Theory of electrophoretic separations. Part I: Formulation of a mathematical model

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.

Electrophoresis simulations using Chebyshev pseudo-spectral method on a moving mesh

We present the implementation and demonstration of the Chebyshev pseudo-spectral method coupled with an adaptive mesh method for performing fast and highly accurate electrophoresis simulations. The Chebyshev pseudo-spectral method offers higher numerical accuracy than all other finite difference methods and is applicable for simulating all electrophoresis techniques in channels with open or closed boundaries. To improve the computational efficiency, we use a novel moving mesh scheme that clusters the grid points in the regions with poor numerical resolution. We demonstrate the application of the Chebyshev pseudo-spectral method on a moving mesh for simulating nonlinear electrophoretic processes through examples of isotachophoresis (ITP), isoelectric focusing (IEF), and electromigration-dispersion in capillary zone electrophoresis (CZE) at current densities as high as 1000 A/m 2 . We also show the efficacy of our moving mesh method over existing methods that cluster the grid points in the regions with large concentration gradients. We have integrated the adaptive Chebyshev pseudo-spectral method in the open-source SPYCE simulator and verified its implementation with other electrophoresis simulators.

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.

Operations Research Model Formulation for Road Maintenance Case

Operations research can be used to apply analytical methods that help make precise and reasonable decisions. In road maintenance, basic principles of operations research are used to create model formulation that could help lower costs in case of an inaccurately made decision. First, the paper provides a literature review on different model formulations. Afterward, hypotheses are proposed regarding the model formulation, and then the model that minimises total generalised costs from wrong duty orders for road maintenance is offered. In conclusion, the paper evaluates the hypotheses and the process of improving the mathematical model.

Simulation and experiment of asymmetric electrode placement for electrophoretic exclusion in a microdevice

Electrophoretic exclusion (EE) is a counterflow gradient technique that exploits hydrodynamic flow and electrophoretic forces to exclude, enrich, and separate analytes. Resolution for this technique has been theoretically examined and the smallest difference in electrophoretic mobilities that can be completely separated is estimated to be 10^-13  cm² /Vs. Traditional and mesoscale systems have been used, whereas microfluidics offers a greater range of geometries and configurations towards approaching this theoretical limit. To begin to understand the impact of seemingly subtle changes to the entrance flow and the electric field configurations, three closely related microfluidic interfaces were modeled, fabricated, and tested. These interfaces consisted of systematically varying placement of an asymmetric electrode relative to a channel entrance: leading electrode placed outside the channel entrance, leading electrode aligned with the channel, and leading electrode placed within the channel. A charged fluorescent dye is used as a sensitive and accurate probe for the model and to test the concentration variation at these interfaces. Models and experiments focused on visualizing the concentration profile in areas of high temporal dynamics, thus providing a severe test of the models. Experimental data and simulation results showed strong qualitative agreement. The complexity of the electric and flow fields about this interface and the agreement between models and testing suggests the theoretical assessment capabilities can be used to faithfully design novel and more efficient interfaces.

Electrohydrodynamic instability of ion-concentration shock wave in electrophoresis

Capillary electrophoresis techniques often involve ion-concentration shock waves in an electrolyte solution, propagating under the effect of an external electric field. These shock waves are characterized by self-sharpening gradients in ion concentrations and electrical conductivity that are collinear with the electric field. The coupling of electric field and fluid motion at the shock interface sometimes leads to an undesirable electrohydrodynamic (EHD) instability. Using linear stability analysis, we describe the motion of small-amplitude disturbances of an electrophoretic shock wave. Our analysis shows that the EHD instability results due to the competition between destabilizing electroviscous flow and stabilizing electromigration of the shock wave. The ratio of timescales corresponding to electroviscous flow and electromigration yields a threshold criterion for the onset of instability. We present a validation of this threshold criterion with published experimental data and also describe the physical mechanism underlying the EHD instability of the electrophoretic shock wave.

Theory of multi-species electrophoresis in the presence of surface conduction

Electrophoresis techniques are characterized by concentration disturbances (or waves) propagating under the effect of an electric field. These techniques are usually performed in microchannels where surface conduction through the electric double layer (EDL) at channel walls is negligible compared with bulk conduction. However, when electrophoresis techniques are integrated in nanochannels, shallow microchannels or charged porous media, surface conduction can alter bulk electrophoretic transport. The existing mathematical models for electrophoretic transport in multi-species electrolytes do not account for the competing effects of surface and bulk conduction. We present a mathematical model of multi-species electrophoretic transport incorporating the effects of surface conduction on bulk ion-transport and provide a methodology to derive analytical solutions using the method of characteristics. Based on the analytical solutions, we elucidate the propagation of nonlinear concentration waves, such as shock and rarefaction waves, and provide the necessary and sufficient conditions for their existence. Our results show that the presence of surface conduction alters the propagation speed of nonlinear concentration waves and the composition of various zones. Importantly, we highlight the role of surface conduction in formation of additional shock and rarefaction waves which are otherwise not present in conventional electrophoresis.

Stability of electrophoretic transport of ions

We present an investigation of instability during electrophoretic transport of ions in a class of electrolytes called oscillating electrolytes. We analyze the onset of instability in electrophoretic transport in a binary electrolyte by modeling growth of small concentration disturbances over a base state with uniform acid and base concentrations. Our linear stability analysis shows that the growth rate of low wave-number concentration disturbances increases with an increase in wave number. Whereas, the growth rate of high wave-number disturbances decreases with increasing wave number due to the stabilizing effect of molecular diffusion. Our analysis also yields the scaling for growth rates and the wave number of most unstable mode with electric field. In addition, we show that the electrophoretic system exhibits instability only for a certain range of species concentrations. We also discuss the physical mechanism underlying the instability of transport process. We show that the instability is exhibited by those binary electrolytes that consist of a multivalent species with unusually high electrophoretic mobility in higher ionization states. Throughout, we provide verification of our linear stability analysis with full nonlinear simulations.

Effect of Joule heating on isoelectric focusing of proteins in a microchannel

Electric field-driven separation and purification techniques, such as isoelectric focusing (IEF) and isotachophoresis, generate heat in the system that can affect the performance of the separation process. In this study, a new mathematical model is presented for IEF that considers the temperature rise due to Joule heating. We used the model to study focusing phenomena and separation performance in a microchannel. A finite volume-based numerical technique is developed to study temperature-dependent IEF. Numerical simulation for narrow range IEF (6 < pH < 10) is performed in a straight microchannel for 100 ampholytes and two model proteins: staphylococcal nuclease and pancreatic ribonuclease. Separation results of the two proteins are obtained with and without considering the temperature rise due to Joule heating in the system for a nominal electric field of 100 V/cm. For the no Joule heating case, constant properties are used, while for the Joule heating case, temperature-dependent titration curves and thermo-physical properties are used. Our numerical results show that the temperature change due to Joule heating has a significant impact on the final focusing points of proteins, which can lower the separation performance considerably. In the absence of advection and any active cooling mechanism, the temperature increase is the highest at the mid-section of a microchannel. We also found that the maximum temperature in the system is a strong function of the [Formula: see text] value of the carrier ampholytes. Simulation results are also obtained for different values of applied electric fields in order to find the optimum working range considering the simulation time and buffer temperature. Moreover, the model is extended to study IEF in a straight microchip where pH is formed by supplying H(+) and OH(-), and the thermal analysis shows that the heat generation is negligible in ion supplied IEF.

Theory of ion transport with fast acid-base equilibrations in bioelectrochemical systems

Bioelectrochemical systems recover valuable components and energy in the form of hydrogen or electricity from aqueous organic streams. We derive a one-dimensional steady-state model for ion transport in a bioelectrochemical system, with the ions subject to diffusional and electrical forces. Since most of the ionic species can undergo acid-base reactions, ion transport is combined in our model with infinitely fast ion acid-base equilibrations. The model describes the current-induced ammonia evaporation and recovery at the cathode side of a bioelectrochemical system that runs on an organic stream containing ammonium ions. We identify that the rate of ammonia evaporation depends not only on the current but also on the flow rate of gas in the cathode chamber, the diffusion of ammonia from the cathode back into the anode chamber, through the ion exchange membrane placed in between, and the membrane charge density.

Electromigration dispersion in capillary electrophoresis

In a previous paper (Ghosal and Chen in Bull. Math. Biol. 72:2047, 2010), it was shown that the evolution of the solute concentration in capillary electrophoresis is described by a nonlinear wave equation that reduced to Burger's equation if the nonlinearity was weak. It was assumed that only strong electrolytes (fully dissociated) were present. In the present paper, it is shown that the same governing equation also describes the situation where the electrolytic buffer consists of a single weak acid (or base). A simple approximate formula is derived for the dimensionless peak variance which is shown to agree well with published experimental data.

10,000-fold concentration increase in proteins in a cascade microchip using anionic ITP by a 3-D numerical simulation with experimental results

This paper describes both the experimental application and 3-D numerical simulation of isotachophoresis (ITP) in a 3.2 cm long "cascade" poly(methyl methacrylate) (PMMA) microfluidic chip. The microchip includes 10 × reductions in both the width and depth of the microchannel, which decreases the overall cross-sectional area by a factor of 100 between the inlet (cathode) and outlet (anode). A 3-D numerical simulation of ITP is outlined and is a first example of an ITP simulation in three dimensions. The 3-D numerical simulation uses COMSOL Multiphysics v4.0a to concentrate two generic proteins and monitor protein migration through the microchannel. In performing an ITP simulation on this microchip platform, we observe an increase in concentration by over a factor of more than 10,000 due to the combination of ITP stacking and the reduction in cross-sectional area. Two fluorescent proteins, green fluorescent protein and R-phycoerythrin, were used to experimentally visualize ITP through the fabricated microfluidic chip. The initial concentration of each protein in the sample was 1.995 μg/mL and, after preconcentration by ITP, the final concentrations of the two fluorescent proteins were 32.57 ± 3.63 and 22.81 ± 4.61 mg/mL, respectively. Thus, experimentally the two fluorescent proteins were concentrated by over a factor of 10,000 and show good qualitative agreement with our simulation results.

Nonlinear waves in capillary electrophoresis

Electrophoretic separation of a mixture of chemical species is a fundamental technique of great usefulness in biology, health care, and forensics. In capillary electrophoresis, the sample migrates in a microcapillary in the presence of a background electrolyte. When the ionic concentration of the sample is sufficiently high, the signal is known to exhibit features reminiscent of nonlinear waves including sharp concentration "shocks." In this paper, we consider a simplified model consisting of a single sample ion and a background electrolyte consisting of a single coion and a counterion in the absence of any processes that might change the ionization states of the constituents. If the ionic diffusivities are assumed to be the same for all constituents the concentration of sample ion is shown to obey a one dimensional advection diffusion equation with a concentration dependent advection velocity. If the analyte concentration is sufficiently low in a suitable nondimensional sense, Burgers' equation is recovered, and thus the time dependent problem is exactly solvable with arbitrary initial conditions. In the case of small diffusivity, either a leading edge or trailing edge shock is formed depending on the electrophoretic mobility of the sample ion relative to the background ions. Analytical formulas are presented for the shape, width, and migration velocity of the sample peak and it is shown that axial dispersion at long times may be characterized by an effective diffusivity that is exactly calculated. These results are consistent with known observations from physical and numerical simulation experiments.

Simulation-based analysis of fluid flow and electrokinetic phenomena in microfluidic devices

Recent advances in microfabrication techniques, sensing methods, and miniaturization have enabled automated analysis of samples using microfluidic systems. Each unique application requires successful custom development of integrated lab-on-a-chip devices. This involves design, analysis and characterization of individual components, (pumps, valves, mixers, separators, sensors) and the integrated system. In this regard, first-principle-based simulations of the underlying complex multiphysics phenomena can provide detailed understanding of device function. An overview of modeling and simulation-based analysis for the design and development of microfluidic devices is presented. In particular, the authors highlight some key factors affecting the performance of lab-on-a-chip systems such as surface tension effects, analyte dispersion, Joule heating, and mass transport limitations, and delineate the parameters that influence them. The limitations of these modeling techniques and future needs are discussed.

Computer simulation of different modes of ACE based on the dynamic complexation model

Several modes of the often used ACE processes are simulated based on the principle of dynamic complexation of interacting species in a capillary column. The model is built on the mass transfer equation, to provide insight into the detailed analyte migration and interaction processes in CE. Normal ACE, Hummel-Dreyer method, vacancy affinity CE, vacancy peak method, and CE frontal analysis are simulated based on typical ACE conditions, and the results are compared with the detector responses of real CE processes using BSA and warfarin as a model system. Remarkable resemblance between the simulated results and the experimental observations was demonstrated for well-buffered ACE systems.