Injection Technique for Generating Accurate Adsorption Isotherm Data

With the use of the elution by characteristic points (ECP) method, the adsorption isotherms are rapidly determined from the diffusive part of an overl...
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Anal. Chem. 2008, 80, 7887–7893

Injection Technique for Generating Accurate Adsorption Isotherm Data Using the Elution by Characteristic Points Method Jo¨rgen Samuelsson and Torgny Fornstedt* Department of Physical and Analytical Chemistry, BMC Box 577, SE-751 23 Uppsala, Sweden With the use of the elution by characteristic points (ECP) method, the adsorption isotherms are rapidly determined from the diffusive part of an overloaded elution profile. However, very large injection volumes are required, which lead to extremely tailed rears of the injection profiles. The ECP method is based on a theory assuming rectangular injection profiles and can therefore not account for such profiles. Consequently, the use of the ECP method in the traditional way, with classical full-loop injections, results in serious errors on the determined adsorption isotherm parameters which has not been demonstrated until this study. Therefore, we developed and validated a new experimental injection method, here denoted the “cutinjection” technique where instead nearly rectangular injection profiles are generated. The result convincingly shows that adsorption isotherms acquired by using the new cut-injection technique nearly totally coincide with adsorption isotherms determined using the accurate reference methods. Determination of adsorption isotherms is important for characterization of stationary phases intended both for analytical as well as for preparative use. Many new stationary phases for rapid high-efficiency separations have been developed most lately.1,2 It is important to follow up this trend by developing or improve correspondingly rapid methods for proper determination of both the thermodynamics and the kinetics of the new phase systems. Recently, a new promising evaluation tool was introduced into the chromatographic community, the calculation of the adsorption energy distribution (AED) directly from the raw adsorption isotherm data (i.e., the real data points). The aim of AED is to ascertain the degree of heterogeneity of the adsorption, before the model selection is made.3-5 Adsorption isotherm data for AED calculations could also be determined, e.g., using frontal analysis (FA),6 the perturbation peak (PP) method,7,8 or the tracer peak * Corresponding author. Tel: +46 18 471 48 79. Fax: +46 18 55 50 16. E-mail: [email protected]. (1) Swartz, M. E. J. Liq. Chromatogr. 2005, 28, 1253–1263. (2) Cunliffe, J. M.; Maloney, T. D. J. Sep. Sci. 2007, 30, 3104–3109. (3) Samuelsson, J.; Franz, A.; Stanley, B. J.; Fornstedt, T. J. Chromatogr., A 2007, 1163, 177–189. (4) Stanley, B. J.; Krance, J. J. Chromatogr., A 2003, 1011, 11–22. (5) Stanley, B. J.; Guiochon, G. Langmuir 1994, 10, 4278–4285. (6) Seidel-Morgenstern, A. J. Chromatogr. A 2004, 1037, 255–272. (7) Lindholm, J.; Forsse´n, P.; Fornstedt, T. Anal. Chem. 2004, 76, 4856–4865. (8) Lindholm, J.; Forsse´n, P.; Fornstedt, T. Anal. Chem. 2004, 76, 5472–5478. 10.1021/ac8010999 CCC: $40.75  2008 American Chemical Society Published on Web 09/10/2008

method.9-11 However, the serious drawbacks of these methods are the large consumption of both time and substance. The most recently developed method is the numerically based inverse method (IM). Here, the adsorption isotherms parameters are estimated from an iterative fitting process to a few overloaded peaks.12 However, the IM does only deliver parameter estimates of the best isotherm model and not the real data points of the adsorption isotherm necessary for AED calculations. In the elution by characteristic points (ECP) method,13 the adsorption isotherm data points are simply generated from the diffusive part of an overloaded elution profile. The advantages are several: the method is very simple, straightforward, and rapid and has high precision because of a large amount of raw isotherm data points are produced by just one injection. Moreover, as for the IM, the ECP method requires very small amounts of sample and is therefore most suitable for life science studies. The ECP calculations are based on the ideal model assuming infinite column efficiency; therefore, it is restricted to be used only for highly efficient separation systems. The error in the adsorption isotherm for different column efficiencies have been systematically investigated using elution profiles generated by the equilibrium-dispersive model for adsorption described by Langmuir models14 and bi-Langmuir models.15 In the Langmuir cases, N higher than 2000 is required for an error less than 3%14 whereas in the bi-Langmuir cases generally N higher than 5000 is required for an error of less than 5%.15 However, the ECP method also assumes rectangular injection profiles and surprisingly, this important issue have never been considered. Another major problem with conducting adsorption isotherm studies using both the ECP method and the IM is that a high maximum eluted concentration is required because the adsorption data are generated from the eluted peak(s). The fulfillment of this requirement is counteracted by the fact that a sample zone is more or less diluted during its migration through the column. One solution to this problem might be, from a theoretical point of view, to inject low volumes of highly concentrated sample solutions. But this strategy is seldom fruitful due to the solutes low solubility. (9) Samuelsson, J.; Forsse´n, P.; Stefansson, M.; Fornstedt, T. Anal. Chem. 2004, 76, 953–958. (10) Samuelsson, J.; Arnell, R.; Diesen, J. S.; Tibbelin, J.; Paptchikhine, A.; Fornstedt, T.; Sjo ¨berg, P. J. R. Anal. Chem. 2008, 80, 2105–2112. (11) Arnell, R.; Fornstedt, T. Anal. Chem. 2006, 78, 4615–4623. (12) Felinger, A.; Cavazzini, A.; Guiochon, G. J. Chromatogr., A 2003, 986, 207– 225. (13) Glueckauf, E. Nature 1945, 156, 748–479. (14) Guan, H.; Stanley, B. J.; Guiochon, G. J. Chromatogr., A 1994, 659, 27–41. (15) Ravald, L.; Fornstedt, T. J. Chromatogr., A 2001, 908, 111–130.

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To bypass this problem, larger volumes must instead be injected using a more diluted sample solution. Unfortunately, for large injection volumes, the experimental injection profile deviates considerably from the rectangular injection profile.16,17 Quin ˜ones et al.17 found that the simulated elution profiles (using rectangular injection profiles) poorly accounts for the rear part of the experimental eluted peaks. In the case of the IM it is possible to implement the experimentally acquired injection profile in the calculations.12,18 However, this strategy is indeed not possible using the ECP method. The aim of this study is to introduce and validate a completely new injection technique for generating “rectangular” injection profiles and thus decrease considerably the greatest remaining source of error using the ECP method with today’s highly efficient separation system.

where VR(C) is the elution volume corresponding to the mobile phase concentration C. Vinj is the injected volume, V0 is the holdup volume and Va is the stationary phase volume in the column. dq/dC is the slope of the adsorption isotherm. The adsorption isotherm can be determined by integrating the rear of a profile according to

THEORY Adsorption Isotherm Models. An adsorption isotherm describes the distribution of the solute in stationary and mobile phases at equilibrium at a specific and constant temperature. The Langmuir adsorption isotherm is the simplest model in which the solute reversibly is adsorbed at a limited number of identical adsorption sites. The n-Langmuir adsorption isotherm model simply assumes n independent different adsorption sites, and the equation becomes

where m is the data point corresponding to elution concentration C and ∆Cj is chosen so that Σ∆Cj ) C. The selection of the starting point where the concentration really is zero is impossible due to the large system noise at extremely low concentrations. In this study we selected the starting point by visual inspection of the chromatograms, in accordance with previous studies.14,15

n

q)

Kiqs,iC

∑ 1+KC

(1)

i

i)1

where Ki and qs,i is the association equilibrium constant and the monolayer saturation capacity for the ith site, respectively. If n ) 1, the model is the Langmuir and if n ) 2 the bi-Langmuir, etc. The initial slope of the adsorption isotherm gives the sum of the distribution coefficients for all sites, a, which is equivalent to the sum of the products of qsK, for all sites. The surface coverage (θi) for the ith adsorption site is defined as

θi )

KiC 1 + KiC

(2)

Determination of Adsorption Isotherms. Determination of adsorption isotherms can be done using several different methods.6 In this study, adsorption isotherms acquired with the ECP are compared with the adsorption isotherms determined using the PP and FA methods. The FA and PP methods were used as reference methods because these methods are considered to be highly accurate.7 The diffuse rear of an overloaded peak using the ideal model and assuming a convex adsorption isotherm and rectangular injection profile is described by

(

VR(C) ) Vinj + V0 1 +

Va dq V0 dC

)

(3)

(16) Katti, A.; Czok, M.; Guiochon, G. J. Chromatogr. 1991, 556, 205–225. (17) Quin ˜ones, I.; Ford, J. C.; Guiochon, G. Anal. Chem. 2000, 72, 1495–1502. (18) Arnell, R.; Forsse´n, P.; Fornstedt, T. J. Chromatogr., A 2005, 1099, 167– 174.

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q(C) )

1 Va

∫

C

0

(VR(C) - V0 - Vinj) dC

(4)

Equation 4 is discretized to a sum to cover the experimental data: m

q(C) )

∑

1 (V (C ) - V0 - Vinj)∆Cj Va j)0 j j

(5)

EXPERIMENTAL SECTION Equipment and Chemicals. An Agilent 1100 chromatographic system from Agilent Technology (Palo Alto, CA) was used, equipped with a binary pump and a diode array UV detector. For larger injection volumes than 900 µL, a Rheodyne 7125 (Cotati, CA) injector with a 4980 µL injection loop was used (all injection volumes were estimated by integrating the injection profile for a sample containing 0.25 mM methyl mandelate). A Kromasil C18 (nominal particle size 5 µm; 4.6 mm × 150 mm i.d.) column from Eka Chemicals AB (Bohus, Sweden) was placed in a laboratoryassembled column jacket and was temperature controlled (25.0 °C) using a LAUDA type B circulating water bath (Ko¨ningshofen, Germany). The solutes were methyl mandelate (99%) (MM) and propranolol hydrochloride (99%) (PR). The hold-up time marker was uracil (99%), and the organic modifier was methanol (CHROMASOLV). All chemicals were bought from Sigma-Aldrich. The water was from a Milli-Q water purification system ZMQS 5000Y from Millipore (Molsheim, France). Procedures. The eluent for the MM-system was 30/70 (v/v) methanol/water and for the PR-HCl system 45/55 (v/v) methanol/ 60 mM acetate buffer (pH ) 4.74 in the pure buffer and 5.46 in the eluent, using aqueous pH references standard solutions). The eluent flow rate was 0.70 mL/min. The addition of PR to the eluent did not change the pH in the eluent up to 10 mM of PR. The column hold-up time in the MM-system was determined to 2.21 min. (RSD ) 0.04%; n ) 5). In the PR-HCl-system, the hold-up time was 2.13 min. (RSD ) 0.02%; n ) 5). The hold-up volume from the PEEK-tee connecting the pumps to the detector (excluding the columns dead volume) was determined with pump pulses of uracil to 2.03 mL (RSD ) 0.6%; n ) 5). The column efficiency was determined for PR to N ) 9182 (RSD ) 1.9%; n ) 4) and for MM to N ) 10 532 (RSD ) 1.6%; n ) 4) using the moment method. Adsorption isotherms for reference purposes were determined for both MM and PR using frontal analyses in the staircase mode and the perturbation peak method. The adsorption models were analyzed before selection using a four-step methodol-

Figure 1. Experimental injection profiles for (a) different injection volumes between 50 and 900 µL aliquots and for (b) a 4980 µL traditional full-loop injection overlaid with the Cut-injection. For other experimental conditions, see the Experimental Section.

ogy: (1) Scatchard plots (2) AED (3) statistical evaluation of the fit to rival model, and finally (4) the model’s ability to predict experimental profiles (data not shown). In this work the equilibrium dispersion model was used as column model and numerically solved using a modified Rouchon algorithm.19 The detector responses were transformed to concentrations by fitting the detector responses from the FA experiments with a third degree polynomial. RESULTS AND DISCUSSION In this study, we present and validate a new experimental injection technique producing almost rectangular injection profiles for generating accurate adsorption isotherm data with the ECP method. Two components were used as model compounds; methyl mandelate (MM) showing a homogeneous adsorption behavior and propranolol (PR) showing a typical heterogeneous interaction, respectively. Determination of Injection Profiles and Development of the “Cut-Injection” Technique. In a separation system, the total dead volume will contribute to band broadening and result in a lowering of the efficiency. Not only the column have dead volumes but also the connections between the injector and the detector. The dead volume connections are rather easily minimized. The injection loop itself is also a mixing chamber but acts somewhat differently compared to other dead volumes. During an injection, the first portion leaving the injection loop will have experienced less time in the injection loop compared with the last portion. This will lead to heterogeneous injection profile distributions. In Figure 1a, experimental injection profiles are presented for different injection volumes. One can note that the tail of the injection profile gets more pronounced as the injection volume increases. At very large injection volumes, a stable concentration plateau is established (between 250 and 900 µL in Figure 1a and 4980 µL in Figure 1b). Figure 1a,b demonstrates clearly that large injection volumes generate experimental injection profiles with extreme deviations from the rectangular injection profiles. (19) Forsse´n, P.; Arnell, R.; Fornstedt, T. Comput. Chem. Eng. 2006, 30, 1381– 1391.

To improve the injection profile, we stopped the injection before the dispersed injection-profile tail has entered the column through bypassing the injection loop. Figure 2 shows a detailed schematic picture of this procedure. At t ) 0 min (Figure 2a), a loaded injection loop is switched to the inject mode. As fast as the sample is injected into the column, the rear of the sample plug in the injection loop starts to become diluted, see Figure 2b. At t ) 4.44 min, which is before the disperse sample zone in the injection loop has reached the column, the injector is reswitched to its load position, see Figure 2c2. Figure 2c1 shows the resting situation if the reswitch was not done; in this standard full-loop injection, the mixed zone in the injection loop is also injected into the column. In Figure 1b, the injection profiles from the Cut and full-loop injections (4980 µL) are overlaid, and the improvement with the new injection technique is very convincing. In the following, we call ECP performed after a Cut-injection and traditional full-loop injection for “Cut-ECP” and “traditional-ECP”, respectively. Determination of the Degree of Improvement Using the Cut-ECP Method As Compared to the Traditional-ECP Method. The degree of improvement on the determined adsorption isotherm using the Cut-ECP method compared with the traditional-ECP method was investigated. As reference methods, the classical FA method and PP methods were used. The adsorption process for MM was described by a Langmuir adsorption isotherm with a large saturation capacity and that of PR was described by a bi-Langmuir adsorption isotherm. The top of Table 1 shows the Langmuir adsorption isotherm parameters for the model component MM and the top of Table 2 shows the bi-Langmuir parameters for PR, as determined by the reference methods FA and PP methods, respectively. The adsorption isotherm parameters determined with FA and PP gave very similar results for both model components (cf. Tables 1 and 2, in line with a previous PP method validation study.7 The averaged parameters from the FA and PP methods were used as reference when calculating the error of the ECP methods (in Tables 1 and 2). The determined adsorption isotherm parameters for MM using ECP method from the 4980 µL full-loop (traditional-ECP) injection and the Cut-injection are presented in Table 1. For the traditionalECP method, the errors of the Langmuir adsorption isotherm parameters were more or less tremendous; the error of the distribution coefficient, the a term, was +12.3% while the error of the equilibrium constant, the K term, was +40.4%. The reason for the larger error in the latter case is probably because the value of the K term depends strongly on the curvature of the rear of the eluted peak. The Cut-ECP method in contrary, showed very small errors of -1.8% for the a term and K term. These errors are only slightly larger than the variation between the FA and PP methods and why the Cut-ECP method is an excellent alternative to PP and FA for determination of adsorption isotherms; it covers all necessarily concentration ranges and has also the advantage of a close-to-rectangular injection profile. In addition, the ECP method is much quicker than the FA and PP methods. The reference methods as well as the ECP method were determined up to a surface coverage of around 40% (cf. Table 1). The lower part of Table 1 shows a traditional-ECP adsorption isotherm generated by injection volumes of 50, 100, 250, 500, and Analytical Chemistry, Vol. 80, No. 20, October 15, 2008

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Figure 2. A schematic representation of the events during the Cut-injection technique (a, b, and c2) compared with classical full-loop injection (a, b, and c1). Table 1. Adsorption Isotherm Determined for Methyl Mandelate (MM) Using the Cut-ECP Method and the Traditional-ECP Method, with Various Injection Volumes, Compared with the Average of the Reference Methodsa Cmaxb method [mM] FA PP CUT 4980 50 100 250 500 900

96 94 10 15 26 38 53

θ [%]

a

37.0 37.0 36.1 35.6 5.56 8.11 13.3 18.3 23.8

7.83 7.77 7.66 8.76 7.78 7.87 7.97 8.21 8.44

error [%]

qs [M]

error [%]

-1.8 12.3 -0.3 0.9 2.2 5.3 8.2

1.32 1.33 1.33 0.01 1.06 -20.0 1.28 -3.4 1.33 0.4 1.28 -3.4 1.24 -6.4 1.21 -8.7

error K [M-1] [%] 5.93 5.84 5.78 8.26 6.07 5.93 6.24 6.63 6.96

-1.8 40.4 3.1 0.8 6.0 12.7 18.3

a FA and PP, listed at top in italic. The adsorption isotherms are determined using a sample of 100 mM MM and injecting 50, 100, 250, 500, 900, and 4980 µL for the traditional-ECP method. For experimental conditions, see the Experimental Section. b Maximum concentration used to determine the adsorption isotherm and the surface coverage.

900 µL. An adsorption isotherm needs to be determined over a broad concentration range in order to make a good estimation of the adsorption isotherm parameters. The low concentration data in the linear range gives the initial slope of the adsorption isotherm (the a term) whereas the concentrations corresponding to the nonlinear part of the adsorption isotherm gives the association equilibrium constant. Estimating the adsorption isotherm using 50 µL injections resulted in small errors of the a term (-0.3%) because it is associated with the low-concentration data. At low injection volumes, the experimental injection profiles effect on the eluted peak is not as serious as for larger volumes. This result in smaller deviation between experimental peaks and peaks originated from rectangular injection profiles. However, small injection volumes lead to low apex concentrations and low surface coverage used to determine the adsorption isotherm. For the 50 µL 7890

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injection, the surface coverage is only 5.6% and why the value of the monolayer capacity term qs has no reliability (although the nominal error of -3.4% in Table 1 is small). Increasing the injection volume leads to larger surface coverages; however, unfortunately the injection profiles deviation from the rectangular injection profile also increases. As a consequence, the rear part of the profile is affected and this will lead to reduced accuracy in the determined adsorption isotherm parameters. In Figure 3, the determined adsorption isotherm for ECP using 50, 250, 900, and 4980 µL and Cut-ECP injection is plotted and compared with the average values of the reference methods. The adsorption isotherm from the Cut-injection nicely overlaps the reference isotherm, which is indeed not the case for the traditional injections. The larger the injection volume, the larger is the deviation from the reference method (cf. Figure 3). In the right figure, the initial part of the adsorption isotherm is zoomed in. Here we can see that small injection volumes and the Cut-injection predict the initial slope much better as compared to larger injection volumes (used with the traditional-ECP). To illustrate the improvement of the Cut-injection, experimental profiles from 100 mM injections of MM using a 4980 µL full-loop injection and a Cut-injection, respectively, are compared with the corresponding simulated profiles using the equilibrium dispersion model (see Figure 4). Experimental data are open circles, and simulated results are solid lines. Figure 4a shows an overlay of elution profiles from experimental full-loop injection and corresponding simulated profile assuming rectangular injection profile. The agreement of the experimental profile with the calculated one is indeed very poor over the whole concentration range. The experimental diffuse rear starts to appear earlier, and the final part is eluted later as compared to the simulated profile. The poor agreement between the diffuse tail of the eluted profile and the simulated peak tail will absolutely lead to large errors in the adsorption isotherm determination using traditional-ECP. Figure

Table 2. Adsorption Isotherms Determined for Propranolol (PR) Using the Cut-ECP Method and the Traditional-ECP Method, With Various Injection Volumes, Compared with the Average of the Reference Methodsa method FA PP CUT 4980 50 100 250 500 900

Cmaxb [mM]

θ [%] c

7.6 7.5 0.33 0.58 1.4 2.9 4.5

d

15.9, 90.1 15.9,c 90.1d 12.5,c 88.7d 12.2,c 88.5d 0.60,c 25.8d 1.10,c 37.8d 2.50,c 59.3d 5.10,c 75.1d 7.80,c 82.5d

aI

qs,I [M]

KI [M-1]

aII

qs,II [mM]

KII [mM-1]

3.59 3.60 3.51 (-2.4) 4.78 (+33.1) 3.43 (-4.5) 3.55 (-1.3) 3.64 (+1.2) 3.28 (-8.8) 5.16 (+43.4)

0.176 0.208 0.188 (-2.2) 0.0790 (-58.8) 4.34 (+2161) 12.4 (+6345) 13.7 (+7030) 0.620 (+223) 0.0730 (-62.0)

20.4 17.3 18.7 (-0.9) 60.5 (+220.9) 0.791 (-95.8) 0.287 (-98.5) 0.266 (-98.6) 5.29 (-72.0) 70.6 (+275)

6.85 6.79 6.66 (-2.4) 7.09 (+3.9) 7.03 (+3.1) 6.94 (+1.7) 7.00 (+2.6) 7.47 (+9.6) 6.32 (-7.3)

6.81 6.12 7.30 (+12.9) 6.49 (+0.3) 7.15 (+10.6) 7.08 (+9.5) 7.04 (+8.9) 9.26 (+43.3) 4.83 (-25.4)

1.01 1.11 0.912 (-13.8) 1.09 (+3.2) 0.984 (-7.0) 0.980 (-7.4) 0.994 (-6.0) 0.806 (-23.8) 1.31 (+23.9)

a FA and PP, listed at top. The adsorption isotherms are determined using a sample of 10 mM PR and injecting 50, 100, 250, 500, 900, and 4980 µL for the traditional-ECP method. The errors are declared in parenthesis after each parameter value calculated; the values are calculated as compared to the average of the reference methods at top (FA and PP). For experimental conditions, see the Experimental Section. b Maximum concentration used to determine the adsorption isotherm and the surface coverage. c Surface coverage of site I. d Surface coverage of site II.

Figure 3. Adsorption isotherm determined for methyl mandelate using the ECP methods (traditional-ECP with various loop volumes and Cut-ECP) as compared with the average isotherms using the reference methods. The left figure shows the whole adsorption isotherm range and the right figure shows only the low concentration region. For more information, see Table 1.

4b shows the comparison between the experimental eluted profiles using Cut-injection with the calculated band profile using rectangular injection profile; the Cut-injection technique agrees well with the calculated elution profile. This demonstrates convincingly that peaks originated from Cut-injections are much more accurate than those originating from full-loop injections, for the purpose of ECP calculations. The Cut-injection and the traditional 4980 µL injection are calculated at such a maximal concentration so that the surface coverage of the high capacity site is around 13% and that of the low capacity site is around 89%. Table 2 shows that the Cut-ECP predicts the PR adsorption isotherm with very small errors as compared with the traditional ECP methods where the error of the high capacity site was extremely large (33.1% for the a term and 221% for the K term). The errors of the parameters using the Cut-ECP were also in this case in the same size as the small variations between the two reference methods FA and PP, respectively. However, as noted before,15 for a bi-Langmuir adsorption isotherm the error is generally larger on the determined adsorption isotherm as compared with a Langmuir model.

Figure 4. Eluted chromatogram for a 100 mM injection of methyl mandelate resulting after (a) a 4980 µL full-loop injection and (b) a Cut-injection. The symbols are experimental data and the lines are simulations using a rectangular injection profile.

This is why the PP and FA methods vary more between each other for the bi-Langmuir data as compared to the Langmuir data, as seen by comparing Table 2 and 1 (top lines). The resulting parameters from traditional ECP measurements for PR, generated by smaller injection volumes than 4980 µL are presented in Table 2. For the 50 µL injection, only the parameters of the high-energy site (site II with low capacity) is estimated rather well, the low-energy site (site I with high capacity) has very large errors. Here, the maximal surface coverage for sites I and II are 0.6% and 25.8%, respectively. In this case only the low and medium concentration ranges are covered, which is the explanation for the isotherm parameters badly predicted for the low-energy sites (site I). Because of the low surface coverage of the low-energy site, the prediction is poor for injections smaller than 500 µL. This agrees well with another study, where the authors used FA and IM to determine adsorption isotherms and found that the low surface coverage leads to large errors in the estimated adsorption isotherm parameters.12 The authors concluded that the determined saturation capacities are always more or less extrapolated independent of the method used for adsorption isotherm acquisition. However, for 50 µL injections, the initial slope (the sum of the a terms) is predicted to 10.46 which is almost identical to the sum of the reference values which is 10.42 (from Analytical Chemistry, Vol. 80, No. 20, October 15, 2008

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0 where δinj is the error in the volumetric injection and Vinj is the true injection volume. Inserting eq 6 in eq 3 and then in eq 4 gives

q)

1 Va

∫

C

0

(

0 Vinj + V0 + Va

)

dq0 0 - Vinj - δinj - V0 dC dC

(7)

where q0 is the true stationary phase concentration. By integration of this equation, the following more simple relation is obtained:

q ) q0 -

Figure 5. Adsorption isotherm determined for propranolol using the ECP methods (traditional-ECP with various loop volumes and CutECP) as compared with the average isotherms using the reference methods. The left figure shows the whole adsorption isotherm range; the right figure shows only the low concentration region. For more information, see Table 2.

top lines of Table 2); the reason is that the injection profile of 50 µL is not so wide and therefore gives a correct determination of the initial slope. For the 500 µL injection, we have a 5.1% surface coverage of site I and a 75.1% surface coverage of site II; these conditions should give more reliable parameters of the low-energy site (site I). However, the increased effect of the dispersed injection profile counteracts this improvement, resulting in an overall larger parameter estimation error (cf. Table 2). These counteracting factors are probably the reason why the error of the capacity of the high-capacity site (qs,I) first increases (to around 7000% error) when going from 50 to 250 µL and then decreases as the injection volume is increased further; at the injection volume of 900 µL, the error of the qs,I term is -62% (see Table 2). At this injection volume, the error of the equilibrium constant of the weak site is even higher (275%) and in a similar magnitude as for the 4980 µL injection because of the spread out of the rear of the injection profile demonstrated in Figure 1. In Figure 5, some adsorption isotherms are compared using the Cut-ECP and traditional-ECP methods, respectively. As for the MM cases (cf. Figure 3), the adsorption isotherm from the Cut-ECP perfectly coincides with the average isotherm using the reference methods and the deviation increases with increasing injection volume using the traditional-ECP method. Error Analysis. Classically, full-loop injections are conducted to increase the reproducibility of the column load in analytical chromatography. However, the new Cut-injection is based on the injection loop being manually returned to the load position at a selected time; this will introduce an error in the injection volume as compared to calibrated full-loop injections. Below, we investigate the consequence of this error on the adsorption isotherm determination. First, we assume that an error is made in the nominal injection volume Vinj why the injected volume can be described by 0 Vinj ) Vinj + δinj

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(6)

δinj C Va

(8)

Thus, a linear error term is subtracted from the true adsorption isotherm. To deduce the influence that the error term have on the determined adsorption isotherm, we estimated its size. Assume that we are conducting experiments on a normal bore column (4.6 mm × 150 mm) with a stationary phase volume (Va) of 1.0 mL. We could estimate that the time error made during the Cutinjection experiments are less than 0.5 s at a flow rate of 1 mL/ min, which will lead to an absolute δinj/Va value of only around 0.008. This error could be related to the error made while using injection loops that are not calibrated. Generally injection loops are constructed using tubing made of PEEK or stainless steel. The desired injection loop is constructed by the geometric constructing method, calculating the length of tubing needed from its nominal inner diameter. Stainless steel tubings with inner diameters of 0.5-1 mm has an inner diameter tolerance of 0.025 mm.20 The error in the injection volume is related to this inner diameter tolerance. As mentioned above, the ECP measurements usually require rather large injection volumes. Let us therefore assume that we make 100, 200, 500, and 1000 µL injection loops using the geometric construction method and we further assume conventional size inner diameter of the stainless steel tubing 0.5, 1, 1, and 1 mm, respectively.21 This will lead to an absolute δinj/ Va value between 0.01 and 0.05. As can be seem from the calculations above, the linear extra term is rather small and will generally not affect the estimation of the initial slope of the adsorption isotherm. In this study, the maximum error in the initial slope using the Cut-injection is around 0.1%. At the maximum concentration used for the adsorption isotherm acquisition in this study, the error in the determined stationary phase concentration is below 0.2%. Both these errors are small as compared to the error in the adsorption isotherm determination using methods like FA and PP and could therefore be neglected. CONCLUSIONS In this paper, a new injection technique for ECP (the “Cutinjection technique”) was introduced for more accurate determination of adsorption isotherms. A sharp slice was made on the rear of the injection zone before it exited the injection loop by returning the position of the injector valve to load at an exact time before the whole loop content had entered the column (cf. Figure 2). The ECP method is derived using the ideal model, i.e., infinite column efficiencies and assuming rectangular injection profiles. Today’s columns have reached such high efficiencies that we can (20) http://www.vici-jour.se (accessed March 11, 2008). (21) Personal communication with Dr. Anders Medin, Scantec Laboratory.

say that the deviation from the ideal model has more or less disappeared for overloaded injections. The Cut-injection technique eliminates the most important remaining source of error, namely, that large injection loop profiles deviate considerably from a rectangular shape. Our Cut-ECP method should therefore be one of the prime choices for rapid thermodynamic characterizations of the new modern stationary phases using the new sub-2 µm1 and fused core particles.2 One major problem with conducting adsorption isotherm studies using the ECP method is that the concentration in the sample needs to be high because the adsorption isotherm is calculated from the eluted peak (which dilutes around 5-20 times during travel in a conventional column). To get good adsorption isotherm data suitable for all types of adsorption parameters, the concentration range needs to be sufficiently wide. Therefore, larger volumes must be injected. However, then other problems arise, namely, that the injection profile deviates more or less strongly from the ideal rectangular injection profile; the larger the volume injected the worse the deviation (cf. Figure 1). This results in tremendous errors of the adsorption parameters, in this study an error of just below 200% was obtained at most when using

the traditional-ECP method with a 4980 µL injection for determining the heterogenous adsorption of propranolol on a C18 column (cf. Table 2). The Cut-ECP method on the other hand could deliver adsorption isotherm parameters with a maximal error of around 10% for the same system (cf. Table 2). The Cut-ECP method involves a certain uncertainty in the injection volume as compared with calibrated injection loops. We therefore made an error analysis on this issue, using the ideal model. The errors on the adsorption isotherm parameters, because of the uncertainty in the injection volume using the new Cutinjection technique, were found to be negligible. ACKNOWLEDGMENT The work was supported by a grant from the Swedish Research Council (VR) for the Project “Fundamental Studies on Molecular Interactions Aimed at Preparative Separations and Biospecific Measurements”. Received for review May 30, 2008. Accepted August 12, 2008. AC8010999

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