Simulation of electrophoretic separations by the flux-corrected transport method

Isotachophoresis: Theory and Microfluidic Applications

Isotachophoresis (ITP) is a versatile electrophoretic technique that can be used for sample preconcentration, separation, purification, and mixing, and to control and accelerate chemical reactions. Although the basic technique is nearly a century old and widely used, there is a persistent need for an easily approachable, succinct, and rigorous review of ITP theory and analysis. This is important because the interest and adoption of the technique has grown over the last two decades, especially with its implementation in microfluidics and integration with on-chip chemical and biochemical assays. We here provide a review of ITP theory starting from physicochemical first-principles, including conservation of species, conservation of current, approximation of charge neutrality, pH equilibrium of weak electrolytes, and so-called regulating functions that govern transport dynamics, with a strong emphasis on steady and unsteady transport. We combine these generally applicable (to all types of ITP) theoretical discussions with applications of ITP in the field of microfluidic systems, particularly on-chip biochemical analyses. Our discussion includes principles that govern the ITP focusing of weak and strong electrolytes; ITP dynamics in peak and plateau modes; a review of simulation tools, experimental tools, and detection methods; applications of ITP for on-chip separations and trace analyte manipulation; and design considerations and challenges for microfluidic ITP systems. We conclude with remarks on possible future research directions. The intent of this review is to help make ITP analysis and design principles more accessible to the scientific and engineering communities and to provide a rigorous basis for the increased adoption of ITP in microfluidics.

Dynamic computer simulations of electrophoresis: 2010-2020

The transport of components in liquid media under the influence of an applied electric field can be described with the continuity equation. It represents a nonlinear conservation law that is based upon the balance laws of continuous transport processes and can be solved in time and space numerically. This procedure is referred to as dynamic computer simulation. Since its inception four decades ago, the state of dynamic computer simulation software and its use has progressed significantly. Dynamic models are the most versatile tools to explore the fundamentals of electrokinetic separations and provide insights into the behavior of buffer systems and sample components of all electrophoretic separation methods, including moving boundary electrophoresis, CZE, CGE, ITP, IEF, EKC, ACE, and CEC. This article is a continuation of previous reviews (Electrophoresis 2009, 30, S16–S26 and Electrophoresis 2010, 31, 726–754) and summarizes the progress and achievements made during the 2010 to 2020 time period in which some of the existing dynamic simulators were extended and new simulation packages were developed. This review presents the basics and extensions of the three most used one‐dimensional simulators, provides a survey of new one‐dimensional simulators, outlines an overview of multi‐dimensional models, and mentions models that were briefly reported in the literature. A comprehensive discussion of simulation applications and achievements of the 2010 to 2020 time period is also included.

High-resolution numerical simulations of electrophoresis using the Fourier pseudo-spectral method

We present the formulation, implementation, and performance evaluation of the Fourier pseudo-spectral method for performing fast and accurate simulations of electrophoresis. We demonstrate the applicability of this method for simulating a wide variety of electrophoretic processes such as capillary zone electrophoresis, transient-isotachophoresis, field amplified sample stacking, and oscillating electrolytes. Through these simulations, we show that the Fourier pseudo-spectral method yields accurate and stable solutions on coarser computational grids compared with other nondissipative spatial discretization schemes. Moreover, due to the use of coarser grids, the Fourier pseudo-spectral method requires lower computational time to achieve the same degree of accuracy. We have demonstrated the application of the Fourier pseudo-spectral method for simulating realistic electrophoresis problems with current densities as high as 5000 A/m² with over tenfold speed-up compared to the commonly used second-order central difference scheme, to achieve a given degree of accuracy. The Fourier pseudo-spectral method is also suitable for simulating electrophoretic processes involving a large number of concentration gradients, which render the adaptive grid-refinement techniques ineffective. We have integrated the numerical scheme in a new electrophoresis simulator named SPYCE, which we offer to the community as open-source code.

Instabilities of the pH gradient in carrier ampholyte-based isoelectric focusing: Elucidation of the contributing electrokinetic processes by computer simulation

Electrokinetic processes that lead to pH gradient instabilities in carrier ampholyte-based IEF are reviewed. In addition to electroosmosis, there are four of electrophoretic nature, namely (i) the stabilizing phase with the plateau phenomenon, (ii) the gradual isotachophoretic loss of carrier ampholytes at the two column ends in presence of electrode solutions, (iii) the inequality of the mobilities of positively and negatively charged species of ampholytes, and (iv) the continuous penetration of carbonate from the catholyte into the focusing column. The impact of these factors to cathodic and anodic drifts was analyzed by simulation of carrier ampholyte-based focusing in closed and open columns. Focusing under realistic conditions within a 5 cm long capillary in which three amphoteric low molecular mass dyes were focused in a pH 3-10 gradient formed by 140 carrier ampholytes was investigated. In open columns, electroosmosis displaces the entire gradient toward the cathode or anode whereas the electrophoretic processes act bidirectionally with a transition around pH 4 (drifts for pI > 4 and pI < 4 typically toward the cathode and anode, respectively). The data illustrate that focused zones of carrier ampholytes have an electrophoretic flux and that dynamic simulation can be effectively used to assess the magnitude of each of the electrokinetic destabilizing factors and the resulting drift for a combination of these effects. Predicted drifts of focused marker dyes are compared to those observed experimentally in a setup with coated capillary and whole column optical imaging.

High-resolution dynamic computer simulation of electrophoresis using a multiphysics software platform

The modeling and simulation software COMSOL Multiphysics^® was recently extended with an electrophoretic transport interface. Its performance was investigated by comparison to results obtained using the 1D dynamic electrophoresis simulators GENTRANS and SIMUL5. Simulations of zone electrophoresis, isotachophoresis, isoelectric focusing and of an oscillating electrolyte system were performed. Smooth profiles were essentially identical indicating that the COMSOL electrophoretic transport interface is able to reproduce results of the 1D simulators. Differences in the way the respective numerical schemes handle steep concentration gradients and associated instabilities were observed. The COMSOL electrophoretic transport interface is expected to be useful as a general model for simulations in 1D, 2D or 3D geometries, as well as for simulations combining electrophoresis with other physical phenomena.

Approximation of weak solution for the problem of a pH-gradient creation in isoelectrofocusing

The mathematical model describing the stationary natural pH -gradient arising under the action of an electric field in an aqueous solution of ampholytes is constructed and investigated. The model is a part of a more general model of the isoelectrofocusing process. Investigation is based on the approximation of a weak solution by the piecewise continuous non-smooth functions. The method can be used for solving classes of problems for ordinary differential equations with a small parameter at the highest derivatives and the turning points.

Prediction and understanding system peaks in capillary zone electrophoresis

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.

High-Resolution Computer Simulations of Stacking of Weak Bases Using a Transient pH Boundary in Capillary Electrophoresis. 1. Concept and Impact of Sample Ionic Strength

The dynamics of focusing weak bases using a transient pH boundary was examined via high-resolution computer simulation software. Emphasis was placed on the mechanism and impact that the presence of salt, namely, NaCl, has on the ability to focus weak bases. A series of weak bases with mobilities ranging from 5 x 10(-9) to 30 x 10(-9) m2/V x s and pKa values between 3.0 and 7.5 were examined using a combination of 65.6 mM formic acid, pH 2.85, for the separation electrolyte, and 65.6 mM formic acid, pH 8.60, for the sample matrix. Simulation data show that it is possible to focus weak bases with a pKa value similar to that of the separation electrolyte, but it is restricted to weak bases having an electrophoretic mobility of 20 x 10(-9) m2/V x s or quicker. This mobility range can be extended by the addition of NaCl, with 50 mM NaCl allowing stacking of weak bases down to a mobility of 15 x 10(-9) m2/V x s and 100 mM extending the range to 10 x 10(-9) m2/V x s. The addition of NaCl does not adversely influence focusing of more mobile bases, but does prolong the existence of the transient pH boundary. This allows analytes to migrate extensively through the capillary as a single focused band around the transient pH boundary until the boundary is dissipated. This reduces the length of capillary that is available for separation and, in extreme cases, causes multiple analytes to be detected as a single highly efficient peak.

Numerical simulation of DNA sample preconcentration in microdevice electrophoresis

A numerical model is presented for the accurate and efficient prediction of preconcentration and transport of DNA during sample introduction and injection in microcapillary electrophoresis. The model incorporates conservation laws for the different buffer ions, salt ions, and DNA sample, coupled through a Gaussian electric field to account for the field modifications that cause electromigration. The accuracy and efficiency required to capture the physics associated with such a complex transient problem are realized by the use of the finite element-flux corrected transport (FE-FCT) algorithm in two dimensions. The model has been employed for the prediction of DNA sample preconcentration and transport during electrophoresis in a double-T injector microdevice. To test its validity, the numerical results have been compared with the corresponding experimental data under similar conditions, and excellent agreement has been found. Finally, detailed results from a simulation of DNA sample preconcentration in electrophoretic microdevices are presented using as parameters the electric field strength and the other species concentrations. The effect of the Tris concentration on sample stacking is also investigated. These results demonstrate the great potential offered by the model for future optimization of such microchip devices with respect to significantly enhanced speed and resolution of sample separation.

Change of migration time and separation window accompanied by field-enhanced sample stacking in capillary zone electrophoresis

When field-enhanced sample stacking was used in capillary zone electrophoresis (CZE) analysis of cations, the decrease of migration time and the reduction of separation window was observed with increase of sample plug length. A simple equation expressing the migration velocity in the stacking process was derived to explain the above phenomenon. From experiments and theoretical consideration, we confirmed that this effect was caused by the higher potential gradient and larger eletroosmotic flow (EOF) mobility at the sample plug than those at the supporting electrolyte. A mathematical model appropriate for the computer simulation of such a system was studied considering the experimental results, and it was concluded that electroosmotic velocity (v(eof)) should be introduced to the equation of continuity as a constant.