Numerical solution of the complete mass balance equation in chromatography

A new approach to simulate the movement of bands through a chromatographic column is presented. Similar to the Craig distribution model, the mass balance equation is divided into two equations describing two successive processes. The first equation includes two effects: solute diffusion in the mobile phase and migration of the solute band with the mobile phase. The second equation deals with the distribution of the solute between phases, i.e., the adsorption isotherm. The partial differential equations are integrated numerically over time and space using two methods. The first approach is a finite difference method. In the second approach, the propagation operator is expanded in a Chebyshev series, where large time steps can be used. The rate of adsorption and desorption is determined by the size of the time increment. By varying the size of the time step, it is possible to study kinetic effects. The influences of sample size, injection width, rate of mass transfer, and mobile phase velocity on the elution profile were studied. Simulations using the modified Craig approach with either of the two numerical procedures showed that the solutes behaved in the chromatographically expected manner. Moreover, with linear adsorption isotherms, direct relationships between HETP, as well as retention times, and experimental parameters could be established.     

An adjoint method in inverse problems of chromatography

How to determine adsorption isotherms is an issue of significant importance in chromatography. A modern technique of obtaining adsorption isotherms is to solve an inverse problem so that the simulated batch separation coincides with actual experimental results. In this work, as well as the natural least-square approach, we consider a Kohn–Vogelius type formulation for the reconstruction of adsorption isotherms in chromatography, which converts the original boundary fitting problem into a domain fitting problem. Moreover, using the first momentum regularizing strategy, a new regularization algorithm for both the Equilibrium-Dispersive model and the Transport-Dispersive model is developed. The mass transfer resistance coefficients in the Transport-Dispersive model are also estimated by the proposed inverse method. The computation of the gradients of objective functions for both of the two models is derived by the adjoint method. Finally, numerical simulations for both a synthetic problem and a real-world problem are given to show the robustness of the proposed algorithm.     

Free surface energy changes in graphite/n-hexadecane system

Applying the modified elution gas chromatography method for determination of adsorption isotherms, the adsorption isotherms of n-octane and n-propanol on a bare graphite surface as well as on that precovered with n-hexadecane were determined. From these isotherms the extrapolated film pressure π_max values were calculated from the integrated Gibbs′ adsorption equation, integrating up to saturated vapour pressure. Equating the π_max values of n-octane to the work of wetting, the dispersion free surface energy of graphite was calculated, as a function of surface coverage with n-hexadecane. The obtained values were then discussed in relation to possible structures of hydrocarbon layers. The spreading films of n-octane seemed the most probable in the investigated systems. The dispersion free surface energy of bare graphite was estimated to be 119.5 mJ/m², which was comparable to the values reported in the literature. The deposited n-hexadecane layers seemed to reduce the surface energy of graphite to 68–70 mJ/m². Polar interactions of the bare graphite surface were also found.     

Validation of the tracer-pulse method for multicomponent liquid chromatography, a classical paradox revisited

The tracer-pulse method was extended and validated for the determination of multicomponent adsorption isotherms in liquid chromatography. Competitive adsorption isotherms can be determined for any number of solutes, up to the column resolution limit. The basic principle is to equilibrate the column with an eluent containing a mixture of the solutes and then measure the migration velocity of each of them through the column. It is easy to calculate the stationary phase concentrations from these velocities, given the eluent composition. As in frontal analysis, real competitive isotherm data are measured using this method, unlike other methods, which only produce parametric estimates. The method was used to measure the binary isotherms of beta-blockers on a Kromasil C8 column. The data were fitted to competitive bi-Langmuir adsorption isotherm functions and was found to agree well with the results of frontal analysis and the perturbation method. Computer simulations based on the isotherm parameters were performed and displayed very good agreement with the experimental chromatograms. An intriguing and seemingly paradoxical property is visualized and discussed: the fact that the injected molecules are not found in the detected peaks.     

Validation of the accuracy of the perturbation peak method for determination of single and binary adsorption isotherm parameters in LC

An analytical validation of the precision and accuracy of the perturbation peak (PP) method for determination of single and competitive thermodynamic isotherm parameters was performed using frontal analysis as a reference. The isotherm parameters of 11alpha-hydroxyprogesterone were determined in an achiral system and the isotherm parameters of (+)-methyl L-mandelate and (-)-methyl D-mandelate were determined in a chiral system, both for the single components and for the competitive binary mixture. The experimental errors in the PP method using different injection techniques were investigated, and we devised a new injection technique for the determination of competitive isotherm parameters that considerably reduced the experimental errors and also made both perturbation peaks detectable. We showed that the PP method with the new injection technique can be used to determine isotherm parameters directly from a racemic mixture. These parameters agreed well with those determined using several enantiomer ratios. Elution-band profiles simulated using the isotherm parameters showed excellent agreement with experimental profiles.     

Validation of the accuracy of the perturbation peak method for determination of multicomponent adsorption isotherm parameters in LC

The isotherm parameters were for the first time determined for a quaternary mixture. This was done by the perturbation peak (PP) method using racemic mixtures of methyl and ethyl mandelate enantiomers. One complication with the PP method is that the traditional blank injection technique makes all perturbation peaks, except one, vanish at moderately nonlinear concentration plateaus. Therefore, we devised a new injection technique that made all four peaks on a quaternary component concentration plateau detectable, thereby making the determination of multicomponent competitive isotherm parameters possible. The measured quaternary perturbation data fitted well to the bi-Langmuir isotherm model and excellent agreement was found between experimental and simulated single-component and multicomponent profiles, thus validating the method and the determined isotherm parameters. The method (i) is valuable for computer-assisted optimization of preparative chiral chromatography and (ii) opens the possibility of quantifying competitive drug-target interactions for chiral drugs directly on racemic mixtures, which today is impossible with any nonlabeled technique including surface plasmon resonance technology.     

Modeling of the separation of the enantiomers of 1-phenyl-1-propanol on cellulose tribenzoate.

The competitive adsorption isotherms of rac-1-phenyl-1-propanol on cellulose tribenzoate were measured by competitive frontal analysis. The experimental data were fitted to four different isotherm models: Langmuir, Bilangmuir, Langmuir-Freundlich, and Tóth. The fittings of the experimental data to all four models were satisfactory. It was excellent in the case of the Langmuir-Freundlich and the Tóth models. Overloaded elution profiles calculated with the Tóth isotherm were in good agreement with the experimental profiles in all the different experimental conditions investigated. This work extends to the case of binary mixtures the equivalence between the general rate and the lumped pore diffusion models already demonstrated for pure compounds when the ratio between the Stanton and the Biot numbers exceeds 5. The adsorption energy distribution for the Tóth isotherm was also calculated.     

Determination of adsorption isotherms for 1-heptene adsorbed on synthetic NaX- and NaY-zeolites from n-heptane solutions

This work presents the adsorption isotherms of 1-heptene, obtained from 10% solutions in n-heptane, on NaX and NaY zeolites. The adsorption isotherms were determined by column liquid chromatography (l.c.) by the frontal technique, followed by repetitive gas chromatographic analyses of the l.c. eluate fractions.     

Numerical solutions of the adsorption integral equation utilizing the spline functions

Detailed studies dealing with numerical solutions of the fundamental integral equation representing the overall adsorption isotherms are carried out utilizing different types of spline function. The successive use of the smoothing cubic spline functions is recommended to calculate the energy distribution function from the adsorption isotherm. A new computer program EDCAIS (energy distribution computation from adsorption isotherm utilizing the smoothing spline functions) is developed for quasi-Langmuir local isotherms involving multilayer correction. EDCAIS is tested using the theoretical adsorption isotherms relating to five types of energy distribution: constant distributions and distributions composed of one, three and five gaussian peaks.     

Adsorption equilibria of benzodiazepines on a hybrid polymeric chiral stationary phase

The chromatographic behavior of a series of racemic benzodiazepines was evaluated under linear and nonlinear conditions on a new hybrid polymeric (DACH-ACR) chiral stationary phase (CSP). Differently substituted benzodiazepines were employed as probes to make hypotheses concerning possible molecular interaction mechanisms originating between target compounds and active sites on the CSP. Hydrogen bonds were found to be pivotal for chromatographic retention and chiral selectivity. The competitive effect from a mobile-phase (MP) modifier able to interact with the CSP through H-bonds was investigated. The performance of the polymeric DACH-ACR CSP for preparative purposes was also evaluated. The competitive adsorption isotherms of two benzodiazepines, lorazepam and temazepam, were measured at different MP compositions through the so-called inverse method. The adsorption data were fitted with a competitive bi-Langmuir adsorption isotherm. Enantiomeric separations under nonlinear conditions were modeled by using the equilibrium dispersive (ED) model of chromatography. Theoretical overloaded band profiles (obtained by solving the system of partial differential equations described by the ED model) matched, in a significantly accurate way, the profiles experimentally measured.     

Electrophoretic effects of the adsorption of anionic surfactants to poly(dimethylsiloxane)-coated capillaries

Poly(dimethylsiloxane) (PDMS) is one of the most convenient materials to construct capillary electrophoresis microchips. Even though PDMS has many advantages, its use is often limited by its hydrophobicity. Although it is well-known that the surface properties of PDMS can be modified by anionic surfactants, very little is known regarding the driving forces or the electrophoretic consequences of the adsorption of anionic surfactants. In this work, the adsorption of alkyl surfactants on PDMS was studied by performing electroosmotic flow (microEOF) measurements. In order to mimic the behavior of PDMS microchannels, fused-silica capillaries were coated with PDMS and used for the microEOF measurements. This approach allowed using standard CE instrumentation and provided significant advantages over similar experiments performed on microchips. The change in the microEOF in the presence of surfactants was correlated to the surfactant adsorbed amount which, plotted versus surfactant concentration, gives an adsorption isotherm. The adsorption isotherms were obtained using alkyl surfactants with different chain lengths and head groups. According to our results, the interaction of alkyl surfactants with the PDMS surface is determined by a combination of hydrophobic and electrostatic interactions, where the former is more significant than the latter. The affinity of each surfactant for the PDMS surface was calculated by fitting the adsorption profiles with a Langmuir equation and, in the case of single-charged surfactants, correlated to the corresponding cmc value.     

Prediction of analyte retention for ion chromatography separations performed using elution profiles comprising multiple isocratic and gradient steps

This study addresses the simulation of ion chromatographic (IC) separations performed under conditions where the elution profile consists of a sequence of isocratic and gradient elution steps (referred to as "complex elution profiles"). First, models for prediction of retention under gradient elution conditions in IC were evaluated using an extensive database of gradient elution retention data. It is shown that one such model is preferred on the basis that it can be used to predict gradient retention times on the basis of isocratic input data. A method is then proposed for using this model for complex elution profiles whereby each step of the elution profile is treated separately and analyte movement through the column is mapped. An empirically based algorithm for predicting peak width under complex elution conditions is also proposed. Evaluation of the suggested approaches was undertaken on a set of 24 analyte anions and 13 analyte cations on 5 different Dionex columns using a range of 5-step complex elution profiles that gave R2 values for correlations between predicted and observed retention times of 0.987 for anions and 0.997 for cations. The simulation of separations of anions and cations using a 3-step complex elution profile is demonstrated, with good correlation between observed and predicted chromatograms. The proposed approach is useful for the rapid development of separations when complex elution profiles are used in IC.     

Interaction of silver nanoparticles (SNPs) with bacterial extracellular proteins (ECPs) and its adsorption isotherms and kinetics

Indiscriminate and increased use of silver nanoparticles (SNPs) in consumer products leads to the release of it into the environment. The fate and transport of SNPs in environment remains unknown. We have studied the interaction of SNPs with extracellular protein (ECP) produced by two environmental bacterial species and the adsorption behavior in aqueous solutions. The effect of pH and salt concentrations on the adsorption was also investigated. The adsorption process was found to be dependent on surface charge (zeta potential). The capping of SNPs by ECP was confirmed by Fourier transform infrared spectroscopy and X-ray diffraction. The adsorption of ECP on SNPs was analyzed by Langmuir and Freundlich models, suggesting that the equilibrium adsorption data fitted well with Freundlich model. The equilibrium adsorption data were modeled using the pseudo-first-order and pseudo-second-order kinetic equations. The results indicated that pseudo-second-order kinetic equation would better describe the adsorption kinetics. The capping was stable at environmental pH and salt concentration. The destabilization of nanoparticles was observed at alkaline pH. The study suggests that the stabilization of nanoparticles in the environment might lead to the accumulation and transport of nanomaterials in the environment, and ultimately destabilizes the functioning of the ecosystem.     

Gravimetric measurements of gas adsorption on zeolites at low temperatures

An apparatus to measure adsorption isotherms of molecular sieves at low temperatures by a gravimetric method is described. Results of measurements on the adsorption at 77 K of nitrogen on the zeolites 13 X, 5 A, and 3 A, and that at 20 K of helium on 13 X and 5 A are given. The results are in agreement with those obtained from volumetric measurements; some new isotherms are presented. The conditioning effect was also observed.     

Development of the tracer-pulse method for adsorption studies of analyte mixtures in liquid chromatography utilizing mass spectrometric detection

The tracer-pulse method provides the real adsorption data points directly from simple, straightforward calculations and is therefore a superior method for multicomponent adsorption isotherm determination in HPLC. Only one important problem has restricted its use so far: the tracer peaks are invisible using any conventional detection principle. We present a solution to this problem with an approach with a firm base in analytical chemistry, utilizing stable isotopes and mass spectrometric detection. The new approach was used for the determination of binary adsorption isotherms, and a systematic investigation was made of its main sources of error. With this modification, the tracer method can be a prime choice for future characterizations of multicomponent separation systems and of competitive drug binding studies.     

A comparison of basic dye adsorption onto zeolitic materials synthesized from fly ash

This investigation reveals the adsorption characteristics of two basic dyes, thionine (TH) and safranine T (ST), onto fly ash (FA) and its three zeolitized products prepared at different hydrothermal conditions. Typical two-step isotherms were observed for TH adsorption onto four adsorbents, whereas the isotherms of the larger ST molecules were S-shaped. The adsorption capacities of the zeolitized fly ash (ZFA) estimated from the first plateau region of the TH isotherms was nearly twice the FA capacity. The capacities increased by up to five times in the second plateau region. The adsorption capacity of FA for ST is equivalent that of TH, whereas the capacities of ZFA are lower than those found for TH. The equilibrium results were well-described by the Freundlich isotherm model. The kinetic data obtained in the temperature range of 298-318 K was analyzed using Paterson's and Nernst Plank's approximations based on the homogeneous surface diffusion model (HSDM). The thermodynamic functions for the transition state were evaluated from the temperature-dependence of the surface diffusion coefficients by applying the Eyring model.     

The removal of phenolic compounds from aqueous solutions by organophilic bentonite

The adsorption of p-chlorophenol (p-CP) and p-nitrophenol (p-NP) on organophilic bentonite (dodecylammonium bentonite, DDAB) was studied as a function of solution concentration and temperature. The observed adsorption rates were found to be equal to the first-order kinetics. The rate constants were calculated for temperatures ranging between 25.0-35.0 degrees C at constant concentration. The adsorption energies, E and adsorption capacity, (qm), for phenolic compounds adsorbed to organophilic bentonite were estimated by using the Dubinin-Radushkevic equation. Thermodynamic parameters from the adsorption isotherms of p-CP and p-NP on organophilic bentonite were determined. These isotherms were modeled according to Freundlich and Dubinin-Radushkevic adsorption isotherms and followed the V-shaped isotherm category with two steps. The amount of adsorption was found to be dependent on the relative energies of adsorbent-adsorbate, adsorbate-solvent and adsorbate-adsorbate interaction.     

Detection of flaws in sandwich plates

A technique is presented to characterize rectangular flaws in sandwich plates. The scattered wave fields in the frequency domain for sandwich plates with rectangular flaws are calculated by using the strip element method. In the present technique, the length of a rectangular flaw in a sandwich plate is determined from the displacement distribution curve on the upper surface of the plate. The damage factor of the flaw is determined from the mean value of the peaks of the additional oscillations on the displacement distribution curves. An approximate polynomial formula of degree three for the prediction of the damage factor of flaws in sandwich plates is also proposed. Numerical experiments are made to demonstrate the present technique.     

Study on adsorption refrigeration cycle utilizing activated carbon fibers. Part 1. Adsorption characteristics

Thermal heat driven adsorption systems using natural refrigerants have been focused on the recent energy utilization trend. However, the drawbacks of these adsorption systems are their poor performance in terms of system cooling capacity and coefficient of performance (COP). The objective of this paper is to improve the performance of thermally powered adsorption cooling system by selecting new adsorbent–refrigerant pair. Adsorption capacity of adsorbent–refrigerant pair depends on the thermophysical properties (pore size, pore volume and pore diameter) of adsorbent and isothermal characteristics of the pair. In this paper, the thermophysical properties of two PAN types of activated carbon fibers (FX-400 and KF-1000) are determined from the nitrogen adsorption isotherms. The standard nitrogen gas adsorption/desorption measurements on various adsorbents at liquid nitrogen of temperature 77.3 K were performed. Surface area of each adsorbent was determined by the Brunauer, Emmett and Teller (BET) plot of nitrogen adsorption data. Pore size distribution was measured by the Horvath and Kawazoe (HK) method. As of the adsorption/desorption isotherms, FX-400 shows very small hysteresis when the value of P/P_o exceeds 0.4, while KF-1000 has no hysteresis in the whole range of P/P_o. The adsorption capacity of FX-400 is about 30% higher than that of KF-1000. The adsorption equilibrium data of activated carbon fiber (ACF)-methanol are presented and correlated with simple equations. The adsorption equilibrium data of ACF (KF-1000)-water also presented in order to facilitate comparison with those of ACFs-methanol pair. The results will contribute significantly in designing the adsorber/desorber heat exchanger for thermally driven adsorption cooling system.     

Fast GCxGC with short primary columns

A novel approach to comprehensive two-dimensional gas chromatography (GCxGC) separations is presented, which operates in a new region of the "GCxGC optimization pyramid". The technique relies on the use of short primary columns to decrease elution temperatures (Te) of analytes from the primary column, with a Te reduction of up to 50 degrees C illustrated. This in turn has implications that will expand the areas where GCxGC can be used, as decreased elution temperatures will allow GCxGC to be applied to mixtures of less volatile compounds or permit the use of less thermally stable stationary phases in the column ensemble. As well, it will allow GCxGC to be applied to thermally labile compounds through a reduction in elution temperature. With short primary columns, resolution and efficiency in the first dimension is sacrificed, but speed is gained; however, the second column in GCxGC provides additional resolution and separation of compounds of differing chemical properties. Thus, it is possible to recover some of the analytical separation power of the system to provide resolution of target analytes from sample impurities. As an example, a case study using short primary columns for the separation of natural pyrethrins, which degrade above 200 degrees C, is described. Even with the sacrifices of overall separation power that are made, there is still sufficient resolution available to separate the six natural pyrethrins from each other and the complex chrysanthemum extract matrix. The use of cold-on-column injection, a short primary column, and a high carrier gas flow rate allow the pyrethrins to be eluted below 200 degrees C, with separation in 17 min and complete resolution from sample matrix.