Computer-aided simulation of electromigration

Simulation and Verification of the Direct Current Electric Field on Fabricating High Porosity f-MWCNTs Thin Films by Electrophoretic Deposition Technique

Electrophoretic deposition (EPD) is the potential process in high porosity thin films' fabrication or complex surface coating for perovskite photovoltaics. Here, the electrostatic simulation is introduced to optimize the EPD cell design for the cathodic EPD process based on functionalized multiwalled carbon nanotubes (f-MWCNTs). The similarity between the thin film structure and the electric field simulation is evaluated by scanning electron microscopy (SEM) and atomic force microscopy (AFM) results. The thin-film surface at the edge has a higher roughness (Ra) compared to the center position (16.48 > 10.26 nm). The f-MWCNTs at the edge position tend to be twisted and bent due to the torque of the electric field. The Raman results show that f-MWCNTs with low defect density are more easily to be positively charged and deposited on the ITO surface. The distribution of oxygen and aluminum atoms in the thin film reveals that the aluminum atoms tend to have adsorption/electrostatic attraction to the interlayer defect positions of f-MWCNTs without individually depositing onto the cathode. Finally, this study can reduce the cost and time for the scale-up process by optimizing the input parameters for the complete cathodic electrophoretic deposition process through electric field inspection.

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.

A Semiempirical Approach for a Rapid Comprehensive Evaluation of the Electrophoretic Behaviors of Small Molecules in Free Zone Electrophoresis

A phenomenological model is proposed for the evaluation of relative electrophoretic migration of charged substances present in mixtures and for the rapid pH optimization prior CZE method development. The simple and robust model is based on the Offord model that takes account of the chemical structure. The effective charge and the molecular mass of the molecule are needed; the charge can easily be calculated from pK a obtained from known sources or simulated with existing pK-calculation programs. A first example was chosen with the separation of hydroxy-s-triazines to illustrate the applicability of this simple approach for determination of the first buffer-pH conditions prior experimental method optimization when separation of different ions is needed. In a second example, the confirmation of aminoalcohols in the CZE method development of unsaturated hexahydro-triazines and oxasolidines.

Electromigration oscillations occurring in ternary electrolyte systems with complex eigenmobilities, as predicted by theory and ascertained by capillary electrophoresis

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 appear 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.

A semi-empirical approach for a rapid comprehensive evaluation of the electrophoretic behaviors of small molecules in free-zone electrophoresis

A phenomenological model is proposed for the evaluation of relative electrophoretic migration of charged substances present in mixtures and for the rapid pH optimization prior to capillary zone electrophoresis method development. The simple and robust model is based on the Offord model, which takes account of the chemical structure. The effective charge and the molecular mass of the molecule are needed; the charge can easily be calculated from pKa obtained from known sources or simulated with existing pK-calculation programs. A first example was chosen with the separation of hydroxy-s-triazines to illustrate the applicability of this simple approach for determination of the first buffer-pH conditions prior experimental method optimization when separation of different ions is needed. In a second example, the confirmation of aminialcohols in the CZE method development of unsaturated hexahydro-triazines and oxasolidines.

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.

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.

Kohlrausch regulating function and other conservation laws in electrophoresis

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.

Simul 5 – Free dynamic simulator of electrophoresis

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.

System zones in capillary zone electrophoresis

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.

Position and intensity of system (eigen) peaks in capillary zone electrophoresis

The intensity of system (or eigen) peaks encountered in capillary zone electrophoresis (CZE) can be predicted by considering mass balances for each of the analyte constituents and each of the constituents in the background electrolyte (BGE). As a result of coherence, in each zone the proportions in which the constituent concentrations vary are fixed; they are determined by the composition of the BGE and the nature of the analyte constituent (if present) and described as eigenvectors of a transport matrix. Considering the effect of an injection, the mass balances for all constituents can be satisfied only via the intensity of each zone. This leads to an n-equations, n-unknowns problem, with the intensities as the unknowns and the mass balances as equations. The latter can be easily solved to obtain the intensities. of the zones, of analytes as well as of system peaks. In this work the approach has been applied to CZE systems with two co-ions in the BGE, and experimental results have been compared to the predictions obtained from the model. Agreement was seen to be reasonable, but the quantitative comparison often failed, probably due to experimental difficulties.

Optimization of background electrolytes for capillary electrophoresis I. Mathematical and computational model

A mathematical and computational model is introduced for optimization of background electrolyte systems for capillary zone electrophoresis of anions. The model takes into account mono- or di- or trivalent ions and allows also for modeling of highly acidic or alkaline electrolytes, where a presence of hydrogen and hydroxide ions is significant. At maximum, the electrolyte can contain two co-anions and two counter-cations. The mathematical relations of the model are formulated to enable an easy algorithmization and programming in a computer language. The model assesses the composition of the background electrolyte in the analyte zone, which enables prediction of the parameters of the system that are experimentally available, like the transfer ratio, which is a measure of the sensitivity in the indirect photometric detection or the molar conductivity detection response, which expresses the sensitivity of the conductivity detection. Furthermore, the model also enables the evaluation of a tendency of the analyte to undergo electromigration dispersion and allows the optimization of the composition of the background electrolyte to reach a good sensitivity of detection while still having the dispersion properties in the acceptable range. Although the model presented is aimed towards the separation of anions, it can be straightforwardly rearranged to serve for simulation of electromigration of cationic analytes. The suitability of the model is checked by inspecting the behavior of a phosphate buffer for analysis of anions. It is shown that parameters of the phosphate buffer when used at neutral and alkaline pH values possess singularities that indicate a possible occurrence of system peaks. Moreover, if the mobility of any analyte of the sample is close to the mobilities of the system peaks, the indirect detector signals following the background electrolyte properties will be heavily amplified and distorted. When a specific detector sensitive on presence of the analyte were used, the signal would be almost lost due to the excessive dispersion of the peak.

Simulation of electrophoretic separations: effect of numerical and molecular diffusion on pH calculations in poorly buffered systems

A poorly buffered cationic isotachophoresis separation, first simulated by Reijenga and Kasica, has been revisited to demonstrate that an inconsistent description of solute and charge transport can lead to significant errors in the pH calculation. The separation is first simulated using a second-order finite difference scheme to show that omission of molecular diffusion from the charge balance results in a pH profile with spurious dips in the steady-state zone boundaries. The separation is also simulated using two first-order methods that employ numerical diffusion to stabilize solutions against spatiotemporal oscillations. Similar pH dips are generated by these first-order schemes, even when molecular diffusion is included in the charge balance, if numerical diffusion is not considered amongst the charge transport mechanisms. When numerical diffusion, inherent in the discretization of the component balances, is introduced to the charge balance, the spurious pH dips are eliminated. The results indicate that (i) pH dips originally reported by Reijenga and Kasicka are merely artifacts of their numerical model, and (ii) nonoscillatory numerical techniques, such as upwinding and flux limiters, should incorporate artificial transport mechanisms in the charge as well as the solute balances.

Discontinuities of pH at zone boundaries in isotachophoretic systems with poorly buffering leading electrolytes

Discontinuities of pH at zone boundaries of strong ion sample zones in isotachophoretic systems with poorly buffering leading electrolytes have been discovered by dynamic computer simulations of isotachophoretic separations with uncommon electrolyte systems. When using a salt solution containing a strong cation (anion) as leading ion and a weak anion (cation) as counterion for the isotachophoretic separation of strong cations (anions), severe pH discontinuities are present at zone boundaries of these cations (anions). The magnitude of these pH discontinuities has been investigated as a function of several parameters, namely counter ion/leading ion concentration ratio of the leading electrolyte, pK values, and mobilities of sample ions and counterion.

Isotachophoresis at pH extremes: theory and experimental validation

The evolution of an isotachophoresis (ITP) system in acidic or basic pH ranges can be quite different from that predicted by the existing theory. It was found theoretically and proved experimentally that the contribution of hydrogen or hydroxyl ion to conductivity of solution and/or its net charge changes the behavior of the ITP system, creating in the terminating electrolyte an additional zone close to the initial interfaces between electrolytes (leader and terminator). One boundary of the zone, being either sharp or dispersed, moves toward the leader; the other is always sharp and stationary and coincides with initial electrolytes' discontinuity. The latter can be registered in the presence of electroosmotic flow which delivers it to the detection point. In order to describe the dynamics of the ITP system at pH extremes an algorithm of analytical solution was developed, based on the revised Kohlrausch theory. Its predictions coincide well with computer simulations and experimental data. The results presented can help in a correct analysis of ITP data and explain some confusing phenomena which were considered to be artifacts.

Training software for electrophoresis

Computer programs, simulating electrophoretic separations, were evaluated and discussed with respect to their suitability for training purposes. Quite a number of them, mainly those dealing with steady-state phenomena, are sufficiently fast and user-friendly for the purpose of visualization of electrophoretic principles. Transient-state or dynamic models, however, are more suitable for the advanced user, mainly because of their inherent complexity and long calculation times.

Computer simulation of transient states in capillary zone electrophoresis and isotachophoresis

Transient states in the evolution of electrophoretic systems comprising aqueous solutions of weak monovalent acids and bases are simulated. The mathematical model is based on the system of nonstationary partial differential equations, expressing the mass and charge conservation laws while assuming local chemical equilibrium. It was implemented using a high resolution finite-difference algorithm, which correctly predicted the behavior of the concentration, pH and conductivity fields at low computational expense. Both the regular and the irregular modes of separation in capillary zone electrophoresis and isotachophoresis are considered. It is shown that the results of separation, particularly zone order, strongly depend on pH distribution. Simulation data as well as simple analytical assessments may help to predict and correctly interpret the experimental results.