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.     

Estimation of single solute adsorption isotherms applying the nonlinear frequency response method using non-optimal frequencies

In order to estimate single solute adsorption isotherms, the nonlinear frequency response (FR) of a chromatographic column is analyzed experimentally and evaluated using the concept of higher order frequency response functions (FRFs) based on the Volterra series and generalized Fourier transform. In this case study, it has been investigated the adsorption of ethyl benzoate on octadecyl silica from a mixture of methanol and water (60:40) as a solvent. Experiments are performed using a standard gradient HPLC unit. For estimation of adsorption isotherms by the nonlinear FR method the column inlet concentration is changed in a nearly sine waveform around several steady-state concentrations. Using this method the first three local derivatives of a single solute adsorption isotherm are estimated from the low frequency asymptotes of the corresponding functions, i.e. the phase and first order derivative of the FRFs. For an accurate estimation of isotherm coefficients periodical experiments should be preformed for frequencies below a certain critical frequency. This is the frequency needed for approaching the low frequency asymptotic behaviour of the corresponding functions close enough, so that errors due to the non-feasibility of experiments with zero frequency can be neglected. Unfortunately, depending on the properties of the system, it can happen (as for the system investigated here) that experiments for the critical frequency would be too long and cannot be realized. In order to study the loss of accuracy of the nonlinear FR method, when it is applied for non-optimal frequencies, experiments are performed for frequencies approximately one order of magnitude higher than the critical frequency required to evaluate the FRF phases. The obtained isotherm model coefficients are compared with the ones estimated using conventional frontal analysis as a reference method. The isotherms determined by two methods are similar, however a closer look reveals that peaks predicted under overloading conditions differ.     

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.     

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.     

乙烯在丝光沸石和改性丝光沸石孔道内的吸附行为

采用频率响应(FR)法研究乙烯在丝光沸石和经CuO和Cs+离子改性的丝光沸石上的吸附机理. 分别测得和解析了252和273 K、压力在26.6-3990 Pa范围内的FR谱图, 发现乙烯在丝光沸石上吸附的速控步骤是传质过程, 同时存在两个不同的吸附过程. 这两个过程分别归属于乙烯在质子酸吸附中心上的吸附(低频吸附)和Na+吸附位上的吸附(高频吸附), 252 K时两个吸附位的吸附值分别是0.692和0.828 mmol·g-1. CuO分子进入分子筛孔道后, 位于质子酸吸附位之间不但使得低频吸附位值增加而且还覆盖了Na+吸附位, 体系中以化学吸附过程为主; Cs+离子的引入使得高频吸附位值增加但中和了质子酸吸附位, 体系中以物理吸附过程为主. CuO的最佳用量是5%. 将FR法与吸附等温线及Langmuir模型相结合能够深入研究乙烯在分子筛上的平行吸附过程.     

Determination of competitive isotherms of enantiomers by a hybrid inverse method using overloaded band profiles and the periodic state of the simulated moving-bed process

A procedure for determination of adsorption isotherms in simulated moving-bed (SMB) chromatography is presented. The parameters of a prescribed adsorption isotherm model and rate constants are derived using a hybrid inverse method, which incorporates overloaded band profiles of the racemic mixture and breakthrough data from a single frontal experiment. The latter are included to reduce the uncertainty on the estimated saturation capacity, due to the dilution of the chromatograms with respect to the injected concentrations. The adsorption isotherm model is coupled with an axially dispersed flow model with finite mass-transfer rate to describe the experimental band profiles. The numerical constants of the isotherm model are tuned so that the calculated and measured band profiles match as much as possible. The accuracy of the isotherm model is then checked against the cyclic steady state (CSS) of the target SMB process, which is readily and cheaply obtained experimentally on a single-column set-up. This experiment is as expensive and time consuming as just a few breakthrough experiments. If necessary, the isotherm parameters are adjusted by applying the inverse method to the experimental CSS concentration profile. The method is successfully applied to determine the adsorption isotherms of Tr?gers base enantiomers on Chiralpak AD/methanol system. The results indicate that the proposed inverse method offers a reliable and quick approach to determine the competitive adsorption isotherms for a specific SMB separation.     

Adsorption Isotherms of Quercetin and Catechin Compounds on Quercetin-MIP

A molecular imprinted polymer(MIP)was prepared with quercetin as the template and methacrylic acid(MAA)as the functional monomer.Acetonitrile and methanol were used as the porogen with ethylene glycol dimethacrylate(EGDMA)as the crosslinker and 2,2'-azobis(isobutyronitrile)(AIBN)as the initiator.The experimental parameters of the equilibrium isotherms were estimated via linear and nonlinear regression analyses.The linear equation as the functions of the adsorption concentration of the single compound in its solution and the competitive adsorption of the single compound in its mixed compounds solution was then expressed,and the adsorption equilibrium data were correlated to Langmuir and Freundlich isotherm models.The mixture compounds show competitive adsorption on the specific binding sites of quercetin-MIP.Furthermore,the competitive Langmuir isotherms were applied to the mixture compounds.The adsorption concentrations of quercetin,( )catechin( C),and(-)epicatechin(EC)on the quercetin molecular imprinted polymer were compared.The quercetin-imprinted polymer shows extraordinarily higher adsorption ability for quercetin than for the two catechin compounds that were also assessed.     

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.     

Comparison of Dinitrogen,Methane, Carbon Monoxide,and Carbon Dioxide Mass‐Transport Dynamics in Carbon and Zeolite Molecular Sieves

Equilibrium (based on Henry constants) and kinetic (based on relaxation‐time constants or rather macropore transport diffusivities) selectivities for commercial zeolite and carbon‐molecular‐sieve (CMS) adsorbents were compared. Adsorption isotherms were recorded at ?20°. The frequency‐response (FR) sorption‐rate spectra were determined in the range of ?78 and 70° at 133 Pa. In particles of a larger size than 1.0 mm, macropore diffusion governed the rate of sorption mass transport in both types of microporous materials. The differences in the intercrystalline diffusivities established the kinetic separation of the gases notwithstanding the essential importance of interactions in the micropores. Zeolites seem to be more advantageous for a dynamic separation of CO₂ and CH₄ than CMS 4A. With the CO₂ and CO pair, the CMS is characterized by short characteristic times which, together with a good separation factor, is a double advantage in a short‐cycle adsorption technology. Upon comminution of the carbon pellets, intercrystalline‐diffusion resistance can be completely removed by using CMS 4A adsorbent particles with a diameter smaller than 1 mm. The carbonization of spruce‐wood cubes resulted in an excellent carbon honeycomb structure, which seems to be ideal from a dynamic point of view for applications in short‐cycle adsorption‐separation technologies. In the development of adsorbents, the use of the FR method can be beneficial.     

A systematic investigation of algorithm impact in preparative chromatography with experimental verifications

Computer-assisted optimization of chromatographic separations requires finding the numerical solution of the Equilibrium-Dispersive (ED) mass balance equation. Furthermore, the competitive adsorption isotherms needed for optimization are often estimated numerically using the inverse method that also solves the ED equations. This means that the accuracy of the estimated adsorption isotherm parameters explicitly depends on the numerical accuracy of the algorithm that is used to solve the ED equations. The fast and commonly used algorithm for this purpose, the Rouchon Finite Difference (RFD) algorithm, has often been reported not to be able to accurately solve the ED equations for all practical preparative experimental conditions, but its limitations has never been completely and systematically investigated. In this study, we thoroughly investigate three different algorithms used to solve the ED equations: the RFD algorithm, the Orthogonal Collocation on Finite Elements (OCFE) method and a Central Difference Method (CDM) algorithm, both for increased theoretical understanding and for real cases of industrial interest. We identified discrepancies between the conventional RFD algorithm and the more accurate OCFE and CDM algorithms for several conditions, such as low efficiency, increasing number of simulated components and components present at different concentrations. Given high enough efficiency, we experimentally demonstrate good prediction of experimental data of a quaternary separation problem using either algorithm, but better prediction using OCFE/CDM for a binary low efficiency separation problem or separations when the compounds have different efficiency. Our conclusion is to use the RFD algorithm with caution when such conditions are present and that the rule of thumb that the number of theoretical plates should be greater than 1000 for application of the RFD algorithm is underestimated in many cases.     

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.     

Adsorption isotherms and retention behavior of 1,1'-bis(2-naphthol) on CHIRIS AD1 and CHIRIS AD2 columns

The separation of the atropoisomers of 1,1'-bis(2-naphthol) was studied on CHIRIS AD1 and CHIRIS AD2, two Pirkle-type chiral stationary phases. Satisfactory selectivity was found only on CHIRIS AD2. The ternary mobile phases comprised hexane, dichloromethane and methanol. The effects of their composition and of the temperature on the retention under analytical conditions and on the single-component and competitive isotherms were investigated. The retention of the R- and S-isomers on CHIRIS AD1 and CHIRIS AD2 is controlled by the enthalpic contribution to adsorption, but the effect of the mobile phase on the retention should be attributed mainly to the entropic contribution. The adsorption of the less retained R-isomer is controlled by the achiral interactions, which are the same as for the S-isomer. The single-component and competitive isotherms of the R- and S-isomers are adequately described by the sum of a Langmuir term for the achiral contribution to adsorption and a linear-term characterising the selective or chiral adsorption of the S-isomer in the concentration range experimentally available, i.e. within the solubility limit of 1,1'-bis(2-naphthol).     

Design of optimum response surface experiments for the adsorption of acetic,butyric, and oxalic acids on Amberlyst A21

In this study the adsorption equilibria of acetic, butyric, and oxalic acids onto Amberlyst A21 were investigated experimentally at 25°C. The process was optimized using response surface methodology (RSM). The time to reach the equilibrium state, effects of adsorbent amount, and initial acid concentrations on adsorption efficiency were investigated. Freundlich, Langmuir, and Temkin isotherms were applied to experimental data. The Freundlich isotherm revealed better results than the others. In addition to the main aim of this research, a statistical/mathematical approach – RSM – was utilized to simulate and determine the optimum conditions of acetic, butyric, and oxalic acids removal by Amberlyst A21 using three selected parameters (adsorbent dose, initial dye concentration, and type of acids). The significance of independent variables and their interactions were tested by the analysis of variance (ANOVA). The optimum acid concentration, amount of adsorbent, type of acid, and removal of acid (%) were found by desirability function to be 0.199?mol/L, 1.999?g, butyric acid, and 84.537%, respectively.     

Characterization and determination of the thermodynamic and kinetic properties of p-CP adsorption onto organophilic bentonite from aqueous solution

The characterization of tetraethylammonium bentonite and the adsorption of p-chlorophenol (p-CP) onto organophilic bentonite (tetraethylammonium bentonite) was studied as a function of the solution concentration and temperature. The observed adsorption rates were found to fit first-order kinetics. The rate constants were calculated for temperatures ranging between 15.0 and 35.0 degrees C at constant concentration. The adsorption energy E and adsorption capacity q(m) for the phenolic compound adsorbing on organophilic bentonite were estimated using the Dubinin-Radushkevic equation. Thermodynamic parameters (Deltag(a), Deltah(a), Deltas(a)) were calculated by a new approximation from the isotherms of p-CP adsorption on organophilic bentonite. These isotherms were modeled according to Freundlich and Dubinin-Radushkevic adsorption isotherms. The amount of adsorption of p-chlorophenol on organophilic bentonite was found to be dependent on the relative energies of adsorbent-adsorbate, adsorbate-solvent, and adsorbate-adsorbate interactions.     

Dynamics of sorption of cyclopropane in various H- and Na-zeolites

The frequency response (FR) method has been applied to study the dynamics of cyclopropane adsorption in faujasite, mordenite, beta and ZSM-5 zeolites containing Brönsted or Lewis acid sites in the concentration range of 0.586-0.772 meq/g.     

Competitive ion-exchange adsorption of proteins: competitive isotherms with controlled competitor concentration

The competitive adsorption processes inevitably present in chromatographic separations of complex mixtures have not been extensively studied. This is partly due to the difficulty of measuring true competitive isotherms, in which all system parameters (including competitor concentrations) are held constant. We report a novel approach to determining competitive protein adsorption isotherms in which the competitor concentration is held constant across the entire isotherm. By using the heme prosthetic group in cytochrome b5 as a quantitative spectrophotometric label, competitive isotherms between cytochrome b5 and alpha-lactalbumin can be constructed. Similarly, manganese-substituted protoporphyrin IX heme replacement allows the non-perturbing labeling of individual cytochrome b5 conservative surface charge mutants by replacement of a single atom in the interior of the protein. This labeling allows the study of competition between cytochrome b5 charge mutants of identical size and shape, which differ only in charge arrangement. Using these techniques, the effect of competing species on equilibrium behavior and the apparent heterogeneity of anion-exchange adsorbents in the presence of competitors can be quantitatively studied by fitting the data to two popular single-component binding models, the Temkin and the Langmuir-Freundlich (L-F) isotherms.     

Synergistic effects in competitive adsorption of carbohydrates on an ion-exchange resin

Adsorption of the three carbohydrates sucrose, glucose and fructose from aqueous solutions was investigated on an ion-exchange resin. The adsorption equilibrium of single components, binary and ternary mixtures was quantified by frontal analysis and the adsorption-desorption method. The experiments covered a concentration range up to 600 g/L at 60 degrees C and 80 degrees C. Within this range the adsorption isotherms of carbohydrates exhibited anti-Langmuirian behavior. Data of mixture adsorption revealed reversed competitive (synergistic or cooperative) effects, i.e., an increase of the concentration of one component of the mixture enhanced the adsorption of others. To model such an adsorption behavior the anti-Langmuir model has been used. The isotherm parameters determined for single components were used to simulate the competitive adsorption equilibria through the IAS (ideal adsorbed solution) theory. Finally, dynamic concentration profiles of multicomponent mixtures have been recorded. The shapes of adsorption and desorption curves confirmed the observed competitive effects found in the equilibrium studies. The breakthrough curves measured were simulated using the equilibrium theory as well as a numerical solution of the equilibrium dispersive model.     

Correlation of sorption behavior of nitrogen, oxygen, and argon with Ca2+ locations in zeolite a: a grand canonical Monte Carlo simulation study

The adsorption isotherms for nitrogen, oxygen, and argon in various NaCaA zeolite samples were calculated theoretically using the grand canonical Monte Carlo simulation method. The adsorption capacity, selectivity, and heat of adsorption of nitrogen increase with an increasing number of calcium cations in zeolite A. The heat of adsorption of nitrogen showed a sudden increase when the calcium ion exchange to zeolite A was around 60%. Adsorption isotherms, determined experimentally by the volumetric adsorption method, support theoretically predicted isotherms. These observations have been explained in terms of the interaction of the nitrogen molecule with Ca2+ ions and their locations in zeolite A.     

On the applicability of the theory of ideal adsorption solutions to equilibria in the N2-O2-zeolite NaX system

Experimental data on the joint adsorption of a nitrogen-oxygen gas mixture on zeolite NaX were analyzed using the theory of ideal adsorption solutions. Acceptable accuracy was obtained in calculations of binary isotherms from the adsorption isotherms of the pure gas mixture components. The calculated binary isotherms can be used to determine the parameters of the mathematical models of binary adsorption, for instance, competitive adsorption models.