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Electrokinetic instability in microchannel flows: A review
Mechanics Research Communications 36 (2009) 33–38
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Electrokinetic instability in microchannel flows: A review Hao Lin* Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, 98 Brett Road, Piscataway, NJ 08854, United States
a r t i c l e
i n f o
Article history: Received 24 April 2008 Received in revised form 5 July 2008 Available online 12 August 2008
Keywords: Electrokinetic instability Electrohydrodynamics Microfluidics Mixing
a b s t r a c t Electrokinetic instability is a microscale instability observed during the development of electrokinetic microfluidic applications, and is induced by an interaction between the electric field and fluid motion. In this brief review, the basic mechanism as well as the various aspects of this instability are summarized. These include the effects of field alignment, electroosmotic velocity (convective and absolute instability), channel dimension, periodic forcing, and multiple-species. A brief discussion on the applications of the instability is also presented. Ó 2008 Published by Elsevier Ltd.
1. Introduction During the past decade, electrokinetic (EK) techniques received renewed interest due to extensive research into the design and engineering of micro- and nano-fluidic devices. Many of these applications employ electric fields to realize novel functionalities, and to enable rapid and miniaturized chemical and biochemical analysis and detection (Reyes et al., 2002). Meanwhile, complex flow phenomena and regimes also emerge from these applications which challenge our fundamental understanding and prediction capability (Squires and Quake, 2005). One such example is electrokinetic instability (EKI), a microscale instability resulting from the coupling of the electric field, ion-electromigration, and fluid motion. In this topical review, we briefly discuss the history, basic theory, instability mechanism and behavior, and applications of this multi-physical phenomenon. EKI in the microfluidic context was first observed in various laboratories when pursuing electrokinetic applications such as sample loading and injection, and field-amplified sample stacking (Dang et al., 2003; Ramsey, 2001; Shultz-Lockyear et al., 1999). Meanwhile, various groups also explored the possibility of using this instability for efficient and non-mechanical micro-mixing (El Moctar et al., 2003; Oddy et al., 2001; Tsouris et al., 2003). Although the instability mechanism was not fully understood, a more systematic investigation by Chen and Santiago (2002) established the critical importance of conductivity gradient and high electric field strength in inducing the instability. Further studies by Santiago and co-authors (Chen et al., 2005; Lin et al., 2004) related the instability physics with earlier works on electrohydrodynamic (EHD) instability by Hoburg and Melcher (1976), Hoburg and Melcher (1977) and Baygents and Baldessari (1998). Combining linear analysis and nonlinear numerical simulations, these studies provided a comprehensive model framework for the understanding and prediction of EKI, and the results compared favorably with experimental measurements. A body of work followed to systematically investigate the instability in greater details, including those on the effects of field alignment (Storey, 2005), electroosmotic convection (convective and absolute instability) (Chen et al., 2005; Posner and Santiago, 2006), channel dimension (Chen et al., 2005; Lin et al., 2008; Storey, 2005; Storey et al., 2005), time-periodic forcing (Boy and Storey, 2007; Shin et al.,
* Tel.: +1 732 445 2322; fax: +1 732 445 3124. E-mail address: [email protected] 0093-6413/$ - see front matter Ó 2008 Published by Elsevier Ltd. doi:10.1016/j.mechrescom.2008.07.012
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2005), and multiple-species (Oddy and Santiago, 2005); and those implemented EKI in various designs for mixing applications (Huang et al., 2006; Pan et al., 2007; Park et al., 2005; Shin et al., 2005; Tai et al., 2006). A discussion on two important features helps us relate EKI to proper context and other similar flow phenomena. The first is that EKI is induced by the presence of an electrical conductivity gradient, which under an applied field results in charge separation and an electrical body force (the Maxwell stress). In this regard EKI can be viewed as a special type of EHD flow, the modern theoretical foundation of which was laid by Melcher and Taylor (1969). However, different than the original leakydielectric model, in EKI charge separation occurs in the bulk of a single miscible fluid, not at the interface of two immiscible fluids. Consequently, and as correctly first pointed out by Baygents and Baldessari (1998), ion diffusion in EKI results in a conditional instability. EHD instability of two (or more) immiscible fluids in microfluidic devices is another subject extensively studied by Li et al. (2007), Ozen et al. (2006), and Zahn and Reddy (2006), among others, and is not discussed here. Another feature of EKI is that it is driven by an electrostatic force in the bulk of the liquid away from charged solid–liquid interfaces (and hence away from the electric double layers). This feature distinguishes EKI from other types of instabilities and/or mixing phenomena in electrokinetic flows, such as those induced by modulated electroosmotic flows (Sundaram and Tafti, 2004). In this latter case electrostatic forcing is regarded as absent away from the (often relatively thin) electric double layers. 2. Theoretical formulation The Binary electrolyte model. The basic behavior of EKI can be described with a simplified binary electrolyte (BE) model (Baygents and Baldessari, 1998; Chen et al., 2005; Hoburg and Melcher, 1976; Lin et al., 2004). The governing equations consist of the convective-diffusive equation for the electrical conductivity, the Ohmic current conservation equation, and the Navier–Stokes equations:
or Dþ D ðzþ þ z Þ ; þ u rr ¼ Deff rr; Deff ¼ D þ zþ þ D z ot r rr/ ¼ 0; ou q þ u ru ¼ rp þ lr2 u þ r2 /r/; r u ¼ 0: ot
ð1Þ ð2Þ ð3Þ
Here r is the electrical conductivity, Deff is an effective diffusivity, D and z are the molecular diffusivity and valence number, and the subscripts ‘‘+” and ‘‘” denote the cation and the anion in the BE system, respectively. / is the electric potential, and q, u, p, l and are the fluid density, velocity, pressure, viscosity, and permittivity, respectively. Eq. (1) is derived from the Nernst–Planck equation for ion-electromigration assuming a BE system and that the electroneutrality condition holds (Levich, 1962). The latter is a valid assumption when the concentration mismatch between the cation and anion is small compared with the background ion concentration. The Ohmic Eq. (2) implies that charge relaxation is instantaneous (Hoburg and Melcher, 1976). In arriving at this equation the diffusive current (due to the diffusive transport of charged species) has also been ignored (Oddy and Santiago, 2005). The last term in the momentum Eq. (3), r2 /r/, is the electrical body force derived from the Maxwell stress tensor. During the derivation we have already assumed the fluid permittivity, , to be a constant. Note that Eqs. (1)–(3) can be regarded as a special case of a more generalized multiple-species (MS) model proposed by Oddy and Santiago (2005). Although the BE model is capable of capturing the basic mechanism of EKI, and is quantitatively accurate when the electrolyte system is indeed binary, it does not include some of the subtle yet important effects when more than two species with differing properties are involved. Such effects are discussed below. Non-dimensionalization. The governing Eqs. (1)–(3) are non-dimensionalized using the following scales:
½x ¼ L; ½t ¼ sev ¼ L=U ev ; ½r ¼ r0 ; ½/ ¼ E0 L; ½p ¼ lU ev =L; U ev E20 L=l:
ð4Þ
Here L is a length scale, r0 is a characteristic conductivity, and E0 is taken to be the applied field strength. The electroviscous velocity, U ev , is derived by setting equal the viscous and electrical body forces in the momentum equation (Hoburg and Melcher, 1976). The resulting equations read
or 1 r 2 r; þ u rr ¼ Rae ot r rr/ ¼ 0; ou Re þ u ru ¼ rp þ r2 u þ r2 /r/; ot
ð5Þ ð6Þ
r u ¼ 0:
ð7Þ
Here the electric Rayleigh number and the Reynolds number are defined as Rae U ev L=Deff , and Re qU ev L=l, respectively. In EK systems and in the thin electric double layer limit, Eq. (7) is complemented with an electroosmotic slip condition at the solid boundaries when present. The ratio of the electroviscous velocity, and the characteristic electroosmotic velocity, U eo , defines another parameter, Rv U ev =U eo (Chen et al., 2005; Lin et al., 2004). The effects of these dimensionless numbers will be discussed below.
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3. Instability characteristics Basic mechanism. Based on the simplified equation system (1)–(3) (or the dimensionless form (5)–(7)), a comprehensive understanding of EKI can be derived. The primary driving force of the instability is the electrical body force in the momentum equation. In the bulk of the liquid (away from the electric double layers), charge separation is induced by the existence of a conductivity gradient subject to an applied electric field. Rewriting Eq. (2) making use of the Gauss’ law yields
qE ¼ ð=rÞrr r/:
ð8Þ
Therefore a charge density (denoted by qE ), and hence an electrical body force is generated where the conductivity gradient and the electric field are not orthogonal to each other. The instability velocity follows approximately the electroviscous scaling (U ev , Eq. (4)). This scaling has been confirmed by Lin et al. (2004) using two-dimensional non-linear simulations for a moderate range of applied electric field strengths. For the higher and lower extremes this behavior is modified by the increasing importances of inertia and molecular diffusion, respectively. Note that although U ev predicts well the correlation with respect to the applied field strength, U ev E20 , the absolute amplitude of this velocity tends to be overestimated when compared with that from numerical computation. On the other hand, the instability sets on only when the applied field strength exceeds certain threshold values, suggesting the existence of a stabilizing mechanism. Such mechanism was correctly identified by Baygents and Baldessari to be molecular diffusion of the conductivity (Eq. (5), Baygents and Baldessari, 1998). The existence of the threshold field can be conveniently understood by a timescale argument. On one hand, the instability grows with the electroviscous time scale, 2 sev ¼ L=U ev E2 0 . On the other, molecular diffusion, with a time scale of sd ¼ L =Deff , relaxes the conductivity gradient that is correlated with charge accumulation and the strength of the electrical body force (Eq. (8)). Instability thus occurs only when the disturbance grows fast enough to overcome diffusive relaxation. The ratio of sd and sev gives the electric Rayleigh number, Rae , which is the critical parameter for the onset of instability (Baygents and Baldessari, 1998; Lin et al., 2004). It is interesting to note that in the absence of molecular diffusion, the flow is unstable for all Reynolds numbers (Hoburg and Melcher, 1976), suggesting that viscous diffusion alone does not provide a sufficient stabilizing mechanism. In addition, various researchers used more sophisticated formulations of Rae to best correlate and collapse data. We refer the readers to a detailed comparison and discussion by Posner and Santiago (2006). Field alignment. EKI can occur with two basic types of configurations (Fig. 1). For type I, the electric field is orthogonal to the conductivity gradient, and there is no net charge density in the base state according to Eq. (8). This type of base flow often occurs in the sample loading phase for microfluidic applications. For type II, the electric field is colinear with the conductivity gradient, and net charge density has a non-trivial distribution even in the base state (Baygents and Baldessari, 1998). This type of flow often occurs in the stacking/separation phase for microfluidic applications (Fig. 1, II-a), or in intended micromixing designs (Fig. 1, II-b, El Moctar et al., 2003). Despite the difference in the base states, EKI in type I and II-b configurations share very similar instability mechanism and behavior, and the quantitative threshold is comparable. For example, for a conductivity ratio of 10 and using 2D linear analysis, both Lin et al. (2004) (Fig. 3 therein) and Baygents and Baldessari (1998) (Fig. 6, curve 1 therein) give a critical Rayleigh number around 1:5 103 . (Note that in Baygents and Baldessari (1998), the dimensionless parameter e2 is equivalent to Rae used in Lin et al. (2004), and rigorously speaking the conductivity ratio in the former is 11.) Due to the susceptibility to instability, both types of configurations are explored for mixing applications (El Moctar et al., 2003; Oddy et al., 2001). On the other hand, type II-a configuration in microfluidic applications is speculated to be more stable, evidenced by the lack of experimental reports in the literature. The stabilizing effect is possibly due to the presence of channel walls along the lateral (streamwise) directions. In fact, the major challenge (for stable fluid handling) in this case is not EKI, but convective dispersion (Taylor-Aris dispersion, Ghosal (2003) and Lin et al. (2008)). A rigorous theoretical study of EKI in type II-a configuration requires further investigation with the above effects carefully considered, although undesired instability in this case does not seem to be a concern for technological development. Convective and absolute instability. In the previous discussion EKI is identified as an internally generated instability, characterized by a derived velocity scale, U ev . In EK systems, the instability is also affected by an external and apparent scale, the electroosmotic velocity, U eo . The instability may be either convective or absolute, depending on the ratio of these two velocities, Rv U ev =U eo . A detailed study was first presented by Chen et al. (2005), in which a spatial (instead of temporal) linear analysis was performed, and the results were compared with experiments. A numerical simulation by Kang et al. (2006) also examined the initial and later stages of growth of the convective instability in T-shaped microchannels. Briefly speaking, a temporally unstable system can be either convectively or absolutely unstable. If the electroosmotic convection is faster than
Fig. 1. Typical base states for EKI. Here
rH (brighter) and rL (darker) denote high- and low-conductivity regions, respectively.
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the electroviscous disturbance can grow, then the instability is convective, characterized by the amplification of disturbances along the electroosmotic flow direction. Otherwise the disturbances may travel both in and against the electroosmotic flow direction, and absolute instability occurs. In Chen et al. (2005) (Fig. 18), the instability phase space is mapped with respect to the two governing parameters, Rae and Rv : A critical Rae demarcates the boundary between stable and unstable regions, whereas a critical Rv demarcates the boundary between the convectively and absolutely unstable regions. Following the above efforts, an extensive parametric study for convective instability in cross-shaped channels was performed by Posner and Santiago (2006). Different than the T-shaped geometry used by Chen et al. (2005), in which a single conductivity interface was considered, the authors investigated the stability of a layer of high-conductivity solution sandwiched between two low-conductivity streams. The applied field in this study was orthogonal to the conductivity gradient. Such realistic flow geometry is directly relevant to a variety of EK applications including injections, preconcentration, and mixing processes. An elaborated, modified critical Rayleigh number was derived which demonstrated validity over a wide range of parametric configurations. In addition the authors also presented mixing characterization in the unstable regimes. This work provides a detailed parametric database for the design of EK applications with a generic and general configuration. Three-dimensional effects. Although most EKI experiments are diagnosed with two-dimensional (2D) scalar images, the flow involved is inherently three-dimensional (3D) in nature. Lin et al. showed that when compared with a 2D linear analysis, a 3D analysis provided a critical electric field strength an order of magnitude higher, which was in much closer agreement with experimentally measured values (Lin et al., 2004). In a series of 3D non-linear simulations, Storey showed that instability could occur in the depth direction (perpendicular to the plane of optical access) as well (Storey, 2005). The significant difference in 2D and 3D analyses in Lin et al. (2004) is due to the presence of a ‘‘shallow” channel depth which constraints and stabilizes the flow. Taking advantage of the small channel depth-to-width aspect ratio ðd d=W 1Þ, a depth-averaged equation set can be developed. Following the efforts of Chen et al. (2005) and Storey et al. (2005), and using a complete asymptotic analysis, Lin et al. derived a quasi-2D equation set which effectively accounted for the non-trivial flow physics in the depth direction and which was computationally more efficient compared with full 3D simulations (Lin et al., 2008). The resulting momentum equation is similar to that in the lubrication theories, whereas the convective-diffusion of the conductivity field (Eq. (5)) in the depth-direction manifests itself as Taylor-Aris dispersion (Ghosal, 2003). The asymptotic model was validated with 3D direct numerical simulations, and provided favorable quantitative comparison with experiments. However, as Storey (2005) pointed out, if the channel is ‘‘deep” such that instability can occur even in the depth direction, then the depth-averaged equations are no longer applicable, and 3D numerical simulation is the only viable option for flow modeling. Periodic forcing. The interest for periodic (AC) forcing in EKI primarily comes from the perspective of mixing applications. The use of AC instead of DC field was demonstrated to equally induce or even enhance EKI mixing (El Moctar et al., 2003; Oddy et al., 2001; Shin et al., 2005). However, it was not clear that the enhancement observed by Shin et al. (2005) was indeed due to the frequency dependent bulk electrostatic forcing (the electrical body force in Eq. (3)). Using both analysis and simulation, Boy and Storey studied AC-field-driven EKI with a co-linear (type II-b) configuration, and compared results with DC-field-driven EKI (Boy and Storey, 2007). They found that EKI demonstrated strong frequency dependence only in the regime where diffusion of the base state dominated; in such a regime the instability is weak and not amenable for mixing applications. On the other hand, in the strong instability regime where mixing is effective, the AC case with an RMS voltage behaves the same as the DC case with no frequency dependence. The authors therefore speculated that the mixing enhancement observed in Shin et al. (2005) was possibly not due to periodic bulk electrostatic forcing, but instead the periodic modulation of the flow rates. Despite the discrepancy in understanding the basic mechanism for mixing enhancement, AC-field-driven EKI is a promising approach for the development of mixing strategies (further discussed below). In addition, the use of an AC instead of a DC field provides the benefit of minimized bubble generation. Multiple-species effects. As previously discussed, the BE model is valid only when the electrolyte system is approximately binary, such as in a strong sodium-chloride solution. Oddy and Santiago (2005) suggested that the presence of multiple (defined as more than two) species can have the most interesting effects on EKI. Using a four-species model, they showed that the predicted critical electric field strength was reduced by two orders of magnitude when compared with that from a BE model with identical conductivity ratio (or gradient). This difference is due to the following reason. In the BE model, the effects of ion-electromigration on conductivity transport cancel each other, and the governing equation becomes convectivediffusive as given by (5). In general for a multiple-species system, such simplification cannot be made, and ion-electromigration can have a profound effect on conductivity transport. Depending on the specific ion concentration and property configurations, this effect can be either destabilizing, via enhancing the conductivity gradient against diffusion; or stabilizing, via relaxing the conductivity gradient in addition to diffusion. We refer the readers to the detailed discussions in Oddy and Santiago (2005). This work suggests the intriguing possibility that EKI can be controlled (either to enhance or suppress instability) via ion composition in addition to electric field strength and conductivity ratio. In addition, the authors also showed that the diffusive current, often ignored in previous works, could have a non-trivial albeit secondary effect on the instability behavior. The complexity exhibited in the multiple-species model suggests that EKI, or more generally, EHD instability and flows, are rich multi-physical phenomena we only begin to understand. Applications. The applications of EKI have two aspects. The first is simply to avoid such instability when stable flow handling is intended. In this case the theoretical work provides a guideline for design improvement. For example, by controlling and avoiding EKI, Jung et al. (2003) was able to achieve thousand-fold concentration amplification using field-amplified sample stacking, an order of magnitude improvement from previous microfluidic devices. The second is harnessing EKI for effi-
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cient micromixing. We refer the readers to a more detailed review by Chang and Yang (2007), whereas only a brief discussion is given here. The potential application of the instability for mixing was proposed by the very work of Hoburg and Melcher (1976). Following Oddy et al. (2001) and along with theoretical development, various mixing experiments and designs have been investigated (El Moctar et al., 2003; Tsouris et al., 2003). These works demonstrated EKI as an efficient, non-mechanical means for rapid micro-mixing. Furthermore, extending from the basic design, performance improvement via channel geometry modification (Huang et al., 2006; Park et al., 2005; Tai et al., 2006) and AC electric fields (Shin et al., 2005) has been pursued. On the other hand, as pointed out by Chang and Yang (2007), successful EKI-mixing applications require relatively high-conductivity ratios and applied field strength. These requirements can be possibly addressed or circumvented via the following considerations. (1) Following Oddy and Santiago (2005), the use of different ion configurations may promote instability and significantly lower the requirement for a high-conductivity ratio. For example, using a typical applied field of 0.55 kV/cm, Oddy and Santiago demonstrated instability mixing for a conductivity ratio of mere 1.05. As previous discussed, such effect is possible via conductivity gradient enhancement resulting from multi-ion electromigration. (2) Strategically placing the electrodes close together can reduce the actual voltage requirement while maintaining high field-strength. In El Moctar et al. (2003), a field strength of 2 kV/cm was achieved by applying a voltage of 50 V across the 250 lm-wide channel (instead of along the often much greater streamwise direction). However, we emphasize that the above are theoretical, speculative guidelines, and successful, practical mixing strategies require further development from experimental and engineering perspectives, and are likely application-specific. 4. Conclusion EKI is an interesting example of complex, multi-physical flow phenomena on the micro-scales. In this brief review, we have discussed the basic mechanism and the various aspects of this instability including applications. The knowledge base established during the study of EKI is not limited to the instability itself, but is applicable to other directly related phenomena. Examples including EKI induced by a gradient in colloid volume fraction (Navaneetham and Posner, in press), EHD-driven colloid pattern formation (Trau et al., 1995), bulk electroconvective instability (Storey et al., 2007), among others. Extension of the basic EHD theories into these areas may suggest new and promising opportunities. Acknowledgements The author acknowledges partial support from an NSF CAREER Award (CBET-0747886) with Dr. William Schultz as contract monitor, and helpful discussions with Dr. Jonathan D. Posner. References Baygents, J.C., Baldessari, F., 1998. Electrohydrodynamic instability in a thin fluid layer with an electrical conductivity gradient. Phys. Fluids 10, 301–311. Boy, D.A., Storey, B.D., 2007. Electrohydrodynamic instabilities in microchannels with time periodic forcing. Phys. Rev. E 76 (026304). Chang, C.-C., Yang, R.-J., 2007. Electrokinetic mixing in microfluidic systems. Microfluid. Nanofluid. 3, 501–525. Chen, C.-H., Lin, H., Lele, S.K., Santiago, J.G., 2005. Convective and absolute electrokinetic instability with conductivity gradients. J. Fluid Mech. 524, 263–303. Chen, C.-H., Santiago, J.G., 2002. Electrokinetic instability in high concentration gradient microflows. Proc. IMECE (33563). Dang, F.Q., Zhang, L.H., Jabasini, M., Kaji, N., Baba, Y., 2003. 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Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Electrokinetic instability in microchannel flows: A review Hao Lin* Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, 98 Brett Road, Piscataway, NJ 08854, United States
a r t i c l e
i n f o
Article history: Received 24 April 2008 Received in revised form 5 July 2008 Available online 12 August 2008
Keywords: Electrokinetic instability Electrohydrodynamics Microfluidics Mixing
a b s t r a c t Electrokinetic instability is a microscale instability observed during the development of electrokinetic microfluidic applications, and is induced by an interaction between the electric field and fluid motion. In this brief review, the basic mechanism as well as the various aspects of this instability are summarized. These include the effects of field alignment, electroosmotic velocity (convective and absolute instability), channel dimension, periodic forcing, and multiple-species. A brief discussion on the applications of the instability is also presented. Ó 2008 Published by Elsevier Ltd.
1. Introduction During the past decade, electrokinetic (EK) techniques received renewed interest due to extensive research into the design and engineering of micro- and nano-fluidic devices. Many of these applications employ electric fields to realize novel functionalities, and to enable rapid and miniaturized chemical and biochemical analysis and detection (Reyes et al., 2002). Meanwhile, complex flow phenomena and regimes also emerge from these applications which challenge our fundamental understanding and prediction capability (Squires and Quake, 2005). One such example is electrokinetic instability (EKI), a microscale instability resulting from the coupling of the electric field, ion-electromigration, and fluid motion. In this topical review, we briefly discuss the history, basic theory, instability mechanism and behavior, and applications of this multi-physical phenomenon. EKI in the microfluidic context was first observed in various laboratories when pursuing electrokinetic applications such as sample loading and injection, and field-amplified sample stacking (Dang et al., 2003; Ramsey, 2001; Shultz-Lockyear et al., 1999). Meanwhile, various groups also explored the possibility of using this instability for efficient and non-mechanical micro-mixing (El Moctar et al., 2003; Oddy et al., 2001; Tsouris et al., 2003). Although the instability mechanism was not fully understood, a more systematic investigation by Chen and Santiago (2002) established the critical importance of conductivity gradient and high electric field strength in inducing the instability. Further studies by Santiago and co-authors (Chen et al., 2005; Lin et al., 2004) related the instability physics with earlier works on electrohydrodynamic (EHD) instability by Hoburg and Melcher (1976), Hoburg and Melcher (1977) and Baygents and Baldessari (1998). Combining linear analysis and nonlinear numerical simulations, these studies provided a comprehensive model framework for the understanding and prediction of EKI, and the results compared favorably with experimental measurements. A body of work followed to systematically investigate the instability in greater details, including those on the effects of field alignment (Storey, 2005), electroosmotic convection (convective and absolute instability) (Chen et al., 2005; Posner and Santiago, 2006), channel dimension (Chen et al., 2005; Lin et al., 2008; Storey, 2005; Storey et al., 2005), time-periodic forcing (Boy and Storey, 2007; Shin et al.,
* Tel.: +1 732 445 2322; fax: +1 732 445 3124. E-mail address: [email protected] 0093-6413/$ - see front matter Ó 2008 Published by Elsevier Ltd. doi:10.1016/j.mechrescom.2008.07.012
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2005), and multiple-species (Oddy and Santiago, 2005); and those implemented EKI in various designs for mixing applications (Huang et al., 2006; Pan et al., 2007; Park et al., 2005; Shin et al., 2005; Tai et al., 2006). A discussion on two important features helps us relate EKI to proper context and other similar flow phenomena. The first is that EKI is induced by the presence of an electrical conductivity gradient, which under an applied field results in charge separation and an electrical body force (the Maxwell stress). In this regard EKI can be viewed as a special type of EHD flow, the modern theoretical foundation of which was laid by Melcher and Taylor (1969). However, different than the original leakydielectric model, in EKI charge separation occurs in the bulk of a single miscible fluid, not at the interface of two immiscible fluids. Consequently, and as correctly first pointed out by Baygents and Baldessari (1998), ion diffusion in EKI results in a conditional instability. EHD instability of two (or more) immiscible fluids in microfluidic devices is another subject extensively studied by Li et al. (2007), Ozen et al. (2006), and Zahn and Reddy (2006), among others, and is not discussed here. Another feature of EKI is that it is driven by an electrostatic force in the bulk of the liquid away from charged solid–liquid interfaces (and hence away from the electric double layers). This feature distinguishes EKI from other types of instabilities and/or mixing phenomena in electrokinetic flows, such as those induced by modulated electroosmotic flows (Sundaram and Tafti, 2004). In this latter case electrostatic forcing is regarded as absent away from the (often relatively thin) electric double layers. 2. Theoretical formulation The Binary electrolyte model. The basic behavior of EKI can be described with a simplified binary electrolyte (BE) model (Baygents and Baldessari, 1998; Chen et al., 2005; Hoburg and Melcher, 1976; Lin et al., 2004). The governing equations consist of the convective-diffusive equation for the electrical conductivity, the Ohmic current conservation equation, and the Navier–Stokes equations:
or Dþ D ðzþ þ z Þ ; þ u rr ¼ Deff rr; Deff ¼ D þ zþ þ D z ot r rr/ ¼ 0; ou q þ u ru ¼ rp þ lr2 u þ r2 /r/; r u ¼ 0: ot
ð1Þ ð2Þ ð3Þ
Here r is the electrical conductivity, Deff is an effective diffusivity, D and z are the molecular diffusivity and valence number, and the subscripts ‘‘+” and ‘‘” denote the cation and the anion in the BE system, respectively. / is the electric potential, and q, u, p, l and are the fluid density, velocity, pressure, viscosity, and permittivity, respectively. Eq. (1) is derived from the Nernst–Planck equation for ion-electromigration assuming a BE system and that the electroneutrality condition holds (Levich, 1962). The latter is a valid assumption when the concentration mismatch between the cation and anion is small compared with the background ion concentration. The Ohmic Eq. (2) implies that charge relaxation is instantaneous (Hoburg and Melcher, 1976). In arriving at this equation the diffusive current (due to the diffusive transport of charged species) has also been ignored (Oddy and Santiago, 2005). The last term in the momentum Eq. (3), r2 /r/, is the electrical body force derived from the Maxwell stress tensor. During the derivation we have already assumed the fluid permittivity, , to be a constant. Note that Eqs. (1)–(3) can be regarded as a special case of a more generalized multiple-species (MS) model proposed by Oddy and Santiago (2005). Although the BE model is capable of capturing the basic mechanism of EKI, and is quantitatively accurate when the electrolyte system is indeed binary, it does not include some of the subtle yet important effects when more than two species with differing properties are involved. Such effects are discussed below. Non-dimensionalization. The governing Eqs. (1)–(3) are non-dimensionalized using the following scales:
½x ¼ L; ½t ¼ sev ¼ L=U ev ; ½r ¼ r0 ; ½/ ¼ E0 L; ½p ¼ lU ev =L; U ev E20 L=l:
ð4Þ
Here L is a length scale, r0 is a characteristic conductivity, and E0 is taken to be the applied field strength. The electroviscous velocity, U ev , is derived by setting equal the viscous and electrical body forces in the momentum equation (Hoburg and Melcher, 1976). The resulting equations read
or 1 r 2 r; þ u rr ¼ Rae ot r rr/ ¼ 0; ou Re þ u ru ¼ rp þ r2 u þ r2 /r/; ot
ð5Þ ð6Þ
r u ¼ 0:
ð7Þ
Here the electric Rayleigh number and the Reynolds number are defined as Rae U ev L=Deff , and Re qU ev L=l, respectively. In EK systems and in the thin electric double layer limit, Eq. (7) is complemented with an electroosmotic slip condition at the solid boundaries when present. The ratio of the electroviscous velocity, and the characteristic electroosmotic velocity, U eo , defines another parameter, Rv U ev =U eo (Chen et al., 2005; Lin et al., 2004). The effects of these dimensionless numbers will be discussed below.
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3. Instability characteristics Basic mechanism. Based on the simplified equation system (1)–(3) (or the dimensionless form (5)–(7)), a comprehensive understanding of EKI can be derived. The primary driving force of the instability is the electrical body force in the momentum equation. In the bulk of the liquid (away from the electric double layers), charge separation is induced by the existence of a conductivity gradient subject to an applied electric field. Rewriting Eq. (2) making use of the Gauss’ law yields
qE ¼ ð=rÞrr r/:
ð8Þ
Therefore a charge density (denoted by qE ), and hence an electrical body force is generated where the conductivity gradient and the electric field are not orthogonal to each other. The instability velocity follows approximately the electroviscous scaling (U ev , Eq. (4)). This scaling has been confirmed by Lin et al. (2004) using two-dimensional non-linear simulations for a moderate range of applied electric field strengths. For the higher and lower extremes this behavior is modified by the increasing importances of inertia and molecular diffusion, respectively. Note that although U ev predicts well the correlation with respect to the applied field strength, U ev E20 , the absolute amplitude of this velocity tends to be overestimated when compared with that from numerical computation. On the other hand, the instability sets on only when the applied field strength exceeds certain threshold values, suggesting the existence of a stabilizing mechanism. Such mechanism was correctly identified by Baygents and Baldessari to be molecular diffusion of the conductivity (Eq. (5), Baygents and Baldessari, 1998). The existence of the threshold field can be conveniently understood by a timescale argument. On one hand, the instability grows with the electroviscous time scale, 2 sev ¼ L=U ev E2 0 . On the other, molecular diffusion, with a time scale of sd ¼ L =Deff , relaxes the conductivity gradient that is correlated with charge accumulation and the strength of the electrical body force (Eq. (8)). Instability thus occurs only when the disturbance grows fast enough to overcome diffusive relaxation. The ratio of sd and sev gives the electric Rayleigh number, Rae , which is the critical parameter for the onset of instability (Baygents and Baldessari, 1998; Lin et al., 2004). It is interesting to note that in the absence of molecular diffusion, the flow is unstable for all Reynolds numbers (Hoburg and Melcher, 1976), suggesting that viscous diffusion alone does not provide a sufficient stabilizing mechanism. In addition, various researchers used more sophisticated formulations of Rae to best correlate and collapse data. We refer the readers to a detailed comparison and discussion by Posner and Santiago (2006). Field alignment. EKI can occur with two basic types of configurations (Fig. 1). For type I, the electric field is orthogonal to the conductivity gradient, and there is no net charge density in the base state according to Eq. (8). This type of base flow often occurs in the sample loading phase for microfluidic applications. For type II, the electric field is colinear with the conductivity gradient, and net charge density has a non-trivial distribution even in the base state (Baygents and Baldessari, 1998). This type of flow often occurs in the stacking/separation phase for microfluidic applications (Fig. 1, II-a), or in intended micromixing designs (Fig. 1, II-b, El Moctar et al., 2003). Despite the difference in the base states, EKI in type I and II-b configurations share very similar instability mechanism and behavior, and the quantitative threshold is comparable. For example, for a conductivity ratio of 10 and using 2D linear analysis, both Lin et al. (2004) (Fig. 3 therein) and Baygents and Baldessari (1998) (Fig. 6, curve 1 therein) give a critical Rayleigh number around 1:5 103 . (Note that in Baygents and Baldessari (1998), the dimensionless parameter e2 is equivalent to Rae used in Lin et al. (2004), and rigorously speaking the conductivity ratio in the former is 11.) Due to the susceptibility to instability, both types of configurations are explored for mixing applications (El Moctar et al., 2003; Oddy et al., 2001). On the other hand, type II-a configuration in microfluidic applications is speculated to be more stable, evidenced by the lack of experimental reports in the literature. The stabilizing effect is possibly due to the presence of channel walls along the lateral (streamwise) directions. In fact, the major challenge (for stable fluid handling) in this case is not EKI, but convective dispersion (Taylor-Aris dispersion, Ghosal (2003) and Lin et al. (2008)). A rigorous theoretical study of EKI in type II-a configuration requires further investigation with the above effects carefully considered, although undesired instability in this case does not seem to be a concern for technological development. Convective and absolute instability. In the previous discussion EKI is identified as an internally generated instability, characterized by a derived velocity scale, U ev . In EK systems, the instability is also affected by an external and apparent scale, the electroosmotic velocity, U eo . The instability may be either convective or absolute, depending on the ratio of these two velocities, Rv U ev =U eo . A detailed study was first presented by Chen et al. (2005), in which a spatial (instead of temporal) linear analysis was performed, and the results were compared with experiments. A numerical simulation by Kang et al. (2006) also examined the initial and later stages of growth of the convective instability in T-shaped microchannels. Briefly speaking, a temporally unstable system can be either convectively or absolutely unstable. If the electroosmotic convection is faster than
Fig. 1. Typical base states for EKI. Here
rH (brighter) and rL (darker) denote high- and low-conductivity regions, respectively.
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the electroviscous disturbance can grow, then the instability is convective, characterized by the amplification of disturbances along the electroosmotic flow direction. Otherwise the disturbances may travel both in and against the electroosmotic flow direction, and absolute instability occurs. In Chen et al. (2005) (Fig. 18), the instability phase space is mapped with respect to the two governing parameters, Rae and Rv : A critical Rae demarcates the boundary between stable and unstable regions, whereas a critical Rv demarcates the boundary between the convectively and absolutely unstable regions. Following the above efforts, an extensive parametric study for convective instability in cross-shaped channels was performed by Posner and Santiago (2006). Different than the T-shaped geometry used by Chen et al. (2005), in which a single conductivity interface was considered, the authors investigated the stability of a layer of high-conductivity solution sandwiched between two low-conductivity streams. The applied field in this study was orthogonal to the conductivity gradient. Such realistic flow geometry is directly relevant to a variety of EK applications including injections, preconcentration, and mixing processes. An elaborated, modified critical Rayleigh number was derived which demonstrated validity over a wide range of parametric configurations. In addition the authors also presented mixing characterization in the unstable regimes. This work provides a detailed parametric database for the design of EK applications with a generic and general configuration. Three-dimensional effects. Although most EKI experiments are diagnosed with two-dimensional (2D) scalar images, the flow involved is inherently three-dimensional (3D) in nature. Lin et al. showed that when compared with a 2D linear analysis, a 3D analysis provided a critical electric field strength an order of magnitude higher, which was in much closer agreement with experimentally measured values (Lin et al., 2004). In a series of 3D non-linear simulations, Storey showed that instability could occur in the depth direction (perpendicular to the plane of optical access) as well (Storey, 2005). The significant difference in 2D and 3D analyses in Lin et al. (2004) is due to the presence of a ‘‘shallow” channel depth which constraints and stabilizes the flow. Taking advantage of the small channel depth-to-width aspect ratio ðd d=W 1Þ, a depth-averaged equation set can be developed. Following the efforts of Chen et al. (2005) and Storey et al. (2005), and using a complete asymptotic analysis, Lin et al. derived a quasi-2D equation set which effectively accounted for the non-trivial flow physics in the depth direction and which was computationally more efficient compared with full 3D simulations (Lin et al., 2008). The resulting momentum equation is similar to that in the lubrication theories, whereas the convective-diffusion of the conductivity field (Eq. (5)) in the depth-direction manifests itself as Taylor-Aris dispersion (Ghosal, 2003). The asymptotic model was validated with 3D direct numerical simulations, and provided favorable quantitative comparison with experiments. However, as Storey (2005) pointed out, if the channel is ‘‘deep” such that instability can occur even in the depth direction, then the depth-averaged equations are no longer applicable, and 3D numerical simulation is the only viable option for flow modeling. Periodic forcing. The interest for periodic (AC) forcing in EKI primarily comes from the perspective of mixing applications. The use of AC instead of DC field was demonstrated to equally induce or even enhance EKI mixing (El Moctar et al., 2003; Oddy et al., 2001; Shin et al., 2005). However, it was not clear that the enhancement observed by Shin et al. (2005) was indeed due to the frequency dependent bulk electrostatic forcing (the electrical body force in Eq. (3)). Using both analysis and simulation, Boy and Storey studied AC-field-driven EKI with a co-linear (type II-b) configuration, and compared results with DC-field-driven EKI (Boy and Storey, 2007). They found that EKI demonstrated strong frequency dependence only in the regime where diffusion of the base state dominated; in such a regime the instability is weak and not amenable for mixing applications. On the other hand, in the strong instability regime where mixing is effective, the AC case with an RMS voltage behaves the same as the DC case with no frequency dependence. The authors therefore speculated that the mixing enhancement observed in Shin et al. (2005) was possibly not due to periodic bulk electrostatic forcing, but instead the periodic modulation of the flow rates. Despite the discrepancy in understanding the basic mechanism for mixing enhancement, AC-field-driven EKI is a promising approach for the development of mixing strategies (further discussed below). In addition, the use of an AC instead of a DC field provides the benefit of minimized bubble generation. Multiple-species effects. As previously discussed, the BE model is valid only when the electrolyte system is approximately binary, such as in a strong sodium-chloride solution. Oddy and Santiago (2005) suggested that the presence of multiple (defined as more than two) species can have the most interesting effects on EKI. Using a four-species model, they showed that the predicted critical electric field strength was reduced by two orders of magnitude when compared with that from a BE model with identical conductivity ratio (or gradient). This difference is due to the following reason. In the BE model, the effects of ion-electromigration on conductivity transport cancel each other, and the governing equation becomes convectivediffusive as given by (5). In general for a multiple-species system, such simplification cannot be made, and ion-electromigration can have a profound effect on conductivity transport. Depending on the specific ion concentration and property configurations, this effect can be either destabilizing, via enhancing the conductivity gradient against diffusion; or stabilizing, via relaxing the conductivity gradient in addition to diffusion. We refer the readers to the detailed discussions in Oddy and Santiago (2005). This work suggests the intriguing possibility that EKI can be controlled (either to enhance or suppress instability) via ion composition in addition to electric field strength and conductivity ratio. In addition, the authors also showed that the diffusive current, often ignored in previous works, could have a non-trivial albeit secondary effect on the instability behavior. The complexity exhibited in the multiple-species model suggests that EKI, or more generally, EHD instability and flows, are rich multi-physical phenomena we only begin to understand. Applications. The applications of EKI have two aspects. The first is simply to avoid such instability when stable flow handling is intended. In this case the theoretical work provides a guideline for design improvement. For example, by controlling and avoiding EKI, Jung et al. (2003) was able to achieve thousand-fold concentration amplification using field-amplified sample stacking, an order of magnitude improvement from previous microfluidic devices. The second is harnessing EKI for effi-
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cient micromixing. We refer the readers to a more detailed review by Chang and Yang (2007), whereas only a brief discussion is given here. The potential application of the instability for mixing was proposed by the very work of Hoburg and Melcher (1976). Following Oddy et al. (2001) and along with theoretical development, various mixing experiments and designs have been investigated (El Moctar et al., 2003; Tsouris et al., 2003). These works demonstrated EKI as an efficient, non-mechanical means for rapid micro-mixing. Furthermore, extending from the basic design, performance improvement via channel geometry modification (Huang et al., 2006; Park et al., 2005; Tai et al., 2006) and AC electric fields (Shin et al., 2005) has been pursued. On the other hand, as pointed out by Chang and Yang (2007), successful EKI-mixing applications require relatively high-conductivity ratios and applied field strength. These requirements can be possibly addressed or circumvented via the following considerations. (1) Following Oddy and Santiago (2005), the use of different ion configurations may promote instability and significantly lower the requirement for a high-conductivity ratio. For example, using a typical applied field of 0.55 kV/cm, Oddy and Santiago demonstrated instability mixing for a conductivity ratio of mere 1.05. As previous discussed, such effect is possible via conductivity gradient enhancement resulting from multi-ion electromigration. (2) Strategically placing the electrodes close together can reduce the actual voltage requirement while maintaining high field-strength. In El Moctar et al. (2003), a field strength of 2 kV/cm was achieved by applying a voltage of 50 V across the 250 lm-wide channel (instead of along the often much greater streamwise direction). However, we emphasize that the above are theoretical, speculative guidelines, and successful, practical mixing strategies require further development from experimental and engineering perspectives, and are likely application-specific. 4. Conclusion EKI is an interesting example of complex, multi-physical flow phenomena on the micro-scales. In this brief review, we have discussed the basic mechanism and the various aspects of this instability including applications. The knowledge base established during the study of EKI is not limited to the instability itself, but is applicable to other directly related phenomena. Examples including EKI induced by a gradient in colloid volume fraction (Navaneetham and Posner, in press), EHD-driven colloid pattern formation (Trau et al., 1995), bulk electroconvective instability (Storey et al., 2007), among others. Extension of the basic EHD theories into these areas may suggest new and promising opportunities. Acknowledgements The author acknowledges partial support from an NSF CAREER Award (CBET-0747886) with Dr. William Schultz as contract monitor, and helpful discussions with Dr. Jonathan D. Posner. References Baygents, J.C., Baldessari, F., 1998. Electrohydrodynamic instability in a thin fluid layer with an electrical conductivity gradient. Phys. Fluids 10, 301–311. Boy, D.A., Storey, B.D., 2007. Electrohydrodynamic instabilities in microchannels with time periodic forcing. Phys. Rev. E 76 (026304). Chang, C.-C., Yang, R.-J., 2007. Electrokinetic mixing in microfluidic systems. Microfluid. Nanofluid. 3, 501–525. Chen, C.-H., Lin, H., Lele, S.K., Santiago, J.G., 2005. Convective and absolute electrokinetic instability with conductivity gradients. J. Fluid Mech. 524, 263–303. Chen, C.-H., Santiago, J.G., 2002. Electrokinetic instability in high concentration gradient microflows. Proc. IMECE (33563). Dang, F.Q., Zhang, L.H., Jabasini, M., Kaji, N., Baba, Y., 2003. 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