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Traveling wave electroosmosis: The influence of electrode array geometry Ji í Hrdli ka*, Niketan S. Patel, Dalimil Šnita Department of Chemical Engineering, Institute of Chemical Technology Prague, Technická 3, 16628 Prague, Czech Republic Keywords: AC electroosmosis, traveling wave pump, mathematical modeling *corresponding author E-mail: [email protected] Tel: +420 22044 - 3251 Fax: +420 220 445 018
Received: 06-Dec-2013; Revised: 14-Mar-2014; Accepted: 17-Mar-2014
This article has been accepted for publication and undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading process, which may lead to differences between this version and the Version of Record. Please cite this article as doi: 10.1002/elps.201300614.
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Abstract We used a mathematical model describing traveling-wave electroosmotic micropumps to explain their rather poor ability to work against pressure loads. The mathematical model is based upon the Poisson-Nernst-Planck-Navier-Stokes (PNPNS) approach, i.e. a direct numerical simulation, which allows a detail study of the energy transformations and the charging dynamics of the electric double layers. Using Matlab and COMSOL Multiphysics, we performed a set of extensive parametric studies to determine the dependence of generated electroosmotic flow on the geometric arrangement of the pump. The results suggest that the performance of AC electroosmotic pumps should improve with miniaturization. The AC electroosmosis is likely to be suitable only at sub-micrometer scale, as the pump's ability to work against pressure load diminishes rapidly when increasing the channel diameter.
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Introduction Alternating current electroosmotic (ACEO) pumps are being developed more than a decade. Without moving parts, the ACEO pumps can be relatively easily miniaturized to manipulate electrolytes at the micro- and nano-scale. The prototypes developed and tested up today have very low operating pressures [1]. We strive to quantify the dependence of the work carried out by the pump on the microchannel geometry.
Ramos published an analysis [2], in which the fluid flow stemming from the action of the Coulomb force on the electric double layers had been termed AC electroosmosis. The experimental setup consisted of two finger coplanar electrodes, separated by a narrow gap, subjected to a two-phase alternating current. The generated flow pattern consisted of four counter-rotating rolls (eddies) with zero net velocity (non-directional flow) due to the system symmetry. In order to achieve pumping, the symmetry has to be broken [3]. The ACEO pumps consist of periodical arrays of microelectrodes placed on microchannel walls. There are two kinds: the asymmetric and the traveling wave pumps. The asymmetric arrays [4, 5, 6] consist of pairs of electrodes, one wide and one narrow, separated by intermitting narrow and wide gaps, respectively. Traveling wave arrays [7, 8] involve groups consisting of
electrodes of equal width separated by narrow gaps. AC voltage phases charging the
adjacent electrodes are shifted by 360°/ from one another. The traveling wave electroosmotic (TWEO) pumps usually employ three [9, 10] or four electrodes [11, 12]. The structure of three- or four-phase TWEO is shown in Fig. 1. The use of more electrodes introduces problems with fabrication as the complexity of electronics increases. The mathematical models of the ACEO flow often relied on the thin-double layer approximation introduced in Ref. [13]. This approach is valid when the channel diameter is much higher than several micrometers. Based on the scaling of the motive forces, we assume that the performance of This article is protected by copyright. All rights reserved.
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the ACEO pumps should improve with a reduction in the device scale. The thin double layer approximation assumes that the channel diameter is considerably higher than the thickness of the electric double layer. Although this condition is valid for all devices fabricated up to date, we believe that a further miniaturization is needed. In sub micrometer range, the assumption of the thin double layer approximation becomes questionable. Moreover, the assumption that the electric double layers reach equilibrium can be also not fulfilled at high frequencies, as the ions do not have enough time to respond to the rapidly changing AC electric field [14]. The experimental devices generally use only bottom wall of the microfluidic channel (single-sided arrangements), which leaves the upper wall inactive, posing as a hydrodynamic resistance. We decided to model double-sided arrangements, where the upper electrode array is a mirror image of the bottom array. Utilizing the upper wall should provide additional power, although the fabrication (especially the alignment of the top and bottom electrode arrays) would be more difficult. The performance of the ACEO pumps can be improved using 3D electrodes [15, 16] or tuned by the optimization of the pump geometry and the driving signal parameters. The non-planar electrodes were also implemented to the traveling wave pumps. An order of magnitude improvement of the electroosmotic flow rate was achieved [17, 18]. We focus on the planar traveling wave arrays. The ACEO pumps are usually modeled without pressure force acting against the generated flow [Refs]. To transport electrolytes, the ACEO pump has to overcome the hydrodynamic resistance of the microchannel, as well as a pressure force caused, e.g., by different levels of liquid in inlet and outlet. We define a pumping power as
, where
is the volumetric flow rate and
pressure difference (back pressure) between the inlet and outlet of the microchannel.
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is the
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Figure 1: Periodical segments of traveling wave electroosmotic pumps using three- and four-phase electrode arrays. Here segment;
refers to values of all model quantities and their derivatives at the left edge of the periodical and
denote the values at the right edge of three- and four-electrode periodical segments,
respectively. The AC-voltage phases powering the electrodes are denoted by
, where is the sequential
number of the electrode ranging from one to the number of AC-voltage phases.
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Materials and Methods The mathematical description of a creeping flow of a Newtonian symmetric monovalent electrolyte is based on the Poison-Nernst-Planck-Navier-Stokes (PNPNS) approach, which is described in Ref. [19]. The model is spatially two-dimensional and fully dynamic. The model quantities are the electric potential , the concentrations of cations
and anions
, the velocity vector
and the
pressure . Due to the multi-scale nature of the studied phenomena (because we study the dependence of the generated flow on geometry), all the model quantities are dimensionless. The dimensionless properties are scaled by following set of characteristic values:
where the zero subscript denotes a characteristic value of the respective quantity. is the universal gas constant,
is the Faraday constant,
is absolute temperature, is the electrolyte permittivity and
je
the mean ionic diffusivity. We use the Debye length
as the characteristic dimension. This choice renders the characteristic length independent of geometry setup and yields a simpler dimensionless form of the governing equations. On the other hand, the use of the Debye length as a characteristic dimension introduces the dependence of the length scale on the concentration of the electrolyte. This article is protected by copyright. All rights reserved.
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The governing equations are the Poisson equation (1), the local balances of ions (2), the NavierStokes equation (3), and the continuity equations for mass (4) and electric charge (5) in dimensionless form
(1)
(2)
(3) (4)
(5) The tilde marker denotes a dimensionless quantity. It is convenient to combine the ionic concentrations into the electric charge density
and electrolyte conductivity . For the
characteristic length equal to the Debye length, two dimensionless parameters
and
(6) represent the Rayleigh and Reynolds numbers, respectively. The Reynolds number refers to the flow regime in the microchannel, which is always laminar. The Rayleigh number has the meaning of the ionic drag coefficient [ref Dukhin]. The flux term in the continuity equation of the electric charge is the density of total dimensionless electric current, Faraday electric current and
, where
is the dimensionless
is the dimensionless Maxwell (displacement) electric current. The
Faradaic current is associated with the convection, diffusion and migration of charged particles. The
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Maxwell current is generated as the electric field changes over time, and does not require the presence of mobile charge carriers. We implemented the dimensionless back pressure gradient,
, (which acts against the pump) as a
source term in the Navier-Stokes equation. The value of the source term is specified as a pressure difference per meter. The electric potential is defined as a function of time at the outer electrode edges as (7) where
is the dimensionless amplitude of the applied traveling wave voltage,
is the
dimensionless frequency of the traveling wave and is the sequential number of an electrode within a group of
electrodes. The combination of the applied voltages generates a traveling wave
potential.
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We assume that all quantities and their derivatives are periodical in the direction with period , which is the length of the modeling domain and also the wavelength of the applied traveling wave potential: (8) The solid-liquid interfaces do not permit mass and charge transfer. The velocity satisfies the non-slip boundary condition. (9)
Since the walls of the system do not allow charge transfer, the total electric current across the outer boundaries consists exclusively of the Maxwell current.
Results and discussion We define a dimensionless pumping power density as
as a measure the performance of
ACEO pumps. To illustrate the decrease in pumping power when scaling up, we conducted a comprehensive parametric study. The generated flow rates depend strongly on frequency. To
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exclude this dependency, we found the optimum frequency for each geometry setup (given length and height of a periodic segment and the length of the electrodes). Our simulations suggest that the generated flow rates depend linearly on the applied back pressure. The maximum flow rates are attained at zero back pressure (a case of an unloaded pump). For each geometric setup operating at the optimum frequency, we found the back pressure at which the pump exhibited the maximum pumping power. Each point in the following plots is a result of such optimization. The dependence of the pumping power density on several geometric parameters is presented in the following paragraphs. 1. The effect of traveling wave wavelength The periodical segment length, which has the meaning of the traveling wave wavelength, influences the optimum frequency, the optimum pressure load and the maximum attainable flow velocity. The optimum frequency occurs at unitary traveling wave speed (the speed at which the traveling wave potential passes through the periodic segment). Taking time scaling into account, the relation between averaged pumping velocity and the traveling wave speed is given by (10)
Assuming that the maximum pumping velocity (and pumping power density) occurs at unitary traveling wave speed (i.e., the optimum frequency), the optimum pressure load and the net velocity are inversely proportional to the periodic segment length. Our results agree with the assumption that the pumping power density should be inversely proportional to the square of periodic segment length (See Fig. 2).
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Figure 2: The dependence of the net velocity on the periodic segment length for three- and four-phase traveling wave electroosmotic pumps.
,
,
at the optimal frequencies and back-
pressure gradients.
2. The effect of the channel depth If zero back pressure is applied on a double-sided micropump, the net velocity is zero in a closed channel and gradually approaches a certain value, which remains approximately constant for any higher channel depth. Although this invokes an idea that deeper channels should provide higher flow rates, this remains true only for zero back pressure. The pumping power is proportional to the volume of the microchannel occupied by the electric double layers. The relative volume occupied by the double layers is inversely proportional to the channel depth. The same applies for the hydrodynamic resistance, which counteracts the pressure driven flow. In other words, the extremely shallow channels hinder the pressure driven flow while electroosmotic flow benefits from the higher relative volume occupied by the electric double layers.
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Figure 3: The dependence of the net velocity on the amplitude for three- and four-phase traveling wave electroosmotic pumps.
,
,
at the optimal frequency and zero back-pressure gradients.
The scaling of the pumping power density is analogical to the relation found for the traveling wave wavelength. Although the net velocity approaches a constant value in deeper channels, the optimum pressure loads are inversely proportional to the square of the periodic segment length (Fig. 3). By the inspection of the velocity profiles of traveling wave pumps at zero applied back pressure, one can notice that the electroosmotic flow has a flat front in a deeper channel, but not in the shallower ones. Apparently, the electroosmotic flow velocity profile cannot fully develop in the channels with depth comparable to the Debye length (at a given concentration). The extremely shallow channels allow the merging of the electrical double layers and the formation of a spatial charge within the whole channel so that the Coulomb force can act across the whole cross-section of the microchannel and the performance of the AC electroosmotic micropump is maximal. Moreover, the hydrodynamics resistance increases with the reduction of the cross-section area and ensures the dominance of the viscous dissipation (the stabilizing agent) over the pressure gradient force. Not only it helps the electroosmotic flow by hindering the pressure driven flow caused by applied back-pressure, it also reduces the pressure driven flow rate when the pressure and net Coulomb force act in the same direction, in the case that the flow rate can be precisely controlled.
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When the traveling wave electroosmotic micropump is integrated into a microfluidic loop, the external pressure gradients can be minimized and the height of the channels ceases to be an issue. Then, the increase in channel height has a positive influence on the generated flow rate [Ref, loop pump]. 3. The effect of periodic segment scale and the number of electrodes The dependence of the pumping power density can be estimated using the trends found for the periodic segment length and height (i.e., the traveling wave wavelength and the microchannel depth, respectively).
Figure 4: The dependence of the net velocity on the amplitude for three- and four-phase traveling wave electroosmotic pumps at a constant aspect ratio.
,
,
at the optimal frequency and
zero back-pressure gradients.
Unlike the isolated periodic segment dimensions (depth of channel and traveling wave wavelength/periodical segment length), the decline of the pumping power density also depends on the number of AC phases, see Fig. 4. The four-phase traveling wave pumps exhibit a slower reduction in the pumping power density. As the Coulomb force is bound to the electrode corners, the additional electrodes increase the average Coulomb force.
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The most significant changes to the model quantities occur near the electrode corners. More electrodes mean more highly active regions and higher mean Coulomb force density in the direction, Fig. 5. If the electrodes cover the same percentage of the microchannel walls, the interelectrode gaps are shorter when denser electrode arrays are used. Higher electric field intensities are then generated between the neighboring electrodes at the same voltage amplitudes. The higher values of the Coulomb force density averaged over time and space alone suggest why the pumps with higher number of phases should deliver higher performances.
Figure 5: The distribution of the Coulomb force density at the electrode level (
) in three- and four-phase
traveling wave pumps. There are two consecutive segments to show the distribution across the periodical boundary.
,
,
,
at the optimal frequency and zero pressure load. Phase angle is
.
The three-electrode alternatives provide approximately 75% of the net velocities generated by the four-electrode counterparts over the entire range of simulated frequencies. This ratio of the pumping power densities could be related to the averaged Coulomb forces, where the ratio is (quite) similar, see Fig 5. In order to achieve the maximum flow rates, four-electrode arrays should be preferred over the three-electrode alternatives. We assume that the majority of the electroosmotic flow is generated at the electrode corners (where the Coulomb force is maximal) and that the regions of high Coulomb force have similar size
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under the optimum conditions (for different number of AC phases). In that case, the longer electrodes (i.e., less AC phases at the same coverage) mean more unused electrode surface and a lower efficiency. More AC-phases would therefore mean better used electrodes.
Concluding remarks We present the results of a comprehensive set of parametric studies to quantify the dependence of AC electroosmotic flow on the geometry of a traveling wave electroosmotic pump. Based on the data obtained using a direct numerical simulation, we demonstrate that the traveling wave electroosmotic pumps perform best at scales comparable to the Debye length. When scaled up, the pumping power density decreases rapidly, which translates into lower flow rates and very low applicable back pressures. We use the Debye length as a characteristic length; therefore, the dilution of working fluid has a similar meaning as miniaturization, owing to the dependency of the Debye length on the concentration. ACEO pumps do not perform well against back pressure. Our results suggest that the current devices (with channel diameter larger than ten micrometers) are too large. The height of the microchannel appears to be the critical parameter. According to our simulations, the applicable back pressures are inversely proportional to the square of the channel height. Therefore, a decrease in the channel height below one micrometer could allow the use of back pressures comparable to the atmospheric pressure. Also, the number of AC phases (electrodes per a periodic segment) determines how rapid the decline of the applicable back pressure with increasing scale is. For that reason, four-phase traveling wave pumps should be preferred over three-phase designs. Acknowledgement This work was supported by the Grant Agency of the Czech Academy of Sciences under Grant No. IAA401280904.
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References [1] Bazant, M. Z., Kilic, M. S., Storey, B. D., and Ajdari, A., Adv. in Colloid Interface Sci. 2009, 152, 4888. [2] Ramos, A., Morgan, H., Green, N.G., Castellanos A., J. Colloid Interface Sci. 1999, 217, 420-422. [3] Ajdari, A., Phys. Rev. E 2000, 61, R45-R48. [4] Brown, A. B. D., Smith, C., Rennie , A. R., Phys. Rev. E 2000, 63,016305 1-8. [5] Olesen, L. H., Bruus, H., Ajdari, A., Phys. Rev. E 2006, 73, 056313 1-16. [6] Urbanski, J. P., Levitan, J. A., Burch, D. N., Thorsen, T., Bazant, M. Z., J. Colloid Interface Sci. 2007, 309, 332-341. [7] Cahill, B. P., Heyderman, L. J., Gobrecht, J. Stemmer, A., Phys. Rev E 2004, 70, 036305 1-14. [8] Ramos, A., Morgan, H., Green, N.G., González, A., Castellanos, A., J. Appl. Phys. 2005, 97, 084906 1-8. [9] Belisle, A., Brown, M. M., Hubbard, T., Kujath, M., J. Vac. Sci. Technol., Part A 2006, 24(3), 737741. [10] Moore, T. A., Diploma thesis, Queen's University Kingston, Ontario, Canada 2011. [11] González, A., Ramos, A., Castellanos, A., Microfluid Nanofluidics 2008, 5, 507-515. [12] Yeh, H.-C., Yang, R.-J., Luo, W.-J., Phys. Rev. E 2011, 83, 056326-056335. [13] González, A., Ramos, A., Green, N.G., Castellanos, A., Morgan, H., Phys. Rev. E, 61, 4019-4028. [14] Hrdli ka, J., ervenka, P., P ibyl, M., Šnita, D., J. Appl. Electrochem. 2010, 40, 967-980, [15] Bazant, M. Z., Ben, Y., Lab Chip 2006, 6, 1455-1461. [16] Huang, C.,-C., Bazant, M., Z., Thorsen, T., Lab Chip 2010, 10, 80-85. [17] García-Sánchez, P., Ramos, A., Microfluid Nanofluidics 2008, 5. 307-312. [18] Mehdipoor, M., Vafaie, R., H., Pourmand, A., Poorreza, E., Ghavifekr, H., B., 8th international symposium on mechatronics and its applications 2012, 1-6. [19] Hrdli ka, J., ervenka, P., P ibyl, M., Šnita, D., Phys. Rev. E 2011, 84,016307. This article is protected by copyright. All rights reserved.
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Traveling wave electroosmosis: The influence of electrode array geometry Ji í Hrdli ka*, Niketan S. Patel, Dalimil Šnita Department of Chemical Engineering, Institute of Chemical Technology Prague, Technická 3, 16628 Prague, Czech Republic Keywords: AC electroosmosis, traveling wave pump, mathematical modeling *corresponding author E-mail: [email protected] Tel: +420 22044 - 3251 Fax: +420 220 445 018
Received: 06-Dec-2013; Revised: 14-Mar-2014; Accepted: 17-Mar-2014
This article has been accepted for publication and undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading process, which may lead to differences between this version and the Version of Record. Please cite this article as doi: 10.1002/elps.201300614.
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Abstract We used a mathematical model describing traveling-wave electroosmotic micropumps to explain their rather poor ability to work against pressure loads. The mathematical model is based upon the Poisson-Nernst-Planck-Navier-Stokes (PNPNS) approach, i.e. a direct numerical simulation, which allows a detail study of the energy transformations and the charging dynamics of the electric double layers. Using Matlab and COMSOL Multiphysics, we performed a set of extensive parametric studies to determine the dependence of generated electroosmotic flow on the geometric arrangement of the pump. The results suggest that the performance of AC electroosmotic pumps should improve with miniaturization. The AC electroosmosis is likely to be suitable only at sub-micrometer scale, as the pump's ability to work against pressure load diminishes rapidly when increasing the channel diameter.
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Introduction Alternating current electroosmotic (ACEO) pumps are being developed more than a decade. Without moving parts, the ACEO pumps can be relatively easily miniaturized to manipulate electrolytes at the micro- and nano-scale. The prototypes developed and tested up today have very low operating pressures [1]. We strive to quantify the dependence of the work carried out by the pump on the microchannel geometry.
Ramos published an analysis [2], in which the fluid flow stemming from the action of the Coulomb force on the electric double layers had been termed AC electroosmosis. The experimental setup consisted of two finger coplanar electrodes, separated by a narrow gap, subjected to a two-phase alternating current. The generated flow pattern consisted of four counter-rotating rolls (eddies) with zero net velocity (non-directional flow) due to the system symmetry. In order to achieve pumping, the symmetry has to be broken [3]. The ACEO pumps consist of periodical arrays of microelectrodes placed on microchannel walls. There are two kinds: the asymmetric and the traveling wave pumps. The asymmetric arrays [4, 5, 6] consist of pairs of electrodes, one wide and one narrow, separated by intermitting narrow and wide gaps, respectively. Traveling wave arrays [7, 8] involve groups consisting of
electrodes of equal width separated by narrow gaps. AC voltage phases charging the
adjacent electrodes are shifted by 360°/ from one another. The traveling wave electroosmotic (TWEO) pumps usually employ three [9, 10] or four electrodes [11, 12]. The structure of three- or four-phase TWEO is shown in Fig. 1. The use of more electrodes introduces problems with fabrication as the complexity of electronics increases. The mathematical models of the ACEO flow often relied on the thin-double layer approximation introduced in Ref. [13]. This approach is valid when the channel diameter is much higher than several micrometers. Based on the scaling of the motive forces, we assume that the performance of This article is protected by copyright. All rights reserved.
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the ACEO pumps should improve with a reduction in the device scale. The thin double layer approximation assumes that the channel diameter is considerably higher than the thickness of the electric double layer. Although this condition is valid for all devices fabricated up to date, we believe that a further miniaturization is needed. In sub micrometer range, the assumption of the thin double layer approximation becomes questionable. Moreover, the assumption that the electric double layers reach equilibrium can be also not fulfilled at high frequencies, as the ions do not have enough time to respond to the rapidly changing AC electric field [14]. The experimental devices generally use only bottom wall of the microfluidic channel (single-sided arrangements), which leaves the upper wall inactive, posing as a hydrodynamic resistance. We decided to model double-sided arrangements, where the upper electrode array is a mirror image of the bottom array. Utilizing the upper wall should provide additional power, although the fabrication (especially the alignment of the top and bottom electrode arrays) would be more difficult. The performance of the ACEO pumps can be improved using 3D electrodes [15, 16] or tuned by the optimization of the pump geometry and the driving signal parameters. The non-planar electrodes were also implemented to the traveling wave pumps. An order of magnitude improvement of the electroosmotic flow rate was achieved [17, 18]. We focus on the planar traveling wave arrays. The ACEO pumps are usually modeled without pressure force acting against the generated flow [Refs]. To transport electrolytes, the ACEO pump has to overcome the hydrodynamic resistance of the microchannel, as well as a pressure force caused, e.g., by different levels of liquid in inlet and outlet. We define a pumping power as
, where
is the volumetric flow rate and
pressure difference (back pressure) between the inlet and outlet of the microchannel.
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is the
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Figure 1: Periodical segments of traveling wave electroosmotic pumps using three- and four-phase electrode arrays. Here segment;
refers to values of all model quantities and their derivatives at the left edge of the periodical and
denote the values at the right edge of three- and four-electrode periodical segments,
respectively. The AC-voltage phases powering the electrodes are denoted by
, where is the sequential
number of the electrode ranging from one to the number of AC-voltage phases.
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Materials and Methods The mathematical description of a creeping flow of a Newtonian symmetric monovalent electrolyte is based on the Poison-Nernst-Planck-Navier-Stokes (PNPNS) approach, which is described in Ref. [19]. The model is spatially two-dimensional and fully dynamic. The model quantities are the electric potential , the concentrations of cations
and anions
, the velocity vector
and the
pressure . Due to the multi-scale nature of the studied phenomena (because we study the dependence of the generated flow on geometry), all the model quantities are dimensionless. The dimensionless properties are scaled by following set of characteristic values:
where the zero subscript denotes a characteristic value of the respective quantity. is the universal gas constant,
is the Faraday constant,
is absolute temperature, is the electrolyte permittivity and
je
the mean ionic diffusivity. We use the Debye length
as the characteristic dimension. This choice renders the characteristic length independent of geometry setup and yields a simpler dimensionless form of the governing equations. On the other hand, the use of the Debye length as a characteristic dimension introduces the dependence of the length scale on the concentration of the electrolyte. This article is protected by copyright. All rights reserved.
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The governing equations are the Poisson equation (1), the local balances of ions (2), the NavierStokes equation (3), and the continuity equations for mass (4) and electric charge (5) in dimensionless form
(1)
(2)
(3) (4)
(5) The tilde marker denotes a dimensionless quantity. It is convenient to combine the ionic concentrations into the electric charge density
and electrolyte conductivity . For the
characteristic length equal to the Debye length, two dimensionless parameters
and
(6) represent the Rayleigh and Reynolds numbers, respectively. The Reynolds number refers to the flow regime in the microchannel, which is always laminar. The Rayleigh number has the meaning of the ionic drag coefficient [ref Dukhin]. The flux term in the continuity equation of the electric charge is the density of total dimensionless electric current, Faraday electric current and
, where
is the dimensionless
is the dimensionless Maxwell (displacement) electric current. The
Faradaic current is associated with the convection, diffusion and migration of charged particles. The
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Maxwell current is generated as the electric field changes over time, and does not require the presence of mobile charge carriers. We implemented the dimensionless back pressure gradient,
, (which acts against the pump) as a
source term in the Navier-Stokes equation. The value of the source term is specified as a pressure difference per meter. The electric potential is defined as a function of time at the outer electrode edges as (7) where
is the dimensionless amplitude of the applied traveling wave voltage,
is the
dimensionless frequency of the traveling wave and is the sequential number of an electrode within a group of
electrodes. The combination of the applied voltages generates a traveling wave
potential.
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We assume that all quantities and their derivatives are periodical in the direction with period , which is the length of the modeling domain and also the wavelength of the applied traveling wave potential: (8) The solid-liquid interfaces do not permit mass and charge transfer. The velocity satisfies the non-slip boundary condition. (9)
Since the walls of the system do not allow charge transfer, the total electric current across the outer boundaries consists exclusively of the Maxwell current.
Results and discussion We define a dimensionless pumping power density as
as a measure the performance of
ACEO pumps. To illustrate the decrease in pumping power when scaling up, we conducted a comprehensive parametric study. The generated flow rates depend strongly on frequency. To
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exclude this dependency, we found the optimum frequency for each geometry setup (given length and height of a periodic segment and the length of the electrodes). Our simulations suggest that the generated flow rates depend linearly on the applied back pressure. The maximum flow rates are attained at zero back pressure (a case of an unloaded pump). For each geometric setup operating at the optimum frequency, we found the back pressure at which the pump exhibited the maximum pumping power. Each point in the following plots is a result of such optimization. The dependence of the pumping power density on several geometric parameters is presented in the following paragraphs. 1. The effect of traveling wave wavelength The periodical segment length, which has the meaning of the traveling wave wavelength, influences the optimum frequency, the optimum pressure load and the maximum attainable flow velocity. The optimum frequency occurs at unitary traveling wave speed (the speed at which the traveling wave potential passes through the periodic segment). Taking time scaling into account, the relation between averaged pumping velocity and the traveling wave speed is given by (10)
Assuming that the maximum pumping velocity (and pumping power density) occurs at unitary traveling wave speed (i.e., the optimum frequency), the optimum pressure load and the net velocity are inversely proportional to the periodic segment length. Our results agree with the assumption that the pumping power density should be inversely proportional to the square of periodic segment length (See Fig. 2).
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Figure 2: The dependence of the net velocity on the periodic segment length for three- and four-phase traveling wave electroosmotic pumps.
,
,
at the optimal frequencies and back-
pressure gradients.
2. The effect of the channel depth If zero back pressure is applied on a double-sided micropump, the net velocity is zero in a closed channel and gradually approaches a certain value, which remains approximately constant for any higher channel depth. Although this invokes an idea that deeper channels should provide higher flow rates, this remains true only for zero back pressure. The pumping power is proportional to the volume of the microchannel occupied by the electric double layers. The relative volume occupied by the double layers is inversely proportional to the channel depth. The same applies for the hydrodynamic resistance, which counteracts the pressure driven flow. In other words, the extremely shallow channels hinder the pressure driven flow while electroosmotic flow benefits from the higher relative volume occupied by the electric double layers.
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Figure 3: The dependence of the net velocity on the amplitude for three- and four-phase traveling wave electroosmotic pumps.
,
,
at the optimal frequency and zero back-pressure gradients.
The scaling of the pumping power density is analogical to the relation found for the traveling wave wavelength. Although the net velocity approaches a constant value in deeper channels, the optimum pressure loads are inversely proportional to the square of the periodic segment length (Fig. 3). By the inspection of the velocity profiles of traveling wave pumps at zero applied back pressure, one can notice that the electroosmotic flow has a flat front in a deeper channel, but not in the shallower ones. Apparently, the electroosmotic flow velocity profile cannot fully develop in the channels with depth comparable to the Debye length (at a given concentration). The extremely shallow channels allow the merging of the electrical double layers and the formation of a spatial charge within the whole channel so that the Coulomb force can act across the whole cross-section of the microchannel and the performance of the AC electroosmotic micropump is maximal. Moreover, the hydrodynamics resistance increases with the reduction of the cross-section area and ensures the dominance of the viscous dissipation (the stabilizing agent) over the pressure gradient force. Not only it helps the electroosmotic flow by hindering the pressure driven flow caused by applied back-pressure, it also reduces the pressure driven flow rate when the pressure and net Coulomb force act in the same direction, in the case that the flow rate can be precisely controlled.
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When the traveling wave electroosmotic micropump is integrated into a microfluidic loop, the external pressure gradients can be minimized and the height of the channels ceases to be an issue. Then, the increase in channel height has a positive influence on the generated flow rate [Ref, loop pump]. 3. The effect of periodic segment scale and the number of electrodes The dependence of the pumping power density can be estimated using the trends found for the periodic segment length and height (i.e., the traveling wave wavelength and the microchannel depth, respectively).
Figure 4: The dependence of the net velocity on the amplitude for three- and four-phase traveling wave electroosmotic pumps at a constant aspect ratio.
,
,
at the optimal frequency and
zero back-pressure gradients.
Unlike the isolated periodic segment dimensions (depth of channel and traveling wave wavelength/periodical segment length), the decline of the pumping power density also depends on the number of AC phases, see Fig. 4. The four-phase traveling wave pumps exhibit a slower reduction in the pumping power density. As the Coulomb force is bound to the electrode corners, the additional electrodes increase the average Coulomb force.
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The most significant changes to the model quantities occur near the electrode corners. More electrodes mean more highly active regions and higher mean Coulomb force density in the direction, Fig. 5. If the electrodes cover the same percentage of the microchannel walls, the interelectrode gaps are shorter when denser electrode arrays are used. Higher electric field intensities are then generated between the neighboring electrodes at the same voltage amplitudes. The higher values of the Coulomb force density averaged over time and space alone suggest why the pumps with higher number of phases should deliver higher performances.
Figure 5: The distribution of the Coulomb force density at the electrode level (
) in three- and four-phase
traveling wave pumps. There are two consecutive segments to show the distribution across the periodical boundary.
,
,
,
at the optimal frequency and zero pressure load. Phase angle is
.
The three-electrode alternatives provide approximately 75% of the net velocities generated by the four-electrode counterparts over the entire range of simulated frequencies. This ratio of the pumping power densities could be related to the averaged Coulomb forces, where the ratio is (quite) similar, see Fig 5. In order to achieve the maximum flow rates, four-electrode arrays should be preferred over the three-electrode alternatives. We assume that the majority of the electroosmotic flow is generated at the electrode corners (where the Coulomb force is maximal) and that the regions of high Coulomb force have similar size
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under the optimum conditions (for different number of AC phases). In that case, the longer electrodes (i.e., less AC phases at the same coverage) mean more unused electrode surface and a lower efficiency. More AC-phases would therefore mean better used electrodes.
Concluding remarks We present the results of a comprehensive set of parametric studies to quantify the dependence of AC electroosmotic flow on the geometry of a traveling wave electroosmotic pump. Based on the data obtained using a direct numerical simulation, we demonstrate that the traveling wave electroosmotic pumps perform best at scales comparable to the Debye length. When scaled up, the pumping power density decreases rapidly, which translates into lower flow rates and very low applicable back pressures. We use the Debye length as a characteristic length; therefore, the dilution of working fluid has a similar meaning as miniaturization, owing to the dependency of the Debye length on the concentration. ACEO pumps do not perform well against back pressure. Our results suggest that the current devices (with channel diameter larger than ten micrometers) are too large. The height of the microchannel appears to be the critical parameter. According to our simulations, the applicable back pressures are inversely proportional to the square of the channel height. Therefore, a decrease in the channel height below one micrometer could allow the use of back pressures comparable to the atmospheric pressure. Also, the number of AC phases (electrodes per a periodic segment) determines how rapid the decline of the applicable back pressure with increasing scale is. For that reason, four-phase traveling wave pumps should be preferred over three-phase designs. Acknowledgement This work was supported by the Grant Agency of the Czech Academy of Sciences under Grant No. IAA401280904.
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References [1] Bazant, M. Z., Kilic, M. S., Storey, B. D., and Ajdari, A., Adv. in Colloid Interface Sci. 2009, 152, 4888. [2] Ramos, A., Morgan, H., Green, N.G., Castellanos A., J. Colloid Interface Sci. 1999, 217, 420-422. [3] Ajdari, A., Phys. Rev. E 2000, 61, R45-R48. [4] Brown, A. B. D., Smith, C., Rennie , A. R., Phys. Rev. E 2000, 63,016305 1-8. [5] Olesen, L. H., Bruus, H., Ajdari, A., Phys. Rev. E 2006, 73, 056313 1-16. [6] Urbanski, J. P., Levitan, J. A., Burch, D. N., Thorsen, T., Bazant, M. Z., J. Colloid Interface Sci. 2007, 309, 332-341. [7] Cahill, B. P., Heyderman, L. J., Gobrecht, J. Stemmer, A., Phys. Rev E 2004, 70, 036305 1-14. [8] Ramos, A., Morgan, H., Green, N.G., González, A., Castellanos, A., J. Appl. Phys. 2005, 97, 084906 1-8. [9] Belisle, A., Brown, M. M., Hubbard, T., Kujath, M., J. Vac. Sci. Technol., Part A 2006, 24(3), 737741. [10] Moore, T. A., Diploma thesis, Queen's University Kingston, Ontario, Canada 2011. [11] González, A., Ramos, A., Castellanos, A., Microfluid Nanofluidics 2008, 5, 507-515. [12] Yeh, H.-C., Yang, R.-J., Luo, W.-J., Phys. Rev. E 2011, 83, 056326-056335. [13] González, A., Ramos, A., Green, N.G., Castellanos, A., Morgan, H., Phys. Rev. E, 61, 4019-4028. [14] Hrdli ka, J., ervenka, P., P ibyl, M., Šnita, D., J. Appl. Electrochem. 2010, 40, 967-980, [15] Bazant, M. Z., Ben, Y., Lab Chip 2006, 6, 1455-1461. [16] Huang, C.,-C., Bazant, M., Z., Thorsen, T., Lab Chip 2010, 10, 80-85. [17] García-Sánchez, P., Ramos, A., Microfluid Nanofluidics 2008, 5. 307-312. [18] Mehdipoor, M., Vafaie, R., H., Pourmand, A., Poorreza, E., Ghavifekr, H., B., 8th international symposium on mechatronics and its applications 2012, 1-6. [19] Hrdli ka, J., ervenka, P., P ibyl, M., Šnita, D., Phys. Rev. E 2011, 84,016307. This article is protected by copyright. All rights reserved.
