An Approach to the Concept of Resolution Optimization through

Anal. Chem. 1999, 71, 2623-2632

An Approach to the Concept of Resolution Optimization through Changes in the Effective Chromatographic Selectivity Jianhong Zhao and Peter W. Carr*

Department of Chemistry, University of Minnesota, Smith and Kolthoff Hall, 207 Pleasant Street SE, Minneapolis, Minnesota 55455

It is very common chromatographic practice to optimize resolution by making changes in selectivity by systematically varying key retention controlling factors. In many instances, a change in conditions merely results in monotonic, systematic variation in the relative retention of all pairs of peaks. Useful or “effective” changes in selectivity generally result when we see peak crossovers, changes in elution order or differential changes in band position of three or more peaks upon changing some operating condition. In this work, we demonstrate that changes in what we now call the effective selectivity can only take place when retention depends on a minimum of two solute molecular properties and further the dependencies must differ for the two sets of conditions. To verify our concept, real chromatographic data are examined from the viewpoint of linear solvation energy relationships (LSERs) and linear solvent strength theory. Five different RPLC stationary phases in different eluents are compared to elucidate the similarities and differences in their effective selectivities. Of major importance is our finding that the effective selectivity can only be understood when it is viewed in terms of the ratios of systemdependent interaction coefficients, such as the LSER coefficients, and not merely the absolute values of the coefficients. We confirm, both theoretically and experimentally, that a change in mobile-phase volume fraction and in column temperature is not as powerful a mechanism for tuning the effective selectivity as is a change in stationary-phase type. Optimization of chromatographic selectivity is one of the key steps in developing a separation method. Selectivity in RPLC can be varied in many ways,2-10 for example, by changing the eluent (either its volume fraction (φ) or the type of organic modifier * Correspondence author: [email protected]. (1) Snyder, L. R.; Kirkland, J. J.; Glajch, J. L. Practical HPLC Method Development, 2nd ed.; John Wiley & Sons: New York, 1997. (2) Antle, P. E.; Goldberg, A. P.; Snyder, L. R. J. Chromatogr. 1985, 321, 1. (3) Jandera, P.; Churacek, J. J. Chromatogr. 1974, 91, 207. (4) Martire, D. E.; Boehm, R. E. J. Phys. Chem. 1983, 87, 1045. (5) Melander, W. R.; Stoveken, J.; Horvath, C. J. Chromatogr. 1980, 199, 35. (6) Melander, W. R.; Horvath, C. Chromatographia 1984, 18, 353. (7) Scott, R. P. W.; Kucera, P. J. Chromatogr. 1977, 142, 213. (8) Snyder, L. R.; Quarry, M. A.; Glajch, J. L. Chromatographia 1987, 24, 33. (9) Snyder, L. R. J. Chromatogr., B 1997, 689, 105. 10.1021/ac981321k CCC: $18.00 Published on Web 06/09/1999

© 1999 American Chemical Society

(methanol, etc.)), the type of stationary phase (octyl, phenyl, etc.), or the column temperature.1 There are a number of very significant studies concerning the effect of solvent strength, solvent type, and temperature on selectivity.8-13 Recently Snyder critically analyzed, summarized, and compared the relative ability of different variables to change selectivity.9 He concluded that altering the solvent type from either acetonitrile or methanol to tetrahydrofuran is pragmatically the optimum procedure. He also indicated that changes in the stationary-phase type from nonpolar to relatively polar can be very useful. On the other hand, as we will show for nonelectrolyte solutes, solvent composition (φ) and temperature are actually not as effective variables as the solvent type and the stationary-phase type but for practical reasons these two variables are to be preferred in the initial phases of method development. The chemical nature of the stationary phase can have a major effect on selectivity. Horvath et al. used κ-κ plots (defined as plots of logarithmic retention factors measured on column pairs with different solutes in the same eluents) to study stationary-phase effects in RPLC.5 However, the comparisons of the energetics of retention they made were focused mainly on alkyl-bonded silica phases. Snyder et al. also studied the retention characteristics of various silica-based phases.2 In our previous paper,14 we used κ-κ plots and linear solvation energy relationships (LSERs) to compare the retention characteristics of a more varied set of stationary phases including aromatic and aliphatic phases based on both organic or inorganic substrates. Abraham et al.15 characterized some polymer-based stationary phases by LSERs and their results agree very well with our results. Additionally, Abraham introduced the idea of comparing different phases through the ratio of their LSER coefficients. The idea of using the ratio of the LSER coefficients for comparison stems from the fact that assessing selectivity by examining the selectivity among an entire set of solutes at one time (we call this a composite measure of selectivity) versus (10) Zhu, P. L.; Snyder, L. R.; Dolan, J. W.; Djordjevic, N. M.; Hill, D. W.; Sander, L. C.; Waeghe, T. J. J. Chromatogr., A 1996, 756, 21. (11) Valko, K.; Snyder, L. R.; Glajch, J. L. J. Chromatogr., A 1993, 656, 501. (12) Zhu, P. L.; Dolan, J. W.; Snyder, L. R. J. Chromatogr., A 1996, 756, 41. (13) Zhu, P. L.; Dolan, J. W.; Snyder, L. R.; Djordjevic, N. M.; Hill, D. W.; Lin, J.-T.; Sander, L. C.; Van Heukelem, L. J. Chromatogr., A 1996, 756, 63. (14) Zhao, J.; Carr, P. W. Anal. Chem. 1998, 70, 3619. (15) Abraham, M. H.; Chadha, H. S.; Leitao, R. A. E.; Mitchell, R. C.; Lambert, W. J.; Kaliszan, R.; Nasal, A.; Haber, P. J. Chromatogr., A 1997, 766, 35.

Analytical Chemistry, Vol. 71, No. 14, July 15, 1999 2623

examining individual responses, that is, pairs of peaks, can be very misleading. As an example, suppose we take a number of solutes most of which differ only in their nonpolarity, but perhaps a few differ in their dipolarity or in their hydrogen-bonding (HB) acidity or basicity. If one uses a composite comparison of similarity of two stationary or mobile phases, such as Horvath’s κ-κ plot correlation coefficient, where the conditions do differ in polarity or HB acidity/basicity, one will necessarily find a high correlation coefficient (r2) indicating a small difference in selectivity because the data set is dominated by analytes having differences only in their nonpolar properties. As additional solutes that are polar or strong hydrogen bond bases or acids are added, r2 will decrease. Our point is that a composite or single-parameter measure of selectivity is very sensitive to the specific solute chosen and the combination of the specific solutes. Now consider the use of LSER to assess differences in selectivity of two different chromatographic conditions. As is well-known, the LSER approach is based on dissecting retention in chromatography into a variety of intermolecular interactions.16,17 The general LSER equation15 is,

log k′ ) log k′0 + vV2 + sπ2* + a

∑R

H

2

+b

∑β

H

2

+ rR2 (1)

where V2, π2*, ΣR2H, Σβ2H, and R2 are solute parameters, and v, s, a, b, and r are the corresponding regression coefficients, which are related to the chemical nature of the mobile and stationary phases. In the LSER approach, once the solvatochromic fitting coefficients are established by including only a few solutes of welldefined polarity and HB acidity/basicity, then the ratios of LSER coefficients are known. Since the ratios of LSER coefficients will not (or at least ought not) vary as additional solutes with high polarity or HB acidity/basicity are added to the data set, we should be able to detect those solute selectivities that are represented by the ratios of LSER coefficients. Thus we can compare different stationary or mobile phases more effectively. This point will be tested both theoretically and experimentally in this work. We caution the reader that the LSER model as applied to, for example, RPLC is only approximate. It does not do at all well at encoding molecular shape information nor does it capture the very important effect of secondary chemical interactions on retention including interactions with surface silanol groups and electrostatic processes. It is a general chemical model that misses important “fine” structure that is important at the practical method development level (∆log R ∼ 0.02). Conventionally, chromatographic selectivity (R) is defined as the ratio of two retention factors. In method development, we are concerned not only about changing selectivity but more importantly differentially changing the band spacing of various pairs of peaks as the chromatographic conditions are changed. For example, changes in band spacing of a homologue series are always the same in any chromatographic system and therefore as shown in Figure 1a for a homologue series, a κ-κ plot is a perfect straight line. In this case, the change in log R is the same for all pairs of homologous solutes and thus σlog R defined as the scatter in the κ-κ plot is zero. We call this kind of change (16) Kamlet, M. J.; Taft, R. W. J. Am. Chem. Soc. 1976, 98, 377. (17) Taft, R. M.; Kamlet, M. J. J. Am. Chem. Soc. 1976, 98, 2886.

2624 Analytical Chemistry, Vol. 71, No. 14, July 15, 1999

Figure 1. Plots of log k′ for system 1 versus log k′ for system 2 as the solute is varied: (a) “monochromatic” selectivity resulting from homologous solutes; (b) “polychromatic” selectivity resulting from nonhomologous solutes.

in selectivity “monochromatic selectivity”. It could just as well be called “univariate selectivity” (see below). It results when the solutes differ from one another in the value of only a single retention determinant, solute-dependent, molecular variable. This single molecular variable could be the number of methylene groups in a homologue series, the number of repeat units in a homopeptide, etc. Monochromatic or univariate selectivity is not fundamentally effective since it cannot produce scatter in a κ-κ plot (Figure 1). On the other hand, a κ-κ plot of a nonhomologous series of solutes shows substantial scatter (see Figure 1b) since the band spacing of these solutes can be changed to different extents upon changing the chromatographic conditions. In this case, the solutes differ from one another in terms of two or more retention determinant properties. This kind of change in selectivity is called “polychromatic or multivariate selectivity” wherein σlog R is not equal to zero. Polychromatic selectivity can give an effective change in selectivity; therefore, it is more useful in chromatographic optimization. In this work, we use the term effective selectivity instead of polychromatic selectivity to emphasize its importance to optimization and method development. The existence of polychromatic selectivity is a necessary but not sufficient condition to induce scatter in a κ-κ plot. As will be proven below, monochromatic selectivity can never suffice to induce scatter in a κ-κ plot. Further, the conditions under which polychromatic selectivity exists may not induce sufficient scatter in a κ-κ plot to allow a practical separation.

This work discusses effective selectivity under different chromatographic conditions using simple theoretical models and provides useful guidelines for the selection of effective chromatographic variables for method development.

(log k′2,i)/b1 - (log k′1,i)/b2 ) a1/b1 - a2/b2 + (c1/b1 - c2/b2)Yi (8) When c1/b1 ) c2/b2, the last term in eq 8 disappears and we obtain

THEORETICAL MODELS In this section we propose a simple model, for relating retention to a solute’s molecular properties. We derive some special cases to describe the factors that control retention and relate the model to the effective selectivity. Furthermore, we use a very common retention model, linear solvation energy relationships, to associate the theoretical model with real chromatographic experiments. Case 1: A Single Dominant Solute-Dependent Retention Controlling Factor. Consider two different chromatographic systems denoted as 1 and 2. For both systems, we assume that retention is dominated by a single solute-dependent factor. For example, in gas chromatography this might be the London (dispersive) interactions between a set of nonpolar solutes and two different nonpolar stationary phases (squalane and Apolane 87). We denote the solute-dependent property as Xi thus:

log k′1,i ) a1 + b1Xi

(2)

log k′2,i ) a2 + b2Xi

(3)

Trivial rearrangement shows that

log k′1,i ) a1 - a2b1/b2 + b1/b2 log k′2,i

(4)

It is evident that a plot of measured log k′1,i versus measured log k′2,i will be absolutely linear and any variance in the plot will be due to only experimental uncertainty and not chemistry. In view of Snyder’s seminal work,1 there will be no real difference in the effective selectivity of the two systems even if the intercepts and slopes differ. By combining the conventional definition of selectivity (R) and eqs 2 and 3, we easily arrive at the result,

log R1,ij ) (b1/b2) log R2,ij

(5)

It is evident that R1,ij ) R2,ij only when b1 ) b2. In this case, the two systems have identical chromatographic selectivities. Snyder’s criterion for selectivity, which we here rename the effective selectivity, is not that R1,ij ) R2,ij, but only that there is a tight linear relationship between log k2,i and log k′1,i.9 Case 2: Two Dominant Solute-Dependent Retention Controlling Factors. If log k′ is controlled by two solutedependent properties (Xi and Yi), the situation is considerably more complicated. A solute (i) in two different systems (1 and 2) will have the following retention factors,

log k′1,i ) a1 + b1Xi + c1Yi

(6)

log k′2,i ) a2 + b2Xi + c2Yi

(7)

Combining eqs 6 and 7 gives,

log k′1,i ) a1 - a2b1/b2 + b1/b2 log k′2,i

(9)

which is identical to eq 4. Consequently, when c1/b1 ) c2/b2, there is no difference in the effective selectivity of the two systems. However, if c1/b1 * c2/b2, we may no longer observe a tight correlation between the retention data in the two systems and thus the effective selectivity might vary sufficiently upon changing from system 1 to system 2 (see Figure 2), provided that with the set of solutes used both Xi and Yi make substantial contributions to retention, so that a useful change in R results. Equations for situations where retention is controlled by more than two solute-dependent properties can also be derived and thus we come to the conclusion that differences in the effective selectivity only exist when there are two or more solute properties and the ratios of the susceptibilities differ. Again whether the changes in band spacing upon changing from system 1 to system 2 will suffice to allow resolution of the critical (least well separated) pair of solutes depends on the magnitude of Xi and Yi and on how different c1/b1 is from c2/b2. A net change in log R from system 1 and system for the critical pair of 0.02-0.03 is often satisfactory. Retention in RPLC can be described by the LSER model, which includes five solute-dependent properties (see eq 1). As a result, when we compare the effective selectivity of a set of solutes under different chromatographic conditions, we should analyze the ratios of LSER coefficients, s/v, a/v, b/v, and r/v, under those conditions. In other words, when two systems give somewhat different LSER coefficients (v, s, a, b, r), this does not mean that they must have different effective selectivities. Given that the LSER ratios of the two systems are distinctive, we will definitely obtain different effective selectivities. Relationship between the Retention Factor and the Chromatographic Variables. The retention factor in RPLC can also be related to certain types of chromatographic variables (Ψ) by the following approximation,

log k′ ) A + BΨ

(10)

where B is the susceptibility of retention to changes in the chromatographic variable Ψ, e.g., volume fraction of organic modifier in the eluent or inverse temperature (1/T).3,9,11 More specifically in RPLC, if Ψ is taken as the mobile-phase composition (φ) then eq 10 becomes the Snyder equation3,11

log k′ ) log k′w - Sφ

(11)

In this case log k′w is the hypothetical retention factor in pure water and S is the susceptibility of retention to changes in the mobile-phase composition. If Ψ ) 1/T 9 then eq 10 becomes

log k′ ) A + (B/T)

(12)

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2625

Figure 2. Plots of log k′ (system 1) versus log k′ (system 2) as the solute is varied. The data correspond to Case 2 with two solutedependent properties governing retention: (a) c1/b1 ) c2/b2 ) 0.5. (b) c1/b1 ) 0.5; c2/b2 ) 0.2.

Figure 3. Plots of log k′ (φ ) 0.8) versus log k′ (φ ) 0.5) using simulated solute parameters. The data were computed according to eqs 17 and 18: (a) cw/bw ) cs/bs ) 1. (b) cw/bw ) 1; cs/bs ) 1.5.

It is evident that if log k′ is controlled by some set of solute properties, then A and B in eq 10 must also be controlled by these same solute properties. For the sake of simplicity, let us assume we are looking at the solvent strength effect but the same logic will apply to the temperature effect. Applying case 1 (a single solute-dependent factor) to log k′w and S of a solute (i) and eq 11 results in a new expression for log k′,

factor controlling retention through its influence on both log k′w and S, changes in φ will not alter the effective selectivity of a set of solutes. This is clearly evident for the case of a homologous series, where log k′w and S are linearly correlated18 by the following equation,

log k′i ) aw - asφ + (bw - bsφ)Xi

(13)

For case 2, when two solute properties govern log k′w and S, at two solvent compositions (φ1 and φ2), we obtain,

Here the subscripts w and s designate the fitting coefficients for log k′w and S, respectively. In two different mobile-phase compositions, where φ ) φ1 and φ ) φ2, eq 13 becomes

log k′1,i ) aw - asφ1 + (bw - bsφ1)Xi + (cw - csφ1)Yi (17)

log k′1,i ) p1 + q1Xi

(14)

log k′2,i ) p2 + q2Xi

(15)

after further simplification by defining p1 ) aw - asφ1, q1 ) bw bsφ1, p2 ) aw - as φ2, and q2 ) bw - bsφ2. Equations 14 and 15 are exactly the same as eqs 2 and 3 and we obtain the same conclusion by a different approach. When there is only one solute-dependent 2626 Analytical Chemistry, Vol. 71, No. 14, July 15, 1999

log k′wi ) aw - asbw/bs + (bw/bs)Si

(16)

log k′2,i ) aw - asφ2 + (bw - bsφ2)Xi + (cw - csφ2)Yi (18) As shown in Figure 3, we plot simulated data in which the ratios of coefficients are the same (cw/bw ) cs/bs) and different (cw/bw * cs/bs) at two mobile-phase compositions. We observe an exact linear relationship (r2 ) 1.000) between the data sets when the ratios are exactly the same. In contrast, we obtain a lower correlation coefficient (r2 ) 0.926) when the ratios are slightly different (1.0 and 1.5). Further arrangement of eqs 17 and 18 when (18) Tan, L.; Carr, P. W. J. Chromatogr., A 1993, 656, 521.

cw/bw ) cs/bs also gives eq 16, indicating a linear relationship between log k′w and S. In summary, only when the dependencies of both log k′w and S on solute properties are different will we have any chance of optimizing selectivity by varying the volume fraction of the organic modifier. Thus, whenever log k′w and S are tightly correlated, variation in φ will only have minor effects on the effective selectivity. The system is essentially monochromatic (univariate). The same thought indicates that if there is strong enthalpy-entropy compensation then temperature will have only minor effects on the effective selectivity. However, in a very special case, when |bw| . |bs| and |cw| . |cs|, the difference between φ1 and φ2 will not significantly affect the coefficients of Xi and Yi in eqs 17 and 18. In other words, if the solute dependence on log k′w is much stronger than that on S, the mobile-phase composition will not affect the effective selectivity. General Comparison among Different Stationary Phases. Above, we only considered a single stationary phase under different chromatographic conditions. We can use a similar approach to compare two stationary phases. Let us assume that we have two stationary phases (I and II) and log k′w and S are controlled by two solute properties. Equation 11 gives,

Table 1. Solutes and Solute Descriptorsa no.

solutes

V2

π2*

ΣR2H

Σβ2H

R2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

benzene toluene ethylbenzene p-xylene propylbenzene butylbenzene naphthalene p-dichlorobenzene bromobenzene nitrobenzene p-nitrotoluene anisole benzonitrile p-nitrobenzyl chloride methyl benzoate acetophenone benzophenone 3-phenylpropanol benzyl alcohol N-benzylformamide phenol p-chlorophenol

0.7164 0.8573 0.9982 0.9982 1.1391 1.2800 1.0854 0.9612 0.8914 0.8906 1.0315 0.9160 0.8711 1.1539 1.0726 1.0139 1.4808 1.1978 0.9160 1.1137 0.7751 0.8975

0.52 0.52 0.51 0.52 0.50 0.51 0.92 0.75 0.73 1.11 1.11 0.75 1.11 1.34 0.85 1.01 1.50 0.90 0.87 1.80 0.89 1.08

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.30 0.33 0.40 0.60 0.67

0.14 0.14 0.15 0.16 0.15 0.15 0.20 0.02 0.09 0.28 0.28 0.29 0.33 0.40 0.46 0.48 0.50 0.67 0.56 0.63 0.30 0.20

0.610 0.601 0.613 0.613 0.604 0.600 1.340 0.825 0.882 0.871 0.870 0.708 0.742 1.080 0.733 0.818 1.447 0.821 0.803 0.990 0.805 0.915

a Values of V , π *, ΣR H, Σβ H, and R were obtained from refs 2 2 2 2 2 26-28.

log k′I,i ) aIw - aIsφ + (bIw - bIsφ)Xi + (cIw - cIsφ)Yi (19) log k′II,i ) aIIw - aIIs φ + (bIIw - bIIsφ)Xi + (cIIw - cIIsφ)Yi (20) Algebra shows that only when

cIw/bIw ) cIs/bIs ) cIIw/bIIw ) cIIs/bIIs

(21)

can the following equation result in all mobile-phase compositions.

(cIw - cIsφ)/(bIw - bIsφ) ) (cIIw - cIIsφ)/(bIIw - bIIsφ) (22)

In this situation, the two stationary phases have identical effective selectivities. We can use LSERs to describe the dependence of log k′w and S on the solute properties as follows,

log k′w ) log k′0w + vwV2 + swπ2* + awΣR2H + bwΣβ2H + rwR2 (23) S ) log k′0s + vsV2 + ssπ2* + asΣR2H + bsΣβ2H + rsR2 (24)

These two equations are inherent properties of the stationary/ mobile-phase pairs that determine the overall retention process for all mobile-phase compositions. On the other hand, the LSER coefficients obtained from eq 1 in a given φ only describe the apparent properties of the stationary/mobile-phase pair at that φ value. Therefore to globally compare differences in selectivities of two different stationary phases we should compare the LSER coefficients of log k′w and S for the two stationary phases. In particular, we should consider the ratio of the LSER coefficients for log k′w and S. If two phases have a different ratio of LSER

coefficients for log k′w and S it will be beneficial to test both columns to optimize the selectivity. EXPERIMENTAL SECTION To probe the above concepts, we have gathered experimental data from a variety of stationary phases. The following stationary phases are compared in this paper: Zorbax Stable-Bond octyl and octadecyl silica-bonded phases (C8-SiO2 and C18-SiO2, respectively) were from Hewlett-Packard (Wilmington, DE), a Alltima phenyl silica phase (Ph-SiO2) was from Alltech (Deerfield, IL), a Hamilton poly(styrene-divinylbenzene) column (PRP-1) was from Chrom Tech (Apply Valley, MN), a polybutadiene-coated zirconia (PBD-ZrO2) column was from ZirChrom (Anoka, MN), and a polystyrene-coated zirconia (PS-ZrO2) column was made and packed in this laboratory.19 The retention data used for the κ-κ plots were taken from refs 20 and 21. The LSER coefficients were also reported in a previous paper.14 Table 1 lists solute descriptors for 22 solutes which were chosen because they reproduce the LSER coefficients obtained with a much larger data set.20 The S and log k′w of the 22 solutes for five different stationary phases (see Table 2) were calculated from retention data, which were measured over a reasonably wide range of mobile-phase compositions in 10% increments.20,21 The linear regression results of log k′ versus φ indicate very good fitting with correlation coefficients ranging from 0.995 to 1. The coefficients of the LSER equations for log k′w and S (eqs 23 and 24) are summarized in Table 3 using multivariable regression of log k′w and S of these 22 solutes against the solute descriptors V2, π2*, ΣR2H, Σβ2H, and R2. The simulation data were generated by Microsoft Excel spreadsheets using a random number generator. (19) Zhao, J.; Carr, P. W. Anal. Chem., submitted. (20) Tan, L. Ph. D., University of Minnesota, Minneapolis, 1994. (21) Zhao, J. Ph.D., University of Minnesota, Minneapolis, 1999.

Analytical Chemistry, Vol. 71, No. 14, July 15, 1999

2627

Table 2. log k′w and S of Different Solutes for Five Different Stationary Phases log k′w solutes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

a

benzene toluene ethylbenzene p-xylene propylbenzene butylbenzene naphthalene p-dichlorobenzene bromobenzene nitrobenzene p-nitrotoluene anisole benzonitrile p-nitrobenzyl chloride methyl benzoate acetophenone benzophenone 3-phenylpropanol benzyl alcohol N-benzylformamide phenol p-chlorophenol

b

S c

d

C8-SiO2

PBD-ZrO2

Ph-SiO2

PS-ZrO2

PRP-1e

1.83 2.38 2.78 2.77 3.33 3.62 2.84 2.91 2.5 1.73 2.21 1.85 1.56 2.37 1.87 1.52 2.87 1.68 0.84 0.85 1.03 1.68

1.13 1.58 1.98 1.97 2.44 2.88 2.04 2.13 1.83 0.92 1.26 1.07 0.61 1.49 0.95 0.52 1.77 0.64 0.035 0.093 0.033 1.001

1.3 1.6 1.92 1.89 2.25 2.58 1.98 1.95 1.74 1.3 1.58 1.33 1.1 1.79 1.29 0.99 1.97 1.02 0.5 0.47 0.73 1.22

1.03 1.41 1.83 1.81 2.27 2.69 2.22 2.08 1.75 1.19 1.6 1.17 0.91 1.9 1.23 0.86 2.23 1.02 0.33 0.42 0.59 1.45

1.87 2.22 2.42 2.43 2.7 3.02 2.63 2.54 2.40 1.76 2.07 1.86 1.48 2.24 1.7 1.32 2.48 0.97 0.57 0.34 0.87 0.97

a

b

C8-SiO2

PBD-ZrO2

2.87 3.61 4.00 3.99 4.70 4.84 4.23 4.25 3.72 2.88 3.52 2.99 2.8 3.81 3.16 2.84 4.56 3.47 2.22 2.54 2.26 3.09

2.85 3.41 3.80 3.74 4.3 4.69 3.89 3.81 3.6 2.88 3.21 2.93 2.68 3.74 2.91 2.56 3.86 2.91 2.37 2.81 2.63 3.16

Ph-SiO2c PS-ZrO2d PRP-1e 2.15 2.54 2.95 2.91 3.39 3.82 3.05 3.02 2.73 2.23 2.59 2.24 2.01 2.9 2.24 1.9 3.09 2.13 1.45 1.55 1.69 2.34

2.65 3.13 3.68 3.64 4.31 4.86 4.18 4.31 3.52 2.95 3.49 2.9 2.71 3.91 3.14 2.74 4.35 3.19 2.37 2.78 2.46 3.40

2.11 2.36 2.50 2.48 2.68 2.92 2.53 2.48 2.43 2.15 2.38 2.14 1.96 2.65 2.03 1.77 2.58 1.68 1.36 1.35 1.72 2.06

a Calculated from log k′ in 20/80, 30/70, 40/60, and 50/50 acetonitrile/water by log k′ ) log k′ - Sφ. b Calculated from log k′ in 20/80, 30/70, w 40/60, and 50/50 acetonitrile/water by log k′ ) log k′w - Sφ. c Calculated from log k′ in 40/60, 50/50, 60/40, and 70/30 acetonitrile/water by log d k′ ) log k′w - Sφ. Calculated from log k′ in 20/80, 30/70, 40/60, and 50/50 acetonitrile/water by log k′ ) log k′w - Sφ. e Calculated from log k′ in 50/50, 60/40, 70/30, and 80/20 acetonitrile/water by log k′ ) log k′w - Sφ.

Table 3. Coefficients of LSER Equations for log k′w and Sa column log k′w

S

d

C8-SiO2 PBD-ZrO2 Ph-SiO2 PS-ZrO2 PRP-1 C8-SiO2 PBD-ZrO2 Ph-SiO2 PS-ZrO2 PRP-1

log k′0 0.01(0.12d

-0.72(0.13 0.26(0.15 -0.79(0.17 0.65(0.16 0.50(0.22 0.94(0.19 0.99(0.20 0.26(0.18 1.26(0.20

v

s

a

b

r

SDb

r2c

3.39(0.14 3.47(0.15 1.89(0.18 2.94(0.19 2.10(0.18 3.98(0.26 3.79(0.22 2.02(0.24 3.82(0.21 1.47(0.23

-0.51(0.10 -0.51(0.11 -0.19(0.13 -0.26(0.14 -0.40(0.13 -0.50(0.19 0.08(0.16 -0.07(0.17 -0.14(0.16 -0.02(0.17

-0.55(0.10 -0.33(0.11 -0.45(0.13 -0.17(0.14 -1.15(0.13 -0.24(0.19 -0.02(0.16 -0.22(0.17 0.23(0.16 -0.44(0.17

-3.02(0.15 -3.73(0.17 -2.06(0.19 -2.87(0.21 -2.87(0.20 -2.72(0.28 -3.14(0.24 -2.04(0.20 -2.84(0.23 -1.94(0.25

0.14(0.14 0.20(0.15 0.09(0.17 0.45(0.19 0.53(0.18 0.28(0.25 -0.26(0.21 -0.01(0.23 0.24(0.21 0.12(0.22

0.09 0.09 0.11 0.12 0.11 0.16 0.13 0.14 0.13 0.14

0.990 0.990 0.962 0.974 0.982 0.968 0.970 0.924 0.972 0.916

a Regression results of log k′ or S against the solute descriptors, V , π *, ΣR H, Σβ H, and R . b Average residual of the fit. c Correlation coefficient. w 2 2 2 2 2 Standard deviation.

RESULTS AND DISCUSSIONS Comparison of LSER Ratios for Different Stationary Phases in 50/50 Acetonitrile/Water. We compared the LSER ratios for six different stationary phases in Table 4 in 50/50 acetonitrile/water. C8-SiO2, C18-SiO2, PBD-ZrO2, and Ph-SiO2 phases have very similar LSER ratios, indicating that these phases are comparable in terms of effective selectivity in this eluent. On the other hand, PS-ZrO2 and PRP-1 phases are somewhat different. The PRP-1 phase has the largest a/v, b/v, and r/v ratios. Its a/v ratio is twice as large as the other phases. This is in agreement with Abraham’s work on PRP-1.15 We believe that the large a/v ratio on PRP-1 is mainly due to the weak interaction of the mobile phase with the polymeric substrate.14 PS-ZrO2 and PBD-ZrO2 have very similar s/v, a/v, and b/v ratios but different r/v ratios. As the r coefficient accounts for the additional π-π interactions between the solute and the mobile/stationary phase,15 we expect that the aromatic PS-ZrO2 will have a higher r/v ratio than does the aliphatic PBD-ZrO2. Furthermore, the r/v of PS2628 Analytical Chemistry, Vol. 71, No. 14, July 15, 1999

ZrO2 is rather comparable to that of PRP-1 due to the similar aromaticity of these two phases. The similarities and differences among these stationary phases are confirmed by the κ-κ plot of each pair of stationary phases.14 In Figure 4, we show only a few examples. Note that we exclude solute 22, p-chlorophenol: because p-chlorophenol, is a hard Lewis base which we suspect interacts with the hard Lewis acidity of the zirconia-based stationary phase.22 We believe that the scatter in Figure 4 is not due to random experimental error in the measurement of k′ but is due to differences in the chemical properties of the various stationary phases. First, the standard deviation of triplicate determination of k′ of each solute is less than 0.1%. Second, the solid circles denoting alkylbenzenes in Figure 4 indicate a very strong correlation and good linearity between all stationary-phase pairs, where retention depends solely on the molecular size. If two phases are linearly correlated over (22) Li, J.; Carr, P. W. Anal. Chim. Acta 1996, 334, 239.

Table 4. LSER Coefficients and Ratios for Different Stationary Phasesa phase

v

s

a

b

C8-SiO2 C18-SiO2 PBD-ZrO2 Ph-SiO2 PS-ZrO2 PRP-1

1.47(0.03b

-0.25(0.03 -0.32(0.03 -0.42(0.04 -0.15(0.05 -0.20(0.09 -0.39(0.06

-0.41(0.04 -0.54(0.04 -0.40(0.04 -0.34(0.05 -0.27(0.09 -0.94(0.06

-1.71(0.04 -1.77(0.06 -2.01(0.07 -0.99(0.07 -1.48(0.14 -1.89(0.08

a

1.62(0.05 1.58(0.06 0.83(0.07 1.09(0.13 1.35(0.08

r

s/v

a/v

b/v

r/v

0c 0.17(0.06 0.09(0.07 0.34(0.12 0.48(0.07

-0.17(0.02 -0.20(0.02 -0.27(0.03 -0.18(0.06 -0.18(0.09 -0.29(0.05

-0.28(0.03 -0.33(0.03 -0.25(0.03 -0.41(0.07 -0.25(0.09 -0.70(0.06

-1.16(0.04 -1.11(0.05 -1.26(0.07 -1.19(0.13 -1.36(0.21 -1.40(0.10

0 0 0.11(0.04 0.11(0.08 0.31(0.12 0.36(0.06

0c

Based on retention data in 50/50 acetonitrile/water. b Standard deviation. c Statistically insignificant.

Figure 4. Plots of log k′ of one stationary phase versus log k′ of another stationary phase as the solute is varied. The mobile phase is 50/50 acetonitrile/water. The Solid line denotes the least-squares line of alkylbenzenes, and the dash line denotes the least-squares line of all solutes. Symbols: (b) alkylbenzenes; (O) nonalkylbenzenes.

the whole set of solutes, the scatter of the data caused by random error should center evenly on the line where the alkylbenzenes are correlated. However, this is not what we observe in Figure 4. As demonstrated in Figure 4a, C18-SiO2 and PBD-ZrO2 exhibit very similar selectivity in 50/50 acetonitrile/water since the κ-κ plot has a strong correlation (r2 ) 0.9870, SD ) 0.06) and good linearity, and as expected, the regression line for all data points (the dotted line) nearly overlaps that of the alkylbenzenes (the solid line). On the other hand, the correlation between PRP-1 and C18-SiO2 is relatively weaker (r2 ) 0.9610, SD ) 0.11), and the regression line for all the data lies away from that of the alkylbenzenes (Figure 4b). PS-ZrO2 is not well correlated to any aliphatic phase, such as PBD-ZrO2 (Figure 4c), but it has a relatively stronger correlation with the other aromatic phase, PRP-1 (Figure 4d). Because the LSER ratios of none these

stationary phases are very different, we do not see very poor correlation coefficients (r2 < 0.8).5,14 From the above comparisons, qualitatively we can say that, with a 50/50 acetonitrile/water mobile phase, these solutes will not show large changes in elution order upon changing from C18SiO2 to PBD-ZrO2 or to Ph-SiO2; on the other hand, there will be larger changes in the effective selectivity when PRP-1 or PSZrO2 phases are used. However, under different mobile phase conditions, the above statement might not be true. To compare intrinsic differences in these stationary phases in a more universal way, we need to examine the ratios of the LSER coefficients of log k′w and S for these phases. Effect of Mobile-Phase Composition on the Effective Selectivity. As discussed above, if for a given stationary phase the ratios of LSER coefficients of log k′w and S are exactly the Analytical Chemistry, Vol. 71, No. 14, July 15, 1999

2629

Figure 5. Plots of log k′w versus S for different stationary phases as the solute is varied. The straight line is the least-squares line: (a) C8-SiO2; (b) PBD-ZrO2; (c) Ph-SiO2; (d) PS-ZrO2; (e) PRP-1.

Table 5. Ratios of the LSER Coefficients for log k′w and S

log k′w

S

a

column

-s/v

-a/v

-b/v

r/v

C8-SiO2 PBD -ZrO2 Ph-SiO2 PS-ZrO2 PRP-1 C8-SiO2 PBD -ZrO2 Ph-SiO2 PS-ZrO2 PRP-1

0.15(0.03a 0.15(0.04 0.10(0.05 0.08(0.05 0.19(0.06 0.13(0.05 0.02(0.06 0.03(0.05 0.04(0.05 0.01(0.09

0.16(0.03 0.10(0.04 0.23(0.05 0.06(0.05 0.55(0.08 0.06(0.05 -0.01(0.05 0.11(0.05 -0.06(0.04 0.30(0.13

0.89(0.06 1.07(0.08 1.09(0.10 0.98(0.10 1.37(0.15 0.68(0.08 0.83(0.11 0.99(0.09 0.74(0.07 1.32(0.27

0.04(0.04 0.06(0.05 0.05(0.06 0.15(0.07 0.25(0.09 0.07(0.06 0.07(0.08 0.00(0.05 0.06(0.05 0.08(0.14

Standard deviation.

same or, in other words, log k′w is strongly linearly correlated with S, it is impossible to adjust the effective selectivity by changing the mobile-phase composition. First, we examined the correlation between log k′w and S for the five different stationary phases studied here. As shown in Figure 5, log k′w and S are most strongly correlated for Ph-SiO2 (r2 ) 0.9814) and they are less correlated for the other four phases. Second, we compare the ratios of LSER coefficients of log k′w and S for the five phases in Table 5. For all phases except PRP-1, the b/v ratio is the largest while the other ratios are nearly negligible. This is consistent with our finding in a previous study;14 that is, the b and v coefficients are the two major factors that contribute to retention in most RPLC phases. Therefore it is reasonable, except for PRP-1, to compare only the b/v ratio of log k′w and S. None of the first four phases in the Table 5 has equal values of b/v for log k′w and S. However, the difference in b/v between log k′w and S is smaller for Ph-SiO2 than for the other three phases. This might explain the phenomenon observed above, that the correlation between log k′w and S is much stronger for Ph-SiO2 than for the other three phases. 2630 Analytical Chemistry, Vol. 71, No. 14, July 15, 1999

Even though the PRP-1 phase has very different s/v, a/v, and r/v ratios for log k′w and S, the b/v ratios are remarkably similar. Additionally, the r/v and s/v ratios are opposite in sign but similar in size and it is possible that their effect on the effective selectivity might cancel. Consequently, the difference in the a/v ratio of log k′w and S might be the main reason that we do not see good correlation between log k′w and S for PRP-1. Overall, by comparing the LSER ratios of log k′w and S and the correlation between them, for Ph-SiO2 we predict that changes in selectivity upon changes in the mobile-phase composition will be minimal. However, for the other phases the effect of mobile phase will be substantial. To confirm this speculation, we further examined the correlation coefficients of log k′ at different mobile-phase compositions against one mobile-phase composition, for all five stationary phases (Table 6). As the mobile-phase compositions increasingly differ, for C8-SiO2, PBD-ZrO2, and PS-ZrO2, the correlation coefficients become much worse than those for Ph-SiO2 and PRP-1. For example, for C8-SiO2 the correlation coefficient changes from 0.998 to 0.933 as φ changes from 0.4 to 0.2. On the other hand, for Ph-SiO2 the correlation coefficient changes from 0.997 to 0.980 as φ changes from 0.6 to 0.4. Except for PRP-1, the results for the other phases in Table 6 agree very well with the LSER ratios and correlations of log k′w and S. The result for PRP-1 is puzzling at first, but it becomes reasonable when we take a closer look at the LSER coefficients of log k′w and S for PRP-1. As discussed in the theoretical section, if the absolute values of the LSER coefficients for log k′w are much larger than those of S, the mobile-phase composition will not greatly affect the effective selectivity (see eqs 17 and 18). As shown in Table 3, PRP-1 has very different LSER coefficients compared to the other phases. For PRP-1, the absolute values of LSER coefficients of log k′w are all substantially larger than those of S. Because of the dominant affect of log k′w on retention on PRP-1, changes in φ are less important and thus we observe minimal changes in effective selectivity. We do not understand why PRP-1

Table 6. Summary of Regression Results of log k′ (at O1) versus log k′ (at O2) on the Same Stationary Phase correlation coefficient (r2) and standard deviationa C8-SiO2 50/50 ACNb/water PBD-ZrO2 50/50 ACN/water Ph-SiO2 70/30 ACN/water PS-ZrO2 50/50 ACN/water PRP-1 80/20 ACN/water a

40/60 ACN/water 0.998(0.02) 40/60 ACN/water 0.997(0.03) 60/40 ACN/water 0.997(0.01) 40/60 ACN/water 0.998(0.02) 70/30 ACN/water 0.999(0.02)

30/70 ACN/water 0.987(0.05) 30/70 ACN/water 0.989(0.07) 50/50 ACN/water 0.992(0.02) 30/70 ACN/water 0.984(0.06) 40/60 ACN/water 0.994(0.04)

20/80 ACN/water 0.933(0.12) 20/80 ACN/water 0.970(0.11) 40/60 ACN/water 0.980(0.05) 20/80 ACN/water 0.964(0.10) 50/50 ACN/water 0.991(0.05)

Standard deviations are given in parentheses. b ACN, acetonitrile.

Figure 6. Plots of S versus log k′ in 50/50 acetonitrile/water for different stationary phases as the solute is varied. The straight line is the least-squares line: (a) C8-SiO2; (b) PBD-ZrO2; (c) Ph-SiO2; (d) PS-ZrO2; (e) PRP-1.

is so different from the other phases. However, in a previous paper we hypothesized that the retention on PRP-1 might be more adsorption-like.14,23 As stated by Snyder,9 if S values correlate with retention time in a given certain eluent composition, a change in mobile-phase composition will not be able to change the selectivity. Figure 6 shows S vs log k′ (φ ) 0.5) for all five phases. We observe very poor correlation for C8-SiO2, PBD-ZrO2, and PS-ZrO2 where r2 is 0.81, 0.78, and 0.76, respectively. However, for Ph-SiO2 and PRP-1, the correlation between S and retention is relatively stronger (r2 ) 0.91). In conclusion, for Ph-SiO2 and PRP-1 phases, changing the mobile-phase composition is not a good strategy for optimizing the effective selectivity. A better approach is to change the type of organic modifier or switch to a new stationary phase. For C8SiO2, PBD-ZrO2, and PS-ZrO2, a large change in φ (from 0.5 to 0.2) may result in an effective change in selectivity. However, we must maintain adequate retention (k′ >1) to achieve resolution. General Comparison of Different Stationary Phases. As discussed above, two stationary phases will have identical effective selectivities when the ratios of the LSER coefficients for log k′w and S are as given in eq 21. We compare the ratios of LSER coefficients of log k′w and S for five different stationary phases (see Table 5). The PRP-1 phase shows very different and much (23) Li, J.; Cantwell, F. F. J. Chromatogr., A 1996, 726, 37.

larger b/v and a/v ratios for both log k′w and S than do the other four phases, indicating that the overall effective selectivity of PRP-1 should be different from the other phases. However, it seems difficult to compare the rest of the phases because the ratios are similar. If two stationary phases are statistically similar, then a change in φ should not alter their effective selectivity. In Figure 7 we plot all the log k′ data collected at four different mobilephase compositions using C8-SiO2 against all such data on one of the other four phases. We observe that the PBD phase is well correlated with the C8-SiO2 phase with r2 equal to 0.973 (Figure 7a) On the other hand, all three aromatic phases have weak correlations with C8-SiO2 as shown in Figure 7b-d. In summary, we have shown that PBD-ZrO2 is very similar to C8-SiO2 in terms of its effective selectivity, but PS-ZrO2 is not the same as PBD-ZrO2 and exhibits a different effective selectivity. Additionally, PRP-1 is a very different phase from any of the other phases; however, on a PRP-1 phase, it is not generally useful to change mobile-phase composition to optimize effective selectivity. From a practical viewpoint, if you cannot separate some unknown compounds on a C8-SiO2 or PBD-ZrO2 column at any accessible mobile-phase composition and you want to switch to a different kind of phase, Ph-SiO2, PS-ZrO2, and PRP-1 are better candidates than any other nonpolar aliphatic phase. We caution that these conclusions are limited to solutes that do not differ greatly in molecular shape and to those that are not involved in Analytical Chemistry, Vol. 71, No. 14, July 15, 1999

2631

Figure 7. Plots of log k′ of one phase versus log k′ of another phase using all mobile phases studied. The straight line is the least-squares line: (a) PBD-ZrO2 versus C8-SiO2; (b) Ph-SiO2 versus C8-SiO2; (c) PS-ZrO2 versus C8-SiO2; (d) PRP-1 versus C8-SiO2.

any secondary chemical interactions. It is well known that even seemingly minor changes in such factor can suffice to bring out a good separation on a C18-SiO2 column versus a second. CONCLUSIONS By using some simple theoretical models, we demonstrate that changes in effective selectivity upon changing the chromatographic conditions (eluent, temperature, stationary phase) can only take place when retention is controlled by two or more solutedependent retention determinant properties and that the dependencies must differ for the two sets of conditions. We further verify our concept by using the LSER model to exam real chromatographic data. We demonstrate that the ratios of the LSER coefficients and not the absolute values of the LSER coefficients must be used to compare different stationary phases. For some stationary phases, such as PRP-1 and Ph-SiO2, a change in the concentration of organic modifier is not effective for optimizing resolution. Furthermore, a change from an aliphatic-type stationary phase (C18-SiO2 or PBD-ZrO2), to an aromatic-type phase (PRP-1 or PS-ZrO2) will alter the effective selectivity significantly. Stationary phases not investigated here, such as cyano- and amino-

2632 Analytical Chemistry, Vol. 71, No. 14, July 15, 1999

bonded silica phases and carbon-coated zirconia phases,24 can provide distinctive selectivity. We did not compare the effect of organic modifier type in this work. However, on the basis of Tan and Carr’s study25 on the LSER coefficients in different organic modifiers, we conclude that use of tetrahydrofuran as an organic modifier will show very different effective selectivity as compared to the use of methanol as an organic modifier. This conclusion agrees with Snyder’s analysis9 of the effect of solvent type on selectivity. Temperature effects on the effective selectivity were not investigated experimentally in this study. However, a similar approach can be taken. Even though the stationary-phase type proved to be a rather effective way to alter selectivity, it is not a convenient or practical approach to optimization in HPLC. Unlike solvent strength and temperature, the stationary phase can only be varied discontinuously. Work is in progress in this laboratory to achieve continuously adjustable changes in stationary phase in RPLC. Theoretically we can compare different stationary phases by using the LSER equations for log k′w and S. However, there are some difficulties. One of the major problems is the accuracy of the log k′w and S values for all solutes since the linearity of eq 11 is limited to a relatively narrow range of mobile-phase compositions and also by the accessibility of measurements in extreme mobile-phase compositions. The other major problem is the limitation of the LSER theory. LSER is a general but not subtle model and it excludes some interactions, such as ion exchange, Lewis acid/base interactions, and shape recognition. As a result, it is very important to choose the right set of solutes covering a very wide range of general molecular properties to obtain valid coefficients. For example, it is not appropriate to select only polyaromatic hydrocarbons or only acidic solutes to test the chromatographic system and demonstrate its effective selectivity. In addition to the LSER model, we can also apply our simple approach to other models, such as principal component analysis (PCA). ACKNOWLEDGMENT The authors acknowledge the financial support of the National Institute of Health (Grant GM-54585). Received for review December 1, 1998. Accepted April 5, 1999. AC981321K (24) Weber, T. P.; Carr, P. W. Anal. Chem. 1990, 62, 2620. (25) Tan, L. C.; Carr, P. W. J. Chromatogr., A 1998, 799, 1. (26) Abraham, M. H.; McGowan, J. C. Chromatographia 1987, 23, 243. (27) Abraham, M. H. Chem. Soc. Rev. 1993, 22, 73. (28) McGowan, J. C. J. Chem. Technol. Biotechnol. 1984, 34A, 38.