Microelectronic Engineering 86 (2009) 1333–1336
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Numerical study on AC electroosmosis in microfluidic channels Petr Cˇervenka, Michal Prˇibyl *, Dalimil Šnita Institute of Chemical Technology, Prague Department of Chemical Engineering, Technická 5, 166 28 Praha 6, Czech Republic
a r t i c l e
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Article history: Received 29 September 2008 Received in revised form 20 January 2009 Accepted 20 January 2009 Available online 27 January 2009 Keywords: AC electroosmosis Mathematical modeling Microfluidic chip Electroosmotic micropumps Non-equilibrium model Flow reversals
a b s t r a c t This paper deals with parametrical studies focused on determination of selected characteristics of AC electroosmotic micropumps in non-linear (high-amplitude) regimes in the interdigitated arrangement. The net velocity and current–voltage phase shift are computed. Non-equilibrium Navier–Stokes–Poisson–Nernst–Planck model that describes the transport processes in the entire time and spatial domains including electric double layers (EDLs) was derived and analyzed. The obtained results were qualitatively interpreted by means of a simple relation derived for RC circuit in series. Locations of the high net velocity regions in the parametric space were found. The model thus can contribute to development and optimization of microfluidic devices driven by AC electric fields. Flow reversals were observed in frequency and geometric scale characteristics. We suggest that the flow reversals do not necessarily rely on the electric charge injection or the condensed layer formation. The direction of the net flow probably depends only on an interplay of the electric, viscous and pressure forces. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Ajdari recently proposed a design of the AC electrokinetic micropumps based on arrays of asymmetric pairs of co-planar microelectrodes [1]. It was expected that the asymmetry of an electric field will lead to a net flow of the electrolyte. His predictions were verified by several experimental and theoretical works [2–5]. In the co-planar interdigitated arrangement of the electrodes, Fig. 1, an AC electric field has the tangential and the normal components. The normal component induces electrode polarization via the coulombic force (capacitive charging). Then, the tangential component of the electric field forces the accumulated electric charge to move along the electrodes. The combination of coulombic, pressure and viscous forces in the liquid results in formation of eddies above the electrodes and the net flow. The net velocity is dependent, e.g. on the frequency of the applied AC electric field, fAC, the characteristic dimension of electrodes L, and the electrolyte concentration c0. The characteristic time scale of the electrode polarization can be estimated from a simple relation sCH kD L=D that can be derived from the analysis of a RC circuit representing the electric double layer (EDL) capacitor and the electrolyte resistor in series [6,7]. Here, D and kD denote diffusivity of ions and Debye length (thickness of the electric double layer capacitor), respectively. The Debye length is proportional [8]. If the characteristic to the electrolyte concentration kD c0:5 0 time of the AC electric field sAC ¼ ð2p f AC Þ1 substantially exceeds * Corresponding author. Tel.: +420 2 2044 3168; fax: +420 2 2044 4320. E-mail address:
[email protected] (M. Prˇibyl). 0167-9317/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2009.01.045
or lags the charging time, the system behaves as a pure capacitor or a resistor, respectively. In order to maximize the net flow (by other words, in order to obtain a sufficient gain of the RC circuit) the following relation should be satisfied
ð2p f AC Þ1 kD L=D:
ð1Þ
No comprehensive numerical study of the AC electrokinetics in microchannels has been reported. There are particularly focused studies mostly relying on the slip approximations [3,9–11] or the Poisson–Boltzmann equation [12] but parametric studies based on mathematical models containing the non-equilibrium EDL description are rare [13]. In this work, we numerically analyze behavior of the proposed electroosmotic micropump on three model parameters: frequency of the electric field fAC, electrolyte concentration c0 and the micropump geometric scale k. 2. Mathematical model and numerical analysis The studied microfluidic pump can be represented by a single segment (30 lm long, 10 lm deep) with periodic boundary conditions, Fig. 1. This design corresponds to a system with co-planar interdigitated electrodes periodically deposited along the entire microchannel. The electrodes are 5 lm and 3 lm long and separated by 2 lm and 20 lm gaps. We call the above mentioned dimensions ‘‘reference dimensions”. When effects of the segment size on the pump behavior were studied, new dimensions were computed as the reference dimensions multiplied by the scaling factor k.
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[30; 10]
Dielectric wall [0; 10] [0; 0]
(i)
(ii)
Microfluidic channel
[15; 0]
(iii) (iv)
(v)
[17; 0]
y
x
[10; 0]
~
[20; 0]
[30; 0]
AC voltage source
Fig. 1. Scheme of one segment of the AC electroosmotic micropump. The point coordinates are in microns. The symbols (i)–(v) denote compartments of the segment. Dashed-dotted lines and dotted lines indicate the periodic boundary conditions and boundaries of ‘‘the internal compartments”, respectively.
The mathematical model is based on the Poisson equation, the Navier–Stokes equation with electric volume force, the continuity equation, the molar balances of ions (a symmetric mono-monovalent electrolyte was considered) and the non-slip boundary conditions [14]. The model equations and model parameters are listed in the Supplementary material. The developed mathematical model enables dynamical analysis of transport processes in the entire model domain including EDLs. The numerical analysis was carried out in Comsol Multiphysics software. Strongly non-equidistant and anisotropic meshes of finite elements had to be used due to formation of thin EDLs. A set of auxiliary variables to characterize the AC micropump was defined, see the Supplementary material. The net velocity hvxiy, the time-averaged net velocity hv x iy; t , the electric current I, current–voltage phase shift h, and the horizontal components of the electric force Fe,x and the pressure force Fp,x in each compartment.
Fig. 3. Dependencies of the net velocity (A) and the current–voltage phase shift (B) on the electrolyte concentration; solid line – fAC = 2 103 Hz, k = 1; dashed line – fAC = 2 104 Hz, k = 1; dotted line – fAC = 2 103 Hz, k = 0.1.
The obtained dependencies of the averaged net velocity on the AC frequency are generally non-linear, Fig. 2. Flow reversals can be observed at certain frequency values. This agrees with experimental observations [5,15,16]. The highest velocities are attained when
the current–voltage phase shift is close to 30°. For given arrangement, the net velocity can be higher than 100 lms1. It can be seen that for a smaller system (k = 0.1) the maximal net velocity is found at higher frequencies, according to Eq. (1), and the absolute value of the net velocity grows due to an increase of the horizontal component of the electric field strength at the electrodes (the amplitude is always 1 V). A low electrolyte concentration (c0 = 1 10 3 mol m3) leads to an increase of the EDL dimension. Hence, the high velocity regime is observed for a bit lower frequencies according to Eq. (1), compare solid and dotted lines in Fig. 2. However, the concentration effect is not so strong because kD c00:5 . If the concentration grows, the EDLs shrink and thus the characteristic time of the electrode polarization decreases. Hence, the high velocity region of high frequency regimes is typically located in an interval of high values of electrolyte concentrations (Fig. 3, solid and dashed lines). According to Eq. (1), a decrease of the EDL thickness can be also compensated by an increase of the system size. However, the dynamical simulations showed that small characteristic size of the system (dotted line) enables to
Fig. 2. Dependencies of the net velocity (A) and the current–voltage phase shift (B) on the AC frequency; solid line – c0 = 1 102 mol m3, k = 1; dashed line – c0 = 1 102 mol m3, k = 0.1; dotted line – c0 = 1 103 mol m3, k = 1.
Fig. 4. Dependencies of the net velocity (A) and the current–voltage phase shift (B) on the geometric scale; solid line – c0 = 1 102 mol m3, fAC = 2 103 Hz; dashed line – c0 = 1 102 mol m3, fAC = 2 104 Hz; dotted line – c0 = 1 103 mol m3, fAC = 2 103 Hz.
3. Parametrical studies
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Fig. 5. One period of a stable periodic regime, k = 0.251 (point I in Fig. 4.), c0 = 1 102 mol m3, fAC = 2 103 Hz. The x-component of the integral electric force Fe,x and the integral pressure force Fp,x were computed in the compartments (ii) – circle, (iv) – x-mark, (i + iii + v) – diamond, (ii + iv) – square, (i + ii + iii + iv + v) – triangle.
pump both the electrolytes of low (expected) and high (not expected) concentrations probably due to the increase of the electric field strength. The parametric study plotted in Fig. 4 can be also interpreted in the view of Eq. (1). A high net velocity in a system with small characteristic dimensions can be attained when a higher frequency (dashed line) and/or an electrolyte of a lower concentration (dotted line) are used. However, the concentration effect is not so significant. The dependencies in Fig. 4 reveal flow reversals at certain values of the geometric scale. It can be seen that the flow reversals are gradual processes in the parametric space. To study the origin of the flow reversals, we compared two stable periodic regimes (highlighted by the symbols I and II in Fig. 4) with opposite direction of the net flow, see Figs. 5 and 6 and animations in the Supplementary material. 4. Stable periodic regimes and flow reversals Both the periodic regimes plotted in Figs. 5 and 6 can be characterized by alternations of the net velocity direction during one period of the electric field (‘‘clonic” flow) and by an inharmonic electric current response. The orientation of the averaged net flow velocity is the main difference between these two analyzed regimes. It was expected that the flow orientation results from the
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Fig. 6. One period of a stable periodic regime, k = 0.794 (point II in Fig. 4.). The other parameters are the same as in Fig. 5.
distribution of the horizontal component of the electric force. Hence, we computed the electric force in all the compartments. The results show that the time course of the total horizontal electric force (triangles) does not coincide with the flow orientation. The total electric force is always negative (for the presented set of parameters), while the net velocity alters its orientation. In the case I, (Fig. 5, k = 0.251), the time courses of the horizontal electric force above the smaller electrode (x-mark) and the horizontal pressure force above the bigger electrode (circle) are qualitatively similar to the time course of the net velocity. In the case II, (Fig. 6, k = 0.794), only the pressure force above the bigger electrode (circle) coincides with the time course of the net flow. From this point of view, the horizontal pressure force above the bigger electrode seems to be a good indicator of the net velocity orientation in this particular case. The pressure is locally increased at the electrode surfaces (especially at their boundaries, see the Supplementary material) in order to equilibrate the normal component of the electric force that is proportional to the local concentration of electric charge. The electric charge distribution is strongly affected by the convective flow of the electrolyte which is a result of an interplay of the electrical, the viscous, the pressure and also the inertial (only the dynamical part) forces. 5. Conclusions Selected characteristics of the proposed AC electroosmotic micropump were computed by means of dynamical numerical
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analysis of the non-equilibrium model in the entire spatial and time domains. Locations of the high net velocity regions in the parametric space can be estimated with the use of Eq. (1). Faradaic interactions at the electrodes are not considered in the mathematical model and thus AC micropumps relying on the Faradaic polarization cannot be studied using it [17]. It was experimentally found that intensive Faradaic interactions in AC micropumps usually occur above 1.5 Vrms, e.g. [4,5,12]. Kilic and Bazant [18,19] pointed out that the classical Poisson–Boltzmann theory predicts physically inadmissible concentration of ions at the polarized surfaces when a higher voltage is applied. For our system, the steric effect can be important if the difference between the surface potential and the bulk potential exceeds about ±0.4 V. Here, we apply 0.707 Vrms, which means that the steric limit on one electrode is not attained. However, if one will use the model for higher voltage regimes, its modification should be considered [16]. We compared the obtained results with those published in experimental papers [2,4,5]. Although we have found relatively good agreement with some data, discrepancies were also observed. Experimental parameters (electrode dimensions, used electrolytes, applied voltage) are not exactly the same as in our numerical studies. For example, the frequency characteristics predicted by the model and published in [5] are quite similar. In the simulations, the flow reversals are shifted to low frequencies and the velocity magnitude is higher. These disagreements are probably caused by the fact that the electrolyte concentration in experiment is by one order of magnitude higher, see the concentration effect in Eq. (1). Other discrepancies result from: (i) the facts that experimental electrodes rise above dielectric walls and are not perfectly smooth, (ii) the precision of experimental measurements, (iii) the precision of numerical approximations, etc. The relative numerical errors between our non-equilibrium model and the equilibrium model [3] are usually about 1% when low amplitudes (in the linear limit) and low frequencies (when the EDLs are equilibrated) are considered. In selected non-equilibrium regimes, we estimated the errors of numerical approximations by using various spatial discretizations. It was found that qualitative behavior remains unaffected by the numerical errors and the relative error usually does not exceed 10%. Our results show that the flow reversals do not necessarily rely on the electric charge injection [1,12] or the condensed layer formation [16]. There is a preferred direction of the net flow in the low-amplitude regimes. According to the published experiments and our model analysis we suggest that both directions are possible in non-equilibrium regimes. The resulting direction depends on
the established velocity field pattern that is given by interplay of the electric, viscous and pressure forces. They are affected by a particular set of parameters (amplitude, frequency, geometry, concentration, etc.) and dynamical behavior of the electric double layer (the Boltzmann distribution is not always applicable). The introduced mathematical model and the obtained results should be a useful contribution to the area of microfluidic devices driven by AC electric fields. Acknowledgements The authors thank for the support of the research by the grant of the Grant Agency of Czech Academy of Science KAN208240651 and by the grant of Ministry of Education, Youth and Sport of the Czech Republic MSM 6046137306. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.mee.2009.01.045. References [1] A. Ajdari, Physical Review E 61 (2000) R45. [2] P. Garcia-Sanchez, A. Ramos, G. Green, H. Morgan, IEEE Transactions on Dielectrics and Electrical Insulation 13 (2006) 670. [3] N.G. Green, A. Ramos, A. Gonzalez, H. Morgan, A. Castellanos, Physical Review E 66 (2002) 026305. [4] M. Mpholo, C.G. Smith, A.B.D. Brown, Sensors and Actuators B-Chemical 92 (2003) 262. [5] V. Studer, A. Pepin, Y. Chen, A. Ajdari, Analyst 129 (2004) 944. [6] M.Z. Bazant, T.M. Squires, Physical Review Letters 92 (2004) 066101. [7] T.M. Squires, M.Z. Bazant, Journal of Fluid Mechanics 509 (2004) 217. [8] R.F. Probstein, Physicochemical Hydrodynamics: An Introduction, second ed., Wiley and Sons, New York, 1994. [9] B.J. Kim, S.Y. Yoon, H.J. Sung, C.G. Smith, Journal of Applied Physics 102 (2007) 074513. [10] N. Loucaides, A. Ramos, G.E. Georghiou, Microfluidics and Nanofluidics 3 (2007) 709. [11] A. Ramos, A. Gonzalez, A. Castellanos, N.G. Green, H. Morgan, Physical Review E 67 (2003) 056302. [12] L.H. Olesen, H. Bruus, A. Ajdari, Physical Review E 73 (2006) 056313. [13] N.A. Mortensen, L.H. Olesen, L. Belmon, H. Bruus, Physical Review E 71 (2005) 056306. [14] M. Pribyl, V. Knapkova, D. Snita, M. Marek, Microelectronic Engineering 83 (2006) 1660. [15] F.O. Morin, F. Gillot, H. Fujita, Applied Physics Letters 91 (2007) 064103. [16] B.D. Storey, L.R. Edwards, M.S. Kilic, M.Z. Bazant, Physical Review E 77 (2008) 036317. [17] D. Lastochkin, R.H. Zhou, P. Wang, Y.X. Ben, H.C. Chang, Journal of Applied Physics 96 (2004) 1730. [18] M.S. Kilic, M.Z. Bazant, A. Ajdari, Physical Review E 75 (2007) 021502. [19] M.S. Kilic, M.Z. Bazant, A. Ajdari, Physical Review E 75 (2007) 021503.