Anal. Chem. 1991, 63, 833-839
LITERATURE CITED
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(14) Lipnick, R. L.; Bichings, C. K.; Johnson, D. E.; Eastmond, D. A.; Aquatic Toxicology and Hazard Assessment-, ASTM Technical Publication 153; Hansen, D. J., Ed.; 1981. (15) Perrin, D. D.; Dempsey, B.; Sergeant, E. P. pKa Prediction for Organic Acids and Bases Chapman and Hall: London, 1981. (16) Khaledi, M. G.; Breyer, E. D. Anal. Chem. 1989, 81, 1040. (17) Khaledi, M. G. Trends Anal. Chem. 1988, 7, 293. (18) Schultz, W.; Rlggln, G. W.; Wesley, S. K. QSAR in Envloronmental Toxicology-II-, Kaiser, K. L., Ed.; Reidel Publishing Co.; Dordrecht, The Netherlands, 1987; p 333. (19) Hansch, C.; Leo, A. Substituent Constants for Correlation Analysis In Chemistry and Biology, J. Wiley and Sons: New York, 1974; p 45. (20) Khaledi, M. G.; Peuler, E.; Ngeh-Ngwainbi, J. Anal. Chem. 1987, 59, 2738.
(1) Blrge, W. J.; Black, J. A. Aquatic Toxicology and Hazard Assessment-, ASTM Special Technical Publication 891; Bahner, R. C., Hansen, D. J., Eds.; April, 1948; p 51. (2) Richet, M. C. C. R. Seances Soc. Biol. Ses. Fit. 1893, 45, 775. (3) Meyer, H. Zur Theorle der Alkolnarkose I. Welche Eigenschaft der Anesthetlca bedlng Ihre narkotlsche wlrkung. Arch. Exp. Pathol. Pharmakol. 1899, 42, 109. (4) Overton, E. Z. Phys. Chem. 1897, 22, 189. (5) Lien, E. J.; Quo, Z.-R.; Li, R.-L.; Su, C.-T. J. Pharm. Sci. 1982, 71, 641. (6) Morlguchl, I.; Kanada, Y.; Katsuichlro, K. Chem. Pharm. Bull. 1976, 24, 1799. (7) Cascorbi, I.; Ahlers, J. Toxicology 1989, 58, 197. (8) Hansch, C. Acc. Chem. Res. 1969, 2, 232. (9) Hansch, C. Nature 1962, 194, 178. (10) Kaliszan, R. Quantitative Structure Chromatographic Retention Relationships-, J. Wiley and Sons: New York, 1987. (11) Valko, K. Trends Anal. Chem. 1987, 6, 214. K. J. Uq. Chromatogr. 1984, 7, 1405. Valko, (12) (13) Konemann, H.; Musch, A. Toxicology 1981, 19, 223.
-,
Received for review October 30,1990. Accepted January 18, 1991. We gratefully acknowledge the funding of this project by the National Institutes of Health (FIRST Award, Grant GM 38738).
Determination of Isotherms from Chromatographic Peak Shapes Eric V. Dose, Stephen Jacobson, and Georges Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, and Analytical Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6120
nificant advantages in measurement speed when it could be
A new method for determining equilibrium isotherms from chromatographic peak shapes Is presented. It differs from alternative methods like elution by characteristic point in that the present method accounts for the finite efficiency of real chromatographic columns, and so the new method Is used to Its best advantage in systems of low plate number. In the new method, Initial estimates of the parameters of an Isotherm equation are refined by finding simulated chromatographic peak shapes that most nearly approximate the experimental peak shapes. A series of example determinations demonstrate the method’s utility.
used. Several methods have been practiced over the previous 40 years including frontal analysis (FA) (6-11), frontal analysis by characteristic points (FACP) (12, 13), peak-maxima
methods (14), elution by characteristic point (ECP) (15-19), and step-and-pulse (minor disturbance) methods (20-24). These methods have been reviewed by others (25-27). A recurring problem in determining isotherms by chromatography is the approximation caused by the assumption that the adsorption system is always at equilibrium. While the intimate contact of the equilibrated phases yields equilibration rates far exceeding those of static measurement systems, the former rates are not infinite, and the chromatographic peak or front shapes differ from ideal (equilibrium) chromatographic peak shapes. The differences are frequently small (8-17), and approximate peak shape corrections have been proposed (28). It is very difficult to demonstrate the validity of these correction methods, and they are not always used in practice. Chromatographic nonideality (low column efficiency) is important, especially in the case of biopolymer separations. Thus, we feel the development of a fast method permitting the use of small amounts of material is warranted. The development of numerical chromatographic simulation methods (29, 30) that explicitly include the effects of finite efficiency (measured as the chromatographic plate number) suggests a different path. A method of successive approximation to the physical isotherm can be designed given a means of computing chromatographic peak shapes from a given isotherm and plate number, provided by the semi-ideal (finite efficiency) model (29); a score measuring the similarity of this simulated peak shape to the experimental shape; and an algorithm that generates new isotherms with increasing similarity scores and that recognizes convergence of the series of new isotherms to a single best isotherm. This last task may be performed by modified simplex optimization methods (31-33). In this article, we describe how to determine iso-
INTRODUCTION The equilibrium isotherm describes how an adsorbate concentration on an adsorbent surface depends on the adsorbate chemical potential in either the stationary phase or the fluid, mobile phase in contact with the stationary phase. The isotherm plot deviates from linear behavior due to surface saturation, surface heterogeneity, and/or adsorbate-adsorbate interactions at the surface. Conversely, measured isotherms can be used to test hypotheses about the extent of heterogeneity and adsorbate-adsorbate interactions exhibited by an adsorbate-adsorbent system (1). Though the equilibrium isotherm is formally defined in terms of the adsorbate chemical potential at the interface, in practice one usually substitutes adsorbate concentration for the chemical potential, and we follow this practice herein. The most straightforward isotherm determination methods are static in nature. Static methods consist of near-equilibrium measurements of the extent of adsorption at specified, constent adsorbate concentrations. The extent of adsorption has been measured by methods including microbalance gravimetry (2), infrared absorption (3), and thermogravimetry (3-5). It was recognized very early that chromatography offered sig0003-2700/91/0363-0833502.50/0
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1991 American Chemical Society
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therms from low-efficiency chromatographic peak shapes by using a new successive-approximation method. We apply the method to experimental chromatograms of low column efficiency (700-1000 theoretical plates).
EXPERIMENTAL SECTION IV-Benzoyl-L-alanine (LBA), /V-benzoyl-D-alanine (DBA), and iV-benzoyl-L-phenylalanine (LBP) were purchased from Sigma Chemical. N-Benzoyl-D-phenylalanine (DBP) was prepared by treating D-phenylalanine (Sigma) with benzoyl chloride (Aldrich Chemical) (34). 1-Propanol (Mallinckrodt) was HPLC grade. Frontal analysis and elution studies were performed on a Hewlett-Packard Model 1090 liquid chromatograph equipped with a Hewlett-Packard Model 1040A diode-array UV detector. For frontal analysis, solvent and sample solutions were mixed at the low-pressure pumps in programs designed to form the concentration steps required. We used an Alltech Resolvosil-BSA-7 (BSA-7) chromatographic column 15-cm length, 4-mm inside diameter, and stationary phase consisting of bovine serum albumin (BSA) bound covalently to porous silica particles. The mobile phases were 3% 1-propanol in 10 mM phosphate buffer (DBA and LBA experiments) and 7% 1-propanol in 100 mM phosphate buffer (DBP and LBP), all at pH 6.8. Injections for the elution experiments were made by the chromatograph’s autosampler in the normal manner. Flow rates were 1 mL/min, and UV absorbance at the detector was recorded every 640 ms. Experiments were conducted at ambient temperatures near 25 °C. Chromatographic plate numbers were estimated on the same chromatographic systems by moment measurements on peak shapes resulting from very small (analytical) injections. These estimated plate numbers were 1000 plates for DBA and LBA data and 700 plates for DBP and LBP data. Numerical computation was performed to double precision (about 16 decimal digits) by using IBM VS FORTRAN 2.4 on an IBM 3090 vector processor ((4-5) X 107 floating point instructions/s). The software is available from the authors. Most simulations required about 0.3-3 s CPU time, requiring about 1-5 CPU min per isotherm determination. For given column dimensions and isotherm, the CPU time per simulation is proportional to the square of the plate number. Routines computing the Akima spline function (35) are from the IMSL Mathematical Library (Houston, TX).
RESULTS AND DISCUSSION Method Description. This isotherm determination method requires that one complete four tasks: collecting experimental chromatographic peak shape data, selecting an appropriate model isotherm equation, simulating the chromatogram by using roughly estimated isotherm parameters, and adjusting those isotherm parameter values to give the best agreement between the simulated and experimental chromatographic peak shapes. This approach is similar to that of a dectector-calibration method described earlier (36). In the following paragraphs, we develop each of the four method tasks.
The chromatographic data required for this method are similar to those needed for ECP isotherm determination, that is, one or more chromatographic peaks whose detected concentration range spans the range of mobile-phase concentrations in the desired isotherm plot. As we show below, cautious extrapolation upwards in mobile-phase concentration is feasible, up to 2-5 times the maximum detected concentration. The isotherm must be sampled in the nonlinear region. This means that sufficient sample must be injected to cause the peaks to assume non-Gaussian shape. The method requires data in concentrations, so calibration must be performed on the raw detector responses. In the present case, a linear calibration curve proved sufficient, but where the curve is nonlinear, the same peak shape data used for isotherm determinations could be used to determine the calibration curve by methods we have presented earlier (36). Required data for each chromatogram comprise the column’s length, diameter, and phase ratio; the mobile-phase flow rate; the
volume and concentration of the sample; and a table of time and concentration data representing the chromatogram. The data collection step is the least demanding, consistent with our aim, to shift the effort from largely experimental to largely computational, in direct contrast to previous isotherm determination methods. Selecting an appropriate isotherm model equation is a recurring problem in surface science and nonlinear chromatography (9,11, 17, 26, 37-40). In this work, we have used the Langmuir C8
ACm
=
1
(1)
+BCm
and two-site Bilangmuir A2Cm 1
+ -BxCm
1
+ B2Cm
(2)
equations, where C8 is the local stationary-phase (adsorbent) concentration of adsorbate, Cm is the local mobile-phase concentration of adsorbate, and A, and B, are parameters describing the properties of site i. These models are useful in a variety of nonlinear adsorption situations, are easy to calculate, and do not typically yield maxima or minima (which cannot exist in equilibrium isotherms). Computational ease is crucial to the present work since a given isotherm determination may require that the right-hand side of one of the above equations be evaluated 107-108 times. The simulation method used in this work is adapted from Godunov methods described previously (29, 41). The semiideal model of chromatography allows solution of the chromatographic conservation equation (42-45) u0
dCm dCm -^ + (1 + fe') dz dt
-f
=
0
(3)
the mobile-phase linear velocity, Cm is the local concentration in the mobile phase, z is the displacement along the column axis, t is the experimental time, and k' is u0 is
k'
(4)
where F is the ratio of the stationary-phase volume to the mobile-phase volume and C8 is the local concentration in the stationary phase. The numerical diffusion caused by the finite-difference solution models the apparent physical diffusion caused by finite chromatographic efficiency (46). The Bilangmuir isotherm equation is generally well behaved but suffers in the present method from strong interdependence of the four parameters. For example, while a perturbation in the value of may perturb the shape of the resulting isotherm plot, there probably exist other parameter changes that can nearly totally compensate for the initial perturbation. This dependence makes it very hard to determine just to what extent each parameter contributes to the chromatogram peak shape. A solution to this problem is a transformation of the A and B parameters into a parameter set P of which some parameters dominate the shape of the isotherm in the concentration range of interest. The optimization procedure then begins by altering the values of only the most independent parameters, fixing the others to reasonable values. Later in the procedure, all the transformed parameters are optimized, and finally the A and B parameters are extracted from the transformed parameters. The advantage of this approach is that the most influential parts of the isotherm parameter set are optimized first, and adjustments to the complete parameter set are then performed only in the vicinity of the global optimum parameter set where the parameters are presumably well behaved. We transformed the Bilangmuir parameters as follows:
ANALYTICAL CHEMISTRY, VOL. 63, NO. 8, APRIL 15, 1991
Pi
Ax + A2
~
The transformed parameters P are much better behaved numerically than the raw A and B parameters, and they correspond to the following physical concepts: Px is proportional to the slope of the isotherm at the origin, in analogy to A in the simpler Langmuir isotherm; P2 is the total capacity of the stationary phase in analogy to the capacity A/Bin the Langmuir isotherm; P3 is the ratio of the two contributions to the isotherm slope at the origins, that is, the ratio of the sites’ low-concentration affinities; and P4 is the excess capacity of site 1. This last parameter measures the deviation of the ratio of the capacities from a rather arbitrary assumption that the two sites’ capacities are inversely proportional to their affinities. The advantage of defining P4 in this way is that its value is frequently nearly one. The transformed parameters P are easier to estimate from chromatographic data than are the raw parameters A and B. We normally begin with the following initial estimates: r
fc-o
-
1
Ft0
P2
=
P3
=
P4
clim
2-6 =
(6)
1
where tc-0 is the time at which the concentration of the peak tail reaches zero, t0 is the column void time, and CUm is the estimated stationary-phase concentration at the limit of very high mobile-phase concentration. The value of P3 is initially
estimated to be 2-3 for nearly triangular chromatographic peaks and is estimated to be 6 for peaks that are strongly concave upwards on the tail, indicating that the adsorbate affinity of one site is much stronger than that of the other. We begin with an initial estimate of 1 for P4. What remains is to adjust the transformed isotherm parameters so that the resulting simulated chromatograms match the experimental chromatograms and then to solve for the raw parameters A,- and B, as Ai
—
PiP3 1 + P3
Pi A2
+ P3 PXP3(P3 + P4) P2P4(1 + P3)
1
1
Pi(P3 + Pd
(7)
P2(l + P3)
The criterion we use for judging the merit of a given set of parameter values P is a weighted sum-of-squares error M
E(P)
=
Zwj j=l
TT
£
Njk=i
(C0,*
_
Ca,k)2
(8)
where M is the number of chromatograms, Wj is the relative weight assigned to chromatogram j (normally 1), Nj is the number of data points representing chromatogram j, Co k is
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835
the observed concentration of time point k, and C8i* is the simulated concentration at the same time point. E(P) is well behaved near the optimum P, but its dependence on P3 and P4 far from the optimum is very complicated. Fixing P3 and P4 to their initial estimated values until Px and P2 are nearly stationary followed by adjusting all four P parameters to minumum error results in an orderly minimization of E(P). We used a modified simplex search method (31-33) for optimization of the parameter values. Numerous attempts to minimize E(P) by analytical or numerical matrix methods, including Gauss-Newton, failed due both to the extreme irregularity of the E(P) surface far from the optimum and to the difficulty of obtaining initial estimates of P3 and P4 close to their optimum values. Gauss-Newton methods were successful in determining the best values of the one-site Langmuir parameters A and B but gave no advantage in computation time over the simplex methods. We thus used simplex methods throughout this work for consistency. The simplex method used searches for the parameter values with minimum (E(P) by generating a sequence of trial values. This search must be constrained by rejecting all trial P that fail certain tests of legality. For example, since isotherms with negative values of C8 or negative slope do not exist in nature, we reject each trial P that causes these conditions. It is sufficient for the present method to require that all four P parameters be positive. Our means of recovery from illegal trial P values by trial vector truncation have been given earlier (33). We note here that good initial estimates of P, especially along the lines of those in eq 6, minimize both the computation time and the number of illegal trial parameter sets. The raw result of the simulation method we use consists of a table of times and concentrations whose time interval between points depends on the isotherm slope at the origin. This dependence arises from the need to satisfy the Courant condition (47) and from our desire to minimize the computation time. However, since the times in the experimental and simulated time-concentration tables differ, the error function in eq 8 cannot be applied directly. To solve this problem, we select in advance a series of equally spaced times and interpolate both the experimental and simulated concentrations at those times. Evaluating the right-hand side of eq 8 using this series of times is then straightforward. HPLC Test Results. To test the proposed method’s ability to determine isotherms from chromatographic data, we collected HPLC chromatograms and frontal analysis data on four compounds eluted through a reversed-phase column. The behavior of the chromatographic system employed has been described previously (48, 49). We determined Bilangmuir isotherm parameters from the chromatographic peaks and then compared the resulting isotherm plots to the isotherm points obtained from frontal analysis. The results for all fits are given in Table I. It is important to recognize that one wants to compare isotherms, which describe directly the chemical systems at equilibrium, and not necessarily parameter values like Ax and Bx. Frequently large Bilangmuir parameter value changes result in only very minor changes in the isotherm plot shape. This is true because the parameters are codependent, and it is the reason why we emphasize comparisons of isotherm plots (Figures 1-11) rather than of parameter values (Table I) herein. Because we wanted to test the present method against independently measured frontal analysis data, we chose to perform the test by using HPLC. HPLC methods allow chromatographic peak shapes and frontal data to be obtained directly. However, the present method will be most useful in just those situations where frontal analysis and static methods are expensive, difficult, or inaccurate. This means that the present method may be used to its greatest advantage
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2
4
6
8
10
12
14
16
Time (min)
Figure 1. Example fit of simulated to experimental chromatograms. Curves: simulated chromatograms from best Bilangmuir isotherm determined by the present method. Points: experimental chromatograms of V-benzoyl-c-phenylalanine, obtained as described in the Experimental Section. Concentration 0 009 89 M in each case: injected volumes 150, 100, and 50 jiL in order of descending peak concentration.
Figure 3. Fit of simulated (Bilangmuir, curves) to experimental (points) chromatograms. N-Benzoyt-o-phenylalanine, concentration 0.009 38 M, injected volumes 150, 100, and 50 /iL in order of descending peak
concentration.
Figure 4. Experimental and determined Bilangmuir isotherm for DBP, for Figure 2.
as
Figure 2. Superposition of experimental (frontal analysis, points) Bilangmuir isotherm for LBP with that determined from experimental chromatograms of Figure 1 by the present method. Isotherms displayed to about 5 times the maximum chromatogram (mobile-phase) concentration.
in measuring gas-liquid or gas-solid isotherms by gas chromatography, where static methods are slow, and frontal methods, which require that one modify the chromatograph and then produce gas mixtures of constant and known composition and flow rates. The present method takes advantage of what chromatographs do best—making accurate injections of solutions and recording the peak shapes. There may also be uses for the present method in overload, nonlinear (preparative) liquid chromatography especially when adequate material for frontal analysis cannot be spared. If the adsorbate is expensive, the present method may be much less expensive than the alternatives, especially for preparative chromatographs with their typically high flow rates. We note that the present method accounts for kinetic broadening of the experimental chromatographic signal while frontal analysis does not. Figures 1 and 2 illustrates the results of our determination of the isotherm of LBP on BSA-7. In Figure 1, the best fit to the chromatogram peaks is very close. The method locates the peak fronts very accurately, and the peak shapes are also closely fit. The slight underestimation of the curvature at the peak maximum indicates that the column efficiency is mis-
estimated
or
that
some
extracolumn broadening is
unac-
counted for by the simulation model used. In Figure 2, the resulting isotherm curve is superimposed on the isotherm points from frontal analysis. The plot is extended to 5 times the peak-maximum concentration (that is, 5 times 950 j
