Anal. Chem. 1989, 61, 2373-2380
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Theoretical Study of System Peaks and Elution Profiles of High Concentration Bands of Binary Mixtures Eluted by a Binary Eluent Containing a Strongly Retained Additive Sadroddin Golshan-Shirazi and Georges Guiochon*
Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, and Division of Analytical Chemistry, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 -6120 A theoretkal investlgatlon Is done of the phenomena that take place when a large sample of a binary mlxture is eluted by a moblie phase containing a strong solvent or additlve which Is nearly as strongly or more strongly retalned as the sample components. For this study we have assumed the competitive equlHbrknn isotherms of the sample components and the strong solvent to be given by a Competitive Langmulr model. The main parameter of the model Is the adsorption strength of the additive compared to that of the sample components. At low additive strength, the band profiles are very slmllar to those found with a pure mobile phase whlch would have the same elution strength. At hlgh additive strength, the band protiles are reversed. They e x h M a smooth, dtffuse front and a steep rear shock layer. When the two component bands interfere, two new effects are observed, a retalnment effect and a pull-back effect. These effects are analogous to the displacement and the tag-along effects, respectlveiy, whlch are observed with pure mobile phases but act In the opposite direction. Then the chromatogram looks as if the first component band displaces the second one and/or tags along it. At very hlgh addttive strength, very complex and unusual band profiles are observed, e.g., two interfering bands exhibitlng three maxlma. If the additive strength Is equal to the strength of one of the components, the band of this component remains nearly Gaussian, while the band of the other one becomes unsymmetrlcai and a shock layer eventually appears on the side of the band dlrected toward the other band.
INTRODUCTION The injection of a sample in a chromatographiccolumn may result in more peaks than there are compounds in the mixture, if the mobile phase contains one or several additives that are retained by the stationary phase. These additional peaks result from the perturbation of the additive equilibrium between the two phases which is caused by the injection of a sample. Since all phase equilibria are competitive processes whenever more than one compound is involved in addition to the weak solvent, the presence of the migrating bands of the sample components creates a local perturbation of the additive equilibrium between the two phases. This perturbation propagates for its own sake. Because of the competitive nature of phase equilibra, this phenomenon takes place even if the sample components are dissolved prior to injection in a solution identical with the mobile phase used. The perturbation of the distribution equilibrium of the mobile phase additives between the mobile and the stationary phases of the chromatographic system results in two sets of concentration signals at the column exit, one positive, one negative. There is a positive peak for each additive. This peak corresponds to the amount of additive expelled from the stationary phase when the new equilibrium between the mobile phase, modified by the introduction of the sample components, and the stationary phase takes place. Because *Author to whom correspondence should be sent, at the University of Tennessee.
of the competition with the sample components, the additive has become less retained. If the sample size is small, e.g., under linear conditions, this band is eluted at the retention time of the additive (which is a function of the concentration of this additive in the mobile phase, through its adsorption isotherm). If the sample size is large, an overloaded band is recorded. Since the injection of the sample has caused a certain amount of each additive to be expelled from the stationary phase, a return of the chromatographic system to equilibrium requires that the stationary phase be replenished. Therefore, for each component of the sample, one negative concentration band of each additive will be eluted. Each of these negative bands correspond to the amount of additive transferred from the mobile phase to the stationary phase, for the requilibration of the system, when the corresponding component is eluted. Thus, the positive band and the sum of all the negative concentration bands of an additive correspond to the same amount. The negative band is often eluted at the retention time of the corresponding solute. Sometimes, however, it is eluted later. This phenomenon has received various names but the additional, perturbation peaks are generally d e d system peaks. System peaks have been discovered long ago ( I ) . They have been investigated in detail by Valentin in gas chromatography (2,3)and by Levin and Grushka (4) in liquid chromatography. Their properties have been used for the determination of isotherms ( 2 , 3 , 5 )and of equilibrium constants (5). They are widely known at present because of an important analytical method permitting the UV detection of compounds that do not have the proper chromophores (6, 7). In this method, a mobile phase containing a constant concentration of a UV absorbing, retained additive is pumped through the column, and the components that do not absorb UV radiation at the wavelength at which the detector is set are detected by the signal change which results from the concentration band of the additive accompanying the elution of each compound out of the column (8). In adsorption chromatography, this is usually a negative concentration band. Explanations of system peaks have been varied, including solvent displacement, solvophobic interactions in the mobile phase, preferential evaporation of one component of the mixed eluent (9),and preferential solvation (10). The perturbation of the equilibrium of the additive between the two phases of the chromatographicsystem, described above, is now generally accepted. The validity of this assumption is demonstrated by the agreement observed between the predictions of the nonlinear theory of chromatography and experimental results ( 1 1 , 12). Most of the previous work on system peaks has dealt with the linear aspects of the phenomenon, those which are related to the sample components. Certainly, system peaks cannot be dealt with exactly within the framework of linear chromatography. Several important aspects of nonlinear chromatography are involved in the present instance: (i) the concentration of the strong solvent or the additive(s) in the stationary phase at equilibrium is not proportional to its
0003-2700/89/0361-2373$01.50/00 1989 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 21, NOVEMBER 1, 1989
concentration in the mobile phase (its equilibrium isotherm between the two phases is usually not linear a t the concentrations used); (ii) there is competition between the strong solvent or the additive and the sample components for interaction with the stationary phase; (iii) the retention times of a small pulse of the sample components, of the strong solvent, or of an additive depend certainly on the concentration of the strong solvent and/or of the additive in the mobile phase, which is the reason why strong solvent and additives are used in the first place, and why the additive concentration has to be optimized. Nevertheless, in analytical applications, when a small sample size is used, e.g., under linear conditions, the perturbation itself can be considered as linear. The competitive isotherms of the sample components can be expanded around the local composition of the mobile phase and considered as locally linear because of the small concentrations involved. Thus, the treatment of system peaks in analytical chromatography as a linear phenomenon is justified. Indeed, under standard analytical conditions, the retention time of a pulse of any compound does not depend on the sample size injected when the sample amount is small. Under these conditions, the retention times of the components of the sample do not depend on the nature or on the concentration of the other components introduced in the same time. In a previous publication, we have extended the theory of nonlinear chromatography to the prediction of the elution profiles of the system peaks observed when a large concentration band of a pure compound is injected in a chromatographic system where the mobile phase is a binary mixture containing a strongly sorbed additive or a strong solvent (11). We have also discussed the conditions under which the phenomenon is negligible in practice, because the strong solvent or additive interacts too weakly with the stationary phase (13). Finally, we have presented an experimental analysis of the factors influencing the band shape (12). The experimental results, obtained with 2-phenyl-1-ethanol,2-phenyl-1-propanol, 3-phenyl-1-propanol,and acetophenone on silica with mixtures of dichloromethane or n-hexane and light alkanols, were in excellent agreement with the theoretical predictions. Conditions were found under which all the phenomena predicted by the theory were observed. The theoretical results also account for a number of observations reported in the literature and which had remained unexplained (14-16). The essential results of that previous work can be summarized as follows. Large samples of a pure compound are injected in a proper chromatographic system, which uses a binary solvent mixture as mobile phase, and where the strong solvent is more strongly adsorbed than the investigated compound. When the concentration of the strong solvent is increased progressively, the elution profiles of the bands obtained are first, a t low strong solvent concentrations, of the type associated with a Langmuir isotherm, with a very steep front and a slanted rear. At higher strong solvent concentrations, the retention time is, of course, smaller, but the direction of the band asymmetry is reversed, the front is slanted while the rear is very steep, a profile normally associated with an anti-Langmuir or S-shaped isotherm. In both cases, however, the series of profiles obtained for samples of increasing sizes differ markedly from those obtained with such isotherms when the mobile phase is a pure solvent: the slanted part of the profile of the larger samples does not appear to be the natural extension of the slanted part of profides obtained with smaller amounts. In an intermediate range, the band profile is very unusual and seems to be the composite of two or three different, more conventional band profiles. A t very large strong solvent concentrations, a hump appears on the front side of the compound band. This hump may sometimes turn into a second peak (11, 13). The aim of the present paper is to extend the theoretical
results previously obtained in the case of a pure compound to the case of a binary sample and to discuss the problems arising from the interference between the system peaks associated to these two components. A companion paper discusses experimental results (17).
THEORY We must consider a three-component model. One of them is the strong solvent, the other two are the components of the binary mixture. Accordingly, we need to use the differential mass balance equations for these three compounds and their competitive isotherms. Since we are dealing here with the most popular applications of high-performance liquid chromatography, the column efficiency is high, usually well exceeding several thousand theoretical plates. Thus, we use the semiideal model, assuming the axial dispersion in the mass balance equation to be negligible and the two phases to be constantly in equilibrium (18). This model assumes the column efficiency to be infinite and the concentration in the stationary phase, in a slice of column, is related to the concentration in the mobile phase by the isotherm equation. Finally, the model corrects for the column efficiency of real columns, which is finite, by adjusting the space and time increment used in the numerical integration of the system of partial differential equations which expresses the model mathematically (18). In such a numerical calculation, truncation errors take place. It can be shown that, under linear conditions, they are equal to an apparent axial dispersion term, directly related to the values of the space and time increments. By taking proper values of these increments, the truncation error can be made to account exactly for the dispersion caused by the finite column efficiency (19). The mass balance equations of the components of the sample are written as follows:
aci( 1 + F-W) + u-aci = 0 (i = 1, 2) at
(la)
ax
where t and x are the time and the abscissa along the column, respectively, Ci and Qi are the concentrationsof the component i in the mobile phase and stationary phase, respectively, u is the mobile phase velocity, and F is the phase ratio. The mass balance of the strong solvent is identical
In this theoretical investigation, we assume the equilibrium isotherms of the three compounds involved to be given by the classical Langmuir competitive isotherm equation
where Qois the column saturation capacity, assumed to be the same for all compounds, and the bis are numerical coefficients, characterizing each compound. The same equation applies to the strong solvent. An equal column saturation capacity for all the compounds involved is required for thermodynamical consistency of the model; otherwise, the Gibbs-Duhem relationship would not be valid (20). So far, there is no discrimination between the two components of the sample mixture on one hand and the strong solvent on the other. The difference between them comes in the way they are introduced in the column, that is in the boundary condition, which characterizes the nature and the size of the perturbation we make at the column inlet. The two components are injected as a rectangular pulse of duration t, where the component concentrations are CIo and C20,respectively. The strong solvent is introduced with the mobile phase, at constant concentration. Accordingly, the boundary conditions selected are as follows:
ANALYTICAL CHEMISTRY, VOL. 61, NO. 21, NOVEMBER 1, 1989
Ci(0,t) =
c) 0
0
cio
Ci(0,t) = 0
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0 < t I t,
