Nonlinear chromatography

In linear chromatography the adsorption isotherm is linear and the amount adsorbed on the stationary phase at equilibrium is proportional to the solute concentration in the mobile phase. Thus, the adsorption isotherm is a straight line starting out at the origin [13] and the retention time and individual band shapes are independent of the sample composition and amount. In analytical chromatography linear conditions often prevail. 

In nonlinear chromatography the concentration of a component in the stationary phase at equilibrium is no longer proportional to its concentration in the mobile phase [13], Under such conditions, the adsorption isotherms also depend on the concentrations of all other components in the sample mixture. Thus, the retention time, peak height and band shape depend both on the sample composition and amount [13], This is the situation found in practically all preparative applications [13], Nonlinear chromatography is extremely complex due to the interdependence of the individual band profiles that are caused by the dependence of the amount of any component adsorbed on the stationary phase on all the components in solution [13], 

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G. Guioehon, S. Golshan-Shirazi, A. Katti, Eundamentals of preparative and nonlinear chromatography., Aeademie Press, New York 1994. 

Mazzotti M., Storti G., Morbidelli M. (1997) Optimal Operation of Simulated Moving Bed Units for Nonlinear Chromatographie Separations, J. Chromatogr. A 169 3-24. 

The steps when designing a SMB which would allow one to process a given amount of feed per unit time have been described in detail [46, 57]. The procedure described was based on modeling of nonlinear chromatography. The procedure is rigorous, versatile and mainly requires the determination of competitive adsorption isotherms. If the adequate tools and methods are used, the procedure is not tedious and requires less work than is sometimes claimed. The procedure is briefly described below. 

Conditions of nonlinear chromatography prevail, and the set of Equations (15-19) leads to the following optimum reduced flowrates ... 

Guiochon, G., Shirazi, S.G., and Katti, A.M., Fundamentals of Preparative and Nonlinear Chromatography, Academic Press, Boston, MA, 1994. 

Golay equation 21, 611 gradient (LC) 490 height equivalent to a theoretical plate 11 longitudinal diffusion 16 mass transfer resistance 17 nonlinear chromatography SOS plate model 14 rate theory IS reduced parameters 78, 361, 611... 

More recently, there have been attempts to study band patterns as they are affected by shock layers in nonlinear chromatography.42 Shock layers are steep boundaries that develop when the boundary front of an elution band becomes very steep and self-sharpening at high concentrations. While comparison of predicted and experimental data was promising, this study, like the others mentioned above, was done with single-component samples and awaits further analysis with the kinds of multi-component feeds more frequently encountered in process purifications. 

Prediction of multicomponent nonlinear chromatography accounting for rate factors requires numerical solution (see Guiochon et al., gen. refs., and "Numerical Methods and Characterization of Wave Shape in "Fixed Bed Transitions ). 

Guiochon G, Golshan Shirazi S, Katti AM (1994) Fundamentals of preparative and nonlinear chromatography, Academic Press, Boston... 

Two principal approaches with a different degree of complexity and differentiation have been pursued Thermodynamics investigations under linear and nonlinear chromatography conditions. 

On the contrary, a more advanced methodology makes use of nonlinear chromatography experiments If the adsorption isotherms are measured under variable temperatures, the corresponding thermodynamic parameters for each site can be obtained in view of the van t Hoff dependency (site-selective thermodynamics measurements) [51,54]. Thus, the adsorption equilibrium constants of the distinct sites bi a = ns, s) are related to the enthalpy (A// ) and entropy (A5j) according to the following equation [54] ... 

A. M. Fundamentals of Preparative and Nonlinear Chromatography, Academic Press, Toronto, 1994. 

Gritti, F., Guiochon, G. Critical contribution of nonlinear chromatography to the understanding of retention... 

When the amount of the sample is comparable to the adsorption capacity of the zone of the column the migrating molecules occupy, the analyte molecules compete for adsorption on the surface of the stationary phase. The molecules disturb the adsorption of other molecules, and that phenomenon is normally taken into account by nonlinear adsorption isotherms. The nonlinear adsorption isotherm arises from the fact that the equilibrium concentrations of the solute molecules in the stationary and the mobile phases are not directly proportional. The stationary phase has a finite adsorption capacity lateral interactions may arise between molecules in the adsorbed layer, and those lead to nonlinear isotherms. If we work in the concentration range where the isotherms are nonlinear, we arrive to the field of nonlinear chromatography where thermodynamics controls the peak shapes. The retention time, selectivity, plate number, peak width, and peak shape are no longer constant but depend on the sample size and several other factors. 

The purpose of this chapter is to illustrate the fundamental phenomena occurring in nonlinear chromatography. Since nonlinear isotherms play a key role in the behaviors we observe in nonlinear chromatography, emphasis is put on the description of nonlinear isotherms and their determination. A complete treatment of the theory of nonlinear chromatography is discussed elsewhere [1]. Here, we furnish a brief summary of the fundamentals. 

The equilibrium models of nonlinear chromatography assume that there always is an instantaneous equilibrium between the mobile phase and the stationary phase. That model is widely applied for the separation of small molecules, when mass transfer or diffusion in the stagnant pores of the mobile phase does not have a significant impact on the band profile. 

Accordingly, two major parameters affect the band profiles in nonlinear chromatography the column efficiency, and the amount of sample injected or loading factor. Parameter F (phase ratio) depends on the total porosity of the packing and cannot be changed in practice. 

The solution of the simplest kinetic model for nonlinear chromatography the Thomas model [9] can be calculated analytically. The Thomas model entirely ignores the axial dispersion, i.e., 0 =0 in the mass balance equation (Equation 10.8). For the finite rate of adsorption/desorption, the following second-order Langmuir kinetics is assumed... 

The assumption of linear chromatography fails in most preparative applications. At high concentrations, the molecules of the various components of the feed and the mobile phase compete for the adsorption on an adsorbent surface with finite capacity. The problem of relating the stationary phase concentration of a component to the mobile phase concentration of the entire component in mobile phase is complex. In most cases, however, it suffices to take in consideration only a few other species to calculate the concentration of one of the components in the stationary phase at equilibrium. In order to model nonlinear chromatography, one needs physically realistic model isotherm equations for the adsorption from dilute solutions. 

In this chapter, we provided a general overview of nonlinear chromatography. In nonlinear chromatography, the shape of the nonlinear isotherms and the molecular interactions leading to a... 

New concepts presented in this edition include monolithic columns, bonded stationary phases, micro-HPLC, two-dimensional comprehensive liquid chromatography, gradient elution mode, and capillary electromigration techniques. The book also discusses LC-MS interfaces, nonlinear chromatography, displacement chromatography of peptides and proteins, field-flow fractionation, retention models for ions, and polymer HPLC. 

Peak Tailing and Some Basics of Nonlinear Chromatography. 270... 

This chapter does not intend to be another exhaustive review of all work done to date or in recent years. We rather attempt to direct attention to still existing problems and hope to contribute to a better appreciation of these problems. We believe that the slow progress of MIP methods in the direction of practical applications has been partly due to the lack of a clear and systematic discussion of some of the problems encountered. Two important topics of this kind are the peak shapes in MIP HPLC, SPE, and CEC, and the problems with defining MIP selectivity. Discussion of these topics requires appreciation of some elements of nonlinear chromatography, which will be presented in this chapter. 

It is worth mentioning here that comparisons between the efficiency of different MIP separation systems like two HPLC systems or two CEC systems, or an HPLC system with a CEC system, are quite difficult when the adsorption isotherms are nonlinear. One of the typical difficulties is that the phase ratios in the two systems may be different. The effect of phase ratio on the separation and particularly on the achievable optimum separation is a complex question even in linear chromatography. In nonlinear chromatography this is really difficult and also burdened by the differences between the isotherms of the two compounds to be separated. The complexity of this matter has been mostly overlooked in the MIP literature and the visual comparison of two separations in rather different systems, operated under very different conditions, has frequently lead to statements declaring one technique better than the other. 

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