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Importance of the Accuracy of Experimental Data in the Nonlinear Chromatographic Determination of Adsorption Energy Distributions Brett J. Stanley and Georges Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, and Chemical and Analytical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6120 Received June 15, 1994. In Final Form: August 11, 1994@ Adsorption energy distributions (AEDs)are calculatedfrom the classical, fundamental integral equation of adsorption using adsorption isothermsand the expectation-maximizationmethod of parameter estimation. The adsorptionisotherms are calculatedfrom nonlinear elution profiles obtained from gas chromatographic data using the characteristic points method of finite concentration chromatography. Porous layer open tubular capillary columns are used to supportthe adsorbent. The performance ofthese columns is compared to that of packed columns in terms of their ability to supply accurate isotherm data and AEDs. The effect of the finite column efficiency and the limited loading factor on the accuracy of the estimated energy distributions is presented. This accuracy decreases with decreasing efficiency, and approximately 5000 theoretical plates are needed when the loading factor,Lf, equals 0.56 for sampling of a unimodal Gaussian distribution. Increasing Lf hrther increases the contribution of finite efficiency to the AED and causes a divergence at the low-energy endpoint if too high. This occurs as the retention time approaches the holdup time. Data are presentedfor diethyl ether adsorption on porous silica and its C-18-bondedderivative. Both the frontal-analysis-by-characteristic-points(FACP) and the elution-by-characteristic-points(ECP) methods of gas chromatography are presented. The FACP experiment yielded divergent results at high column loadings for this system. The results indicate that the accuracy of estimated AEDs with respect to the true, underlying distribution of the surface is poor for most studies of this nature.
Introduction The characteristic points methods of isotherm determination were suggested by Glueckauf40 years ago.' Their adaptation to gas chromatography (GC) was presented by Conder and Purne1125 years ago as an effectivemethod of obtaining a complete adsorption isotherm with the data taken from one chromatogram over the concentration region sampled by that c h r ~ m a t o g r a m Most . ~ ~ ~notable of these methods are the frontal-analysis-by-characteristicpoints (FACP) and elution-by-characteristic-points(ECP) methods. The ECP method has been used extensively for isotherm measurement ever since. The consideration of heterogeneous adsorption was accelerated by Adamson and Ling when they showed that a n adsorption energy distribution (AED) function could be obtained from a heterogeneous isotherm if a local site model of adsorption was assumed.* The potential utility of this advance was immediately recognized, and a plethora of work on this approach has r e ~ u l t e d . Rudz~,~ inski et al. showed how the energy distribution function could be directly related to the retention volume of an eluting solute, as a function of its partial pressure, using GC.' Cooper et al. utilized this method with several systems, obtaining the distribution function from one chromatogram.8-10 Previously, the Rudzinski- Jagiello method was shown to be an effectiveapproximate solution Abstract published inAdvance ACSAbstracts, October 1,1994. (1)Glueckauf, E . Trans. Faraday SOC.1966,51,1540. (2)Conder, J. R.; Purnell, J. H. Trans. Faraday SOC.1969,65,824. (3) (a) Conder, J. R.; Young, C. L. Physicochemical Measurement by Gas Chromatography; Wiley: New York, 1979; Chapter 9. (b) Katti, A. M.; Guiochon, G. Adv. Chromatog. 1991,31,1. (4) Adamson, A. W.; Ling, I. Adv. Chem. Ser. 1981,33,51. ( 5 ) Rudzinski, W.;Everett,D. H.AdsorptionofGases on Heterogeneous Surfaces; Academic Press: New York, 1992. (6)Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, The Netherlands, 1988. (7)Rudzinski, W.;Waksmundzki,A.; Leboda, R.; Suprynowicz,Z. J. Chromatogr. 1974,92, 25. (8) Cooper, W. T.; Haynes, J. M. J . Chromatogr. 1984,314,111. @
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to the energy distribution by representing the distribution function as a Taylor series,ll and then later Jagiello et al. chose a virial expansion to represent the isotherm data.I2 This method was then combined with the ECP method and applied to alkane adsorption on silicas and alkyl grafted ~i1icas.l~ Most previous work with AED calculations from isotherm data has utilized some form ofindirect extrapolation in order to circumvent the problems with isotherm measurement at high loading factors. These extrapolations typically take form by fitting the isotherm to a function or power series and then basing the AED calculation on the parameters of this fit. Once these parameters are obtained, the AED may be calculated over whatever region is deemed necessary. Another form of indirect extrapolation is to simply calculate the AED using an extended energy range, thus force-fitting the highpressure data to a particular region of the AED domain without regard to what type of information (i.e. perhaps lower energy adsorption) may exist at these higher pressures. The benefit of ECP isotherm analysis is indeed noteworthy. It minimizes the time required to obtain a complete isotherm, and hundreds of data points may be obtained from one single chromatogram. Moreover, the amount of solute required is minimized. However, the method is based on the ideal model of chromatography, which assumes an infinite column efficiency. The effect of finite efficiency upon the accuracy of ECP isotherm determination has been long k n o ~ n . ~It ~was J ~recently (9) Boudreau, S. P.; Smith, P. L.; Cooper, W. R. Chromatogr. Mag.
1987,2,31. (10)Boudreau, S. P.; Cooper, W. R. Anal. Chem. 1989,61,41. (11) Rudzinski, W.; Jagiello, J.; Grillet, Y. J . Colloid Interface Sci. 1982,87,478. (12)Jagiello, J.; Czerpirski, L. Chem. Eng. Sci. 1989,44,797. (13)Jagiello, J.;Ligner, G.; Papirer,E . J. Colloid Interface Sci. 1990,
137,128. (14) Huber, J. F. K.; Gerritse, R. E. J . Chromatogr. 1971,58, 138. (15)Gerritse, R. E.; Huber, J. F . K . J . Chromatogr. 1972,71,173.
0 1994 American Chemical Society
Langmuir, Vol. 10, No. 11, 1994 4279
Adsorption Energy Distributions investigated quantitatively for the case of single Langmuir isotherms, and it was shown that approximately 2000 theoretical plates are needed to estimate the a parameter (the initial slope) to within 1%error.16 Therefore, this finding must have important repercussions on the singlepeak, finite concentration GC work used to estimate adsorption energy distributions, as reported above. The aforementionedstudies have all utilized the packed chromatographic column to obtain the required data and have typically been studies of particles greater than 50 pm in diameter. The study of fine particles less than 10 pm in diameter results in column inlet pressures that are too high for adequate study with standard chromatographs. More importantly, the James-Martin correction for the retention volumes becomes too large under such conditions and may no longer be valid with the assumption of a homogeneous packing along the column. Under these conditions short column lengths and/or low flow rates must be used because of the practical limitations of the chromatograph, and the efficiencyof the chromatographic band profiles decreases as a result. For the studies of larger diameter particles where the pressure drop is alleviated, column efficiency is still an issue as the efficiency is inversely proportional to the square of the particle diameter. The objective of the present paper is to demonstrate some of the problems associated with AED calculation from nonlinear chromatographic data as it is currently practiced. The calculation of AEDs from raw isotherm data is an ill-posed problem (discussed further below), and this precludes an exhaustive quantitative study of the effects of systematic error on the estimation of AEDs within a single article. The effects of either the column efficiency or the range of adsorption energies (parameters which are limited experimentally) on the estimation of the +4E!D are of primary concern here and have not been previously considered in a serious manner. To help achieve this end, the AEDs reported are obtained by strictly using only the data which are available. With this idea in mind, a transform from pressure to energy is used to calculate the minimum and maximum adsorption energies that are able to be estimated from the data. These are defined by the maximum and minimum pressures measured respectively. Also, only the data taken from the detector are used in the calculations; i.e. no smoothing or splines are applied to the chromatographicband profile or resulting isotherm. Use of the expectation-maximization (EM)method of parameter estimation allows a robust AED estimate and minimizes artifactual information from the numerical standpoint.17J8This approach provides the least biased solution available and allows a study of the accuracy of AED calculation with respect to only that information about the AED which is encoded in the actual data measured. The energy distributions for diethyl ether adsorption on porous silicas are considered in this work. Diethyl ether is a slightly basic, relatively small and nonpolar molecule, with a low boiling point, and as such is a good probe of acidic silanol adsorption sites. Further work describing the surface chemistryofvarious modified silicas with this and other molecular probes will be forthcoming.lg
Theoretical Section Finite Dilution Chromatography. In gas chromatography, the mobile phase is compressible, and a proper (16) Guan, H.;Stanley, E. J.; Guiochon, G. J. Chromatogr. 1994, 659, 27. (17) Stanley, E.J.;Bialkowski, S. E.; Marshall, D. B. Anal. Chem. 1993, 65, 259; (18) Stanley, B. J.; Guiochon, G. J. Phys. Chem. 1993,97,8098. (19) Stanley, E.J.;Guiochon, G. In preparation.
correction is r e q ~ i r e d . Furthermore, ~ for accurate determination of retention data, the gas phase cannot be considered as ideal, and a correction based on the virial equation of state is n e c e ~ s a r y The . ~ characteristic points method of gas chromatographyyields the global isotherm q(p), with the following relation^:^,^
where
and
are the James-Martin coefficient^.^ In the above equations, M,is the mass of stationary phase, y is the mole fraction of solute at the column outlet, c is the solute concentration corresponding toy, Pi and Po are the inlet and outlet column pressures, B22 is the second virial coefficient for the solute, ko' is the capacity factor at infinite dilution, R is the ideal gas constant, and T i s the column temperature. The flow rate is corrected for the sorption e f f e ~ t ~ and , ~ O is calculated as
(6) The retention volume is then obtained at each y by multiplying the correspondingretention time, tR, by Fb). The Adsorption Energy Distribution (AED). The AED, RE),is defined from the adsorption isotherm using the fundamental integral equation:21
(7) where 8(E,p)is the local model of adsorption. The selection of this function is a prerequisite of the entire analysis and has been a matter of long debate. Suffice it to say that a number of choices can be made and have been offered. Many studies have been completed concerning the ramifications of this choice, notably in the work done by Jaroniec.6 For the purposes of our work, the Langmuir equation was chosen on the basis of its ubiquity and simplistic assumptions:22 (20) Bosanquet, C.H.; Morgan, G. 0. In Vapour Phase Chromatog raphy; Desty, D. H., Ed.; Butterworths: London, U.K., 1957; p 35. (21) Sips, R.J. Chem. Phys. 1948, 16, 490. (22) Langmuir, I. J. Am. Chem. SOC.1918,40, 1361.
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Vol. 10,No. 11, 1994
where K is the preexponential factor and is dependent on the molecular partition functions of the solute in both the mobile and stationary phases.23 We have decided to use the formulation of J a r o n i e ~ : ~ ~
where P,is the vapor pressure of the solute at the column is its heat of vaporization. temperature and Evap From eq 8, the end points of the estimated energy distribution are estimated from the minimum and maximum solute pressures at which measurements were acquired. According to the condensation approximation, 0 goes from 0 to 1at a characteristic concentration; at this concentration 0 = V2. Solving eq 8 for V 2 , gives pi = K,,(-EiRT), and the end points of the estimated energy distribution correspond to
rim)
E,, = -RT In and
E,, = -RT Numerical Calculation of the AED. The AED function, RE), must be obtained by inversion of the isotherm equation, eq 7. This equation belongs to the class of linear Fredholm integral equations of the first kind. The inversion process is ill-posed mathematically, which means that a unique solution cannot be obtained in the presence of any amount of experimental error. Only an estimate may be obtained. Adamson and Ling originally obtained an estimate with a graphical p r ~ c e d u r e . ~ Considerablework has been undertaken subsequently on efficient and robust numerical methods of solution of eq 7. A review of these methods can be found in ref 5. Notable of these methods is House and Jaycock’s HILDA (heterogeneity investigated at Loughborough by a distribution analysis) programFe but it has recently been recognized that a more robust method may be needed that is able to calculate the AED without any preprocessing of the isotherm data, and that a statistical approach is desirable.26J8 The most popular statistical approach to these types of problems is the regularized least-squares approach, recently applied by Heuchel et al. t o AED solution.27 Regularized least-squares seeks t o minimize an object function which increases when certain specified conditions occur in the solution: i.e. it solves
V(a)= IWY2 (q - J O f d,)l l2
+ alIGOll2 = minimum (11)
(23)Gawdzik,J.;Suprynowicz,Z.; Jaroniec,M. J.Chromatogr. 1976, 121,185. (24)Jaroniec, M.Su$. Sci. 1975,50,553. (25)House, W.A.;Jaycock, M. J. J. Colloid Polym. Sci. 1978,256, 52. (26)Golshan-Shirazi, S.;Guiochon, G. J.Chromatogr. 1994,670,l. (27)Heuchel, M.;Jaroniec, M.; Gilpin, R. K. J. Chromutogr. 1993, 628,59.
where M is an arbitrary weight matrix, a is the regularization parameter, G is a function offlE) that imposes the regularization, and II 11 denotes any Euclidean norm. ORen this function is the second derivative of the solution, or more simply, ccf,=f(ridge regression). Either of these choices effectively punishes the solution for excess curvature, thus yielding the smoothest estimate consistent with the data. The parameter a is the rheostat to adjust the magnitude of regularization desired and constitutes an estimation problem itself. Recently we implemented an iterative maximumlikelihood method called expectation-maximization(EM) to obtain a statistical, robust estimate of flE).18 The algorithm updates eachflEi)point with the correction step
where qe, is the experimental isotherm data, qcal is the estimated data at iteration k via eq 7 and appropriate quadrature, AE is the grid spacing around each energy point i , and the sum is over the pressure points j . The initial guess is taken as a uniform distribution. This method was shown to yield smoother estimates than those corresponding to two versions of “optimal”regularization for Fredholm integrals corresponding to first-order rate constant distributions (i.e.Laplace transforms).17 Without prior knowledge of the error distribution, the EM solution appears to be the most prudent choice in terms of guarding against artifactual information. However, a disadvantage of the technique can be prohibitively long computation times if a fine energy grid is used in conjunction with hundreds of data points and several thousand iterations. Under the limitations of computing power, the regularization method becomes the preferred method. In particular, the method and program of Provencher are particularly appealingaZ8
Experimental Section
Numerical Simulations. All simulations were performed by assuming a “true”distribution function, calculating the “true” isotherm with eq 7,calculating the diffuse rear chromatographic band profile by either the ideaP9,30or the equilibrium-dispersive m0de1~~932 of chromatography, and then calculating the ECP isotherm using the characteristic point calculation (eq 11, and finally estimating the distribution function with the EM algorithm (eq 12). This estimate is then compared with the original distribution. For the sake of simplicity,the simulations proceeded as if in liquid chromatography; i.e. gas phase compressibility was not modeled in the calculation of the band profile. Only the effects of efficiency were intended to be investigated in these simulations. The equilibrium-dispersive band profile calculation is based on a numerical algorithm utilizing the finite difference method of numerical a n a l y ~ i s . 3 ~ In J ~ this algorithm the apparent dispersion of the band profile, which is specified by the column efficiency, is uniquely and accurately approximated by the numerical diffusion of the routine. The ideal model does not consider any band dispersion and may be implemented with the analytical solution. The original distributions were constructed by assuming a 1000 point distribution of b parameters in reference to the classical Langmuir equation (28)Provencher, S.W.Comput. Phys. Commun. 1982,27,213. (29)DeVault, D.J . Am. Chem. SOC.,1943,65,532. (30)Guiochon, G.; Golshan-Shirazi, S.; Katti, A. M. Fundamentals ofPreparative and Nonlinear Chromatography;AcademicPress: Boston, MA, 1994;Chapter 7. (31)Guiochon, G.; Golshan-Shirazi, S.; Katti, A. M. Fundamentals ofpreparative and Nonlinear Chromatography;Academic Press: Boston, MA, 1994;Chapter 10. (32)Golshan-Shirazi, S.;Guiochon, G. J. Chromatogr. 1990,506, 495.
Adsorption Energy Distributions
e=-
bC
1
+ bC
Langmuir, Vol. 10, No. 11, 1994 4281
(13)
where C denotes the equilibrium mobile phase concentration. The distributions were constructed in logarithmic space in keeping with the notion of energy distributions, eq 8; i.e. we considerf(1og b). One thousand points were also calculated for the true isotherms. The EM estimation was performed over 500 energy points for 20 000 iterations. All calculations were carried out in double precision on a VAX 7640 computer. Column experiments. Experimental data were acquired on both packed and open tubular columns. The experiments considered here are with silica particles
