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Journal PaperTrends in Analytical Chemistry, Vol. 165, Published 2023

The theory of movement of charged particles in solutions is based on fundamental physico-chemical equations, which form an inherently nonlinear mathematical model. An alternate approach is the formulation of the approximate linear model for capillary zone electrophoresis. The nonlinear and linear models of electromigration are implemented in two computer programs we have developed, Simul and PeakMaster, respectively. Both of them serve for method development in electromigration separation methods.

PeakMaster is based on the linear model and was designed for the computer optimization of background electrolytes for capillary zone electrophoresis to reach higher efficiency and better selectivity of separation. Simul is based on the numerical solution of the nonlinear mathematical model of electromigration and serves mainly for inspecting disturbing phenomena in electromigration. Both PeakMaster and Simul in version 6 have been launched as free software available for download at https://echmet.natur.cuni.czJournal PaperELECTROPHORESIS, Volume 44, Issue 7-8, pp.667-674, Published APR 2023

In 1961, Svensson described isoelectric focusing (IEF), the separation of ampholytic compounds in a stationary, natural pH gradient that was formed by passing current through a sucrose density gradient-stabilized ampholyte mixture in a constant cross-section apparatus, free of mixing. Stable pH gradients were formed as the electrophoretic transport built up a series of isoelectric ampholyte zones-the concentration of which decreased with their distance from the electrodes-and a diffusive flux which balanced the generating electrophoretic flux. When polyacrylamide gel replaced the sucrose density gradient as the stabilizing medium, the spatial and temporal stability of Svensson's pH gradient became lost, igniting a search for the explanation and mitigation of the loss. Over time, through a series of insightful suggestions, the currently held notion emerged that in the modern IEF experiment-where the carrier ampholyte (CA) mixture is placed between the anolyte- and catholyte-containing large-volume electrode vessels (open-system IEF)-a two-stage process operates that comprises a rapid first phase during which a linear pH gradient develops, and a subsequent slow, second stage, during which the pH gradient decays as isotachophoretic processes move the extreme pI CAs into the electrode vessels. Here we trace the development of the two-stage IEF model using quotes from the original publications and point out critical results that the IEF community should have embraced but missed. This manuscript sets the foundation for the companion papers, Parts 2 and 3, in which an alternative model, transient bidirectional isotachophoresis is presented to describe the open-system IEF experiment.

Journal PaperELECTROPHORESIS, Volume 44, Issue 7-8, pp.675-688, Published APR 2023

The carrier ampholytes-based (CA-based) isoelectric focusing (IEF) experiment evolved from Svensson's closed system IEF (constant spatial current density, absence of convective mixing, counter-balancing electrophoretic and diffusive fluxes yielding a steady state pH gradient) to the contemporary open system IEF (absence of convective mixing, large cross-sectional area electrode vessels, lack of counter-balancing electrophoretic- and diffusive fluxes leading to transient pH gradients). Open system IEF currently is described by a two-stage model: In the first stage, a rapid IEF process forms the pH gradient which, in the second stage, is slowly degraded by isotachophoretic processes that move the most acidic and most basic CAs into the electrode vessels. An analysis of the effective mobilities and the effective mobility to conductivity ratios of the anolyte, catholyte, and the CAs indicates that in open system IEF experiments a single process, transient bidirectional isotachophoresis (tbdITP) operates from the moment current is turned on until it is turned off. In tbdITP, the anolyte and catholyte provide the leading ions and the pI 7 CA or the reactive boundary of the counter-migrating H3O+ and OH- ions serves as the shared terminator. The outcome of the tbdITP process is determined by the ionic mobilities, pK(a) values, and loaded amounts of all ionic and ionizable components: It is constrained by both the transmitted amount of charge and the migration space available for the leading ions. tbdITP and the resulting pH gradient can never reach steady state with respect to the spatial coordinate of the separation channel.

Journal PaperELECTROPHORESIS, Volume 44, Issue 7-8, pp.689-700, Published APR 2023

In modern isoelectric focusing (IEF) systems, where (i) convective mixing is prevented by gels or small cross-sectional area separation channels, (ii) current densities vary spatially due to the presence of electrode vessels with much larger cross-sectional areas than those of the gels or separation channels, and (iii) electrophoretic and diffusive fluxes do not balance each other, stationary, steady-state pH gradients cannot form (open-system IEF). Open-system IEF is currently described as a two-stage process: A rapid IEF process forms the pH gradient from the carrier ampholytes (CAs) in the first stage, then isotachophoretic processes degrade the pH gradient in the second stage as the extreme pI CAs are moved into the electrode vessels where they become diluted. Based on the ratios of the local effective mobilities and the local conductivities (mu(eff)(L)(x)/kappa(x) values) of the anolyte, catholyte, and the CAs, we pointed out in the preceding paper (Vigh G, Gas B, Electrophoresis 2023, 44, x x x) that in open-system IEF, a single process, transient, bidirectional isotachophoresis (tbdITP) operates from the moment current is turned on. In this paper, we demonstrate some of the operational features of the tbdITP model using the new ITP/IEF version of Simul 6.

Journal PaperELECTROPHORESIS, Volume 42, Issue 12-13, Page 1291-1299, Published JUL 2021

Simul 6 is a 1D dynamic simulator of electromigration based on the mathematical model of electromigration in free solutions. The model consists of continuity equations for the movement of electrolytes in a separation channel, acid-base equilibria of weak electrolytes, and the electroneutrality condition. It accounts for any number of multivalent electrolytes or ampholytes and provides a complete picture about dynamics of electromigration and diffusion in the separation channel. The equations are solved numerically using software means which allow for parallelization and multithreaded computation. Simul 6 has a user-friendly graphical interface. It is typically used for inspection of system peaks (zones) in electrophoresis, stacking and preconcentrating analytes, optimization of separation conditions, method development in either capillary zone electrophoresis, isotachophoresis, and isoelectric focusing. Simul 6 is the successor of Simul 5, and has been launched as a free software available for download at https://simul6.app

Journal PaperELECTROPHORESIS, Volume 42, Issue 7-8, Page 881-889, Published: APR 2021

The structure of the double layer on the boundary between solid and liquid phases is described by various models, of which the Stern-Gouy-Chapman model is still commonly accepted. Generally, the solid phase is charged, which also causes the distribution of the electric charge in the adjacent diffuse layer in the liquid phase. We propose a new mathematical model of electromigration considering the high deviation from electroneutrality in the diffuse layer of the double layer when the liquid phase is composed of solution of weak multivalent electrolytes of any valence and of any complexity. The mathematical model joins together the Poisson equation, the continuity equation for electric charge, the mass continuity equations, and the modified G-function. The model is able to calculate the volume charge density, electric potential, and concentration profiles of all ionic forms of all electrolytes in the diffuse part of the double layer, which consequently enables to calculate conductivity, pH, and deviation from electroneutrality. The model can easily be implemented into the numerical simulation software such as Comsol. Its outcome is demonstrated by the numerical simulation of the double layer composed of a charged silica surface and an adjacent liquid solution composed of weak multivalent electrolytes. The validity of the model is not limited only to the diffuse part of the double layer but is valid for electromigration of electrolytes in general.

Journal PaperELECTROPHORESIS, Volume: 41, Issue: 7-8, Pages: 493-501, Published: APR 2020

Thermodynamic acidity constants (acid or acid-base dissociation constants, sometimes called also as ionization constants) and limiting ionic mobilities (both of them at defined temperature, usually 25 degrees C) are the fundamental physicochemical characteristics of a weak electrolyte, that is, weak acid or weak base or ampholyte. We introduce a novel method for determining the data of a weak electrolyte by the nonlinear regression of effective electrophoretic mobility versus buffer composition dependence when measured in a set of BGEs with various pH. To correct the experimental data for zero ionic strength we use the extended Debye-Huckel model and Onsager-Fuoss law with no simplifications. Contrary to contemporary approaches, the nonlinear regression is performed on limiting mobility data calculated by PeakMaster's correction engine, not on the raw experimental mobility data. Therefore, there is no requirement to perform all measurements at a constant ionic strength of the set of BGEs. We devised the computer program AnglerFish that performs the necessary calculations in a user-friendly fashion. All thermodynamic pKa values and limiting electrophoretic mobilities for arbitrarily charged substances having any number of ionic forms are calculated by one fit. The user input consists of the buffer composition of the set of BGEs and experimentally measured effective mobilities of the inspected weak electrolyte.

Journal PaperELECTROPHORESIS, Volume: 41, Issue: 7-8, Pages: 514-522, Published: APR 2020

Fourteen low molecular mass UV absorbing ampholytes containing 1 or 2 weakly acidic and 1 or 2 weakly basic functional groups that best satisfy Rilbe's requirement for being good carrier ampholytes (Delta pK(a) = pKa(monoanion) - pKa(monocation) < 2) were selected from a large group of commercially readily available ampholytes in a computational study using two software packages (ChemSketch and SPARC). Their electrophoretic mobilities were measured in 10 mM ionic strength BGEs covering the 2 < pH < 12 range. Using our Debye-Huckel and Onsager-Fuoss laws-based new software, AnglerFish (freeware, ), the effective mobilities were recalculated to zero ionic strength from which the thermodynamic pK(a) values and limiting ionic mobilities of the ampholytes were directly calculated by Henderson-Hasselbalch equation-type nonlinear regression. The tabulated thermodynamic pK(a) values and limiting ionic mobilities of these ampholytes (pI markers) facilitate both the overall and the narrow-segment characterization of the pH gradients obtained in IEF in order to mitigate the errors of analyte ampholyte pI assignments caused by the usual (but rarely proven) assumption of pH gradient linearity. These thermodynamic pK(a) and limiting mobility values also enable the reality-based numeric simulation of the IEF process using, for example, Simul (freeware, ).

Journal PaperELECTROPHORESIS, Volume: 41, Issue: 7-8, Pages: 536-544, Published: APR 2020

We present a new theoretical approach for calculating changes in the physico-chemical properties of BGEs for measurements by CZE due to the electrolysis in electrode vials (vessels). Electrolysis is an inevitable phenomenon in any measurement in CZE. Water electrolysis, which occurs in most measurements, can significantly alter the composition of the BGE in electrode vials and in the separation capillary and has a negative influence on the robustness and quality of separations. The ability to predict changes in the composition of the BGE is important for evaluation of the suitability of the BGEs for repeating electrophoretic runs. We compared theoretically calculated changes in the physico-chemical properties (pH, conductivity) with those measured using pH-microelectrode and contactless conductivity detection of the BGE after the electrophoretic run. We confirmed the validity of our theoretical approach with a common BGE composed of acid-base pair, where one constituent is fully dissociated while the second constituent is dissociated by only half, and with Good's buffer. As predicted by theoretical approach, the changes in the physico-chemical properties of the Good's buffer after the electrophoretic run were several times lower than in the case of a common BGE composed of a weak acid - strong base pair

Journal PaperELECTROPHORESIS, Volume: 41 Issue: 7-8, Pages: 481-492, Published: APR 2020

Peak shapes in electrophoresis are often distorted from the ideal Gaussian shape due to disturbing phenomena, of which the most important is electromigration dispersion. For fully dissociated analytes, there is a tight analogy between nonlinear models describing a separation process in chromatography and electrophoresis. When the velocity of the separated analyte depends on the concentration of the co-analyte, the consequence is a mutual influence of the analytes couples, which distorts both analyte zones. In this paper, we introduce a nonlinear model of electromigration for the analysis of two co-migrating fully dissociated analytes. In the initial stages of separation, they influence each other, which causes much more complicated peak shapes. The analysis has revealed that the two most important phenomena-the displacement and the tag-along effects-are common both for nonlinear chromatography and electrophoresis, though their description is partly based on rather different phenomena. The comparison between the nonlinear model of electromigration we describe and the numerical computer solution of the original continuity equations has proven an almost perfect agreement. The predicted features in peak shapes in initial stages of separation have been fully confirmed by the experiments.

Journal PaperELECTROPHORESIS, Volume: 40, Issue: 5, Pages: 668-682, Published: MAR 2019

The continuity equations that describe the movement of ions in liquid solutions under the influence of an external stationary electric field, as it is utilized in electrophoresis, were introduced a long time ago starting with Kohlrausch in 1897. From that time on, there have been many attempts to solve the equations and to discuss the results. In electrophoresis, special attention has always been devoted to the peak shapes obtained by the detector since the shapes have a tight connection with the phenomena taking place during electromigration and influence the efficiency and selectivity of the separation. Among these phenomena, the most important is electromigration dispersion. In this commented review paper, we compare various models of electromigration, try to find points that connect them, and discuss the range of their validity in light of the linear and nonlinear theory of electromigration.

Journal Paper ELECTROPHORESIS, Volume: 39, Issue: 11, Pages: 1390-1398, Published: JUN 2018

Conductivity detection is a universal detection technique often encountered in electrophoretic separation systems, especially in modern chip-electrophoresis based devices. On the other hand, it is sparsely combined with another contemporary trend of enhancing limits of detection by means of various preconcentration strategies. This can be attributed to the fact that a preconcentration experimental setup usually brings about disturbances in a conductivity baseline. Sweeping with a neutral sweeping agent seems a good candidate for overcoming this problem. A neutral sweeping agent does not hinder the conductivity detection while a charged analyte may preconcentrate on its boundary due to a decrease in its effective mobility. This study investigates such sweeping systems theoretically, by means of computer simulations, and experimentally. A formula is provided for the reliable estimation of the preconcentration factor. Additionally, it is demonstrated that the conductivity signal can significantly benefit from slowing down the analyte and thus the overall signal enhancement can easily overweight amplification caused solely by the sweeping process. The overall enhancement factor can be deduced a priori from the linearized theory of electrophoresis implemented in the PeakMaster freeware. Sweeping by neutral cyclodextrin is demonstrated on an amplification of a conductivity signal of flurbiprofen in a real drug sample. Finally, a possible formation of unexpected system peaks in systems with a neutral sweeping agent is revealed by the computer simulation and confirmed experimentally.

Journal Paper Electrophoresis, Volume 30, Issue S1,June 2009, Pages S7-S15

Electrophoresis utilizes a difference in movement of charged species in a separation channel or space for their spatial separation. A basic partial differential equation that results from the balance laws of continuous processes in separation sciences is the nonlinear conservation law or the continuity equation. Attempts at its analytical solution in electrophoresis go back to Kohlrausch's days. The present paper (i) reviews derivation of conservation functions from the conservation law as appeared chronologically, (ii) deals with theory of moving boundary equations and, mainly, (iii) presents the linear theory of eigenmobilities. It shows that a basic solution of the linearized continuity equations is a set of traveling waves. In particular cases the continuity equation can have a resonance solution that leads in practice to schizophrenic dispersion of peaks or a chaotic solution, which causes oscillation of electrolyte solutions.

Journal Paper Electrophoresis, Volume 28, Issue 1‐2, January 2007, Pages 3-14

The Kohlrausch regulating function (KRF) is a conservation law (conservation function), which is held in electrophoresis and which enables calculation of the so‐called adjusted concentrations of constituents. The KRF is not the only conservation function and, depending on the complexity of the electrophoretic system, other conservation laws may be obeyed having a broader range of applicability. The conservation laws are tightly related to system eigenmobilities and system zones (system peaks). In principle, no system eigenmobility is exactly zero, but in most practical cases at least one system's eigenmobility is close to zero. The existence of the close‐to‐zero eigenmobility inherently points to the existence of a conservation function and a system zone which is stationary. The stationary system zone is called injection zone, stagnant zone, water peak, or solvent dip. Electrophoretic (electromigration) systems can be divided into two types: (i) conservation systems, in which the absolute value of at least one system eigenmobility is close to zero and where at least one conservation law is obeyed and (ii) nonconservation systems, where no system eigenmobility is close to zero and no conservation law is obeyed. The paper reviews work dealing with conservation functions in electromigration, derives some “historical” conservation functions in a new way, derives several conservation functions for systems of multivalent electrolytes, and discusses electrophoretic systems that have nonconservation behavior. In some typical instances, the conservation functions are simulated by means of a dynamic simulation tool and depicted graphically.

Journal Paper Electrophoresis, Volume 27, Issue 5‐6, March 2006, Pages 984-991

We introduce the mathematical model of electromigration of electrolytes in free solution together with free software Simul, version 5, designed for simulation of electrophoresis. The mathematical model is based on principles of mass conservation, acid–base equilibria, and electroneutrality. It accounts for any number of multivalent electrolytes or ampholytes and yields a complete picture about dynamics of electromigration and diffusion in the separation channel. Additionally, the model accounts for the influence of ionic strength on ionic mobilities and electrolyte activities. The typical use of Simul is: inspection of system peaks (zones), stacking and preconcentrating analytes, resonance phenomena, and optimization of separation conditions, in either CZE, ITP, or IEF.

Journal Paper Journal of Chromatography A, Volume 960, Issues 1–2, 25 June 2002, Pages 187-198

A background electrolyte system for capillary zone electrophoresis which is composed of three strong univalent ionic constituents is investigated. The ion I is considered as a counter-ion and two ions, 2 and 3, are considered as co-ions in relation to the analyte ion 4. We investigate the linearized model of electromigration in such a system and calculate the eigenvalues of a corresponding matrix. The model is formulated in such a way that the eigenvalues of the system are certain mobilities, which we call eigenmobilites, which characterize specific features of the electrophoretic migration. One of the eigenmobilities is the system eigenmobility u(S) causing the rise of the system peak, called here the system eigenpeak. A situation when the analyte has the same mobility as the system eigenmobility, u(4) = u(S), is analyzed in detail. We show that it leads to the resonance-the mutual jump in the concentration profile of both co-ions, 2 and 3, has a shape of the spatial derivation of the originally sampled analyte profile and, moreover, it grows linearly with time. After a sufficiently long time it can be "amplified" to any value. The resonance has then a great impact on signals of indirect detection methods, like indirect UV detection or conductivity detection. In the framework of the linearized model the relative velocity slope S, a measure of electromigration dispersion, is expressed as S-X = F(u(1) + u(4))(u(2) - u(4))(u(3) - u(4))/[u(4)(u(s) - u(4))], where u(1) is the mobility of the ith ion and F is the Faraday constant. As in practice the concentration of the analyte is not infinitely small and has a certain finite value, the analyte will be at the resonance severely dispersed to a much broader spatial interval. When a specific detector is used, the signal of such an analyte can apparently be missed without any notice.

Journal Paper Electrophoresis, Volume 24, Issue 3, No. 3, February 2003, Pages 536-547

A mathematical model of capillary zone electrophoresis (CZE) based on the conception of eigenmobilities, which are the eigenvalues of a matrix M tied to the linearized governing equations is presented. The model considers CZE systems, where constituents, either analytes or components of the background electrolyte (BGE), are weak electrolytes – acids, bases, or ampholytes. There is no restriction on the number of components nor on the valence of the constituents nor on pH of the BGE. An electrophoretic system with N constituents has N eigenmobilities. In most BGEs one or two eigenmobilities are very close to zero so their corresponding eigenzones move very slowly. However, there are BGEs where no eigenmobility is close to zero. The mathematical model further provides: the transfer ratio, the molar conductivity detection response, and the relative velocity slope. This allows the assessment of the indirect detection, conductivity detection and peak broadening (distortion) due to electromigration dispersion. Also, we present a spectral decomposition of the matrix M to matrices allowing the assessment of the amplitudes of system eigenpeaks (system peaks). Our model predicted the existence of BGEs having no stationary injection zone (or water zone, gap, peak, dip). A common practice of using the injection zone as a marker of the electroosmotic flow must fail in such electrolytes.

Journal Paper Electrophoresis, Volume 25, Issue 18‐19, October 2004, Pages 3071-3079

A mathematical model of capillary zone electrophoresis (CZE) based on the conception of eigenmobilities, which are the eigenvalues of a matrix M tied to the linearized governing equations is presented. The model considers CZE systems, where constituents, either analytes or components of the background electrolyte (BGE), are weak electrolytes – acids, bases, or ampholytes. There is no restriction on the number of components nor on the valence of the constituents nor on pH of the BGE. An electrophoretic system with N constituents has N eigenmobilities. In most BGEs one or two eigenmobilities are very close to zero so their corresponding eigenzones move very slowly. However, there are BGEs where no eigenmobility is close to zero. The mathematical model further provides: the transfer ratio, the molar conductivity detection response, and the relative velocity slope. This allows the assessment of the indirect detection, conductivity detection and peak broadening (distortion) due to electromigration dispersion. Also, we present a spectral decomposition of the matrix M to matrices allowing the assessment of the amplitudes of system eigenpeaks (system peaks). Our model predicted the existence of BGEs having no stationary injection zone (or water zone, gap, peak, dip). A common practice of using the injection zone as a marker of the electroosmotic flow must fail in such electrolytes.

Journal Paper Electrophoresis, Volume 25, Issue 18‐19, No. 18‐19, October 2004, Pages 3080-3085

We are introducing a computer implementation of the mathematical model of zone electrophoresis (CZE) described in Štědrý, M., Jaroš, M., Hruška, V., Gaš, B., Electrophoresis 2004, 25, 3071–3079 program PeakMaster. The computer model calculates eigenmobilities, which are the eigenvalues of the matrix tied to the linearized continuity equations, and which are responsible for the presence of system eigenzones (system zones, system peaks). The model also calculates other parameters of the background electrolyte (BGE) – pH, conductivity, buffer capacity, ionic strength, etc., and parameters of the separated analytes – effective mobility, transfer ratio, molar conductivity detection response, and relative velocity slope. This allows the assessment of the indirect detection, conductivity detection and peak broadening (peak distortion) due to electromigration dispersion. The computer model requires the input of the BGE composition, the list of analytes to be separated, and the system instrumental configuration. The output parameters of the model are directly comparable with experiments; the model also simulates electropherograms in a user‐friendly way. We demonstrate a successful application of PeakMaster for inspection of BGEs having no stationary injection zone.

Journal Paper Electrophoresis, Volume 27, Issue 23, No. 23, December 2006, Pages 4610

We present a mathematical model of CZE based on the concept of eigenmobilities – the eigenvalues of matrix **M** tied to the linearized governing equations of electromigration, and the spectral decomposition of matrix **M** into matrices of amplitudes **P**j. Any peak in an electropherogram, regardless of whether it is an analyte peak or a system peak (system zone), is matched with its matrix Pj. This enables calculation of the peak parameters, such as the transfer ratio and the molar conductivity detection response (which give the indirect detection signal and the conductivity detection signal, respectively), when the initial disturbance caused by the injection of the sample is known. We also introduce new quantities, such as the generalized transfer ratio and the conductivity response of system zones, and show how the amplitude (intensity, area) of the analyte peaks and the system peaks can be calculated. We offer a free software, PeakMaster, which yields this information in a user‐friendly way.

Journal Paper Electrophoresis, Volume 33, Issue 6, March 2012, Pages 923-930

We extended the linearized model of electromigration, which is used by PeakMaster, by calculation of nonlinear dispersion and diffusion of zones. The model results in the continuity equation for the shape function ϕ(x,t) of the zone: ϕt = −(v0 + vEMDϕ)ϕx + δϕxx that contains linear (v0) and nonlinear migration (vEMD), diffusion (δ), and subscripts x and t stand for partial derivatives. It is valid for both analyte and system zones, and we present equations how to calculate characteristic zone parameters. We solved the continuity equation by Hopf–Cole transformation and applied it for two different initial conditions—the Dirac function resulting in the Haarhoff‐van der Linde (HVL) function and the rectangular pulse function, which resulted in a function that we denote as the HVLR function. The nonlinear model was implemented in PeakMaster 5.3, which uses the HVLR function to predict the electropherogram for a given background electrolyte and a composition of the sample. HVLR function also enables to draw electropherograms with significantly wide injection zones, which was not possible before. The nonlinear model was tested by a comparison with a simulation by Simul 5, which solves the complete nonlinear model of electromigration numerically.

Journal Paper Electrophoresis, Volume 33, Issue 6, March 2012, Pages 931-937

We introduce a computer implementation of the mathematical model of capillary zone electrophoresis described in the previous paper in this issue (Hruška et al., Electrophoresis 2012, 33), the program PeakMaster 5.3. The computer model calculates eigenmobilities, which are the eigenvalues of the Jacobian matrix of the electromigration system, and which are responsible for the presence of system eigenzones (system zones, system peaks). The model also calculates parameters of the background electrolyte: pH, conductivity, buffer capacity, ionic strength, etc., and parameters of the separated analytes: effective mobility, transfer ratio, molar conductivity detection response, and relative velocity slope. In addition to what was possible in the previous versions of PeakMaster, Version 5.3 can predict the shapes of the system peaks even for a complex injected sample profile, such as a rectangular plug. PeakMaster 5.3 can replace numerical simulation in many practically important configurations and the results are obtained in a very short time (within seconds). We demonstrate that the results obtained in real experiments agree well with those calculated by PeakMaster 5.3.

Journal Paper Journal of Chromatography A, Volume 1267, December 2012, Pages 102-108

We introduce a new nonlinear electrophoretic model for complex-forming systems with a fully charged analyte and a neutral ligand. The background electrolyte is supposed to be composed of two constituents, which do not interact with the ligand. In order to characterize the electromigration dispersion (EMD) of the analyte zone we define a new parameter, the nonlinear electromigration mobility slope, SEMD,A. The parameter can be easily utilized for quantitative prediction of the EMD and for comparisons of the model with the simulated and experimental profiles. We implemented the model to the new version of PeakMaster 5.3 Complex that can calculate some characteristic parameters of the electrophoretic system and can plot the dependence of SEMD,A on the concentration of the ligand. Besides SEMD,A, also the relative velocity slope, SX, can be calculated. It is commonly used as a measure of EMD in electrophoretic systems. PeakMaster 5.3 Complex software can be advantageously used for optimization of the separation conditions to avoid high EMD in complexing systems. Based on the theoretical model we analyze the SEMD,A and reveal that this parameter is composed of six terms. We show that the major factor responsible for the electromigration dispersion in complex-forming electrophoretic systems is the complexation equilibrium and particularly its impact on the effective mobility of the analyte. To prove the appropriateness of the model we showed that there is a very good agreement between peak shapes calculated by PeakMaster 5.3 Complex (plotted using the HVLR function) and the profiles simulated by means of Simul 5 Complex. The detailed experimental verification of the new mode of PeakMaster 5.3 Complex is in the next part IV of the series.

Journal Paper Electrophoresis, Volume 31, Issue 9, May 2010, Pages 1435-1441

It has been reported many times that the commercial mixtures of chiral selectors (CS), namely highly sulfated β‐CDs (HS‐β‐CDs), provide remarkable enantioselectivity in CZE when compared with single‐isomer CDs, even single‐isomer HS‐β‐CDs. This enhanced enantioselectivity of multi‐CS enantioseparative CZE is discussed in the light of multi‐CS model that we have introduced earlier. It is proposed on a theoretical basis and verified experimentally that the two enantiomers of a chiral analyte under interaction with a mixture of CSs are very likely to differ in their limit mobilities, which is opposite to single‐CS systems where the two limit mobilities are likely to be the same. Thus while the enantioseparation is usually controlled by different distribution constants between the two enantiomers and CS used in single‐CS systems, an additional, electrophoretic, enantioselective mechanism resulting from different limit mobilities may play a significant role in multi‐CS systems. This additional mechanism generally makes the multi‐CS systems more selective than the single‐CS systems. The possible inequality of limit mobilities is also significant for optimization of separation conditions using mixtures of CSs. A practical example supporting our considerations is shown on enantioseparation of lorazepam in the presence of a commercial mixture of HS‐β‐CDs and a single‐isomer HS‐β‐CD, heptakis(6‐O‐sulfo)‐β‐CD.

Journal PaperElectrophoresis, Volume 27, Issue 3, February 2006, Pages 513-518

Chemical oscillations are driven by the gradient of the chemical potential so that they can appear in systems where the substances are not in chemical equilibrium. We show that under the influence of the electric field, concentrations of electrically charged substances in solutions can oscillate even if the system is in chemical equilibrium. The driving force here is not the gradient of the chemical potential but rather the gradient of the electric potential. Utilizing CE we found periodic structures invoked by the application of a constant driving voltage in BGEs possessing complex eigenmobilities. By analogy with the behavior of dynamic systems, complex eigenmobilities implicate that the system will be unstable. Instead of forming system zones (system peaks) in the separation channel (capillary) the originally uniform concentration of electrolyte constituents becomes periodically disturbed when the electric current passes through it.

Journal PaperJournal of Physical Chemistry B, Volume 113, Issue 37, SEP 2009, Pages 12439-12446

Chemical oscillations are driven by a gradient of chemical potential and can only develop in systems where the substances are far from chemical equilibrium. We have discovered a new analogous type of oscillations in ternary electrolyte mixtures, which we call electromigration oscillations. They appeal in liquid solutions of electrolytes and are associated with the electromigration movement of ions when conducting an electric current. These electromigration oscillations are driven by the electric potential gradient, while the system can be close to chemical equilibrium. The unequivocal criterion for the instability of the electrolyte solution and its ability to oscillate is the existence of complex system eigenmobilities. We show how to calculate the system eigenmobilities by utilizing the linear theory of electromigration and how to identify the complex system eigenmobilities to predict electromigration oscillations. To experimentally prove these electromigration oscillations, we employ a commercially available instrument for capillary electrophoresis. The oscillations start a certain period of time after switching on the driving electric current. The axial concentration profiles of the electrolytes in the capillary attain a nearly periodic pattern with a spatial period in the range of 1-4 mm, with almost constant amplitude. This periodic pattern moves in the electric field with mobility that is equal to the real part of the complex eigenmobility pair. We have found several ternary oscillating electrolytes composed of a base and two acids, of which at least one has higher valence than one in absolute value. All the systems have three system eigenmobilities: one is real and close to zero, and the two others form the complex conjugate pair, the real part of which is far from zero.

Journal PaperElectrophoresis, Volume 24, Issue 3, FEB 2003, Pages 498-504

The principle of an on-line preconcentration method,for capillary zone electrophoresis (CZE) named electrokinetic supercharging (EKS), is described and based on computer 2 simulation the preconcentration behavior of the method is discussed. EKS is an electrokinetic injection method with transient isotachophoretic process, is a powerful preconcentration technique for the analysis of dilute samples. After filling the separation capillary with supporting electrolyte, an appropriate amount of a leading electrolyte was filled and the electrokinetic injection was started. After a while, terminating electrolyte was filled subsequently and migration current was applied. This procedure enabled the introduction of a large amount of sample components from a dilute sample without deteriorating separation. Computer simulation of the electrokinetic injection revealed that EKS was effective for the preconcentration of analytes with wide mobility ranges by proper choice of transient isotachophoresis (ITP) system and electroosmotic flow (EOF) should be suppressed to increase injectable amount of analytes under constant voltage mode. A test mixture of rare-earth chlorides was used to demonstrate the uses of EKS-CZE. When a 100 muL sample was used, the low limit of detectable concentration was 0.3 mug/L (1.8 nm for Er), which was comparable or even better than that of ion chromatography and inductively coupled plasma-atomic emission spectrometry (ICP-AES).

Journal PaperAnalytical Chemistry, Volume 65, Issue 15, AUG 1993, Pages 2108-2115

To increase the injection volume in capillary zone electrophoresis, a preconcentration step can be carried out. In this way a 50 times higher sample load is easily reached. Two types of such stacking systems are compared by computer simulation and are optimized with respect to pH, effective mobility of the constituents, and lengths of leading and stacking zones. The results of the computer simulation are compared with experiment, and good agreement is found. The application of such an optimized system to the separation of tryptic peptides is demonstrated. High efficiency and good reproducibility are obtained.

Journal PaperElectrophoresis, Volume 26, Issue 10, May 2005, Pages 1948-1953

A simple rule stating that the signal in conductivity detection in capillary zone electrophoresis is proportional to the difference between the analyte mobility and mobility of the background electrolyte (BGE) co‐ion is valid only for systems with fully ionized electrolytes. In zone electrophoresis systems with weak electrolytes both conductivity signal and electromigration dispersion of analyte peaks depend on the conductivity and pH effects. This allows optimization of the composition of BGEs to give a good conductivity signal of analytes while still keeping electromigration dispersion near zero, regardless of the injected amount of sample. The demands to achieve minimum electromigration dispersion and high sensitivity in conductivity detection can be accomplished at the same time. PeakMaster software is used for inspection of BGEs commonly used for separation of sugars (carbohydrates, saccharides) at highly alkaline pH. It is shown that the terms direct and indirect conductivity detection are misleading and should not be used.

Journal PaperElectrophoresis, Volume 25, Issue 23‐24, December 2004, Pages 3901-3912

When working with capillary zone electrophoresis (CZE), the analyst has to be aware that the separation system is not homogeneous anymore as soon as a sample is brought into the background electrolyte (BGE). Upon injection, the analyte creates a disturbance in the concentration of the BGE, and the system retains a kind of memory for this inhomogeneity, which is propagated with time and leads to so‐called system zones (or system eigenzones) migrating in an electric field with a certain eigenmobility. If recordable by the detector, they appear in the electropherogram as system peaks (or system eigenpeaks). However, although their appearance can not be forecasted and explained easily, they are inherent for the separation system. The progress in the theory of electromigration (accompanied by development of computer software) allows to treat the phenomenon of system zones and system peaks now also in very complex BGE systems, consisting of several multivalent weak electrolytes, and at all pH ranges. It also allows to predict the existence of BGEs having no stationary injection zone (or water zone, EO zone, gap, dip). Our paper reviews the theoretical background of the origin of the system zones (system peaks, system eigenpeaks), discusses the validity of the Kohlrausch regulating function, and gives practical hints for preparing BGEs with good separation ability not deteriorated by the occurrence of system peaks and by excessive peak‐broadening.

Journal Paper Journal of Separation Science, Volume 30, Issue 10, Jul 2007, Pages 1435-1445

Introduction of a sample into the separation column (microchip channel) in capillary zone electrophoresis (microchip electrophoresis) will cause a disturbance in the originally uniform composition of the background electrolyte. The disturbance, a system zone, can move in some electrolyte systems along the separation channel and, on reaching the position of the detector, cause a system peak. As shown by the linear theory of electromigration based on linearized continuity equations formulated in matrix form, the mobility of the system zone – the system eigenmobility – can be obtained as the eigenvalue of the matrix. Progress in the theory of electromigration allows us to predict the existence and mobilities of the system zones, even in very complex electrolyte systems consisting of several multivalent weak electrolytes, or in micellar systems (systems with SDS micelles) used for protein sizing in microchips. The theory is implemented in PeakMaster software, which is available as freeware (www.natur.cuni.cz/gas). The linearized theory also predicts background electrolytes having no stationary injection zone (water zone, water gap, water dip, EO zone) or unstable electrolyte systems exhibiting oscillations and creating periodic structures. The oscillating systems have complex system eigenmobilities (eigenvalues of the matrix are complex). This paper reviews the theoretical background of the system peaks (system eigenpeaks) and gives practical hints for their prediction and for preparing background electrolytes not perturbed by the occurrence of system peaks and by excessive peak broadening.

Journal PaperElectrophoresis, Volume 23, Issue 20, October 2002, Pages 3520-3527

Two constructions of the high‐frequency contactless conductivity detector that are fitted to the specific demands of capillary zone electrophoresis are described. The axial arrangement of the electrodes of the conductivity cell with two cylindrical electrodes placed around the outer wall of the capillary column is used. We propose an equivalent electrical model of the axial contactless conductivity cell, which explains the features of its behavior including overshooting phenomena. We give the computer numerical solution of the model enabling simulation of real experimental runs. The role of many parameters can be evaluated in this way, such as the dimension of the separation channel, dimension of the electrodes, length of the gap between electrodes, influence of the shielding, etc. The conception of model allows its use for the optimization of the construction of the conductivity cell, either in the cylindrical format or in the microchip format. The ability of the high‐frequency contactless conductivity detector is demonstrated on separation of inorganic ions.

Journal PaperElectrophoresis, Volume 18, Issue 12‐13, 1997, Pages 2123-2133

A review on peak broadening in capillary zone electrophoresis in free solutions is given which covers a selection of the literature published on this topic over the period mainly between 1992 and the beginning of 1997 (consisting of 71 publications). The contributions to peak dispersion from extracolumn effects (e.g. due to the finite length of the injection zone, or the aperture of the detector), from longitudinal diffusion, Joule heating, electromigration dispersion (concentration overload), a different path length of the solute ions, wall adsorption, laminar flow and the (longitudinally) homogeneous or nonhomogeneous electroosmotic flow are described. The latter may also occur when a longitudinally nonhomogeneous radial electric field is applied. Peak dispersion is depicted either by the plate‐height model, or the concentration of the solute as a function of space and time is calculated either analytically or numerically by solving the equation of continuity with appropriate initial and boundary conditions and possibly completed by equations governing further quantities.

Journal PaperElectrophoresis, Volume 16, Issue 1, 1995, Pages 958-967

A mathematical model is described for the simulation of peak profiles in capillary zone electrophoresis taking wall adsorption into account. It is based on such physico‐chemical relations as mass balance equations, sorption rate equations and appropriate boundary conditions. The numerical solution of the model is carried out, allowing the depiction of the concentration profile of the sample in the capillary as a result of the dynamics of the processes involved, which depend on a number of input parameters: rate constants, adsorption isotherms, column dimensions, field strength etc. Four cases are discussed in detail, namely those with either slow or fast adsorption kinetics and linear or nonlinear isotherms.

Journal PaperElectrophoresis, Volume 16, Issue 1, 1995, Pages 2027-2033

An exact analysis of the unsteady axial dispersion of an analyte, undergoing a linear adsorption at the column wall in capillary zone electrophoresis, is presented. A system of partial differential equations – in which the radial coordinate is one of the independent variables – is taken as a model for linear wall adsorption. It is shown that the dispersion is a sum of two terms, one which depends linearly on time and whose exact form is generally known, and a nonlinear one. The most interesting result of this work is that it derives another system of differential equations, which this nonlinear term is to satisfy. It makes it possible to present a closed formula for the asymptotic value of the nonlinear term, i.e., its limit for large time. Its behavior for times close to zero is also studied.

Journal PaperJournal of Chromatography A, Volume 709, Issue 1, August 1995, Pages 63-68

An expression is derived which gives the plate height contribution caused by an electroosmotic flow (EOF) in a cylindrical capillary with longitudinally uniform zeta potential. The derivation is made in terms of effective thickness of electric double layer (an analogue to the Debye length). Typical values of the effective thickness calculated for common situations are given. The resulting expression for the plate height, Heo = β2veo/D, enables one to calculate the plate height simply as a function of the electroosmotic velocity, veo, and the thickness of the electric double layer, β. The impact of peak broadening by the EOF is compared with that from longitudinal diffusion and extra-column effects for solutes with widely varying diffusion coefficients.

Journal PaperJournal of Chromatography, Volume 709, Issue 1, AUG 1995, Pages 51-62

The influence of the longitudinally non-uniform zeta potential on processes in capillary zone electrophoresis was studied. The velocity field of the electroosmotic flow in capillary tubes is modelled by the Navier-Stokes equations. Their stationary solution represents convective transport of a solute which is taken into account in the continuity equation for the concentration distribution. All equations are studied numerically. The results represent the time evolution of initial forms of sample peaks. These are presented in graphical form for several cases of zeta potentials which are either instructive or closely related to situations encountered in practice. It is shown that plug-like flow in the capillary cannot be expected and that a non-uniform zeta potential generally leads to significant dispersion of peaks.

Journal PaperJournal of Chromatography A, Volume 545, Issue 2, Jun 1991, Pages 225-237

Equations that describe the electrophoretic migration of monovalent ionic substances in solution with a significant presence of H+ or OH− ions are formulated. A derivation of the Kohlrausch regulating function for these conditions is presented. The model of electromigration consists of a set of continuity equations, together with a set of algebraic equations describing the chemical equilibria involved, and is implemented on a personal computer. Simulation of some experimental phenomena in electrophoretic methods, e.g., the sharpening effect in capillary zone electrophoresis or anomalous spikes in isotachophoretic systems, is presented.

Journal PaperJournal of Chromatography A, Volume 644, Issue 1, July 1993, Pages 161-174

A model of electrophoretic migration that is influenced by generated Joule heat is presented. The model takes into account the axial flux of the heat. It is shown that the mutual influence of non-equilibrium fluxes of mass and heat may lead to new phenomena: oscillation of the concentration profile on concentration boundaries and changes in concentration of the electrolytes (either an increase or a decrease) appearing at sites of jumps of the radial heat flux of the capillary tube. The theoretical results are supported by experiments.

Journal PaperJournal of Chromatography A, Volume 628, Issue 2, Jan 1993, Pages 283-308

Qualitative and quantitative isotachophoretic indices of 73 amino acids, dipeptides and tripeptides were simulated under 24 leading electrolyte conditions covering the pH range 6.4–10. The RE values and time-based zone lengths are tabulated together with the absolute mobility (mo) and pKa values used. The leading electrolyte used was 10 mM HCl and the pH buffers were imidazole, tris(hydroxymethylamino)methane, 2-amino-2-methyl-1,3-propanediol and ethanolamine. The simulated indices will be useful in the assessment of the separability and determination of the listed and related compounds.

Journal PaperJournal of Chromatography A, Volume 192, Issue 2, May 1980, Pages 253-257

A new detection system for isotachophoresis, the high-frequency contactless conductivity detector, is described. This detector has a high resolving power and gives good reproducibility.