A neural network model for rapid prediction of analyte focusing in isotachophoresis.

We present the development and demonstration of a neural network (NN) model for fast and accurate prediction of whether or not a chosen analyte is focused by an isotachophoresis (ITP) buffer system. The NN model is useful in the rapid evaluation of possible ITP chemistries applicable to analytes of interest. We trained and tested the NN model for univalent species based on extensive data sets of over 10,000 anionic and 10,000 cationic ITP simulations. The NN model uses as inputs the mobilities and the acid dissociation constants of leading electrolyte ion, trailing electrolyte ion, counterion, and a single analyte as well as the leading-to-counterion concentration ratio of the leading zone. The output then indicates whether the chosen electrolyte system yields stable ITP focusing of the analyte. The prediction accuracy of the NN model is over 97.7%. We demonstrate the applicability of the NN by validating its predictions with reported experimental data for anionic and cationic ITP. We have packaged the NN model in a free, web-based application named IONN (isotachophoresis on neural network), which can be used to rapidly screen ITP electrolyte systems.

Lattice Boltzmann finite-difference-based model for fully nonlinear electrohydrodynamic deformation of a liquid droplet.

Electric field-induced flows involving multiple fluid components with a range of different electrical properties are described by the coupled Taylor-Melcher leaky-dielectric model. We present a lattice Boltzmann (LB)-finite difference (FD) method-based hybrid framework to solve the complete Taylor-Melcher leaky-dielectric model considering the nonlinear surface charge convection effects. Unlike the existing LB-based models, we treat the interfacial discontinuities using direction-specific continuous gradients, which prevents the miscalculation arising due to volumetric gradients without directional derivatives, simultaneously maintaining the electroneutrality of the bulk. While fluid transport is recovered through the LB method using a multiple relaxation time (MRT) scheme, the FD method with a central difference scheme is applied to discretize the charge transport equation at the interface, in addition to the electric field governing equations in the bulk and at the interface. We apply the developed numerical model to study the different regimes of droplet deformation due to an external electric field. Similar to the existing analytical and other numerical models, excluding the surface charge convection (SCC) term from the charge transport equation, the present methodology has shown excellent agreement with the existing literature. In addition, the effect of SCC in each of the regimes is analyzed. With the present numerical model, we observe a strong presence of SCC in the oblate deformation regime, contrary to the weak effect on prolate deformations. We further discuss the reason behind such differences in the magnitude of nonlinearity induced by the SCC in all the regimes of deformation.

Physics of moderately stretched electrified jets in electrohydrodynamic jet printing.

Electrohydrodynamic (EHD) jet printing involves the deposition of a liquid jet issuing from a needle stretched under the effect of a strong electric field between the needle and a collector plate. Unlike the geometrically independent classical cone-jet observed at low flow rates and high applied electric fields, at a relatively high flow rate and moderate electric field, EHD jets are moderately stretched. Jetting characteristics of such moderately stretched EHD jets differ from the typical cone-jet due to the nonlocalized cone-to-jet transition. Hence, we describe the physics of the moderately stretched EHD jet applicable to the EHD jet printing process through numerical solutions of a quasi-one-dimensional model of the EHD jet and experiments. Through comparison with experimental measurements, we show that our simulations correctly predict the jet shape for varying flow rates and applied potential difference. We present the physical mechanism of inertia-dominated slender EHD jets based on the dominant driving and resisting forces and relevant dimensionless numbers. We show that the slender EHD jet stretches and accelerates primarily due to the balance of driving tangential electric shear and resisting inertia forces in the developed jet region, whereas in the vicinity of the needle, driving charge repulsion and resisting surface tension forces govern the cone shape. The findings of this study can help in operational understanding and better control of the EHD jet printing process.

Rapid prototyping of polydimethylsiloxane (PDMS) microchips using electrohydrodynamic jet printing: Application to electrokinetic assays.

Polydimethylsiloxane (PDMS) based microfluidic devices have found increasing utility for electrophoretic and electrokinetic assays because of their ease of fabrication using replica molding. However, the fabrication of high-resolution molds for replica molding still requires the resource-intensive and time-consuming photolithography process, which precludes quick design iterations and device optimization. We here demonstrate a low-cost, rapid microfabrication process, based on electrohydrodynamic jet printing (EJP), for fabricating non-sacrificial master molds for replica molding of PDMS microfluidic devices. The method is based on the precise deposition of an electrically stretched polymeric solution of polycaprolactone in acetic acid on a silicon wafer placed on a computer-controlled motion stage. This process offers the high-resolution (order 10  µ $\umu$ m) capability of photolithography and rapid prototyping capability of inkjet printing to print high-resolution templates for elastomeric microfluidic devices within a few minutes. Through proper selection of the operating parameters such as solution flow rate, applied electric field, and stage speed, we demonstrate microfabrication of intricate master molds and corresponding PDMS microfluidic devices for electrokinetic applications. We demonstrate the utility of the fabricated PDMS microchips for nonlinear electrokinetic processes such as electrokinetic instability and controlled sample splitting in ITP. The ability to rapid prototype customized reusable master molds with order 10  µ $\umu$ m resolution within a few minutes can help in designing and optimizing microfluidic devices for various electrokinetic applications.

A simplified model of oscillating electrolytes.

Unstable electrophoretic transport leading to oscillations in concentration profiles occur in certain electrolyte systems known as oscillating electrolytes whose eigenmobilities are complex valued. The study of the nonlinear behavior of such systems is of great interest but is constrained due to a high degree of complexity in the governing equations. Here we present a simplified model of unstable electrophoretic transport in a binary system that reduces the governing equations to two partial differential equations only and does away with other equations that characterize acid-base dissociation reactions and electroneutrality. We present analytical expressions for electromigration fluxes and validate the model with full nonlinear simulations. The model exhibits similar nonlinear behavior as the actual unstable electrophoretic system under various initial disturbances. For comparison, we also show that similar modeling for a stable system predicts concentration profiles that quantitatively agree with its nonoscillating dynamics. Moreover, the unique feature of electromigration flux in oscillating electrolytes that unfolds from the modeling led us to find an elegant explanation of the instability mechanism. Our theory gives a qualitative understanding of the existence and growth of large oscillation patterns in oscillating electrolytes.

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$^{2}$ . 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.

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.

An inexpensive microfluidic device for three-dimensional hydrodynamic focusing in imaging flow cytometry.

We present design, characterization, and testing of an inexpensive, sheath-flow based microfluidic device for three-dimensional (3D) hydrodynamic focusing of cells in imaging flow cytometry. In contrast to other 3D sheathing devices, our device hydrodynamically focuses the cells in a single-file near the bottom wall of the microchannel that allows imaging cells with high magnification and low working distance objectives, without the need for small device dimensions. The relatively large dimensions of the microchannels enable easy fabrication using less-precise fabrication techniques, and the simplicity of the device design avoids the need for tedious alignment of various layers. We have characterized the performance of the device with 3D numerical simulations and validated these simulations with experiments of hydrodynamic focusing of a fluorescently dyed sample fluid. The simulations show that the width and the height of the 3D focused sample stream can be controlled independently by varying the heights of main and side channels of the device, and the flow rates of sample and sheath fluids. Based on simulations, we also provide useful guidelines for choosing the device dimensions and flow rates for focusing cells of a particular size. Thereafter, we demonstrate the applicability of our device for imaging a large number of RBCs using brightfield microscopy. We also discuss the choice of the region of interest and camera frame rate so as to image each cell individually in our device. The design of our microfluidic device makes it equally applicable for imaging cells of different sizes using various other imaging techniques such as phase-contrast and fluorescence microscopy.

A fully coupled hybrid lattice Boltzmann and finite difference method-based study of transient electrokinetic flows.

Transient electrokinetic (EK) flows involve the transport of conductivity gradients developed as a result of mixing of ionic species in the fluid, which in turn is affected by the electric field applied across the channel. The presence of three different coupled equations with corresponding different time scales makes it difficult to model the problem using the lattice Boltzmann method (LBM). The present work aims to develop a hybrid LBM and finite difference method (FDM)-based model which can be used to study the electro-osmotic flows (EOFs) and the onset of EK instabilities using an Ohmic model, where fluid and conductivity transport are solved using LBM and the electric field is solved using FDM. The model developed will be used to simulate three different problems: (i) EOF with varying zeta-potential on the wall, (ii) similitude in EOF, and (iii) EK instabilities due to the presence of conductivity gradients. Problems (i) and (ii) will be compared with the analytical results and problem (iii) will be compared with the simulations of a spectral method-based numerical model. The results obtained from the present simulations will show that the developed model is capable of studying transient EK flows and of predicting the onset of instability.

Effect of surface conduction-induced electromigration on current monitoring method for electroosmotic flow measurement.

Current monitoring method for measurement of EOF in microchannels involves measurement of time-varying current while an electrolyte displaces another electrolyte having different conductivity due to EOF. The basic premise of the current monitoring method is that an axial gradient in conductivity of a binary electrolyte in a microchannel advects only due to EOF. In the current work, using theory and experiments, we show that this assumption is not valid for low concentration electrolytes and narrow microchannels wherein surface conduction is comparable with bulk conduction. We show that in presence of surface conduction, a gradient in conductivity of binary electrolyte not only advects with EOF but also undergoes electromigration. This electromigration phenomenon is nonlinear and is characterized by propagation of shock and rarefaction waves in ion concentrations. Consequently, in presence of surface conduction, the current-time relationships for forward and reverse displacement in the current monitoring method are asymmetric and the displacement time is also direction dependent. To quantify the effect of surface conduction, we present analytical expressions for current-time relationship in the regime when surface conduction is comparable to bulk conduction. We validate these relations with experimental data by performing a series of current monitoring experiments in a glass microfluidic chip at low electrolyte concentrations. The experimentally validated analytical expressions for current-time relationships presented in this work can be used to correctly estimate EOF using the current monitoring method when surface conduction is not negligible.

Scaling behavior in on-chip field-amplified sample stacking.

Field amplified sample stacking (FASS) uses differential electrophoretic velocity of analyte ions in the high-conductivity background electrolyte zone and low conductivity sample zone for increasing the analyte concentration. The stacking rate of analyte ions in FASS is limited by molecular diffusion and convective dispersion due to nonuniform electroosmotic flow (EOF). We present a theoretical scaling analysis of stacking dynamics in FASS and its validation with a large set of on-chip sample stacking experiments and numerical simulations. Through scaling analysis, we have identified two stacking regimes that are relevant for on-chip FASS, depending upon whether the broadening of the stacked peak is dominated by axial diffusion or convective dispersion. We show that these two regimes are characterized by distinct length and time scales, based on which we obtain simplified nondimensional relations for the temporal growth of peak concentration and width in FASS. We first verify the theoretical scaling behavior in diffusion- and convection-dominated regimes using numerical simulations. Thereafter, we show that the experimental data of temporal growth of peak concentration and width at varying electric fields, conductivity gradients, and EOF exhibit the theoretically predicted scaling behavior. The scaling behavior described in this work provides insights into the effect of varying experimental parameters, such as electric field, conductivity gradient, electroosmotic mobility, and electrophoretic mobility of the analyte on the dynamics of on-chip FASS.

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.

Mechanism of electrohydrodynamic instability with collinear conductivity gradient and electric field.

We describe the physical mechanism responsible for electrohydrodynamic (EHD) instability of a fluid layer with collinear conductivity gradient and electric field. In particular, we resolve the ambiguity in literature regarding the cause for switching between stationary and oscillatory modes of EHD instability. Using linear stability analysis, we show that a small perturbation in conductivity field perturbs the local electric field and also induces a perturbation charge. The coupling of base-state electric field with the perturbation charge leads to a force which causes overstability. Whereas, the coupling of base-state free charge with perturbation electric field leads to a force which causes EHD instability via a stationary mode. The proposed mechanism correctly explains the existence of stationary and oscillatory modes for varying conductivity gradients and wave number of disturbances, depending upon the relative magnitude of these two forces.

Capillary Displacement of Viscous Liquids in Geometries with Axial Variations.

Axial variations in geometry and presence of viscous displaced fluid are known to alter the diffusive-dynamics of capillary imbibition of a wetting liquid. We here show that the coupled effect of axially varying capillary geometry and finite viscosity of the displaced fluid can lead to significant variations in both short and long time dynamics of imbibition. Based on a theoretical model and lattice Boltzmann simulations, we analyze capillary displacement of a viscous liquid in straight and diverging capillaries. At short times, the imbibition length scales proportionally with time as opposed to the diffusive-dynamics of imbibition of a single wetting liquid. Whereas, at long times, geometry-dependent power-law behavior occurs which qualitatively resembles single liquid imbibition. The distance at which the crossover between these two regimes occurs depends strongly on the viscosities of the imbibing and the displaced liquid. Additionally, our simulations show that the early time imbibition dynamics are also affected by the dynamic contact angle of the meniscus.

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.

Uncertainty quantification in modeling of microfluidic T-sensor based diffusion immunoassay.

Comparison of experimental data with modeling predictions is essential for making quantitative measurements of species properties, such as diffusion coefficients and species concentrations using a T-sensor. To make valid comparisons between experimental data and model predictions, it is necessary to account for uncertainty in model predictions due to uncertain values of model parameters. We present an analysis of uncertainty induced in model predictions of a T-sensor based competitive diffusion immunoassay due to uncertainty in diffusion constants, binding reaction rate constants, and inlet flow speed. We use a non-intrusive stochastic uncertainty quantification method employing polynomial chaos expansions to represent the dependence of uncertain species concentrations on the uncertainty in model parameters. Our simulations show that the uncertainties in model parameters lead to significant spatially varying uncertainty in predicted concentration. In particular, the diffusivity of fluorescently labeled probe antigen dominates the overall uncertainty. The predicted uncertainty in fluorescence intensity is minimum near the centerline of T-sensor and relatively high in the regions with gradients in fluorescence intensity. We show that using centerline fluorescence intensity instead of first derivative of fluorescence intensity as the system response for measuring sample antigen concentration in T-sensor based competitive diffusion immunoassay leads to lower uncertainty and higher detection sensitivity.

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.