923-932. (7) Blu ¨ mel, C.; Hugo, P.; Seidel-Morgenstern, A. J. Chromatogr., A 1999, 865, 51-71. (8) Heuer, C.; Ku ¨ sters, E.; Plattner, T.; Seidel-Morgenstern, A. J. Chromatogr., A 1998, 827, 175-191. (9) Forsse´n, P.; Lindholm, J.; Fornstedt, T. J. Chromatogr., A 2003, 991, 3145. (10) Lindholm, J.; Forsse´n, P.; Fornstedt, T. Anal. Chem. 2004, 76, 4856-4865. 10.1021/ac0522308 CCC: $33.50 Published on Web 03/10/2006

© 2006 American Chemical Society

and areas of small analytical perturbation peaks are important since they hold information that can be used for adsorption isotherm determination11 and indirect detection.12 Large perturbation peaks are used when the inverse method is applied on one or two plateaus for increased accuracy at adsorption isotherm determination.13 Unfortunately, the perturbation peak may deform the coeluting solute peaks, causing them to assume irregular forms. In this work, we will refer to all deviations from classical Gaussian, Langmuirian (steep front, diffuse rear), and antiLangmuirian (diffuse front, steep rear) shapes as “deformed peaks”. In ion-pair chromatography, such a deformation-inducing additive can be the ion-pairing agent,4,5 and in normal-phase chromatography (polar stationary phase combined with nonpolar eluent), it can be the polar additive.1 However, the organic modifiers used in reversed-phase systems (nonpolar stationary phase combined with polar eluent) do not adsorb strongly enough to produce such effects.3 It was recently demonstrated that this effect can also occur in a modern system intended for chiral separation.14 This is because modern chiral systems are often normal-phase systems using a mobile phase in which hexane is the main solvent and a polar component is the additive causing the effect. The deformations are especially extreme with preparative injections, since large solute injections create large perturbation peaks. These effects were systematically investigated in one study in which computer simulations were combined with experiments, and the additive signal was also studied.2,3 The prerequisite for the deformation was that the additive should be more strongly retained than the solute is in the pure, main (weak) eluent. The peak shape of the injected solute also depends on the relative retention, R, between the perturbation and the solute peak. The overloaded solute peak shape changes from a Langmuirian shape (tailing) at values of R greater than unity to an anti-Langmuirian shape (leading) at values of R less than unity; at values of R close to unity, round peak deformations may arise. In the above twocomponent1-3 and three-component cases,15 the solute deformations were always visible. It has long been believed that peak deformations can only occur in multicomponent systems. In this work, however, we will (11) Tondeur, D.; Kabir, H.; Lou, L. A.; Granger. J. Chem. Eng. Sci. 1996, 51, 3781-3799. (12) Arvidsson, E.; Crommen, J.; Schill, G.; Westerlund, D. J. Chromatogr. 1989, 461, 429-41. (13) Arnell, R.; Forsse´n, P.; Fornstedt, T. J. Chromatogr., A 2005, 1099, 167174. (14) Lindholm, J.; Fornstedt, T. J. Chromatogr., A 2005, 1095, 50-59. (15) Golshan-Shirazi, S.; Guiochon, G. Anal. Chem. 1989, 61, 2373-2380.

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show that deformed profiles may be found also in the singlecomponent case, where the additive and analyte have identical adsorption properties. In such situations, the injected molecules push the additive (plateau) molecules in front, like a fast-moving wave crest giving rise to a visible perturbation peak. The injected molecules are left behind, filling up the wave trough (negative plateau perturbation), and will elute later (mass peak), though invisible to a standard detector. We will show that the mass peak can be deformed, whereas the perturbation peak is always normal in shape. Such deformations can occur in any chromatographic system displaying a convex adsorption isotherm (i.e., the most common Langmuirian type). In fact, the effects are not restricted to adsorption systems, but can occur in any system with any partition function, as long as the partitioning displays the slightest degree of convexity. Deformations may also arise with concave (i.e., anti-Langmuirian type) and S-shaped adsorption isotherms, but how these deformations appear is beyond the scope of this study. These deformed peaks have never been reported as they are invisible to a standard detector. In this work, the enantiomer tracer-pulse method was used to detect the mass peaks,16 a method in which one enantiomer is used as the additive and the other is the injected solute. Provided an achiral column is used, the enantiomers will display identical adsorption isotherms and the system can be considered a one-component system. By analyzing the chiral composition at the column outlet, one can distinguish the sample peak from the plateau perturbations. This method was recently developed and has been used to prove that, for analytical injections the sample molecules are not always in the detected peak,16 in accordance with a previous prediction.17 By measuring the retention time of the mass peak (injected molecules) at different plateau concentrations, the adsorption isotherm can be determined, without having to make prior assumptions regarding the adsorption model. This study systematically examines these invisible deformations, enabling the introduction of a new injection strategy having the potential to produce more accurate adsorption isotherms than previously possible. THEORY Computer simulations are necessary for the theoretical study of chromatographic peak deformations, since the equations involved cannot be solved analytically. Assuming rapid mass transfer and reasonably high column efficiency, the migration of solutes through a chromatography column can be described by the equilibrium-dispersive model:18

∂qi(C) ∂Ci ∂2Ci ∂Ci +F +u )D 2 ∂t ∂t ∂x ∂x

0 e x e L, t g 0 (1)

C is the mobile phase concentration vector, at different positions and times (x, t). In this work, C ) (C1, C2), since we are considering the adsorption of two enantiomers. F is the volumetric (16) Samuelsson, J.; Forsse´n, P.; Stefansson, M.; Fornstedt, T. Anal. Chem. 2004, 76, 953-958. (17) Helfferich, F.; Peterson, D. L. Science 1963, 142, 661-662. (18) Guiochon, G.; Shirazi, S. G.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: New York, 1994.

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Figure 1. Schematic representation of the relationship between the retention times of analytical perturbation and mass peaks (top) and the adsorption isotherm (bottom). For an explanation of the relationships and symbols, see the Theory section.

ratio between the stationary and mobile phases, u is the linear velocity of the eluent, L is the column length, and D is the lumped axial dispersion coefficient proportional to N, the number of theoretical plates. In eq 1, qi(C) is described by an adsorption isotherm model, which defines the equilibrium between the adsorbed and desorbed states of the two components. In competitive systems, the stationary phase concentration of a component is given by the mobile phase concentration of all components, since the molecules compete for the same binding sites. There are plenty of adsorption isotherm models of varying degrees of complexity.18 One common model is the Langmuir isotherm, which will be used in this work

qi(C) )

aiCi 1 + b1C1 + b2C2

i ) 1,2

(2)

Here qi and Ci are the stationary and mobile phase concentrations of component i, ai is a dimensionless constant, and bi is the association equilibrium constant. The model assumes ideal solutions and monolayer adsorption. Equation 2 can also be expressed as fractional surface coverage, θ ) bC/(1+ bC). When it is very low (θ , 1%), the adsorbed concentration is proportional to the mobile phase concentration. Higher values of θ correspond to the nonlinear part of the adsorption isotherm.18 In this study, the two enantiomers have identical adsorption properties, since an achiral stationary phase is used. Consequently, both components have identical adsorption isotherm parameters, a and b. The numerical method will treat this as a two-component system, although the column cannot distinguish between the two enantiomers. When a column is equilibrated with an eluent containing a solute and then a sample containing a slightly higher concentration of the same component is injected, one visible perturbation peak will be detected.16 Although there is only one component in the system, two peaks elute, as shown in Figure 1. The visible perturbation peak (solid line) will elute at the following retention time:

(

dq tR,1 ) t0 1 + F dC

| ) C)C0

(3)

where t0 is the column holdup time and C0 is the solute concentration in the eluent. It has previously been shown16 that the injected molecules are not present in this peak; instead they are eluted as a separate peak, the mass peak (dash-dotted line), at

(

)

q(C0) tR,2 ) t0 1 + F C0

(4)

For Langmuirian-type adsorption isotherms (i.e., convex), the mass peak will elute after the perturbation peak. The mass peak is canceled by a plateau vacancy peak (dotted line), so it is invisible when using a normal detection principle. The perturbation peak and the negative vacancy peak are referred to as the “plateau perturbation”. The equations show that the retention time of the faster propagating perturbation peak (eq 3) is proportional to the tangential slope of the adsorption isotherm, while the slower mass peak retention time (eq 4) is proportional to the chord of the adsorption isotherm. This is shown graphically in the bottom portion of Figure 1. In the case of anti-Langmuirian (i.e., concave) adsorption isotherms, the elution order between the analytical mass and perturbation peaks will be reversed. This is because the tangent is then steeper than the chord. The linear relationship described by eqs 3 and 4 is only valid if the perturbation is very small, i.e., the sample excess and sample volume are both small. For larger perturbation injections, it is well known that the perturbation peak will display characteristic Langmuirian tailing.19 This visible peak will propagate faster than the injected molecules do, analogous to the situation depicted in Figure 1. The front of the perturbation peak will propagate at a velocity given by the chord of the adsorption isotherm between the actual plateau concentration and the maximum peak concentration. Assuming the Langmuir adsorption isotherm, the perturbation front will move at linear velocity uf:

uf )

u ∆q 1+F ∆C

u

) 1+

Fa (1 + bC0)(1 + bCf)

C0 e Cf e Cinj (5)

where Cf is the concentration maximum of the front. Initially, Cf is equal to Cinj, the injected concentration, but it decreases during the passage through the column. Consequently, the velocity of the perturbation front will also decrease. The concentrations in the diffuse rear of the peak will propagate at linear velocity uc:

uc )

u dq 1+F dC

u

) 1+

Fa (1 + bC)2

C0 e C e Cinj

(6)

The velocities of the concentrations and fronts of large perturbations have previously been studied and described in detail.19 What happens to the injected molecules in the case of large perturbations has not yet been studied. All solute molecules will travel at (19) Zhong, G.; Fornstedt, T.; Guiochon, G. J. Chromatogr., A 1996, 734, 6374.

Figure 2. Association of each concentration on the diffuse rear of the overloaded perturbation profile (solid line) with two linear velocities: the concentration velocity, uc (dashed line), and the solute velocity, um (dash-dotted line). Because of this velocity difference, injected solutes will elute later than the perturbation peak. For details, see the Theory section.

the linear velocity um:

um )

u u ) Fa q(C) 1+ 1+F 1 + bC C

C0 e C e Cinj

(7)

as dictated by the local concentration, C. The velocity of the injected molecules will obviously vary inside the column. Solutes in the perturbation zone (C0 < C e Cf) will travel faster than those that have been left behind in the concentration plateau (C ) C0). Furthermore, the characteristic velocity of a concentration in the perturbation zone will be higher than the corresponding solute velocity. The velocity distribution of the solute molecules and concentrations is shown in Figure 2. Assuming that the Langmuir adsorption isotherm applies, uc/um will always be larger than unity for C > 0. The extreme point of a function is where its derivative is equal to zero. By solving

d(uc/um)/dC ) 0

(8)

one can show that the ratio will reach a maximum at

C ) x(Fa + 1)/b2

(9)

Both perturbation peak and mass peak elution profiles can be calculated by numerically solving eq 1 with appropriate boundary and initial conditions. EXPERIMENTAL SECTION The details of the experiments have been described previously, so here we present only a brief summary. As solutes, (+)-methyl L-mandelate (LM) and (-)-methyl D-mandelate (DM) were used. Two different LC systems were used, an achiral and a chiral system. The achiral system was a 150 × 4.6 mm, 5-µm Kromasil C18 column (Eka Chemicals, Bohus, Sweden) with a 35% (v/v) methanol in water eluent. Methanol was chosen as the organic modifier since it competes very little for the stationary phase surface.20 The adsorption isotherm of methyl mandelate in the achiral system was determined using frontal analysis in the (20) Gritti, F.; Guiochon, G. Anal. Chem. 2005, 77, 4257-4272.

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Figure 3. Comparison of experimental (symbols) and simulated (solid lines) large perturbation peaks, mass peaks (dashed-dotted lines), and plateau perturbations (dotted lines). Circles and squares represent the experimental mass peak and plateau perturbation, respectively. The concentration plateaus and samples were performed as follows: (a) plateau 5 mM DM, sample 5 mM DM + 50 mM LM, (b) plateau 10 mM DM, sample 10 mM DM + 500 mM LM, (c) plateau 15 mM DM, sample 15 mM DM + 750 mM LM, and (d) plateau 35 mM DM, sample 35 mM DM + 500 mM LM. For other conditions, see the Experimental Section.

staircase mode, and the fitted Langmuirian parameters were a ) 7.04 and b ) 5.43 M-1. The phase ratio (F) of the column was 0.770, and the column temperature was 27 ((0.1) °C. The chiral system was a 100 × 4.6 mm, 5-µm CHIRAL-AGP column (ChromTech AB, Ha¨gersten, Sweden) with which an aqueous acetate buffer, pH 6.0, containing 0.25% (v/v) 2-propanol was used as the eluent. The flow was set to 0.70 mL min-1 for both systems. The UV detector was set to 274.5 nm for the achiral and 260 nm for the chiral system. The achiral column was equilibrated with an eluent containing DM, establishing a concentration plateau. Injection samples consisted of LM added to the eluent (containing DM). We will refer to this as an injected LM excess. The sample volume was 50 µL. Fractions were taken every 6 s at the column outlet; the fractions were injected into the chiral system, and the molar fractions of the two enantiomers were calculated using the peak areas of the chromatograms. The molar fractions were used to calculate the individual elution concentrations of DM and LM in the achiral chromatogram. The LM profile is the mass peak and the DM profile is the plateau perturbation; the sum of the contributions indicates the perturbation peak. The simulations were performed using the equilibrium-dispersive model. RESULTS AND DISCUSSION In a previous study,16 small-sized excesses were injected at different concentration plateaus, and both the visible perturbation peaks and the invisible mass peaks were Gaussian. The present study, however, investigates the effects of large-sized excess injections. Mass Peak Deformations as a Function of the Plateau Concentration. Experimental Verification. The achiral C18 column was equilibrated with an eluent containing DM to establish a concentration plateau, and a large excess of LM was injected. The perturbation peak was recorded by the UV detector, while chiral analysis had to be performed to detect the mass peak. Figure 3 2768 Analytical Chemistry, Vol. 78, No. 8, April 15, 2006

Figure 4. Simulations of large perturbation and mass peaks at increasing plateau concentration levels: (a) 5, (b) 15, (c) 25, and (d) 100 mM DM. The dashed-dotted line represents the UV signal showing the perturbation peak and the solid line represents the mass peak. Excess injections (50 µL) of 50, 100, 250, 500, and 750 mM LM were made on all concentration plateaus.

presents experimental and simulated elution profiles, the lines representing the simulated and the symbols the experimental results. It can be seen that the perturbation peaks are composed of both plateau molecules and injected molecules, the composition depending on the degree of resolution. It is also clear that the shape of the mass peak varies with the resolution. In Figure 3a, an excess of 50 mM LM is injected at a concentration plateau of 5 mM DM, corresponding to near-linear conditions (θ ) 2.6%), and the mass peak displays a slightly Langmuirian shape. The mass peak has a combined elution with almost the whole perturbation peak. In Figure 3b, an excess of 500 mM LM is injected at the somewhat higher concentration plateau of 10 mM DM, corresponding to weakly nonlinear conditions (θ ) 5.1%); here the mass peak is deformed, with a small hump at its rear. In Figure 3c, an excess of 750 mM LM is injected at a concentration plateau of 25 mM DM, which corresponds to a moderately nonlinear case (θ ) 12%); the mass peak is much more rectangular than in (b), and the hump at the rear is raised even more. In Figure 3b and c, the mass and perturbation peaks are still not completely resolved. In Figure 3b, the deformed peak is more Langmuirian, and as the plateau concentration increases (Figure 3c) the mass peak starts to transform in shape to that of an antiLangmuirian peak. In Figure 3d, the concentration plateau is even higher and has increased to 35 mM DM, corresponding to nonlinear conditions (θ ) 16%), with an excess injection of 500 mM LM. Now the mass and perturbation peaks are totally resolved, and the mass peak has assumed a narrow, antiLangmuirian shape. It is clear that the resolution between mass and perturbation peaks increase at high plateau concentrations. Systematic Simulations. The simulations and experimental profiles in Figure 3 are nearly identical, indicating a very good model agreement. Simulation is a more convenient tool for the systematic study of mass peak deformations, so it was used extensively in this work. Figure 4 shows simulated peak profiles originating from large excess injections at four different concentration plateau levels: 5, 15, 25, and 100 mM DM. The following five excesses were injected at each plateau level: 50, 100, 250, 500, and 750 mM LM. Note that the perturbation peak (dasheddotted line) is always an ordinary overloaded Langmuirian-shaped

peak, while the mass peak (solid line) more or less deviates from the classical peak shapes. The plateau perturbation contribution has been excluded for clarity. The 5 mM concentration plateau (Figure 4a) corresponds to near-linear conditions (θ ) 2.6%), and here the mass peak shapes deviate only slightly in shape from the corresponding perturbation peak shapes. The mass peaks are slightly more retained than the perturbation peaks. The shape of the mass peak is still similar to an overloaded Langmuirian profile but is more rounded. This is because at low plateau concentrations (i.e., near-linear conditions) only a small amount of the concentration plateau can contribute to the perturbation peak. In other words, the major parts of the perturbation peaks have a combined elution with the corresponding mass peaks. At even lower plateau concentrations, the difference in retention times between the perturbation and mass peaks decreases further, and at a truly linear plateau, the two peaks coelute completely. At a concentration plateau of 15 mM (Figure 4b), corresponding to weakly nonlinear conditions (θ ) 7.5%), the mass peaks start to show deformations. Here, major parts of the mass peak and the perturbation peak are separated from each other, though the peaks are not resolved from each other. As the excess increases, the contribution of injected (mass) molecules captured inside the perturbation zone increases. At high injection excesses, the fraction of the mass peak that is still in the perturbation zone displays Langmuirian characteristics, while the fraction that elutes after the perturbation starts to exhibit anti-Langmuirian signs. At low injection excesses, such as 50 and 100 mM, the mass and perturbation peaks are almost completely separated, so the mass peak looks mainly anti-Langmuirian. At 250 mM and above injection excesses, the mass peaks display pronounced deformations, as the resolved fraction strives for an anti-Langmuirian shape while the nonresolved fraction strives to keep the Langmuirian shape. At the 25 mM concentration plateau (Figure 4c), corresponding to moderately nonlinear conditions (θ ) 12%), the deformations of the mass peak are transformed into a more pronounced antiLangmuirian shape with low excesses injected and a more rectangular shape with high excesses injected. With 50-500 mM excesses injected, the peak is strictly anti-Langmuirian. When injecting a 750 mM excess, the rectangular mass peak seams to split, with one peak in front and one at the rear. At the 100 mM concentration plateau (Figure 4d), corresponding to strongly nonlinear conditions (θ ) 35%), all excess injections (from 50 to 750 mM) have anti-Langmuirian profiles with a tendency toward a Gaussian form. It is interesting to consider the complete series of peak profiles in Figure 4, profile by profile, focusing on how the shape of the mass peak successively changes as the plateau level increases. Looking at the excess injections made of 250, 500, and 750 mM LM (i.e., the three largest injections depicted in Figures 4a-d), we note that the mass peak shape shifts from Langmuirian to antiLangmuirian as the plateau concentration increases. The original peak maximum at the front becomes more and more eroded, and a hump rises at the disperse rear of the peak. Finally, the rear hump becomes the new peak maximum and the old front is completely eroded, yielding an anti-Langmuirian mass peak. If we instead examine the two smallest excess injections (50 and 100

mM LM), we note that the mass peak displays anti-Langmuirian tendencies over the entire range of plateau level concentrations studied (cf. Figure 4a-d). Explanation of the Deformations. Each concentration on the perturbation peak has a characteristic velocity (eq 6), each of which is greater than the corresponding solute velocity (eq 7). Consequently, the perturbation zone migrates faster than the solutes present in that zone (Figure 2). Thus, the injected molecules in the perturbation zone will be left behind and gradually be replaced by plateau molecules. The velocity ratio between the perturbation and the solute molecules reaches its maximum, 1.77, at C ) 462 mM (eq 9) with the current adsorption isotherm parameters. As we follow the diffusive rear of the Langmuirian perturbation peak from the peak maximum down to the plateau concentration, this velocity ratio gradually decreases. Because of this velocity ratio gradient, the injected molecules will rapidly add up as a hump at the mass peak rear. Meanwhile, dispersion causes a mixing between sample and plateau molecules, resulting in a broadening zone of injected molecules, so that a considerable fraction of the sample will be detained in the faster propagating perturbation zone. This gives rise to the antiLangmuirian tailing of the mass peak. At low plateau concentrations, the symmetry inversion of the mass peak happens very slowly. The perturbation front cannot propagate much faster than the solutes (cf. eq 5), since there are relatively few plateau molecules in the system. At higher plateau concentrations, the perturbation front moves faster, enabling a more rapid separation of the mass and perturbation peaks. We can conclude that a low degree of resolution will yield a mass peak with mainly Langmuirian characteristics, whereas a high degree of resolution will result in an anti-Langmuirian mass peak. Moderate resolution gives rise to deformed profiles. The ideal model of chromatography, which is obtained from eq 1 by assuming D ) 0, fails to predict this behavior of the mass peak. One can show that, according to the ideal model, a rectangular pulse injection will give rise to a Langmuirian mass zone, which is gradually transformed into a rectangular profile as the zones are resolved (not shown). Solute dispersion in the velocity gradient is obviously required for the symmetry inversion to occur. Mass Peak Deformation as a Function of Column Length. The results of both experiments (Figure 3) and simulations (Figure 4) demonstrate that the mass peak deformation depends on the degree of resolution from the visible perturbation peak. This resolution was affected by changes in plateau concentration and the amount of the injected excess. The velocities of the perturbation and mass peaks zones and the velocity ratio increases as the plateau concentration increases. This leads to shorter elution times, peak separation, and more compact peaks. For moderate velocity differences, long columns are required to achieve separation of the perturbation and mass peaks. Simulations were performed for various column lengths, L, assuming 50-µL injections of 500 mM LM excess at a concentration plateau of 35 mM (DM). The resulting peak profiles are presented in Figure 5, using normalized time (t - t0)/t0 and normalized concentration log((C - C0)/C0) to make the plots visible in the same figure. The perturbation peak profiles are indicated by the bold lines, while the dashed bold line shows Analytical Chemistry, Vol. 78, No. 8, April 15, 2006

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Figure 5. Simulated mass peaks (thin solid lines) and perturbation peaks (bold solid lines) versus normalized time and column length. The intersection point between the two peaks is indicated by the dashed line. The concentration plateau contained 35 mM DM, and the sample was 35 mM DM + 500 mM LM. Table 1. Mass Peak Asymmetry, asf50, for Different Column Lengths, L length (cm)

mass peak assym asf50

1.0 2.5 5.0 5.5 6.0 6.5 10 20 50

3.90 1.82 1.15 1.04 0.47 0.44 0.43 0.53 0.70

where the profiles intersect with the corresponding mass peak (thin lines). For short columns, the mass peak has the same characteristics as peaks produced in longer columns by large excess injections at very low concentration plateaus (cf. Figures 3a,b and 4a). As L increases, the mass peak changes from a Langmuirian shape into a deformed rounded and rectangular intermediate peak shape and, finally, into an anti-Langmuirian shape. Once the whole mass peak has resolved from the perturbation peak, no further deformation will occur. Even for very long columns, the mass peak will still be anti-Langmuirian in shape, although the dispersion will smear the peak into a near-Gaussian shape. Table 1 shows the values of the asymmetry factor at halfheight, asf50, of the mass peak at selected column lengths. The peak shows tailing for small values of L and leading at large values, with a sudden change when L is 5.5-6.0 cm. Still, at L ) 50 cm there is an anti-Langmuirian trend (asf50 ) 0.7). This simulation shows that the deformation of the mass peak is a result of the passage through the perturbation zone. For sufficiently long columns, the eluted mass peak will always be anti-Langmuirian in shape; however, before elution, the peak must experience the full range of deformation characteristics as it travels through the column. Flat-Line Chromatography: Invisible Perfectly Gaussian High-Concentration Bands. We demonstrated above that the deformed shapes of the mass peak result from a complex interplay between the velocity gradient and solute dispersion. Once the peaks have separated, the shape of the mass profile will be 2770 Analytical Chemistry, Vol. 78, No. 8, April 15, 2006

Figure 6. Flat-line chromatography: mass peak monitoring with no perturbation. The concentration plateau was 20 mM DM. The sample injected was 10 mM DM + 10 mM LM (dotted lines), 5 mM DM + 15 mM LM (dashed-dotted lines), and 0 mM DM + 20 mM LM (solid lines). The top chromatogram shows the visible signal from a standard detector, while the middle and bottom chromatograms show the calculated and experimental mass peaks, respectively. The experimental fractions were taken every 6 s and analyzed on a chiral column according to the method presented in the Experimental Section.

conserved, except for some smearing. The negative compensation of the plateau perturbation will totally cancel out the positive mass peak, so that only the plateau baseline is observed with a standard detector (cf. Figure 3d). In this zone, consequently, the total solute concentration is equal to the plateau concentration. All solute molecules, i.e., both enantiomers, will therefore experience identical and constant conditions from a thermodynamic point of view. The whole population of molecules in this zone will travel at the same mean velocity, explaining the conserved antiLangmuirian mass peak shape. The limited column efficiency, however, causes the smearing mentioned above. If the injected molecules can be introduced without creating any perturbation, the mass peak should be perfectly Gaussian, whatever the size of the mass injected. This hypothesis was tested using both simulations and experiments. Figure 6 shows the results of three such injections. The column was equilibrated with 20 mM DM (θ ) 10%), and the following three injections were made: (i) 10 mM LM + 10 mM DM, (ii) 15 mM LM + 5 mM DM, and (iii) 20 mM LM + 0 mM DM. The total sample concentration was 20 mM in each case, i.e., the plateau level. All solutes travel at a linear velocity described by eq 7 with C ) 20 mM. The top chromatogram shows the calculated UV signal, which is a clean baseline. Simulated and experimental mass peaks are presented in the middle and bottom figures, respectively. The mass peaks are clearly Gaussian, as predicted. Thus, it is possible to obtain Gaussian peaks under strictly nonlinear operating conditions, although they are usually only observed in linear chromatography. Another observation is that the retention time of the mass peaks is the same independent of injection composition, as long as no perturbation peak is present. The area of the mass peaks corresponds to the amount of injected LM in this case. Figure 6 indicates that there can be no mass peak deformation in the absence of a velocity gradient, i.e., when there is no perturbation. The new injection strategy developed in this work is based on making injections on a plateau, so that the total concentration of the sample is identical to that of the plateau. The

retention time of the mass peak measured under flat-line conditions is then accurately described by eq 4. The accuracy of adsorption isotherms measured using the tracer-pulse method16 should therefore be optimal with this injection strategy. Better adsorption isotherm data are crucial for making more advanced adsorption model analysis, e.g., adsorption energy distribution calculations, where accurate isotherm raw data points are required. For this, one cannot use methods that only give parametric estimations. Furthermore, detection of the mass peak may be simplified considerably, since the injected pulse may be very large. An online chiral detector could then be used to speed up isotherm determination. CONCLUSIONS Invisible zone deformations occur in liquid chromatography, even when it is performed with just one component. The deformations can be observed by selectively monitoring the injected solute molecules in experiments in which identical molecules are present in the eluent. Similar, but only visible, effects have previously only been observed in multicomponent systems when studying overloaded perturbation peaks. Mass peak deformation is caused by a complex interplay between the migration velocity gradient and dispersion inside the column. The shape of the eluted mass peak depends on the degree of resolution between the perturbation peak and the mass peak. If the peaks have separated very little, the mass peak will exhibit Langmuirian characteristics. If resolution is increased, the mass peak will become oddly deformed. At full resolution, the peak will

have an anti-Langmuirian shape. The degree of resolution is determined by the plateau concentration, the size of the injected excess, and the length of the column. For a sufficiently long column, the mass peak will always be anti-Langmuirian in shape. However, this characteristic shape is not constant throughout the entire passage through the column; rather, the peak shifts through many shapes during elution. A new injection strategy was developed, in which the sample concentration equals the plateau concentrationsthat is, no excess is injected. No perturbation will occur, and the mass peak will therefore be perfectly Gaussian. This will be the case, even at concentration plateaus corresponding to the nonlinear part of the adsorption isotherm. This new injection strategy will allow easier mass peak detection and more accurate adsorption isotherm data acquisition, because the tracer concentration can be high and mass peak deformation is entirely avoided. The enantiomer tracer-pulse method is a useful tool for visually revealing some of the few remaining “blind spots” in nonlinear chromatography. Using systematic computer simulations, verified by selected experiments, we have discovered and described a previously unknown chromatographic phenomenon. This finding is an important contribution to the common understanding of chromatography.

Received for review December 16, 2005. Accepted February 13, 2006. AC0522308

Analytical Chemistry, Vol. 78, No. 8, April 15, 2006

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