Anal. Chem. 1999, 71, 3453-3462
Stochastic Theory of Multiple-Site Linear Adsorption Chromatography Alberto Cavazzini,† Maurizio Remelli,† Francesco Dondi,*,† and Attila Felinger‡
Department of Chemistry, University of Ferrara, Via Luigi Borsari, 46, I-44100 Ferrara, Italy, and Department of Analytical Chemistry, University of Veszpre´ m, P.O. Box 158, H-8201 Veszpre´ m, Hungary
The problem of multiple-site adsorption chromatography is connected with a great number of questions of both chromatographic nature (e.g., the effects of surface heterogeneity on the column efficiency and on the peak shape parameters) and physical chemical relevance (e.g., the study of residence time in one sorption step on heterogeneous surfaces). In this study, the multiple-site adsorption, under linear conditions, is considered by using the molecular dynamic theory of chromatography. The probabilistic description is made by means of the characteristic function method and the solution is obtained under the most general conditions of surface heterogeneity. Different cases of surface energy distribution are considered. Relevant chromatographic attributes and the peak shape parameterssskew and excesssare investigated for heterogeneous stationary-phase surfaces. The chromatograms show that slow kinetics and surface heterogeneity have momentous impact on peak tailing. The equivalence of the stochastic model and the lumped kinetic model is demonstrated. Tailing peaks are commonly observed in chromatography although the models of linear chromatography predict Gaussian peak profiles in long-time or long-column approximations. Usually the heterogeneity of the surface of the stationary phase, the heterogeneity of the column packing, column overload, or extracolumn effects are blamed for peak asymmetry. When peaks lose their symmetry, there is (i) an efficiency loss due to an increase of the chromatographic space occupied by a single peak, (ii) a loss of sensitivity in peak detection, (iii) indeterminate errors in peak area/height evaluation, and (iv) an increased contaminating effect of impurities in preparative chromatography. The surface heterogeneity of the stationary phase is of fundamental importance in chiral separations or during the RPLC separation of basic compounds. The nonhomogeneity of the chiral selector stationary phases is usually modeled with the equilibriumdispersive model by using biLangmuir isotherm models.1,2 The biLangmuir model assumes that the stationary phase contains two * Corresponding author: (e-mail)
[email protected]; (fax) +39 0532 240709. † University of Ferrara. ‡ University of Veszpre ´ m. (1) Staerk, D. U.; Shitangkoon, A.; Winchester, E.; Vigh, Gy.; Felinger, A.; Guiochon, G. J. Chromatogr., A 1996, 734, 155-162. (2) Staerk, D. U.; Shitangkoon, A.; Winchester, E.; Vigh, Gy.; Felinger, A.; Guiochon, G. J. Chromatogr., A 1996, 734, 289-296. 10.1021/ac990282p CCC: $18.00 Published on Web 07/10/1999
© 1999 American Chemical Society
types of sites; in chiral separations, one site is regarded as chirally nonselective, the other type being chiral selective. Usually, the concentration of the chiral selective sites is much less than that of the nonselective sites. Accordingly, the selective sites are soon saturated, and for those sites, the nonlinear region of the sorption isotherm is reached. In this case, peak tailing is observed due to a combined effect of surface heterogeneity and overload. Peak tailing, however, can be observed also in the case when the concentration of the different sites is comparable, but the adsorption-desorption or mass-transfer kinetics is much slower on one type of site. This phenomenon is usually modeled either with the transport-dispersive model of chromatography, where rate constants describe the kinetics of mass transfer, or with the reaction-dispersive model, where the finite rate of adsorption and desorption is taken into consideration. Besides the finite rate of mass-transfer and sorption kinetics, both the transport-dispersive and the reaction-dispersive models involve the peak-broadening effects of the mobile phase via an axial dispersion term. In linear chromatography, the liquid film and the solid film transportdispersive models and the reaction-dispersive model are equivalent. It has been demonstrated by Fornstedt and Guiochon that tailing is the most enhanced when the number of slow sites is much fewer than the number of fast sites, requiring that there be several orders of magnitude difference between the rate constants of the two types of sites.3 The same aspect has been showed by Cavazzini et al. by means of the characteristic function (CF) analysis.4 In contrast, when the stationary phase is homogeneous, the mass-transfer kinetics should be extremely slow in order to observe peak asymmetry. Several authors investigated the peaksplitting effect that might occur in extreme conditions. Kinetic peak splitting is observed when a portion of the injected sample leaves the column unretained at the void time, while the rest of sample elutes with an elongated tailing profile.3,5-7 The problem of multiple-site linear adsorption chromatography is investigated here by means of the probabilistic molecular dynamic theory of chromatography developed by Giddings and (3) Fornstedt, T.; Zhong, G.; Guiochon, G. J. Chromatogr., A 1996, 741, 1-12. (4) Cavazzini, A.; Remelli, M.; Dondi, F. J. Microcolumn Sep. 1997, 9, 295302. (5) Lin, B.; Golshan-Shirazi, S.; Ma, Z.; Guiochon, G. Anal. Chem. 1988, 60, 2647-2653. (6) Jaulmes, A.; Vidal-Madjar, C. Anal. Chem. 1991, 63, 1165-1174. (7) Place, H.; Se´bille, B.; Vidal-Madjar, C. Anal. Chem. 1991, 63, 1222-1227.
Analytical Chemistry, Vol. 71, No. 16, August 15, 1999 3453
Eyring.8 This approach has important advantages compared to other fundamental models of chromatography, such as the following: some physicochemical aspects such as the surface heterogeneity, the stationary-phase distribution, and the nonhomogeneous packing are better described by their statistical attributes; the phase-exchange chromatographic process is really a stochastic process; a full description of the peak shape (by means of its statistical attributes or by the inversion of the characteristic function) can be obtained and thus the problem of peak tailing can simply be investigated. The theoretical model here developed includes also the twosite chromatographic case, originally proposed by Giddings8-10 and completely solved by some of the authors of the present study.4 The solution is obtained by means of the CF method.11 THEORY Molecular Dynamic Theory of Chromatography. In the molecular dynamic theory of chromatography, originally proposed by Giddings and Eyring in 1955, the chromatographic process is modeled at the molecular level.8 The effect of axial dispersion is neglected, and it is assumed that the number of adsorption and desorption steps is governed by a Poisson process as the molecule travels along the column. When the stationary phase is homogeneous (single-site case), and ka and kd rate constants give the probability per unit time of adsorption and desorption, respectively, the following band profile is obtained:
f(t) ) (X/2t)e-kdt-kat0I1(X)
(1)
where t0 is the dead time, X ) (4kakdt0t)1/2, and I1 is a modified Bessel function of the first kind and first order. It is important to note that this model gives the corrected retention time, because f(t) is the probability density function describing the time (t) a molecule spends adsorbed in the stationary phase. Giddings and Eyring8 and later Giddings9,10 and McQuarrie12 addressed the issue of two-site adsorption, too, giving rather complex expressions for the peak profile. Nevertheless, these calculations revealed that slow adsorption-desorption kinetics is a major source of peak tailing. CF Method in the Theory of Chromatography. The CF is the inverse Fourier transform of the probability density function.13 For a random variable t (for example, the time a single molecule spends in the column) with the probability density function f(t), the CF is defined as
Φ(ω) )
∫ exp(iωt)f(t) dt
(2)
In the above equation, i is the imaginary unit and ω an auxiliary real variable (frequency). In this approach, f(t) dt represents the infinitesimal probability that a molecule leaves the column with a (8) Giddings, J. C.; Eyring, H. J. Phys. Chem. 1955, 59, 416-421. (9) Giddings, J. C. J. Chem. Phys. 1957, 26, 169-173. (10) Giddings, J. C. Anal. Chem. 1963, 35, 1999-2002. (11) Dondi, F.; Remelli, M. J. Phys. Chem. 1986, 90, 1885-1891. (12) McQuarrie, D. A. J. Chem. Phys. 1963, 38, 437-445. (13) Crame´r, H. Mathematical Methods of Statistics; Princeton University Press: Princeton, NJ, 1974.
3454 Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
retention time between t and t + dt. The analytical inversion of the CF allows one to obtain f(t), i.e.,sin the interpretation of probabilistic chromatographic theorysthe chromatographic peak.11 In the modern theory of probability,13 the CF has a very important role. It is well-known that in order to find the density function of a random variable that is the sum of two or more independent random variables, it is necessary to solve the convolution integral of the density function of the individual random variables. The CF provides an alternative to solve this problem. In this case, the density function can be calculated directly by the inversion of the CF obtained as the product of the CFs of the individual random variables.13 Furthermore, the statistical moments (i.e., the mean, the variance, or the most important peak shape parameters: the skew, S, and excess, Ex) can be obtained directly from the derivatives of the CF.4,11,13,14 Accordingly, a complete description of the peak shape can be obtained also in the case when the mathematical expression of f(t) is not available, because usually the analytical inversion of the CF is rather complicated and can be done only in some very simple cases.15 The number of adsorption-desorption steps executed by a molecule in the chromatographic column is a random variable. In the stochastic interpretation, the history of the molecule is the sum of a random number (number of adsorption-desorption steps) of random variables (times spent on the site). Mathematically, this history will be the convolution integral of the density functions of the time spent on the site. By means of the properties of the CF, it is possible to substitute the convolution integral with the product of the elementary CFs and to obtain both the fundamental peak shape parameters11 and the chromatogram itself, when the inversion of the CF is possible. Single-Site (Homogeneous) Model. When the chromatographic process is envisioned in the same manner as in the molecular dynamic theory of Giddings and Eyring, the following CF is obtained:11
{ [1 -1iωτ - 1]}
Φ(ω) ) exp n
(3)
where Φ(ω) is the CF of the time spent in the stationary phase by the molecule (corrected retention time) and τ and n are the mean time spent on one site and the mean number of adsorptiondesorption steps executed by the molecule in the column, respectively. The above CF and the peak profile calculated in eq 1 are Fourier transform pairs; i.e., they identically represent the peak. This was proved by comparing the moments calculated from the two expressions11 and by calculating the inverse transform of eq 1.15 There exists a straightforward relationship between the rate constants of the Giddings-Eyring model and the above CF. As kd is the probability or rate of desorption in unit time, it is the inverse of the average residence time in one sorption step (14) Dondi, F.; Blo, G.; Remelli, M.; Reschiglian, P. In Theoretical Advancement in Chromatography and Related Separation Techniques; Dondi, F., Guiochon, G., Eds.; NATO ASI Series C, Vol. 383; Kluwer Academic Publishers: Dordrecht, 1992; pp 173-210. (15) Felinger, A. Data analysis and signal processing in chromatography; Elsevier: Amsterdam, 1998.
Table 1. Mean, Variance, Skew, and Excess Obtained by Means of the CF Method for the Homogeneous One-Site Case, the Heterogeneous Two-Site Case, and the General Multiple-Site Case 1-site
2-site
m-site m
mean
n(p1τ1 + p2τ2)
nτ
n
∑p τ
i i
i)1 m
variance
2n(p1τ12 + p2τ22)
2nτ2
2n
∑p τ
2 i 1
i)1
m
skew
3 x2n
p1τ13 + p2τ23
3 x2n (p1τ12 + p2τ22)3/2
∑p τ
3
i i
3
i)1
x2n (
m
∑p τ )
2 3/2
i i
i)1
m
excess
6 n
p1τ14
p2τ24
+ 6 n (p τ 2 + p τ 2)2 1 1 2 2
∑p τ
4
i i
6
i)1
n
{[
m
(
∑p τ )
2 2
i i
(“sojourn” time) on one site: kd ) 1/τ. Furthermore, as the molecule stays for a fixed t0 time in the mobile phase, during which time it performs n adsorption and desorption steps, n ) kat0 since ka is the probability of adsorption in unit time. For the corrected retention time (i.e., for the first moment) we shall get t′R ) nτ. We can re-express the Giddings-Eyring model after these considerations as
xtτn e
-t/τ-n
(x )
I1
4nt τ
{[
p2 p1 + -1 1 - iωτ1 1 - iωτ2
1
∑p 1 - iωτ j
j)1
]}
(6)
j
where the sum is calculated over all the m site types present. This last equation can also be extended to the continuous case, i.e., when the number of different sites present in the column becomes very high and tends to a continuous distribution. In this case one gets
{[
Φ(ω) ) exp n -1 +
∫1 -1iωτf(τ) dτ ∑
]}
(7)
(4)
The moments of the peak profile can simply be calculated after differentiating the CF. The most important moments and peak shape parameters of the single-site model are summarized in the first column of Table 1. Two-Site Model. Cavazzini et al. extended the CF method to describe the chromatographic process on two-site heterogeneous surfaces.4 In contrast to the complexity of the two-site model in the time domain, they obtained the following simple expression for the CF:
Φ(ω) ) exp n
m
Φ(ω) ) exp n -1 +
i)1
f(t) )
J. Calvin Giddings: “One aspect of the stochastic theory which has been pursued from the beginning is the effect of a nonuniform surface with different kinds of adsorption sites. The mathematics rapidly becomes intractable, however, as we pass from the sheltered simplicity of one-site theory”.16 Multiple-Site Model and Extension to the Continuous Case. The multiple-site model of chromatography assumes the column as a random sequence of many different types of sites over which a molecule performs a sequence of entry processes, spending a random sorption time on each local site. The time spent on the different sites will differ according to the physical-chemical properties of the sites and in this model, it is again considered as a random variable. The extension to the discrete multiple-site case can be made by replacing, in eqs 43-45 of ref 4, the binomial coefficient with a multinomial one. In the case when the column is constituted by sites of m different types, each one of given abundance (p1, p2, p3, ..., pm), the expression for Φ(ω) is
]}
(5)
where the subscripts refer to site types 1 and 2, τ1 and τ2 are the mean sorption times for the two sites and p1 and p2 represent their relative abundance, being p1 + p2 ) 1. The moments of the peak profile can again be easily calculated from the derivatives of the CF. The inversion of the above CF back to the time domain is possible numerically by a fast Fourier transform (FFT) routine, whereas the parallel reactive-dispersive or transport-dispersive model can only be solved by numerical integration of partial differential equations. To appreciate the beauty and simplicity of the CF, we call the reader’s attention to the following thoughts of
where f(τ) is the probability density function of the continuous mean time spent on the site. Symbol ∑ indicates the entire range of “sojourn” times over which the integration must be made. This case concerns all the real cases because, when a nearly homogeneous surface is given, a certain dispersion around the mean sorption energy is always present.17 The mathematical forms of eqs 3 and 5-7 hold only for welldefined distributions of the time spent on the site and number of entries, i.e., when the sojourn time of the molecule on one particular site has an exponential distribution and the random number of adsorption-desorption steps follows a Poisson distribution (exponential Poisson models). This case corresponds to the fundamental model of Giddings and Eyring.8 Accordingly, eqs 3 and 5-7 refer only to the contribution of the stationary phase to the observed peak profile (i.e., it gives the corrected retention time), but in a very simple way it is also possible to consider the mobile-phase term or different hypotheses can be postulated about the distribution of time spent on one site or also about the number of entries.11,14 It can be seen that the chromatographic peak (statistically interpreted as the density function of the retention times of the molecules leaving the column) will depend on the statistical attributes of the model of the chromatographic column.14 In (16) Giddings, J. C. Dynamics of Chromatography; M. Dekker: New York, 1965. (17) Roles, J.; Guiochon, G. J. Phys. Chem. 1991, 95, 4098-4109.
Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
3455
practice, the distribution of τ can be related to the thermodynamics of the adsorption-desorption process via the Frenkel-de Boer equation:18
τ ) g0 exp(E/RT)
(8)
where g0 is a configurational factor (approximately 10-14 s in our calculations), E is the (mean) adsorption energy, and R and T are the gas constant per mole and the temperature, respectively. Linearity. In all stochastic models of chromatography, the adsorption isotherm is assumed as linear (infinite dilution models). This ensures that the adsorption-desorption processes follow a rate law of first order, because the linearity of the adsorption isotherm and that of the reaction kinetics will generally cover the same range of concentration.16 Accordingly, the peak shape effects here investigated refer to the so-called kinetic tailing. RESULTS AND DISCUSSION The CF can be used with two different objectives. By means of its derivatives, it is possible to obtain all the relevant peak shape parameters19 and by its analytical or numerical inversion to time domain it is possible to directly obtain the peak shape. In the following we shall show results obtained by means of analysis of both the moments and the peak profiles. Single-Site (Homogeneous) Surface. In the case of homogeneous stationary-phase surface, the stochastic model and the conventional mass balance model give identical results, except for the dispersion occurring in the mobile phase, which factor is neglected by the stochastic model. In the conventional kinetic model, the number of mass-transfer units is expressed as Nm ) kmk′t0 where km is the rate constant of the solid film model, a parameter equivalent to the desorption rate constant (kd) of the reaction-dispersive model, k′ is the capacity factor, and t0 the holdup time of the column. The corrected retention timesi.e., the first moment of the chromatogramscalculated with the stochastic model is t′R ) nτ (see Table 1). The retention time, therefore, can be written as
tR ) t′R + t0 ) t′R +
t′R k′ + 1 ) nτ k′ k′
(9)
The number of mass-transfer units will be Nm ) n because km ) kd ) 1/τ and k′t0 ) t′R ) nτ. It is important to emphasize that the number of mass-transfer units in the lumped kinetic model is equal to the average number of adsorption-desorption steps in the stochastic model. From the stochastic model, we can derive the number of theoretical plates as
N)
n k′ + 1 2 k′
(
2
)
(10)
whereas for the number of theoretical plates in the lumped kinetic (18) De Boer, J. H. The Dynamical Character of Adsorption; Clarendon Press: Oxford, 1968. (19) Dondi, F.; Cavazzini, A.; Remelli, M. In Advances in Chromatography; Brown, P. R., Grushka, E., Eds.; Marcel Dekker: New York,1998; Vol. 38, pp 5174.
3456 Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
Figure 1. Homogeneous one-site model. Different peak shapes have been simulated for different values of the number of the adsorption-desorption steps, n ) 1-5000.
model the following expression is valid:20
2 1 1 k′ + ) N Ndisp Nm k′ + 1
(
2
)
(11)
The above equations demonstrate the equivalence of the two approaches with the only exemption that 1/Ndisp ) 0 in the stochastic model because the dispersion in the mobile phase is neglected. Figure 1 refers to a study of the homogeneous case by means of the CF method. Different peak shapes have been simulated for different values of the number of the adsorption-desorption steps executed by the molecule in the column while migrating (n ) 1, 4, 200, 2000, 5000). The peaks simulated all have unit area and the time scale has been normalized with respect to the corrected retention time. When the mean number of steps, i.e., the number of mass-transfer units, is only 1, the corresponding peak shape is roughly constituted by a high spike located at the origin of the normalized coordinate and of a very long, extremely elongated tail. This result is in agreement with the split-peak effect observed by many researchers.5-7 In this case a fraction of the molecules leaves the column without ever adsorbing on the stationary phase. The stochastic model exactly gives the fraction of the molecules eluting with no adsorption at all: this amount is e-n.8 When there is only one mass-transfer unit, 36.8% of the sample elutes nonadsorbed. The split-peak phenomenon, however, is limited for the cases of extremely slow kinetics. When the number of mass-transfer units is already 4, only 1.8% of the molecules are not at all retained by the column. Therefore, the spike tends progressively to disappear as n is increased (for n ) 4 it is still visible, but for n ) 20, it has already completely disappeared). When n is further increased, the peak becomes more and more symmetrical (tending to a Gaussian limit). When n grows from 200 to 5000, the skew and the excess decrease from 0.15 to 0.03 and from 0.03 to 0.001, respectively. (20) Guiochon, G.; Golshan-Shirazi, S.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: Boston, 1994
Φ(ω) ) exp
[
] [
]
np1 np2 - np1 exp - np2 1 - iωτ1 1 - iωτ2
(12)
The above CF is the product of two CFs written for the one-site model (eq 3). The convolution theorem of Fourier transform states that the multiplication in frequency domain is equivalent to a convolution in time domain. Therefore, the peak shape of the twosite heterogeneous model becomes
f(t) )
(x
(x )) (x
4n1t * τ1
n1 -t/τ1-n1 e I1 tτ1
(x )) 4n2t τ2
n2 -t/τ2-n2 e I1 tτ2
Figure 2. Heterogeneous two-site model. Different peak shapes have been simulated for different values of the number of the adsorption-desorption steps, n ) 100-2000; τ2/τ1 ) 10 and proportion of the stronger site p2 ) 1%, for all the chromatograms.
Figure 3. Heterogeneous two-site model. Different peak shapes have been simulated for different values of the number of the adsorption-desorption steps, n ) 100-5000; τ2/τ1 ) 10 000 and proportion of the stronger site p2 ) 1%, for all the chromatograms.
Two-Site Surface. Figures 2 and 3 present an analogous study but for the heterogeneous two-site model. For this case, it is not possible to invert analytically the CF (eq 5), but the mean and the other fundamental moments of the peak shape have been obtained by the derivatives of the CF (see Table 1).4 In the twosite heterogeneous model, the surface is constituted by two different types of sites (indicated with 1 and 2) having different abundance p1 and p2 and different adsorption energies (E1 and E2) or, which is the same, different mean adsorption times (τ1 and τ2).4,8-10 (In this discussion, the subscript 2 refers to the stronger site). The abundance of the stronger site is only 1% in both Figures 2 and 3. The study of this situation is the most important, because the stronger site must be present only in a very little quantity in order to see a tailing effect.3,4 The peak in the case of the heterogeneous two-site model is interpretable as the convolution of two peaks that correspond to the peaks of the two homogeneous phases (of kinds 1 and 2), with n1 ) np1 and n2 ) np2 as the mean number of adsorption steps, respectively. Equation 5 can be rewritten as follows:
(13)
where the symbol * stands for the operation convolution. The above convolution leads to a cumbersome expression and it is impractical to use it for the calculation of the peak profile, but it clearly demonstrates the connection between the one-site and the two-site models. For the calculation of the peak profile, the numerical inversion of eq 5 or 12 is favored. If, for instance, the mean number of steps is 100 and the abundance of the stronger site is only 1%, then the mean number of sorption steps executed by the molecule on the stronger site will be 1, so the peak will be the result of the convolution of an exponential-like decay function (stronger site) with a narrow, symmetrical peak of 99 sorption steps (weaker site). This case corresponds to the peak with n ) 100 in Figure 2. When n is increased, the peak profile changes its shape and it becomes more symmetrical. Nevertheless, with respect to the homogeneous case, the skew and the excess are now remarkably greater (for n ) 200 the skew and the excess are 1.5 and 2.9 and for n ) 5000 they are about 0.3 and 0.1, respectively). Figures 2 and 3 differ only in the choice of the mean sojourn time ratio (τ2/τ1) which is a direct measure of the surface heterogeneity.4 Note that the mean sojourn time ratio is identical to the inverse of the rate constant ratio of the two types of sites, i.e., τ2/τ1 ) kd,1/kd,2. In Figure 3, this ratio is 100 times greater than in Figure 2. The fundamental conclusions of these two figures are almost the same but, when τ2/τ1 is increased to 10 000, a spike appears at the origin, as in Figure 1, for n ) 100 (it almost disappears at n ) 400). This occurs because the exponential-like decay function (the chromatogram obtained when only the slow site is regarded) corresponding to n ) 100, i.e., to np2 ) 1, goes very quickly to 0 depending on the average sojourn time. In case of heterogeneous surfaces, the small amount of slow sites and the large difference in mass-transfer rate constants (i.e., in average sojourn times) on the two types of sites are the most relevant sources of tailing and loss of column efficiency. The number of theoretical plates for the case of a two-site heterogeneous surface can be expressed as 2
N)
n (p1 + p2(τ2/τ1)) k′ + 1 2 p + p (τ /τ )2 k′ 1
2
2
1
(
2
)
(14)
The ratio of N for the case of a two-site heterogeneous surface over N for the case of a one-site homogeneous surface is plotted Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
3457
Figure 4. Plot of the ratio of N for different heterogeneous models (two-site case with variable relative abundance of the two kinds of site; uniform adsorption energy case; uniform average sojourn time case) over N for the homogeneous one-site model.
in Figure 4. The figure demonstrates that the higher the ratio of the average sojourn times and the lower the proportion of the slow sites, the higher the loss of column efficiency. When there is, for instance, a 500-fold difference in rate constants of the two types of sites and the proportion of the slower site is only 1%, the column efficiency drops to nearly 1% of the efficiency achievable on a homogeneous surface. The relative efficiency always drops to the value of the relative amount of the slow sites (p2) provided that there is a difference of several orders of magnitude between the rate constants of the two types of sites. When τ2/τ1 is increased and the abundance of the slower sites is reduced, both the skew and the excess of the peak profile increase significantly compared to their values observed in the case of a homogeneous surface.4 When n ) 200 and τ2/τ1 ) 10, for instance, S ) 0.42 and Ex ) 0.25, at p2 ) 0.1, but when p2 ) 0.01 and τ2/τ1 ) 100, then S ) 1.48 and Ex ) 2.94. This agrees with the precedent considerations because when S increases, the peak profile becomes more asymmetrical, reducing inevitably the efficiency of the separation system. Nevertheless, when there is a difference of some orders of magnitude between the mean sojourn times, the skew and the excess approach a limit and their values depend only on the abundance of the stronger site (and not on τ2/τ1) and furthermore, naturally, on the number of sorption steps. It is easy to show that for τ2/τ1, the limit of the skew and of the excess are 3/(2np2)1/2 and 6/np2, respectively. When the order of magnitude between the rate constants of the two types of sites is already 100, the skew and the excess do not change significantly if τ2/τ1 is further increased. For example, when τ2/ τ1 ) 10 000, p2 ) 0.01, and n ) 200, the skew and excess are 1.50 and 3.00 respectively, values that are very similar to those for τ2/ τ1 ) 100 with the same p2. When n ) 5000, we have S ) 0.29 and Ex ) 0.11 for τ2/τ1 ) 100 and S ) 0.30 and Ex ) 0.12 for τ2/τ1 ) 10 000. All these results show thatsalthough the relative amount of the slower sites has the most significant impact on the peak width and asymmetrysan almost steady-state situation is reached when the difference of mass-transfer rates is about 100-fold. Further difference in the mass-transfer rates only has a minor influence on the peak width and shape. This effect is illustrated in Figure 4. Multiple-Site Surface. We have extended the CF model of the stochastic theory of chromatography to include any number 3458 Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
Figure 5. Chromatograms calculated in the case of uniform average sojourn time distribution. τ2/τ1 is varied between 10 and 10 000; number of the adsorption-desorption steps n ) 10 for all the cases. The homogeneous one-site case (Giddings-Eyring model) is plotted as a reference.
of different types of sites. The CF of the generic multiple-site surface is given by eqs 6 and 7. In this study, we have investigated the uniform distribution of mean sojourn times, as well as the uniform and normal distribution of sorption energies. Besides these unimodal distributions, we studied the bimodal Gaussian sorption energy distribution. Uniform Distribution of Mean Sojourn Times. When the mean sojourn times are uniformly distributed, any sojourn time between the fastest (τ1) and the slowest (τ2) sites has equal probability. The CF of this configuration can be analytically calculated when one combines eq 7 with the probability density function of the uniform distribution of mean sojourn times (see line 1 in Table 2). From the derivatives of the CF, the moments of the peak profiles can simply be calculated. The CF, the corrected retention time, the variance, the skew, and the excess for the case of uniformly distributed mean sojourn times are also given in Table 2. In Figure 5, chromatograms calculated with the inversion of the CF are presented for this model. The number of mass-transfer units, i.e., the average number of sorption steps is n ) 10 and the ratio of the mass-transfer rate on the slowest and fastest sites is varied between 10 and 10 000. As a reference, the peak profile obtained with the Giddings-Eyring model for the homogeneous surface is also plotted in Figure 5. Heterogeneity introduces some peak shape distortion, but the effect is not as relevant as one would expect. Remember that, in the case of the two-site model, the most enhanced peak shape distortion occurred when the slow site had a marginal abundance only. With the uniform distribution, there are as many slow sites as there are fast ones; therefore, the effect of heterogeneity is modest. In agreement with what we observed with the two-site model, when the difference of mass-transfer rates between the slowest and the fastest sites is over 100, the peak reaches a constant shape in the case of uniformly distributed mean sojourn times, too. In Figure 5, for instance, the chromatograms plotted for τ2/τ1 ) 100 and for τ2/τ1 ) 10 000 are indistinguishable. In Figure 4, the loss of column efficiency is plotted for this model, too. We can see that the maximum loss of efficiency is
Table 2. CF, Mean, Variance, Skew, and Excess for the Heterogeneous Uniform Mean Sojourn Time Model and the Heterogeneous Uniform Adsorption Energy Model uniform mean sojourn time distribution
mean adsorption time distribution
adsorption energy distribution
uniform adsorption energy distribution
{ {
{
1 1 for τ1 e τ e τ2 f(τ) ) τ ln(τ2/τ1) 0 elsewhere 1 for E1 e E e E2 f(E) ) E2 - E1 0 elsewhere
1 for τ1 e τ e τ2 f(τ) ) τ2 - τ1 0 elsewhere E for E e E e E A exp 1 2 RT f(E) ) 0 elsewhere
{
A)
( )
1 RT[exp(E2/RT) - exp(E1/RT)]
{[
(1 - iωτ1) 1 ln -1 iω(τ2 - τ1) (1 - iωτ2)
]}
{[
CF
Φ(ω) ) exp n
mean
n
n
variance
2n 2 (τ + τ1τ2 + τ22) 3 1
n
skew
9 4
3 r 3 + r2 + r + 1 2n (r2 + r + 1)3/2
r)
τ2 τ1
2 r2 + r + 1 xn r + 1
excess
54 r4 + r3 + r2 + r + 1 5n (r2 + r + 1)2
r)
τ2 τ1
6 r2 + 1 ln r n r2 - 1
τ1 + τ2 2
x
Figure 6. Plot of the ratio of the skew and the excess for the uniform mean sojourn time model over the homogeneous one-site case, as a function of the surface heterogeneity.
25% compared to the efficiency we can reach with a homogeneous surface. The increase of skew and excess due to heterogeneity are plotted in Figure 6. Again, we can see that when the difference between the mass-transfer rates of the slowest and the fastest sites is at least 100, a plateau is reached for both skew and excess (and also for the efficiency loss in Figure 4) and the peak shape does not change any longer. Uniform Distribution of Sorption Energies. We can also suppose that the sorption energies follow a uniform distribution between two extremes, E1 the weakest and E2 the strongest sorption energy. By means of the Frenkel-de Boer equation (eq 8) we can relate the sorption energy and mean sojourn time. In this case, the mean sojourn times have a hyperbolic distribution. Similarly to the procedure we followed before, the CF, the corrected
Φ(ω) ) exp n
]}
ln[(1 - iωτ1)/(1 - iωτ2)] ln(τ2/τ1)
τ2 - τ1 ln(τ2/τ1) τ22 - τ12 ln(τ2/τ1)
x
ln r r2 - 1
r)
r)
τ2 τ1
τ2 τ1
Figure 7. Chromatograms calculated in the case of uniform adsorption energy distribution. τ2/τ1 is varied between 10 and 10 000; number of the adsorption-desorption steps n ) 10 for all the cases. The homogeneous one-site case (Giddings-Eyring model) is plotted as a reference.
retention time, the variance, the skew, and the excess can analytically be calculated; these results are also presented in Table 2. In Figure 7, chromatograms calculated with the numerical inversion of the CF are presented for the uniform sorption energy model. The number of mass-transfer units, i.e., the average number of sorption steps is again n ) 10 and the ratio of the masstransfer rate on the slowest and fastest sites is varied between 10 and 10 000. Note that the ratios τ2/τ1 ) 10 and τ2/τ1 ) 10 000 correspond roughly to ∆E ) 1 kcal/mol and ∆E ) 7 kcal/mol sorption energy differences, respectively, provided that the temAnalytical Chemistry, Vol. 71, No. 16, August 15, 1999
3459
energy:
τ* ) τ0 exp(E*/RT)
(17)
Note that as the log-normal distribution is not symmetrical, τ* is not equal to the average sojourn time. The average sojourn time can be expressed as the first moment of the log-normal distribution:15
µ1,τ ) τ*xq
(18)
where, for the sake of simplicity, q is defined as
q ) exp[(σE/RT)2]
Figure 8. Plot of the ratio of the skew and the excess for the uniform sorption energy model over the homogeneous one-site case, as a function of the surface heterogeneity.
perature is about 115 °C. As a reference, the peak profile obtained with the Giddings-Eyring model for the homogeneous surface is again plotted in Figure 7. When surface heterogeneity is modeled with uniform distribution of sorption energies, the peak profile significantly distorts as heterogeneity is increased. Since, in this case, the mean sojourn times follow a hyperbolic distribution, the relative abundance of the slow sites is small. Therefore, the tailing observed is much higher than in the case of uniform distribution of mean sojourn times. We can see in Figure 7 that we are on the brink of the split-peak effect. Were the number of mass-transfer units further reduced, the split-peak phenomenon would be clearly detected. When looking at Figure 4, we can see that the limiting case we observed for the loss of column efficiency with both the twosite model and with the uniform sojourn time distribution is not reached with uniform sorption energy distribution. As the difference in the mass-transfer rates increases, the efficiency keeps decreasing. A similar impact can be observed on the skew and the excess plotted in Figure 8. Both the skew and the excess show an almost linear relationship with ∆E. Normal Distribution Sorption Energies. It is evident to assume that, even for a homogeneous surface, there is some dispersion of the sorption energies.17,21 This dispersion can simply be taken into account when we apply the normal distribution for modeling the sorption energies. In this instance, the probability density function of the sorption energy is written as
f(E) )
1
x2πσE
[
exp -
]
(E - E*)2 2σ2E
(15)
where E* is the average energy and σE is the standard deviation of the distribution. When the sorption energy follows a normal distribution, the average sojourn time is characterized with the following log-normal distribution:
f(τ) )
1
x2π(σE/RT)τ
[
exp -
]
(ln τ - ln τ*)2 2(σE/RT)2
(16)
where τ* is the sojourn time corresponding to the average sorption 3460 Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
(19)
It is important to emphasize that the spread of both the sorption energies and the sojourn times can be expressed by means of their relative standard deviations (RSDE ) σE/E and RSDτ ) στ/ τ*). Furthermore, the variance of the sojourn times is15
µ′2,τ ) (τ*)2q(q - 1)
(20)
From the above expression it follows that RSDτ ) q - 1. This means that the relative standard deviation (i.e., the spread) of the sojourn times depends only on the standard deviation of the sorption energies; it is independent of the average of either the sorption energy or the sojourn time. For the present model, i.e., for normally distributed sorption energy and for log-normally distributed sojourn time, the CF can be calculated numerically only. The numerical calculation of the CF is possible by eq 6. Although the CF itself cannot be calculated in closed form for normally distributed sorption energies, its derivatives at the origin can be determined; therefore the calculation of the moments of the chromatograms is possible. For the first momentsi.e., for the corrected retention timeswe get t′R ) nτ*(q)1/2. The variance of the peak is µ′2 ) 2nτ*q2. Now we can express the number of theoretical plates as
N)
( ) ( ) [ ( )]
n k′ + 1 2q k′
2
)
n k′ + 1 2 k′
2
exp -
σE RT
2
(21)
From the above equation, it is now obvious that only the standard deviation of the sorption energies affects the number of theoretical plates regardless of the average sorption energy when we depart from the homogeneous case. The skew and the excess of the peak profile can also be calculated for this model and one gets S ) 3q3/2/(2n)1/2 and Ex ) 6q4/n, respectively. Chromatograms calculated with the present model are plotted in Figure 9 for n ) 100 adsorption-desorption steps. The peaks in the main part of the figure were calculated by assuming the same average sorption energy. The dispersion of the sorption energy around the mean was increased, however. It can be seen that when we depart from the homogeneous case, a minor energy (21) Stanley, B. J.; Guiochon, G. Langmuir 1995, 11, 1735-1743.
Figure 9. Chromatograms calculated with the normal distribution sorption energy model. Number of the adsorption-desorption steps n ) 100 for all the cases.
dispersion already has a significant impact on both the retention time and the peak width. Although the relative dispersion of the sorption energy is between only 2.6 and 4.3% in Figure 9, the impact of heterogeneity on the peaks is enormous. One must remember, however, that it is not the relative spread but the actual spread of the sorption energy that primarily determines the peak shape regardless of the value of the mean sorption energy. To illustrate this aspect, we compare two peaks on a normalized time scale in the inset in Figure 9. For the two peaks, the average sorption energy is 23.2 and 25 kcal/mol, respectively; while the standard deviation is 0.6 kcal/mol in both cases. It is obvious that, due to the difference between the average sorption energies, there is a 10-fold difference between both the retention times and the widths of the two peaks. When the time scale is, however, normalized with respect to the corrected retention time, only a minor difference can be detected between the two peaks. This result shows again that the value of σE is the most relevant factor affecting the peak profile. Bimodal Distribution of Sorption Energies. Stationary phases can frequently be characterized by a bimodal sorption energy distribution. The two-site model is an ideal view of the bimodal sorption energy distribution. From a physical view it is more sound to assume that the sorption energy has some dispersionseven if minorsaround the mean values. Recently, significant effort has been devoted to study the distribution of the sorption energy on heterogeneous surfaces. Very often a bimodal sorption energy distribution was found.17,21 By means of the CF, this energy distribution can also be modeled with ease. For our calculations, we assumed that the energy distribution is the sum of two Gaussian peaks. With the procedure we described above, we can derive the distribution of the average sorption times, which is the sum of two log-normal distributions in this instance. The CF can again be calculated numerically by eq 6. In Figure 10, a series of calculations is presented for a bimodal distribution. The average sorption energies are 22 and 25 kcal/ mol, respectively, and their standard deviation is 0.2 kcal/mol in both cases. The distribution of the sorption energy is plotted in Figure 10a for the case when the relative abundance of the slower sites is 90%. The corresponding distribution of the sojourn times
Figure 10. Bimodal sorption energy distribution model. (a) example of sorption energy distribution; (b) corresponding average sojourn time distribution; (c) chromatograms for different stationary phases, with ratio of the fast and slow sites varied between 92:8 and 10:90%.
is plotted in Figure 10b. Note that, due to the logarithmic transformation, the spread of the sojourn times is quite different on the two types of sites even though the standard deviation of the energies is constant. It is also essential to see in Figure 10b that the distribution of the sojourn time on the slow sites is very broad and flat; the area below the second log-normal function is 9 times larger than that of the first one, though. When the relative abundance of the slow sites is very small, the distribution of the sojourn times will exhibit a rather long tail, which will have a significant tailing effect on the peak profile. The chromatograms in Figure 10c correspond to n ) 100 adsorption-desorption steps on different stationary phases. The ratio of the fast and slow sites was varied between 92:8 and 10:90%. We can see that both the mean and the width of the peaks are greatly affected by the ratio of the different sites. CONCLUSIONS In this study, we have applied the stochastic model of chromatography to heterogeneous adsorption-desorption processes. We have demonstrated that the molecular dynamic theory and the reaction-dispersive or the transport-disperive models of linear chromatography are equivalent. When one applies the stochastic model of chromatography, the CF of the peak profiles can be calculated in closed form for several models of surface heterogeneity; therefore, the only numerical step in our approach is the inverse Fourier transform of the CF back to the time domain Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
3461
in order to calculate the peak profile. Our results indicate that the surface heterogeneity can very simply be taken into account by means of the stochastic model. Even for rather complex, bimodal or multimodal surface energy distributions, by means of the CF method, the chromatograms can be determined with unforeseen simplicity. On heterogeneous surfaces, we experienced the largest loss of column efficiency and the most enhanced peak tailing when the stationary surface contained the slow sites in a minor relative abundance. As the moments of the peak profile are readily available for heterogeneous surfaces, we can rather simply calculate the number of theoretical plates and the peak shape parameters directly from the CF. Our calculations also reveal that when there exists a slight dispersion of sorption energies on an otherwise homogeneous surface, the peak asymmetry may be considerable. This phenomenon may explain the origin of peak tailing observed in several
3462 Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
cases. The determination of the sorption energies is an extremely ill-posed problem.17,21 Therefore, it is almost impossible to experimentally determine whether a surface we assume to be homogeneous is a real one-site surface or there is a slight dispersion of the sorption energies around the mean. When an energy dispersion is present, we shall always have tailing peaks. ACKNOWLEDGMENT This work was supported by research Grants T 025458 from the Hungarian National Science Foundation (OTKA) and FKFP 0609/1997 from the Hungarian Ministry of Education, as well as by the Italian Ministry of the University and the Scientific Research (MURST, 60 and 40%) and the NATO Grant OUTR.LG 971480. Received for review March 15, 1999. Accepted May 22, 1999. AC990282P
