Numerical determination of the competitive isotherm of enantiomers

A numerical method was developed and used to determine adsorption isotherms in chromatography. The numerical parameters of an isotherm model were derived from the recorded band profiles of the racemic mixture of the 1-phenyl-1-propanol enantiomers, by means of a nonlinear least-squares method. We used the equilibrium-dispersive model of chromatography with several isotherm models. The numerical constants of the isotherm models were tuned so that the calculated and the measured band profiles match as much as possible. We show that this numerical inverse method can be applied even without the knowledge of the individual band profile of the pure enantiomers. The isotherms determined from the--usually unresolved--overloaded band profiles matched extremely well the isotherms determined by frontal analysis. Several isotherm models were used and tested--such as Langmuir, biLangmuir, Tóth, Langmuir-Freundlich. The best-fit isotherm was selected by means of statistical evaluation of the results.     

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.     

Effect of temperature on the adsorption behavior of tryptophan in reversed-phase liquid chromatography

Single-component adsorption isotherm data of l-tryptophan on a C(18)-bonded silica column were acquired by frontal analysis (FA), with aqueous mobile phases containing 5% of acetonitrile at five different temperatures between 23 and 62 degrees C. The non-linear fitting of these data provided the bi-Moreau model for all temperatures as the best isotherm model. The inverse method (IM) was used to derive the parameters at these temperatures from the parameters of the 25 degrees C isotherm. The adsorption constants and the saturation capacities of the low and high-energy sites decreases by increasing the temperature, while the adsorbate-adsorbate parameters of both sites increase. An excellent agreement was found between the experimental and calculated overloaded band profiles at all the temperatures used. The breakthrough curves obtained and the overloaded band profiles obtained were found to have different shapes according to the range of concentration studied and the temperatures. At low concentration 0.05-0.5 g/L the breakthrough curves and the overloaded band profiles have a front shock and diffuse rear, which indicates langmuirian behavior, but at intermediate 1-2 g/L and high concentration 8 g/L they start to have diffuse fronts and shocks at the rear or more than one shock at the rear which indicates non-langmuirian behavior. At 23 degrees C the isotherm has another langmuirian part, which appears at high concentration. The behavior of the breakthrough curves is explained by the shape of the isotherm in which all of the isotherms have a langmuirian part (the isotherm is concave upward) and an antilangmuirian part (the isotherm is concave downward). The temperature affected the breakthrough curves by decreasing the time of the appearance of the fronts for all concentration ranges studied, and by decreasing the time difference between the highest concentration and lowest concentration of the fronts, especially the low concentration range at 0.5 g/L. The fronts of the breakthrough curves at high concentration seems to be the most affected by temperature.     

Separation mechanism of nortriptyline and amytriptyline in RPLC

The single and the competitive equilibrium isotherms of nortriptyline and amytriptyline were acquired by frontal analysis (FA) on the C18- bonded discovery column, using a 28/72 (v/v) mixture of acetonitrile and water buffered with phosphate (20 mM, pH 2.70). The adsorption energy distributions (AED) of each compound were calculated from the raw adsorption data. Both the fitting of the adsorption data using multi-linear regression analysis and the AEDs are consistent with a trimodal isotherm model. The single-component isotherm data fit well to the tri-Langmuir isotherm model. The extension to a competitive two-component tri-Langmuir isotherm model based on the best parameters of the single-component isotherms does not account well for the breakthrough curves nor for the overloaded band profiles measured for mixtures of nortriptyline and amytriptyline. However, it was possible to derive adjusted parameters of a competitive tri-Langmuir model based on the fitting of the adsorption data obtained for these mixtures. A very good agreement was then found between the calculated and the experimental overloaded band profiles of all the mixtures injected.     

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.     

Effect of the mobile phase composition on the adsorption behavior of tryptophan in reversed-phase liquid chromatography

Single-component adsorption isotherm data of l-tryptophan on a C18-bonded silica column were acquired by frontal analysis (FA), with aqueous mobile phases containing 2.5, 5, and 7.5% of acetonitrile (ACN) or 7, 10, 15, and 20% of methanol (MeOH). Most of these isotherms have two inflection points and three different parts. The low and the high concentration parts exhibit langmuirian behavior. The intermediate part exhibits anti-langmuirian behavior. The inflection points shift toward higher concentrations with increasing mobile phase concentration in ACN or MeOH, which causes the differences in the isotherm profiles. The nature of the organic modifier and its concentration affect only the isotherm profile and the numerical values of its parameters, not the nature of the best model, which is the bi-Moreau model in all cases. The isotherm profiles depend on the experimental conditions because they affect the intensity of the adsorbate-adsorbate interactions. Overloaded band profiles of tryptophan were recorded with the seven mobile phase compositions. They were used to determine the best values of the isotherm coefficients by the inverse method (IM) of chromatography. There is an excellent agreement between the values of these parameters obtained by FA and by IM. Increasing the concentration of either ACN or MeOH in the mobile phase causes a slight decrease in the saturation capacities of the low and the high energy sites, and in the adsorption constant of the low energy sites. The adsorption constant of the high energy sites increases with increasing concentration of either solvent or is little affected. The adsorbate-adsorbate interaction constants of both low and high energy sites increase for both solvents. Saturation capacities of the high energy sites are higher for ACN than for MeOH.     

Accurate and rapid estimation of adsorption isotherms in liquid chromatography using the inverse method on plateaus

The inverse method (IM) is an attractive approach for estimating adsorption isotherm parameters in liquid chromatography (LC), mainly due to its experimental simplicity and low sample consumption. This article presents a new experimental approach, the inverse method on plateaus (IMP), which uses elution profiles on concentration plateaus together with IM. This approach enabled us to obtain very accurate adsorption isotherms that agreed well with those estimated by means of frontal analysis over the entire concentration range under consideration. IMP is recommended when accurate adsorption isotherm estimates are required, and standard IM is insufficient.     

Determination of competitive adsorption isotherm parameters of pindolol enantiomers on alpha1-acid glycoprotein chiral stationary phase

In this paper, inverse method (IM) was used to determine the binary competitive adsorption isotherm of pindolol enantiomers by a least-square fitting of the proposed model to the experimentally measured elution curves of racemic pindolol. The isotherm parameters were determined by minimizing the least-square error using an adaptation of genetic algorithm, non-dominated sorting genetic algorithm with jumping genes (NSGA-II-JG). An equilibrium dispersive (ED) model combined with bi-Langmuir isotherm was used in predicting the elution profiles. The determined parameters show good agreement with the experimental profiles at various experimental conditions such as sample volume, concentration and flow rates of the racemic mixture. Robustness and validity of the isotherm parameters were also verified by frontal analyses at various step inputs. Results from both the pulse tests and the frontal analysis indicate that adsorption isotherm derived from the inverse method is quite reliable. This method requires relatively less number of experiments to be performed and therefore, lower experimental costs confirming that inverse method is an attractive alternative approach of experimental technique in determining the competitive adsorption isotherm for binary systems.     

Numerical determination of competitive adsorption isotherm of mandelic acid enantiomers on cellulose-based chiral stationary phase

The use of inverse method for the determination of competitive adsorption isotherm of mandelic acid enantiomers on cellulose tris(3,5-diethylphenyl carbamate) stationary phase is proposed in this work. Non-dominated sorting genetic algorithm with jumping genes (NSGA-II-JG) was applied to acquire the isotherm parameters by minimizing the sum of square deviations of the model predictions from the measured elution profiles. Three different competitive isotherm models, i.e., Langmuir, biLangmuir and Tóth, combined with transport-dispersive chromatographic model were used in predicting the elution profiles. Orthogonal collocation on finite element (OCFE) method was applied to obtain the calculated elution profiles. Results indicate that biLangmuir isotherm and Tóth isotherm give remarkably similar equilibrium isotherms within the investigated liquid concentration range. Band profiles calculated from both isotherm models are in good agreement with the experimental data. The validity of the determined parameters was verified by comparing the model predictions with experimental elution profiles at various experimental conditions.     

Breakthrough curves and elution profiles of single solutes in case of adsorption isotherms with two inflection points

The shape of breakthrough curves and elution profiles depends strongly on the course of the specific equilibrium functions characterizing the chromatographic system. For a highly efficient system the equilibrium theory provides a methodology how to predict the band profiles. The concept is frequently applied to analyze single component systems characterized by isotherms possessing simple shapes (Langmuir or anti-Langmuir behaviour). However, adsorption isotherms often possess more complicated shapes and have inflection points in their courses. This leads to the development of composite concentration waves and results in complex shapes of breakthrough curves and elution profiles. In this paper, the equilibrium theory is used to predict breakthrough curves for a chromatographic system characterized by an adsorption isotherm with two inflection points. The results obtained are validated by comparing with numerical solutions of the equilibrium dispersive model.     

Competitive effect of glucose–fructose adsorption in a fixed‐bed chromatographic column

A continuous separation system such as a simulated moving‐bed process requires adsorption data with precise equilibrium and kinetic model parameters of a single chromatographic column. The adsorption of glucose and fructose in a fixed‐bed chromatographic column was investigated to determine the competition effect of each component resulting from their initial molar ratios. The model parameters including bed porosity and axial dispersion coefficient were determined using the moment analysis method. The equilibrium isotherm parameters were estimated by conducting experiments at various molar ratios and initial sugar concentrations. The parameters obtained were then used for the simulation of dynamic breakthrough curves of glucose and fructose. The equilibrium isotherms revealed that the linear adsorption pattern provided good prediction for each molar ratio using the Henry equation. In addition, the modified Langmuir model was proposed to account for the competitive adsorption, due to the cooperative competition effect whereby glucose was promoted to the active sites by fructose to a greater degree than vice versa. A good agreement between the experimental and numerical data of the adsorption time profiles was also observed.     

Retention of ionizable compounds in high-performance liquid chromatography. 14. Acid-base pK values in acetonitrile-water mobile phases

Using competitive frontal analysis, the binary adsorption isotherms of the enantiomers of 1-phenyl-l-propanol were measured on a microbore column packed with a chiral stationary phase based on cellulose tribenzoate. These measurements were carried out using only the racemic mixture. The experimental data were fitted to four different isotherm models: Langmuir, BiLangmuir, Langmuir-Freundlich and Tóth. The BiLangmuir and the Langmuir-Freundlich models accounted best for the competitive adsorption data. An excellent agreement between the experimental and the calculated overloaded band profiles for various samples of racemic mixture was obtained when the equilibrium dispersive model of chromatography was used together with the BiLangmuir competitive isotherm. The isotherm parameters measured under competitive conditions were used to calculate the overloaded band profiles of large samples of the pure S- and R-enantiomers, too. A satisfactory agreement between the experimental and calculated band profiles was observed when using in the computation the corresponding single component BiLangmuir isotherm derived from the binary isotherm previously determined. Thus oniy data derived from the racemic mixture are required for computer optimization of the preparative chromatography separation of the enantiomers.     

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.     

Systematic errors in the measurement of adsorption isotherms by frontal analysis Impact of the choice of column hold-up volume, range and density of the data points

Besides the accuracy and the precision of the measurements of the data points, several important parameters affect the accuracy of the adsorption isotherms that are derived from the data acquired by frontal analysis (FA). The influence of these parameters is discussed. First, the effects of the width of the concentration range within which the adsorption data are measured and of the distribution of the data points in this range are investigated. Systematic elimination of parts of the data points before the calculation of the nonlinear regression of the data to the model illustrates the importance of the numbers of data points (1) within the linear range and (2) at high concentrations. The influence of the inaccuracy of the estimate of the column hold-up volume on each adsorption data point, on the selection of the isotherm model, and on the best estimates of the adsorption isotherm parameters is also stressed. Depending on the method used to measure it, the hold-up time can vary by more than 10%. The high concentration part of the adsorption isotherm is particularly sensitive to errors made on t(0,exp) and as a result, when the isotherm follows bi-Langmuir isotherm behavior, the equilibrium constant of the low-energy sites may change by a factor 2. This study shows that the agreement between calculated and experimental overloaded band profiles is a necessary condition to validate the choice of an adsorption model and the calculation of its numerical parameters but that this condition is not sufficient.     

Adsorption Henry constants calculated from the entire isotherm

A method is introduced to derive adsorption Henry constants from experimental isotherm data regardless of whether the measurements extend into the Henry law region. The method uses the intersection between Henry’s law and the Gaussian isotherm model. In this way, the Henry constant can be calculated from Gaussian model parameters obtained from a fit of the model to the entire isotherm. In addition, the Gaussian model is shown to closely agree with Henry’s law over a wide range. To demonstrate the accuracy of the method, Henry constants are derived from 63 experimental isotherms collected from the literature for supercritical n alkanes on 5A zeolite. These Henry constants compare well to values found in the literature.     

Choice of Model for Estimation of Adsorption Isotherm Parameters in Gradient Elution Preparative Liquid Chromatography

The inverse method is a numerical method for fast estimation of adsorption isotherm parameters directly from a few overloaded elution profiles and it was recently extended to adsorption isotherm acquisition in gradient elution conditions. However, the inverse method in gradient elution is cumbersome due to the complex adsorption isotherm models found in gradient elution. In this case, physicochemically correct adsorption models have very long calculation times. The aim of this study is to investigate the possibility of using a less complex adsorption isotherm model, with fewer adjustable parameters, but with preserved/acceptable predictive abilities. We found that equal or better agreement between experimental and predicted elution profiles could be achieved with less complex models. By being able to select a model with fewer adjustable parameters, the calculation times can be reduced by at least a factor of 10.

     

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.     

New thermodynamically consistent competitive adsorption isotherm in RPLC

A new equation of competitive isotherms was derived in the framework of the ideal adsorbed solution (IAS) that predicts multisolute adsorption isotherms from single-solute isotherms. The IAS theory makes this new isotherm thermodynamically consistent, whatever the saturation capacities of these single-component isotherms. On a Kromasil-C(18) column, with methanol-water (80/20 v/v) as the mobile phase, the best single-solute adsorption isotherm of both toluene and ethylbenzene is the liquid-solid extended multilayer BET isotherm. Despite a significant difference between the monolayer capacities of toluene (370 g/l) and ethylbenzene (170 g/l), the experimental adsorption data fit very well to single-component isotherms exhibiting the same capacities (200 g/l). The new competitive model was used for the modeling of the elution band profiles of mixtures of the two compounds. Excellent agreement between experimental and calculated profiles was observed, suggesting that the behavior of the toluene-ethylbenzene adsorbed phase on the stationary phase is close to ideal. For example, the concentrations measured for the intermediate plateau obtained in frontal analysis differ by less than 2% from those predicted by the IAS model.     

Adsorption behavior and prediction of the band profiles of the enantiomers of 3-chloro-1-phenyl-1-propanol. Influence of the mass transfer kinetics

The single-component and competitive adsorption isotherms of the enantiomers of 3-chloro-1-phenyl-1-propanol were measured by frontal analysis. The stationary phase was a cellulose tribenzoate coated on silica, the mobile phase an n-hexane-ethyl acetate (95:5) solution. The adsorption data measured fitted well to the Langmuir isotherm model. The band profiles of single components and of their mixtures were calculated using the equilibrium-dispersive model. These profiles were found to match quite satisfactorily the experimental band profiles. However, the agreement between calculated and experimental band profiles was significantly improved when a more complex model taking into account the mass transfer kinetics was used. The mass transfer rate coefficients, k(f), for both single components were determined by using the transport-dispersive model of chromatography. The coefficients obtained were used to predict the band profiles of mixtures of the two enantiomers to good agreement.     

Modeling of the adsorption behavior and the chromatographic band profiles of enantiomers : Behavior of methyl mandelate on immobilized cellulose

The adsorption isotherms of (−)- and (+)-methyl mandelate from a hexane-isopropanol (90:10) solution were measured on a chromatographic column packed with 4-methylcellulose tribenzoate coated on silica. These isotherms are accounted for by a bi-Langmuir isotherm model, the two Langmuir terms having widely different initial slopes and saturation capacities, but each term having the same saturation capacity for the two enantiomers. The competitive isotherms were also measured. They are in excellent agreement with the prediction of a competitive bi-Langmuir model based on the single-component isotherms. The individual band profiles are in agreement with the profiles calculated from these isotherms. Thus, a simplified competitive isotherm can be used to model a separation on a chiral stationary phase the recognition mechanism of which is not well identified and the adsorption behavior of which is certainly not ideal.