Role of Joule heating in dispersive mixing effects in electrophoretic cells: convective-diffusive transport aspects

This contribution addresses the problem of solute dispersion in a free convection electrophoretic cell for the batch mode of operation, caused by the Joule heating generation. The problem is analyzed by using the two-problem approach originally proposed by Bosse and Arce (Electrophoresis 2000, 21, 1018-1025). The approach identifies the carrier fluid problem and the solute problem. This contribution is focused on the latter. The strategy uses a sequential coupling between the energy, momentum and mass conservation equations and, based on geometrical and physical assumptions for the system, leads to the derivation of analytical temperature and velocity profiles inside the cell. These results are subsequently used in the derivation of the effective dispersion coefficient for the cell by using the method of area averaging. The result shows the first design equation that relates the Joule heating effect directly to the solute dispersion in the cell. Some illustrative results are presented and discussed and their implication to the operation and design of the device is addressed. Due to the assumptions made, the equation may be viewed as an upper boundary for applications such as free flow electrophoresis.     

Batch electrophoretic cells with Eyring fluids: analysis of the hydrodynamics

Mixing and dispersion phenomena caused by the carrier fluid in an electrophoretic cell is the main subject of this study. In particular, the effects of Joule heating on temperature and velocity profiles for Eyring-model fluids (EMF) are studied. The heat transfer is sequentially coupled with momentum transfer to derive an analytical expression for both the temperature and the velocity profiles. These results are then used to show the hydrodynamic behavior of the fluid in a batch electrophoretic cell. Furthermore, the results obtained are useful to compare with the fluid behavior of other carriers of different rheology, such as Newtonian fluids, power-law fluids, and viscoelastic fluids that obey the CEF model. The results show that EMF are potentially good carriers for relatively high Joule heat generation and therefore good candidates to control mixing inside the electrophoretic cell.     

Dispersive mixing in a batch electrophoretic cell with Eyring fluids

The main objective of this study is analysis of dispersive mixing inside a batch electrophoretic cell due to Joule heating, especially for the case of non-Newtonian carriers. To this end, a carrier fluid that follows the Eyring rheological model is used in the analysis of the species convective-diffusive equation that describes the solute motion inside the device. The hydrodynamic problem (Bosse, M. A. et al., Electrophoresis 2002, 23, 2149-2156) of the electrophoretic cell is sequentially coupled to this equation. Then, by following a procedure based on the area-averaging method, an effective diffusion coefficient is obtained. This equation is the first a priori design equation for devices such as the ones analyzed in this contribution. It is useful in determining mixing conditions for the values of the relevant parameters of the physical system. The results of this analysis are used to study the cell behavior under several conditions imposed by their main parameters. Finally, some suggestions are offered about the use of Eyring fluids as potential carriers useful for controlling dispersive mixing in batch electrophoretic cells.     

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.     

Joule heating induced transient temperature field and its effects on electroosmosis in a microcapillary packed with microspheres

The Joule heating induced transient temperature field and its effect on the electroosmotic flow in a capillary packed with microspheres is analyzed numerically using the control-volume-based finite difference method. The model incorporates the coupled momentum equation for the electroosmotic velocity, the energy equations for the Joule heating induced temperature distributions in both the packed column and the capillary wall, and the mass and electric current continuity equations. The temperature-dependent physical properties of the electrolyte solution are taken into consideration. The characteristics of the Joule heating induced transient development of temperature and electroosmotic flow fields are studied. Specifically, the simulation shows that the presence of Joule heating causes a noticeable axial temperature gradient in the thermal entrance region and elevates a significant temperature increment inside the microcapillary. The temperature changes in turn greatly affect the electroosmotic velocity by means of the temperature-dependent fluid viscosity, dielectric constant, and local electric field strength. Furthermore, the model predicts an induced pressure gradient to counterbalance the axial variation of the electroosmotic velocity so as to maintain the fluid mass continuity. In addition, under specific conditions, the present model is validated by comparing with the existing analytical model and experimental data from the literature.     

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.     

Assessment of Joule heating and its effects on electroosmotic flow and electrophoretic transport of solutes in microfluidic channels

Joule heating is inevitable when an electric field is applied across a conducting medium. It would impose limitations on the performance of electrokinetic microfluidic devices. This article presents a 3-D mathematical model for Joule heating and its effects on the EOF and electrophoretic transport of solutes in microfluidic channels. The governing equations were numerically solved using the finite-volume method. Experiments were carried out to investigate the Joule heating associated phenomena and to verify the numerical models. A rhodamine B-based thermometry technique was employed to measure the solution temperature distributions in microfluidic channels. The microparticle image velocimetry technique was used to measure the velocity profiles of EOF under the influence of Joule heating. The numerical solutions were compared with experimental results, and reasonable agreement was found. It is found that with the presence of Joule heating, the EOF velocity deviates from its normal "plug-like" profile. The numerical simulations show that Joule heating not only accelerates the sample transport but also distorts the shape of the sample band.     

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.     

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.     

Effect of the Joule heating and of the material voids on free-convective transport in porous or fibrous media with applied electrical fields

The effect of the geometry of the soil in electrokinetic application has been studied by using capillary models of annular geometry. The Joule heating generation has been included as a primary effect of temperature development leading to buoyancy flows. The heat transfer model has been formulated for conduction-dominated regime. The results of this model have been coupled with the motion equation to obtain the analytical hydrodynamic velocity profile. Numerical illustrations, demonstrating the effect of the cross-sectional area of the annular region on the velocity field, have been included. It is observed that a substantial effect on the magnitude of such velocity field for different parameters of the system. The results are useful to obtain better understanding of the role of the soil geometry in potential soil cleaning field operations.     

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.     

Numerical modeling of the Joule heating effect on electrokinetic flow focusing

In electrokinetically driven microfluidic systems, the driving voltage applied during operation tends to induce a Joule heating effect in the buffer solution. This heat source alters the solution's characteristics and changes both the electrical potential field and the velocity field during the transport process. This study performs a series of numerical simulations to investigate the Joule heating effect and analyzes its influence on the electrokinetic focusing performance. The results indicate that the Joule heating effect causes the diffusion coefficient of the sample to increase, the potential distribution to change, and the flow velocity field to adopt a nonuniform profile. These variations are particularly pronounced under tighter focusing conditions and at higher applied electrical intensities. In numerical investigations, it is found that the focused bandwidth broadens because thermal diffusion effect is enhanced by Joule heating. The variation in the potential distribution induces a nonuniform flow field and causes the focused bandwidth to tighten and broaden alternately as a result of the convex and concave velocity flow profiles, respectively. The present results confirm that the Joule heating effect exerts a considerable influence on the electrokinetic focusing ratio.     

Joule heating monitoring in a microfluidic channel by observing the Brownian motion of an optically trapped microsphere

Electric fields offer a variety of functionalities to Lab‐on‐a‐Chip devices. The use of these fields often results in significant Joule heating, affecting the overall performance of the system. Precise knowledge of the temperature profile inside a microfluidic device is necessary to evaluate the implications of heat dissipation. This article demonstrates how an optically trapped microsphere can be used as a temperature probe to monitor Joule heating in these devices. The Brownian motion of the bead at room temperature is compared with the motion when power is dissipated in the system. This gives an estimate of the temperature increase at a specific location in a microfluidic channel. We demonstrate this method with solutions of different ionic strengths, and establish a precision of 0.9 K and an accuracy of 15%. Furthermore, it is demonstrated that transient heating processes can be monitored with this technique, albeit with a limited time resolution.     

Light‐induced self‐thermophoresis of Janus spheroidal nanoparticles

A theoretical framework is provided for determining the self‐thermophoretic velocity of a light irradiated spheroidal Janus nanoparticle consisting of symmetric dielectric and perfectly conducting semi‐spheroids. The analysis is based on solving the linearized Joule heating problem due to uniform laser irradiance and on explicitly determining the temperature fields inside and outside the particle. We employ the thermoelectric (Peltier‐ Seebeck) methodology to find the surface self ‐ induced temperature gradient and the related slip velocity which determines the autonomous phoretic (self ‐ propulsion) mobility of the Janus particle. Simplified explicit expressions for the self ‐ thermophoretic velocities of spheroidal (prolate and oblate) Janus particles in terms of their aspect ratios are found and few practical limiting cases (i.e., sphere, disk and needle) are also discussed.     

Joule heating and determination of temperature in capillary electrophoresis and capillary electrochromatography columns

This article reviews the progress that has taken place in the past decade on the topic of estimation of Joule heating and temperature inside an open or packed capillary in electro-driven separation techniques of capillary electrophoresis (CE) and capillary electrochromatography (CEC), respectively. Developments in theoretical modeling of the heat transfer in the capillary systems have focused on attempts to apply the existing models on newer techniques such as CEC and chip-based CE. However, the advent of novel analytical tools such as pulsed magnetic field gradient nuclear magnetic resonance (NMR), NMR thermometry, and Raman spectroscopy, have led to a revolution in the area of experimental estimation of Joule heating and temperature inside the capillary via the various noninvasive techniques. This review attempts to capture the major findings that have been reported in the past decade.     

Infrared-based temperature measurements in capillary electrophoresis

In capillary electrophoresis (CE), the temperature inside the capillary is one of the most important parameters. In a concept for Analytical Instrument Qualification (AIQ) of CE systems, the temperature accuracy and stability have to be included. This fact requires an accurate look at the measurement of temperature which is generated by the applied electrical power. The generation of Joule heating is measured on the outside of the capillary using an infrared (IR) thermometer. The thermometer linearity is demonstrated over a wide range of various electrical field strength, buffer systems, and different capillary inner diameters. A slope of 6.3 °C m/W was found for the optimal thermometer capillary distance of 8 mm. Furthermore, the temperature measurements are highly precise, depending almost solely on the current variability. The proposed method is compared with three methods calculating the temperature from the conductance, the electroosmotic velocity, or the current. These indirect methods estimate slopes ranging from 7 to 10 °C m/W. In addition, the maximal suitable electrical power per unit length is estimated. Joule heating can often be tolerated up to 4 W/m. However, sensitive analytes can already be affected by using more than 1 W/m. In conclusion, the consideration of the temperature is essential for not only Analytical Instrument Qualification, but also certainly useful for method optimisation and method transfer.     

Analytical and numerical study of Joule heating effects on electrokinetically pumped continuous flow PCR chips

Joule heating is an inevitable phenomenon for microfluidic chips involving electrokinetic pumping, and it becomes a more important issue when chips are made of polymeric materials because of their low thermal conductivities. Therefore, it is very important to develop methods for evaluating Joule heating effects in microfluidic chips in a relatively easy manner. To this end, two analytical models have been established and solved using the Green's function for evaluating Joule heating effects on the temperature distribution in a microfluidic-based PCR chip. The first simplified model focuses on the understanding of Joule heating effects by ignoring the influences of the boundary conditions. The second model aims to consider practical experimental conditions. The analytical solutions to the two models are particularly useful in providing guidance for microfluidic chip design and operation prior to expensive chip fabrication and characterization. To validate the analytical solutions, a 3-D numerical model has also been developed and the simultaneous solution to this model allows the temperature distribution in a microfluidic PCR chip to be obtained, which is used to compare with the analytical results. The developed numerical model has been applied for parametric studies of Joule heating effects on the temperature control of microfluidic chips.     

A model for Joule heating-induced dispersion in microchip electrophoresis

This paper presents an analytical and parameterized model for analyzing the effects of Joule heating on analyte dispersion in electrophoretic separation microchannels. We first obtain non-uniform temperature distributions in the channel resulting from Joule heating, and then determine variations in electrophoretic velocity, based on the fact that the analyte's electrophoretic mobility depends on the buffer viscosity and hence temperature. The convection-diffusion equation is then formulated and solved in terms of spatial moments of the analyte concentration. The resulting model is validated by both numerical simulations and experimental data, and holds for all mass transfer regimes, including unsteady dispersion processes that commonly occur in microchip electrophoresis. This model, which is given in terms of analytical expressions and fully parameterized with channel dimensions and material properties, applies to dispersion of analyte bands of general initial shape in straight and constant-radius-turn channels. As such, the model can be used to represent analyte dispersion in microchannels of more general shape, such as serpentine- or spiral-shaped channels.