Determination of competitive adsorption isotherms applying the nonlinear frequency response method: Part I. Theoretical analysis

In this work adsorption equilibria of binary mixtures are quantified analyzing the nonlinear frequency response of a chromatographic column. Local partial derivatives of an isotherm model can be estimated for certain steady-states from the low frequency asymptotes of the corresponding frequency response functions (FRFs). The required FRFs correspond to two different compounds and the type of the imposed inlet concentration changes, e.g. periodical inlet concentration changes of only one compound or of both of them. For an accurate determination of isotherm parameters, it is necessary to approach as close as possible the low frequency asymptotic behaviour of these functions. Based on principles valid for the FRFs corresponding to the adsorption of a single solute, frequencies needed to reach the low frequency asymptotes of the functions of interest for estimating competitive isotherms are defined in this paper. The relation between the accuracy of the isotherm parameters determined and numbers and types of periodical inlet concentration changes and steady-states analyzed is also evaluated.     

Determination of competitive adsorption isotherms applying the nonlinear frequency response method: Part II. Experimental demonstration

This paper demonstrates an experimental application of the nonlinear frequency response (FR) method extension to determine adsorption isotherms of binary mixtures. This method, based on the analysis of the response of a chromatographic column subjected to the sinusoidal inlet concentration changes, is shown to be an alternative for isotherm determination. The critical issue related to the successful application of the method is to reach experimentally the low frequency asymptotic behaviour of the corresponding frequency response functions (FRFs). Although, there are different possibilities to perform periodical inlet concentration changes, in this paper only simultaneous changes for both components were considered. The adsorption of phenol and 2-phenylethanol on octadecyl silica was analyzed experimentally using a mixture of methanol and water as a solvent. Parameters of competitive isotherms were also estimated for comparison using the classical perturbation method. Despite certain differences between competitive isotherms estimated with the two methods that were found, the obtained results show the potential of the nonlinear FR method for measuring competitive isotherms.     

Use of Nonlinear Frequency Response for Discriminating Adsorption Kinetics Mechanisms Resulting with Bimodal Characteristic Functions

One of the characteristic examples of the inability of the classical linear frequency response (FR) method to identify the correct kinetic mechanism is adsorption of some substances (p-xylene, 2-butane, propane or n-hexane) on silicalite-1. The linear FR resulted with bimodal FR characteristic functions, which fitted equally well to three different kinetic models: nonisothermal micropore diffusion, two independent isothermal diffusion processes, and an isothermal diffusion-rearrangement process. We show that the second order frequency response functions (FRFs), obtained from the nonlinear FR, can be used for discrimination among these three mechanisms. Starting from the nonlinear models, we derive the theoretical expressions for the first and second order FRFs corresponding to these three mechanisms and show that different shapes of the second order FRFs are obtained for each mechanism. This would enable identification of the real mechanism from nonlinear FR data.     

Discrimination between adsorption isotherm models based on nonlinear frequency response results

A number of criteria are established for distinguishing between different adsorption isotherm types. These criteria are defined based on the adsorption isotherm derivatives up to the third order, which, on the other hand, can be estimated from nonlinear frequency response data. The criteria for five favourable (Langmuir, Freundlich, Sips, Toth and Unilan) isotherms and two complex isotherms (BET and quadratic) are presented. These criteria enable unique identification of the underlying adsorption isotherm relation if the values of the local first, second and third order isotherm derivatives at several points are known. The method is applied to experimental data from our previous publications, for one case of a favourable and one case of a complex isotherm.     

Impact of an error in the column hold-up time for correct adsorption isotherm determination in chromatography I. Even a small error can lead to a misunderstanding of the retention mechanism

The impact of a realistic error in the column hold-up time on the determination of the adsorption isotherm model was systematically investigated. Frontal analysis and the inverse method were used for the accurate determination of the adsorption isotherm. The true retention times of the breakthrough curves were used with a known hold-up time as reference. The adsorption isotherms were calculated using the same procedure that is used for real experimental adsorption isotherms, where the true hold-up time is unknown. The raw data were analyzed with calculations of adsorption energy distributions (AEDs), Scatchard plots, fitting to different rival adsorption models and finally their ability to predict true profiles. The results show that for a true Langmuir and bi-Langmuir model with an underestimated hold-up time the error may lead to a more heterogeneous model and for overestimated cases false adsorption processes like multi-layer adsorption or solute-solute interaction are assumed. The Scatchard plots for data obtained using a Langmuir adsorption isotherm are nonlinear and the AEDs show clear deviations from Langmuir behavior already at small deviations from the true hold-up time at a moderate surface coverage. The inverse method confirms the result that was obtained from the frontal analysis procedure.     

Determination of the adsorption isotherm of fluoxetine by adjustable factor h-root method (AF-HRM)

Summary Conventional methods to determine competitive adsorption isotherm are tedious, and require large quantities of pure isomers. In some cases pure enantiomers are not available, thereby making the full experiments to determine the competitive adsorption isotherm of enantiomers impossible. A new method, adjustable factor h-root method (AF-HRM), had been presented for the measurement of langmuir equilibrium coefficients. In the new approach, the retention factors and the frontal retention factors are required to determine the parameters of isotherms. As the ultimate goal is to separate enantiomers of fluoxetine on a preparative scale, a convenient method to determine the adsorption isotherm is essential. In this paper, we have demonstrated that AF-HRM can be applied to determine the parameters in the extended Langmuir model. Experimental results agree well with the prediction.     

Adsorption-desorption isotherm hysteresis of phenol on a C18-bonded surface

Single component adsorption and desorption isotherms of phenol were measured on a high-efficiency Kromasil-C18 column (N = 15000 theoretical plates) with pure water as the mobile phase. Adsorption isotherm data were acquired by frontal analysis (FA) for seven plateau concentrations distributed over the whole accessible range of phenol concentration in pure water (5, 10, 15, 20, 25, 40, and 60 g/l). Desorption isotherm data were derived from the corresponding rear boundaries, using frontal analysis by characteristic points (FACP). A strong adsorption hysteresis was observed. The adsorption of phenol is apparently modeled by a S-shaped isotherm of the first kind while the desorption isotherm is described by a convex upward isotherm. The adsorption breakthrough curves could not be modeled correctly using the adsorption isotherm because of a strong dependence of the accessible free column volume on the phenol concentration in the mobile phase. It seems that retention in water depends on the extent to which the surface is wetted by the mobile phase, extent which is a function of the phenol concentration, and of the local pressure rate, which varies along the column, and on the initial state of the column. By contrast, the desorption profiles agree well with those calculated with the desorption isotherms using the ideal model, due to the high column efficiency. The isotherm model accounting best for the desorption isotherm data and the desorption profiles is the bi-Langmuir model. Its coefficients were calculated using appropriate weights in the fitting procedure. The evolution of the bi-Langmuir isotherm parameters with the initial equilibrium plateau concentration of phenol is discussed. The FACP results reported here are fully consistent with the adsorption data of phenol previously reported and measured by FA with various aqueous solutions of methanol as the mobile phase. They provide a general, empirical adsorption model of phenol that is valid between 0 and 65% of methanol in water.     

Comparison of Adsorption Isotherms of Methylxanthines on C18 Column

A comparison of the adsorption isotherms of caffeine, theophylline and theobromine and the competitive adsorption of the three compounds on a C 18 column were investigated. The experimental parameters of the equilibrium isotherms were estimated by linear and nonlinear regression analyses. The linear equation as a function of the adsorption concentration of the single compound in its solution and the competitive adsorption of a single compound in a mixed solution were then determined. The adsorption equilibrium data were then correlated to the linear, Langmuir, Freundlich, Langmuir-Freundlich and stoichiometric displacement theory for adsorption(SDT-A) isotherm models. The mixed compounds of the three compounds were competitively adsorbed on the C 18 particles. The expression of stoichiometric displacement theory for adsorption was found to be more suitable for adsorption of methylxanthines on a C 18 column.     

Determination of the single component and competitive adsorption isotherms of the 1-indanol enantiomers by the inverse method

The inverse method of isotherm determination consists in calculating the numerical values of the coefficients of an isotherm model that give a set of chromatographic profiles in best possible agreement with the set of experimental profiles available. This method was applied to determine the adsorption isotherms of the 1-indanol enantiomers on a cellulose tribenzoate chiral stationary phase. Both single-component and competitive isotherms were determined by using no more than one or two overloaded band profiles. The isotherms determined from the overloaded band profiles agreed extremely well with the isotherms determined by frontal analysis. Several isotherm models were used and tested. The best-fit isotherm was selected by means of statistical evaluation of the results. The results show that the adsorption is best characterized with a model describing heterogeneous adsorption with bimodal adsorption energy distribution.     

New theoretical expressions for the five adsorption type isotherms classified by BET based on statistical physics treatment

New theoretical expressions to model the five adsorption isotherm types have been established. Using the grand canonical ensemble in statistical physics, we give an analytical expression to each of five physical adsorption isotherm types classified by Brunauer, Emett, and Teller, often called BET isotherms. The establishment of these expressions is based on statistical physics and theoretical considerations. This method allowed estimation of all the mathematical parameters in the models. The physicochemical parameters intervening in the adsorption process that the models present could be deduced directly from the experimental adsorption isotherms by numerical simulation. We determine the adequate model for each type of isotherm, which fixes by direct numerical simulation the monolayer, multilayer, or condensation character. New equations are discussed and results obtained are verified for experimental data from the literature. The new theoretical expressions that we have proposed, based on statistical physics treatment, are rather powerful to better understand and interpret the various five physical adsorption type isotherms at a microscopic level.     

一种测量超临界条件下苯酚吸附等温线方法

超临界吸附相平衡是超临界吸附／色谱分离过程设计的基础,通常,研究超临界吸附相平衡的实验不仅需要在高压下操作,而且需使用耐高压的检测器,本文提出一种测量超临界条件下吸附相平衡关系的“扩容减压吸收法”方法,它不需要耐高压检测器,以“苯酚－活性炭－超临界二氧化碳流体”为体系,测定了苯酚在活性炭－超临界二氧化碳流体之间的吸附相平衡关系,测定了苯酚在两种活性炭上的超临界吸附等温线,比较了苯酚在超临界条件和常     

Separation of the First Adsorbed Layer from Others and Calculation of the BET Compatible Surface Area from Type II Isotherms

In the previous paper it has been proven that a BET compatible specific surface area, asc(N2, 77), can be calculated from any Type I isotherm measured below the critical temperature. In this paper it is proven that the same calculation can be performed from any Type II isotherms if the isotherm has a pure monolayer domain. In order to distinguish the mono- and multilayer adsorption the relative free energy of the surface as a function of the adsorbed amount, pir(ns), and the functions psi(pr) and psi(ns) are applied, both defined by the differential expression (ns/pr)(dpr/dns). When the multilayer adsorption becomes the dominant process then the function pir(ns) has a point of inflexion and functions psi(pr) and psi(ns) have maximum values. It has been demonstrated that in most of the Type II isotherms the mono- and multilayer domains can be separated, so the monolayer component isotherm can be calculated by the T (Tóth) equation. Therefore, it is possible to calculate the BET compatible specific surface area discussed in detail in the previous paper. It has also been proven that there are Type II isotherms which describe only multilayer adsorption; i.e., the functions psi(pr) and psi(ns) do not have maximum values. In these cases the Harkins-Jura equation should be applied. Copyright 1999 Academic Press.     

Theoretical study of the accuracy of the pulse method,frontal analysis,and frontal analysis by characteristic points for the determination of single component adsorption isotherms

The adsorption isotherms of selected compounds are our main source of information on the mechanisms of adsorption processes. Thus, the selection of the methods used to determine adsorption isotherm data and to evaluate the errors made is critical. Three chromatographic methods were evaluated, frontal analysis (FA), frontal analysis by characteristic point (FACP), and the pulse or perturbation method (PM), and their accuracies were compared. Using the equilibrium-dispersive (ED) model of chromatography, breakthrough curves of single components were generated corresponding to three different adsorption isotherm models: the Langmuir, the bi-Langmuir, and the Moreau isotherms. For each breakthrough curve, the best conventional procedures of each method (FA, FACP, PM) were used to calculate the corresponding data point, using typical values of the parameters of each isotherm model, for four different values of the column efficiency (N=500$N=500$, 1000, 2000, and 10,000). Then, the data points were fitted to each isotherm model and the corresponding isotherm parameters were compared to those of the initial isotherm model. When isotherm data are derived with a chromatographic method, they may suffer from two types of errors: (1) the errors made in deriving the experimental data points from the chromatographic records; (2) the errors made in selecting an incorrect isotherm model and fitting to it the experimental data. Both errors decrease significantly with increasing column efficiency with FA and FACP, but not with PM.     

Modeling of adsorption in hydrophobic interaction chromatography systems using a preferential interaction quadratic isotherm

A preferential interaction quadratic isotherm model for hydrophobic interaction chromatographic systems is presented in this paper. In this isotherm, the nonlinear effect of salt on the capacity factor is described using the preferential interaction model developed by Perkins et al. [J. Chromatogr. A, 766 (1997) 1]. This is then coupled with a quadratic nonlinear isotherm to describe nonlinear adsorption behavior at high solute concentrations. The resulting preferential interaction quadratic isotherm is examined for its ability to describe solute adsorption behavior under both linear and nonlinear conditions over a wide range of salt concentrations in HIC systems. The results indicate that this isotherm is well suited for predicting nonlinear adsorption behavior in HIC systems for both proteins and low-molecular mass HIC displacers.     

Solving the adsorption integral equation with Langmuir-kernel and the influence of temperature on the stability of the solution

Adsorption energy distribution functions can be calculated from measured adsorption isotherms by solving the adsorption integral equation. In this context, it is common practice to use general regularization methods, which are independent of the kernel of the adsorption integral equation, but do not permit error estimation. In order to overcome this disadvantage, we present in this paper a solution theory which is tailor-made for the Langmuir kernel of the adsorption integral equation. The presented theory by means of differentiation and Fourier series is the basis for a regularization method with explicit terms for error amplification. By means of simple and complicated adsorption energy distribution functions we show for ideal gas adsorption isotherms without measurement error that reliable distribution functions can be obtained from the isotherms. Furthermore we show how the stability of the solution depends on temperature.     

Use of the BET adsorption isotherm equation to examine styrene adsolubilization by nonionic surfactants at the water-silica interface

A modified BET adsorption isotherm equation was used to explain adsolubilization at the water-silica interface, producing a method for understanding adsolubilization that is independent of admicelle structure. The two constants k1 and k2 provide a measure of the surfactant-solute and solute-solute interactions, respectively. As predicted, k1 changed with various factors including surfactant structure, the presence and quantity of added lipophilic linker and hexane. On the other hand, k2 was fairly constant for all examined systems having the same solute. Commonly used partition coefficients can be easily related to the model and adsolubilization isotherms may be reproduced by the developed equation. Inferences about solute distribution at the interface and its location within the admicelle may be extracted from the results. The model is useful in explaining trends in adsolubilization and in predicting adsolubilization behavior for similar systems.     

Role of association on protein adsorption isotherms. Beta-lactoglobulin A adsorbed on a weakly hydrophobic surface

This paper explores the role of association on the adsorption isotherms of beta-lactoglobulin A on a weakly hydrophobic stationary phase at 4 degrees C and mobile phases of 0.85 M and 1 M ammonium sulfate, pH 4.5. The isotherms, obtained by frontal analysis, show an S-shape and the corresponding Scatchard plots indicate positive cooperativity. The slopes and intercepts of the Scatchard plots at low solute concentration are analyzed in terms of two species--a promoter and a higher order stronger adsorbing species. An explicit equation of the isotherm is developed based on this model, and this expression is shown to reproduce the isotherm shape using the appropriate derived parameters. It is further shown from this equation that a Langmuir-shaped adsorption isotherm can be obtained if the higher order associate or aggregate binds weaker to the support than the promoter. These results indicate that protein-protein interactions and the formation of associates can play a significant role on the shape of the isotherm and ultimately on the behavior of the species in preparative scale chromatography.     

Adsorption of selected herbicides from aqueous solutions on activated carbon

The adsorption of MCPA and 2,4-D on the activated carbon Filtrasorb 300 was studied. The adsorption isotherms of herbicides from aqueous solutions were measured over a wide range of solute concentrations and at different temperatures. The experimental equilibrium data were analyzed by the Langmuir–Freundlich isotherm taking into account the energetic heterogeneity of adsorption system. The effect of temperature and herbicide properties on its uptake was discussed. The thermal analysis was applied in order to find the differences in herbicide interactions with carbon surface. The kinetic dependences were measured and the relations between solute properties and adsorption rate were discussed.