Heterogeneous Adsorption of 1-Indanol on Cellulose Tribenzoate and

Anal. Chem. 2004, 76, 197-202

Heterogeneous Adsorption of 1-Indanol on Cellulose Tribenzoate and Adsorption Energy Distribution of the Two Enantiomers Gustaf Go 1 tmar,† Dongmei Zhou,† Brett J. Stanley,‡ and Georges Guiochon*,†

Department of Chemistry, The University of Tennessee, Knoxville, Tennessee, 37996-1600 and Division of Chemical and Analytical Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, and Department of Chemistry, California State University, San Bernardino, California 92407-2397

The distributions of the adsorption energies (AED) of two enantiomers, (R)-1- indanol and (S)-1-indanol, on a chiral stationary phase were measured and the results are discussed. The chiral phase used is made of cellulose tribenzoate coated on porous silica. The AEDs were determined using the expectation maximization method, a numerical method that uses directly the raw experimental isotherm data, inverts this set of data into an AED, and introduces no arbitrary information in the calculation. However, it uses the Langmuir equation as the local isotherm. The experimental data fit very well to the biLangmuir isotherm model for the more retained enantiomer. Our results show that the AEDs of these two enantiomers have no energy modes that would be identical (same mean energy, mode profile, and mode area), in contrast to numerous cases previously studied, e.g., that of the β-blockers on a Cel7A column. This indicates a significantly different retention mechanism. The investigation of the adsorption behavior of two enantiomers on a chiral selective stationary phase (CSP) provides unique opportunities for the physical chemist. After the Pasteur principle, the behaviors of these two compounds are identical except for what is related to their specific interactions with the chiral selective sites. Provided the contributions of the nonselective interactions are carefully separated from those of the enantioselective interactions, the latter can be accurately determined and the influence of the experimental conditions assessed. This approach has proven fruitful in the investigation of the separation mechanism on numerous CSPs.1 Two types of CSPs have been identified: Those for which the enantioselective sites are remote from each other, interact little or not, and on which there are no adsorbateadsorbate interactions (case I CSPs), and those for which these conditions are not fulfilled (case II CSPs). To the former type belong proteins chemically bonded to silica supports,2-5 imprinted * To whom correspondence should be addressed: (fax) 1-865-974-2667; (e-mail) [email protected]. † The University of Tennessee, and Oak Ridge National Laboratory. ‡ California State University. (1) Fornstedt, T.; Sajonz, P.; Guiochon, G. Chirality 1998, 10, 375. (2) Fornstedt, T.; Zhong, G.; Bensetiti, Z.; Guiochon, G. Anal. Chem. 1996, 68, 2370. (3) Fornstedt, T.; Sajonz, P.; Guiochon, G. J. Am. Chem. Soc. 1997, 119, 1254. 10.1021/ac030174+ CCC: $27.50 Published on Web 11/25/2003

© 2004 American Chemical Society

polymers,6,7 and, probably, the Pirkle bonded phases.8 Most cellulose-based CSPs1,9 and other phases with a high concentration of chiral carbons,10 belong to the latter type (case II CSPs). The adsorption of the two enantiomers on the first type of CSP follows usually a bi-Langmuir isotherm behavior,1,11 one of the Langmuir terms being identical for the two enantiomers and accounting for the nonselective contributions to adsorption and the other being markedly different in either its adsorption constant or its saturation capacity and accounting for the enantioselective interactions.1-5,7 A considerable amount of energy has been devoted to the study of the heterogeneity of adsorbent surfaces.12-15 One of the best measures of the degree of this heterogeneity is the adsorption energy distribution (AED) for probe compounds. Unfortunately, there are no truly homogeneous surfaces, nor any surfaces with an independently known AED. So, the different methods used to derive the AED cannot be compared, nor can the performance of these methods be assessed, and it is difficult to decide on a best approach. There is no direct method of validation of AED measurements. The exceptional properties of enantiomers may offer an attractive solution to this serious problem. In two recent studies, we used the expectation maximization (EM) method to calculate the AED of pairs of enantiomers on two different CSPs that both belong to the first group described earlier.16,17 The EM method15,18-20 is based on a robust algorithm (4) Fornstedt, T.; Go ¨tmar, G.; Andersson, M.; Guiochon, G. J. Am. Chem. Soc. 1999, 121, 1164. (5) Go ¨tmar, G.; Fornstedt, T.; Guiochon, G. Anal. Chem. 2000, 72, 3908. (6) Sellergren, B. Chirality 1989, 1, 63. (7) Chen, Y.; Kele, M.; Sajonz, P.; Sellergren, B.; Guiochon, G. Anal. Chem. 1999, 71, 928. (8) Pirkle, W. H.; Pochapsky, T. C. Chem. Rev. 1989, 89, 347. (9) Charton, F.; Jacobson, S. C.; Guiochon, G. J. Chromatogr. 1993, 630, 21. (10) Burger, D.; Neumu ¨ ller, R.; Yang, G.; Engelhardt, H.; QuiZones, I.; Guiochon, G. Chromatographia 2000, 51, 517. (11) Graham, D. J. Phys. Chem. 1953, 57, 665. (12) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992. (13) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, The Netherlands, 1988. (14) Umpleby, R. J., II; Baxter, S. C.; Chen, Y.; Shah, R. N.; Shimizu, K. D. Anal. Chem. 2001, 73, 4584. (15) Stanley, B. J.; Krance, J.; Roy, A. J. Chromatogr., A 1999, 865, 97. (16) Stanley, B. J.; Szabelski, P.; Chen, Y.-B.; Sellergren, B.; Guiochon, G. Langmuir 2003, 19, 772. (17) Go ¨tmar, G.; Stanley, B. J.; Fornstedt, T.; Guiochon, G. Langmuir 2003, 19, 6950. (18) Stanley, B. J.; Guiochon, G. J. Phys. Chem. 1993, 97, 8098.

Analytical Chemistry, Vol. 76, No. 1, January 1, 2004 197

that uses directly the raw experimental isotherm data and inverts this set of data into a distribution of adsorption constants without introducing any arbitrary information, except for the classical assumption that the local adsorption isotherm is Langmuir.12,13 In each case studied, the equilibrium isotherm data were best modeled by the bi-Langmuir isotherm and one of the two Langmuir terms was identical for both enantiomers. In the first case, the CSP was a polymer imprinted for L-phenylalanine anilide and the two probes were L- and D-phenylalanine anilide.16 In the second case, the probes were the enantiomers of two β-blockers, alprenolol and propranolol, and the CSP was a cellulase, Cel7A, immobilized on silica.17 In all cases, the AEDs calculated with the EM method give bimodal energy distributions, as expected for compounds exhibiting bi-Langmuir isotherm behavior. Importantly, there was excellent agreement between the values of the modes of the AEDs and their changes and the values and trends of the adsorption behavior. More specifically, for example, the intensity and average energy of the two modes of the AED of the enantiomers of alprenolol on Cel7A change with changing eluent pH exactly as do the saturation capacity and adsorption constant of the corresponding Langmuir terms.5,17 The difference between the energies of the high-energy modes of the two enantiomers of alprenolol is consistent with the difference between their respective binding energies with Cel7A as measured by microcalorimetry.17,21 The goal of this paper is to extend this type of investigation to a different case, one belonging to the other group of CSPs. We determined the adsorption equilibrium data of the two enantiomers of 1-indanol on cellulose tribenzoate and used the EM method15 to derive their AEDs. The single-component and competitive adsorption behavior of the enantiomers of 1-phenyl-1-propanol and 1-indanol on cellulose tribenzoate was previously studied using analytical and microbore columns.22,23 The results depend on the mobile-phase composition. For example, the adsorption behavior of the enantiomers of 1-phenyl-1-propanol is best accounted for by a competitive Langmuir isotherm if the mobile phase is made of hexane and ethyl acetate (95:5)22 and by a bi-Langmuir model if the mobile phase is a solution of 2-propanol (3%) in hexane.23 Recently, it was shown that the adsorption of 1-indanol, a structurally related compound, on the same column, also follows bi-Langmuir isotherm behavior.24 In this case, however, the two Langmuir terms of the isotherms of the two enantiomers are markedly different. Therefore, it is not possible in this case to identify one type of site giving nonselective interactions and another type giving enantioselective interactions as was the case for the adsorption of the β-blockers on Cel7A, where enantioselective and nonselective types of adsorption sites could be isolated.3-5 (19) Stanley, B. J.; Guiochon, G. Langmuir 1994, 10, 4278. (20) Stanley, B. J.; Bialkowski, S. E.; Marshal, D. B. Anal. Chem. 1993, 65, 259. (21) Hedeland, M.; Henriksson, H.; Ba¨ckman, P.; Isaksson, T.; Petersson, G. Thermochim. Acta 2000, 356, 153. (22) Khattabi, S.; Cherrak, D. E.; Fischer, J.; Jandera, P.; Guiochon, G. J. Chromatogr., A 2000, 877, 95. (23) Cavazzini, A.; Felinger, A.; Kaczmarski, K.; Szabelski, P.; Guiochon, G. J. Chromatogr., A 2002, 953, 55. (24) Zhou, D.; Cherrak, D. E.; Kaczmarski, K.; Cavazzini, A.; Liu, X.; Guiochon, G. Chem. Eng. Sci. 2003, 58, 3257.

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Analytical Chemistry, Vol. 76, No. 1, January 1, 2004

THEORETICAL SECTION Isotherm Models. The Langmuir isotherm is the simplest nonlinear expression relating the concentrations in the mobile phase, C, and that in the stationary phase, q. It corresponds to the case of an analyte that has a limited number of (identical) adsorption sites and undergoes reversible adsorption

q)

qsbC aC ) 1 + bC 1 + bC

(1)

In this equation, qs is the saturation capacity and b the equilibrium constant of adsorption. b is exponentially related to the energy of adsorption, E

b ) b0eE/RT

(2)

b0 is a preexponential factor, R the universal gas constant, and T the temperature (K). In the Langmuir isotherm (eq 1), a () bqs) is the Henry constant (i.e., the ratio of the concentrations of the solute in the stationary and the mobile phases at infinite dilution) or initial slope of the isotherm. At infinite dilution, the retention factor, k, is

k ) Fa ) Fbqs

(3)

where F is the phase ratio, Vs/Vm ) (VG - Vm )/Vm ) (1 - )/, where Vs, Vm, and VG are the volumes occupied by the solid phase, the mobile phase, and the whole column packing (i.e., the geometrical volume of the column), respectively, and  is the total porosity of the column. Assuming that there are two types of adsorption sites on the stationary phase and that the analyte can interact independently with either type, the following isotherm model is obtained, by adding the two Langmuir expressions corresponding to the two types of sites

q)

aIC aIIC qI,sbIC qII,sbIIC + ) + 1 + bIC 1 + bIIC 1 + bIC 1 + bIIC

(4)

where I and II denotes the two different types of sites. This model (bi-Langmuir) was previously used for the adsorption of different β-blockers on Cel7A.3-5 The separation factor under analytical conditions is the ratio of the sum of the Henry constants of adsorption of the two enantiomers (i.e., the ratio of their retention factors):

R)

∑a ) ∑b q ∑a ∑b q 2

2 s,2

1

1 s,1

(5a)

Obviously, the origin of enantioseparation might be found either in a difference of interaction energy on at least one type of site (bi,j) or to a different number of sites (qs,i,j) that can interact with each of the two enantiomers. Adsorption Energy Distribution. The experimental isotherm results from the summation of the contributions to the global isotherm of the adsorption of the compound studied onto all the

sites available on the surface. The distribution of the equilibrium constants, b, in the distributed b-space is given by a Fredholm integral of the first kind12,13

q(C) )

∫

lnbmaxf(ln

lnbmin

b)bC d ln b 1 + bC

(6)

where f(ln b) is the distribution function that indicates the number of adsorption sites on which the equilibrium constant is equal to b and is interpreted as qs(ln b) or qs(b). The EM method calculates the isotherm data, qcal(C), from the adsorption model f(ln b) by replacing the integral with a sum across a grid of ln b values:15,18-20 bmax

qcal(Cj) )

∑f(ln b )θ(C ,b )∆ln b

(7a)

b iC j 1 + biCj

(7b)

i

j

i

bmin

where

θ(Cj,bi) )

The solution, f(ln b), is updated iteratively through the relationship Cmax m+1

qs

(bi) ) f(ln bi)

qexp(Cj)

∑ θ(C ,b )∆ln b q j

Cmin

j

(8)

cal(Cj)

where qexp(C) is the experimental isotherm data and m is the iteration number. The EM algorithm is an iterative procedure that must start with an initial estimate of the adsorption distribution. By starting with a guess of maximum ignorance, i.e., a constant distribution across all possible values, the minimum amount of arbitrary information is introduced in the calculation and the data dictate the shape of the numerical solution to the maximum possible extent. The application of the EM method in chromatography has been previously described.16-19 Its successful application requires that the isotherm data acquired scan a range of several orders of magnitude of the solute concentration. This requirement is most important because the equilibrium constant, bi, may exist over a wide range of values and the sampling of the sites having different values of b by the analyte occurs in inverse proportion to its concentration in the mobile phase, C. To obtain a meaningful AED, the isotherm data should extend to concentrations low enough for the highest energy sites (which are the first ones to be occupied) to operate under quasi-linear conditions. At the other end of the range, the highest concentrations should be sufficient for the sites with low b values, the lowest energy sites, to operate under nonlinear conditions. EXPERIMENTAL SECTION Equipment. All the measurements used in this work were carried out with a HP 1090 liquid chromatograph (Agilent Technologies, Palo Alto, CA), equipped with a multisolvent

delivery system, an automatic injector, a column oven, a diodearray detector, and a data acquisition system.24 Chromatographic System. This system was previously described.24 The column used for the experiment was a 200 × 10 mm column, packed in-house with Chiracel OB (cellulose tribenzoate coated on a silica support; Daicel, Tokyo, Japan). The mobile phase was a solution of n-hexane and 2-propanol (92.5:7.5, v/v). All the experimental data were measured at room temperature, with a 2.5 mL/min mobile-phase flow rate. Procedures. Frontal Analysis. The adsorption isotherms of the 1-indanol enantiomers were measured by frontal analysis in the traditional mode, as described previously.22-24 The signal of the UV detector was recorded at 280 nm. The concentration range investigated was 0.18-18 g/L for (R)-1-indanol and 0.21-21 g/L for (S)-1-indanol (i.e., the ratio of the highest and lowest concentrations was 100 for both enantiomers), using 19 data points. The upper limit was close to the solubility of 1-indanol in the mobile phase. The best numerical values of the bi-Langmuir model were estimated by fitting the experimental adsorption data to the model equation, using the least-squares Marquardt method modified by Fletcher.25 Procedures. Expectation Maximization. Three hundred points in the ln b-space were used to digitize it, with a constant spacing ∆ln b, in the range between bmin ) 0.0025 L/g and bmax ) 5.55 L/g. The differential saturation capacity at each point, qs(ln bi)

qs(bi) ) f(ln bi) ∆(ln bi)

(9)

is then plotted against ln bi as the adsorption constant distribution. The corresponding curves are reported later as the results of this work. The values comprising one mode in these distributions can be added and give an estimate of the saturation capacity of the corresponding type of sites. The average adsorption energy of these sites is the corresponding mean b-value. It is not possible accurately to calculate information beyond the range corresponding to the experimental data. The range of equilibrium constants sampled is the range between b′min and b′max, with b′min ) 1/Cmax and b′max ) 1/Cmin. It defines the adsorption region that is effectively studied. The lowest values of b, those that correspond to the highest concentrations for which isotherm data were measured, were not adequately saturated under the conditions of the experiment (measurements could not be performed at higher concentrations for the lack of solubility of 1-indanol in the mobile phase). This limitation results in a divergence of the distribution at bmin. Decreasing bmin in the calculation circumvented this divergence for the second eluted enantiomer, S, but not for the other one. Decreasing bmin further than shown in this paper resulted in a shift of the divergence (accompanied with an increase of qs). However, even for (R)-1indanol, decreasing bmin does help to deconvolute the information at low b-values away from the higher energy b-sites without affecting the quality of the information at those higher b-values. Therefore, bmin was lowered until no further improvement on the overall appearance of the distribution was noted (bmin ≈ b′min/20). (25) Fletcher, R. A modified Marquardt sub-routine for nonlinear least squares, AERE-R6799-Harwell.

Analytical Chemistry, Vol. 76, No. 1, January 1, 2004

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At bmax, the data are better sampled and few divergences were observed (they are totally absent for (S)-1-indanol). Thus, bmin and bmax () b′max for (R)-1-indanol) are used to calculate the distribution with the EM algorithm and b′min and b′max define the effective range within which the distribution may be interpreted accurately. In the figures, the whole range (bmin to bmax) is graphed but b′min and b′max are indicated. One or 20 million iterations were performed by the algorithm. RESULTS AND DISCUSSION The adsorption isotherms of the enantiomers of 1-indanol were measured on cellulose tribenzoate coated on silica, using a mixture of hexane and 2-propanol as the mobile phase. The adsorption data acquired by FA were fitted to different isotherm models, using a nonlinear regression.24 The experimental data fitted best to the bi-Langmuir isotherm model. Calculations carried out using the best isotherm model gave an excellent agreement between the experimental and theoretical overloaded chromatographic bands, as had been the case also for 1-phenyl-1-propanol using a mixture of hexane and 2-propanol as the mobile phase.23 Analysis of the set of best numerical values of the parameters showed that the CSP studied interact with the enantiomers at the two different types of sites. Single-component isotherms revealed that neither the adsorption energy nor the saturation capacity of either type of site is the same for the two enantiomers.24 Investigating the AED of these enantiomers affords an interesting alternative to analyze the experimental adsorption isotherm data. The EM method allows the direct determination of the AED of the enantiomers studied from the experimental adsorption data.15,18,19 If the actual surface contains only one type of site and if these sites have a reasonable degree of homogeneity, the AED has a single mode. It has two modes if the surface is occupied by two different types of sites but each type of sites is relatively homogeneous. Figure 1 shows the experimental isotherm data (symbols) obtained for 1-indanol and the best isotherm models derived by curve fitting (solid lines). The (S)-enantiomer being more retained than the (R)-enantiomer, its isotherm is higher (a larger stationaryphase concentration is achieved at equilibrium with a given mobilephase concentration). Note that practically the same solid lines are obtained if the isotherms are calculated from the adsorption energy distributions given by the EM method. It is impossible to discriminate between the EM fit and the analytical bi-Langmuir fit; they coincide for both enantiomers. The lack of high concentration data is obvious. The slopes of the isotherms at the highest concentrations are still significant. Consequently, the values of the saturation capacity of the lower energy sites are relatively inaccurate since the isotherms are ended too far from saturation. However, as noted earlier, it was impossible to increase this range due to solubility problems. The best values of the numerical coefficients of the isotherms are reported in Table 1. We note that the largest values of the product bIC for the (R)- and (S)enantiomers are 0.22 and 0.42, values that are, respectively, insufficient and barely sufficient for an accurate estimate of the corresponding saturation capacity4 (see later, Figure 4). Figure 2 shows the AED derived from the simulated and the experimental isotherms of the 1-indanol enantiomers, using 1 million iterations. For the simulated AEDs, the values of the solidphase concentrations at equilibrium were calculated from the best bi-Langmuir isotherm equations and the numerical values of the 200 Analytical Chemistry, Vol. 76, No. 1, January 1, 2004

Figure 1. Single-component equilibrium isotherms for (R)- and (S)1-indanol. Experimental conditions: column, 200 × 10 mm; stationary phase, immobilized cellulose tribenzoate on silica; mobile-phase flow rate, 2.5 mL/min. Symbols: experimental data, (O) (R)-enantiomer, (/) (S)-enantiomer. Lines: best calculated bi-Langmuir isotherms (parameters in Table 1), dotted lines and best fits corresponding to the AED model, dashed lines. Eluent: hexane/2-propanol 92.5:7.5. Main figure: high concentration range, up to ∼21 g/L. Insets: low concentration range, up to 0.9 g/L. (a) Number of iterations 1 000 000 (106); (b) number of iterations 20 000 000 (2 × 107).

isotherm parameters in ref 24, for each of the mobil-phase concentrations used in the measurement of the experimental isotherm data. Thus, the mobile-phase concentrations and the number of data points (19) were the same for the calculated data as for the experimental isotherms. By calculation, the data points so obtained fit exactly to the bi-Langmuir model. They were used for the calculation of an AED, following the EM method. For (S)1-indanol, the AEDs from the experimental and the simulated isotherms coincide exactly whereas the situation is different for the (R)-enantiomer. For the latter, the “shoulder” of the simulated isotherm is wider than the experimental one and it appears at slightly higher b-values. However, attempts at using a Toth isotherm model would be unsuccessful. Although the adsorption data would fit reasonably well to this isotherm model, the AED corresponding to a Toth model has a single energy mode, which is clearly ruled out by the AED derived from the experimental isotherm data. Calculations carried out with the EM program to

Table 1. Estimated Parameter Values for the Different Sites, Comparison between EM and Bi-Langmuir Fitting

bR,I (L/g) qR,I,s (g/L) bR,II (L/g) qR,II,s (g/L) bS,I (L/g) qS,I,s (g/L) bS,II (L/g) qS,II,s (g/L)

EMa

BLb

0.072 26 0.018 95 0.36 8.6

0.012 106 0.11 14 0.020 89 0.36 8.6

a Values obtained from Figure 3. q - and b-values from the zero and s first moments, respectively. b Bi-Langmuir parameter values from ref 24.

Figure 3. As for Figure 2, but number of iterations 20 000 000 (2 × 107). Distributions from b ) 0.0025 (ln b ) -6.0) to b ) 5.55 (ln b ) 1.7).

Figure 2. Experimental and simulated AEDs of (R)- and (S)-1indanol. Simulated isotherms calculated from the bi-Langmuir parameters of ref 1. (O) (R)-experimental + (R)-simulated; (/) (S)experimental; (3) (S)-simulated. R(E) denotes distribution from experimental isotherm, R(S) distribution from simulated isotherm, and (S) distributions for the (S)-enantiomer. The number of iterations is 1 000 000 (106) and 300 grid points. Distributions from b ) 0.0025 (ln b ) -6.0) to b ) 5.55 (ln b ) 1.7). The (vertical) dashed lines denote b′min (1/Cmax), i.e., lowest bonding constants, and b′max (1/ Cmin), i.e., highest bonding constants, actually measured. Within this range, the determination of the energy distribution can be considered to be accurate. The values beyond those values are more or less uncertain.

derive a hypothetical AED corresponding to a simulated Toth isotherm model for (R)-1-indanol do not converge after 1 million iterations. There is a plateau at low energies and a tail extending beyond b ) 0.4 L/g (not shown), which does not agree with the experimental results. Figure 3 shows the same AEDs, derived from the simulated and the experimental isotherms as in Figure 2, but after a 20 times larger number of iterations. The AEDs obtained for the (S)enantiomer with either the experimental or the simulated isotherms are practically identical. The low-energy mode of the former one is very slightly wider, but this is hardly noticeable and the position of its mode maximum is the same as that of the latter (i.e., simulated) one (cf. Figure 3). This result is in strong support of the earlier observation that the bi-Langmuir model accounts best for the adsorption behavior of this enantiomer on the stationary phase.24 Again, the findings for (R)-1-indanol are

different. Whereas the AED derived from the simulated isotherm shows a first, unresolved energy mode at a value intermediate between the two modes of (S)-1-indanol, the AED derived from the experimental isotherm gives a well-resolved high-energy mode butsmost significantlysa stronger divergence at the low-energy end of the distribution (cf. Figure 3). These results do not seem to be consistent with the earlier finding that a bi-Langmuir model accounts well for the adsorption behavior of the (R)-enantiomer.24 Again, as in the discussion of Figure 2, a Toth isotherm model must be ruled out since the AED derived from this simulated model has no common features with the experimental AEDsit exhibits a very wide mode with a maximum at b ∼ 0.2 (not shown). Although it seems that the adsorption behaviors of the two enantiomers are completely different, this conclusion cannot be made definitive. As noted earlier, the largest values of the products bIC at which experimental data were acquired for the (R)- and the (S)-enantiomers are 0.22 and 0.42, respectively, meaning that the nonlinear range of the first (low-energy) mode of the bi-Langmuir model is sampled in a far narrower range for the former enantiomer than for the latter, which may explain why the AED of the (R)-enantiomer does not converge, even after 20 million iterations, while that of the (S)-enantiomer does (cf. Figures 3 and 4). Nevertheless, the relative positions of the two distributions confirm the results of the earlier study,24 that there is no “common” or “nonselective” type of site for the two enantiomers. This is confirmed by the strong differences between the best values obtained for the coefficients of the bi-Langmuir isotherms of the two compounds. The value of bS,I is almost twice as large as that of bR,I, bS,II is more than 3 times as large as bR,II, and bR,II is more than 5 times larger than bS,I. Figure 4 illustrates why it is important to collect data at sufficiently high concentrations. The same AED calculations as reported above were made using the experimental and simulated isotherms of (S)-1-indanol used previously but after omitting the five highest concentrations. In this truncated isotherm, the largest value of the product bS,IC is 0.21, a value close to the one obtained for (R)-1-indanol (0.22). The AEDs were calculated based on those isotherms, using 20 million iterations in the EM program. The comparison of the AEDs obtained for (S)-1-indanol in Figures 2 (complete isotherm, 1 million iterations), 3 (complete isotherm, Analytical Chemistry, Vol. 76, No. 1, January 1, 2004

201

Figure 4. As for Figure 3, but only for (S)-1-indanol with the isotherm truncated at C ) 10.6 g/L, corresponding to bS,IC ) 0.21.

20 million iterations), and 4 (truncated isotherm, 20 million iterations) is striking. The most obvious feature is that the experimental AED in Figure 4 shows divergence at low energies, like the AED for the (R)-isomer in Figure 3. Also, the excellent agreement between the calculated and the simulated distributions has vanished. The simulated AED, at low energies, is closer to the one observed at 1 million iterations, Figure 2. The experimental one, at low energies, is similarly close to the simulated AED for (R)-1-indanol at 20 million iterations, Figure 3. Although the high-energy mode has a maximum at the same value as the complete isotherm of (S)-1-indanol, the resolution between the two modes is destroyed. From Figures 3 and 4, it can be concluded that an approximate value of 0.2 for the product bIC is insufficient to allow convergence of the calculation of the AED. A value of 0.4 seems to be a threshold limit above which convergence is probable. This result suggests a semiquantitative estimate of the highest concentration required to obtain a satisfactory result in the determination of the AED. Another interesting observation is that the accuracy for the high -energy mode of the (S)-enantiomer in Figure 4 is better than that for the (R)-enantiomer in Figure 3 because the ratio bII/bI or the value of bIICmax is larger for the former (18 and 4, respectively) than for the latter (9 and 2, respectively). So, the high-energy mode is better resolved and better sampled for the (S)- than for the (R)-enantiomer. In summary, for the more retained (S)-enantiomer, the conclusions derived from the AED calculated by the EM method are in excellent agreement with those resulting from the analysis of the characteristics of the best bi-Langmuir model. For the (R)enantiomer, the experimental data were acquired in too narrow a range of concentrations to sample sufficiently the nonlinear behavior of the low-energy sites and not even a high number of iterations and an expanded window at the low-energy side could overcome the divergence. Nevertheless, the limited results obtained suggest that this enantiomer also has a bimodal distribution of adsorption energies. CONCLUSION Our results confirm the importance of the determination of the affinity energy distribution, which is calculated from the same

202 Analytical Chemistry, Vol. 76, No. 1, January 1, 2004

raw adsorption isotherm data as used for the modeling of the isotherm. The expectation maximization method gives these distributions for the two enantiomers studied, knowing only the range of b-values and assuming a Langmuir model for the local isotherm. The distributions obtained are fully consistent with the results of the analysis and modeling of the adsorption equilibrium data, but they inform in the choice of the best isotherm model and in the interpretation of the results. The only restriction is that the EM method requires that a large amount of data be available at high concentrations; otherwise, convergence of the numerical calculations toward the AED cannot take place. The agreement confirms that the EM method gives AEDs that are physically meaningful and are not artifacts of numerical calculations. In the case of cellulose tribenzoate coated over porous silica, both the AED and the modeling of adsorption data give results that are remarkably consistent for (S)-1-indanol on a chiral selective phase that contains a high density of chiral carbons. The results are more ambiguous for the less retained enantiomer (R)1-indanol. Whereas the agreement between the results of the EM method and those obtained by direct fitting of the bi-Langmuir model to the data is good for the properties of the high-energy sites, the lack of sufficient adsorption data for the low-energy sites (i.e., at high concentrations) prevents the convergence of the EM calculations in the corresponding range of adsorption constants. This constitutes the essential limitation of the method. The AED results show also that a two-site model accounts for the isotherm data, for both the (S)- and the (R)-enantiomers. Since the stationary phase used in this work is a case II CSP, as noted in the introduction, this did not have to be so because the assumptions of the bi-Langmuir model do not hold for case II CSPs. Yet, the bi-Langmuir model does hold for the (S)-enantiomer. By contrast, although a two-site model is also apparent for the (R)-enantiomer, it does not appear that a bi-Langmuir model accounts well for the AED results. The distributions calculated from the isotherm, R(S), and shown in Figures 3 and 4, suggest that the AED predicts accurately the value of the parameter bII. However, if the mean or maximum value is accurate, the two modes are unresolved in Figure 3 for the (R)-enantiomer. Then, if a bi-Langmuir model would describe the data, the experimental value for the bII peak in the AED, R(E), should be the same as R(S). Since the two values are significantly different, this suggests that the AED does not agree with the bi-Langmuir model, despite the effect of the undersaturation of bI. Therefore, the more retained enantiomer may exhibit biLangmuir model adsorption behavior on case II CSPs as on case I CSPs, but a different retention mechanism holds for the less retained enantiomer, although it is likely a two-site type of model. ACKNOWLEDGMENT This work was supported in part by Grant DE-FG05-88-ER13869 of the U.S. Department of Energy and by the cooperative agreement between the University of Tennessee and Oak Ridge National Laboratory. Received for review May 1, 2003. Accepted October 8, 2003. AC030174+