Analytical study of Joule heating effects on electrokinetic transportation in capillary electrophoresis

Electric fields are often used to transport fluids (by electroosmosis) and separate charged samples (by electrophoresis) in microfluidic devices. However, there exists inevitable Joule heating when electric currents are passing through electrolyte solutions. Joule heating not only increases the fluid temperature, but also produces temperature gradients in cross-stream and axial directions. These temperature effects make fluid properties non-uniform, and hence alter the applied electric potential field and the flow field. The mass species transport is also influenced. In this paper we develop an analytical model to study Joule heating effects on the transport of heat, electricity, momentum and mass species in capillary-based electrophoresis. Close-form formulae are derived for the temperature, applied electrical potential, velocity, and pressure fields at steady state, and the transient concentration field as well. Also available are the compact formulae for the electric current and the volume flow rate through the capillary. It is shown that, due to the thermal end effect, sharp temperature drops appear close to capillary ends, where sharp rises of electric field are required to meet the current continuity. In order to satisfy the mass continuity, pressure gradients have to be induced along the capillary. The resultant curved fluid velocity profile and the increase of molecular diffusion both contribute to the dispersion of samples. However, Joule heating effects enhance the sample transport velocity, reducing the analysis time in capillary electrophoretic separations.     

Band-broadening in capillary zone electrophoresis with axial temperature gradients

It is widely accepted that Joule heating effects yield radial temperature gradients in capillary zone electrophoresis (CZE). The resultant parabolic profile of electrophoretic velocity of analyte molecules is believed to increase the band-broadening via Taylor-Aris dispersion. This typically insignificant contribution, however, cannot explain the decrease in separation efficiency at high electric fields. We show that the additional band-broadening due to axial temperature gradients may provide the answer. These axial temperature variations result from the change of heat transfer condition along the capillary, which is often present in CZE with thermostating. In this case, the electric field becomes nonuniform due to the temperature dependence of fluid conductivity, and hence the induced pressure gradient is brought about to meet the mass continuity. This modification of the electroosmotic flow pattern can cause significant band-broadening. An analytical model is developed to predict the band-broadening in CZE with axial temperature gradients in terms of the theoretical plate height. We find that the resultant thermal plate height can be very high and even comparable to that due to molecular diffusion. This thermal plate height is much higher than that due to radial temperature gradients alone. The analytical model explains successfully the phenomena observed in previous experiments.     

Numerical analysis of the thermal effect on electroosmotic flow and electrokinetic mass transport in microchannels

Joule heating is present in electrokinetically driven flow and mass transport in microfluidic systems. Nowadays, there is a trend of replacing costly glass-based microfluidic systems by the disposable, cheap polymer-based microfluidic systems. Due to poor thermal conductivity of polymer materials, the thermal management of the polymer-based microfluidic systems may become a problem. In this study, numerical analysis is presented for transient temperature development due to Joule heating and its effect on the electroosmotic flow (EOF) and mass species transport in microchannels. The proposed model includes the coupling Poisson-Boltzmann (P-B) equation, the modified Navier-Stokes (N-S) equations, the conjugate energy equation, and the mass species transport equation. The results show that the time development for both the electroosmotic flow field and the Joule heating induced temperature field are less than 1 s. The Joule heating induced temperature field is strongly dependent on channel size, electrolyte concentration, and applied electric field strength. The simulations reveal that the presence of the Joule heating can result in significantly different characteristics of the electroosmotic flow and electrokinetic mass transport in microchannels.     

Electroosmotic flow with Joule heating effects

Electroosmotic flow with Joule heating effects was examined numerically and experimentally in this work. We used a fluorescence-based thermometry technique to measure the liquid temperature variation caused by Joule heating along a micro capillary. We used a caged-fluorescent dye-based microfluidic visualization technique to measure the electroosmotic velocity profile along the capillary. Sharp temperature drops close to the two ends and a high-temperature plateau in the middle of the capillary were observed. Correspondingly, concave-convex-concave velocity profiles were observed in the inlet-middle-outlet regions of a homogeneous capillary. These velocity perturbations were due to the induced pressure gradients resulting from axial variations of temperature. The measured liquid temperature distribution and the electroosmotic velocity profile along the capillary agree well with the predictions of a theoretical model developed in this paper.     

Estimation of Joule heating effect on temperature and pressure distribution in electrokinetic-driven microchannel flows

In this study we present simple analytical models that predict the temperature and pressure variations in electrokinetic-driven microchannel flow under the Joule heating effect. For temperature prediction, a simple model shows that the temperature is related to the Joule heating parameter, autothermal Joule heating parameter, external cooling parameter, Peclet number, and the channel length to channel hydraulic diameter ratio. The simple model overpredicted the thermally developed temperature compared with the full numerical simulation, but in good agreement with the experimental measurements. The factors that affect the external cooling parameters, such as the heat transfer coefficient, channel configuration, and channel material are also examined based on this simple model. Based on the mass conservation, a simple model is developed that predicts the pressure variations, including the temperature effect. An adverse pressure gradient is required to satisfy the mass conservation requirement. The temperature effect on the pressure gradient is via the temperature-dependent fluid viscosity and electroosmotic velocity.     

Numerical modeling of Joule heating-induced temperature gradient focusing in microfluidic channels

Temperature gradient focusing (TGF) is a recently developed technique for spatially focusing and separating ionic analytes in microchannels. The temperature gradient required for TGF can be generated either by an imposed temperature gradient or by Joule heating resulting from an applied electric field that also drives the flow. In this study, a comprehensive numerical model describing the Joule heating induced temperature development and TGF is developed. The model consists of a set of governing equations including the Poisson-Boltzmann equation, the Laplace equation, the Navier-Stokes equations, the energy equations and the mass transport equation. As the thermophysical and electrical properties including the liquid dielectric constant, viscosity, and electric conductivity are temperature-dependent, these governing equations are coupled, and therefore the coupled governing equations are solved numerically by using a CFD-based numerical method. The numerical simulations agree well with the experimental results, suggesting the valid mathematical model presented in this study.     

Influence of moderate Joule heating on electroosmotic flow velocity, retention, and efficiency in capillary electrochromatography

The influence of Joule heating on electroosmotic flow velocity, the retention factor of neutral analytes, and separation efficiency in capillary electrochromatography was investigated theoretically and experimentally. A plot of electrical current against the applied electrical field strength was used to evaluate the Joule heating effect. When the mobile phase concentration of Tris buffer exceeded 5.0 mM in the studied capillary electrochromatography systems using particulate and monolithic columns (with an accompanying power level of heat dissipation higher than 0.35 W/m), the Joule heating effect became clearly noticeable. Theoretical models for describing the variation of electroosmotic flow velocity with increasing applied field strength and the change of retention factors for neutral analytes with electrical field strength at higher Tris buffer concentrations were analyzed to explain consequences of Joule heating in capillary electrochromatography. Qualitative agreement between experimental data and implications of the theoretical model analysis was observed. The decrease of separation efficiency in capillary electrochromatography with macroporous octadecylsilica particles at high buffer concentration can be also attributed to Joule heating mainly via the increased axial diffusion of the analyte molecules and dispersion of solute bands by a nonuniform electroosmotic flow profile over the column cross-section. However, within a moderate temperature range, the contribution of the macroscopic velocity profile in the column arising from radial temperature gradients is insignificant.     

Towards high concentration enhancement of microfluidic temperature gradient focusing of sample solutes using combined AC and DC field induced Joule heating

It is challenging to continuously concentrate sample solutes in microfluidic channels. We present an improved electrokinetic technique for enhancing microfluidic temperature gradient focusing (TGF) of sample solutes using combined AC and DC field induced Joule heating effects. The introduction of an AC electric field component services dual functions: one is to produce Joule heat for generating temperature gradient; the other is to suppress electroosmotic flow. Consequently the required DC voltages for achieving sample concentration by Joule heating induced TGF are reduced, thereby leading to smaller electroosmotic flow (EOF) and thus backpressure effects. As a demonstration, the proposed technique can lead to concentration enhancement of sample solutes of more than 2500-fold, which is much higher than the existing literature reported microfluidic concentration enhancement by utilizing the Joule heating induced TGF technique.     

A simple method for assessment and minimization of errors in determination of electrophoretic or electroosmotic mobilities and velocities associated with the axial electric field distortion

It is commonly accepted that the modern CE instruments equipped with efficient cooling system enable accurate determination of electrophoretic or electroosmotic mobilities. It is also often assumed that velocity of migration in a given buffer is constant throughout the capillary length. It is simultaneously neglected that the noncooled parts of capillary produce extensive Joule heating leading to an axial electric field distortion, which contributes to a difference between the effective and nominal electric field potentials and between velocities in the cooled and noncooled parts of capillary. This simplification introduces systematic errors, which so far were however not investigated experimentally. There was also no method proposed for their elimination. We show a simple and fast method allowing for estimation and elimination of these errors that is based on combination of a long‐end and short‐end injections. We use it to study the effects caused by variation of temperature, electric field, capillary length, and pH.     

Theoretical and numerical analysis of temperature gradient focusing via Joule heating

We present a detailed theoretical and numerical analysis of temperature gradient focusing (TGF) via Joule heating-an analytical species concentration and separation technique relying upon the dependence of an analyte's velocity on temperature due to the temperature dependence of a buffer's ionic strength and viscosity. The governing transport equations are presented, analyzed, and implemented into a quasi-1D numerical model to predict the resulting temperature, velocity, and concentration profiles along a microchannel of varying width under an applied electric field. Numerical results show good agreement with experimental trials presented in previous work. The model is used to analyze the effects of varying certain geometrical and experimental parameters on the focusing performance of the device. Simulations also help depict the separation capability of the device, as well as the effectiveness of different buffer systems used in the technique. The analysis provides rule-of-thumb methodology for implementation of TGF into analytical systems, as well as a fundamental model applicable to any lab-on-a-chip system in which Joule heating and temperature-dependent electrokinetic transport are to be analyzed.     

Joule heating in electrokinetic flow

Electrokinetic flow is an efficient means to manipulate liquids and samples in lab-on-a-chip devices. It has a number of significant advantages over conventional pressure-driven flow. However, there exists inevitable Joule heating in electrokinetic flow, which is known to cause temperature variations in liquids and draw disturbances to electric, flow and concentration fields via temperature-dependent material properties. Therefore, both the throughput and the resolution of analytic studies performed in microfluidic devices are affected. This article reviews the recent progress on the topic of Joule heating and its effect in electrokinetic flow, particularly the theoretical and experimental accomplishments from the aspects of fluid mechanics and heat/mass transfer. The primary focus is placed on the temperature-induced flow variations and the accompanying phenomena at the whole channel or chip level.     

毛细管反相电色谱法分离行为的研究

对乙睛-水-磷酸二氢销体系毛细管反相电色谱分离行为进行了研究。采用柱上紫外检测,在75μmi.d.×30cm的毛细管ODS（3μm）填充柱上获得了小于2.0的折合培板高度。同时还研究了乙睛的比例、电解质的浓度和电场强度等因素对电渗流和往效的影响。     

Electrokinetic flow in a capillary with a charge-regulating surface polymer layer

An analytical study of the steady electrokinetic flow in a long uniform capillary tube or slit is presented. The inside wall of the capillary is covered by a layer of adsorbed or covalently bound charge-regulating polymer in equilibrium with the ambient electrolyte solution. In this solvent-permeable and ion-penetrable surface polyelectrolyte layer, ionogenic functional groups and frictional segments are assumed to distribute at uniform densities. The electrical potential and space charge density distributions in the cross section of the capillary are obtained by solving the linearized Poisson-Boltzmann equation. The fluid velocity profile due to the application of an electric field and a pressure gradient through the capillary is obtained from the analytical solution of a modified Navier-Stokes/Brinkman equation. Explicit formulas for the electroosmotic velocity, the average fluid velocity and electric current density on the cross section, and the streaming potential in the capillary are also derived. The results demonstrate that the direction of the electroosmotic flow and the magnitudes of the fluid velocity and electric current density are dominated by the fixed charge density inside the surface polymer layer, which is determined by the regulation characteristics such as the dissociation equilibrium constants of the ionogenic functional groups in the surface layer and the concentration of the potential-determining ions in the bulk solution.     

Electrochromatography and micro high-performance liquid chromatography with 320 μm I.D. packed columns

Electrochromatography is a chromatographic method in which the mobile phase (liquid or supercritical fluid) is “pumped” through a stationary phase in a microbore or capillary column by electroosmosis using an electric field. The technique permits separation of charged and uncharged compounds with higher resolution and superior efficiency when compared with micro-HPLC with an identical column. It is desirable to work with packed capillary columns with wide diameter in electrochromatography in order to improve detectability and column loadability. This study shows that we have moved a step forward towards this goal in spite of problems and difficulties, due to Joule heating, frit making and column packing in using wide-diameter columns. The paper demonstrates that the pressure pump of micro-HPLC with a commercially available 320 μm I.D. column can be replaced by the electroosmotic “pump” of capillary zone electrophoresis. Experiments were carried out in a chromatographic system under both electroosmosis and pressure-driven flow with 320 and 50 μm I.D. columns packed with 3- and 5-μm ODS. The advantage of electrochromatography over conventional micro-HPLC is shown.     

Joule heating effects on electrokinetic flows with conductivity gradients

Instability occurs in the electrokinetic flow of fluids with conductivity and/or permittivity gradients if the applied electric field is beyond a critical value. Understanding such an electrokinetic instability is significant for both improved transport (via the suppressed instability) and enhanced mixing (via the promoted instability) of liquid samples in microfluidic applications. This work presents the first study of Joule heating effects on electrokinetic microchannel flows with conductivity gradients using a combined experimental and numerical method. The experimentally observed flow patterns and measured critical electric fields under Joule heating effects to different extents are reasonably predicted by a depth-averaged numerical model. It is found that Joule heating increases the critical electric field for the onset of electrokinetic instability because the induced fluid temperature rise and in turn the fluid property change (primarily the decreased permittivity) lead to a smaller electric Rayleigh number.     

Rapid concentration of deoxyribonucleic acid via Joule heating induced temperature gradient focusing in poly-dimethylsiloxane microfluidic channel

This paper reports rapid microfluidic electrokinetic concentration of deoxyribonucleic acid (DNA) with the Joule heating induced temperature gradient focusing (TGF) by using our proposed combined AC and DC electric field technique. A peak of 480-fold concentration enhancement of DNA sample is achieved within 40 s in a simple poly-dimethylsiloxane (PDMS) microfluidic channel of a sudden expansion in cross-section. Compared to a sole DC field, the introduction of an AC field can reduce DC field induced back-pressure and produce sufficient Joule heating effects, resulting in higher concentration enhancement. Within such microfluidic channel structure, negative charged DNA analytes can be concentrated at a location where the DNA electrophoretic motion is balanced with the bulk flow driven by DC electroosmosis under an appropriate temperature gradient field. A numerical model accounting for a combined AC and DC field and back-pressure driven flow effects is developed to describe the complex Joule heating induced TGF processes. The experimental observation of DNA concentration phenomena can be explained by the numerical model.     

Diffusioosmosis of electrolyte solutions in a fine capillary tube

A theoretical study is presented for the steady diffusioosmotic flow of an electrolyte solution in a fine capillary tube generated by a constant concentration gradient imposed in the axial direction. The capillary wall may have either a constant surface potential or a constant surface charge density of an arbitrary quantity. The electric double layer adjacent to the charged wall may have an arbitrary thickness, and its electrostatic potential distribution is determined by an analytical approximation to the solution of the Poisson-Boltzmann equation. Solving a modified Navier-Stokes equation with the constraint of no net electric current arising from the cocurrent diffusion, electric migration, and diffusioosmotic convection of the electrolyte ions, the macroscopic electric field and the fluid velocity along the axial direction induced by the imposed electrolyte concentration gradient are obtained semianalytically as a function of the radial position in a self-consistent way. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential (or surface charge density) of the wall, the properties of the electrolyte solution, and other relevant factors. For a prescribed concentration gradient of an electrolyte, the magnitude of fluid velocity at a position in general increases with an increase in its distance from the capillary wall, but there are exceptions. The effect of the radial distribution of the induced tangential electric field and the relaxation effect due to ionic convection in the double layer on the diffusioosmotic flow are found to be very significant.     

Frequency-dependent laminar electroosmotic flow in a closed-end rectangular microchannel

This article presents an analysis of the frequency- and time-dependent electroosmotic flow in a closed-end rectangular microchannel. An exact solution to the modified Navier-Stokes equation governing the ac electroosmotic flow field is obtained by using the Green's function formulation in combination with a complex variable approach. An analytical expression for the induced backpressure gradient is derived. With the Debye-Hückel approximation, the electrical double-layer potential distribution in the channel is obtained by analytically solving the linearized two-dimensional Poisson-Boltzmann equation. Since the counterparts of the flow rate and the electrical current are shown to be linearly proportional to the applied electric field and the pressure gradient, Onsager's principle of reciprocity is demonstrated for transient and ac electroosmotic flows. The time evolution of the electroosmotic flow and the effect of a frequency-dependent ac electric field on the oscillating electroosmotic flow in a closed-end rectangular microchannel are examined. Specifically, the induced pressure gradient is analyzed under effects of the channel dimension and the frequency of electric field. In addition, based on the Stokes second problem, the solution of the slip velocity approximation is presented for comparison with the results obtained from the analytical scheme developed in this study.     

Electrokinetic flow and electric conduction of salt-free solutions in a capillary

The electrokinetic flow and accompanied electric conduction of a salt-free solution in the axial direction of a charged circular capillary are analyzed. No assumptions are made about the surface charge density (or surface potential) and electrokinetic radius of the capillary, which are interrelated. The Poisson–Boltzmann equation and modified Navier–Stokes equation are solved for the electrostatic potential distribution and fluid velocity profile, respectively. Closed-form formulas for the electroosmotic mobility and electric conductivity in the capillary are derived in terms of the surface charge density. The relative surface potential, electroosmotic mobility, and electric conductivity are monotonic increasing functions of the surface charge density and electrokinetic radius. However, the rises of the relative surface potential and electroosmotic mobility with an increase in the surface charge density are suppressed substantially when it is high due to the effect of counterion condensation. The analytical prediction that the electroosmotic mobility grows with increases in the surface charge density and electrokinetic radius agrees with the experimental results for salt-free solutions in circular microchannels in the literature.     

Joule heating effects on separation efficiency in capillary zone electrophoresis with an initial voltage ramp

An analytical model is developed to quantify the Joule heating effects on the separation efficiency in CZE with an initial voltage ramp. This model considers the temporal variations of nonuniform temperature and flow fields in the course of voltage ramping. The temperature dependence of electrical conductivity, dynamic viscosity, and mass density of the fluid is also taken into account. We demonstrate that the application of an initial voltage ramp delays the development of pressure-driven flows induced passively by the axial temperature gradients. The thermal dispersion is thus significantly reduced, resulting in a higher theoretical plate number in CZE. Such improvement in the separation efficiency is apparent in noncoated capillaries at high electric fields with an appropriate voltage ramp-up time. These predictions are consistent with previous observations in both aqueous and nonaqueous CZE that took place in uncoated capillaries. In coated capillaries where the EOF is suppressed, however, our model predicts a lower plate number in the presence of an initial voltage ramp.